Electronic States in Semiconductor Heterostructures

Electronic States in Semiconductor Heterostructures

SOLID STATE PHYSICS, VOLUME 44 Electronic States in Semiconductor Heterostructures G. BASTARD,? J. A. BRUM,*AND R. FERREIRA? TDipartement de Physique...

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SOLID STATE PHYSICS, VOLUME 44

Electronic States in Semiconductor Heterostructures G. BASTARD,? J. A. BRUM,*AND R. FERREIRA? TDipartement de Physique de I'Ecole Normale Superieure Paris, France *Departamento de Fisica Universidude Estadual de Campinas Szlo Paulo, Brazil

I. Introduction. .............................................................. 11. The Envelope Function Approximation: A Summary .... la. Bulk Band Structures ................................... .......................... ,236 lb. Building Heterostructure States 111. Subband Edges in Quantum Wells and Superlattices.. . . 2. Quantum Wells.. ........................................ 3. Superlattices . . . . . . . . . . . . . . . . ........................... ,260 IV. In-Plane Dispersion Relations . . . . . . . . . . . . . . . . . 4. Conduction Subbands .................................... 5. Valence Subbands . . . . . . . . . . V. Stark Effects in Semiconductor Q 6. Introduction . . . . . . . . . . . . . . . 7. Electric Field Effects in Isolated Quantum Wells.. ............. 8. Electric Field Effects in Double Quantum Wells ....................... .310 9. Electric Field Effects in Superlattices . . . . . . . . . . ..........,313 VI. Broadening of Heterostruc 10. Introduction ........... .......................... ,337 turbation. ........... 11. Carrier-Impurity and Ca Hamiltonians .......... 12. Electron Scattering and 13. Resonances in the Hole Transfer Times.. ............................ ,359 VII. Excitons in Semiconductor Heterostructures ......................... ,364 . . . . . . . . .366 14. Excitonic Effects in the Decoupled Approximation 15. Coupled Excitons in Quantum Wells.. .............................. .373 . . . . . . . . . . . . . 381 VIII. Quasi-One-Dimensional Systems ...................... . . . . . . . . . . . . . 382 16. Quasi-Decoupling of the Wire Eigenstates . . . . . . . . . . 17. One-Side n-Spike- and Modulation-Doped Quantum 18. Magnetoelectric Subbands in a Quasi-1D Electron Gas . . . . . . . . . . . . . . . ,393 19. Valence Band States and Dispersion Relations . . . . . . . . . . .399 20. Quantum Wires with Lateral Defects. ..................... 407 Acknowledgments.. ................................................................. 415

229

Copyright 0 1991 by Academic Press, Inc. All rights of reproduction in any form reserved. ISBN 0-12-607744-4 ISBN 0-12-606044-4 (pbk.)

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G. BASTARD et al.

I. Introduction

The last 15 years or so have witnessed a worldwide effort toward the realization of artificial semiconductor heterostructures. Starting from onedimensional quantum wells and superlattices,' which are best grown by modern epitaxial techniques, the field of heterostructures is progressively shifting by a combination of epitaxy, etching, and lithography toward quantum wires and boxes where the carrier motion is artificially confined in two and three dimensions, respectively. Our understanding of the electronic properties of these man-made heterolayers has steadily progressed over the years. Experimental techniques, such as transport and optics, have brought, and keep bringing, a wealth of detailed information on the near band edges electronic states of the heterostructures. In addition, progress in the techniques of fabrication have opened new areas of research: the outcome of the quantum Hall effect, integer' or fra~tional,~ or, more recently, of the quantization of resistance due to narrow constriction^^*^ are topics that have emerged only because it is now possible to fabricate high-mobility quasibidimensional electron gases and high-quality narrow wires, respectively. Band structure models are required to interpret the experimental findings. They were substantially developed in the early 1980s, where the envelope function approximation,6- empirical tight-binding,9*'0and pseudopotential formalisms' ' - ' were set up to describe the electronic states in superlattices (quasi-three-dimensional materials), quantum wells (quasi-bidimensional), wires (quasi-one-dimensional), and boxes (quasi-zero-dimensionalmaterials). There are several reviews on the electronic properties of semiconductor heterostructures, starting with the celeberated Ando, Fowler, and Stern review14 and continuing with summer or winter schools," conference lectures,16 book chapters, or textbook^.'^-'^ This Chapter will certainly overlap others. We shall, however, attempt to minimize overlap and discuss 'L. Esaki and R. Tsu, IBM J . Res. Dev. 14,61 (1970). 'K. von Klitzing, M. Pepper, and G. Dorda, Phys. Rev. Lett. 45,494 (1980). 3D. C. Tsui, H. L. Stormer, and A. C. Gossard, Phys. Rev. Lett. 48, 1559 (1982). 4B. J. van Wees, H. van Houten, C. W. J. Beenarkker, J. G. Williamson, L.P. Kouwenhoven, D. van der Marel, and C. T. Foxon, Phys. Rev. Lett. 60, 848 (1988). 5D. A. Wharam, T. J. Thornton, R. Newbury, M. Pepper, H. Ahmed, J. E. F. Frost, D. Hasko, D. C. Peacock, D. A. Ritchie, and J. A. C. Jones, J. Phys. C 21,L209 (1988). 6G.Bastard, Phys. Rev. B 24, 5693 (1981);25,7584 (1982). 'S. White and L. J. Sham, Phys. Rev. Lett. 47, 879 (1981). 'M.Altarelli, Phys. Rev. B 28, 842 (1983). 'J. N. Schulman and Y. C. Chang, Phys. Rev. B 24,4445 (1981);31,2056 (1985). loY. C. Chang and J. N. Schulman, Appl. Phys. Lett. 43,536(1983);Phys. Rev. B 31,2069(1985). "M. Jaros, K. B. Wong, and M. Cell, Phys. Rev. B 31, 1205 (1985). I'D. Ninno, K. B. Wong, M. A. Cell, and M. Jaros, Phys. Rev. B 32,2700 (1985). 13M. A. Cell, K. B. Wong, D. Ninno, and M. Jaros, J. Phys. C 19,3821 (1986). I4T.Ando, A. Fowler, and F. Stern, Rev. Mod. Phys. 54, 437 (1982).

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23 1

features that have not been heavily exposed elsewhere (at least by us). Because the field is by now too large to be covered in its entirety in one chapter, we shall omit many interesting topics-for instance, strain” and many-body effectsz1 We feel confident that recent reviews more than compensate for this omission. We shall use the envelope function description6 - 2 6 of electronic states. This is a versatile method that has proved to be accurate. Like any approximation it has drawbacks, but in our opinion they are more than compensated by the method’s flexibility and simplicity. Since the envelope function scheme has already been reviewed, we shall only give a brief summary of its formal machinery (Section 11) and devote the rest of the chapter to specific applications. We start with a simple problem, evaluating the subband edges in quantum wells and superlattices under flat-band conditions (Section 111). In the latter instance we shall find situations where the envelope function scheme fails even qualitatively (short-period GaAs-A1As superlattices). Section IV will deal with the in-plane dispersion of semiconductor heterolayers, particularly the valence subbands, which are more complicated. In Section V we shall examine linearly biased single, double, and multiple quantum wells, and Section VI will present some scattering times of the heterostructure eigenstates due to imperfections and electron-phonon interaction. In Section VII

-’”’

I5See, for example, the NATO AS1 Series B 170, 179, 189, and E 87, edited by Plenum Press (New York) and Martinus Nijhoff (Dordrecht), respectively. See also Springer’s Series in Solid State Sciences,volumes 67,83 (Springer-Verlag, Berlin), and Springer Proceedings in Physics, volume 13 (Springer-Verlag, Berlin). Also see “Heterojunctions and Semiconductor Superlattices”, Springer-Verlag, Berlin, 1986. I6See the “Proceedings of the Electronic Properties of Two Dimensional Systems” published biannually in Surface Science; see also the “Proceedings of the Modulated Semiconductor Structures” biannual conferences. L7‘‘HeterojunctionBand Discontinuities: Physics and Device Applications” (F. Capasso and G. Magaritondo, eds.). North-Holland, Amsterdam, 1987. ”G. Bastard, “Wave Mechanics Applied to Semiconductor Heterostructures.” Les Editions de Physique, Les Ulis, 1988. “G. Bastard, C. Delalande, Y. Guldner, and P. Voisin in “Advances in Electronics and Electron Physics,” Academic Press, New York, 1988. ”“See “Semiconductors and Semimetals,” vol. 32. Academic Press, New York, 1990, particularly the article by J. Y. Marzin, J. M. Gkrard, P. Voisin, and J. A. Brum. 20bE.P. OReilly, Semicond. Sci. Technol. 4, 121 (1989). Schmitt-Rink, D. S. Chemla, and D. A. B. Miller, Adv. Physics 38, 89 (1989). ’IbG. E. W. Bauer, Phys. Rev. Lett. 64, 60 (1990). 22M.F. H. Schuurmans and G . W. t’Hooft, Phys. Rev. B 31, 8041 (1985). 23M.Kriechbaum in “Two Dimensional Systems: Physics and New Devices,” p. 120. SpringerVerlag, Berlin, 1986. 24W. Potz, W. Porod, and D. K. Ferry, Phys. Rev. B 32, 3868 (1985). 25D.L. Smith and C . Mailhiot, Phys. Rev. B 33, 8345, 8360 (1986). 26D. L. Smith and C. Mailhiot, to be published in Rev. Mod. Phys.

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,&+A-1 2

L A I t

_

k//fl11]

A

X

-

k//[100]

FIG.1. Band structure of GaAs. After Ref. 27.

we shall discuss the exciton eigenstates in quantum wells and superlattices, including the effect of an electric field. Finally, Section VIII will deal with the eigenstates of quantum wires, including constrictions. II. The Envelope Function Approximation: A Summary

la. BULKBANDSTRUCTURES Computational techniques allow fairly accurate calculations of the oneelectron band structure of crystalline semiconductors, although there still exist difficulties in predicting the exact zone center band gap. In particular, semiempirical techniques (pseudopotential, tight binding) give a good description of the energy bands over the entire Brillouin zone (for an example see Fig. 1). On the other hand, such complete knowledge is seldom of much use since many of the electronic properties of semiconductors are governed by electronic states that are close to the lowest-lying conduction band and topmost-lying valence band edges. By close, we mean a fraction of l e v , which is much smaller than typical bandwidths of groups IV, 111-V,and 11-VI semiconductors. A local description of the band structure is enough for such electronic states. The k p (or effective mass) method provides dispersion relations, which in the vicinity of an edge located at ko are parabolic in terms of k - ko:

-

7Landolt-Bdrnstein,“Numerical Data and Functional Relationships in Science and Technology” (0. Madelung, ed.), Group 111, vol. 17. Springer-Verlag, Berlin, 1982.

ELECTRONIC STATES IN SEMICONDUCTOR HETEROSTRUCTURES

233

where the effective mass tensor is expressible in terms of matrix elements of the p operator between Bloch functions at k,: 1 1 -=-a,,+--, m(n) a,

2 mo

"'0

c (nlPalm) En(k0)

m*n

-Em(M

'

(1.2)

Refinements can be included within the k * p method to account for departures from quadratic dispersion relations (the band nonparabolicity) when the kinetic energy in the nth band, En(k) - &,(k,), is not very small compared with the k, band gaps En(k0) - cm(k0).This, of course, happens more often in narrower-band-gap semiconductors (e.g., InSb, InAs, Hg, -,Cd,Te). is the most commonly used to account for nonparaboThe Kane licity effects. For zinc blende 111-V and 11-VI materials, the fundamental band gap occurs at the point (k, = 0).The Kane model is an exact diagonalization of the (h/mo)k * p perturbation in the (truncated) basis generated by the eight zone center Bloch functions of the conduction band (2), topmost valence band (4), and split-off valence band (2). These wave functions are listed in Table I, where S , X, I:2 denote periodic functions that transform like s, x, y, z atomic functions under the symmetry operations that map the local tetrahedron onto itself. The dispersion relations are each found to be doubly degenerate and the solutions of

A(A

+ &,)(A + + A) = h2k2P2(A+ 80 + 2A/3), E,

A=

-& 0

TABLE I. PERIODIC PARTS OF THE BLOCHFUNCTIONS AT THE r6,r,, AND rs EDGES,RESPECTIVELY. ul0 = IS, 1/2, 1/2) = l i s t ) u30 =

lp, 3/2, 3/2)

4

+ iY)t)

1 + -l(X +iY)J) & 1 1 = IP, 1/2, 1/2) = -I(X + i Y ) J )+ -127) = IP,

u70

1

= -I(X

3/2, 1/2)

uzo = IS, 4'2,

= -B

I Z ? )

J5

- 1/2) =

J5

181) 1

ud0 = IP, 3/2, -3/2)

= -I(X

u60 = IP, 3/2, - 1/2)

=

4

- iY)J)

1 -- I(X - i Y ) t ) - J2/3IZJ)

& 1

us0 = IP, 1/2, - 1/2) = - -I(X - i Y ) t )

J5

'"E. 0.Kane, J . Phys. Chem. Solids 1, 249 (1957).

1

+ -IZJ)

J5

(1.3a) (1.3b)

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G . BASTARD et al.

where

R = E - hzk2/2mo.

(1.4)

In Eqs. (1.3) the energy origin is taken at the l-6 (S-like)edge, E~ = Er6 - Ey, is the fundamental band gap, A = &r8- Ey, is the zone center spin orbit coupling, and

is the Kane matrix element. The r6 effective mass mr6 is isotropic (like any other effective masses in this description) and is such that 1

-=-

1

+ 2P2(&,++2A/3) A) ’

mr6 mo ‘O(‘O where usually the second term largely dominates the free-electron one. Equation (1.6) reciprocally allows a determination of P 2 if m,,, c0, and A are known, which is usually the case. The eight zone center Bloch functions can be labeled in terms of the eigenvalues of a pseudoangular momentum J and of its projection along an axis J , . r6 corresponds to J = 3 and J , = r, to J = 3 and J , = k+, and r7to J = and J , = Since the resulting dispersion relations are isotropic, this axis can be chosen at will-either along a crystallographic direction, say [OOl], or along the k vector. In the latter case, the eigenvalues of Eq. (1.3a) all correspond to J , = 3, while the dispersionless branch [Eq. (1.3b)l corresponds to J, = k 5. The latter is the heavy hole branch, and it is clear that k * p interaction between the (r6,r7,r,)and remote edges has to be taken into account if one wants to describe soundly the hole dispersions. When this is done, the E,(k) relationships can no longer be obtained analytically. In addition, they are no longer isotropic (band warping). It is often justified and useful to simplify again the complicated 8 x 8 hamiltonian to emphasize a property. For instance, in GaAs one may wish to describe fully the topmost hole kinematics. To do so, and provided the hole energy is much smaller than the zone center spin orbit coupling, one folds the 8 x 8 hamiltonian onto the 4 x 4 T, subspace to obtain the Luttinger hamiltonian”

4

3fr8 = Ak2

-

++.

++,

[B2k“ + C2(kIkg

+*,

+ k2k: + k,2kz)]”2.

In Si or Si-Ge alloys it is not possible to get rid of the r7 band (since As.o. 44meV in Si), and one ends up with a 6 x 6 (r, + r,) hole hamil29J. M. Luttinger, Phys. Rev. 102,1030(1956);see also G. L. Bir and G. E. Pikus “Symmetry and Strain-Induced Effects in Semiconductors.” Wiley, New York, 1974.

ELECTRONIC STATES IN SEMICONDUCTOR HETEROSTRUCTURES

1

1

- GaAs

1

1

I

l

l

I

1

--

l

InSb

l

1

I

I

I

I

235

-

-

0 0

W \

4

0,” -

--

-1 -

-

\

.<

-

-5

I

I

I

I

I

I

I

I

I

I

tonian. For the GaAs conduction states the most accurate nonparabolic hamiltonian is that derived by Braun and Ro~sler.~’ It is written Sr6 = h2k2/2m,,

+ a0k4 + /?,(k;k; + k,”k,’ + kzk;).

(1.8)

In bulk materials, the wave vector k has to be real to comply with the integrability of the wave functions. In heterostructures, where layers are at most semi-infinite, imaginary k values become relevant since, for a given energy, the heterostructure states can be built out of propagating Bloch states in one layer and of evanescent Bloch states in another layer. One should thus know the dispersion relations of these evanescent states. In quantum wells and superlattices, the heterostructure is translationally invariant in the layer plane. Thus, only the k component along the growth axis can be imaginary. In quantum wires k can have two imaginary components, and in quantum dots the wave vector can display three imaginary components. Figure 2 displays the real and imaginary dispersion relations for the light particle states [i.e., solutions of Eq. (1.3a)l in the Kane model. One sees the clear correspondance between the conduction and light hole states: a propagating light hole state becomes evanescent and progressively transforms in the gap into an evanescent conduction state that finally gives rise to a propagating conduction state. 30M.Braun and U. Rossler, J . Phys. C 18, 3365 (1985); U. Rossler, Solid State Commun. 49,943 (1984).

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G . BASTARD et al.

1 b. BUILDING HETEROSTRUCTURE STATES

We shall discuss the electronic states found in heterostructures made out of lattice-matched semiconductors. This is the exception rather than the rule. However, the strained-layer materials’ electronic states can as well be described by the same sort of formalism (for recent reviews see Refs. 20 and 26) as the one we shall use. Since we shall only marginally discuss the strain effects, we can as well discard this complication at the moment. Modern epitaxy techniques (molecular beam epitaxy, metal organic chemical vapor deposition) allow (in principle) the growth of defect-free, atomically sharp interfaces between two lattice-matched homopolar semiconductors (e.g., IIIV on 111-V or 11-VI on 11-VI). Thus an ideal interface between an AC semiconductor and a BC semiconductor will be a plane of C atoms, each having half of their bonds with A atoms and half with B atoms. It is thus obvious that an interface is a somewhat fuzzy concept, since at best it extends over two halves of a monolayer (one on each side of the C plane) and may eventually be thicker if there exist long-range potentials due to charge redistribution at the interface. This is the electronic rearrangement at the interfaces that fix the magnitude of the band offset (i.e., the energy by which the zero of energy of the BC band structure should be shifted with respect to that of the AC band structure). Only elaborate microscopic models can tackle the band offset problem. The emergence of heterostructure physics has been a strong impetus to research in band lineup, and significant progress has been achieved on the reliability of the first principle approaches. Although still questionable on the absolute value scale (the accuracy of a given band offsetis seldom better than 0.1 eV), the first principle predictions are unquestionably trustworthy when trends are concerned. For recent reviews on the band lineup problem, see Refs. 17 and 31. Once one band offset is known, all the others follow from the knowledge of the magnitude of the band gaps in the bulk AC and BC semiconductors. Thus we know which bulk eigenstates are going to enter into the heterostructure wave function. For a given energy E this wave function can be projected on the complete orthonormal set of bulk eigenstates at this particular energy:

In Eq. (1.9) the summation runs over all the bands of the AC and BC host layers; kLA),kiB)are the propagating (or evanescent) wave vector solutions of the implicit equations, (1.10)

ELECTRONIC STATES IN SEMICONDUCTOR HETEROSTRUCTURES

237

and $):,! are the Bloch functions for the nth band and wave vector k. Finally, Y(z E AC) denotes the step function, which is equal to 1 in the AC layers and 0 elsewhere. The coefficients a,, b,, c,, d, are determined by imposing the continuity of $ and a$/az across the interfaces. The allowed energy E in the heterostructure is obtained by demanding a given asymptotic behavior of $(r) at large distances. For instance, in a superlattice structure we require $ to satisfy the Bloch theorem: $(p, z

+ d ) = eiqd$(p, z).

(1.11)

In a quantum box we require that $ -+ 0 far away from the box, and so forth. The general solution given in Eq. (1.9) is totally useless because the Bloch functions are seldom known for all the bands of a crystal. Thus, we need to truncate the summation in Eq. (1.9) by using some physical reasoning. The key remarks on which envelope function type of models are based is that the band offsets are usually a small fraction of the host bandwidths and that the band structures of the host layers constituting the heterostructures are usually very similar. The first remark means that within a fraction of 1eV around the conduction band edge of the AC material one likely finds the conduction band edge of the BC material. The second remark implies that these edges are often of the same symmetry (say r6).Allowed heterostructure levels in the energy range CECA', E:'] that are predominantly built out of, say, propagating r6-related states in the AC layers and of evanescent r6-related states in the BC layers should thus exist. These states in both layers will be described in terms of the k p expansion around the r point. Our task will finally reduce to a generalization of the k p hamiltonian, which will account for the band offsets. As we shall see, this will amount to adding diagonal contributions to the k p matrix. This favorable situation occurs in Hg, -,Cd,Te-Hg, -,Cd,Te, InAs-GaSb, GaAs-Gal _,Al,As (x 5 0.4), Ga,.,,In,.,,As-InP-that is, in the vast majority of heterostructures. On the other hand, if the edges are of different symmetry, say r6 in AC and X , in BC, no simple treatment will be possible. Suppose then the band lineup is such that edges of the same symmetry are close in energy, and assume for definiteness that these are the r edges (the model works as well for L-related states as found in PbTe-Pb, _,Sn,Te heterolayers or X-related states in GaAs-A1As superlattices). Within the framework of the Kane model we shall write, as a truncation of Eq. (1.9),

-

-

a

$(r) =

I=1

+

(ul~)(r)[al(klA))e'"*"' b1(k'A))e 1 -ikiA'*r]

Y(z E AC) (1.12)

for heterostructures with a unidirectional modulation (z-axis) and under flat-

238

G.BASTARD et aI.

band conditions. We can exploit the translational invariance in the layer plane to assert that the in-plane projection of the electron wave vector

k,

= (4,ky)

(1.13) (1.14a) (1.14b)

We now make the ansatz underlying the envelope function scheme, which is that the zone center periodic parts of the Bloch functions are the same in the AC and BC layers:

uft)(r) = uif)(r) = ulo(r).

(1.15)

This identity is, of course, not exactly true, but it is very reasonable in view of, for example, the constancy of the Kane matrix element (Slp,lX) across the 111-V or 11-VI family. Smith and M a i l h i ~ t ~ ' ,have ' ~ developed an envelope function scheme that does not require Eq. (1.15) to be fulfilled. It turns out that the differences in the heterolayer energy levels calculated by assuming, or not assuming, Eq. (1.15)to be valid are small. Thus, we shall keep Eq. (1.15) as a good trade-off between accuracy and simplicity. Equations (1.14) and (1.15) allow us to rewrite Eq. (1.9)in the simple form Il/(r) = eik, 'rl

8

1 Ulo(r)flA%),

I=1

(1.16)

wheref$A,B)(z) is an envelope function that is slowly varying on the scale of the host periodicity. If, as assumed so far, there is no band bending in the heterostructure, f;'."(z) is a linear combination of plane waves, as seen by comparing Eqs. (1.12) and (1.16). If, on the other hand, there exists a slowly varying band bending (because there are charges), f;'~"will be more complicated. Let us now establish the effective hamiltonian acting on the envelope functions. To do so, we write the hamiltonian in the form Yt? = p2/2Wto

VA(T)Y(ZEAC)

+ VB(r)Y(Z€BC)+ Kxt(Z),

(1.17)

where I/ext(z)is the band-bending term, assumed to depend only on the z coordinate (generalization to more complicated cases-e.g, a coulombic potential due to an impurity does not cause any problem). The ulo(r) are, by definition, solutions of [p2/2mo

+ VA(r)Y(z€AC)+ ~B(r)Y(ZEBC)]ulo(r) = [&~*)Y(Z E AC)

+ &jB)Y(zE BC)]ulo(r).

(1.18)

ELECTRONIC STATES I N SEMICONDUCTOR HETEROSTRUCTURES

239

Letting A? act on Eq. (1.16), multiplying by u:o(r), integrating over a unit cell, and making use of the different lengths of variations of the envelopesf, and the cell periodic parts, we get

gf

(1.19)

= Ef,

where f is an 8 x 1 column vector whose componentsf, are solutions of the coupled differential system [&iA)Y(zE AC) + E ~ ) YE( BC) z

L

+ KXt(z)+ 2m0

2m0 dz

9 is thus the effective hamiltonian we were looking - for. All the microscopic

information about the rapidly varying functions has explicitly disappeared from 9.It survives, however, through effective parameters such as the r band gaps and the Kane matrix element. Notice also that in the envelope function hamiltonian the potential steps at the interfaces &IA) - 4’) appear only in the diagonal. This results from the assumption Eq. (1.15) and is more related to a symmetry argument than to an assumption of a slow variation. (The latter is certainly invalid since the band discontinuities take place on an atomic scale.) In the following we shall replace cIA)Y(zE AC) + &IB)Y(zE BC) by &IA) [email protected]), where v ( z ) are step functions that are zero in the AC layers and equal to elB) - 6:”’ in the BC layers. Thus they are the algebraic energy shifts of the Ith edge when going from the AC layer to the BC layer. If E ~ A, A , A B are the r 6 - Ts bandgaps and zone center spin orbit coupling in the AC and BC layers, respectively, there exists a single unknown, say the l-6 offset (hereafter termed V,), and the other edge offsets (V, and V,, respectively) are expressible in terms of V,, cA, cB, A A , A5 (see Fig. 3). Depending on the relative signs of V,, V,, V,, there exist several kinds of heterostructures that have significantly different electronic properties (see Fig. 4).Type I heterostructures are the most common. They correspond to V, > 0, V, < 0 and are therefore such that one kind of layer confines both the electrons and the holes, a very favorable situation for photodetectors and lasers. GaAs-Gal -,Al,As (x 5 0.4),Gao,,In,~,,As-InP, GaSb-AlSb, and CdTe-Cd, -,Mn,Te are type I heterostructures. Type I1 heterostructures (also called staggered or misaligned) are such that V , > 0 and V, > 0. Thus, one kind of layer confines the electrons but is a barrier for holes that are confined in the other kind of layer. A10,,81no,,2As-InP and InAs-GaSb are examples of type I1 heterostructures. In the latter case V, is so large (-0.56 eV) that it exceeds and actually the top of the GaSb valence band lies higher in energy than the bottom of the InAs conduction band, which

+

240

G. BASTARD et a/.

A

B

FIG3. Relationship between V,, V,, V, and &A,

U

In As- Gash

E ~ AA, ,

Ae.

c

r6

tYPe

FIG.4. Ts and Tsedge profiles in type I, type 11, InAs-GaSb, and type 111 quantum wells.

ELECTRONIC STATES IN SEMICONDUCTOR HETEROSTRUCTURES

24 1

gives rise to a number of interesting physical situations such as occurrence of an apparent semimetal-to-semiconductor transition, formation of interface excitons, and coexistence of electrons and holes. Finally, a peculiar type 111 band ordering exists in some 11-VI heterostructures involving Hg: Hg,-,Cd,Te-CdTe (x 5 0.16) and Hg, _,Mn,Te-CdTe (x 5 0.07). It corresponds to V, > 0 and V, < 0, but the difference with the usual type I structures is that c0 < 0 in Hg, -,Cd,Te (x 5 0.16) and Hg, _,Mn,Te (x 5 0.07). This negative c0 implies that these materials are zero-gap semiconductors due to symmetry-induced (r,)band degeneracy between the heavy hole band and the light electron band. The r6band is now a light hole band (instead of being a conduction band as in GaAs or CdTe) to comply with the fact that cr6 < cr,. What makes HgTe-CdTe such peculiar heterostructures is the sign reversal of the light Ts particles across the interfaces, since they are electrons in one kind of layer (HgTe) and light holes in the other kind of layer (CdTe). Notice that the type 111 configuration should also be found in HgSebased heterostructures, but they have not been much studied so far. Before discussing Eq. (1.20), we point out that the envelope function scheme is not restricted to the r-related extrema. It describes as well states originating from the L-extrema, as shown by Kriechbaum for PbTe-based heterostru~tures,~~ or from the X-extrema. A given heterostructure does have electronic states that are derived from different extrema in the hosts’ Brillouin zone. Usually, they do not interfere, there being enough energy separation. Under some circumstances (small well thicknesses or external hydrostatic pressure) electron states originating from, say, r-and X-extrema, are made to overlap on the energy scale. The envelope function models, applied separately to r-related and to X-related states, predicts that these states cross when the parameter that controls the T-X energy separation changes. These crossings do not exist and are actually replaced by anticrossings. To describe the latter, we should either modify the envelope function framework or, more accurately, perform a full band structure calculation, starting from a global description of the electronic states over the hosts’ entire Brillouin zone. The most notable situations where the heterostructure eigenstates involve the coupling of different Brillouin zone extrema are (i) the GaAs- AlAs shortperiod superlattices where the r-related states are made to anticross the X , related states by decreasing the GaAs well thickness, (ii) the GaSb- AlSb narrow quantum wells, where the r-related conduction states anticross the Lrelated levels, and (iii) the Ge-Si superlattices where the X-related states of Ge are hybridized with the L-related states of Si to form the superlattice eigenstates. The Q l l , matrix elements can be explicitly evaluated from Eq. (1.20) and Table I. One finds

0

0

0

0

0

0

-J2/3Phk,

0

0

0

J2/3Phk-

0

0

Phk+/,,h

Phk..

EA-AA+

V,(Z)

+ h2k2/2m, +K X d 4

0

- J2/3PhkZ

J2/3Phk+

0

0

0

0

0

0

0

0

h2k2/2m,

+K(4 +K X d 4

Phk-/,,h Phk +

(1.21)

ELECTRONIC STATES IN SEMICONDUCTOR HETEROSTRUCTURES

where the energy zero has been taken at the k, = 2-'I2(k,

k

= (kl,

243

r6edge of the AC material:

ikJ,

(1.22) (1.23)

-i(d/dz)),

and P is the Kane matrix element defined in Eq. (1.5). Equation (1.21) is useful in order to calculate the I-,-related dispersion relations of heterostructures as well as the edges (k, = 0) of the r7-and r8 ( J , = f 112)-related states. It has, however, a severe drawback, namely the inaccurate description of the heavy hole-related state and more generally of the r8-r7 kinematics. It is well known that the heavy hole effective mass does not originate from the direct k * p coupling between the r6, r7,and Ts edges. Rather, it arises from the indirect (second-order) k p coupling between these edges due to virtual transitions to more remote band edges of the crystal. The indirect k p matrix is complicated but well known.29 In the ulo basis of Table I the indirect k * p matrix consists of a diagonal contribution F(h2k2)/2moin the r,-related diagonal block, no coupling term between T6 and r7,r8, and a 6 x 6 submatrix in the r8 x r7 block. Except for Si-Ge heterostructures where AA is small (AA = 44 meV in Si), the zone center spin-orbit coupling is usually large compared with the energies of the valence subbands of practical relevance. Thus, it suffices to describe the r8-related kinematics and to discard the r8 couplings with r,. In such a case, the l-8 block of the secondorder k * p interaction takes the form

-

-

1312,312)

1312, - 112)

1312, 112)

1312, - 312)

244

G. BASTARD et al.

(1.27) (1.28) and yl, yz, and y 3 are the modified Luttinger parameters of the valence bands-that is, those computed by taking into approximate account all the hosts bands but r6.These parameters are, like K ( z ) , V,(z), and G(z),step functions of the electron position along the growth axis with values yIA), ,)":y and y$*' (respectively, yIB), ybB), and yhB1) in the AC (respectively, the BC) material. This position dependence of the y parameters leads to the symmetrization procedures apparent in Eqs. (1.25)-(1.28). For instance, an expression of the form h2(y, - 2y2)k,2in a host bulk has to be replaced by pz(yl - 2y2)p, in the heterostructure to comply with the hermiticity requirement for the hamiltonian. Usually, the fact that the y parameters are positiondependent is not quantitatively very important for the evaluation of the heterostructure eigenstates. This is due to the heavy localization of the Tarelated eigenstates in one kind of layer. A counterexample of this statement is provided by the type I l l heterostructures: in these materials the y parameters change sign from one layer to the other, which leads to wave functions peaking at the interfaces. To find the eigenstates of our complete effective hamiltonian, we need to derive some connection rule at the interfaces for the envelope wave functions and to impose boundary conditions at large z on these envelopes. Since the effective hamiltonian is quadratic in p z , the envelope wave function f should be continuous. The second rule (the equivalent of the a$/& continuity in vacuum) is fulfilled by demanding the continuity of Af, where A is a n 8 x 8 matrix with elements All, equal to SdzD,,.. Since the first-order k - p interaction only gives rise to gar[. terms that are linear in a/&, its contribution to the Af continuity does not provide any new information inasmuch as thef,'s have to be continuous from the first connection rule. New information arises from the second-order k p terms. For instance, the restriction of A to the Ta subspace is Eq. 1.29 where Ac3Y(z)is the jump yiB) - yhA) of the y 3 parameter when going from the AC to the BC material across the z = O interface, and where other terms have been dropped since they amount only to the f continuity. The off-diagonal Af terms are proportional to k , and k, and therefore couple the (3/2, & 3'2)envelopes to the 13/2, & 1/2) envelopes only if k, # 0. The boundary conditions at large z depend on the problem at hand. For a quantum-well problem when an AC layer is surrounded by two semi-infinite BC layers, there will exist bound states in an energy segment corresponding to propagating states in the AC layer and evanescent states in the BC ones.

-

0

246

G. BASTARD et al.

To calculate these bound states, we shall require that f(z) vanish at z = _+ co. Under flat-band conditions when no other z-dependent potential Kxt(z), exists, (r/ext(z)= 0) there are 8 x 2 independent solutions per bulk layer corresponding to a given energy E. Within each host layer the general solution of the effective hamiltonian is a linear combination of these 16 independent solutions. In the BC layers, however, half of these solutions have to be discarded since they would correspond to exponentially growing f either when z + + m or when z + -a. Thus, altogether we have 16+%16+ 16) = 32 unknown coefficients. The boundary conditions at the two AC-BC interfaces provide us with 2 x 8 + 2 x 8 independent relations, which is exactly equal to the number of unknowns. Thus, the bound states of a quantum well can only occur for discrete energies. For a superlattice we use the BIoch theorem. lfdis the superlattice period (d = LA + LB)we demand that f(z

+ d ) = ei4df(z),

(1.30)

where q is the superlattice wave vector along the growth axis that we can restrict to the first superlattice Brillouin zone:

-n/d

< q < +n/d.

(1.31)

Suppose we take for the superlattice unit cell two consecutive AC and BC layers. Denote by z = 0, z = LA, and z = d the locations of the AC-BC interfaces. There are 16 + 16 unknown coefficients to be determined. The continuity conditions at the inner (z = LA)AC-BC interface provide us with 16 independent conditions. Those at the right-hand-side boundary ( z = d ) give us 16 others. To write f at z = d in the AC layer, one uses the Bloch theorem [f(z = d’) = eiqdf(z= O’)] and finally ends up with a 32 x 32 homogeneous set of equations: for each q (and k,, of course) value there exist discrete allowed superlattice energies that correspond to the various superlattice minibands. If we let q describe the segment [-n/d, +n/d], the miniband is generated by continuity. If the in-plane wave vector is set equal to zero, a considerable simplification occurs: an inspection of Eqs. (1.21)-(1.24) reveals that (i) Thefis associated with the heavy hole band edges )3/2, & 312) become decoupled from those associated with the light particle band edges IS, &$),

1% +t>> It

+3>.

(ii) The eigenstates are twice (Kramers) degenerate.

”S. Baroni, R. Resta, A. Baldereschi, and M. Peressi in “Spectroscopy of Semiconductor Microstructures” (G. Fasol, A. Fasolino, and P. Lugli, eds.). NATO AS1 Series B, Physics, Vol. 206, p. 251. Plenum Press, New York, 1989.

247

ELECTRONIC STATES I N SEMICONDUCTOR HETEROSTRUCTURES

Indeed the complete effective hamiltonian splits into two decoupled and identical blocks:

D = [ ” +0

‘1,

D-

(1.32)

where

(ism

v,(4 + v,,I(Z)

(312,3121

0

-&A+

vp(z)

-PAY1

(312, 1/21

- [email protected]

0

+P , F P , / ~ ~ ~

+

PP.IJ5

0

0

-2Yz)Pz/2mo

-PP,@

0

vp(z)+ XI(^) +2Y,)Pz/2mo Jz,,YaP.lmo

-&A+

fiPzYZPz/mO

-PAY1

(112, 1/21

PPSJ3

0

V,(Z)+ -PzY,Pzl2mo

-&A-AA+

v,&)

(1.33) The second row and second column, which are uncoupled with the others, refer to the heavy hole states. The first row and column are associated with the r6 edge, while the third and fourth rows and columns refer to the I$,$) and I%,4) edges, respectively. The energy zero is again taken at the bottom of the r6 edge in the A material. The eigenstates at k, = 0 thus split into two categories:

f=

[6.],

I;[

(1.34)

f- =

where f, and f - are now 4 x 1 column vectors that are, respectively, eigenstates of D + and D - . The boundary conditions are that f + and f - are continuous and that A + f + and A -f- are continuous at the heterointerfaces, where A+=A-=

0

0

0

-

0

0

0

0

0

h2JZ

a

m, az

(1.35)

248

G. BASTARD et al.

Before presenting numerical results in the following sections, let us derive some explicit results that follow from Eqs. (1.34) and (1.35) after some simplifications. First, the heavy hole states (at k, = 0) are obtained by solving a Ben Daniel-Duke equation3’ for their envelope functions:

The y l , y 2 jumps (or, in other words, the heavy hole effective mass jumps) at the interfaces change the usual continuity rule of dfhh/dz into that of m i ,(Z)(dfhh/dZ) at the interfaces. This is consistent with the requirement of the conservation of the probability current. Second, if one neglects all the second-order k - p terms in the D,, D matrices, it is possible to express all the components of the light particle f,, fin terms of a single one, say the r6-relatedf,, in which case one finds that the remaining envelopef, is a solution of an energy-dependent Ben Daniel-Duke problem: (1.37) where

(1.38) As in the usual Ben Daniel-Duke problem,32 one finds that f , and p - ‘ ( E , z)(df,/dz) have to be continuous at the interfaces. The effective mass in Eq. (1.38) is position- and energy-dependent. The energy dependence, absent in the hole Ben Daniel-Duke problem [see Eq. (1.36)], arises from nonparabolicity effects-that is, from the fact that the energy E is not assumed to be A A . Notice that in the negligible with respect to the band gaps presence of band bending there is an explicit position dependence of &,z) even in the AC material where Vp(z) and &(z) are zero. This is because the nonparabolicity effects are local, and what matters is the energy measured from the local r6 edge-that is, E - V,,,(z)-compared with the band gaps + A A . This implies that nonparabolicity corrections can be nontrivial. If one expands the denominators in Eq. (1.38) in ascending powers of [ E - Kxt(Z)]/[&A - Vp(z)], and similarly for the spin-orbit terms, one finds first-order corrections to the kinetic energy terms that involve crossed contributions like pzKxt(z)pz.These sorts of terms are reminiscent of the

+

32D.J. Ben Daniel and C. B. Duke, Phys. Rev. 152, 682 (1966).

ELECTRONIC STATES I N SEMICONDUCTOR HETEROSTRUCTURES

249

relativistic corrections to the Schrodinger equation when one expands the Dirac equations in powers of vlc, the band gaps cA, + A A playing here the part of moc2. The simplicity of Eqs. (1.37)-( 1.38) compared to the original hamiltonian Eq. (1.33) has been obtained at the expense of the continuity of all thefi’s. In fact, since f , and ,t-l(&,z)(df,/dz) are continuous from Eq. (1.33), the components

cannot be simultaneously continuous, except if the spin-orbit splitting is zero or infinite in both bands. These two conditions ensure that ,u(E,z) is proportional - Vp(z)+ K,,(z). In actual heterostructures the assumption of infinite spin-orbit coupling is more realistic, and, in fact, it is often an acceptable (good) approximation in arsenide (antimonide, telluride)-based heterostructures. The full hamiltonian Eq. (1.33) is not free of difficulties. The second-order k * p terms bring in the continuity of all the fi’s. However, it carries over a “wing bands” problem, first pointed out by White and Sham’ and lucidly discussed by Schuurmans and t’Hooft.’’ When the small F, yl, y z terms are neglected, thef, are discontinuous. When they are taken into account, the f, continuity is restored. Since the F , yl, y z terms are small, they do not play any significant part far away from the interfaces, where the envelopes are built out of plane waves with small wave vectors. To ensure thef, continuity at the interfaces, the F , yl, y z terms should be added tofi’s plane waves with large, evanescent wave vectors. Indeed, when the evanescent solutions of Eq. (1.33) are calculated for each bulk layer, they fall into two categories. The first one involves small wave vectors and is already present in the Kane model. The solutions of the first kind are stable with respect to small changes in F , yl, y z parameters. The second set of evanescent solutions of Eq. (1.33) involves very large wave vectors, eventually exceeding the size of the Brillouin zone. They are unstable with respect to small changes of the F, yl, y z parameters. These are the “wing bands,” to which the continuity of all thef, solutions of Eq. (1.33) is warranted. Thef,’s are continuous but vary very rapidly (on the scale of a host unit cell) in the vicinity of the interfaces. Away from the interfaces, the wing bands have died out and thef, vary gently. One may wonder if the existence of wing bands is not contradictory with the envelope function scheme. The eigenenergies of Eq. (1.33) are insensitive to these wing bands, and most of the observable quantities, such as oscillator strength for optical

250

G. BASTARD et al.

transitions, are also unaffected by the details of the envelope functions near the interfaces. Apart from empirical tight-binding and pseudopotential approaches (which also have some degree of arbitrariness in their description of the interfaces), there have been several attempts to improve or to justify the envelope function framework. The improvements aim at relaxing the stringent conditions imposed on the cell periodic parts of the Bloch functions.25,26,33,34Smith and Mailhiot’ 5 , 2 6 have developed a generalized envelope function approach and applied it to a variety of heterostructures. These authors and Ando et have found that, usually, the results of their more elaborate models are close to those of the simpler envelope function scheme used here. 111. Subband Edges in Quantum Wells and Superlattices

In the following we assume that the heterostructures are under flat-band conditions [i.e., l/ex,(z) = 01. 2. QUANTUM WELLS

The subband edges of a rectangular quantum well are found by solving Eq. (1.36) for the heavy hole states and Eqs. (1.37)-(1.38) for the light particle states. The most salient effect is the nonvanishing confinement energy experienced by a carrier when it is placed in a quantum-well structure compared with its lowest energy (the band edge) when it is in the bulk wellacting material. It can be shown that this excess kinetic energy is of quantum origin insofar as it is a direct consequence of the Heisenberg inequality Az Ap, 2 h/2, where Az = and similarly for Apz. If the particle is known to be in the well, the uncertainty on its position is L (i.e., Az L), where L is the quantum-well thickness. The Heisenberg inequality implies that Ap, R h/2L. Since the lowest state in the well is bound, ( p , ) vanishes and, therefore, ( p i ) k h2/4L?, which implies that the average kinetic energy is nonvanishing and about h2/8m,L2. It is only in the bulk ( L -+a) that the extra kinetic energy vanishes and that the ground state of the particle may coincide with the r6 band edge of the A material. The eigensolutions of the rectangular quantum-well problem are well known. Let us discuss the case of the light particle states [Eqs. (1.37)-(1.38)]. Since the potentials r/s(z), V,(z), and &(z) are even in z, one may choosef,(z) as

-

d

w

,

-

”M. G. Burt, Semicond. Sci. Technol. 3, 739 (1988); Semicond. Sci. Technol. 3, 1224 (1988); in “Band Structure Engineering in Semiconductor Microstructures” (R. A. Abram and M. Jaros, eds.), p. 99. Plenum, New York, 1989. 34T.Ando, S. Wakahra, and H. Akera, preprint, 1989; T. Ando and H. Akera, preprint, 1989.

25 1

ELECTRONIC STATES IN SEMICONDUCTOR HETEROSTRUCTURES

even or odd in z. Equation (1.39) shows thatf, andf, will thus be odd or even. For even f c we take = A,cos

k,z,

d L/2, = B, expr - L/2)1, z 3 L/2, = BcexPC~c(z+ L/2)1, z d -L/2, while for odd f , we take

i

IZI

f , ( z ) = A , sin k,z, f c ( 4 = B, exPC - K,(Z

L(4 = -BcexpC+

where E(E

(E -

V,)(E - V ,

+

&A)(&

IZI

- L/2)1,

+ L/2)1,

d L/2,

z 3 L/2,

+ EB)(E

-

E

(2.2)

z d -L/2,

+ + AA) = h2k:P2 ( + + EA

(2.1)

EA

V , + EB + AB) = - h 2 ~ , 2 P 2 E

(

-

3

V , + EB + - . 2AB> 3

(2.3) The boundary conditions at z = L/2 yield implicit equations whose roots are the allowed energies for the even and odd states: kcL kc . kcL cos - --sin -= 0, 2 PA(&) ICc

. kcL sin-

kc PB(&) kcL +-cos 2 ~

PA(&)

= 0.

ICc

(2.4) (2.5)

It is easily shown that there always exists one bound (even) solution for all L. The confinement energy of this ground bound state approaches V , only when L + 0, while the (n + 1)th bound state (n = 1 , 2 , . ..) coincides with the top of the well = V,) when

kc(V,)Ln= n ~ . (2.6) The quantum-well continuum is not structureless; that is, the transmission coefficient of a particle with energy E > V, impinging from the left on the quantum well is not unity but exhibits oscillations. To find the continuum solutions of Eq. (1.37), we write = eikb(Z+L/2) + r(

-ikb(Z+L/2)

4e = a cos kcz + fl sin k,z,

, z < -LIZ I4 < L/2, z 2 L/2,

(2.7)

252

G . BASTARD et al.

where k, is related to (E

-

E

by Eq. (2.3) and kb by

K)(E- V, + EB)(E

-

V, + EB + AB)

= h2k2P2(&-

V , + EB + 2AB/3). (2.8)

The transmission amplitude is found to be equal to 1 t(&)= cos k, L - (i/2)(5

+ 5 - ‘)sin k, L ’

(2.9)

and the transmission coefficient T(E), which is equal to It(&)IZ, takes the simple form

T(E)=

1 + &C

1

+ 5-l)sin2 k,L ’

(2.10)

where

5 = kbpAfpBkc*

(2.1 1)

The transmission is zero at the edge of the quantum-well continuum (since k, and diverge). Going farther in energy from this edge, T exhibits oscillations, reaching unity when k,L

=pz.

(2.12)

These transmission resonances correspond to virtual bound states of the quantum well35characterized by a piling up of the wave functions in the well when compared with the nonresonant situation. This virtual bound states can also be seen as resulting from a Fabry-Perot effect for the electron waves in the quantum-well slab.36There is in fact a very close analogy between the Schrodinger equation and that describing the propagation of an electromagnetic wave in a medium of piecewise varying refractive index. When approaching the edge of the quantum-well continuum, the wavefunction inside the quantum well at the transmission resonance looks more and more like a true bound state eigenfunction. Indeed, when p = 2j in Eq. (2.12), where j = 0, 1, . . . , one finds that the coefficients tl and /l in Eq. (2.7) go to (- 1)’ and 0, respectively, when the 2jth resonance approaches the top of the well, which means that the 2j th resonance matches the even (2j + 1)th bound state of the well when E z j + l coincides with V,. Symmetrically, the (2j + 1)th resonance ( j = 0, 1, . . .) corresponds to tl and /l coefficients, which go to 0 and (- l)j, respectively, when it approaches the continuum edge. Thus, this resonance matches the (2j + 2)th quantum well bound state at this edge. 35G.Bastard, Phys. Rev. B 30, 3547 (1984). 36D.Bohm, “Quantum Theory,” Prentice-Hall, Englewood Cliffs, N.J., 1951.

ELECTRONIC STATES IN SEMICONDUCTOR HETEROSTRUCTURES

253

FIG.5. Calculated bound (solid lines) and virtually bound (dashed lines) conduction states in lattice-matched Ga(1n)As-InP single quantum wells.

Examples of calculated energy levels in semiconductor quantum wells using Eqs. (1.36)-(1.37) and (2.12) are shown in Figs. 5-8 for the conduction (T,-related) states of Ga,,,,In,, ,As-InP, Gao,471no.3As-Alo.481no.5,As (Figs. 5, 6), and of GaSb-InAs-GaSb double heterostructures (Fig. 8). The first two systems are of type I and potentially interesting for long-wavelength communications, since Ga0~,,1n,,,,As is a relatively narrow gap material ( E ~ 0.8 11eV at low temperature). Thus, one may easily create situations (narrow quantum wells) where the electron confinement energy is sizable with respect to the band gap (i.e., where nonparabolicity effects are significant). The comparison (Fig. 7) between the energies of the fundamental optical transition3, (HH, -+ El) of Ga,~4,1n,,,,As-InP and Ga,.,,In,,,3AsN

37Theexperimental data are taken from (a) W. Stolz, K. Fujiwara, L. Tapfer, H. Oppolzer, and K. Ploog in “GaAs and Related Compounds” (B. de Crtmoux and Adam Hilger, eds.) p. 139. Bristol, 1985 and references cited therein for Ga(1n)As-AI(1n)As. (b) D. Gershoni, H. Temkin, and M. B. Panish, Phys. Reo. B 38, 7870 (1988); D. Moroni, J. P. Andre, E. P. Menu, Ph. Gentric, and J. N. Patillon, J . Appl. Phys. 62, 2003 (1987) for Ga(1n)As-InP.

FIG. 6. Calculated conduction bound states in Ga(1n)As-Al(1n)As quantum wells latticematched to InP. L

(A, *

GdIn)As-AL(In)As

FIG.7. Calculated energy shift of the band-to-band fundamental absorption edge versus the Ga(1n)Asslab thickness in Ga(1n)As-Al(1n)As (lower horizontal scale) and Ga(1n)As-InP (upper horizontal scale) quantum wells. The symbols correspond to various experimental data.37

ELECTRONIC STATES IN SEMICONDUCTOR HETEROSTRUCTURES

255

Alo,481no.52As, where HH, is the ground heavy hole bound state, is interesting since the well (Gao,471no,53As) is the same in both systems. The barriers are also similar, but the band offsets are quite different (see Table I1 for the assumed band parameters). Thus, Fig. 7 is an illustration of the effect of the band offset on the position of the fundamental band-to-band absorption edge (or on the energy of the intrinsic photoluminescence line). It is evident in Fig. 7 that a reliable determination of the band offset from the sole knowledge of the El-HH, transition energy is almost impossible, which in particular rules out photoluminescence as a useful tool to determine the band offset. The case of GaSb-InAs-GaSb type I1 double heterostructures is peculiar, owing to the overlap of the GaSb valence band with the InAs conduction band38-41(see Table I1 for the assumed band parameters). For large enough InAs slab thicknesses, the ground (and eventually some excited) r,-related InAs bound state will be degenerate with the GaSb light hole continuum (since at k, = 0 the heavy hole states do not mix with the light particle states and can be discarded from the present analysis). This state will actually be virtually bound. Finding its energy location is equivalent to looking for the TABLEIIa. FUNDAMENTAL r BANDGAP(E,), r SPIN-ORBITENERGY(A), r6 BAND EDGE EFFECTIVEMASS (n,,, IN UNITSOF THE FREE ELECTRON MASS)FOR SOME111-V COMPOUNDS.

AND

(eV) A (ev)

E,,

m:6/m,

GaAs

GaSb

InAs

InP

AlSb

1.5192 0.341 0.067

0.81 1 0.752 0.041

0.418 0.38 0.023

1.4236 0.108 0.079

2.3 0.75 -

TABLE IIb. LISTOF SOME A-B TsBANDOFFSETS BETWEEN Two A AND B MATERIALS USEDIN THE CALCULATIONS. GaAs-Ga,, 7Al,,. ,As In As-GaSb Ga,,,71n,,5,As-InP Gan.47Ino.5 3As-Al0.4aIno. 5 A s Hgo.wCdn.,iTe-CdTe GaSb- AlSb

0.25 eV 0.96 eV 0.24 eV 0.4-0.5 eV 1.33 eV 1.19 eV

38G.A. Sai-Halasz, R. Tsu, and L. Esaki, Appl. Phys. Lett. 30,651 (1977). 39G.A. Sai-Halasz, L. L. Chang, J. M. Walter, C. A. Chang, and L. Esaki, Solid State Commun. 27, 935 (1978). 40L. L. Chang, N. Kawai, G. A. Sai-Halasz, R. Ludeke, and L. Esaki, Appl. Phys. Lett. 35,939 (1979). 41L. Esaki, IEEE J . Quantum Electron. QE22, 1611 (1986) and references cited therein.

256

G. BASTARD et al.

FIG.8. Calculated bound states (solid lines) and virtually bound states (dashed lines) in GaSbIn As-GaSb double heterostructures versus the InAs slab thickness. The energy zero is taken at

the Tsedge of InAs.

transmission resonances of a light hole impinging from one GaSb layer, tunneling through InAs (as a propagating electron), and finally escaping into the other GaSb layer as a propagating light hole state.42 For that purpose one only has to use Eq. (2.12). We see in this example one advantage of the multiband envelope function approximation, which allows one to conveniently handle the apparent change of the nature of a particle (a r,-related light hole into a r6-related electron) at the interfaces. Actually, there is no abrupt change in the envelope column vector f. The transmission at the interface is made possible because the f in the GaSb layer displays a nonvanishing r,-related component and the f i n the InAs layer displays a nonvanishing r,-related component-in other words, because of nonparabolicity effects. As seen in Fig. 8, the transmission resonances fulfilling Eq. (2.12) match the InAs true bound states when the well is narrow enough to make their energy exceeding the top of the GaSb valence band. Since the GaSb layers, which are potential wells for holes, are assumed to be infinitely thick, there exists only a heavy hole continuum extending from large negative energies to the top of the GaSb valence band. Finally, when the carrier energy exceeds the onset of the GaSb conduction band, the bound states of the GaSb-InAs-GaSb again become transmission resonances. Notice that for 42G. Bastard, Surf: Sci. 170, 426 (1986).

ELECTRONIC STATES IN SEMICONDUCTOR HETEROSTRUCTURES

257

such a large kinetic energy, the envelope function of the InAs conduction electron has nearly as large r,-related components as r,-related ones. The spatial localization of the carrier is better ascertained by calculating the expectation value over the nth eigenstate of the standard deviation Anz of the carrier position A,z

= J(z’)~

(2.13)

- (2):.

For a square well, (z), vanishes since the eigenstates are of definite parity. Figure 9 shows the dependence of the ground bound state confinement energy E l and of Alz upon the quantum-well thickness L in a simplified case: the nonparabolicity and effective mass mismatch have been neglected. The parameters chosen in Fig. 9 are not too different from those pertaining to actual GaAs-Ga(A1)As quantum wells (m*= 0.07rn0, barrier heights vb ranging from 83.5 to 334 meV). The confinement energy decreases monotonically with L approaching the behavior h2n2/2rn*L2at large L. A1(z) shows a minimum that shifts toward lower L’s with increasing vb. Actually if vb is infinite, Alz should vary linearly with L: AIZ(vb = a)= L

I

I

,

/

I

m

.

(2.14)

1

FIG.9. The confinement energies (dashed lines) and the root-mean-square deviation on the electron position with respect to the center of the well (solid lines) are plotted versus the well thickness L for several values of V, Curves 1, 2, 3, 4, respectively, correspond to V, = 83.5 meV, 167.1 meV, 250.6 meV, 334.2 meV, respectively. m*=0.07m0.

258

G. BASTARD et al.

The curves A l z versus L at finite V, all stay above this asymptote due to the nonvanishing penetration of the carrier in the barrier. We know that at vanishing L, Alz will become infinite. Thus, the Alz curves should have a minimum. It is for this L value that the maximum spatial confinement is achieved. Another quantity of interest is the integrated probability P b of finding the carrier in the barrier when its quantum state is El. Pb decreases with L from 100% at L = 0 to 0 at infinite L. It is not difficult to show algebraically that Pb decreases like L - 3 at large L which is a relatively slow decay. The same L dependence is found for the El deviation to the infinite well result. There has been recently a very fine demonstration that the envelope wave functions of actual quantum wells (GaAs-Gal -,Al,As) do behave like the predictions discussed p r e v i ~ u s l y . To ~ ~ *probe ~ ~ the quantum-well eigenstates, Marzin and Gerard intentionally introduced during the growth planes of isovalent impurities (In or Al) at the Ga sites. The isovalent impurities are of short (atomiclike) range. Thus, on the scale of the envelope functions they behave like delta potentials that are either attractive (In) or repulsive (Al). These perturbations lead to energy shifts of the quantum-well eigenstates that are proportional to the envelope probability densities of finding the carrier at the impurity planes. By optically detecting the shifts of the band-to-band transitions for several locations of the impurities, Marzin and Gerard were able to reconstruct the envelope probability densities for several eigenstates. An example of their findings is shown in Fig. 10. It is clear that these results support the contention that the envelope function framework is consistent with the experimental observations. One may wonder that actual quantum wells are characterized by finite barrier thicknesses, being limited by the sample surface on the one hand and by the buffer layer on the other hand (Fig. 11). Thus, a carrier initially placed in the quantum well may finally escape in the buffer layer (rather than outside the sample due to the larger barrier that occurs in the vicinity of the surface). It is clear that if the barrier is of finite thickness, the quantum well no longer admits true bound states but only resonant states. A semiclassical analysis helps to find a criterion that defines the minimal barrier thickness beyond which the lifetime of such resonant states is long enough. In this semiclassical reasoning, each time the particle hits the barrier it has a finite probability Po of escaping. Thus, the escape frequency w is w = z-'Po,

(2.15)

where z is the period of the classical motion. But Po coincides with the 43J. Y. Marzin and J. M. GCrard, Phys. Rev. Lett. 62, 2172 (1989), 44J. M. GPrard and J. Y. Marzin, Phys. Rev. B 40, 6450 (1989).

ELECTRONIC STATES IN SEMICONDUCTOR HETEROSTRUCTURES

259

Experimental Envelope Probability Densities

0.15-

0

80

160

0.10-

0.H)

0.05 -

0.05

0.00-

0

a0 Z A

o

MO

0.00

0

80

160

z(i)

In

A A1

FIG.10. Calculated and measured envelope probability densities versus the electron position for the three lowest bound states of a GaAs-Ga(A1)As quantum well. After Ref. 43.

transmission of the finite barrier for the carrier, which has energy El corresponding to the classical period z. Thus, a stability criterion can be set by demanding that oz << 1-in other words, that the transmission of the finite barrier be low. Since the transmission decays exponentially with 21cbh, where &, is the evanescent wave vector in the barrier at the energy El and h is the barrier thickness, one should have icbh >> 1. Notice that, as expected, the criterion is more stringent for narrower wells or, for a given well thickness, for excited states. The criterion oz << 1 refers to the existence of the quantum level E , versus its decay in the continuum of states to which it is coupled by the imperfect barrier opacity. It may happen that the time required to observe the level, say through its participation in an optical transition, is considerably longer, because the

FIG.11. Sketch of the relationship between the escape frequency of an electron in a virtually bound state of a quantum well and the period of the classical motion T and the transparency of the barrier T(E,).

260

G. BASTARD et ul.

coupling constant involved in the observation is small. In such a case the carrier in such a quantum state will be ‘‘lost’’ for that measurement, even though the quantum state was stable against tunneling through the finite barrier. In a double-barrier structure, for instance, the lifetime of a virtual bound state may be long (say 50ps) with respect to z (a few ps) but much shorter than a typical recombination time (say 500 ps). The radiative recombination of the electron in the virtual bound state at resonance with eventual holes will thus be very difficult to observe under C.W.conditions. 3. SUPERLATTICES A periodic superlattice is an infinite array of layers such that one may define an elementary cell generating the whole lattice by successive translations zd of the fundamental period along the growth axis. The simpler case is the binary superlattice AB whose elementary cell comprises two undoped layers of A and B materials with thicknesses L A and L,, respectively. This is the practical realization (and generalization) of the Kronig-Penney model, where the potential is piecewise constant and exhibits steps V,, V,, V,, and so on, at the interfaces. Doping superlattices (“ni-pi”) may even involve a single host material, the periodic modulation being achieved by doping. Recent advances in planar (or spike) doping have allowed the growth of high-quality doping superlattices (for reviews see Refs. 45 and 46). In a superlattice, the periodicity of the superlattice potential implies that the eigenstates fulfill the Bloch theorem:

where q is the superlattice wave vector that belongs to the segment [ - n/d,n / d ] .In addition, a binary superlattice AB under flat-band conditions admits two sets of inversion centers: if C A , C , denote any of the centers of each A or B layer, the superlattice potential V ( z) is even with respect to z - CA,z - C,. Let R,, R , denote the reflection operators with respect to a particular pair C,, C,, where C , - C , = d/2. We know that

[z, RA] = [z, R B ] = 0, [Ix,

Fdl

= O,

(3.2) (3.3)

where X is the superlattice hamiltonian and [A, B ] denotes the commutator of A and B operators. Moreover, one can readily verify that

45G. H. Dohler, I E E E J. Quantum Electron. QE22, 1682 (1986). 46E. F. Schubert, SurJ Sci. (1990) in press.

ELECTRONIC STATES IN SEMICONDUCTOR HETEROSTRUCTURES

Thus,

RA

261

and R , in general do not commute since

[Re, RA]

=F

d - F - d .

(3.5)

However, when applied to a Bloch state, Eq. (3.5) has an interesting consequence, namely LRB, R A 1 f , ( z )

= (2isin qd)f,(’).

(3.6)

Thus, both at the zone center (q = 0) and zone extrema (q = +n/d), [RB, R A ] f q vanishes. This implies that one may label the Bloch eigenstates corresponding to these two 4 values according to their symmetry with respect to R A and RB.47 Moreover, it follows from Eq. (3.4) that iff, is even in z - CA it will be even (odd) in z - CBif q = 0 (q = fn/d).Symmetrically, iff, is odd in z - CA, it will be odd (even) in z - C , if q = 0 ( q = fn/d). Notice that these q values correspond to standing Bloch waves since the group velocity of the carrier vanishes for these two values. These symmetry properties are helpful in explicit evaluation of the wave functions at q = 0 and q = f~ / d , 4 ~ particularly for the superlattice eigenstates that correspond to propagating states in one kind of layer (say the A layers) and to evanescent states in the other hand of layer (say the B layers) and thus can be viewed as arising from the hybridization of isolated wells eigenstates. Before giving the explicit superlattice dispersion relations of an AB superlattice, let us recall that the shapes of such dispersion relations are necessarily of the form cos qd

(3.7)

=f ( E )

irrespective of the band edge profile inside the superlattice unit cell. Actually, Eq. (3.7) is obtained by making use of the Bloch theorem and by expressing the fact that inside a unit cell the general (superlattice) solution is a linear combination of the two independent solutions of the Schrodinger equation.48 The functionf(s), unfortunately, depends on the exact shape of the band edge profile inside the unit cell, which practically limits the usefulness of Eq. (3.7) for applications to actual cases of interest. In the case of flat-band conditions, one has to solve Eq. (1.36) for heavy holes and Eqs. (1.37)-(1.38) for the light particle states with &(z) = 0 and V,(z

+ d) = T/S(z),

Vp(z

+ d) = VP(z).

(3.8)

In the case of propagating states in both kinds of layers and of the matching 47P.Voisin, G. Bastard, and M. Voos, Phys. Reo. B 29,935 (1984). 48See N. W. Ashcroft and N. D. Merniin, “Solid State Physics.” Holt, Rinehart and Winston, New York. 1976.

262

G. BASTARD et al.

of f,(z) and p-’(~,z)(df,/dz) at the heterointerfaces, one finally ends up with the light particle dispersion relations cos qd

= cos kALA

where E(E

(E

-

+

-

21 (5 + 5 - ’)Sin kALA Sin kBLB,

+ + A,) = h2k:P2(& + + 2AJ3), - V, + + AB) = h2kgP2(E V, + EB + 2A,/3), c = - -kA P B

&AXE

V,)(E- V, + E B X E

Cos k&,

EA

EB

PA

(3.9)

EA

(3.10)

-

(3.11) (3.12)

kB’

and p - ’ ( & , z) is defined in Eq. (1.38). Equations similar to Eqs. (3.9)-(3.12) hold for the heavy hole superlattice states except that their dispersions are parabolic in kA and k,:

(3.13) = mi:);

pLg = mi:).

(3.14)

Equations (3.9)-(3.14) are very similar to the textbook Kronig-Penney result except that the carrier effective mass in the heterostructure is liable to be position- and energy-dependent. Previous dispersions have been obtained by assuming that the eigenstates were propagating in both kinds of layers; in other words, E < V, - E, for heavy hole states and E > V, or E < V , - E, or E < V, - 6, - A, for light particles in type I heterostructures. These unconfined states are, however, not the most commonly encountered in many experiments since they are never the ground superlattice states. The latter states are instead obtained by hybridizing propagating states in one kind of layer (A for definiteness) with evanescent states in the other kind of layer (B). The dispersion relations of these eigenstates are simply obtained by changing kB to kBin Eqs. (3.9) and (3.11) and, thus, 5 to -9, where

(3.15) This changes Eq. (3.9) to

(3.16) When energy increases from 0 to V,f ( ~ ) the , right-hand side of Eq. (3.16) exhibits damped oscillations. The allowed superlattice states correspond to

ELECTRONIC STATES IN SEMICONDUCTOR HETEROSTRUCTURES

263

the energy segment(s) where l f ( ~ ) I < 1. Sincef(E) > 0 in the vicinity of E = 0, the ground superlattice state for r,-related states in type I superlattices corresponds to q = 0, and the extremity of the first allowed energy segment, if it exists, corresponds to & n/d,wherej(E) = - 1. If LA is large enough, there may be a second allowed energy segment below V, separated from the first one by a forbidden energy gap. Its lower and upper bounds correspond to q = +-z/d and q = 0, respectively. The set of the allowed energy segments contains the superlattice minibands that are energy separated by minigaps. For given LA and L, the minibands (minigaps) are broader (narrower) with increasing energy. This feature is a consequence of the well-known fact that a carrier with increasingly large kinetic energy is less and less scattered by a potential of fixed range and strength or, equivalently, that a carrier with increasingly shorter de Broglie wavelength is less and less diffracted by a grating of fixed period. To establish a direct relationship among the superlattice states, solutions of Eq. (3.16) for a given LA and wide L g , and the bound states of the isolated wells with thickness L A , it is useful to rewrite the right-hand side of Eq. (3.16) in the form (3.17)

f

f*(&) = $[cos k,LA

7

$(z

-

9-l)sin k A L A ] .

(3.18)

In the limit of wide barriers the eKBLB term blows up. Sincef(E) should be smaller than 1 to get an allowed superlattice state, the superlattice minibands should necessarily be located around the zeros off+(&).It is easily checked that these zeros coincide with the bound states El, E,, .. . of an isolated BAB quantum well. Actually 2f+(~)is nothing but the product of Eqs. (2.4)-(2.5). To an accuracy of e-2KBLE,one may drop the f - contribution to f ( E ) and expandf+(E) to the lower order in E - E,, to get the approximate dispersion relations (3.19) (3.20)

These dispersions relations are the result of the tight-binding analysis performed on the envelope functions?' Owing to the assumptions underlying Eqs. (3.19)-(3.20), the tight-binding result is expected to be more valid for the ground superlattice subband. Equation (3.19) shows that the levels of weakly coupled quantum wells hybridize to form subbands that are symmetrically placed around the levels of the isolated wells. Allowing for a better approximation (i.e., taking into account thef- term) gives rise to a shift S , of the center of gravity of the subbands with respect to E n . The main result,

264

G . BASTARD et al.

however, is still that the bandwidth A, for a given LA should decrease exponentially with barrier thickness LB.This tight-binding prediction for the ground subband is excellently confirmed by an exact computation of At by means of Eq. (3.16) as seen in Fig. 12 for the case of GaAs-Ga0,,A1,,,As superlattices down to periods equal to 60 A. It is actually possible (see, e.g., Ref. 18) to establish directly Eqs. (3.19)-(3.20) by starting from the wave functions of the isolated quantum wells and retaining only nearest-neighbor couplings. Such a procedure permits calculation of A, solely in terms of isolated wells parameters and proves that the shifts S , are always smaller than A,, since their dependence upon the barrier thickness is exp(-2KB(En)LB). When the wells are weakly coupled, the bandwidths are small and the carrier superlattice effective mass for its motion along the growth axis is heavy and increases almost exponentially with L, for a given LA. In the vicinity of the ground superlattice state [q = 0, n = 1 in Eq. (3.19)], one may expand E up to the second order in q to get E =

E l - At/2

+ h2q2/2mll,

mil/m, = 2h2/m,d2A,,

-

(3.21)

-

(3.22)

where m, is the free-electron mass. As seen from Fig. 12, mll/m, 0.15 for a 50 A-50 A GaAs-Ga0,,A1,~,As superlattice, but 0.78 for a 50 A-90 A structure. This occurs in spite of the fact that the carrier effective mass in both host layers is less than O.lm,, A heavy superlattice effective mass merely

FIG. 12. Calculated El bandwidth (solid lines, left scale) and carrier effective mass for its motion along the superlattice (dashed lines, right scale) in GaAs-Ga(A1)As superlattices. The well width is kept constant: LA= 30 A, 50 A, 100 A, and the barrier thickness L, is varied.

ELECTRONIC STATES IN SEMICONDUCTOR HETEROSTRUCTURES

265

means that the spatial localization of the carrier inside a superlattice period is very unevenly shared between wells and barriers-in other words, that the carrier motion is almost bound in the A layer. Since a bound state is characterized by a vanishing average velocity, its effective mass for such a periodic motion is infinite. The tendency of a superlattice effective mass to increase with increasing LE merely restates that the flattening subbands approach the bound states of the isolated wells. Once we have established a proper connection between the superlattice eigenstates and the bound states of the isolated wells, we may use the symmetry arguments outlined previously to simplify Eq. (3.16) for q = 0 and q = +n/d and to split this equation into two decoupled groups of two equations that deal with eigenstates that are even or odd in the A layer. We know from the tight-binding model that the n = 1, 3, ... subbands are built from the hybridization of the isolated well eigenstates that are even, while the n = 2, 4, . . . subbands arise from hybridizing the odd eigenstates of the isolated wells. Thus, for the n = 1, 3, . . . subbands we write =

(3.23) The resulting dispersion relations are cos(F)sinh( ~ A L A

cos( I ) c o s h (

F) F)

- sin( T)cosh( ~ALA ~ALA - sin( ?)sinh(

F)

-2-> K&B

= 0,

= 0,

4

= 0,

n:

=d'

(3.24)

(3.25)

For the subbands n = 2, 4, . . . we write

fq(4= A Sin k A Z ,

Izl

L A <2 '

(3.26)

266

G. BASTARD et al.

r?)(KF)

and the resulting dispersion relations are

sinh - = 0,

sin(F)cosh(F)

+tcos

sinr+)sinh(F)

+zcos(y)cosh(F)

~

= 0,

q

= 0, 71

q = 2.

(3.27) (3.28)

It is clear from Eqs. (3.24) to (3.25) and Eqs. (3.27) to (3.28) that the eigenstates of a given subband converge for large LBto the same equation, which is that of either the even or the odd bound states of the isolated wells. Figure 13 presents the superlattice band structure for heavy holes in GaAs-Gao,,A1o,,As superlattices versus the period d and equal layer thicknesses. For hole energies smaller than IVJ, the height of the hole barrier, one recognizes the result of the hybridization of the isolated well bound states and the broadening of these levels into minibands. These minibands are narrower than the electron ones at a given d (see Fig. 14 for a comparison between electron and hole ground subband width). The “unconfined” hole states (i.e., those corresponding to hole energies larger than I VJ) display a richer behavior. In particular, one notices the existence of discrete d’s, where the minigap between two subbands is exactly zero. These d values are such that there exists a Fabry-Perot effect inside each layer (i.e., such that k,L, = Zz,k,L, = mz). So far, these superlattice states have not given rise to any detectable effects.

FIG. 13. Calculated heavy hole superlattice states in GaAs-Ga(A1)As superlattices. x =0.3. LA= L,. k,=O. The allowed states are hatched.

267

ELECTRONIC STATES IN SEMICONDUCTOR HETEROSTRUCTURES

0

100

50

LB ( % I

FIG.14. Comparison between the E , and HH, bandwidths in GaAs-Ga(A1)As superlattices. The well width LA is kept constant: LA= 30 A, 50A, loo& and the barrier thickness L, is varied.

When d goes to zero, one notices in Fig. 13 that the lowest-lying hole states

in the superlattice converges to I Vp/,1/2-that is, to the value one would naively affix to the top of the valence band of the virtual crystal Gao.s5Al,.,5A~.This

feature is a general property of the modified Kronig-Penney model. Let US start from Eq. (3.16) and search for its solution for electrons when LA,L,, d go to zero, keeping the ratio 2 = LB/d

fixed. At q

=0

(3.29)

one finds immediately that the solution of Eq. (3.16) is

E,

=

En.

(3.30)

To find the carrier effective mass in the vicinity of q = 0, one expands cos qd up to q2 and finds that the solution of Eq. (3.16) can be written as E(q) = El

+ h2q2/2M,

(3.31)

where

M

= %l(E1)

+ (1

- %@I).

(3.32)

Equations (3.30)-(3.32) are the pseudoalloy limit of the superlattice eigen-

268

G. BASTARD et al.

states when the period is short: the ground superlattice state extrapolates to the bottom of the conduction band of the virtual crystal, and the electron effective mass is the average of the well (pA)and barrier ( p B )effective masses weighed by the equivalent “molar” fraction of A (LA/d)and B (LB/d) materials. One associates superlattice eigenstates that are uniformly spread over the period (if L A = L B ) . with the notion of pseudoalloy. One may therefore check if actual superlattices do fulfill this expectation. The integrated probability P, of finding the electron in the A layers when the superlattice eigenstate is In, q ) , where n is the subband index, is the expectation value of the operator Y(lz1 < L A / 2 ) .Since In,q ) admits a decomposition on the periodic parts of the Bloch function of the form (3.33) one finds that (3.34) where we have made use of the different scale of variations of the uIoandf‘ functions. By expressing the valencelike f”s in terms of the r,-related f’and by using Eq. (3.23) at q = 0, one finds Pw

= Qw/(Q,

+ Qb),

(3.35) (3.36)

where P b is the integrated probability of finding the particle in the barrier of one superlattice period. The first two terms in the brackets of Eqs. (3.37)(3.38) arise from the r,-related f ’ , while the remaining terms are the

ELECTRONIC STATES I N SEMICONDUCTOR HETEROSTRUCTURES

1oc

I

I

I

--/ I

GaAs- ALAS

L1, = L g

269

I

h

s

Y

a" 51 3

a

c

0

I

10

I

1

20

30

d

I

40

(i)

I

50

FIG.15. The calculated probability of finding the electron in the well-acting material ( P w ,solid lines) or in the barrier-acting material (Pb,dashed lines) is plotted versus the superlattice period d for several superlattices. E , subband. q = 0.

consequence of the conduction band nonparabolicity. In fact, they vanish if E << and J E - V,l << E ~ In . practice, we have found that they are not very important in III-V-based superlattices. They can bring noticeable contributions when very narrow gap materials are host layers in the superlattice (e.g., Hg,,,Cd,,,Te-CdTe). Figure 15 displays the d dependence of P , and P , for the ground r,-related state of several superlattices with short periods (d < 50 A). It is seen that the Ga(1n)As-InP superlattices are closer from an equirepartition of the electron wave function between wells and barriers. This is due to both a relatively light mass ( 0.05rnO)and a small conduction band offset (0.244eV), which favor spatial delocalization. In spite of the lighter InAs effective mass, the InAs-GaSb superlattices more readily depart from the ideal pseudoalloy behavior. This is associated with the larger V, (0.961eV). Finally, the GaAs- AlAs superlattices more quickly deviate from the pseudoalloy behavior. This results from a large V, (1.08eV) and a heavier conduction mass (0.067rn0)at the GaAs r6conduction band edge. Besides their academic interest, short-period superlattices, whose growth is difficult, were also studied in the hope that they can advantageously replace the equivalent alloy. If the band structure of a short-period superlattice is comparable to that of the equivalent alloy, the former structure is, in principle, free of alloy scattering, which is detrimental to the mobility of the N

270

G. BASTARD et al.

latter structure. Thus, InAs-GaAs short-period superlattices would conveniently replace Gao,4,1no,,,As alloys. However, the considerable lattice mismatch between InAs and GaAs (7%) is a formidable obstacle to such a replacement. In addition, the enormous elastic energy that is stored in such a highly strained material destabilizes the whole lattice, to the extent that first principle calculation^^^ point to the instability of (InAs), -(GaAs), superlattices. In addition to these fundamental limitations, the actual growth difficulties (In segregation) conspire to deny to InAs-GaAs short-period superlattices a better quality than that found in the ternary random alloy.50 Most likely, the alloy scattering has been replaced by interface roughness scattering even in the best-grown samples. The GaAs- AlAs short-period superlattices have met a somewhat brighter fate. The Gal -,AI,As random alloys often contain deep traps (DX centers) and a significant amount of residual charged impurities. The deep traps give rise to the phenomenon of persistent photoconductivity (quasi-irreversible photoionization of the traps at low temperature). It is not clear that GaAsAlAs short-period superlattices are free of this effect. However, their overall quality seems to be sufficient to compete with the alloys, for instance, in the optical properties. The ternary Gal-,Al,As random alloys undergo a direct to indirect crossover near x = 0.42 (i.e., the virtual crystal hamiltonian is such that the lowest-lying conduction state is at the r point if x < 0.42 and near the X or L point for larger A1 contents). The AlAs binary material has its (six) lowestlying conduction band states located near the X points, while the r,-related minimum is far above in energy ( E -~ E, 0.9 eV). As in all 111-V materials, the top of the valence band is at the r point. The GaAs conduction band edge is at the point, while the six equivalent X minima lie -0.47eV higher in energy.51 Thus, if the GaAs-A1As conduction band discontinuity is large enough, the GaAs layers are potential wells (barriers) for the r-related ( X related) electron states in the GaAs-AIAs superlattices (see Fig. 16). The rrelated states are describable in the same fashion as outlined previously. The envelope function approximation can also be used to describe the X-related states apart from the r-related ones. The bulk isoenergy surfaces near one of the six equivalent X minima are ellipsoids of revolution with a longitudinal mass that is much heavier than the transverse one (mil 1.98m0, rn, 0.32m0 in GaAs). In the superlattices, when grown along the [OOl]

-

-

-

49R. M a g i and C. Calandra, Superlattices and Microstructures 5, 1 (1989); P. Boguslawski and A. Baldereschi, Phys. Rev. B 39, 8055 (1989). 'OJ. M. Gkrard and J. Y. Marzin, C. d'Anterroches, B. Soucail, and P. Voisin, Surf: Sci. 229,456 (1990).

5'A. Onton, R. J. Chicotka, and Y. Yacoby, Proc. 11th Int. Con$ Semiconductors. Warsaw 1972. Polish Scientific Publishers, Warszawa 1972. p. 1023.

ELECTRONIC STATES IN SEMICONDUCTOR HETEROSTRUCTURES

27 1

-r6 0.902eV

---- X 0.177eV

-k Ga A s

ALAS

FIG.16. Assumed Ta and X level ordering in bulk GaAs and AIAs.

direction, the sixfold degeneracy of the X-related minima will be lifted by the superlattice potential into a doublet and a quadruplet, respectively labeled X, and Xx,,. The carrier effective mass entering into the Schrodinger equation for the associated envelope functions will be mll for the X,-related states and rn, for the X,,,-related states. Since mll >> m,, one expects for a given superlattice period that the former states will lie below the latter ones. This simple reasoning bypasses the effect of the (small) lattice mismatch between GaAs and AlAs (6a/a 0.2%). For samples grown on GaAs substrates, the AlAs layers experience a tetragonal distortion that forces AlAs to acquire the same in-plane lattice parameter as the GaAs one. The stress effect pushes the X,,,-related states below the X,-related ones. This small effect competes with the size quantization in both limits of very thin and very thick superlattice periods, where the pseudoalloy regime, on the one hand, and the disappearance of the size quantization, on the other hand, bring both sorts of Xrelated states into coincidence. A recent pseudopotential c a l c ~ l a t i o nhas, ~~ however, shown that the stress effect cannot revert the ordering of the Xx,,and X,-related levels. Whether the X,-related states lie lower or higher in energy than the r6related eigenstates critically depends on the ratio between the GaAs and AlAs layers. Previous comments on the pseudoalloy regime of a short-period superlattice would lead us to conclude that if 2 = LB/(LA LB)5 0.42, where LA stands for the GaAs layer thickness, the q = 0 r,-related state (El) is the absolute ground conduction state of the superlattice, while if 2 k 0.42 the ground superlattice conduction state is an X,-related one. Obviously, if LA is

-

+

52L.D. L. Brown, M. Jaros, and D. J. Wolford, Phys. Rev. B 40 6413 (1989).

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large enough, the ground superlattice state will be r-related, irrespective of L,: not only should the short-period superlattice be AlAs rich, but it should also be such that E , lies above the AlAs X point. Thus, one expects that if, say, LA = L, (i.e., 2 = O S ) , there will exist a critical GaAs thickness L, beyond which E l is the absolute ground conduction state, while if LA < L, the ground state is X,-related. Owing to the r-related nature of the heavy hole states, they are preferentially localized in the GaAs layers of all LA. Thus, one expects a type I to type I1 transition to occur in GaAs-A1As superlattices with decreasing d and LA = L,. This has important consequences on the optical properties of these super lattice^.^^-^^ Although in both cases optical transitions are allowed (vertical transitions in the superlattice Brillouin zone), they should be much weaker in type I1 configurations than in type I configurations, because of the spatial separation between electrons and holes and because the conduction (valence) states are X,- (r-) related, which in the bulk leads to forbidden transitions. These considerations have been fully supported by experiments. When E is the lower conduction state, one finds optical transitions that are allowed, as evidenced by a short photoluminescence decay time, and a near-energy coincidence between photoluminescence and absorption. On the other hand, in the type I1 configuration, where only pseudodirect transitions take place, one finds photoluminescence decay times that are much longer and a large energy offset between the photoluminescence lines and the absorption edge. As for the transitions ending in the X x , y conduction states, they are strongly indirect (and thus forbidden in the absence of external agents such as defects and phonons), since at the edge the hole in-plane wave vectors is zero while that of the XX,?electron is 2n/a,, where a,, is the GaAs lattice constant. Once these qualitative considerations have set the general frame, one can %. Danan, B. Etienne, F. Mollot, R. Planel, A. M. Jean Louis, F. Alexandre, B. Jusseraud, G. Le Roux, J. Y. Marzin, H. Savary, and B. Sermage, Phys. Rev. B 35, 6207 (1987). 54J. Nagle, M. Garriga, W. Stolz, T. Isu, and K. Ploog, J. Phys. (Paris)Colloq. 29, C5-495 (1987). 55E.Finkman, M. D. Sturge, and M. C. Tamargo, Appl. Phys. Lett. 49, 1299 (1986). 56K.J. Moore, P. Dawson, and C. T. Foxon, Phys. Rev. B 38, 3541 (1988). 57E. Finkman, M. D. Sturge, M. H. Meynadier, R. E. Nahory, M. C. Tomargo, D. M. Hawang, and C. C. Chang, J . Lum. 39, 57 (1987). "K. J. Moore, G . Duggan, P. Dawson, and C. T. Foxon, Phys. Rev. B 38, 5535 (1988). 59H. Kato, Y. Okada, M. Nakayama, and Y. Watanabe, Solid State Commun. 70, 535 (1989). 60R. Cingolani, L. Baldassarre, M. Ferrara, and M. Lugara Phys. Rev. B 40,109 (1989). 6'D. Scalbert, J. Cernogora, C. Benoit a la Guillaume, M. Maaref, F. F. Charfi, and R. Planel, Solid State Commun. 70, 945 (1989) and Surt Sci. (1990) in press. 62J. Feldman, R. Sattmann, E. Gobel, J. Kuhl, J. Hebling, K. Ploog, R. Muralidharan, P. Dawson, and C. T. Foxon, Phys. Rev. Lett. 62, 1892 (1989). 63J. Feldmann, R. Sattmann, G. Peter, E. 0. Gobel, J. Nunnenkampf, J. Kuhl, J. Hebling, K. Ploog, R. Cingolani, R. Muralidharan, P. Dawson, C. T. Foxon, Surf. Sci. (1990) in press.

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proceed with more detailed studies. From a theoretical point of view, the band structure of GaAs-A1As superlattices has been calculated by using a variety of methods,64- 7 3 from the simple envelope function frame~o~k53.54~58 to more microscopic approaches based on tight-binding,9*70*71 empirical, or self-consistent pseudo potential^,^^ - 66 Wannier orbitals,67 local density,68 or augmented spherical wave methods.69 The outcomes of these calculations conflict as far as extremely short-period superlattices are concerned. In particular, the status of the (GaAs),-(AlAs), superlattice is theoretically unclear. Some authors conclude that it is a pseudodirect 3,58.64.70 with the lowest-lying conduction state being the folded X,-related levels, while some others conclude that it is indirect.66 There are also reports that the (GaAs),-(AlAs), band gap is smaller than that of (GaAs),-(AIAs),, while others conclude that the (GaAs),-(AIAs), band gap is a monotonically decreasing function of n. All the calculations, however, agree that if n is large enough, the (GaAs), - (AlAs), superlattices are direct-bandgap materials with the lowest-lying conduction state derived from the l-6 extremum. Experimentally, the crossover between the type I to type I1 transition is also disputed, values of n = 10-13 in (GaAs),-(AlAs), having been quoted. The experimental situation of the (GaAs),-(AlAs),, n = 1,2, 3, superlattices is as much a subject of controversy as the magnitude of the shrinkage or nonshrinkage of the band gap when n increases. One should realize that the growth of such short periods is an extraordinarily demanding task, for one knows that the growth of a layer is never perfectly bidimensional. In the case of monolayer superlattices, this implies that there is a fraction of the sample area where there is no GaAs layer (or no AlAs layer) at all. In the following, we present a few results to summarize GaAs-A1As shortperiod superlattices. The calculations have been performed under the envelope function approximation, taking a GaAs- AlAs conduction (valence) and band offset of 1.08eV (0.53eV), as established by Danan et neglecting stress effects. Our purpose is to substantiate the qualitative ideas outlined previously and to discuss to what extent the notion of pseudoalloy 64W. Andreoni and R. Car, Phys. Rev. B 21, 3334 (1980). 65M. A. Gell, D. Ninno, M. Jaros, and D. C. Herbert, Phys. Rev. B 34, 2416 (1986). 66T. Nakayama and H. Kamimura, J . Phys. Soc. Japan 54,4726 (1985). 67D.Z.-Y. Ting and Y. C. Chang, Phys. Rev. B 36,4359 (1987). 68D.M. Bylander and L. Kleinman, Phys. Rev. B 36, 3229 (1987). 69R. Eppenga and M. F. H. Schuurmans, Phys. Rev. B 38, 3541 (1988). "Y.-T. Lu and L. J. Sham, Phys. Rev. B 40,5567 (1989). 71J. Ihm, Appl. Phys. Lett. 50, 1068 (1987). "L. Brey and C. Tejedor, Phys. Rev. B 35, 9112 (1987). 73N.E. Christensen, E. Molinari, and G. B. Bachelet, Solid State Commun. 56, 125 (1985).

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w

0

10

20

30

40

h

50

60

70

Bo

I I

,

-

GaAs-AIAs

-

I

L,=L,

AEV=O53eV

w

-

I

I

I

FIG.17. The calculated lowest-lying superlattice bands (hatched areas) for electrons and holes in GaAs-A1As superlatticesare plotted versus the period d. Equal layer thicknesses are assumed. The energy zeros for electrons (holes) are taken at the Ts(r,)points of bulk GaAs.

applies to GaAs- AlAs superlattices. Figure 17 presents the calculated subbands of r6-related (HH,) states versus the superlattice period d of GaAs- AlAs superlattices when L A = L,. The (GaAs), -(AIAs), superlattice corresponds to d = 5.66 A. When d vanishes, both X , and X x , yextrapolate to 0.322eV, the midpoint between XGaAs and XAIAs, while El and HH, extrapolate to 0.54 eV and 0.265 eV, respectively. At large periods the bandwidths decrease to zero and X,,X x , yconverge to XAIAs, while El and HH, converge to the r6 and Ts edges of GaAs, respectively. One notices that the X x , ybandwidth is larger than that of X , for a given d, a consequence of the lighter effective mass (mlversus m,,) entering into the size quantization. The type I to type I1 transition is calculated to occur near d = 70A, which corresponds to 12 < n < 13 in the formula (GaAs),-(AIAs),. For such “thick” layers these superlattices are not pseudoalloys or even true superlattices, inasmuch as the spatial localization of the carrier is very unevenly shared by both kinds of layers: at d = 70A we calculate that the probability of finding the carrier in GaAs is 92.4% (El), 93.9% (HH,), and 0.8% ( X , ) . Equivalently, the bandwidths are very small: AE, 2.1 meV; AHH,, A X , , 5 10-’meV. The very notion of Bloch wave becomes very hazy for such weakly coupled quantum wells, for any fluctuation of one monolayer in layer thickness produces energy shifts that are larger than the bandwidths in the perfect

-

74J. Y. Marzin, 1987, unpublished.

ELECTRONIC STATES IN SEMICONDUCTOR HETEROSTRUCTURES

275

FIG.18. The energy of the ground ( q = O ) states of the lowest lying r,-related (El, solid lines) and X,-related (Xzl, dashed lines) bands is plotted versus the equivalent A1 content ZA, for several superlattice periods d. Z,, =L,,,As/d.

system and thus give rise to a disorder-induced l o c a l i ~ a t i o n . ~Figure ~ - ~ ~18 is a plot of the lowest-lying eigenstates ( q = 0) for the lowest r-related and X,-related subbands for fixed superlattice periods but varying the well and barrier thicknesses as expressed by the equivalent A1 mole fraction ZA, (ZA1 = L,/d). The black circles correspond to the crossings of El and X , , l that is, to the location of the type I to type I1 transition that is found to increase with increasing superlattice period. The heavy X, effective mass causes the single quantum-well limit to be reached quickly for the X,-related states. Thus, to the extent that the period is large enough and thus that the interwell coupling can also be neglected for the E , state, the location of the type I to type I1 transition is given by El[d(l

- %Al)]

=

% - (&r- &X)AIAs?

(3.39)

"A. Chomette, B. Deveaud, A. Regreny, and G. Bastard, Phys. Rev. Lett. 57, 1464 (1986). 76sL.Pavesi, E. Tuncel, B. Zimmermann, and F. K. Reinhart, Phys. Rev. B 39, 7788 (1989). 76bP.W. Anderson, Phys. Rev. 109, 1492 (1958). 77S.Das Sarma, A. Kobayashi, and R. E. Prange, Phys. Rev. Lett. 56, 1280 (1986). 78J. B. Sokoloff, Phys. Rev. B 22, 5823 (1980). 79J. D. Dow, J . Phys. (Paris), Colloq. 45, (25-525 (1984). ''5. M. Ziman, "Models of Disorder." Cambridge University Press, Cambridge, 1979.

276

G. BASTARD et al.

which is a convenient way of measuring V,. In fact, plots similar to Fig. 18 have already been obtained for the optical transition energies, and good agreement between the calculated and observed type I to type I1 transition has been found for V, 1 eV. A recent summary5* of the calculated and measured optical transitions versus d is shown in Fig. 19a for GaAs-AIAs superlattices with equal layer thicknesses. The type I to type I1 transition occurs near n = 11-12 for V, 1eV. The situation for small periods is experimentally unclear. We have already mentioned that it is also very much disputed theoretically. In Fig. 19b we show the calculated and measured transition energies for d = 40A and 20A, respectively, plotted versus the equivalent A1 mole fraction One again finds a good qualitative description of the experimental data for V, 1 eV. We pointed out in the introduction that the GaAs-A1As short-period superlattices are good examples of situations where the envelope function approximation qualitatively breaks down. Indeed, by applying this method to X-related and r-related states separately, we neglected any couplings between them due to the superlattice potential. It has been shown that there exist ways to generalize the envelope function approximation to account for T-X However, a priori unknown parameters enter into this generalized scheme. It seems to us more reasonable to resort to more microscopic approaches to handle such situations where the coupling between different hosts' valleys is expected to play a significant part. Empirical tight binding and pseudopotential and Wannier orbital methods, which account for the whole Brillouin zone of the hosts, have been applied to short-period superlattices. Lu and Sham7' have given a fairly complete analysis of the crossing or anticrossing behavior between X,-related and rrelated states. They pointed out via a careful parity analysis of the X,-related states that in the (GaAs),-(AlAs), superlattice each r-related level (Ei)retains the same symmetry as n varies but each X level alternates for successive n's. (Notice that in a tight-binding analysis the parity properties concern the whole wave function, not only the envelope function). Thus, when a r-related level approaches an X,-related one, the actual crossing or anticrossing depends not only on the level index (ground, first excited) but also on the number of monolayers n where the crossing should occur. For instance, Xlz, which has the same parity as n anticrosses El near n = 12, while E , and X,, do cross at n = 16 since they are of opposite parities. Quantitatively speaking, Lu and Sham found the lower-lying anticrossing (between El and Xlz) to be about 1.7meV. This value is close in magnitude to an El-X1, anticrossing determined experimentally by Meynadier et (2.5 meV), although a strict

-

-

-

"M. H. Meynadier, R. E. Nahory, J. M. Worlock, M. C. Tamargo, J. L. de Miguel, and M. D. Sturge, Phys. Reo. Lett. 60, 1338 (1988).

ELECTRONIC STATES IN SEMICONDUCTOR HETEROSTRUCTURES

0

r-r ref

13 X-Tref 13

277

0 ref 16

B ref 12 o ref 14 A

r - T this work X-T this work

-

*

1.8 -

2

I

I

I

I 10

I

I

* I

* I

6 14 18 Superlattice period ( Monolayers)

I

FIG. 19a. A compilation of the calculated (lines)and measured (symbols) transition energies in GaAs-AIAs superlattices versus the superlattice period (in monolayers). One monolayer is 2.83 A thick. Equal layer thicknesses are assumed. After Ref. 58.

%A,

FIG. 19b. Calculated (lines) and measured (symbols) transition energies in GaAs-A1As shortfor d = 40 A and 20 8,respectively. period superlattices versus the equivalent A1 mole fraction After Ref. 53.

278

G. BASTARD et al.

comparison is difficult since the calculated value is obtained for a (GaAs),(AlAs), superlattice with n = rn = 12, while the experimental one refers to an induced crossing (by an electric field) in a sample with n g 12 and m z 28. In any event, the mixing is fairly weak. Thus, we can safely conclude that, despite the qualitative breakdown of the envelope function approximation when a rX coupling takes place, this computational scheme proves to be quantitatively satisfactory, at least in the specific case of GaAs-AlAs. There is an increasing effort in the growth of 11-VI superlattices. Wide-gap superlattices have, roughly speaking, the same sort of band structure as 111-V compounds with some extra complications due to lattice mismatches between the host materials. We shall here briefly discuss two kinds of II-VIbased superlattices that display some “exotic” electronic properties. The CdTe-Cd(Mn)Te superlattices8’ are peculiar in that the barrieracting materials Cd(Mn)Te contain localized magnetic moments arising from the half-filled d shells of the MnZ+ ions. Tetrahedral bonds are formed between Cd and Te or Mn and Te. The bulk materials retain the zinc blende structure up to xMln 0.7. The d electrons weakly hybridize with the s and p electrons, and the electronic properties resemble those of the nonmagnetic wide-gap materials at a zero magnetic field. In the presence of a field, and neglecting in a first approximation any orbital effects (Landau levels), the Mn moments line up. The conduction electrons (or holes) interact, via the Heisenberg exchange interaction, with the localized moments. In a mean field approximation this is describable in terms of a giant g* factora3(in excess of lo’), which gives rise to interesting optical properties (giant Faraday rotation, closing of the band gap with increasing B, etc.). In a CdTe-Cd(Mn)Te superlattice, one thus deals with magnetically tunable barriers, which decrease with increasing B. If the magnetic-field-induced lowering of the barrier becomes comparable with the barrier height, the electron or hole eigenstates, although preferentially localized in the CdTe layers at zero field, may eventually completely delocalize on the barrier. There is (presently) agreement on the very uneven apportionment of the band gap energy difference between the conduction and valence bands, the valence band offset being small (typically 10-20% of the band gap difference).Thus, the electrons that have a wider offset and a smaller exchange interaction will remain localized in CdTe, while the holes, eventually helped by stress effects, may in a strong field transfer to the Cd(Mn)Te layers (Fig. 20). A particular doublequantum-well structure has been predicted to undergo a type I to type I1 c~nfiguration.’~A recent magnetooptical study of a CdTe-Cd(Mn)Te superlattice has evidenced a magnetic field-induced type I to type I1

-

”See L. A. Kolodziejski, R. L. Gunshor, N. Otsuka, S. Datta, W. M. Becker, and A. V . Nurmikko, I E E E J . Quantum Efectron. QE22, 1666 (1986). 83J. A. Gaj, R. Planel, and G. Fishman, Solid State Commun. 29,435 (1979). 84J. A. Brum, G. Bastard, and M. Voos, Solid State Commun. 59 569 (1986).

ELECTRONIC STATES IN SEMICONDUCTOR HETEROSTRUCTURES

TYPE I

0-0

TYPE

279

n

increasing B

FIG.20. Sketch of the magnetic-field-induced type I to type 11 transition in a CdTe-Cd(Mn)Te quantum well.

transition.248.Some anisotropy in the B )I z and B Iz configurations has been ascribed to the formation of bound magnetic polarons-that is, the bound state that a carrier may form by lining up the Mn spins in its orbit.82 The dynamics of spin transfer between the mobile carriers and the localized d moments have been studied on a femtosecond time scale by Awschalom et al. and have clearly demonstrated the influence of the size quantization in CdTe-Cd(Mn)Te multiple quantum wells.85In ZnSe/Zn, -,Fe,Se superlattices, Liu et al.249have reported magnetic field-induced type I to type I1 transition. Another type of 11-VI superlattice is found in the HgTe-CdTe system. The peculiarities of these heterostructures arise from the inverted band structure of HgTe. In this material the r6edge lies below the Tsedge by 0.3 eV. Thus, the HgTe conduction band is of Ts symmetry at k = 0 and is degenerate with the topmost (heavy) valence band (Fig. 21). CdTe, on the other hand is a regular (GaAs-like)semiconductor. There has been a considerable debate on the magnitude of the HgTe-CdTe valence band offset. Currently it is believed (from XPS,86987 magnetooptical measurements, and theoretical c o n s i d e r a t i ~ n s that ~ ~ "$tTe ~ ~ ~-~ ~ 0.35 eV. There are no particular difficulties in applying the envelope function framework to the HgTe-CdTe superlattices. At k, = 0 the heavy hole and light particle states are decoupled. The HgTe layers are deep potential wells for the heavy holes whose eigenstates are thus largely localized in them. The situation for light particle

-

&;tTe

"D. D. Awschalom,J. Warnock, J. M. Hong, L. L. Chang, M. B. Ketchen, and W. J. Gallagher, Phys. Reu. Lett. 62, 199 (1989). P. Kowalczyk, J. T. Cheung, E. A. Kraut, and R. W. Grant, Phys. Reo. Lett. 56,1605 (1986). "Tran Minh Duc, C. Hsu, and J. P. Faurie, Phys. Rev. Lett. 58, 1127 (1987). ssaJ. M. Perez, R. J. Wagner, J. R. Meyer, J. W. Han, J. W. Cook, and J. F. Schetzina, Phys. Reo. Lett. 61, 2269 (1988). 88bN.F. Johnson, P. M. Hui, and H. Ehrenreicb, Phys. Rev. Lett. 61, 1993 (1988). "J. M. Berroir, Y. Guldner, J. P. Vieren, M. Voos, X. Chu, and J. P. Faurie, Phys. Rev. Lett. 62, 2024 (1989).

280

CdTe

G . BASTARD et al.

HgTe

FIG.21. Comparison between the band structures of HgTe and CdTe near the r point.

states is more interesting for we have to hybridize an electronlike Tsband in HgTe with a light holelike T, band in CdTe. Besides, since the l-8 offset is of the same order of magnitude as the r8-l-6 energy separation in HgTe, one has to expect a strong admixture between the r,-related and r6-related eigenstates, a situation reminiscent of that met in InAs-GaSb heterolayers and that one can deal with because the nonparabolicity effects are naturally embodied in the multiband envelope function approach. The mass reversal effect implies the existence of interface states42.90*91 in the energy segment [$Fe- E F ; ~ ~By ] . interface state we mean a superlattice eigenstate built from evanescent states in the HgTe and CdTe layers. In a regular superlattice (say GaAs-Gal -,Al,As) no state exists below (above) the GaAs conduction (valence)band edge, for it would violate the continuity of p - ‘(8, z)(dfc/dz).In HgTe-CdTe materials at the interface, it is possible to build, say, a nodeless state, which is an hyperbolic cosine in both HgTe and CdTe and therefore lies in the [E:? -$fTe] energy segment due to the sign reversal of p(&, z). This state corresponds to q = 0. Depending on the layer thicknesses, other superlattice “interface” states may be found. They correspond to eigenfunctions that are either hyperbolic cosines or sines in each layer. Because the hyperbolic functions have at most one zero, it is impossible to build more than two interface subbands, even in the thick layer limit. This contrasts with regular superlattices, which accommodate an arbitrarily large number of subbands lying in the energy segment [O, &I. Figure 22 shows a plot92 of the calculated subband structure of HgTe-CdTe superlattices with equal layer thicknesses and a Ts valence band offset 1 V,l = 0.35eV. Only one heavy hole subband (HH has been represented for clarity. The lowest-lying hole state (q = 0) extrapolates to the mid-distance between the two r8 points, an expected pseudoalloy limit. The light particle states consist of: “Y. C. Chang, J. N. Schulman, G. Bastard, Y. Guldner, and M. Voos, Phys. Reo. B 31, 2557 (t985).

“Y. R. Lin Liu and L. J. Sham, Phys. Rev. B 32, 5561 (1985).

”J. M. Berroir, 1985, 1989, unpublished.

ELECTRONIC STATES IN SEMICONDUCTOR HETEROSTRUCTURES

28 1

0.1

2

0

v

>

a -0.1

% z w

-o1.2P,F!21 -0.3

,

-0.4

0

200

$00

600

d (A) FIG.22. Calculated band structure of HgTe-CdTe superlatticesversus the superlattice period d. Equal layer thicknessesare assumed in the calculations. The allowed energy states are hatched or black. After Ref. 92.

(1) Two interface bands, labeled I , and I,,, that extrapolate to E , = -lVpl/(l mlh/mc) at infinite d, where mth and m, are the CdTe light hole and HgTe conduction masses at the energy Em. The ground interface band I , is such that its T, envelope at q = 0 corresponds to hyperbolic cosines in either layers. The I,, band is such that its Ts envelope at q = 0 corresponds to hyperbolic sines in each layer. The q = 0 state of the I , band converges to -0.175 eV at zero period. This is in some sense the equivalent of the pseudoalloy limit for the ground superlattice state in regular superlattices. The I,, band enters in the HgTe conduction band continuum near d = 180A. Its energy increases with decreasing d. (2) Two anticrossing subbands S, and LH2 in the vicinity of the HgTe Ts edge and CdTe r8edge. S, is mostly HgTe- and r,-related and converges to Er, HgTe at large periods. LH, is a propagating, mostly CdTe, r,-related subband. It extrapolates to &yeat large period. Since ImF,BTel< ImFyl, the confinement effect pushes S, near LH,. Because of the band nonparabolicity,

+

282

G . BASTARD et al.

the r,-related and r,-related states interact and the two subbands repel each others. Notice that S,, . .. should also anticross LH2 for thicker periods, in close analogy with the multiple anticrossing experienced by E l and LH, in GaSb-InAs superlattices. From a practical viewpoint (e.g., magnitude of the fundamental optical band gap) what matters is the relative position of the ZAS and HH, subbands. The crossing of these two subbands near 195A would lead us to conclude that a semiconductor (d < 195 A) to semimetal (d > 195 A) transition takes place at 195 A. Actually, the hybridization between the heavy hole and light particle states at nonzero in-plane wave vectors strongly modifies this simple picture.

IV. In-Plane Dispersion Relations

When the carrier in-plane wave vector k, is nonzero, the decoupling between the light particles and heavy hole states becomes impossible. In addition, for heterostructures with noncentrosymmetric band edge profiles, the twofold degeneracy of the levels at a given k, # 0 is lifted. This results from the nonvanishing hosts' spin-orbit coupling. These effects therefore are larger for the valence than for the conduction subbands, where they are induced by the band nonparabolicity. In the first part of this section we present some results for the conduction subbands while the second part is devoted to the valence subbands. Unless otherwise specified, the heterostructures are assumed to obey flat-band conditions. 4. CONDUCTION SUBBANDS

For the conduction subbands it is still useful and reasonably accurate to restrict ourselves to Eq. (1,21) (without the free-electron term) and to project the 8 x 8 coupled differential system onto the 2 x 2 basis corresponding to the two S-like edges. This leads to

where

ELECTRONIC STATES IN SEMICONDUCTOR HETEROSTRUCTURES

283

(4.3) Hereff (fl)is the envelope function for the z motion associated with the S.t (SJ)edge and [ A , B ] denotes the commutator of the A and B operators. The projected hamiltonian immediately reveals that H,, arises from the spin orbit of the host materials. If both A, and AB vanish, H,,is identically zero [V,(z) = l/a(z)], each eigenenergy is twice degenerate, and the S-like envelopes are unadmixed. The latter situation, of course, prevails if k, = 0. Finally, Eq. (4.3) clearly shows that H,, is associated with the conduction band nonparabolicity since it is proportional to the commutators of p z with [E EA - V,(Z)] - and [ E AA - l/a(z)] - Actually, H, can be rewritten as

+

,

+ +

1

[E

aVP --

+ EA - VP(z)l2 dz

[E

+ + EA

1 AA - V,(Z)]~

+

dz (4.4)

A,.,)-’ than Hd. It Thus, H,, clearly involves a higher power in E; and (E, is also apparent in Eq. (4.4) that H,, is an interface-related phenomenon, since in our flat-band assumption Vp(z) and V,(z) are staircase-like functions. We may therefore anticipate that the effects associated with Hsf will be the larger for the more excited subbands of a given heterostructure and for the smaller E, and E* if the subband index is fixed. Had we included a z-dependent bandbending Kxl(z), diagonal in the edge index, then one would have replaced in Eq. (4.4) Vp(z) and %(z) by Vp(z) + Kx&)and V,(z) + KxI(z),respectively. The H,, effects would thus not only be interface-related but would also involve contributions from the bulk of the host layer^.'^.^^ We recognize in the particular combination of energy denominators in H , the definition of the position- and energy-dependent effective mass p ( ~z), introduced previously. One notices that the effective barrier height &(z) + h2k:/2p(&,z ) in Eq. (4.2) becomes explicitly k,-dependent, a feature that may have some important consequences in quantum wells based on narrow-gap hosts, for it may 93U.Rossler, F. Malcher, and G. Lommer in “Proceedings of the International Conference on the Physics of Semiconductors in High Magnetic Fields” (G. Landwehr, ed.). Springer Series in Solid State Sciences, Springer-Verlag, Berlin, in press. 94M. Dobers, J. P. Vieren, Y. Guldner, P. Bove, F. Omnes, and M. Razeghi, Phys. Rev. B 40, 8075 (1989); see also D. Stein, C. Ebert, K. von Klitzing, and G. Weimann, Surj Sci. 142,406 (1984).

284

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happen that a given quantum well binds more (less) states at finite k, than at vanishing k, .95 The boundary conditions that arise from the integration of Eq. (4.1)across an interface are (i) f 7;

f l continuous.

continuous, where q(z) =

=( + 3

E

1 EA

-

V,(Z)

1

E

+ + A, EA

-

).

V,(Z)

(4.5)

In addition, one has to impose boundary conditions for theft, f l behavior at large lzl; that is, f f a n d f l decay far away from a quantum wel1,ff andfl fulfill the Bloch theorem in a superlattice, and so on. For rectangular quantum wells, V,(z), V,(z), and V,(z) are even in z. Since H , and H,, are, respectively, even and odd in z, there exist two sets of eigenfunctions of Eq. (4.1), which have a definite parity. For the bound states in the well they can be written f t = A COS kAZ,

fl

f

f 1 = A COS kAZ,

=

C sin kAz,

= C sin kAz

if IzI

L < -, 2

L if IzI < -, 2

ELECTRONIC STATES IN SEMICONDUCTOR HETEROSTRUCTURES

285

and mr, is the Ts band edge mass in the A material. The two solutions, Eqs. (4.6)-(4.7),are degenerate in energy, and their associated eigenvalues are the roots of 1 COSkAL + - SinkAL 2

(4.10)

The last term in Eq. (4.10) is a direct consequence of the H,, term. If it were zero, then Eq. (4.10)would be the straightforward generalization of the bound state equations of a quantum well to the case where kA, xg depend on k, via Eqs. (4.8)-(4.9), and there would be no explicit k, dependence in this equation. The spin-orbit splitting instead, via a nonvanishing HSf,leads to an explicit k, dependence in Eq. (4.10). As seen from Eqs. (4.8)-(4.10),the eigenvalues only depend on kf, that is, they can be rewritten as E,(k,)

= En

+ h2kf/2rn,*(k,),

(4.11)

which defines the electron effective mass for its in-plane motion in the nth subband whose edge is located at En.Notice that Eq. (4.11) is a definition, but m,* is not necessarily a quantity that is directly measurable. For instance, the density of states associated with the nth subband is pn(E) =

Sm:(k,) nh (1

dm*)-19

-

dk,

(4.12)

where the wave vector k, is the root of E = E,(k,). Thus the density of state effectivemass is m,*/[1 -(k,/2rn,*)(dm,*/dkt)]. The fact that rn,* differs from the bulk band edge mass m,, of the well-acting material can be termed a nonparabolicity effect in quantum wells, a point thoroughly discussed by Ekenberg.96 Notice, however, that the in-plane nonparabolicity as expressed by Eq. (4.11) is not as simple a phenomenon as found in bulk materials. In the latter case specification of the extra kinetic energy in the band, whatever the k orientation, suffices to calculate the carrier effective mass. This arises from the isotropicity of the r6band (in our Kane-like treatment). In quantum wells instead, the in-plane nonparabolicity not only depends upon k, but also, in an implicit way, upon the longitudinal motion (through the subband index, the boundary conditions, etc.). In Figs. 23-27 we present a set of calculated curves m*/rng versus L and kl --* 0 or versus kf for several values of L in different quantum wells. Here rn* and rnt stand for m: and rn,,, respectively. More precisely we have plotted in these figures [El(k,)- El(0)]- 'h2kf/2mr, versus L (k, -+ 0) or versus kf. As expected, from nonparabolicity effects rn:(O) increasingly deviates from rn,, when E l increases; this happens when the 96U.Ekenberg, J . Phys. (Paris) 48, C5-207 (1987).

286

G. BASTARD et al.

FIG.23. The in-plane effective mass m* at the E , subband edge divided by the bulk conduction band edge mass of the well-acting material m; (right scale) or the relative increase Am*/m,*=m*/mQ - 1 (left scale) are plotted versus the well thickness L for different quantum wells. Notice that the Hg(Cd)Te-CdTe curve refers to the upper horizontal axis.

wells get narrower or, for a fixed L, when the well-acting material has a smaller band gap and thus a lighter conduction band mass. Depending on the magnitude of m:/mr6, we have either directly plotted this ratio against k: or the fractional change Arn*/rnz [ =m:(k,)/rn,, - 11 against k:. The most direct determination of the in-plane effective mass is from cyclotron resonance experiments. Compared with the vast literature devoted to the effective mass measurements in modulation-doped single heterostructures, relatively little data on doped quantum wells have been published. Two sets of experiments performed by Nicholas et d9’and Singleton et involved significant nonparabolicity effects. In rather heavily doped (n, = 8 x lo1’cm-2) Gao,4,1no,,,As-Alo,4,1n,,52As quantum wells (L = 150A), Nicholas et d9’have reported effective masses extrapolating to -0.0456mo at zero frequency. We interpret this value as the in-plane effective mass on the Fermi circle (kF 2.24 x lo6cm- I). Our calculation with V, = 0.5 eV gives rn*(k,) = 0.0454m0. In a narrow (L = 22 A) nominally

-

97R. J. Nicholas, M. A. Hopkins, M. A. Brummell, and D. R. Leadly in “Interfaces, Quantum Wells and Superlattices”, edited by C. Richard Leavens and Roger Taylor, NATO ASZ Series 179, p. 243 (Plenum Press, New York, 1988). 98J. Singleton, R. J. Nicholas, D. C. Rogers, and C. T. B. Foxon, SurJ Sci. 196,429 (1988).

ELECTRONIC STATES IN SEMICONDUCTOR HETEROSTRUCTURES

287

-

undoped GaAs-Gao.64A10,,,As quantum well, Singleton et have measured under illumination an effective mass that extrapolates to 0.085m0 at zero frequency. If the population in the well is negligible, this mass should be interpreted as the carrier in-plane effective mass at the El subband edge. Our calculations with V, = 296meV give m*(O) = 0.079m0. Thus, one can only claim a qualitative agreement with experiments. When the quantum well is asymmetrical (see Fig. 28, there is a lifting of the two fold degeneracy for a given k,. Under flat-band conditions the only possibility of achieving such an asymmetry is to surround a well by two nonequivalent barriers. It is possible to derive the bound states equation for such a well by writing that the eigenstates are of the 2 x 1 form (4.13) (4.14) (4.15) Using the k,-dependent boundary conditions at both the one arrives at

L/2 interfaces,

(4.16) where

1 + + + AA’ + 2

a, = E

EA

E

EA

(4.18)

288

G. BASTARD et al.

FIG. 24. The relative increase Am*/mg for the E, subband is plotted versus k: for GaAsGa,,,Al,.,As quantum wells with different thicknesses.

FIG.25. Same as Fig. 24 but for Ga,,,,ln,,,,As-InP

quantum wells.

ELECTRONIC STATES IN SEMICONDUCTOR HETEROSTRUCTURES

21 0

I

I

t:

I

I

10

289

20

1

( 10l2

FIG. 26. The ratio of the in-plane effective mass m* in the El subband to the bulk InAs conduction band edge mass is plotted versus k: for several GaSb-Id-GaSb double heterostructures.

-CdTe

Hg,,Cd,,Te 0

I

1

El I

FIG.27. Same as Fig. 26 but for Hg,,,,Cd,,,,Te-CdTe

I

quantum wells.

290

G. BASTARD et a1

and K ~ , ~ , Alqr,and q,rare the barriers' evanescent wave vectors, r band gap, r spin-orbit energies, and conduction band offset in the left- and righthand-side barriers, respectively. Equation (4.16) nicely shows that the origin of the lifting of the twofold degeneracy arises from the asymmetry of the structure: if the two barriers are identical, the right-hand side of Eq. (4.16) identically vanishes and one can check that Eq. (4.16) reduces to Eq. (4.10). Also, we see from Eq. (4.16) that both a finite in-plane wave vector and a nonvanishing spin-orbit coupling are necessary to lift the twofold degeneracy. Finally, due to the energy denominators appearing in the bl,r and b, expressions, the effects should be more pronounced for quantum wells based on narrower-band-gap materials with the larger spin-orbit energies and the smaller thicknesses (to increase k , and decrease E - LQ. We illustrate these points by showing in Figs. 29 and 30 the quantity A E , = E : ( k , ) - E ; ( k , ) versus k , for several well thicknesses in GaAs-Gal -,Al,As (xleft= 0.1, xright = 0.3) and InP-Gao,471n,,,,As-Alo,481no,52As quantum wells. The effects are very small in GaAs quantum wells but sizable in Ga,,471n,,,,Asbased heterostructures. The same procedure as applied to quantum wells can be used to find the k , dependence of the superlattice eigenstates. Besides the problem of finding the superlattice in-plane effective mass and its dependence upon k,, these calculations are useful in checking to what extent the assumption that the longitudinal and transverse electronic motions are decoupled in actual superlattices is reasonable. In fact, most of the transport properties, which involve scatterings between superlattice eigenstates, are analyzed in terms of simplified dispersion relations of the form

4% k,)

= h2k:/2m*

+ E,M,

(4.23)

that is, they assume a complete decoupling between the z and x, y motions. This assumption, convenient as it is, however, deserves to be more critically analyzed. The superlattice dispersion relations at finite k , that correspond to evanescent waves in the barrier for the z motion are simply obtained from Eq. (4.10) by writing cosqd = g(&),

(4.24)

ELECTRONIC STATES IN SEMICONDUCTOR HETEROSTRUCTURES

291

InP - G a ( 1 n ) A r - A l ( I n ) A r V, = 0.5eV

2

-W

a

Ga,,ALo, As-GaAs -Gao +lo 3As 0

002

001

003

k,

k l (I!-')

FIG.29. The calculated splitting of the E , state AEl is plotted versus k , for asymmetrical GaAs-Ga(AI)As quantum wells with different thicknesses.

FIG.30. The calculated splitting of the E l state AEI is Plotted versus k , for InPG % 47In,.s,As~Al,.4,ln,.s~AS quantum wells with different thicknesses.

where d is the superlattice period and g(E) is the left-hand side of Eq. (4.10), with cos kAL(sin k,L) being multiplied by cosh K B L B (sinh K B L B ) , where LBis the barrier thickness. We have calculated the lowest-lying solutions [E,(q = 0), E,(q = n/d)] of Eq. (4.24) and plotted the ground superlattices bandwidth A, versus k:. A simplified dispersion relation like Eq. (4.23) predicts that A, is k,-independent. The presence of an explicit k, dependence in Eq. (4.10), and thus in Eq. (4.24),immediately shows that Eq. (4.23)cannot be rigorously exact. However, it is an excellent approximation in the case of GaAs-Ga,.,Al,,,As superlattices (see Fig. 31) where the large band gaps and the similarities between the well and barrier effective masses ensure that both the r6edges move nearly parallel with increasing k,, which means that the superlattice eigenstates are nearly the same when k, increases. A similar feature is found in Ga(1n)As-InP superlattices, where A 1 decreases from 105.1 to 101.1meV when k, increases from 0 to 2 x lo6 cm-' if L A = L, = 30w, from 8.68 to 7.65meV when k, increases from 0 to 4.5 x 106cm-' if LA = L, = 60& and from 0.77 to 0.64meV when k, increases from 0 to 4.5 x lo6cm-' if LA = L B = 90A. Figure 32 presents the k, dependence of A, in InAs-GaSb superlattices. The contrast with Fig. 31 is striking. In InAs-GaSb superlattices one finds that Eq. (4.23) is usually a poor approximation: A1 varies by 50% in a 20 A-20 A superlattice when k, increases from 0 to 4.5 x 106cm - by a factor of 3 in a 40 A - 40 A material and by a factor of - 8 in a 60A-60A superlattice. In contrast with the two previous cases, the k, increase strongly alters the host eigenstates out of which the superlattice states are built. Increasing k, amounts to suppressing

-

292

G . BASTARD et al

X = 0.3 0

5

'4,

kf(10 cm- 2)

15

FIG.31. The calculated E l bandwidth A l is plotted versus kf for three GaAs-Ga,,,Al,.,As superlattices.

FIG.32. The calculated E , bandwidth A, is plotted versus k: for three InAs-GaSb superlattices.

the overlap between the InAs conduction band and GaSb valence band. It is thus natural that the states that are the more affected by this change are those where the El subband is energeticallyclose from the overlap region (i.e., those with the larger period). The strong decrease of the El bandwidth with k, leads one to speculate about the effect of unidimensional disorder, for example that produced by deliberately randomizing the well widths of a s u p e r l a t t i ~ e , ~on ~ *the ~ ~nature " of the electronic transport along the growth axis. This transport is usually inhibited by disorder, irrespective of k, in type I superlattices [because Eq. (4.23) is a fair approximation in these materials]. If one adopts the criterion that fluctuations in the isolated well eigenstates of - A l cause all the states to become localized for the z motion, it might be possible in the case of a 60 a-60 InAs-GaSb superlattice to get delocalized superlattice states near k, = 0 and localized states when k, increases, which is a rather uncommon feature. 5 . VALENCE SUBBANDS

Section 4 has shown that the in-plane dispersion relations of the conduction subbands are relatively simple and that complications can arise from band-mixing effects. Except in a few cases (quantum wells with infinite barriers," and even in the parabolic approximation, there is no way to 99S.S. Nedorezov, Sou. Phys. Sol. State 12, 1814 (1971).

ELECTRONIC STATES IN SEMICONDUCTOR HETEROSTRUCTURES

293

obtain the in-plane dispersion relations of the valence subbands analytically. This is because the bulk valence edge is fourfold degenerate and that the effective valence hainiltonian is not a scalar but a 4 x 4 matrix, whose simplest (i.e., spherical) approximation can be writtenz9 as i%? = akz

+ P(k - J)',

(5.1)

where a and P are two constants, expressible in terms of the light and heavy hole effective masses, and J is the 4 x 4 matrix of a pseudo-spin 312. Equation (5.1) assumes the hole kinetic energy is much smaller than the rspin-orbit energy. This approximation will be retained from now on and is usually a satisfactory one in 111-V and 11-VI heterolayers. If the J quantization axis is fixed, say along the growth axis (%axis), then the decoupling between light and heavy holes is possible only if k 11 J. In the bulk materials it is always possible to line up J and k by choosing the J quantization axis along k. In heterolayers this is usually impossible because if such a rotation is done in the A material, it will bring J along k, but not along k, unless k, and k, are parallel to 2, which only happens if k, = 0. Thus one has to resort to numerical diagonalization of Eq. (1.24),using Alterelli's boundary conditions given by the conservation of the M matrix Eq. (1.29) across an These valence band intricacies were neglected for some time until it was realized,"'- l o * after the pioneering works of Altarelli et u1.,8*100,109-113 that there was a significant physical effect with many implications buried in the complicated algebra. The key feature, now well, if not easily, established, is the band-mixing effect that takes place when k, # 0 -that is, the fact that the very notion of heavy and light hole states applies only at k, = 0, whereas if k, # 0 the valence eigenstates are of mixed nature. For heterostructures under flat-band conditions an exact solution can be found. It amounts to finding all the propagating and evanescent solutions in 'OM. Altarelli in "Heterojunctions and Semiconductor Superlattices." Springer-Verlag, Berlin, 1986. '*'G. Bastard and J. A. Brum, I E E E J . Quantum Electron. QE22,1625 (1986);see also J. A. Brum, These de doctorat, Paris, 1987, unpublished. Io2J. M. Berroir and J. A. Brum, Superlattices and Microstructures 3, 239 (1987). lo3T. Ando, J . Phys. Soc. Japan 54, 1528 (1985). lo4E. Bangert and G. Landwehr, Superlattices and Microstructures 1, 363 (1985). lo5D.Broido and L. J. Sham, Phys. Rev. B 31, 888 (1985). lo6A. Hernandez-Cabrera and P. Aceituno, Solid State Commun. 65, 1451 (1988). lo7L. C. Andreani, A. Pasquarello, and F . Bassani, Phys. Rev. B 36, 5887 (1987). "'L. J. Sham, Superlattices and Microstructures 5, 335 (1989). "'M. Altarelli, J . Luminescence 30, 472 (1985). "OU. Ekenberg and M. Altarelli, Phys. Rev. B 30, 3369 (1984). "lM. Altarelli, U. Ekenberg, and A. Fasolino, Phys. Rev. B 32, 5138 (1985). I1'A. Fasolino and M. Altarelli, Surf: Sci. 142, 322 (1984). 113G.Plater0 and M. Altarelli, Phys. Rev. B 36, 6591 (1987).

294

G . BASTARD et al.

FIG. 33. Calculated in-plane dispersion relations for the valence subbands of GaAsGa,,,A1,,,As quantum wells with L= lOOA and L = lSOA. The dashed lines correspond to the diagonal approximation of the Luttinger hamiltonian.

each host layer and matching them through the conservation of the matrix equation (1.29). Another method that works well consists in finding all the k, = 0 solutions whose energies are within the bottom and the top of the valence well and to diagonalize the off-diagonal, k,-dependent terms of the 9 matrix equation (1.24). This is an approximate method because the continuum solutions are not included in the basis. In addition, it experiences difficulty in handling materials with very different Luttinger parameters. The reason is that one cannot ensure the continuity of the M matrix equation (1.29) at k, # 0. For many heterostructures this is unimportant, for the y parameters are quite similar and because the eigenstates are usually fairly well localized in one of the layers. A counterexample is provided by the HgTe-CdTe superlattices, where this method does not provide sensible results compared with the exact solutions. Most of the results shown here have been obtained by neglecting the inplane anisotropy of the valence subbands. This axial approximation is obtained by replacing y z and y 3 in Eq. (1.27) by their arithmetical average. Figure 33 shows the in-plane dispersion relations of the valence subbands for two GaAs-Ga0,,A1,,,As quantum wells with L = l O O A and l50A, respectively (IVJ = 0.15 ev)."' The dashed lines correspond to calculations that neglect the off-diagonal terms of the valence hamiltonian (diagonal approximation). They exhibit the so-called mass-reversal effect, which corresponds to having heavy hole states along the growth axis, which are lighthole-like in their in-plane motion, and vice versa for the light hole states. Thus, in the diagonal approximation, where heavy hole states are purely

ELECTRONIC STATES IN SEMICONDUCTOR HETEROSTRUCTURES

295

FIG. 34. Calculated k, dependence of for the valence subbands of two GaAsGao.,A1o.,As quantum wells with L = 100 A and L= 150 A.

m, = _+ 3/2 whatever k,, and light hole states are purely m, = f 112 whatever k,, the heavy and light hole subbands should cross for some k, value. In the

full calculation (solid lines in Fig. 33) these crossings are replaced by anticrossings. The dispersion relations become strongly nonparabolic. In particular, one notices that the ground light hole subband acquires an electronlike dispersion near k, # 0. This happens because HH2 repels LH, upward in energy more than HH, does downward. This camelback shape for the LH, dispersion disappears if the wells are very thin. The strong nonparabolicity of the valence subbands has its counterpart in a wave function admixture. We have found it convenient to express this mixing”’ by calculating the k , dependence of where J t is a diagonal matrix in the Ts basis with matrix elements equal to 9/4 (twice) and 1/4 (twice). The ( ) symbol means that an average of J t is taken over the k,-dependent eigenstates. Figure 34 shows the k , dependence of for the two quantum wells discussed previously. If the valence hamiltonian were pure diagonal, the heavy (light) hole subbands would all be characterized by = 3/2 (1/2) no matter what the value of k,. Thus, the deviations from the two horizontal lines 3/2 and 1/2 measure the band-mixlpg effects. Figure 34 shows that they are quite large, especially for LH,. Although we keep labeling the valence subbands according to their k , = 0 nature, Fig. 34 clearly shows that the notion of light and heavy hole at k , # 0 is very fuzzy.

m,

296

G. BASTARD et a/.

I

lo+-

r

GaAs - Ga(AL)Ar

1

1

401



I

n

i

E w

4

-

10-

0-

FIG.35. Calculated k , dependence of the ground valence subband width in GaAsGao.,A1,,,As superlattices with equal layer thicknesses and d = 60 A,80 A,100 A.



L

FIG. 36. Calculated k , dependence of the ground valence subband width in GaAsGa,,Al,,,As superlattices with a fixed GaAs slab thickness (40 A) and different barrier thickness.

The band-mixing effect also has a strong consequence on the superlattice valence eigenstates. We saw in Section 4a that the band mixing on the superlattice conduction eigenstates were usually weak in the sense that it was a reasonable approximation in GaAs-Ga(A1)As and Ga(1n)As-InP superlattices to consider the z and x, y motions as decoupled, as witnessed by the small decrease of the ground superlattice bandwidth with increasing k,. Figures 35- 36 show the k , dependence of the ground hole superlattice bandwidth for several superlattice periods and equal well and barrier thicknesses (Fig. 35) and for a fixed well width but increasing barrier thicknesses (Fig. 36). One notices that, in contrast with the electronic subbands, the hole bandwidth increases with k,. This bandwidth is defined as the energy separation between the lowest-lying q = n/d and the lowest-lying q = 0 hole states. At k , = 0 this energy separation is the heavy hole HH, bandwidth. One notices that for narrow enough barriers, the hole bandwidth increases by more than a factor of 2 when k , increases. The physical interpretation for such a trend is that the heavy and light hole states get admixed when k , increases, which for the z motion means that the heavy hole acquires a light hole behavior (i.e., a broader bandwidth). Such a feature can be translated in terms of a hopping time from one well to the other, z = h/AE. Figures 35-36 thus show that z(k,) decreases with k,, which may have some implications on our current understanding of the vertical hole transport in superlattices. The latter is often considered as inefficient, if possible at all, due

ELECTRONIC STATES IN SEMICONDUCTOR HETEROSTRUCTURES

n=l

n:2

nr3

297 n:b

FIG.37. Calculated subband dispersions at k, = 0 (left panel) and in-plane dispersions at different superlattice wave vectors q (q=O, n/2d, nld) for a 50A-50A GaAsGa,,,Al,,,As superlattice. IV,l is taken equal to 135meV, and the energy zero corresponds to the bulk Ga,,,AI,,,As valence band edge. After Ref. 102.

to the implicit statement that the holes are always heavy, irrespective of their in-plane wave vector. For thicker wells and barriers, the HH, subband is almost dispersionless along the growth axis and to a good approximation the decoupling between the z and (x,y) motions is a satisfactory assumption. This is illustrated in Fig. 37 for a 50A-50A GaAs-Gao~,A1o,,As superlattice (IVJ = 135meV).'OZ Although negligible for HH,, the band-mixing effects are crucial for LH,, whose shape is strongly altered when k , increases at a fixed 4. For the LH, subband the decoupling between the z and (x,y)motions is a very poor approximation. The pseudoalloy regime for the valence levels in GaAs-A1As superlattices,102illustrated in Fig. 38, is only reached for the (GaAs),-(AIAs), monolayer superlattice. Only in this ultimate case does one find that HH, and LH, are nearly degenerate at k , = 0 (the residual splitting is only 2 meV) and that the effective masses for their in-plane motion are very close from the ones in the Ga0.,A1,,,As bulk random alloy (in the virtual crystal approximation). For a (GaAs),-(AIAs), superlattice the k , = 0 splitting is already very large (>20 mev), and the eigenstates are heavily localized in the GaAs layers. In many respects this short-period superlattice already behaves like a multiple quantum well, as a result of the large valence offset and of a relatively heavy hole mass.

298

G . BASTARD et al

b)

dl

FIG.39. Band edge profiles of some heterostructures lacking for inversion symmetry.

When the quantum well is asymmetrical (see Fig. 39 for a few examples), the twofold degeneracy of the eigenstates for a given k, is lifted. This is analogous to the situation found for the electron states. The effects are, however, larger for the valence subbands since, even in the parabolic approximation, the spin-orbit coupling is already fully operative. An example of calculated in-plane dispersion relations for the valence subbands is shown in Fig. 40 in the case of a 100-A-thick GaAs-Ga,,,Al,,,As quantum well. The asymmetry in the band edge profile is provided by a static electric field F (F = lo5 V/cm) applied parallel to the growth axis. We find again that the lifting of the twofold degeneracy for a given k, requires the simultaneous presence of a spin-orbit coupling, a nonzero k, (the states are twofold degenerate at k, = 0), and an asymmetrical band edge profile. (In Fig. 33 the states are twofold degenerate whatever k, is.) We discuss the valence subbands in biased multiple quantum wells in the next section. It is clear that the in-plane dispersions are, to a large extent, governed by the k, = 0 level scheme. After all, one can envision, as Sham does,lo8 that these dispersion relations result from a “mini k p interaction” between the k, = 0 heavy and light hole edges. Thus, any external agent that alters the k, = 0 level scheme is likely to affect the in-plane dispersions strongly as well. Uniaxial stress, either externally applied or built into lattice-mismatched heterostructures,*’ is a powerful way to control the k, = 0 level scheme and,

-

ELECTRONIC STATES IN SEMICONDUCTOR HETEROSTRUCTURES

299

thus, to study the in-plane dispersion relations. The stress effects in heterostructures have been recently reviewed, and we refer the reader to these reviews for a thorough discussion.'' In narrow-band-gap materials the r6 band may eventually have a strong effect on the valence dispersion relations. Two heterolayers are particularly sensitive to r6-r8interactions: the InAs-GaSb and HgTe-CdTe (or Hg, -,Mn,Te-CdTe or Hg, _,Zn,Te-CdTe) superlattices. In InAs-GaSb the top of the GaSb valence band lies above the bottom of the InAs conduction band by 0.15 eV. Thus, in superlattices with equal layer thicknesses, there should exist a critical period d, below which the ground r6related subband (E,), which is predominantly InAs-like, lies at higher energy than the ground heavy hole r,-related subband, which is predominantly GaSb-like. For d > d,, E , is partially or completely below HH, (see Fig. 41). At k, = 0 these two subbands can cross, while we expect a strong mixing between r6- and r,-related states if k, # 0, changing the crossings into anticrossings. Figures 42-43 present the in-plane dispersion relations of a 70A-70A and l O O A - l O O A InAs-GaSb superlattices at 4 = 0.92In the 70 A-70 A superlattice El is still above HH, at k, = 0. Thus, this particular superlattice is a very narrow gap semiconductor ( E < ~ 20meV). One notices a strong repulsion between El and HH, whose dispersions are nearly mirrorlike near k, = 0. The fast decrease of the HH, subband edge with increasing k, due to the k * p interaction with El would cause it to cross HH,, which is nearly dispersionless near k, = 0. The valence band mixing enters again into play to produce a strong anticrossing between HH, and HH,. The lOOAl O O A superlattice is such that El < HH,, HH, at k, = 0. If there was no band mixing (diagonal approximation) El would move up in energy, cross both HH, and HH,, and keep increasing with increasing k,. In this approximation, the heavy hole subbands would move downward in energy. One would thus have a true semimetallic configuration with an equal number of holes in the GaSb layers and of electrons in the InAs layers. The valence band mixing induces strong anticrossings between the E l , HH,, and HH2 subbands, preventing the formation of a true semimetallic phase since there is always a finite (but small) energy gap between the hybridized subbands. This does not mean that there is no charge redistribution in a superlattice unit cell. However, it should occur in such a way that the Fermi level at T = 0 K in an undoped sample lies in the gap between the last holelike subband (having a negative curvature) and the first electronlike subband (having a positive curvature), as in NaCl crystals that are insulators with a complete charge transfer inside a unit cell. Notice that Fig. 43 does not include the effect of charge redistribution within a unit cell. The experimental situation38- 4 1 appears to favor a real semimetallic phase since the conductivity and magnetoconductivity measurements show evidence of a fairly large increase

300

G. BASTARD et al.

XzO.3

- 5 O l L

, \qH2 1 0.5

0

1

0

k l ( J t x lo6 ern-’)

50

100

150

200

250

d ti)

FIG.40. Calculated in-plane dispersion relations of a 100-A-thick GaAsGa,,,AI,,,As quantum well subjected to an external electric field F I/ i ( F = lo5 V/cm).

Fic. 41. Calculated subbands of InAs-GaSb superlattices with equal layer thicknesses. The energy zero is taken at the bulk r6 edge of InAs. = 0.96 eV. The allowed light particle states are hatched. The allowed heavy hole states are black. Notice the E , - H H , crossing near d = 160A.

7As-GaSb 0i-70i

‘O0!/

-

150

t Y

50

0

1

2

k, ( n / d )

FIG.42. Calculated in-plane dispersion relations of the q = O superlattice states of a 70A-70A InAs-GaSb superlattice. The energy zero is taken at the edge of the InAs bulk conduction band. After Ref. 92.

0

1

2

kl(T[/d)

FIG.43. Same as in Fig. 42 but for a 100A- 100 A InAs-GaSb superlattice. Flat-band conditions are assumed in the calculations. After Ref. 92.

ELECTRONIC STATES IN SEMICONDUCTOR HETEROSTRUCTURES

301

HgTe - CdTe

1008 -36H

I

jo0!

I

FIG.44.Calculated subband dispersions at k, = O (left panel) and in-plane dispersion relations at q = 0 for a 100A-36A HgTe-CdTe superlattice. The valence band offset A is taken equal to 0.3 eV. The energy zero coincides with the top of the valence band of bulk CdTe. After Ref. 92.

of free carriers when E , passes below HH, (at k, = 0). However, the InAsGaSb superlattices have so far contained a significant number of impurities cm-3 does not seem to be an unrealistic number). Besides, it has been shown”4 possible to reconcile the experimental findings in GaSb-InAsGaSb double heterostructures with a theoretical model that does take the anticrossings and the presence of impurities into account. In our opinion, the fascinating InAs-GaSb system deserves more attention, especially near the semiconductor to semimetal transition. Figure 44 shows the calculated92 in-plane dispersion relations of a 1WA36A HgTe-CdTe superlattice at q = 0 (right panel) and the q dispersion relations at k , = 0 (left panel) for a large valence band o f f ~ e t ~ ~ - ~ ~ (A = E F : ~-~$ye= 0.3eV). For such a large offset the ground interface subband I s is deeply buried in the Ts offset. I,, has a significant q dispersion (- 20 meV). It lies below the HH1 subband and is crossed by HH2 (such a crossing would be replaced by an anticrossing, even at kl = 0, if the inversion asymmetry splitting of the host materials were taken into account).g0If there ‘I4M. Altarelli and J. C. Maan quoted in M. Altarelli in “Optical Properties of Narrow-Gap Low-Dimensional Structures (C. M. Sotomayor Torres, J. C. Portal, J. C. Maan, and R. A. Stradling, ed.), p. 15. Plenum Press, New York, 1987.

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were no couplings between these subbands at k , # 0, the superlattice would be a semimetal. The "mini k * p" interaction as well as the direct k p coupling between S-like and P-like edges leads to a strong repulsion between the q = 0 levels. The topmost level HH, acquires a light electronlike curvature, HH2 displays a camelback shape, and I,, quickly becomes heavy-hole-like. For q # 0 the HH, in-plane dispersion becomes lighter (since the k , = 0 levels get closer in energy). The situation for HH, and I,, is more complex at small k,. Both of them, however, end up as holelike at large k,. The semimetallic phase is thus suppressed. This superlattice should be a narrow-gap semiconductor at T = OK without doping and with a Fermi level located between lAs(q= n/d)and HH,(q = 0). Actual samples may significantly deviate from such a behavior since the doping of 11-VI superlattices cannot yet be properly controlled.

-

V. Stark Effects in Semiconductor Quantum Wells and Superlattices

6. INTRODUCTION The previous sections have been devoted to a presentation of the electronic structure of heterolayers under flat-band conditions. In this section we consider the case where an external and constant electric field is superimposed to the band edge profile of the heterostructure. We confine our discussion to a theoretical description of Stark shifts in semiconductor quantum wells and superlattices. In the first kind of structures, the field leads to a red shift of the ground electron and hole states and, thus, of the fundamental band-to-band transition.' 5 * 1 By contrast, in superlattices the of the band-to-band absorption edge. electric field leads to a blue shift117y118 The opposite signs of the two effects stem from their different physical origin. In quantum wells there is a field-induced polarization'" of the bound eigenstates and, thus, a nearly quadratic energy shift due to the interaction of the induced dipole with the field. In superlattices, the field suppresses the tunneling between the consecutive wells' and thus isolates the different wells from each other. Thus, the lowest-lying optical transition occurs in a strong field between the levels of quasi-independent wells, while at zero field it takes place between the lowest-lying eigenstates of interacting wells, which, as

'

'15G. Bastard, E. E. Mendez, L. L. Chang, and L. Esaki, Phys. Rev. B 28, 3241 (1983). 'I6D. A. B. Miller, D. S. Chemla, T. C. Damen, A. C. Gossard, W. Wiegmann, T. H. Wood, and C. A. Burrus, Phys. Rev. Lett. 53, 2173 (1984) and Phys. Rev. B 32, 1043 (1985). "'5. Bleuse, G. Bastard, and P. Voisin, Phys. Reu. Lett. 60, 220 (1988). '18E. E. Mendez, F. Agullo-Rueda, and J . M. Hong, Phys. Rev. Lett. 60,2426 (1988).

ELECTRONIC STATES IN SEMICONDUCTOR HETEROSTRUCTURES

303

shown in the previous sections, occur at an energy lower than roughly half of the electron or hole subband widths. The Stark effect in isolated wells or superlattices has stimulated a significant amount of research. They are basic to novel optoelectronic devices (e.g., fast electrooptical modulators).'lg - 12' In the following we shall first describe the Stark effect in isolated quantum wells, then in double quantum wells, which are the shortest possible superlattices, and finally in superlattices. 7. ELECTRIC FIELDEFFECTS IN ISOLATED QUANTUM WELLS As pointed out in the introduction, there is a rich technological potential in applications of a longitudinal ( F 11 z) electric field to quantum wells. The usefulness of these structures lies in their capability of withstanding large fields ( F d lo5V/cm), which produce large Stark shifts, while still displaying quasi-discrete bound states. We shall assume that the field is uniform over the whole structure, as approximately realized when the quantum well is inserted in the intrinsic part of a reverse-biased pin junction or in the depletion length of a reverse-biased Schottky diode. In the case of multiple quantum wells, the barrier separating two consecutive wells will be assumed thick enough to prevent any sizable coupling between their eigenstates. Thus, neglecting band-mixing effects, the Schrodinger equation we have to investigate is

-h2 d 2

-k vb(Z) -k e F z

1

x ( Z ) = EX(Z).

In Eq. (7.1) the in-plane motion has been dropped, v&) is the potential energy profile of a single rectangular quantum well, and the electrostatic potential e F z has been set equal to zero at the center of the quantum well. It may sometimes be more convenient to take its origin on the left-hand side (1.h.s.) corner of the well. A constant electric field leads to pathological behavior of the eigenstates of Eq. (7.1) versus F . At zero field the Schrodinger equation admits at least one bound state ( E c vb), while an arbitrarily small F is sufficient to transform the allowed energy spectrum into a continuum. This occurs because the potential energy is arbitrarily large and negative at large and negative z. Despite the lack of true bound states, we expect, if F is not too large, that there will exist in the continuous spectrum some particular energies where the carrier wave function piles up in the quantum we11."5*'22 Miller, J. S. Weiner, and D. S. Chemla, IEEE J. Quantum Electron. QE22,1816 (1986). '*OK. Wakita, Y. Kawamura, Y. Yoshikuni, H. Aoahi, and S. Uehara, I E E E J. Quantum Electron. QE22, 1831 (1986). I2'D. A. B. Miller, D. S. Chemla, T. C. Damen, A. C. Gossard, W. Wiegmann, T. H. Wood, and C. A. Burrus, Appl. Phys. Lett. 45, 13 (1984). Iz2E.J. Austin and M. Jaros, Appl. Phys. Lett. 47, 274 (1985). 'I9D. A. B.

304

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Moreover, these particular energies will smoothly extrapolate to the true quantum well bound states when F + 0. In fact, we do know from experiments that, over a significant field range (typically F < 105V/cm), the quantum well structures support states that behave as if they were truly bound. Thus, it is worth trying to convince ourselves that some stationary eigenstates of Eq. (7.1) are indeed peculiar in that they display an accumulation of their wave functions in the well. An alternative description, which is often more revealing, is to consider them as metastable (i.e., nonstationary) solutions of the time-dependent tilted quantum-well problem. This kind of description is relevant when the decay time of the quasi-bound states is long. Hereafter, we shall also denote the peculiar stationary solutions of Eq. (7.1)as the metastable state, since it can be shown that the real part of the complex eigenenergies of the time-dependent problem coincide with the (real) energies of the peculiar solutions of Eq. (7.1). Apart from the exact, Airy-like solutions, there are various ways to find the metastable states of Eq. (7.1). First, one may cutoff the electric field at some large distance from the investigated quantum well (Fig. 45) and impose the existence of an infinite barrier. The problem becomes that of finding true bound states in a complicated band edge profile. The outcome of such calculations is that there exist a large number of states that are essentially localized in the large triangular well and show a very small probability of being in the well. A small number of states are found to display an enhanced probability of being in the well. These are the metastable states whose energy positions extrapolate smoothly to those of the zero-field bound states of the well.

FIG.45. Two approximate ways to calculate the metastable states of a quantum well tilted by an electric field.

ELECTRONIC STATES IN SEMICONDUCTOR HETEROSTRUCTURES

305

One may also cut the electric field at some large distances on both sides of the well and investigate the transmission coefficient T(E)= It(~)f’of a plane wave with unit amplitude impinging at z = -9on the barrier and being finally transmitted at z = + 9, T(E)is very low (because 9is large) except in the vicinity of some energies that belong to the segment [ q 1 , q 2 ] , where it displays sharp peaks (T < 1). As usual, these transmission resonances”6 correspond to the trapping of the particle inside the quantum well. If the resonances are narrow, this means that the corresponding trapping times ztrap are long: z,,,,AE > h/2, where AE is the width of the resonance. Another way to depict these resonances is to consider them as virtual (or metastable) bound ~ t a t e s ,that ~ ~is,, as ~ ~bound ~ states of the quantum well with a complex energy E, - ih/2ztrap. The reason why the energies of these states have to be complex and not purely real is that the solutions of the Schrodinger equation Eq. (7.1) with the boundary conditions corresponding to a piling up of the wave function at t = 0 in the quantum well and to an outgoing wave at z = - co do not fulfill the requirement of the probability current conservation; that is, they are not stationary. On the other hand, it can be shown36 that such solutions are approximate eigenstates of Eq. (7.1) provided their energies are complex. In order to set up a criterion allowing us to neglect, in a first approximation, the escape of the particle outside the quantum well, the condition eFtcC’ <<

vb

- El

(7.2)

was proposed some time where El is the confinement energy in the quantum well, and IC;’ is the characteristic decay length of the bound state wave function at zero field. The meaning of Eq. (7.2) is that the lowering of the barrier by the field over the characteristic decay length of the zero-field wave function must remain negligible with respect to the effective F = 0 barrier height v b - E l . One can derive a similar criterion by looking at the problem from a different perspective. Suppose we know that we have built up at t = 0 a quasibound state El, which, if the 1.h.s. barrier were flat, would be truly bound. Correspondingly, the particle would (classically) oscillate back and forth in the well forever with a period T(E,).Since, however, the 1.h.s. barrier is tilted, the electron that oscillates in the well progressively escapes to infinity. Using the same semiclassical approach as in Section 111, we get 1/%c

(7.3)

= D(El)/T(El),

where D ( E , ) is the transmission of the tilted barrier. In the semiclassical approximation, D(E,) is given by D(E,) = Do exp{ -(4/3eF)(2m*/h2)”’(Vb

-

(7.4)

306

G . BASTARD et al

with Do = 1. As for the evaluation of T(E,),one gets

T(E,) = 4/eF(m*E1/2)”’ = 4/eF(m*E1/2)’/’[(1

+ (1 - eFL/E,))’/’]

if El < eFL,

(7.5)

if El 2 eFL.

(7.6)

The lifetime z(E,) will be long if the argument of the exponential in Eq. (7.5) is large and negative, which means vb

- E l >> 3eFlc;’.

(7.7)

Apart from the factor 3/4, this is just the criterion that we derived previously on the smallness of the barrier lowering by the field over the distance t c i 1 , where the wave function is important in the barrier. Assuming that the lower bound for >> is 10 and taking v b - El = 0.125 eV, m* = 0.07m0, we find that Eq. (7.7) is satisfied if F < 7.93 x lo4 V/cm. If, in addition, L = 1008, Do 2 1, and El = 70meV, we get h/2zesc(E1)z 2.5 meV and zesc(El)z 1.3 ns. For these typical sample parameters the level broadening is still small with respect to the confinement energy. The escape lifetime is relatively long but is not much longer than a recombination lifetime ( z1ns). The capture of the carrier by some shallow impurity may eventually be more efficient than the field-induced tunneling. Notice finally the strong dependence [Fexp(-FJF)] of; ; ,7 with the field. It arises from the transparency coefficient equation (7.4). If instead of applying 8 x 104V/cm we only apply 50 kV/cm, z(E,) increases by several decades, and the field-induced tunneling becomes a completely negligible effect with respect to recombination or capture phenomena. Figure 46 shows the calculated zest versus F for El, HH,, and LH, in GaAs-Ga(A1)As quantum wells with thicknesses 30di, 608, and 908. ,z, is found to vary considerably (from 1ps to 1s) in the investigated field range. Moreover, Fig. 46 clearly shows that the fieldinduced escape of the carrier outside the quantum well is practically

10

40

F (kV/cm)

70

100

FIG. 46. Semiclassical estimates of the escape time of an electron (in the E , state) a light hole (LH, state) and a heavy hole (HH, state) out of quantum wells (L=30A, 60A, 90A) tilted by an electric field. For the electron, light hole, or heavy hole the escape time at a given field increases monotonically with L.

ELECTRONIC STATES IN SEMICONDUCTOR HETEROSTRUCTURES

307

negligible when L >, l00.k Most of the quantum-well structures that have been so far investigated (e.g., in electroabsorption) have shown excellent performances(i.e., pronounced exciton peaks) in fields up to N lo5V/cm. The previous considerations have aimed to show that this is reasonable in spite of the field-induced tunneling. From now on we shall neglect this effect and investigate the second useful feature of the electric field effects on quantumwell states, which is the existence of large Stark shifts. The zero-field eigenstates of Eq. (7.1) may be classified according to the parity operator. Since the eFz is odd in z, it only couples the zero-field eigenstates of opposite parities. Since we neglect field-induced tunneling, we can use perturbation or variational approaches' to calculate the field dependence of the eigenenergies. The basic physics is that the quantum-well bound states become polarized by the electric field, which in turn lead to a shift of their eigenenergies by $D, F, where D, is the average value of the dipole operator in the nth state. In the low-field limit D, = a,F, where a, is a c-number. This results in a quadratic (Stark) shift upon F . For larger fields, a, becomes F-dependent and displays some saturation: D z eL/2. This is the carrier accumulation regime where the wave function piles up near the 1.h.s. of the well for electron and the right-hand site (r.h.s.)for holes (see Fig. 47).This carrier accumulation is a useful feature for electrooptical devices: by inhibiting the carrier escape toward +a (which is unavoidable in bulk materials), the quantum-well walls give access to large ( N 30 meV) energy shifts while preserving the carrier localization and thus provide beneficial action of enhanced excitonic binding."6,'23 This effect has lead to a number of electrooptical devices, fast modulators,' 2o for example. In the small-field limit we can use a second-order perturbation approach to obtain 159116

-

''J

where In) is the nth zero-field bound eigenstate with energy E,(O) and where the summation over m runs over the zero-field bound and unbound states. The latter give a small contribution and are usually neglected. From Eq. (7.8)we see that the ground state (n = 1)experiences a red shift, as always. Since En - Em z L - 2 and (nlzll) z L, E,(F) - E,(O) scales like I?F2m*. But the domain of validity of Eq. (7.8), which is that the field-induced shift remains small with respect to the unperturbed energy splittings, narrows like m*FL3 = constant. Once the field is too large to use Eq. (7.8), one may use variational approaches. A linear variational treatment consists of expanding ~ ( zin ) the incomplete basis spanned by the zero-field bound eigenstates of Eq. lt3J.

A. Brum and G. Bastard, Phys. Reo. B 31,3893 (1985).

308

G. BASTARD et al.

1.5

-

1

N

Y

ru-

N \

0.5

J

0

-

z/L

2.5 2 h

N Y

1.5

01-

X N \

I

0.5 0 -1

-0.5

0

2

/L

05

FIG.47. Calculated envelope functions for electrons (El state) or holes (HH, state) in a GaAsGa,,,AI,,,As quantum well ( L = 200 A) for several electric field strengths (in kVjcm).

(7.1). In this way, one obtains the field dependencies of all the states. In Fig. 48 we show the outcome of such a calculation. If one is interested in the ground state only (as often in device applications), a nonlinear variational wave function like

x ( 4 = x‘O’(4

exp( - B 4

(7.9)

is quite accurate, since it describes the tendency toward accumulation (p > 0 for electron; j? < 0 for holes) and is simple to use. In Eq. (7.9), ~ “ ’ ( z )is the ground bound solution of Eq. (7.1) at zero field. The wave function given in Eq. (7.9) also contains the signature of significant field-induced tunneling: as 101increases with F , it happens that it becomes larger than q,(O), the zero-field wave vector characterizing the evanescent wing of the ground bound state. The minimization procedure becomes impossible, and one may rightfully

ELECTRONIC STATES IN SEMICONDUCTOR HETEROSTRUCTURES 151

-1 5 0

I

I

I

50 F ( kV/cm

309

100

1

FIG.48. Calculated energy shifts of the conduction and valence levels in a 100-&thick GaAsGa,,,AI,.,As quantum well versus the electric field strength.

consider that the very notion of quasi-discrete bound state fades away. To exemplify the tunability range of energy level shifts upon the electric field, we show in Figs. 49 and 50 the calculated field dependence of the E,-HHl interband energy for various well thicknesses L in GaAs-Ga(A1)As- and Ga(1n)As-based quantum wells. It is seen that considerable shifts can be produced for L > lWA. The data obtained in GaAs quantum wells124are very well described by the calculations. The electric-field-induced polarization of the carrier wave functions suppresses their parity properties. Thus, optical transitions that were parityforbidden at zero field become allowed at non vanishing F. Their growth occurs at the expense of the F = 0 parity-allowed transitions. Miller et have given a nice discussion of the sum rules associated with optical transitions in biased quantum wells as well as the progressive evolution of the optical absorption line shape from that of biased quantum wells to the Franz-Keldysh effect in thick, bulklike structures (L 2 500 A). 124L. Vina, E. E. Mendez, W. Wang, L. L. Chang, and L. Esaki, J. Phys. C.Solid State Phys. 20, 2803 (1987). '25D.A. B. Miller, D. S. Chemla, and S. Schmitt-Rink, Phys. Rev. B 33, 6976 (1986).

3 10

G. BASTARD et al.

\ mj

1

-60 Vs:0.264tV

a

Vpz036StV \?OOA Ga(1n)At - A L ( l n ) A r _ _ _ . - 6 0 1V~=D~4eV,V;:0169eV

,

STARK SHIFT

0

20

FIELD (V/cm) FIG. 49. Electric field dependence of the fundamental optical transitions in GaAs-Ga(A1)As quantum wells with different thicknesses. Solid: line theory. Symbols: experiments. After Ref. 124.

40

60

80

100

F (kV/crn ) FIG. 50. Calculated Stark shift of the fundamental optical transition ( E , -HH,) in Ga(1n)As-InP and Ga(1n)As-Al(1n)As quantum wells with different thicknesses.

8. ELECTRIC FIELDEFFECTS IN DOUBLE QUANTUMWELLS The eigenstates of double quantum wells can be accurately obtained by diagonalizing the eFz term between all the zero-field bound states of the problem (these states are analytically known). The physics is, however, more transparent if one attempts a tight-binding expansion of the eigenstates in a basis spanned by all the zero-field bound states of each well when considered as isolated.'26 For symmetrical double quantum wells one would write where d = L + h is the period of the associated superlattice, L(h) is the quantum well (intermediate barrier) thickness, and &:(z - zo) is the nth bound state wave function of the quantum well centered at zo when considered as isolated and under flat-band condition. To keep the matter simple, let us assume that the 4{:: are orthonormalized:

(4{:b(z - id/2)lc#@(z - i'42)) = ~ 3 ~ , ~ . 6 ~ , , i,, ; i'

=

& 1.

(8.2)

ELECTRONIC STATES IN SEMICONDUCTOR HETEROSTRUCTURES

31 1

Then, in the case where the unbiased, isolated wells admit two bound states ( E , and E,), the hamiltonian takes the matrix form

1

El -eFd/2

I’

eFZ12

eFz1,

11

El +eFd/2 -eFzlz

eFz,, E,-eFd/2

eFZ12

eFZ1, eFz12

eFZ12

I2

I2

E,

+ eFd/2

I

(8.3)

where some terms have been dropped since they do not produce essential physical features. Z , , (respectively, z , , ) are the matrix elements of the z operator between 4;:; and 4;::, which are centered in the same well (respectively, in different wells). The two diagonal 2 x 2 blocks express the couplings between the same sort of states ( E l or E,) in either well, while the eFZ,, term is just the intrawell field-dependent polarization (the one that gives rise to the Stark effect in isolated wells), and the e F z , , term is the interwell field-dependent polarization. z l , is, in general, smaller or much smaller than Z , , since the wave functions involved in the former matrix element peak in different wells, while those in the latter matrix element are both centered in the same well. A, and I, are the nearest-neighbor transfer integrals between the wells corresponding to the ground and excited states, respectively. If one neglects the off-diagonal blocks, which amounts to performing two one-band tight-binding analyses separately, one immediately gets El,* = E l E,,+

=

f ,/(eFd/2)’

E , L ,/(eFd/2)’

+ If,

(8.4)

+ I:.

Equations (8.4) to (8.5) are the equivalent of Wannier one-band tight-binding analysis’ 27 for the specific double-well problem of the superlattice eigenstates in an electric field. Equations (8.4) to (8.5) describe the field-induced turningoff of the resonant tunnel coupling between wells and the concomitant localization of the eigenstates. The primary effect of the field on the double well is to misalign the levels of both wells lined up at F = 0 by eFd. If eFd exceeds the coupling terms between the wells (I,for the ground state, I, for the excited state), the tunnel effect between the wells becomes effectively nonresonant and thus weakens to the extent that if F becomes very large there is no more coupling between the wells: the eigenenergies respectively ’%. Bastard, C. Delalande, R. Ferreira, and H. W. Liu, J . Luminescence 44, 247 (1990). ”’G. H. Wannier, “Elements of Solid State Theory.” Cambridge University Press, Cambridge, 1959.

3 12

G. BASTARD et al.

F (kV/crn)

A GaAsGa,,,A1,,,As double quantum well versus the electric field strength. The dashed lines are the strong field asymptotes of the decoupled E , and E , levels in each isolated well. FIG.51. Calculated conduction eigenstates (solid lines) of a biased 100A-20 A-100

converge toward El f 1/2eFd and E , f 1/2eFd, while their associated wave functions converge toward c$jAL(z& 4 2 ) and c$itL(z f d/2), respectively. The second important physical feature exhibited by our simple model arises from the effect of the off-diagonal blocks on the eigenstates. When these blocks are absent, there exists a crossing between + and E, - taking place at F = F, (where eF,d E E , - El if eF,d/2 >> Al, A,). This crossing is actually replaced by an anticrossing when the off-diagonal blocks are taken into account. If the field was strong enough to have localized both the E , and E , related states (i.e., if 1/2eF,d >> ,Il, A,) in either well, the eigenstates again delocalize in a field range AF around F,, which strongly depends on the intermediate barrier thickness: the thicker the barrier the smaller are the anticrossing and the departure of the eigenstates from a linear dependence upon F. In the field range F, f 1/2AF a carrier assumed to be in the upper branch will make a very efficient relaxation toward the lower branch by emitting acoustical or optical phonons (energy conservation permitting) or by being elastically scattered in the lower branch and converting some of its longitudinal energy into transverse kinetic energy, this process being followed by a subsequent intralower branch deexcitation.126*128 Of course, if the wells are not identical, the aforementioned anticrossings may as well take place between the ground, now unequivalent, bound states of both wells. Finally, the off-diagonal blocks, mainly via the e F Z , , term, induced a deviation from lZsR. F. Kazarinov and R.A. Suris. Fiz. Tekl. Poluprovodn. 6,148 (1972) [Sou. Phys.-Semicond. 6, 120 (1972)l.

ELECTRONIC STATES IN SEMICONDUCTOR HETEROSTRUCTURES

3 13

a straight-line dependence at large F ( F >> F,) of the eigenstates that is quadratic upon F. This is nothing but the intrawell Stark effect. Figure 51 displays the results of the field-dependent energies for a 100A-20 A- 100 A GaAs-Ga,.,Al,,,As double quantum well (barrier height: 213 meV, rn* = 0.07mo).The electrostatic potential is taken to vanish at the center of the middle barrier, and the energy zero is taken at the bottom of the GaAs conduction band edge at F = 0. The states moving up (down) in energy are those that are mostly localized in the right-hand side (left-hand side) well. Figure 52 shows a plot of the spatial dependence of the four lower eigenstates of that structure for four different electric field strengths. At F = 0 the four levels are delocalized over the whole heterostructure. When F increases, the eigenstates become progressively localized in a given well, except near F , where an anticrossing take place between el + and E~ _.When this anticrossing is passed through, there is an interchange between the main spatial localizations of the second and third states, respectively.

9. ELECTRIC FIELDEFFECTSm SUPERLATTICES

An early motivation for the growth of semiconductor superlattices was the hope of achieving a negative differential resistance by realizing a Bloch oscillator.' The Bloch oscillator, suggested in 1927,'29 results from a semiclassical analysis of the electron motion in a potential, which is the sum of a periodic term V(z)and a linearly varying term eFz. F is considered as so small that its effects can be treated semiclassically on the band structure defined by V(z).The fully quantum mechanical version of the Bloch oscillator was derived by Wannier in the late 1 9 5 0 ~ . ' ~ ' Wannier - ' ~ ~ showed that a oneband analysis of the problem leads to the replacement of the quasicontinuous band spectrum of the crystalline solid (F = 0) by a ladder of bound levels evenly spaced by eFd, where d is the period of the lattice along the electric field (Wannier-Stark ladder). Wannier's results were long - 13' on the ground that the electric-field-induced interband mixing (Zener breakdown) would unavoidably wash out the ladders. Careful lZ9F.Bloch, Z. Phys. 52, 555 (1928). 13"G.H. Wannier, Rev. M o d . Phys. 34, 645 (1962). 131J. Zak, Phys. Rev. Lett. 20, 1477 (1968); Solid State Physics," Vol. 27, p. 1. Academic Press, New York, 1972. I3'J. B. Krieger and G. J. lafrate, Phys. Rev. B 33, 5494 (1986). 133D.Emin and C. F. Hart, Phys. Rev. B 36, 7353 (1987). '34Qian Niu, Phys. Rev. B 40, 3625 (1989). 135J. Callaway, Phys. Rev. 130, 549 (1963). IJ6M. Luban and J. H. Luscombe, Phys. $ev. B 34, 3674 (1986). 13'F. Bentosela, V. Grecchi, and F. Zironi, Phys. Rev. Lett. 50, 84 (1983); J . Phys. C 15, 7119 (1982). '38M. J. Saitoh, J . Phys. C 5, 914 (1972).

314

G. BASTARD et al.

c

X

I

-120

FIG. 52a. Position dependence of the envelope function of the first eigenstate in a loo& 20A-lWA GaAs-Ga, ,AI,,,Al,.,As biased double quantum wells at four different electric field strengths: F = 0, 60 kV/cm, 90 kV/cm, 120 kV/cm.

I

I

-60

W I , 0

z (A,

1

60

8

1

120

FIG.52b. Position dependence of the envelope function of the second eigenstate in a 100 A-20 A-loo A GaAs-Ga,,,AI,,,As biased double quantum wells at four different electric field strengths: F=O, 60 kV/cm, 90 kV/cm, 120 kV/cm.

12

-

N L

0

'4

oe 04 0

I

2 -04 \ VI" -I2

-120

-60

0

60

120

2 ( % I FIG.52c. Position dependence of the envelope function of the third eigenstate in a lOOA20 A-lOOA GaAs-Ga,,,AI,,,As biased double quantum wells at four different electric field strengths: F = 0, 60 kV/cm, 90 kV/cm, 120 kV/cm.

z (8, FIG.52d. Position dependence of the envelope function of the fourth eigenstate in a 100 '4-20 A-loo A GaAs-Ga,,,Al, ,As biased double quantum wells at four different electric field strengths: F=O, 60 kV/cm, 90 kV/cm, 120 kV/cm.

ELECTRONIC STATES IN SEMICONDUCTOR HETEROSTRUCTURES

3 15

theoretical and numerical analysis,'37 however, showed that Wannier's findings were essentially correct, the interband couplings leading to a broadening of the bound levels into virtually bound ones. The widths of the virtual bound states were estimated to be much smaller than the spacing eFd under many circumstances, thereby leading to the conclusion that the Wannier-Stark ladders could be observable in principle. In actual bulk samples, however, it was quickly realized that the time needed to complete a period of the semiclassical bound motion was considerably longer than any realistic collision time due to impurities, defects, phonons, and so on, thus leaving only a faint hope for observing this effect. In fact, on the experimental side, the search for Wannier-Stark ladders proved to be elusive, although there have been reports on their observation in the early 1970s in wide-band-gap semiconductors.' 39 Actually, the dominant effect of an electric field on the absorption edge of a bulk semiconductor is the appearance of the Franz-Keldysh effe~t'~','~'(tail below the F = 0 edge and oscillations above). These effects originate from the field-induced breakdown of the optical selection rules(Ak = 0) along the electric field. As in the Bloch oscillator model, the Franz-Keldysh formulae require that the potential energy drop eFd over a lattice period remains much smaller than the bandwidths (or band gaps), in order to treat the e F z term in a one-band effective mass approximation. Recently, V o i ~ i n 'suggested ~~ a reexamination of the Wannier-Stark problem in semiconductor superlattices from the point of view of the inhibition of the resonant tunnel effect between consecutive wells. He pointed out that this should result in an effective blue shift of the superlattice absorption edge. Bleuse et aL117showed this idea to be algebraically correct, and Mendez e t ~ 1 . " and ~ Voisin e t al.143experimentally demonstrated the existence of a blue shift in GaAs-Ga(A1)As superlattices. Since then, several experiments have confirmed various aspects of the Wannier-Stark q ~ a n t i z a t i o n ,including '~~ a room temperature achievement of an electrooptical switch utilizing the blue shift.'45 In the following we attempt a theoretical survey of the Wannier-Stark quantization in semiconductor superlattices. Efforts will be made to provide a 139R.W. Koss and L. M. Lambert, Phys. Rev. B 5, 1479 (1972). I4OW. Franz, Z . Naturforsch. Teil A 13, 484 (1958). I4'L. V. Keldysh, Zh. Eksp. Teor. Fiz. 34, 1138 (1958) [Sou. P h y s - J E T P 7, 788 (1958)l. 142P.Voisin, "A Superlattice Optical Modulator." French patent, 1986. 143P.Voisin, .I.Bleuse, C. Bouche, S. Gaillard, C. Alibert, and A. Regreny, Phys. Rev. Lett. 61, 1639 (1988). 144J. Bleuse, P. Voisin, M. Allovon, and M. Quillec, Appl. Phys. Lett. 53, 2632 (1988). I 4 5 I. Bar-Joseph, K. W. Goossen, J. N. Kuo, R. F. Kopf, D. A. B. Miller, and D. S. Chemla, Appl. Phys. Lett. 55, 340 (1989).

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comparison between the fully quantum mechanical treatment and the semiclassical one. We shall first discuss the simplest case, which corresponds to the one-band analysis made by Wannier and which is applicable to a good approximation to the conduction subbands of superlattices whose wells, if they were isolated, would only support a single bound state (Section 9a). Section 9a will also include short discussions of the intraband and interband optical properties associated with the Wannier-Stark states. The WannierStark localization for the valence subbands of superlattices is again complicated by the k,-induced couplings between the light and heavy hole dispersions and by the fact that, usually, many subbands come into play due to the heavy hole mass. As a result, no analytical solutions appear thus manageable in the general case. The outcomes of numerical diagonalizations of the valence hamiltonian in biased but finite super lattice^'^^ are, however, sufficiently clear to allow the drawing of reliable conclusions about infinite superlattices. Section 9b will thus be devoted to the multiband effects on the Wannier-Stark quantization both for electrons and holes. a. The Wannier-Stark Ladders We consider an infinite semiconductor superlattices subjected to a constant electric field. For simplicity we assume that the conduction states are built out of a single parabolic band that has the same symmetry (say r6) for both host materials. The hamiltonian is written as H =T

+1 V(z - nd) + eFz, n

(94

where V(z - ad) is the “atomic” square-well potential felt in the superlattice period centered at z = nd. Consider the translation operators r j d

= exp(ipzjd/h).

(9.2)

We find [sjd, H] = F j d H - Hyjd

= jeFdsjd.

(9.3)

Equation (9.3) states that the translation operators are no longer constants of motion if F # 0, a result physically arising from the potential energy drop eFjd between two superlattice periods separated by jd. However, the fact that [ r j d r H I is proportional to Fjd itself implies that this operator evolves harmonically with time. In particular, suppose that at time t = 0 the eigenstate of Eq. (9.1) was also an eigenstate of r i d with the eigenvalue exp(ik$d). Then Eq. (9.3) shows that the eigenstate of Eq. (9.1) will remain an eigenstate of q ‘ d with the eigenvalue exp[ik(t)jd], where k(t) = k, - eFdt/h. 146R. Ferreira and G. Bastard, Phys. Rev. B 38,8406 (1988).

(9.4)

3 17

ELECTRONIC STATES IN SEMICONDUCTOR HETEROSTRUCTURES

This represents a rigorous justification of the “Newton” law of semiclassical mechanics:

(9.5) Notice that this law is exact only because the potential is linearly varying with z. Any other variation would not allow the generalization of Eq. (9.5) to be exactly derivable. The semiclassical equations of motion are entirely determined when Eq. (9.5) is used in conjunction with the definition of the velocity for a semiclassical electron

In Eq. (9.6), E,,(k) is the dispersion relation of the nth subband (at zero field) and n is a constant of motion in the semiclassical analysis. Assuming a decoupling between the z and (x, y ) motion, we obtain

[ ,,(

v.F 1 a u,(t) = __ = - F h ak,

E

ko - __ ]):e

,

(9.7)

which proves that u,(t) and z(t) are periodic functions of time with a period TB equal to

TB= 2nh/eFd = 2 7 1 / 0 B ,

(9.8)

since ~ , ( q )is periodic in q with periodicity 2n/d. The oscillatory motion described by Eqs. (9.7) to (9.8) is the Bloch oscillator, and COB is the Bloch angular frequency. The Bloch oscillator is independent of the exact shape of the dispersion relations en(& To go one step further and obtain more explicit results, we restrict our attention to the ground subband ~ ~ (derived 4 ) from the ground bound state El of the isolated wells. We adopt the simple but accurate tight-binding expression for ~ ~ ( 4 ) : El(q)

(9.9)

= E l - 2(llcos(qd).

The subband width A1 is thus equal to 4(,41. From Eqs. (9.7) we obtain u,(t) = (2111d/h)sin[(ko- eFt/A)d], z(t) = zo

+ (21il/eF)cos[(ko - eFt/h)d],

(9.10) (9.11)

where zo is a constant of integration. The Bloch oscillator executes harmonic motion with period TBand amplitude 21l(/eFaround its equilibrium position z,. This oscillator is, however, peculiar in that its amplitude is unrelated to its total energy Efotsince E,,, is equal to El + eFz,, whereas for a regular

318

G. BASTARD et a1

harmonic oscillator (e.g., a mass m* attached to a spring) one knows that the amplitude a is related to El,, by El,, = 1/2m*a2c02,where w is the angular frequency of the oscillator. The difference between the two kinds of oscillators ultimately rests on the difference between the dependence of their kinetic energies upon the momentum. While the conventional oscillator can display an arbitrarily large kinetic energy, the Bloch oscillator is bounded by El + 2/11. The latter property is closely associated with the counterintuitive response of the electron to an external field in the crystalline solid. While our intuition associates a constant acceleration to the electron motion in a constant field, the electron in the solid actually oscillates. The band structure (or the Bragg reflection effect) is still dominant in the semiclassical equations. Notice finally that the amplitude of the Bloch oscillator blows up when F vanishes. This raises the question of the applicability of Eqs. (9.10) to (9.11) to real superlattice structures, which are finite. The finite size effects could be incorporated in Eq. (9.5) by adding, say, infinitely repulsive potential walls corresponding to both ends of the structure. We immediately see that, for such a finite structure comprising 2N + 1 periods, one should get (zo +_ 2(IZ(/eF(< ( N + 1/2)d to ensure that Eqs. (9.10) to (9.11) are still relevant. This criterion is better fulfilled for oscillators centered as far as possible from the edges (zo = 0) or for strong fields. Otherwise, one should numerically solve Eqs. (9.5) to (9.6) to get the “edge-interacting Bloch oscillator” motions. Actually their period is equal to

xdge

xdge

= (2/w,){arcsin[((N

+

+ 1/2)d - zo)eF/2121]

(9.12) +arcsin[((N 1/2)d + zo)eF/21~11). with the understanding that the arcsine functions have to be replaced by 4 2 when their arguments exceed unity. One notices that the Bloch oscillator centered right on the edge [zo = ( N + 1/2)d] and which does not reflect from the - ( N + 1/2)d interface has a period that is half of that of the bulk Bloch oscillator. A similar result is obtained for a regular harmonic oscillator bouncing from an infinitely repulsive wall placed at its equilibrium position. Let us now examine the quantum treatment of Eq. (9.1), restricting ourselves to a one-band tight-binding expansion. The general form of the spectrum can be obtained without any approximation. Suppose that $ v is a solution of Eq. (9.1) corresponding to the eigenvalue E ~ Consider . the function fd$,,(z) = tjV(z d ) and apply H to it. We find

+

HFd$v(Z)

= (&O

-

eFd)Fd$v(z)?

(9.13)

which shows that F d $ , , ( Z ) is an eigenstate of H corresponding to the energy E~ - eFd. This procedure can be iterated and so can the application of F:( = F - d ) , which translates the energies upward. Finally through the

ELECTRONIC STATES IN SEMICONDUCTOR HETEROSTRUCTURES

successive translations E,

F j d ,

= E~

3 19

j a relative integer, one generates the ladder

+ veFd,

v a relative integer.

(9.14)

Since we have restricted our analysis to a one-band tight-binding description of the problem, there are as many available states as periods in the superlattice (i.e., a countable infinity). Thus, the spectrum given by Eq. (9.14) exhausts all the possible states of the biased superlattice, and we may conclude that in the one-band analysis the Wannier-Stark ladders are evenly spaced. Notice that the one-band assumption is here crucial, insofar as the unknown E~ in Eq. (9.14) is necessarily unique. Had we included several bound states per well or, a fortiori, the continuum states, it would have been impossible to draw any conclusion on the even spacing of the level. In fact, when several bound states are included in the calculations, the spectrum is unevenly spaced. What remains, however, is the existence of a periodicity in eFd on the energy scale because Eq. (9.13) always holds, but the “elementary cell” of this periodic “lattice’ instead of containing a single eigenvalue contains as many of them as the number of states per period that have been retained. The one-band result is called a Wannier-Stark ladder, since Wannier first derived In its derivation, he was able to prove that E~ is actually the center of gravity of the band at zero field: (9.15)

which coincides with El in the simplest tight-binding scheme. The spacing between two consecutive eigenvalues is hw,, which one could have anticipated from the semiclassical result. The Wannier-Stark spectrum recalls in many respects that of the harmonic oscillator. There exist several differences, however. First, the Wannier-Stark spectrum has no ground state, which implies that the usual rules on the nodes of the eigenfunctions do not apply. In fact, each eigenstate of the Wannier-Stark problem has an infinite number of nodes and the $,’s are all but the same wave function whose argument is translated by an integer times d (since Fjdr,hv = $ v - j ) when going from one state to the other. The equivalent of the raising ( c + ) and lowering operator (c) of the harmonic oscillator are the translation operators F: and F d respectively. , However, .Fdand commute, but c and c + do not. This has some important implications when considering the perturbation of a Wannier-Stark ladder by a harmonically varying electric field-that is, when one addresses the question of light absorption or emission by an electron moving in a linearly biased superlattice. We shall return to this point later on. To understand more about the actual shape of the $,’s, we expand the solutions of H in the basis spanned by the &,(z - nd), which are the isolated

320

G . BASTARD et al.

quantum-well eigenstates (confinement energy El) centered at z

= nd:

(9.16) We assume the +‘s are orthonormalized (a convenient oversimplification), neglect couplings between all the neighbors but the nearest, and absorb the shift integral into a redefinition of El to finally end up with the secular equation cnv(E1- E ,

+ e F n d ) - 14(cn+lv +

= 0,

(9.17)

where the identification of the (4,,leFzl4,) matrix elements to eFnds,,, has been used. One recognizes in Eq. (9.17) the recursion relations of the Bessel functions. If the superlattice is infinite, the c,,’s should not blow up, which implies that the divergent Bessel functions (I,) should be eliminated. This leaves us with (9.18) Cnv = J v - n ( - 2/f), E, =

E,

+ veFd,

(9.19)

where f is the dimensionless strength of the electric field:

f = eFd/J/ZJ= 4 e F d / A , .

(9.20)

In finite superlattices one should take the en's as linear combinations of the Jn’s and In’s and determine the eigenenergies by writing appropriate boundary conditions at the superlattice edges (see S a i t ~ h for ’ ~ a~ more complete account of edge effects on Wannier-Stark ladders). Equations (9.18) to (9.20) call for several remarks. First, the one-band tight-binding model provides a universal description of the electric field effects on a superlattice: one only has to scale eFd and E , - E , to the bandwidth. This scaling allows us to understand readily why the observation of the Wannier-Stark quantization is far easier to observe in superlattices than in the bulk materials: a typical bandwidth in a bulk crystal is 2eV, corresponding to a lattice periodicity of 6 A, while the corresponding A 1 and d are typically 70 meV and 60 A in a superlattice. Assuming an electric field strength of lo5V/cm, we getf,,,, s0.012 andf,, s 3.43. In other words, the electric field has a small effect in a bulk crystal compared to that produced by the periodic potential, while in a superlattice the two effects are comparable. Second, the replacement of the extended minibands spectrum by a set of discrete localized levels, as evidenced, by Eqs. (9.19) calls for a physical interpretation. This is the field-induced turning off of the tunnel effect between consecutive wells, which is at the heart of the Wannier-Stark quantization and which again explains why the semiconductor superlattices are ideal candidates for its observation. Consider the biased superlattice (or bulk material) as a collection of localized potential wells whose bound states are misaligned by the electric field and which are nearest-neighbor coupled

ELECTRONIC STATES IN SEMICONDUCTOR HETEROSTRUCTURES

321

with an interaction strength - 121. By extension of the familiar H: molecular ion or the double-well (Section 8) problems, we know that the tunnel coupling between the wells 0 and n will remain operative if the misalignment between the two levels neFd remains smaller than 214 (see Fig. 53).147 This means that the number of sites visited by an eigenstate t,bv on each side of the vth site is 2lAl/eFd = 2/f: Thus, for a typical bulk material, a Wannier-Stark state extends over 2 x 170 periods, while for the same field strength t,bv hardly extends beyond the vth site in a superlattice. This is because the superlattice periods are larger, the minibands are narrower, the misalignment of consecutive energy levels are bigger than the corresponding quantities in bulk materials that the tunnel coupling between the wells of the superlattice can be so effectively turned off. An extra benefit is gained by the 10-fold increase between consecutive eigenstates of the ladder, making the ratio eFd/(h/z)(=w,z, where z is the scattering time) 10 times longer in a superlattice than in a bulk material if the z’s are identical. Notice finally that the argument concerning the number of sites visited by an electron in any of the Wannier-Stark states on each side of the vth site leads to a result ( 2 / f ) that exactly coincides with the amplitude of the Bloch oscillator. A more quantitative assessment of the spatial localization of the $,’s is obtained by computing P,,, the modulus squared of the projection of $, onto the localized wave function C # I , ~ ~ (Z nd). From Eq. (9.16) we see that P,, is equal to (c,,)~. Figure 54 shows a plot of P,, versus Iv - nl for different values off: The increasing localization around the vth period when f increases is evident. This localization is very rapid (faster than exponential) since J ; ( x ) z x ~ P / ( P ! ) ~ if x is small. Another striking property of the Wannier-Stark states (when compared with our intuitive expectation) is the even probability distribution of the $, around the vth period [since J , - , ( x ) = (- l ~ - v J v - , ( x ) ] In . the one-band approximation, there is as much probability for a Wannier-Stark state to be found in regions of lower electrostatic potential energy than in regions of higher energy, while one might have expected a piling up of the eigenstates where this energy is the lower. Such a piling up, for instance, occurs for the lowest state of a biased single quantum well, but the comparison between the two results is, we believe, not pertinent. A better one is made if one compares the spatial distribution of the vth Wannier-Stark state around the vth period with that of the second level of a biased single quantum well that binds at least three levels at zero field. In the latter structure, the E , level shift (and thus the spatial polarization) depends sensitively on the energy separation E3-E2 and E , - E , at zero field: the lower level pushes E , upward while E , counter14’A similar localization occurs due to barrier height or layer thickness fluctuations;see, e.g., R. Lang and K. Nishi, Appl. Phys. Lett. 45,98 (1984)for double quantum wells. For disorderedinduced localization in superlattices see Refs. 75-80.

322

G . BASTARD et al.

/

FIG. 53. Conduction band edge profile of a portion of a biased superlattice. Each eigenstate is coupled to its nearest neighbor (interaction strength A). The delocalization of a particular eigenstate centered at a given site (say 0) is over all the sites whose shaded bands overlap that centered at the 0th site. The eigenstate centered at the 0th site will not extend to the site n=4.

f .2 f = 0.4

fs

1

lO-1

L ' 3

lsite - v I site - V

I

FIG.54. Spatial localization of the vth Wannier-Stark state around the vth site for different reduced field strengths.

ELECTRONIC STATES IN SEMICONDUCTOR HETEROSTRUCTURES

323

balances this trend. In some occasion, there is nearly a cancellation of these two effects, which leads to a small shift of the E , state (ideally a vanishing one). The E , eigenfunction remains then nearly unpolarized, as at zero field, and thus displays an almost equidistribution over the regions of higher and lower electrostatic potential energy. The exactly even spatial distribution of the Wannier-Stark states around their center appears in the light of the previous discussion as resulting from the peculiar situation where equidistant levels (the diagonal terms El neFd in Eq. (9.1711 are symmetrically coupled to their nearest neighbors. To complete our survey of the one-band analysis of the Wannier-Stark ladders, let us summarize the comparison between the quantum and the semiclassical approaches. We have already mentioned that the semiclassical motion occurs with an angular frequency oB,which is exactly that expected from the quantum spacing ho,. Moreover, we found that a rough evaluation of the spatial extent of a quantum eigenstate is just equal to the amplitude of the Bloch oscillator. This agreement can be further quantified by calculating the root-mean-square deviation A, of the particle position around its equilibrium position,

+

A2 = [((z

-

ZO),>

- ( z - zo> 2 1l j 2 9

where (.-.> denotes quantum averages over t+hv (with zo period 27c/oB of the semiclassical motion. We find CAzlBloch

= 2d$ff,

(9.2 1) = vd)

or over the (9.22)

[&I" = (401z21$o> + 2dJZi.f

(9.23)

Both expressions are nearly identical in the limit of smallf (where one indeed expects the semiclassical analysis to be the more valid). For strong fields the field-dependent uncertainty decrease in both cases as both the semiclassical and quantum motions become increasingly restricted. This shrinking can reach a complete collapse in the case of the Bloch oscillator, since both the position and velocity of the Bloch oscillator can be defined with an arbitrary accuracy, whereas in the quantum case there always remains a residual uncertainty (401z21$o) associated with the finite extension of the isolated well eigenstates in the well and barrier layers. An even more demanding comparison between the two descriptions is obtained by plotting the integrated probability of finding the particle in the jth cell versus j when the oscillator is centered at z = vd. The classical probability is equal to Pj, = 2

J

(u,[ - dz jth cell

(9.24)

324

G. BASTARD et a1

if the Bloch oscillator visits the jth cell, and P j = 0 elsewhere. The quantum probability is equal to

(9.25) (9.26) If the “atomic” wave function 410c(z- Id) is well localized in the well centered at z = Id, one gets a good approximation

(9.27)

Pjv = ]CjVl* = J:-j(2/f).

The comparison between the classical and quantum P j v is shown in Fig. 55 for a small value of the reduced electric field. The agreement between both descriptions is excellent and recalls the well-known outcome of a similar comparison for a regular harmonic oscillator. We have previously mentioned that the Wannier-Stark ladder has no ground state and that F: and F dthe , equivalent of the harmonic oscillators raising and lowering operators, commute. These two properties lead to an interesting consequence with respect to the response of the Wannier-Stark I

I

I f

-

1

I

1

f =eFd r0.1

lo-’ -

-20

Ihl

I

-10

0

SITE - v

T.

10

:

I 1

FIG.55. Comparison between the semiclassical (solid line) and quantum (circles) probabilities of finding the electron in the vth Wannier-Stark state at a given site. f = O . l .

ELECTRONIC STATES IN SEMICONDUCTOR HETEROSTRUCTURES

325

state to a sinusoidally varying electric field (parallel to the static one). Let the electromagnetic (e.m.) perturbation be described by eF,,z cos(ot). If F,, is small, one can investigate by perturbation theory the transition rate that an electron in the vth state makes a transition to the pth one. This rate is proportional to ((vl~1p)(~. By using the tight-binding expansion [Eq. (9.16)], we get (vlzb)

=

c ~ l v ~ l ~ p ( 4 1 0 c~~)lzI41o,(z (z r4). I"

-

-

(9.28)

By writing that z = ( z - Id) + Id and by neglecting any integral of the form (&)z - Zdl&d), one finally gets (vlzb) = vd&l

+( m 4 & 1 .

(9.29)

The diagonal term is of no relevance here since it leads to no absorption or emission of energy by the electron. The off-diagonal term displays the expected shape, namely, that a given Wannier-Stark state is coupled by the e.m. wave to its nearest neighbor. The key point is that the coupling is vindependent, a feature clearly associated with the fact that all the WannierStark states are characterized by wave functions that are isomorphic. If the transitions rates (a) between v and v & 1 are the same, it is not very difficult to show that no net energy is absorbed by the electron due to interaction with the oscillating electric field.148 Let f , be the (arbitrary) distribution function of the vth state. The net power absorbed due to the v v k 1 transitions is hoa[f,(l -f,+l) +fv-l(l - f v ) - f v ( l - f v P l ) -f,+ 1(1 - f v ) ] . This is equal to h o a [ f , - - f v + J; that is, the net power does not depend onf,. Instead of considering only the v -+ v f 1 transitions, one may calculate the net power absorbed due to all the transitions between the states - N , -(N - l), . . . , N - 1, N and find that it is equal to h o ~ [ f -fN]. _ ~ Thus, there is no bulk absorption. The only possible absorption inside a Wannier-Stark ladder is therefore due to edge effects-that is, to the fact that any real superlattice is finite. Notice that the previous analysis has neglected spontaneous emission. Had we considered the net power absorbed by a regular harmonic oscillator (say a ladder of Landau levels), the results would have been entirely different. The transition rates n n + 1 (n - 1) would have been proportional to n + 1 (n), where n is the Landau level index, and the net power absorbed by the electron due to the n -+ n k 1 transitions would have been proportional to (n + l)(fn - f , + l ) - f n ) . The total power absorbed due to all the possible transitions would be proportional to C,(n + 1 - n)f, (i.e., to the concentration of electrons as it should for a bulk effect). -+

-+

+

148

G. H. Dohler, private communication, 1988.

326

G . BASTARD et al

The Wannier-Stark states are thus of little interest for intraband transitions. They display, however, rich electrooptical properties that are associated with interband transitions (i.e., to transitions between the valence and conduction ladders). It is not our purpose to give a detailed account of the optical properties of the Wannier-Stark states (for recent reviews see, e.g., Ref. 149.) Let us only sketch the salient features here. Consider the possible optical interband transitions in a finite but thick superlattice. The electromagnetic wave is assumed to propagate along the growth axis. The in-plane wave vector has to be conserved for allowed optical transitions in the dipole approximation. If there is no electric field, the superlattice wave vector q is conserved as well. If the superlattice is biased, the electrostatic potential breaks the q conservation rule. However, we have seen that the superlattice minibands are destroyed and replaced by the WannierStark ladders. For single, parabolic, and nondegenerate conduction and valence bands, the optical matrix elements involving the coupling between the light and the electron is M,,

=

( U ~ J E * ~ ~ U ~ ) ( I ) ( ~ ) J ~ (9.30) (”’),

where uc(uv)is the periodic part of the conduction (valence) zone center Bloch function, E is the light polarization vector, and I)@), $(’) are the envelope wave functions of the conduction and valence subbands, in our case the WannierStark states $!?, $f). Notice that although both the valence and conduction ladders are evenly spaced by the same eFd and are parallel, the eigenstates $f), I)!,“)are not the same, for their projections on the “atomic” wave functions &b(z - nd) and &L(z - nd), namely J v - , , ( - 2 / f c ) and Jv-,,(2/fv), are different due to the different conduction and valence subband widths. The overlap integrals ($(“)I$(’) are thus, in general, nonvanishing, and all the possible transitions between v and ,u become allowed. This result, again, contrasts with the case of magnetoabsorption (i.e., the interband transitions between Landau levels) where, in spite of different ladder spacings (ho,and ho,), the conduction and valence eigenstates are the same for a given n, which leads to the selection rule for the envelope functions An = 0. It is not very difficult to calculate the transition rates for 10, ,u) -,Ic, v) transitions, to integrate over k,, taking into account its conservation in the optical process, and to finally end up with an absorption coefficient that, for a type I superlattice, in the limit of a thick sample is given by117

where 2N

(i)

(9.31) + l)a, 1 Jf Y [ h o (cg + E , + HH, + peFd)], P + 1 is the (large) number of periods in the superlattice, Y ( x )is the

a(w) = ( 2 N

149P.Voisin, Surf: Sci. 1990, in press.

-

ELECTRONIC STATES IN SEMICONDUCTOR HETEROSTRUCTURES

327

step function, E, is the band gap of the well-acting material, E , and HH, are the confinement energies of the ground electron and hole states in the quantum wells when they are isolated, p is a relative integer (- N < p < + N ) , and fcv is the dimensionless electric field strength fc, =

eFd/(l&l

+ 4).

(9.32)

Finally, a,, is the absorption coefficient related to the HH, + E l optical transition in the isolated quantum well (ao 0.6% in 111-V quantum wells). We see that the absorption coefficient is just the sum of staircase absorption edges occurring at the energies E, + E , + HH, + peFd. These staircases have a simple physical explanation: they are associated with the oblique transitions in real space that promote an electron in the 0th-valence WannierStark state to the pth conduction one. These oblique transitions are symmetrically placed on the energy scale with respect to the p = 0 vertical transition. There is therefore no absorption edge for the band-to-band transitions (if NeFd > E~ + El + HH,). However, the oscillator strengths of the oblique transitions sharply drop with p to such an extent that they quickly become unmeasurably small. We illustrate these considerations in Fig. 56, where the absorption coefficient of a 41-period superlattice is plotted versus the reduced photon energy E [ E = {ho- ( E , El HHJ}/(IJ.J + A,)] for several field strengths f,,. At zero field the unperturbed superlattice

+ +

FIG.56. Calculated band-to-band absorption line shape of an infinite superlattice(f = 0 ) and of 41-period superlattice(f #O). For f = 4 the absorption edge practically reduces to two small steps ( f 1 oblique transitions) and a large one located at the edge of the isolated quantum well.

328

G. BASTARD et al.

absorption coefficient is drawn and displays the well-known arccosine shape. When fcv increases, steps corresponding to the oblique (p # 0) and vertical ( p = 0) transitions develop. It is seen that their amplitudes are not monotonic functions off,, (because J; has an infinite number of nodes). However, when f,, > f p , where 2/fp is the smaller zero of J i , the pth oblique transition fades away. For f,, > 4 (which corresponds to a potential energy drop over a period equal to the sum of the conduction and valence subband widths), one is left with a dominant p = 0 vertical step (nearly 90% of the total absorption coefficient) and two small p = k 1 steps, evenly sharing the remaining 10% of the absorption. There is therefore an effective blue shift of the band-to-band absorption edge. This blue shift can be quite large since 1/2(Ac+ A,) can easily reach 30meV. Notice that when the blue shift is significant, the well and barrier thicknesses are usually small ( E 30 A), which implies that the intrawell Stark shift (EsQw)discussed in Section 7 is small ( 1-2 meV). On the other hand, the excitonic effects (to be discussed in Section VII) become larger with increasing f,, because the structure continuously evolves from a quasi-threedimensional material (f,,= 0) to a quasi-bidimensional material (A, 3 4), which enhances the exciton binding energy. Thus, the measurable blue shift Ass is limited to

+

= 1/2(Ac

- (RGW

- RS*L) - ESQW,

(9.33)

where R;Tw - R& is the increase of the exciton binding energy when going from the coupled to the electric-field-isolated quantum-well situation. We show in Figs. 57-59 ail estimate of the periods for which a useful blue shift can be obtained in three different superlattice systems, assuming equal well and barrier thicknesses. The criteria used to define the rectangles in Figs. 57-

\\\b,

-E s,o, '

2-

' u

-50

\

.20

'

1oc

SL

1

zL, 1

0

~,10

b\

, , ,~ 80

d

(A)

2

\\L -ICU

\\\

-5

\\

's:

L G ~ A ~ - ~ G ~ , A Ld ) A F\\\,. T 40

-2 -

120

FIG.57. The critical field needed to achieve the condition f = 4 ( F c , left scale) and the maximum blue shift of the El conduction states (+Ac,, right scale) are plotted versus the superlattice period d Z in GaAs-Ga,,,Al,,,As superlattices with equal layer thicknesses. The rectangle defines the area where F', < lo5 V/cm and *Acl > 10meV.

I5'F. Agullo-Rueda, E. E. Mendez, and M. H. Hong, Phys. Rev. B 40, 1357 (1989). 151J. Barreau, K. Khirouni, Do Xuan Than, T. Amand, M. Brousseau, F. Laruelle, and B. Etienne, Solid State Commun. (1990),in press.

ELECTRONIC STATES IN SEMICONDUCTOR HETEROSTRUCTURES

329

150

FIG. 58. Same as Fig. 57 Ga,,,,AI,,,,As-InP superlattices.

but

for

FIG. 59. Same as Fig. 57 but for Ga,,,In,,,,As-Al, ,,In, szAs superlattices.

59 are that I/ZAc should be larger than 10 meV and that the electric field required to achieve an almost complete Wannier-Stark localization ( f = 4) should be smaller than lo5 V/cm. It is seen that for the three systems, the period d = 70 A, which is not exceedingly demanding from the growth point of view, fulfills both criteria. Figure 60 illustrates the experimental confirmation of the blue shift in a (Ga, 1n)As-(Ga, Al, 1n)As ~ u p e r l a t t i c e 'and ~~ the significant potentialities of this effect for electromodulation. Figure 61 is a "fan" diagram (transition energies versus electric field strength) showing the field dependencies of the vertical and oblique transitions in a GaAs-(Ga, A1)As superlattice. Oblique transitions from p = - 5 up to p = + 3 have been observed, which demonstrates both the relevance of the Wannier-Stark description of a biased superlattice and the fact that the coherence of the conduction states extends at least up to seven periods. (The hole states are quickly field-localized in a given period and, actually, act as markers' 5' of the conduction envelope functions). l'ZNotice that this approach neglects the interference effects between different ~ c a t t e r e r s and '~~ therefore cannot account for the modem scaling theory of one-dimensional conduction. The latter is discussed in Refs. 154-157 following Landauer's pioneering ~ o r k s . ' ~ ~ - ' ~ ' 153G.Bergmann, Phys. Rev. B 28, 2914 (1983); Phys. Rep. 107, 11 (1984). ls4P. W. Anderson, D. J. Thouless, E. Abrahams, and D. S. Fisher, Phys. Rev. B 22,3519 (1980). 155D.J. Thouless, Phys. Rev. Lett. 39, 1167 (1977). 156B. L. Al'tshuler, A. G. Aronow, and B. Z. Spivak, Pis'ma Zh. Eksp. Teor. Fiz. 33, 101 (1981) [ J E T P Lett. 33, 94 (1981)l. '"R. Landauer, IBM J. Rex Dev. 1, 223 (1957). ISsR. Landauer, Philos. Mag. 21, 863 (1970). 159SeeRef. 14, pp. 536-542 and references cited therein.

330

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ELECTRO-MODULATION

Ga( I d A s - (Ga,Al.ln)As p _ i-n I 920

I

structure

1

I

960

I

1000

t i (~m e V ) FIG.60. Measured absorption line shape versus photon energy at two voltages in a 39 A-46 A Ga,.,,In,.,,As-(Ga, Al, 1n)As superlattice (upper scale). Absorption difference versus photon energy (lower scale). Adapted from Ref. 144.

1.72 I

-

1.68

-

%

v

&

E

-

1.64 -

I

I

I

2

'0

I

I

I

I

2

.

I

I

2

-

0

ct,.

1

W

z W

-1 1.60 -2

-3 1.56

I

I

I

I

I

I

I

I

I

-

FIG. 61. Electric field dependence of interband transitions between Wannier Stark states in a 60-A-period GaAs-Ga,~,,A1o,,,As superlattice. After Ref. 150.

ELECTRONIC STATES IN SEMICONDUCTOR HETEROSTRUCTURES

331

b. Multiband Effects on the Wannier-Stark Quantization The previous section has been devoted to a one-band tight-binding analysis of the Wannier-Stark quantization. In reality, any bulk or superlattice band structure displays an infinite number of bands. Let n be the subband index of a superlattice miniband centered at E n . The zeroth order of approximation consists of constructing a Wannier-Stark ladder attached to each subband: E,,

=

+ veFd,

v a relative integer.

(9.34)

For discrete values of the field, there will exist crossings between E,, and [edF,, = ( E , ~- em0)/(p- v)]. In reality these crossings are replaced by anticrossings since there is no reason why the matrix elements (4j:t(z - id)leFzlcjjr?(z - j d ) ) should all be zero if m # n. When one considers the ladders whose energy corresponds to the continuum, the index n or m becomes continuous and, thus, there exists a broadening of the Inv) Wannier-Stark state due to its interaction with the Imp) continuum. These interactions have long cast doubts’ 3 1 - 138 on the very existence of the Wannier-Stark ladders, for it may have happened that the escape could have been faster than the Bloch period TB,invalidating Wannier’s approach. It took some time to establish formally that these effects were, in practice, small, a feature that recalls the findings in the problem of the intrawell Stark effect (Section 7, where one also deals with virtual bound states but where many physical quantities are accurately calculated by models that neglect interaction with the continuum). If one thus forgets about finite lifetime effects, there are still possible interactions between the ladders attached to different subbands generated by the hybridization of different quantum-well bound states. In the vicinity of the FpV’sdefined above, the eigenstates that were field-localized in the oneband approximation delocalize again. This delocalization, similar to the one described in Section 8 for double wells, is very important for the carrier relaxation and transport along the growth axis. Notice, however, that its spatial extent is limited: if only nearest-neighbor couplings are significant, the larger achievable delocalization, out of completely localized states at the zeroth order of approximation, takes place over two superlattice periods at the maximum of the anticrossing between the two interacting ladders. The reasoning we presented in Section 9a about the linear variation of the Wannier-Stark states with F is no longer relevant when several bound states per period are admixed by the field. Instead of dealing with a single c0 appearing as an additive constant to all the energy levels of a ladder that can be absorbed in a redefinition of the zero of energy or be evaluated by some direct calculations (e.g., e0 = center of gravity of the miniband), there are as many E , ~ as bound states per period. They cannot all be eliminated. Moreover, the previous reasoning on F d[Eq. (9.13)] is unable to tell us a E,,,~

332

G. BASTARD et

a!.

10 M

i

100 A - 20 11 wdlr

GaAr-Ga (ALIAS

I

I

30 40 50 60 70 80 90 100 F ( kV/cm 1 FIG.62. Calculated conduction eigenstates in an 11-well 100 A-20A GaAs-Ga,.,AI,,,As superlattice versus electric field. The numbers - 5, -4,. . . label the 1 1 El-originating quantum states while the numbers -5’, - 4 , . . . label the 11 &originating ones. The numbers also correspond to the wells where the eigenstates are mainly localized in strong fields.

0

priori if the E,, are field-dependent, which precludes the assertion that the eigenstates vary linearly with F in the multiband situation. In fact they do not, as shown by the numerical diagonalizations of the superlattice hamiltonian. What remains true, however, is that the sequence E:), where n = 1,2, . . . , M labels the various bound states per period, repeats itself periodically on the energy scale and the wave functions $$‘)(z) generate all the eigenfunctions of the problem by successive applications of F dor 5:. Figure 62 shows the calculated eigenstates of an 11-period (100,4-20,4) GaAs-Ga,,,Al,.,As biased superlattice versus the electric field strength. The well thickness is such that two states are bound at zero field. The eFz term has been diagonalized within the basis spanned by the 2 x 11 bound states at zero field. At zero electric field one sees clearly two minibands centered around 30meV and 117meV, respectively. Since the superlattice is finite, each of the subband continua actually gives rise to 11 levels. At nonvanishing F the actual spectrum is derived from the two interpenetrating Wannier-Stark ladders E , vd, E , p d ( - 5 d v, p d 5) attached to the two zero-field subbands. There exist two kinds of departures from the zeroth-order spectra. The first kind is an adge effect. Since the superlattice is finite and there exist 2 x 11 nondegenerate bound levels at zero field, the small-field behavior (here F ,< 1 kV/cm) should be a quadratic regime analogous to the intrawell Stark effect described in Section 7. Because the “equivalent quantum well” is very thick, the nondegenerate perturbation approach that describes the quadratic

+

+

ELECTRONIC STATES IN SEMICONDUCTOR HETEROSTRUCTURES

333

Stark shift quickly becomes invalid and the levels start varying linearly with F . The departure from linear behavior is less for the central level and greater for the 5 levels, which are more sensitive to the edges (see Section 9a for the related “edge-interacting’’ Bloch oscillators). For field strengths near FBvr there is a second kind of departure from the zeroth order of approximation. This is associated with the field-induced resonant tunneling effects taking place between the different levels of the two ladders that, in the limit of a strong Wannier-Stark localization, would coincide with the isolated well eigenstates. These effects are more pronounced when p - v is smaller (i:l), which happens when FpP+ is equal to 80 kV/cm. For F > 80 kV/cm, edge and anticrossing effects become negligible, and the regime of linearly varying levels should, apart from the intrawell Stark effect, be recovered. When dealing with the Wannier-Stark ladders in the valence band of superlattices, one faces a more difficult numerical problem than that met for the conduction ladders. In the latter case we were able, by implicitly neglecting nonparabolicity effects, to decouple the z and x, y motions and, thus, to deal only with a one-dimensional Schrodinger equation. We have seen in Section IV that such a decoupling is impossible for the valence subbands unless the in-plane wave vector k, vanishes. One must therefore resort to numerical diagonalization of the valence hamiltonian in finite structures to find the in-plane dispersion relations of the valence WannierStark 1adde1-s.’~~ The fact that there exist two categories of hole levels at k, = 0 (heavy and light holes), which in the diagonal approximation of the Luttinger hamiltonian display different in-plane effective masses (massreversal effect), implies that, in addition to the regular multiband anticrossing effects taking place at k, = 0 (similar to those discussed above for electrons), one has also to expect that there will exist anticrossings occurring at k, # 0 that are a consequence of the off-diagonal terms of the Luttinger matrix. Despite this very complex situation, the notion of the Wannier-Stark ladder is still relevant for the valence subbands, at least within the generalized meaning specified above for electrons. In fact, let us consider the valence hamiltonian of a biased superlattice and assume again that one can restrict our considerations to the J = 312 topmost quadruplet. The hamiltonian is now a 4 x 4 matrix, and its eigenstates $(r) are 4 x 1 spinors:

CT + 1 ( W + eFz)l$(r) = M r ) ,

(9.35)

where 1 is the 4 x 4 identity matrix and T is the Luttinger kinetic energy term [Eq. (1.24)]. Since the potential energy depends only on z, $(r) factors into (9.36) $(r) = $k,(z) exP(ik,. r , ) / f i . From now on we drop the r, = (x, y ) dependence and focus on t,bk1(z),which is implicitly k,-dependent since it is the solution of a k,-dependent hamiltonian.

334

G. BASTARD et al.

Suppose that J / ~ , ( z )is an eigenstate of Eq. (9.35) corresponding to an energy c0. Thus, T & k 1 ( z ) is also an eigenstate corresponding to e0 - eFd. There is therefore a Wannier-Stark ladder, but, as in the case of conduction ladders with multisubband effects, this does not imply a linear variation of the eigenenergies with F. Besides, the e0 has not only to be indexed by n, where n = 1, 2, . . . , 2N and N is the number of interacting hole subbands, but also by k, since the hamiltonian changes with k, (and not only by an additive constant). Thus, one can conclude that if one only takes into account the bound hole states, the valence spectrum of a biased infinite superlattice consists of an evenly spaced (by eFd)sequence of groups of states, where each group of states contains twice the number of bound hole states at k, = 0. The extra factor of 2 comes from the strong spin-orbit coupling. Only at k, = 0 does one recover a twofold degeneracy of each state. As mentioned previously for electrons, the coupling of a ladder attached to a bound state with the continuum states leads to a broadening of the Wannier-Stark states. We shall again assume that this broadening is negligible. At k , = 0 the situation is somewhat simpler. The decoupling between the heavy and light hole states (m, = &3/2, m, = 1/2, respectively) is exact. Also the 3/2 (+ 1/2) and - 3/2 (- 1/2) components are uncoupled. Thus, the Wannier-Stark ladders split into two independent categories, and each category displays a twofold degeneracy. The situation is much the same as discussed for the conduction states. Notice in particular that there exist anticrossings between the heavy hole Wannier-Stark states attached to different subbands and between the light hole states. On the other hand, exact crossings take place between a heavy hole and a light hole state. This illustrated in Fig. 63 where the field dependence of the valence energy levels is shown for a 50 A-40 A-50 A GaAs-Gao,7A10.,As symmetrical double well. The primed and unprimed levels, respectively, correspond to states that are mostly localized on the right-hand-side and left-hand-side wells. One notices that the dashed lines (light hole levels) cross the solid lines (heavy hole levels), while two states of the same category anticross (HH, and HH; do anticross at F = 0, but the anticrossing gap is very small). When k , # 0, the heavy and light hole levels interact, and if F # 0 the twofold degeneracy for a given k , is lifted. We therefore expect that all the crossings will be replaced by anticrossings. This is illustrated in the lower panel of Fig. 63 for k , = 2.5 x lo6 cm-'. Clearly the light hole branches now anticross the heavy hole branches and, in fact, the notion of light and heavy hole becomes irrelevant when k , # 0. As we shall see in Section VI the band-mixing effects are very important for understand,ig the assisted hole transfer in biased heterostructures both qualitatively and quantitatively. In superlattices, one expects the numerical outcomes to agree with the general property concerning the existence of an evenly spaced sequence of

+

ELECTRONIC STATES IN SEMICONDUCTOR HETEROSTRUCTURES

335

F ( kV/crn FIG.63. Calculated field dependence of the valence eigenstates of a 50 A-40 A-50 A GaAsGao,,Alo,,As biased double quantum well. k, =O: upper panel. k , =2.5 x lo6at-':lower panel.

groups of states. Since the numerical diagonalization can only deal with finite superlattices, edge effects may perturb the even spacing. These features are illustrated in Figs. 64-66, where we show for F = 40kV/cm the in-plane dispersion relations of a 50-A-thick GaAs-Gao,,A1,,,As single quantum well (Fig. 64) of a three-period (Fig. 65) and a five-period (Fig. 66) 50A-50A GaAs-Gao~,Al0,,As superlattice. At zero field each isolated well supports two heavy hole states (HH,, HH,) and one light hole state (LH,). At k, # 0 the levels are labeled according to their k , = 0 nature and to the well where they are principally localized. One clearly sees the existence of an evenly spaced sequence (HHY),LH';+'), HH';+')) when n, n + I, n + 2 are such that they do not correspond to a terminating well of the finite superlattice. Notice that the members of the group of states generating the sequence change with F . In particular, in the limit of a large field, the group should only contain levels that all are mainly localized in the same well. In this limit the spectrum of the finite superlattice is just a repetition on the energy scale of that of a biased single well. An example of the band-mixing effects is shown in Figs. 67a, b where the

336

G . BASTARD et at. 1

h

I

-

0-

3

\

E

> c3

5 z w

-100-

G a d s - Ga (ALIAS X = 0.3e Lz50A F-40 k V / c m -200

0

1

a01

I 0.02

h

2

5 >-

a

55

z

-100

F I 40 kV/cm -200,

3 wells

I

I

003

ELECTRONIC STATES IN SEMICONDUCTOR HETEROSTRUCTURES

->,

337

0

1 I

E

u

> c3

55

z w

I

-100

-200 0

0.01

a02

FIG.66. Calculated in-plane dispersion relations of five-well 50 8,-50 8, GaAs-Gao,,Alo.,As biased multiple quantum well. F = 40 kV/cm.

is shown for a single 50-&thick GaAsk, dependence of Gao,,A1,,,As quantum well (Fig. 67a) and for the group of states generating a Wannier-Stark ladder in a 50 A-50 A GaAs-Ga0,,A1,,,As superlattice (Fig. 67b). If there were no band-mixing effects, would either be equal to 3/2 (heavy hole state) or 1/2 (light hole state). The departures from these two values arise from the off-diagonal and k,-dependent terms of the Luttinger matrix. The comparison between Fig. 67a and Fig. 67b allows one to specify whether the band mixing is an intrawell effect or arises from the interaction between the heavy and light hole states mainly localized in different wells.

VI. Broadening of Heterostructure Eigenstates 10. INTRODUCTION

Previous sections have been devoted to discussions of the energy levels in ideal quantum wells and superlattices. Actual heterostructures contain impurities and defects, and the carriers interact with the phonons. The carrier-phonon or carrier-impurity hamiltonians break the translational

338

G. BASTARD et al.

0

0.01

k,

0.02

0.03

6-l)

FIG.67a. The quantity is plotted versus k , for the three bound states of a biased GaAs-Ga0,,A1,,,As single quantum well. L= 50A,F=40 kV/cm.

1.4 -

1.2 -

1-

0

0.01

GaAs -Ga(AI)As =! 0.3 5 O A - 50A F = 40 k V h m 3 wells I I 0.02 003

k1(8-’) FIG.67b. The quantity is plotted versus k , for the three central states out of the nine eigenstates of a three-well 50A-50A GaAs-Ga0,,A1,,,As biased superlattice. F = 40 kV/cm.

ELECTRONIC STATES IN SEMICONDUCTOR HETEROSTRUCTURES

339

invariance in the layer plane and along the growth axis in the case of superlattices. Thus, the eigenstates we calculated previously, which are characterized by plane or Bloch waves in the layer plane and for superlattices along the growth axis, are not the eigenstates of the actual hamiltonians. It happens, however, that under many circumstances the differences between the actual and model hamiltonians are small. Thus, these differences can be estimated by perturbation calculations. Since the plane or Bloch waves are associated with continuous-energy spectra, the lowest-order carrierimpurity or carrier-phonon interactions amount to a broadening of the energy levels, which is equivalently expressible in terms of a level lifetime. Let $&be an unperturbed eigenstate associated with the eigenvalue &,k (here k stands for k,, q, or both). To the lowest order its lifetime znk due to the presence of static scatterers is given by the Fermi golden rule

where V(r - Rj) is the potential energy due to thejth scatterer located at the site Rj. In Eq. (10.1)we have neglected any population effect on the initial and final states and assumed that In'k') was empty and that Ink) was populated. This is an approximation that is well suited to undoped heterolayers when one wishes to describe the relaxation of photoexcited carriers. In doped structures, the right-hand side of Eq. (10.1) should be multiplied by fnk(1 - f n j k ) , wherefnk is the occupation function of the Ink) state. Since the scatterers are very numerous (but diluted enough with respect to the number of available sites) and randomly distributed over the whole heterostructure, the / . . . I 2 term in Eq. (10.1) appears like the modulus square of a sum over a large number of random variables. The sample can be decomposed into a number (again large) of subsamples where the number of impurities is fixed but the impurity configuration changes, that is, where a given impurity occupies different sites in different subsamples. Under circumstances when the occupancy of given site by an impurity has no influence on the occupancy of another site by another impurity (i.e., in the limit of infinite dilution), one can show that

I

(nkl

Ri

V(r - Ri)ln'k')

z Nimp(l(nklV(r - Ri)ln'k)lz)av,

(10.2)

where (...),,, denotes a spatial averaging over all the available sites Ri of a scatterer, for example, over the entire barrier volume in the case of a quantum well that has been selectively doped in the barrier or over the entire planes z = zi for delta doping, and so on. Equations (10.1)-( 10.2) will be repeatedly used in the following to evaluate the effect of static scatterers. Notice in particular the linear scaling of l/z with Nimp.

152

340

G . BASTARD et al

In the case of the electron-phonon interaction, the scattering takes place by phonon emission or absorption. Each scattering event can be decomposed into the sum of elemental processes where the phonon wave vector Q, polarization u, and energy R C O Q ~are specified. Equation (10.1) is modified to give

R

-=

znk

2n

1 1 (1 +

n k Qa

nQa)l(% klffe-p,(Q,

U)ln’,

k)126(Enk- En’k‘ - hwQa), (10.3)

where nQa is the occupancy of the (Q,cr) phonon mode, and only emission processes have been considered. For absorption processes 1 nQa has to be replaced by nQ. in Eq. (IV-3) and -ha,, by + haQ, in the argument of the delta function. In the following we shall assume that the phonons are at thermal equilibrium and, therefore,

+

n ~ =, [eXp(flhCOQ,) - I]-’.

(1 0.4)

At low temperatures ( T < 77 K) and for longitudinal optical phonons ( ~ c o Q ,= 36meV in GaAs) nQa is very small ((4 x Thus, the lifetimes associated with absorption processes are considerably longer than those limited by emission processes. In the following absorption will be neglected. In Eqs. (lO.l)-(l0.3) the scattering events either leave the carrier in the same subband (n’= n, intrasubband scattering) or scatter it into a different subband (n’# It, intersubband scattering). In the case of double or multiple quantum wells, the n and n‘ (n # n’) states can be mainly localized in the same wells or in different wells. In the latter case the broadening mechanisms is associated with a spatial transfer from one well to the other. The transfer is assisted by elastic or inelastic scatterers. There are situations where the Born approximation is insufficient. For instance, there is no way one can describe the formation of a bound electron state due to a coulombic impurity in the Born approximation, for a bound state is energetically located below the continuum edge. By definition there is no $nk and &,k corresponding to this energy. Similarly, when the disorder over the well widths of a superlattice becomes too large, all the eigenstates become l~calized,~~ while - * ~ the unperturbed states are all extended. A second kind of failure of Born type of calculations is found when the density of final states is large (or infinite). This happens for the Landau levels in quantum-well structures where the unperturbed density of states is infinite at the Landau level energies or in quantum wires at the subbands’ edges. Under such conditions, one finds that h/rnk diverges. A divergent broadening is, of course, unphysical and arises from our use of the lower Born approximation, as evidenced by the conservation of the unperturbed energies in the argument of the delta functions. To avoid these spurious divergences, one should go beyond the first Born approximation and self-consistently

ELECTRONIC STATES IN SEMICONDUCTOR HETEROSTRUCTURES

341

account for the fact that the states connected by the scattering hamiltonian are already “dressed” by these scatterings. To the lowest order this is the selfconsistent Born appro~imation.’’~In its phenomenological version, it amounts to replacing the delta functions in Eqs. (10.1)-(10.3)by lorentzians, r/z[(&& - &n,k)2 r2]p,where r is a phenomenological damping parameter. In the exact self-consistent Born approximation, r is identified with the broadening itself and thus depends on n and k. Equations (10.1)-(10.3) thus become integral equations that can only be solved numerically, except in the case of short-range scatterers where analytical solutions exist. Notice that in the self-consistent Born approximation, or in any calculations going beyond the first Born approximation, the proportionality between l/z and Nimpis eliminated as is the independence of l / ~ on ” the ~ sign of the scattering potential. The section is organized as follows: In section 11 we shall specify the carrier-impurity and carrier-phonon perturbation hamiltonians. In sections 12 and 13 we present the results of z,k calculations in the Born approximation for electrons and holes in biased single-, double-, and multiple-quantum-well structures. Our goal is twofold. First, we wish to provide Tnk figures in representative cases. Second, we wish to illustrate the significant effects that an electric field has on the intersubband relaxation or interwell transfers. Bound impurity states will not be discussed here. For reviews on singleimpurity effects see Ref. 160. Multi-impurity effects have been recently discussed by Gold et a1.161

+

1 1. CARRIER-IMPURITY AND CARRIERPHONON PERTURBATION HAMILTONIANS

We specify in this section the perturbation hamiltonians we have used to evaluate the broadening effects due to the carrier interaction with static scatterers and phonons. The potential energy of a carrier with charge e interacting with an unscreened coulombic impurity with charge e‘ located at Ri[Ri = (pi, zi)] is ee’ V(r - RJ = - J(p - pi12 K

+ (z - zilZ

where IC is the relative dielectric constant of the heterostructure (image force effects are neglected), S is the sample area, and q is a two-dimensional 160Y.

l6IA.

C. Chang, J . Phys. (Paris) 48, C5-373 (1987). Gold, J. Serre, and A. Ghazali, Phys. Rev. B 37, 4589 (1988).

342

G. BASTARD et al.

wave vector: q = (qx,qy). The matrix elements of V(r - Ri) between two eigenstates that are separable in p and z is thus

( n , kJV(r- Ri)ln’k + q)

= (27lee’/~S(ql)expCiq.PiI

x

(Xnl

expC-Iql

IZ

- zilll~n,). (11.2)

As expected from the long range of the unscreened coulombic potential, its matrix elements blow up when the momentum supplied by the impurity during the scattering vanishes. If there are free carriers in the heterostructures, they screen the impurity p ~ t e n t i a l . ~ ~ ~In. the ’ ~ random ~ . ‘ ~ ~ phase approximation, at T = OK, the screening amounts to replacing 4 ( q = Ik - kl) in the denominator of Eq. (11.1) by q&(q),where &(q)is the dielectric function of the electron g a ~ . ’ ~ When ~ . ’ ~ a~ single subband is occupied by electrons with parabolic in-plane dispersion relations, &(q)is equal to &(q)=

-k (qTF/q)f,(q)S~(q/~~F),

(11.3)

where qTF is the Thomas-Fermi wave vector (qTF = 2/a,*,where a; is the three-dimensional Bohr radius), kF is the Fermi wave vector (kF = 2nn,, where n, is the areal concentration of electrons), f, is a screening form factor that depends on the geometry of the heterostructure, and gS(q/2kF) is a function that takes into account the finite size of three Fermi surfaces:

The screening suppresses the q = 0 divergence of the coulombic matrix elements, thereby enhancing the electronic lifetime near the subband edge. In the case of coulombic scattering between the superlattice states Inqk,), (n‘q’k;), the impurity location along the growth axis zi can be written as Id + ti, where -d/2 < ti < d/2. Thus, one obtains

(11.6) 16*T.Ando, J . Phys. SOC.Japan 51, 3900 (1982). 163A.Gold, Phys. Reo. B 38, 10798 (1988). 164S. Mon and T. Ando, 1.Phys. SOC. Japan 48,865 (1980); G. Fishman and D. Calecki, Phys. Rev. B 29. 5778 (1984); F. Stern in “Physics and Applications of Quantum Wells and Superlattices” (E. E. Mendez and K. von Klitzing, eds.), p. 133. Plenum Press, New York, 1987. 165F.Stern, Phys. Rev. Lett. 18, 546 (1967).

ELECTRONIC STATES IN SEMICONDUCTOR HETEROSTRUCTURES

343

If one uses a tight-binding expansion for xnq and if the minibands are narrow, integral is obtained by identifying xnq a rough estimate of the (xnql ...)., XI and x,,.~.with the isolated quantum-well wave function (P{:, (PI::)centered at the 0th site (apart from a normalization factor). It then follows that the only part played by the minibands spectrum compared with the single quantumwell situation takes place in the dispersion relations that appear in the delta functions ensuring energy conservation in Eq. (10.1).Notice that ~ ~ does ~ k not depend on which superlattice period hosts the impurities (i.e., on l), but only on t i ,the impurity location of these impurities inside a unit cell. This is a direct consequence of the translational invariance of the superlattice eigenstates. The interface defect scattering'62.'66 is specific to heterolayers. It arises from the fact that the ,growth of a layer is never perfectly bidimensional, which, around the location of a nominal interface, results in the existence of local protrusions of the well in the barrier and of the barrier in the well. The shape and dimension of these defects are likely to be dependent upon the growth process, which may explain why relatively little is known on the interface defect scattering and why rough models are still used (and useful) to describe the interface roughness. In the following, we shall adopt the model of gaussian pillbox defects.'68 If z = 0 denotes the nominal boundary between a well-acting and a barrier-acting material, an attractive interface defect centered at pi (well protrusion in the barrier) is specified by its depth b and lateral extent a, which are such that V,,,(p - p i , z ) =

-

V,Y(- z ) Y ( z

+ b)exp[ -(p

- ~ ~ ) ~ / 2 a ' ] (1 . 1.7)

Similarly, for a repulsive defect we would write - pi, Z ) =

+ K Y ( z ) Y ( b - z)exp[-(p

-

pJ2/2a21,

(11.8)

where Vb is the barrier height and Y ( z )is a step function. The matrix elements between two conduction eigenstates are given by

x[ -6u.P

x.*(z)xn,(z)dz+ au,+

lo b

%:o~.,(i)dz], (11.9)

where a stands for attractive (a = -) or repulsive (a = +) defects. Notice that the effective height of the defect (for the in-plane motion) is considerably smaller than V,since the integrals in Eq. (1 1.9)are usually of the order of a few 166R.Gottinger, A. Gold, G. Abstreiter, G. Weimann, and W. Schlapp, Europhys. Lett. 6, 183 (1988).

,

344

G. BASTARD et al.

percent. Thus the interface defects are in fact weak (few meV’s in depth) and well-behaved (no q singularity) scatterers. Notice also that their efficiency increases with the subband index since the x,,’s amplitude at the interfaces increases with n. The depth b is often assumed to be equal to a few angstroms, while the inplane size a varies between a few tens of angstroms to several thousands of arigstroms. Owing to the quick drop of Eq. (1 1.9) with 4, it is clear that very large defects are inefficient scatterers. More precisely, large defects amount to an almost rigid shift of the whole subband dispersion (in the limit where a + co the attractive interface defect is just an unperturbed ideal quantum well with thickness L b instead of L). For large defects it is better to envision the quantum well as a collection of micro quantum wells each having a definite thickness. The physical properties of the quantum well coincide with the sum of those associated with each micro quantum well (the so-called inhomogeneous broadening). The only difficulty is the quantitative assessment of “large.” The in-plane mean free path can be a useful characteristic length. In any event, we shall assume that a 5 200w to be sure that the notion of interface defect is pertinent. For intraband transitions [n’ = n in Eq. (1 1.9)] occurring in deep wells, it is easy to show that (nk,IV&,,,Inkh) behaves like L - 3 , where L is the quantum-well thickness. Thus znkscales like L6 at large L. This shows that the interface roughness scattering can effectively limit the level lifetime only in thin well^'^','^^ (typically L S 30w in GaAs-AIAs quantum wells). When the well and/or the barriers are ternary random alloys, the carriers experience short-range scatterings on the alloy fluctuations. The alloy scattering, analyzed a long time ago by Nordheim, is describable (see, e.g., Ref. 18 for a derivation in heterostructures) in terms of delta like scatterers acting on the carrier envelope functions. This leads to

+

s,

R

(11.10) 4 1 -x)cm2 x,”(z)x,”,( z ) & S where R, is the volume of the host unit cell, x is the molar fraction in the chemical formula A, -,B,C of the alloy, and 9 is the length of the ternary material where the alloy scattering takes place. Finally, [SV] is the average in a, of the A and B potentials weighed by the periodic part of the Bloch function at the conduction band edge of the virtual crystal: (I(~k,l~ll,,l~’k;)12),”. =

.

P

(1 1.1 1) 16’H.

Sakaki, T. Noda, K. Hirakawa, M. Tanaka, and T. Matsusue, Appl. Phys. Lett. 51, 1934

(1987).

‘68G.Bastard, C. Delalande, Y. Guldner, and M. Voos, Rev. Phys. Appl. 24, 79 (1989).

ELECTRONIC STATES IN SEMICONDUCTOR HETEROSTRUCTURES

345

Due to the short-range nature of the alloy fluctuations, the matrix elements of the alloy scattering only depend on k; - k, via the phase factor exp i(k; - k,).p,. Thus, the average (1...12)ay. is k,- and k,-independent, which, in particular, implies that the level lifetime Z&, coincides with the velocity relaxation time for intrasubband scatterings. The latter governs the carrier mobility and is given by an expression similar to Eq. (lO.l), except that 1...1*S[...] is multiplied by 1 - cos 8, where 0 is the angle between k, and k;. Equation (1 1.10) also shows that the level lifetime due to alloy scattering taking place in the barrier is much longer than when it takes place in the well, a feature already discussed for the velocity relaxation time. The electron-phonon interaction is the dominant broadening and relaxation process in heterostructures at room t e m p e r a t ~ r e . ' ~ ~ The reason is that most of the host materials are polar, allowing for the existence of the powerful Frohlich interaction between the carrier and the electric field created by the displacement of the two different atoms in the unit cell (i.e., between the electron and the longitudinal optical (LO) phonon mode). In doped heterostructures, the scattering involves the absorption of an LO phonon, which requires an elevated (T 2 100K) temperature to have these phonons available. The scattering by emission of an LO phonon is more difficult in doped heterostructures, not because of the phonon population effect, since the phonon emission is always possible, but because the electron gas is usually degenerate, which, by the Pauli exclusion principle, prevents the existence of available final states for the transition. Again an elevated temperature is required to obtain enough empty states below the chemical potential at equilibrium. At low temperatures ( T 5 77 K) the LO phonon population is negligible, and only emission is possible, provided the final state occupancy can be neglected. Another kind of electron-phonon scattering is possible that is less energetically demanding, namely interaction with the acoustical modes. Since the energy of acoustic phonons is 5 1 meV the acoustic phonons become populated as soon as T- 10-20K and at room temperature each acoustic mode is heavily populated: n~~ kT/hoQ,. The only electron-acoustic phonon interaction we shall consider is due to the modulation (local

-

169D.K. Ferry, Surf: Sci. 75, 86 (1978). "OK. Hess, Appl. Phys. Lett. 35, 484 (1979). I7'P. J. Price, Ann. Phys. (N.Y)133,217 (1981);Suif Sci. 113, 199 (1982); Phys. Reu. B 30,2234 (1984). 172T.J. Drummond, H. Morkoc, K. Hess, and A. Y . Cho, J . Appl. Phys. 52, 5231 (1981). 173B.K. Ridley, J . Phys. C 15, 5899 (1982). 174W.Walukiewicz, H. E. Ruda, J. Lagowski, and H. C. Gatos, Phys. Rev. B 30,4571 (1984). 175B. A. Mason and S. Das Sarma, Phys. Rev. B 35, 3890 (1987). 176B.Vinter, Appl. Phys. Lett. 45, 581 (1984).

346

G . BASTARD et al.

dilatation or compression of the crystal) of the hosts’ band edges by the longitudinal acoustic (LA) phonons and described by the deformation potential approximation. In heterostructures, the phonon spectra, like the electron spectra, are modified by the size quantization. As for electrons, one finds phonon modes that are spatially extended while others are confined in one kind of layer. Since there are interfaces, there also exist interface phonon modes that have no equivalent in the bulk hosts. The quantization of phonon modes in heterolayers is by now quite well documented thanks to Raman spectroscopy (see, e.g., Refs 177-178). Less well documented is the influence of the phonon quantization on the carrier-phonon interaction. One of the difficulties is that the Frohlich interaction and deformation potential models are formulated in bulk approximate theories that cannot be easily generalized to heterolayers. In order to present orders of magnitude of the electron-phonon scatterings, we shall only use bulklike phonons. This is not much of a problem for the electron-LA phonon interaction because the acoustic phonon branches of different hosts largely overlap: all of them start from a zero frequency at zero bulk wave vector. Furthermore, the sound velocities of all the 111-V materials are of the same order of magnitude. The approximation is certainly more severe for the Frohlich interaction, since there are cases where the LO phonon branches do not overlap, which leads to heterostructure phonon modes that are localized in one kind of layer. However, a recent paper by Mori and and^'^^ has concluded that the differences between the electronLO phonons interaction when one does or does not take into account the phonon quantization are under most circumstances small. Thus, if one adopts bulklike phonon modes we have, in lieu of Eq. (10.2), He-ph(Q,a) = 4Q)e-’Q’’,

(11.12)

with la(Q)IZ= 2 7 t h ~ , , ( e ~ / ~ , Q Q ~ ) ,

(11.13)

- K-1,

(11.14)

E p l = E,1

and 14Q)IZ= (co/WWQ), C,

= D2/2pCz

(11.15) (11.16)

for the interaction with the LO and LA phonons, respectively. In Eqs. ”7M. Klein, IEEE J. Quantum Electron. QE22, 1760 (1986). ”‘B. Jusserand and D. Paquet in “Heterojunctions and Semiconductor Superlattices,”p. 108. Springer-Verlag, Berlin, 1986; J. Sapriel and B. Djafari Rouhani, Su?f Sci. Rep. 10,189 (1989). 17’N. Mori and T. Ando, Phys. Rev. B 40,6175 (1989).

ELECTRONIC STATES IN SEMICONDUCTOR HETEROSTRUCTURES

347

(11.12)-(11.16) hwLois the LO phonon energy (the LO phonon dispersion is neglected), R is the volume of the heterolayer, E , and IC are the high-frequency and static dielectric constants, respectively, C, is the longitudinal sound velocity, D is the conduction band deformation potential, p is the crystal density, and o(Q)= C,Q. In principle, both the LO and LA electronphonon interactions have to be cut off at some appropriate wave vector. However, it turns out that the creation or annihilation of high-Q phonons give a negligible contribution to the Tnk. Using Eqs. (1 1.12)-(11.16), one finds that for the emission of LO phonons in quantum-well structures, 1

--

m*e20L,(l

Q = (kf

s

&2 1 - k2 1

and

(11.17)

2k2E,

znk,

Z""'(Q) =

+ nLo)

+ 'k'

- 2kLk;

+ (2m*/A2)(E,

-

cos

O)li2,

En.- hw,o),

dzdz'X:(z)Xn(z')Xn*.(z)Xn.(z')e- Q l z - z ' l ,

fnn'(ql) =

I

fm

~,*(z)xnr(z)e-iq~~* dz

(11.18) (1 1.19) (11.20)

(1 1.23)

-m

in the case of the emission of LA phonons. In Eqs. (11.17) and (11.21) nLo and nLA denote the occupancy of the LO and LA modes, respectively. AS is apparent from Eqs. (1 1.17)-(11.23), we have neglected any band nonparabolicity effects in the in-plane dispersion of the conduction subbands. The band-mixing effects are unavoidable for the evaluation of the lifetime in the valence subbands. These effects have only been recently discussed for static scatterers,126"80andeven less for the hole-phonon interactions.'*' We shall restrict our considerations to the broadening due to the static scatterers. All the results shown in Section 13 have been obtained by projecting a '*OH. W. Liu, R. Ferreira, G. Bastard, C. Delalande, J. F. Palmier, and B. Etienne, Appl. Phys. Lett. 54, 2082 (1989). "'R. W. Kelsall, R. I. Taylor, A. C. G. Wood, and R. A. Abram, Superlattices and Microstructures 5, 207 (1989).

348

G. BASTARD et al.

valence eigenstate (unk,) on the incomplete basis spanned by the k, = 0 bound solutions. The basis states are labeled Iu,m,,n), where - 312 d m, d 312 and 2 < n < n,.nh, where n, and nh are the number of light (m, = k 112) and heavy (m, = k 3/2) bound solutions at k, = 0 (including Kramers degeneracy). Thus, we have written Iunk,)

=

1 Pnm(m,,k,)lu, m,,

(11.24)

m ) eiki‘PS-1/2.

m,mJ

Instead of scalars the hamiltonians of the static scatterers have to be considered as identity matrices on the l-8 basis [V(r - Ri) + V(r - Ri)l]. Consequently, Eq. (1 1.2) has to be generalized to

x

C

m,m’.m

k,)an’mr(mJ,

k;)(o, m j , nI ew[- Ik, - kiI IZ - ziIIIv, m j , n’>.

(11.25) For the conduction eigenstates we have used the following numerical parameters: D = 8.6eV, p = 5.3 g/cm3, hw,, = 36meV, C , = 3.7 x lo5cm/s, m* = 0.07m0, E , = 10.9, and K = 12.5. 12. ELECTRON SCATTERING AND TRANSFER TIMES Figure 68 presents the intrasubband and intersubband scattering times for electrons’82 in single and unbiased GaAs-Ga,,,Al,,,As quantum wells versus the quantum-well thickness L when the relaxation is due to the emission of optical phonons at T = OK. The Frohlich mechanism is by far the most effective when it is energetically allowed (cf - ci > hw,,). In Fig. 68 the initial energy coincides with the onset of the E , subband, and the initial state is either IE,, k, = 0) (intersubband scattering) or IE,, kL), where h2k:/2m* = E , - El (intrasubband scattering). One sees that the intrasubband scattering is almost L-independent. It is also practically independent of the initial energy in the El subband. The LO phonon emission thus has a threshold when the initial carrier energy is equal to El+ hw,,. The intersubband scattering increases roughly by a factor of 8 when L decreases from its maximal value [E,(L) - El(L) = ho,,]-that is, L = 178 A-to its minimal value, which is reached when E , becomes unbound ( L 55 A). This increase is due to the delocalization of x,(z) in the barrier while xl(z) remains heavily localized in the well. When the barrier height varies, there is an increase of the intrasubband relaxation time with V, (see Fig. 69). However,

-

ISzR. Ferreira and G. Bastard, Phys. Rev. B 40,1074 (1989).

ELECTRONIC STATES IN SEMICONDUCTOR HETEROSTRUCTURES

349

GaAs -Ga(Al)As 16

60

loo

L

CZP

180

FIG. 68. Quantum-well thickness dependence of the intrasubband ( T +J ~ and intersubband relaxation times in single quantum-well structures.The relaxation is associated with LO phonon emission. T = 0 K.

(T~-

this increase is modest ( 550%). Also at fixed V,, one does not recover the Lbehavior expected if the infinite V, approximation were used in the calculations. The electron-LA phonon interaction leads to intrasubband and intersubband relaxation times that at T = 0 K are much longer than those due to the electron-LO phonon interaction. This is illustrated in Fig. 70. Both 72-1 and z increase roughly linearly with L at large L. At low temperatures the scattering by charged impurities or alloy fluctuations can be as effective as that due electron-LA phonon interaction. Figure 71 shows the intersubband scattering time due to the interaction of electrons with 10" cm-, impurities located on a plane zi(0 < zi< L/2).The on-center impurities cannot induce an E , + El transition (by symmetry). When the impurity plane is in the barrier, the scattering is weak. Thus 7 2 + 1 ( Z i ) should display a minimum value when ziis in the quantum well. One notices that this value (- 20 ps if L = 150A) is much shorter than that arising from the electron-LA phonon interaction (at T = 0 K). Figure 72 shows a plot of the L dependence of the intrasubband scattering time due to alloy scattering in Gao.,,Ino,,2As-InP quantum wells. It is seen

1

A

2Lo'L GaAs-Ga(Al)As

GaAs -Ga(Al)As OPTICAL PHONON

Xz03

1

2oot

FO :

ACOUSTICAL PHONON SCATTERING

-

"I 160'

~

v

G=GF

1201

1008

016; 02

l-1-_1 04 06 08

-

80

_-

OL

1

4

X

FIG69 The intrasubband relaxation time z , - ~ due to optical phonon emission at T=OK is plotted versus the A1 content x in GaAs-Gal -.AI,As quantum wells for several well thicknesses. The initial energy coincides with E ,

SINGLE QUANTUM WELL

\

LO

80

120

L

(A,

160

200

FIG.70. Quantum-well thickness dependence of the intrasubband ( z , + ~ ) and intersubband ( T * - I ) relaxation times due to LA phonon emission. T=O K.

n

1508

'J

0 0

01

IMP URlT IES

,

N,,~

F;O

0.2

0.3 Zl/

L

I 04

I

To 0.5

FIG. 71. The quantities h / 2 ~ * +(left , scale) and (right scale) are plotted versus the dimensionless impurity position z l / L for impurity scattering. z , / L = O(i) corresponds to on-center (onedge) impurities. The two horizontal arrows correspond to a uniform distribution of the impurities in the GaAs well. t2-,

-

0

100

200

L d ) FIG. 72. The velocity relaxation time is plotted versus the quantum-well thickness in Ga(1n)As-InP-doped quantum wells (n,= 3 x 10' cm ', T = 0 K) for screened ionized impurity scattering, alloy scattering, and interface defect scattering.

ELECTRONIC STATES IN SEMICONDUCTOR HETEROSTRUCTURES

35 1

that alloy scattering is very efficient. Actually, it dominates the lowtemperature mobility of the Ga,,,,In,,,,As-InP modulation-doped heterostructures. As a result the mobilities are smaller than those of GaAsGa,,,Al,,,As heterojunctions where the alloy scattering, which takes place in the barrier, is inefficient. When single or multiple quantum wells are biased, one deals with significant polarization effects of the eigenstates (see Section V). In single wells xl(z) moves toward an interface, while the xz shift occurs at a much larger field strength. Over a large field range the x1 and xz centers of gravity are thus shifted from one another. This leads to large changes of the intersubband scattering times. For instance, Fig. 73 shows the field dependence of the zz+l relaxation time due to 10" cm-' ionized impurities or interface defects located on one interface ( - L / 2 ) . when F < 0, the xl(z) and x2(z) envelope functions are preferentially localized near z = + L/2. Thus, the scattering is weak and z2+1is long. When F increases, x1 and x2 move toward the - L/2 interface and zz+l decreases. When F is large and positive, one finds ultimately another z2+1increase with F. It is associated with the recovery of a quasi single-heterostructure configuration, where both x1 and xZ are localized near z = - L/2 and where their centers of gravity are increasingly displaced

1

-100

1

-50

I

0

I

50

100

F (kV/cm) FIG.73. The decimal logarithm of the intersubband relaxation time 72-1 (in ps) is plotted versus the electric field F for three quantum-well thicknesses ( L = 6 0 A , 120A, 18OA).Two scattering mechanisms are considered on-edge impurity scattering (solid line) and interface defect scattering (dashed line).

352

G . BASTARD et al. 1

I

I

I

I

SINGLE OUANTUM WELL OPTICAL PHONON SCATTERING -

O L -

0

20

40

60

80

100

F ( kV/cm) FIG. 74. Electric field dependence of the intrasubband ( T ~ - ~ and ) intersubband ( T ~ + ~ ) relaxation times in single quantum wells. LO phonon scattering. T = 0 K. For both zl-+ 1, T,+ 1 the initial energy is E , .

when F increases (at a rate -F1/3). These polarization effects are strongly dependent upon the quantum-well thickness, being almost negligible when L = 60 A. This is to be expected from the Stark effect in a quantum well which predicts that the shift of the center of gravity of the x1 subband with respect to the center of the well roughly scales like L4. One notices in Fig. 73 that the interface defects are much less efficient scatterers than the coulombic impurities when their areal concentration is the same. This is a consequence of the short-range nature (along the growth axis) of the interface defect potential. To be important the latter requires that the envelope functions of the initial and final states be significant near the interfaces, which seldom happens unless the wells are very thin. The long-range nature of the coulombic potential makes it more efficient, even when the impurities are located at the interfaces. Figure 74 shows the field-induced polarization effects on the intrasubband and intersubband LO phonon-mediated relaxation times. One notices that the intrasubband relaxation is practically F-independent, while the intersubband relaxation becomes inhibited by the field when L is large. Notice, however, that the Stark shift may open the LO phonon relaxation channel, which was forbidden at F = 0 ( L = 180A). This occurs because E,(F) decreases more strongly with F than E,(F) does (see Section V).

ELECTRONIC STATES IN SEMICONDUCTOR HETEROSTRUCTURES

353

The field-induced polarization in double or multiple quantum wells corresponds, in general, to a localization of the eigenstates in different wells while they were delocalized at zero field (Wannier-Stark quantization). Over a limited field range, where levels mostly localized in different wells are made to anticross by the field, the eigenstates delocalize again. Thus one may encounter situations where the intersubband, interwell scatterings are weakened by the field-induced localization of the initial and final states in different wells or enhanced when there are field-induced anticrossings. The first kind of effects,'*' met when the isolated wells support a single bound state, is illustrated in Figs. 75-76 in the case of biased symmetrical double quantum wells for electron-impurities, electron-interface defects, and electron-LO phonon interaction. For electron interaction with static scatterers (Fig. 75) there is an increase of more than two orders of magnitude in the interwell relaxation time when F increases from 0 to lo5V/cm. At zero electric field the splitting between the El and E , states of the double well is small and z2+' is very short (< lops). When the field increases ( F > 0), the reduced field strengthf ( = eFd/lAl) becomes comparable to, or larger than, 1, leading to an increasing localization of x1 (x,) in the left-hand-side (right-hand-side) well and thus to a large z,+'. The effect is more pronounced for thicker periods since the f are correspondingly larger for a given F (larger d's and smaller 121's). In Fig. 76 we show that the LO phonon-mediated interwell scattering time increases with increasing field-induced localization, while the intrawell scattering remains unaffected. Notice again that while the E , + El deexcitation by LO phonon emission was impossible at F = 0, it has become allowed when the field eFd 2 ho,,. In biased superlattices one may wonder if the Wannier-Stark quantization (or the Bloch oscillator) survives the broadening of the eigenstates due to the electron interaction with static scatterers or LO p h o n o n ~ . ' ~ ~W- e' ~ ~ show in Fig. 77 a plot of OZ, versus 2 / f (lower horizontal scale) or F (upper horizontal scale) in a 30 A-30 A GaAs-Ga,,,Al,,,As superlattice, where wB is the Bloch frequency (hw, = e F d ) and z the relaxation time of any of the Wannier-Stark states. Static and inelastic scatterings have been included. One finds an increase of OZ, with increasing F , which is superlinear. This is a consequence of the field-induced localization of the initial and final states in different wells. The quantity wBz remains larger than 1, even in the case of LO phonon scattering. Thus, in principle, the scatterings should not prevent the observation of the Wannier-Stark states in superlattices. This is in marked contrast to the bulk situation, where the scattering times are invariably found Ia3R.

Ia4Q. Ia5R.

Ferreira and G. Bastard, Surf. Sci. 229, 424 (1990). H. Dohler, R. Tsu, and L. Esaki, Solid State Commun. 17, 317 (1975). Tsu and G . Dohler, Phys. Reo. B 12, 680 (1975).

354

G. BASTARD et al.

-100

-50

0

100

50

F (kV/cm) FIG.75. The decimal logarithm of the intersubband ( T , + ~ ) relaxation times is plotted versus the electric field strength F in a biased, symmetric, double-well structure. Two scattering mechanisms are considered impurity scattering (solid lines) and interface roughness scattering (dashed lines). The scatterers are assumed to sit on two equivalent interfaces of the double well.

L PHONON SCATTERING

L ._.....d: 60: ---d, 90A

-10

20

FLO( kV/cm 601

80

100

FIG.76. The decimal logarithm of the intersubband ( z ~ - . ~and ) intrasubband ( z ~ - +relaxation ~) times is plotted versus F for two symmetrical double-well structures. LO phonon scattering. T=OK. For both T ~ and - ~ T~~~ the initial energy is E , .

ELECTRONIC STATES I N SEMICONDUCTOR HETEROSTRUCTURES

355

FIG.77. The decimal logarithm of 0 ~ 7 ,where oBis the Bloch frequency and 7 the relaxation time at the edge of any Wannier-Stark state, is plotted versus F (upper horizontal scale) or 2/f (lower horizontal scale), where f =eFd/lll, for a 30 A-30A GaAs-GaO,,A1,.,As superlattice.

to be much shorter than the Bloch period. Finally, one notices the existence of pronounced wBz oscillations in LO phonon scattering. They are periodic in F - l and arise when the condition

haLo = neFd

(12.1)

is satisfied (i.e., any time a new channel for intersubband, interwell relaxation by LO phonon emission becomes available). Their periodicity in F-' is a direct consequence of the even spacing of the Wannier-Stark ladders. They are the counterpart in the electric quantization of the well-known magnetophonon oscillations (hwLo = nhw,) in the magnetic quantization. The tesonances in the assisted interwell transfer' '','"*' *' are, to a large extent, the counterpart of the phenomena analyzed above. They arise unavoidably when two eigenstates mostly localized in different wells of a multiple-quantum-well structure are lined up by the electric field. We first describe a very simplified case, which will enable us to point out analytically the main trends and difficulties of the resonant-assisted transfer. Then we present the results of more elaborate calculations. la6T.Weil and B. Winter, Surf. Sci. 174, 505 (1986).

356

G. BASTARD et ul.

0

0

100

50

F ( kV/crn 1

FIG. 78. Electric field dependence of the eigenstates (upper part) and of u: (lower part) in a biased asymmetrical double quantum well.

Let us first consider a biased asymmetrical double well (Fig. 78). Each well, when isolated, binds one state. El and El are the confinement energies in the wide (thickness L ) and narrow (thickness L') well, respectively. If we perform the simplest tight-binding approach of this biased double-well problem, we obtain

x*(4 = a * d ( z ) + b*cpl(Z), E+

= el

- ( E ~- E ' J ~

a: = ( E * - EI)'/C(E*

(12.2)

& J [ ( E ~ - ~;)/2]' - El)'

+ +eeF(L + h), = E', - $eF(L' + h),

+ a21,

+ ad,

(12.3) (1 2.4)

c1 = El

(12.5)

E',

( 12.6)

(12.7) (12.8) where h is the intermediate barrier thickness, and a and a' are the transfer integrals of the problem (a, a' < 0, la1 > Ia'l). Resonance takes place at the

ELECTRONIC STATES IN SEMICONDUCTOR HETEROSTRUCTURES

3 57

electric field F , such that - E; vanishes. The magnitude of the anticrossing is [email protected] At resonance there is an incomplete delocalization of the xk eigenstates since, when F = F,, u: = a’/(@+ a’) and b: = a/(a a’). If the zero-field detuning is very large ( L >> E), there will remain at resonance a large polarization of the eigenstate toward the wide well. Off resonance, of course, the polarization of the x+ state depends on the eigenstates and can be complete: when F >> F,, a: -+ 0, b: -+ 1 and a! -+ 1, b2 -+ 0, corresponding to E , + E l , E E;. There is a two-dimensional subband attached to the x+, x- states, and defects or phonons induce transitions between the two subbands. It is clear that off resonance (say F >> F,) a transfer from the upper subband to the lower one will be slow because of the increasing spatial separation of the initial ( cpl) and final (- cp;) states. This is just the situation discussed above in the framework of the Wannier-Stark localization. When F decreases to F,, the eigenstates delocalize and the transfer should be faster. There is an ambiguity concerning the notion of interwell transfer in the vicinity of a resonance, precisely because of the spatial delocalization. Actually the curve z(F) depends on which initial state one has prepared. In an optical experiment the electron will be photogenerated. The resonant optical excitation of the wide well is a clean way to prepare electrons selectively in the quantum state that evolves to 9,by continuity. This means that for F > F , the appropriate path is creation in the x+ state ( k , = 0) followed by assisted transfer to the x- state (k‘’ # 0) and eventually by an ad hoc detection in the x- state. For F < F , the same path is impossible at T = 0 K because now the creation in the x- state cannot be followed by an elastic or inelastic emission process to the x+ state. Thus, a curve of the transfer time versus F when the initial state is the edge of the wide-well ground subband stops at F = F,. Actually the cutoff at F = F , should be rounded off because of the broadening of the initial and final states due to the imperfections (second and higher orders of the Born approximation) and/or because of the finite temperature effects. If T # OK, there should exist a thermalization in the upper branch ( E + ) , and consequently one should average the initial state over a distribution function of the in-plane wave vectors. When F < Fo, the tail of this distribution may be small (if kT << E + - L ) , but one fraction of this tail is such that a resonant transfer is possible. Thus, roughly speaking, when F < F,, the transfer rate from the lower branch to the upper one is equal to the transfer rate at resonance weighed by the exponential factor exPC-P(E+ - &-)I. On the other hand, if one is always able to create electrons in the upper branch by any means, the curve z(F) versus F is defined for all F and exhibits a resonance at F = F,. Figure 79 shows the computed transfer times for electrons at T = 0 K in a

+

-+

-

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-100

I 0

I

- 50 F (kV/cm)

50

FIG.79. The decimal logarithm of the interwell transfer time is plotted versus the electric field strength F for several scattering mechanisms in a 47 A-SOA-lOOA GaAs-Ga,,,AI,,,As double quantum well. Solid lines: coulombic impurities (10'' cm-' at two equivalent interfaces).Dashed lines: interface defects. Dashed-dotted lines: LA phonons. Dotted lines: LO phonons.

47 A-50 A-100 A biased GaAs-Ga0,,A1,,,As double well.175 Elastic and inelastic scatterings have been considered. The initial state was chosen at the edge of the x+ state. The LO phonon emission does not play a significant part in this structure because over most of the field range El - E , < rZoLo The . ionized impurity scattering appears to be the dominant transfer mechanism, leading to a pronounced resonance (z, 1ps) in the x+ + x- assisted transfer near -40kV/cm ( E + - E - anticrossing). A second resonance takes place near F , = 35 kV/cm. It corresponds to the anticrossing between c2 and E + (in this field range E + and x+ almost coincide with E; and 9;). Notice the different shapes of the two resonance curves: the E + , E - resonance is symmetrical, while the E + , e2 one is not. In the former case the elastic or LA phonon-assisted x+ + x- transfer is always possible (at the expense of reverting the main spatial localization of the initial and final states), while in the latter case the x+ + x2 is only possible when F > F , . One notices a pseudoresonance in the LO phonon-assisted transfer near F , : the energy differenceE + - E - - ho,, is not zero near F,. Yet, the LO phonon-assisted transfer is accelerated because of the mixing of x+ with x2, which in the vicinity of F , , where x+ and x2 are strongly admixed, transforms the LO phonon-assisted interwell x+ + x- transfer into an intra (wide) well transition. Recent time-resolved luminescence experiments with subpicosecond

-

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359

resolution' have dealt with biasedlE7or unbiased'88 double quantum wells and permitted a clear distinction between the time scale pertinent to the LO phonon-assisted transfer between wells under resonant c2 E , or nonresonant E~ >> E + conditions. A comparison'** between the experimental findings and predictions based on models similar to those discussed in this section leads to excellent qualitative agreement (e.g., on the predicted behavior of the transfer time upon the intermediate barrier thickness), but indicates that the calculated times are shorter by a factor of about 4 than the measured times. The outcome of cw photoluminescence experiments'*' can be summarized in terms of the field dependence of the ratio between the integrated photoluminescences of the narrow well and of the wide well. Pronounced resonances were observed. To explain these resonances, simple rate equations were solved using the calculated transfer times for electrons and holes. Good qualitative agreement was found between the measurements and the model. The most striking feature was the evidence of resonances in the hole transfer time. These features are discussed in the following section. 877188

-

13. RESONANCES IN THE HOLETRANSFER TIMES As mentioned in Section IV, the anticrossings of valence eigenstates are strongly dependent upon the in-plane wave vector. In particular, heavy and light hole subbands cross at k, = 0, but anticross at k, # 0. This strongly influences the assisted hole transfer in biased multiple quantum wells, because even if the initial state is a zone center state (say a heavy hole state) an elastic scattering mechanism can couple to a k , # 0 state belonging to a light hole branch that, because of band-mixing effects, has a nonzero projection on the k , = 0 basis vectors corresponding to the heavy hole states. Band-mixing effects, which as already shown in Section IV lead to an increase of the heavy hole bandwidth in a superlattice when k , increases, are also responsible for an acceleration of the hole-assisted transfers when compared with a model (diagonal approximation of the Luttinger matrix) that assumes that heavy and light holes are always decoupled. This decoupled model has led to the widespread belief that the holes are considerably slower than electrons. When applied to biased multiple quantum wells, this means that the electron transfer time is much faster than that of the hole. We wish to show that this simple rule is most likely wrong in true quantum heterostructures. We have found that the assisted elastic transfer times of electrons and holes are in general comparable. "'D. Y. Oberli, J. Shah, T. C. Damen, T. Y. Chang, C. W. Tu, D. A. B. Miller, J. E. Henry, R. F. Kopf, N. Sauer, and A. E. Digiovanni, Phys. Rev. B 40,3028 (1989). lS8B.Deveaud, F. Clerot, A. Chomette, A. Regreny, R. Ferreira, and G. Bastard, Europhys. Lett. 11, 367 (1990).

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To emphasize the band-mixing effects, we shall assume that the initial valence state of a biased double quantum well is the ground heavy hole state (k, = 0) HH;.ls9 Primed and unprimed notations will refer to the “ground” and “excited well, respectively-that is, to the large and narrow wells in asymmetrical double wells or to the left-hand-side and right-hand-side wells in biased symmetrical structures ( F < 0). Figure 80 shows the electric field dependence of the valence eigenstates at k , = 0 in a 50 A-40 A-50 A GaAsGa,,,Al,.,As double quantum well. The heavy hole states anticross each other, but cross the light hole states. In particular, HH; anticrosses HH, near - 70 kV/cm, crosses LH, near - 36 kV/cm, and anticrosses HH, at F = 0. Consider now the interwell transfer mediated by ionized impurities (assumed to be placed on two equivalent GaAs-Ga,,,Al,,,As interfaces). The transfer is elastic as a whole, which means that the excess hole energy HH; -HH,, HH; - LH,, HH; -HH,, for the z motion is converted into inplane kinetic energy. The larger the excess hole energies, the larger the final in-plane wave vectors and thus the weaker are the corresponding scattering matrix elements. In addition, the initial and final states are heavily localized in the narrow and wide wells, respectively (Wannier-Stark localization) except near the anticrossings, where a delocalization of the eigenstates over the two wells takes place. This is the second important factor that contributes to enhancing the interwell transition near the anticrossings. Starting from - 100kV/cm and increasing F , the HH; -+ HH, transitions (AB in Fig. 80) become more and more probable until F coincides with the anticrossing field F,, where the energy mismatch HH; - HH, is lower. When F > F,, the HH; -+ HH, transitions become impossible and one is left with the nonresonant HH; -+ LH, (EF in Fig. 80) and HH; -+ HH, (EG) transitions. Since the band-mixing effects allow for the HH; -+ LH, transitions, the latter become more and more probable because the energy mismatch HH; - LH, decreases. Only at the crossing field F , does the matrix element (HH;Je-Qlz-zllILH,)vanish. On the other hand, the energy mismatch vanishes and so does the in-plane wave vector q. The q-’ singularity of the coulombic scattering blows up. The net result is a finite transition rate. When F > F , , the HH; -+ LH, transitions become impossible, and one is left with the HH; -+HH, impurity-assisted transitions, which, when F + 0, become resonant. In a simplified scheme, where the heavy and light hole states are always decoupled, the coulombic scattering between HH; and LH, would have always been impossible, thereby leading to an overestimate of the transfer time in the [F,, F , ] field range. This is not a small effect, as evidenced in Fig. 81 where the HH; transfer time z is plotted versus F for the two models: the parabolic lS9R. Ferreira and G. Bastard, Europhys. Lett. 10, 279 (1989), Surf. Sci. 229, 165 (1990).

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4

>

-100

-80

-60

-40

F (kV/crn 1

-20

0

FIG.80. Electric field dependence of the k , = 0 valence eigenstates in a 50 A-40 A-SO A GaAsGa,.,AI,.,As symmetrical double well. Solid lines: heavy holes. Dashed lines: light holes.

FIG.81. The decimal logarithm of the impurity-assisted interwell hole transfer time i for the same structure as in Fig. 80 is plotted against the electric field strength F . Dashed line: decoupled approximation. Solid line: band mixing included. The initial state is HH;.

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G. BASTARD et a1

model (dashed line) and that which takes into account the band-mixing effects (solid line). It is seen that the parabolic model overestimates the heavy hole transfer time by about two orders of magnitude off resonance and more than four orders of magnitude at the F , resonance. We show’*’ in Fig. 82 the field dependence of the hole transfer time in a biased 80&40&80A Gao,,,Ino~,,As-InP double quantum well. The transfer is either assisted by the alloy scattering (solid line) or by interface defect scattering (dashed line). The two curves have been horizontally displaced for clarity. The initial state is HH;. Again, one notices pronounced resonances that correspond either to heavy hole anticrossings or to heavy hole-light hole anticrossings. The interface defects are found to be inefficient scatterers. Since the wells are relatively wide (80A) and deep (0.365 eV), the resonances are very narrow and pronounced as a result of the small anticrossings. In reality, one should expect an “inhomogeneous broadening” of these resonances due to the simultaneous presence in a given heterostructure of several microheterostructures. We believe that little will remain of these pronounced resonances after a simulation of the inhomogeneous broadening. However, even off resonance the hole transfer time falls over a significant field range in the 200-ps range. For the 47 A-50A-100A GaAs-

-I_

-8p

-6r

-4p

Ga (In)A:-InP

D C p L E OUANTUM WELL 80 A - 4 0 k 3 0 A ---roughness scattering I \ az75A ,b,574A, Nde,=lolocm~

\I

-

I

FIG. 82. Calculated interwell transfer time ‘T for holes in a biased 80 A-40A-80A Ga,,,,In,.,,As-InP symmetrical double quantum well versus the electric field strength F . Solid line, lower scale: alloy scattering. Dashed line, upper scale: interface roughness scattering. The initial state is HH;.

ELECTRONIC STATES IN SEMICONDUCTOR HETEROSTRUCTURES

363

FIG.83. Calculated impurity-assisted interwell transfer time for holes in a biased 47 A-50 A100A GaAs-Ga,,,AI,,,As versus the electric field strength F. The impurities (areal concentration 10'om-z) are assumed to sit on two equivalent interfaces of the structure. The initial

state is HHl up to

-+

10kV/cm. For stronger fields the initial state is HH,.

Ga0,,A1,,,As structure discussed in section 12 for electrons, the hole transfer time exhibits many resonances versus F (see Fig. 83), some of which are quite deep (z < 0.1 ps). One can (arbitrarily) decide to cut all the segments below 1 ps. Still Figs. 82 and 83 clearly show that the hole transfer time is often between 1 and 500 ps. If we compare the hole and electron transfer times (Fig. 79), we notice that they are of the same order of magnitude. Figure 84 shows similar results for a 77 A-55 A-35 A GaAs-Ga0.,A1,,,As biased double well. The initial state is the x+ state for electron (i.e., either E l or El) and the HH; state for the holes. The transfer is assisted by impurities located on two equivalent interfaces of the heterostructure. We again find theoretical evidence that the electrons do not always transfer much faster than the holes. Figure 84 shows that this statement is only partially correct. If we neglect the HH; - HH, resonance (on the grounds that it is too narrow), the electrons transfer faster than the holes between - 70 and - 20 kV/cm. In the remaining field range the holes transfer faster than the electrons. Notice that if the band-mixing effects were disregarded (no HH; - LH, contributions), the holes would transfer faster than the electrons only in the [ - 20 kV/cm, 01 segment.

-

-

364

G. BASTARD ef al.

xoL\

-

= 0.3

2a

h

In

n

1.0

&# v

0

- 0 3 -1.0

-2.0

-100

- 50

0

F ( kV/cm) FIG.84. Comparison between the impurity-assisted interwell transfer times for electrons and GaAs-Ga,.,AI,.,As double quantum well versus the electric holes in a biased 78 A-55 A-35 .J% field strength F. The initial state for electron is either E , or E', . The initial state for holes is HH;. The impurities sit on two equivalent interfaces of the structure.

In any event, we believe to have established firmly that the hole transfer time is, in general, overestimated, due to the implicit use of decoupled hole subbands. Recent calculations of Wessel and Altarelli'go of the hole tunnel current in double-barrier structures have also pointed out the crucial part played by the band-mixing effects on a proper estimate of the hole tunneling effects. VII. Excitons in Semiconductor Heterostructures

In previous sections we discussed the single-particle states in heterostructures under different conditions. In general, optical transitions in undoped quantum wells or superlattices are dominated by the correlation between the electron and the hole, that is, excitonic effects. Figure 85 shows the excitation spectrum of the photoluminescence of a biased 47 A-50 A100 GaAs-Ga(A1)As double quantum well."' The existence of pronounced peaks (instead of plateaux) is evidence for the excitonic nature of the various transitions. Thus, transitions between excited subbands are excitonic, in spite of the degeneracy of the related excitons with the continuum of the lower-lying excitons. In addition, the excitons are not dissociated by the 19'K. Wessel and M. Altarelli, Phys. Reo. B 39,12803 (1989).

ELECTRONIC STATES IN SEMICONDUCTOR HETEROSTRUCTURES

365

V z 0.9 Volt

I

1550

1

1600 1650 ENERGY ( meV)

1550

1600

ENERGY (rneV)

1650

FIG. 85. Photoluminescence (dashed lines) and photoluminescence excitation spectra (solid lines) of a biased 47 A-5OA-lOOA GaAs-Ga,,,AI,,,As double quantum well for two different bias. T = 1.7 K.

longitudinal electric field. These features, established on a myriad of samples with different host materials, have given rise to a number of theoretical and offer interesting device prospects.21a The quantum-well confinement breaks the heavy and light hole degeneracy, giving rise to two series of excitons. Because of the complexity of the valence band (see Section IV), the heavy and light hole excitons are coupled. However, these couplings are difficult to describe, and have been neglected in early work on excitons in semiconductor he te r o str ~ c tu r e s. ' ~19' ~ The decoupled exciton models amount to retaining only the diagonal part of the Luttinger hamiltonian. The decoupled approximation is easily tractable and ~

IY1G.Bastard, E. E. Mendez, L. L. Chang, and L. Esaki, Phys. Rev. B 26, 1974 (1982). I9'R. L. Greene, K. K. Bajaj, and D. E. Phelps, Phys. Rev. B 29, 1807 (1984). L93J. A. Brum and G. Bastard, J. Phys. C 18, L-789 (1985). 194A.Chomette, B. Lambert, B. Deveaud, F. Clerot, A. Regreny, and G. Bastard, Europhys. Lett. 4,461 (1987). '95R. C. Miller, D. A. Kleinman, W. T. Tsang, and A. C. Gossard, Phys. Rev. B 22, 1134 (1981). 196T.F. Jiang, Solid State Commun. 50, 589 (1984). 19'C. Priester, G. Allan, and M. Lannoo, Phys. Rev. B 30,7302 (1984). 19*K. S. Chan, J. Phys. C 19 L-125 (1986). 19'G. Duggan, Phys. Rev. B 37,2759 (1988). 'OoG. D. Sanders and Y. C. Chang, Phys. Rev. B 32,5517 (1985); Phys. Rev. B 35,1300 (1987). ''ID. A. Broido and L. J. Sham, Phys. Rev. B 34, 3917 (1986). '"G. E. W. Bauer and T. Ando, Phys. Rev. B 37,3130 (1988); Phys. Rev. B 38,6015 (1988). '03U. Ekenberg and M. Altarelli, Phys. Rev. B 35,7585 (1987). '04B. Zhu and K. Huang, Phys. Rev. B 36,8102 (1987). '05L. C. Andreani and A. Pasquarello, Europhys. Lett. 6,259 (1988).

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works relatively well for the HH, - El: 1s exciton. It predicts correctly that the LH, - E l:1s exciton is more strongly bound than the HH, - E , :1s one; it predicts a reasonable variation of the 1s exciton binding energy with the quantum-well thickness; it accounts for the stability of the HH, - El: 1s exciton against the field-induced ionization ( F parallel to the growth 1 6 , 1 2 3 and it correctly predicts the variation of the HH, - El: 1s excitonic binding in superlattices, both in the unbiasedlg4 (when the superlattice period decreases) and in the biased cases.206*207 These points will be considered in detail in Section 14, where the excitonic states in the decoupled approximation will be discussed. Section 15 will be devoted to a more realistic description of the excitonic states in quantum wells, which takes into account the effects associated with the nonparabolic dispersion of the valence band,198-205and the coupling among the different excitons (Is, 2s, 2p,. . .-like), not considered in the decoupled approach. We shall discuss only excitons in type I heterostructures (i.e., where the electron and the hole are both confined in the well-acting material). A few calculations have addressed the problem of the excitonic states in type I1 heterostructures. 14. EXCITONIC EFFECTSIN THE DECOUPLED APPROXIMATION For excitons in unbiased or biased quasi-bidimensional systems, we face the same difficulties as for 3 D excitons subjected to an inhomogeneous electric field. The existence of unidirectional (in the z-direction) confining and/or electrostatic potentials prevents the existence of analytical solutions (even in the zero-field case). Variational methods have to be used. Let us first consider the decoupled exciton hamiltonian, which in the parabolic effective mass approximation is

where the first four terms account for the center-of-mass, relative electronhole, independent electron, and independent hole z motions, respectively. The last term is the coulombic interaction, with I = Ire - r,l denoting the electronhole distance and K the relative dielectric constant of the heterostructure. The relative ( p ) and z-dependent motions are coupled via the coulombic interaction, while the (in-plane) center-of-mass (free) motion is completely independent from the others and contributes to the energy only by an additive term h2K&/2M,. Since in the electric dipole approximation only the excitons with Kc,M,= 0 are optically detectable, we will consider only these exciton states. '06R. H. Yan, F. Laruelle, and L. A. Coldren, Appl. Phys. Lett. 55, 2002 (1989). 207J. A. Brum and F. Agullo-Rueda, Surf: Sci. 229,472 (1990).

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a. Excitons in Single Quantum Wells Notice that several sets of excitons may be formed with the electron and the hole belonging to the various conduction and valence quantum-well subbands. A single set that consists of the excitons that are formed between electrons and holes both belonging to the ground subbands is truly bound. The other excitons are resonances superimposed on the two-dimensional continua of the lower-lying electron-hole subbands. The broadenings associated with the autoionization of the excited exciton states are small, since it is possible to observe excitonic structures (i.e., peaks) in the absorption or photoluminescence excitation spectra associated with optical transitions between highly excited electron and hole subbands. We shall neglect these broadenings. We also neglect the broadening related to the dissociation of the exciton due to the escape of the electron and/or hole out of the quantum well in a biased system, as discussed in Section V for independent electrons or holes. In a first approach to the exciton problem, and guided by experiments (which show the existence of exciton states primarily associated with one of the various band-to-band edges), we chose trial exciton wave functions of the form $,,, = $ n m j , where n(m) labels the electron (hole) subband, and j labels the Is, 2s, 2p,. . . levels. Consequently, the Rnmjexciton (associated with En - H m : j )binding energy is given by Rnmj

= EG

+ E n + H m - Min<$nmj I Hex I $nmj>,

(14.2)

where E~ is the energy gap of the bulk material of the well and where the minimization is performed over all the variational parameters of the trial wave function. Several $nmj have been used. A very convenient one works rather well over the range of GaAs quantum-well thicknesses, which is the most extensively investigated, namely 0.5af < L < 2a7, where a7 denotes the in-plane effective bulk Bohr radius calculated by using the in-plane reduced mass p1. This simple wave function is $nmj(P? z e ?

zh)

= qn(ze)Xn(Zh)fi(P,

q),

(14.3)

where the qnand xm’s are the En and Hmenvelope eigenfunctions of HJz,) and Hh(z,), respectively, and where we have made use of isotropicity of the coulombic interaction in the layer plane. For the ground (quasi-ls, E n - H,: 1s) exciton state, fls(~,

CP)= JG%Z exp(-P/Anm),

(14.4)

where A,, is a variational parameter. To write Eqs. (14.3)-(14.4) amounts to assuming that the longitudinal electron and hole motions are restricted by

368

G. BASTARD et al.

the quantum-well potential, while the bound states for the in-plane reduced motion are provided by the coulombic electron-hole interaction averaged over the probability densities of q$(z,) and xL(zh) for the electron and hole, respectively. Besides leading to simple algebra, Eqs. (14.3)-( 14.4) have the advantage of providing excited state (n, m > 1) wave-functions that are automatically orthogonal to the lower-lying solutions. More elaborate trial wave functions have been proposed, leading to small improvements over the results derived from Eqs. (14.3)-(14.4). Results of the calculation for the binding energies R,, = Rnm:lsof the quasi-1s excitons in unbiasedlg3 (biasedlz3) isolated quantum wells are presented in Fig. 86a (Fig. 86b, respectively). We show in Fig. 86a the L dependencies of several ground light hole and heavy hole exciton binding energies R,, in GaAs-Ga,.,Al,,,As quantum wells calculated from Eqs. (14.2)-(14.4). Several features are noticeable in Fig. 86a. (1) R,,(L) first increases with decreasing L and then decreases again. This behavior is the same as that found for coulombic impurity states. It reflects the increasing quasi-bidimensionality of the coulombic problem when L decreases until one of the two particles (or both) loses this quasibidimensionality by having a confinement energy that approaches the height

FIG.86. (a) Variational estimates of the En-HH, (solid lines) and Em-LH, (dashed lines) Is exciton binding energies in GaAs-Ga,,,AI,,,As quantum wells versus the GaAs slab thickness. Decoupled approximation. (b) Electric field dependence of the HH, -El :Is exciton binding energy for several GaAs slab thicknesses in GaAs-Ga,,,AI,.,As single quantum wells. Decoupled approximation.

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369

of the confining well. For this reason all the excitons E, - HH,:ls, E, - LH,:ls disappear at small enough L unless n = rn = 1. Notice, however, that it is, in principle, possible to form an exciton between a true bound state (say a valence state) and a virtual bound state (say a conduction state) of the quantum well if the resonant state is narrow enough. The magnitude of the binding energy for such an exciton and its stability against dissociation have, however, as far as we know, never been calculated. (2) The binding energies R,, decrease with increasing n or m,but relatively slowly. In a first approximation, the exciton binding energies R,, are nearly the same for all n (if both the En, HH, or En, LH, pairs of levels are tightly bound in the well). This arises from the weak dependence of the averaged electron-hole interaction upon the electron and hole quantum-well bound state. (3) The ground light hole exciton E , - LH,: 1s is found to be more strongly bound than the heavy hole exciton El - HH,: 1s unless L is very small. This is a consequence of the mass-reversal effect for the hole subbands in the diagonal approximation: the heavy hole subbands have a lighter inplane mass than do the light hole subbands. Thus, the in-plane reduced mass is heavier for electrons and light holes than for electrons and heavy holes. This, together with the insensitivity of the averaged electron-hole interaction to the difference between the xi's of the light hole and heavy hole, implies that El - LH, :1s is more bound than El - HH, :1s unless L is so small that the light hole wave function x1 leaks much more effectively into the barrier than the heavy hole x,. Let us now consider the excitonic states of a biased single quantum well. The electric field polarizes the electron and hole wave functions in opposite directions and therefore weakens the excitonic binding. However, we have seen in Section V that the field-induced carrier escape is considerably inhibited by the presence of the quantum-well barriers. This allows for the persistence of a sizable excitonic binding up to very large fields. In fact, clear excitonic resonances in the absorption coefficient have been observed in GaAs-Gal -,Al,As quantum wells subjected to longitudinal fields in excess of 100 kV/cm. These fields are one order of magnitude larger than the ones that ionize the excitons in bulk GaAs. We have already discussed in Section V the longitudinal Stark effect of free particles. Keeping the same ideas as the ones that led to Eq. (14.3), we notice that the external field will change the electron and hole confinement energies in the well but will affect the excitonic binding only through the modification of the averaged electron-hole coulombic interaction. Thus the electric field affects the excitonic binding in second order. These ideas can be substantiated by detailed variational calculations using Eqs. (14.3)-( 14.4) for the exciton wave function, except that the q l ’ s and xi's now refer to the ground electron and hole envelope functions in a quantum well tilted by the field.116.’23

370

G . BASTARD et al.

We show in Fig. 86b the electric field dependence of the E l - HH,: 1s exciton binding energy in GaAs-Gao.,,A1o,,,As quantum wells of different thicknesses. The most striking feature is that the exciton binding energy varies little with F compared with the E,(F), HH,(F) shifts. At low field the exciton binding energy decreases quadratically with F, and the decrease becomes steeper with increasing quantum-well thickness. At large fields the carrier accumulation regime prevails, and the exciton binding energy varies little with F Ultimately the electric field sweeps the carriers outside the well and the excitonic binding energy should drop to zero. This regime is, however, not describable by the trial wave functions given in Eq. (14.4). When the electric field is applied in the layer plane, it affects the excitonic binding drastically. In fact, the physical situation is much the same as that found in bulk materials when the field-induced energy difference across one Bohr diameter becomes comparable with the zero-field exciton binding energy; the exciton ionizes. Being in a quantum well, the exciton is more bound than in bulk material, and its in-plane effective Bohr radius is smaller. Thus a larger field is needed to ionize an exciton in a quantum well than in the corresponding bulk material. The gain, however, is modest, typically a factor of 2. b. Excitons in Unbiased Superlattices In an unbiased superlattice the quantum-well bound states hybridize to give rise to minibands. As a result the electron (or hole) wave functions become increasingly delocalized when the superlattice period d decreases. Since only the excitons with a zero center of mass momentum are optically active, we write the trial exciton wave function for an E l - HH,:ls exciton as194

where N is a normalization constant, 1 and are variational parameters, qe is the z component of the electron wave vector ( - n / d < qe < n / d ) , and q,.(z,) and x-Jzh) are the E l and HH, electron and hole envelope functions in the superlattice. It can be verified that Eq. (14.5) reduces to the separable exciton envelope function of isolated wells in the multiple-quantum-well limit (i.e., thick barriers) and gives a bulk exciton binding energy that is about 85% of the bulk Rydberg near vanishing d . Figure 87 shows the calculated variation^'^^ of the E l - HH, :Is heavy hole exciton binding energy as a function of the SL period d when the well and barrier widths are equal. When the period increases, the binding energy increases from the GaAs bulk value (d + O ) , reaches a maximum near d z 140A, and then decreases (d -+ 00). The latter behavior is the same as that

ELECTRONICSTATES IN SEMICONDUCTOR HETEROSTRUCTURES

37 1

GaAs -Ga(AL)As

X 0.3 LA :L, U

100

d

(A)

200

FIG. 87. HH, -E,:ls exciton binding energy versus the superlattice period d in GaAsGa0,,A1,,,As superlattices with equal layer thicknesses. Solid line; decoupled approximation. Symbols: experiments. After Ref. 194.

of single QWs. The experimental values are also reported in Fig. 87, and their variations upon d are correctly reproduced by the calculations. Two domains are apparent in Fig. 87: (i) d < 94 (for the chosen set of parameters), where the exciton spreads over several layers and displays a 3D character; and (ii) d > 94 A, where the exciton is found to be confined in the wells (B -,00) and has a 2D character as in MQW structures. The localized-delocalized transition is found to be abrupt in our calculations. In fact, the curves E(A, b) always have two minima that correspond either to a finite b or to a very large one. The transition at d TC 94A results from the interchange of the relative positions of these two minima. A more elaborate trial wave function may eventually smooth the transition. Several experiment^^^^^^^^ have pointed out the existence of optical features that might be associated with the saddlepoint exciton-that is, an exciton formed in the vicinity of the q = n/d states of the El and HH, subbands in superlattices. Detailed numerical calculations210have, however, revealed that the resonances in question exist but are rather built out of the entire superlattice subband. c. Excitons in Biased Superlattices The electric field dependence of the one-particle (electron or hole) states of a superlattice under the action of a longitudinal electric field (Wannier-Stark Deveaud, A. Chomette, F. Clerot, A. Regrkny, J. C. Maan, R. Romestain, G. Bastard, H. Chu, and Y. C. Chang, Phys. Rev. B 40,5802 (1989). *09Y. S. Yoon, P. S. Jung, J. J. Song, J. N. Schulman, C. W. Tu, and H. Morkoc, Bull. Am. Phys. SOC.33, 365 (1988). 'lOH. Chu and Y. C. Chang, Phys. Reo. B 36,2946 (1987). "'B.

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et

al.

ladders) has been discussed in detail in Section V. Recent optical studies performed in high-quality samples have evidenced the excitonic nature of their spectrum. Although many experimental features follow the predictions of band-to-band models, a better quantitative understanding of the experiments requires the consideration of the coulombic coupling of the electron and hole Wannier-Stark states. The Stark ladder excitonic ~ t a t e ~will~be ~considered ~ , ~ here ~ in ~ the , ~ ~ ~ simplest approximation: with each Wannier-Stark valence level, we associate an infinite manifold of independent excitonic states by coupling this hole level to each of the Wannier-Stark levels of the conduction band. As we have seen (Section V), the heavy hole Wannier-Stark states localize rapidly with increasing electric field (F). These heavy hole excitonic states can be visualized as the interaction of one very localized (in the mth well) hole with electrons that are, in the weak-field regime, delocalized over the structure and that localize progressively with increasing F to reach (in the strong F regime) a situation where the nth state is strongly localized in the nth well. Then in the weak-field region (where the hole, but not the electron, is already localized) all the Rnm:lS binding energies should be small (relative to the corresponding isolated quantum-well value) and not so different. Increasing F favors the conduction states that develop their localization around the mth site (small In - ml) by increasing their binding, until the strong F limit is reached where the direct (n = m) exciton should largely predominate over the crossed (n # m) excitons, whose binding energies should saturate to a lower value as a consequence of the increasing localization. Figure 88 the excitonic binding energies as a function of the electric field for a 40-A GaAs-40-A Ga,,,Al,,,As superlattice for heavy (full lines) and light (dashed lines) hole excitons ( n - m < 0) The Rnm:ls= R n P m binding energies have been calculated according to Eqs. (14.2)-( 14.4).The En, qn(z,), H,, xrn(zh)free-electron and free-hole eigenvalues and envelope functions have been numerically obtained (for details see Ref. 207). In particular, they incorporate the interactions between the Wannier-Stark ladder derived from E l (HH,) with the (pseudo) continua of the delocalized states above the confining barriers. Figure 88 shows the principal trends discussed above. In addition, we can see that the heavy and light hole excitons have close binding energies at low fields, with the light exciton curves saturating at higher values of the binding energy at large F . We also notice that the heavy hole exciton binding energy does not increase continuously with F , as expected from previous considerations, but exhibits a maximum versus F This is due to the two concurrent effects of the electric fields: the increasing interwell localization (which increases R,), and the increasing intrawell separation of the electron and hole states (Stark effect, which

ELECTRONIC STATES IN SEMICONDUCTOR HETEROSTRUCTURES

0

10

80

373

120

Electric field (kV/crn)

FIG.88. Electric field dependence of the HH, -E,:ls (solid lines) and LH, -E,:ls (dashed lines) exciton binding energies for different optical transitions between the mth valence Stark ladder and the nth conduction Stark ladder ( m - n < 0). After Ref. 207.

decreases Ro). For F M 40 kV/cm the interwell localization begins to saturate, and the intrawell Stark shift dominates at higher fields. For light hole excitons the Stark shift region is not achieved, since the light hole autoionizes into the superlattice continuum for electric fields larger than F z 60 kV/cm. (The interaction of the Wannier-Stark levels with the continua of levels of the biased superlattice is implicitly considered in the uncorrelated electron and hole motions along the field direction). 15. COUPLED EXCITONS IN QUANTUM WELLS The in-plane dispersion of the valence subbands is quite complicated and far from being parabolic. In particular, the ground light hole subband displays a camelback shape. This peculiar dispersion should certainly affect the exciton binding energy. However, this effect can somehow be masked by the fact that what enters in the exciton problem is the reduced mass, where the electron mass plays the main role since it is lighter. In addition, the existence of coupling between different excitons (for instance, between the 1s

374

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excitons with excited excitonic states like 2p, 3d) allows one to observe transitions originally forbidden in the decoupled approximation. Several paper^'^'-^'^ have been devoted to the study of the effect of the coupling among the different excitons introduced by the valence band complexity. Bauer and Ando202and took that coupling into account by using an extended nonorthogonal basis in a variational calculation. They obtained the binding energy for several excitons. Broido and Sham”’ used a variational calculation to include the strong nonparabolicity of the valence band into the exciton ground state (1s) binding energy. Ekenberg and Altarelli203 developed a perturbation method to include the interaction between the different excitons into the exciton ground state binding energy. They showed that, besides the coupling of the 1s heavy hole exciton with the 2p, 3d,. . . light hole excitons, one has to include the interaction with the exciton continuum. The importance of the El - HH, exciton continuum on the binding energy of the El - LH, : 1s was stressed by Andreani and P a ~ q u a r e l l o . ~ ~ ~ We discuss two approximations to calculate the exciton binding energy, including the coupling among the different excitons. In the first approach we calculate the Is, 2s, 2p, 3d states for the heavy and light hole excitons in the diagonal approximation (that is, taking into account only the diagonal part of the Luttinger hamiltonian, as previously done). We describe the different excitons by variational wave functions and maximize the binding energy with respect to these parameters. In a second step, we project the whole hamiltonian onto this basis. The eigenstates are finally obtained by a numerical diagonalization. In this approach, we do not include the higher exciton states as well as the exciton continuum. Clearly, we do not expect to obtain a full description of the effect but a qualitative picture. In a second approach, each exciton state is described by a series of nonorthogonal variational wave functions of the same type as in the first approach. Again, we project the whole hamiltonian onto this basis, and the eigenstates are obtained by solving the generalized eigenvalue problem numerically. Here, the exciton continuum is simulated by a few discrete states, giving a better quantitative description. This second approach is close to those followed by Bauer and Ando202 and Chan.’98 a. Orthogonal Basis In the effective mass approximation, the exciton hamiltonian can be written as

(.:

+ H&,

-

31

+

1

P ) - {e2/lcCp2 (ze - 2J2I2}1,

(15.1)

ELECTRONIC STATES IN SEMICONDUCTOR HETEROSTRUCTURES

375

where

(15.2)

and

(15.3a) (15.3b)

where p = (x,y) is the in-plane relative coordinate. The center of mass (or its equivalent for degenerate bands) is disregarded in the present analysis. Its motion is free, and we know that only immobile excitons (Kc,M,= 0) are coupled to the light (in the dipdlar approximation). We first neglect the off-diagonal terms and solve only the scalar hamiltonian variationally for several exciton states. The variational wave functions for the excitons related to the nth conduction band and the mth valence band are given by Eq. (14.3)' with

f d P , cp) f2,,(P,

=

J G z exP(-

d = 2 / J K P exP(-

f3d*(P?cp)

=2

/J=LP2

(15.4a)

P/Als), PlA2,*

exp(-dA3,,

fid, f 2id,

(15.4b) (15.4~)

where the A's are the variational parameters. We obtain the exciton binding energies in the diagonal approximation by minimizing the energy as a function of the 2s. Since the variational wave functions are decoupled in the in-plane and z coordinates, we expect to obtain an accurate solution only for quantum wells that are neither very narrow nor very large.19' As mentioned before, to take into account the coupling we project the hamiltonian in the basis described by Eqs. (15.4). By inspecting the exciton hamiltonian and taking into account the z-symmetry of the problem, we observe that the El - HH,: 1s exciton is coupled to the El - LH2:2p exciton

376

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by the term B and to the El - LH,:3d exciton by the term C . In the same way, El - LH,: 1s is coupled to E l - HH2:2p and El - HH,:3d. Depending on the energy difference between HH, and LH,, the E l - LH,:Is exciton might lie in the continuum of the E l - HH, excitons. In this orthogonal basis we neglect any effect of this continuum. We will come back to this point later. For one-photon spectroscopy only the exciton states with a nonzero amplitude at vanishing electron-hole distance are observable (i.e., the ns excitons in the decoupled approximation). It is the coupling among the 1s and 2p, 3d components that allows the 2p and 3d excited exciton states to be observable, since they carry then a part of the ls-type wave function. Notice that care has to be exercised about the valence band mixing-induced observability of excitonic transitions that are forbidden in the decoupled exciton approximation (say a d-like or p-like exciton). A lucid discussion of the oscillator strengths of excitonic transitions taking the band mixing into account has been given in Ref. 205. b. Nonorthogonal Basis As mentioned before, to obtain an accurate description of the quantitative effects of the off-diagonal terms in the Luttinger hamiltonian, we need to include the continuum. This is partially taken into account by simulating this continuum by a few discrete states. In other words, for each exciton state, we expand the in-plane wave functions into a number of nonorthogonal bound states of the type of Eqs. (15.4) with different length parameters (A’s). The inclusion of the 4f exciton states was shown to be unnecessary. We calculated the exciton states by considering the axial approximation [i.e., we put y 2 = y 3 in Eq. (15.3c)l. The parameters varied from 1= 100 to 1 = 300A for the s states, R = 200 to 400 A for the p states, and 1= 400 to 600 A for the d states. We consider four nonorthogonal in-plane states for each En- H,: j exciton, with R varying between the extreme values in geometrical progression. We tested the accuracy of our basis, and we found a precision of at least 0.1 meV. c. Results In Fig. 89 we plot the El - HH, :Is exciton binding energy as a function of the quantum-well thickness, for a GaAs-Gal -,Al,As (x = 0.3) quantum well. The parameters used are m, = 0.067m0, y , = 6.85, y 2 = 2.1, and y 3 = 2.9. The dielectric constant, IC, is 12.5. The barrier potential is given by V,= 1247x meV, with 60% for the conduction band and 40% for the valence band. The dashed-dotted line is the exciton binding energy using the orthogonal basis in the diagonal approximation. The dashed line is obtained by using the nonorthogonal basis in the diagonal approximation. Finally, the dotted lines and the solid lines are the solutions with the full Hamiltonian taken in account and where orthogonal and nonorthogonal bases, respectively, are

ELECTRONIC STATES IN SEMICONDUCTOR HETEROSTRUCTURES

-

-

70

110

L

150

6)

I9O

377

FIG. 89. Calculated HH, - E , : l s exciton binding energy versus the GaAs slab thickness L in GaAs-Ga,,,AI,,,As quantum wells for several approximations. Dashed-dotted line: orthogonal basis, diagonal approximation. Dotted line: nonorthogonal basis, diagonal approximation. Dashed line: orthogonal basis, full hamiltonian. Solid line: nonorthogonal basis, full hamiltonian.

used. The difference between the two diagonal solutions comes from the more flexible solutions allowed by the nonorthogonal basis (four linear variational parameters instead of one nonlinear one). The gain in the binding energy is significantly larger for the nonorthogonal basis than for the orthogonal one. This comes from the discrete states we introduced to simulate the continuum. The gain in the binding energy is, however, not very large, about 0.6 meV. In Fig. 90 we plot the El - LH,: 1s (a) binding energy and (b) the ratio of 1s characters in the solution, using the orthogonal basis; in Figs. 91a and b we use the nonorthogonal basis. In Figs. 90a and 91a the dashed (solid) line is the result obtained in the diagonal approximation (with the full hamiltonian).

Y h

w

1 9

X I 0.3

.u

c

.-

0.6

0.4

za 0.2 70 90 110 130 150 l70 l90

L&

FIG.90. (a) Calculated LH, -El: Is exciton binding energy versus the GaAs slab thickness L in GaAs-Ga,,,AI,,,As quantum wells using the orthogonal basis. Dashed line: diagonal approximation. Solid line: full hamiltonian. There is an anticrossing between the L H , - E , : l s and the HH,-E,:3d excitons. (b) Relative LH,-E,:ls weight in theexciton wavefunction for the two interacting L H , - E , : l s and HH,-E,:3d excitons. The solid and dashed lines, respectively, correspond to the upper and lower branch of the interacting excitons.

378

G. BASTARD et al.

FIG. 91. (a) Calculated L H , - E , : l s exciton binding energy versus the GaAs slab thickness L in GaAs-Ga,.,AI,,,As quantum wells using the nonorthogonal basis. Dashed line: diagonal approximation. Solid lines: full hamiltonian. There are anticrossings between the LH, --E,:ls and the HH,-E,:nd excitons n = 3 , 4 , 5 , 6 . (b) LH, -E,:ls relative weight in the exciton wave functions of the interacting LH, - E , : l s and HH,-E,:nd, n = 3 , 4 , 5 , 6 excitons. The dashed line labeled Is corresponds to the lower branch of the five interacting states. The solid lines labeled 6d, 5d, 4d, 3d refer to the four other interacting states.

Using the orthogonal basis, only the E l - LH,: Is and the El - HH:3d states anticross, the heavy hole exciton continuum not being described. The anticrossing is quite abrupt, with the wave functions being interchanged quite sharply. The situation for the El - LH, :1s exciton state is more complicated when we include a description of the continuum. For small values of the quantumwell width, the El - LH,: Is exciton is degenerate with the continuum of the E , - HH, excitons, although it is still possible experimentally to observe pronounced El - LH,:ls resonances. In our calculations the continuum is simulated by the very weakly bound states, E l - HH,:4d, E , - HH,:Sd, and El - HH1:6d. Since these states are strongly dependent upon the basis we use, the strong mixing between the light hole exciton with them is an artefact of the limited basis. Far from the anticrossing region between these states, the light hole exciton shows a clear behavior. The actual cases should be described with a very large basis of weakly bound and unbound exciton states to obtain an accurate description of the continuum where the light hole is degenerate. However, for numerical reasons, this procedure is not practical. We can get a good approximation of the resonance positions by averaging the binding energy with the oscillator strengthzo5(not shown here). We have found that the anticrossing between the nd and the 1s states is rather smooth.

ELECTRONIC STATES IN SEMICONDUCTOR HETEROSTRUCTURES

379

With a very good sample it is possible to differentiate between both states2" Finally, we observe again that the light hole 1s binding energy increases with the inclusion of the off-diagonal terms. This increase is larger for the light hole exciton than for the heavy hole exciton because of the stronger nonparabolicity of the LH, subband. d. Excitonic Transitions in the Presence of an Electric Field in Single Quantum Wells One can probe the coupling among the different excitonic states by applying external fields. This has been demonstrated by several authors, by applying an electric field,211a magnetic field,'12 and an external stress.213As an example, we discuss the case of an electric field applied along the zdirection. The first effect of the electric field is to weaken the excitonic oscillator strength and to red-shift the transition energy of the ground exciton state, since the electrons and holes are polarized toward opposite directions of the quantum we11.116*123 The second effect is the breaking of the z-symmetry. It induces couplings between other excitonic states besides those described. For instance, the E l - LHl:ls exciton is now coupled with the E l - HHl:p-type exciton states by the B term [Eq. (15.3b)J This adds a further difficulty to our problem, since for certain values of the quantum-well width and/or the electric field the light hole exciton is degenerate with both the p-like and the d-like heavy hole exciton continuum states. However, it is still possible to gain insight into the resonance positions, which provides a good comparison with the experimental results, as shown by Bauer and Ando.'02 Figure 92 shows the excitonic transitions, and Fig. 93 the oscillator strengths as a function of the electric field for a 160-A-wide GaAs-Ga(A1)As quantum well (x = 0.30). The calculations have been performed by using the full exciton hamiltonian and the nonorthogonal basis. All the other parameters are the same as in Fig. 89. As we increase the electric field, the E l - HH,:p-type exciton states increase their coupling with the E l - LH,: Is exciton. They strongly interchange the wave functions (crossing in the oscillator strengths). At even higher electric fields, the light hole exciton gets closer to the 3d heavy hole exciton and finally enters in the heavy hole exciton continuum. Although the anticrossing picture is quite complex, the light hole 1s exciton resonance remains observable. Also, because the 2p and 3d heavy hole excitons keep some of the 1s light hole character, it is possible to observe *"L. Vina, R. T. Collins, E. E. Mendez, and W. I. Wang, Phys. Reu. Lett. 58, 832 (1987). 212WOssau, B. Jakel, E. Bangert, G. Landwehr, and G. Weimann, Surf. Sci. 174, 188 (1986). 'I3C. Jagannah, E. S. Koteles, J. Lee, Y . J. Chen, B. S. Elman, and J. Y . Chi, Phys. Reo. B34,7027 (1986).

380

G. BASTARD et at.

0

I

10

I

20

'.

30

F (kV/cm)

I

I

F (kV/cm)

FIG. 92. Calculated electric field dependence of several excitonic transitions in a 160-A-thick GaAsGa,,Al,,,As quantum well. For such a thickness there exist pronounced anticrossings between the LH,-E,:ls exciton and the H H , :np, md excitons. Nonorthogonal basis, full hamiltonian. The labels HI,, H,,, H,,, H,,, L,, refer to the dominant component of the exciton wavefunction at F = O . The triangles correspond to the experimental data of Ref. 211.

I

FIG.93. Calculated electric field dependence of the oscillator strengths (in arbitrary units) of some of the excitonic branches displayed in Fig. 92. The labels H I , , H,,, H,,, H,,, H,,, L,, refer to the dominant component of the exciton wavefunction at F =O.

ELECTRONIC STATES IN SEMICONDUCTOR HETEROSTRUCTURES

38 1

them, as shown by Vina and co-workers.’” It can be seen that the description of the experimental findings by our theory is satisfactory. Yet, the simulation of the heavy hole continuum by a few discrete states is, we believe, makeshift. In reality, there are two different situations corresponding respectively to the anticrossings between the discrete El- LH,: Is and the E , - HH,:nd (or mp) discrete (but increasingly denser) states, on the one hand, and to the anticrossings between the discrete El - LH, :1s and the E , - HH, dissociated exciton states, on the other hand. Our approximate treatment is acceptable for the discrete part of the spectrum. The fact that it seems to account correctly for the position of the LH-originating resonance in the E , - HH, continuum is somehow unexpected. This might indicate that the shift of the El - LH, :1s resonance due to the E , - HH, continuum is a small effect (compared with the repulsion due to the discrete E , - HH,:nd states) and that the main result of the interaction with the continuum is a broadening of the El - LH, :1s level into a resonance. VIII. Quasi-One-Dimensional Systems

Previously we have considered the carrier confinement effects due to the one-dimensional modulation of the band edge profile in superlattices and quantum wells. The large number of fundamental and applied results has led to the search of carrier confinement in additional directions of the semiconductor crystals. We call these lateral confinements to distinguish them from epitaxial confinements, for which the state of the art is quite advanced. Besides the challenge in itself for crystal growth, promising results already obtained in quasi-2D semiconductor physics allow us to expect several new effects when additional carrier confinement comes into play. Since the first attempt by Petroff et ~ l . , ’many ~ ~ laboratories worldwide have worked on lateral confinement. Different techniques have been employed, such as the A1 diffusion,’15 etching, and cutting te~hniques,’~~-’’~ tilted growth,’” and 214P. M. Petroff, A. C. Gossard, R. A. Logan, and W. Wiegman, Appl. Phys. Lett. 44,635 (1982).

215J. Cibert, P. M. Petroff, G. J. Dolan, D. J. Werder, S. J. Pearton, A. C. Gossard, and J. H. English, Superlattices and Microstructures. 3, 35 (1987). 216M.A. Reed, J. N. Randall, R. J. Aggarwal, R. J. Matyi, T. M. Moore, and A. E. Wetsel, Phys. Rev. Lett. 60, 535 (1988). 217T.P. Smith 111, H. Arnot, J. M. Hong, C. M. Knoedler, S. E. Laux, and H. Schmid, Phys. Reo. Lett. 59, 2802 (1987). 218T.Demel, D. Heitmann, P. Gambow, and K. Ploog, Appl. Phys. Lett. 53, 2176 (1988). 219K.Kash, A. Scherer, J. M. Worlock, H. G. Craighead, and M. C. Tamargo, Appl. Phys. Lett. 49, 1043 (1986).

220M.Tsuchiya, J. M. Gaines, R. H. Yan, R. J. Simes, P. 0. Holtz, L. A. Coldren, and P. M. Petroff, Phys. Rev. Lett. 62,466 (1989); M. Tsuchiya, J. M. Gaines, R. H. Yan, R. J. Simes, P. 0. Holtz, L. A. Coldren, and P. M. Petroff, J. Vac. Sci. Technol. B7,315 (1989).

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G. BASTARD et al.

t' 0

X

FIG.94. Cross section of a quantum wire. The wire axis lays along the y-direction. The inner part of the wire is defined by the intersection of z = O and q ( x , z)=O.

strain confinement.221Although the dimensions and interface quality of the lateral confinement is still not comparable to the epitaxial quality, several pieces of evidence of further confinement have been obtained. According to the type of growth, different spectroscopic techniques, from transport to optics, were employed. Samples where the electrostatic potential originates from the charge transfer have the holes and electrons spatially separated. Transport techniques, like capacitance2l 7 and tunneling,21 or intrasubband optics, like infrared absorption,222are more appropriate. On the other hand, when it is possible to have both carriers close spatially, luminescence and luminescence excitation223and cathodoluminescence214~2'5spectroscopies are the more popular ones. It is not within the scope of this review to describe the extensive research in the growth, characterization, and measurements of laterally confined semiconductor heterostructures. We will focus on a few theoretical aspects and will mention experimental findings whenever a comparison between theory and experiment is possible.

16. QUASI-DECOUPLING OF THE

WIRE

EIGENSTATES

In this section we discuss the effect of lateral confinement in one direction (here, the x-direction). Thus the carrier motion in the y direction will be free and will exhibit a dispersion (see Fig. 94). This sort of geometry will be referred to as quantum wire geometry. It is quite disappointing that the addition of further degrees of confinement to the semiconductor heterostructures adds difficulties in the numerical calculations of the eigenstates for such systems. This can easily be observed "'K. Kash, A. S. Gozdz, D. D. Mahoney, J. P. Harbison, R. Bhat, J. M. Worlock, B. P. Van der Gaag, P. S. D. Lin, A. Scherer, L. T. Florez, and M. Koza, SurJ Sci. (1990), in press; K. Kash, R. Bhat, D. D. Mahoney, P. S. D. Lin, A. Sherer, J. M. Worlock, B. P. Van der Gaag, M. Koza, and P. Grabbe, Appl. Phys. Lett. 55, 681 (1989). 222W.Hansen, M. Horst, J. P. Kotthaus, U. Merkt, Ch. Sikorski, and K. Ploog, Phys. Rev. Lett. 55, 2586 (1987). 223J. S. Weiner, G. Danan, A. Pinczuk, J. Valladares, L. N. Pfeiffer, and K. West, Phys. Rev. Lett. 63, 1641 (1989).

ELECTRONICSTATES IN SEMICONDUCTORHETEROSTRUCTURES

383

even for the simplest cases. Let us, for instance, consider the solutions of the Schrodinger equation with a barrier potential in two directions (z, x):224

Y(x, z) = & Y ( X , z).

(16.1)

We already know that a similar equation in one dimension, although a textbook type, is a good starting point for the description of the energy levels in the presence of a confining potential in the effective mass approximation for quasi-2D heterostructures. However, when the confinement is extended to an additional dimension, we can find an exact solution only in a very few special cases. The whole problem depends on the coupling between the z and x motions by the barrier potential. Even for the simplest case, a square-type confinement, the actual finite values of the potential prevent a search for exact solutions. Only when V(x,z) has very well defined symmetries and infinite height are we able to find the analytical solutions of Eq. (16.1). Some of these exact solutions (for example, of the parabolic potential or the infinite square-well potential) might be useful to obtain preliminary insight into the search for more accurate solutions for actual cases. The rectangular-type potential is an important one for describing the cases where the difference between the epitaxial confinement and the lateral one is large. The parabolic potential, as we will discuss, might be the approximate shape of the confinement when charge transfer plays the crucial role. However, the finite values of the actual potentials prevents the existence of analytical solutions. Full numerical calculations are one way to overcome the difficulties mentioned. This was the approach followed by Laux and Stern.225They solved the 2D and 3D dimensional Schrodinger equation numerically, including also self-consistency for the charge transfer. They found that, for the samples of Ref. 217, the lateral confinement evolves from a parabolic one to a quasi-rectangular one with increasing carrier density in the quantum wire. Although this method is successful, since it is able to simulate different semiconductor structures, it requires a large amount of computational work. Other methods have also been d e ~ e l o p e d . ~As ~ ~mentioned, -~~~ the existence of approximate symmetries suggests the use of the solutions of 224A. Messiah, “Quantum Mechanics.” North-Holland Amsterdam, 1966. 225S. E. Laux and F. Stern, Appl. Phys. Lett. 49, 91 (1986). 226K.Kojima, K. Mitsunaga, and K. Kyuma, Appl. Phys. Lett. 55, 882 (1989). ”’K. B. Wong, M. Jaros, and J. P. Hagon, Phys. Reo. B 35, 2463 (1987). 228Y.-C.Chang, L. L. Chang, and L. Esaki, Appl. Phys. Lett. 47,4324 (1985). 229 J. A. Brum, G. Bastard, L. L. Chang, and L. Esaki in “Proceedings of the 18th International Engstrom, ed.), p. 505. World Scientific, Conference on the Physics of Semiconductors”(0. Singapore, 1987.

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idealized problems as the tools to develop approximate solutions either by diagonalizing the difference between the actual and the model potential in a basis spanned by the solutions of the ideal problem or by generating variational solutions of the actual problem. The latter method is limited by its difficulty in handling the excited states.224 Since these states are more important here than in the quasi-2D case, due to the weakness of the lateral confinement, the former approach looks more promising. Most of the structures so far available consist of a strong epitaxial confinement in one direction (z) on which is superimposed a weak and not always well-defined confinement in the x direction. We should expect that both directions create quantitatively different constraints to the carrier motion. We can then redefine our problem as consisting of a confinement along the z direction plus some perturbative confinement which is z and x dependent:228- 29 V(X, Z) = Vb(z) + AW(X, Z).

(16.2)

The solution will be assumed to have the form (16.3) where xn(z) is the nth solution of the one-dimensional Schrodinger equation for the z motion, and ky is the wave vector characterizing the free motion along the wire axis. a,(x) is the solution of the set of coupled Schrodinger equations

(16.4) where En are the z-related eigenvalues. With this approach we can, in principle, solve for any shape of two-dimensional confinement. The success of the method depends on the relative strengths of the confinement along z and x. Basically, the approximation will work the better when Aw(x, z) is small compared with V, in the (x,z) regions where Vb(z)is large. As a first step, we can solve the above equation in zeroth order in Aw(x, z). In this case, each z-related subband generates a family of x-related solutions that are the solutions of the effective hamiltonian: H.

= En

+ T, +

s

x;(z)Aw(x, z) dz

+ hzk:/2m*.

The eigenvalues of the whole hamiltonian

(16.5)

are thus given by

ELECTRONIC STATES IN SEMICONDUCTOR HETEROSTRUCTURES

+

385

+

tn,,,(a)(k,,)= En &,,,(a) h2k,2/2m*. Although the x-related subbands are implicitly dependent on the z-related motion, the eigenfunctions are factorized in x and z. Eventually, some accidental degeneracies between 1D subbands associated with different z-related levels [ E , E , , ( ~ ) = E j E , , , ~ ) ] show up. They are lifted, symmetry permitting, by the inclusion of the coupling among different subbands: (i, n(i) I Aw(x, y ) I j , m ( j ) ) . As expected from our initial assumption, we find that our decoupling procedure will be more valid when the size quantization along x is smaller than along z. We have checked the validity of the decoupled approximation by considering the case of rectangular quantum wires (see Fig. 95). The barrier potential is

+

V(X,

[

z ) = Vb 1 - Y(?

-

2 ) Y($

-

;.)I,

+

(16.6a) (16.6b) (16.6~)

where vb is the barrier height, Y(x) is the step function (Y(x) = 1 if x > 0, Y(x) = 0 if x < 0). In zeroth order the x-dependent solutions are given by

+ Vk"lf(x)I~rn(x)= Em(n)arn(x)>

CTx

where Vk")(x)=

v,

i s 1-

IZI bL,P

&z) dz} Y(x2

-?).

(16.7)

(16.8)

We plot in Fig. 96 the zeroth-order eigenstates as a function of the lateral barrier width, L,, for a rectangular wire of GaAs surrounded by Ga,,,A10,2As and with L, = lWA. We observe clearly the existence of two families of solutions that are associated with each of the z-related subbands (for the parameters we used there are only two bound states in the z direction). This

c _ _

L, V(1,Z)

=

Vb(2)

+

w(v)

FIG.95. Splitting of the rectangular wire problem into the sum of a quantum-well problem and of a small (x, z)-dependent potential. The hatched areas correspond to a vanishing potential energy.

386

G. BASTARD et al.

200

-5

"b

u

t

a

f5100 z W

FIG. 96.

Energy levels of GaAsrectangular wires versus L;LZ = loo.&. Decoupled approximation. The two horizontal dashed lines correspond to the energy location of the E , and E , bound states at infinite L,. Ga,.,AI,,As

0 100

50t

1000

Lx ( A )

zeroth-order approximation is excellent whenever L, << L,. The Schrodinger equation is effectively separable. Because of the symmetry of the potential, the degeneracies shown in Fig. 96 remain even beyond the zeroth-order approximation. Further improvements in the eigenstates are obtained by including the solutions related to the continuum of the z-motion. When L, z L,,we should go beyond the zeroth order of approximation. Let us consider now the density of states (DOS) for these structures: g(E)=2

(16.9)

2m*

Here, the factor 2 accounts for the spin degeneracy. If we consider energies smaller than E , , the DOS becomes g(4 = 2 71h2 m

,*, 2m*L; L x (E L El y- E,) ~ Y ( E/- E~l - E,),

(16.10)

where L, is the wire length and n replaces n(1). The total number of states existing below E is

N ( E )= 4

m

~

~

~

L

y - El ~

-

6,). /

~(16.11) Y

When the number of bound states is large (in practice, L, + 00 and E, + 0), we can derive an expression for the 1D DOS related to the 2D DOS with the

(

~

ELECTRONIC STATES IN SEMICONDUCTOR HETEROSTRUCTURES

387

( E - E 1 )(rneV) FIG. 97. Calculated energy dependence of the one-dimensional density of states in a flat

(L,>> L,) quantum wire for three values of the phenomenological broadening parameter m,. e l = 1meV.

help of Poisson’s summation formula. In this case, for E, z h2x2n2/2m*LI= ElnZ, we have

E

>> e l , where

(1 6.12)

where gZD is the two-dimensional density of states xh2] and

r

I

k2-2 I1 IL

I

[g2D

= m*L,L,/

(16.13)

There is an analogy between Eq. (16.12) and the Fourier expansion of the magnetoconductivity tensor, J , playing the part of a cosine. We can pursue this analogy by introducing a phenomenological damping and write

To illustrate this approach, we plot in Fig. 97 the value of g/gZD as a function of the energy for = 1 meV and three values of the damping

388

G. BASTARD et al.

parameter m,. For rn, = 00 we have the well-known 1D DOS. As the broadening increases, by decreasing m, , the strong singularities are smeared out. For m, = 1, the singularities are quite rounded off, and the 1D DOS shows only weak oscillations around the 2D DOS. The above simple models have introduced the problem of twodimensional confinement. For the conduction band levels we have used a parabolic bulk dispersion. As discussed previously, the nonparabolicity plays a more important role for higher-energy states. For present actual structures, the states of interest are close to the first z-related subband, and the parabolic approximation is quite justified. The more complex case of the valence band will be considered later (Section 19). We now have a method to calculate the eigenstates for a quasi-1D electron gas with a parabolic band. It can be employed to study more complex situations such as those created by charge transfers in modulation-doped structures, external fields, and other situations. This framework is the one we shall use for studying the laterally confined structures. 17. ONE-SIDE ESPIKE-AND MODULATION-DOPED QUANTUM WIRES

The study of the charge transfer in low-dimensional structure is an important one. First, we can confine the carriers laterally by transferring the charge spatially, as in the quantum wires grown by the combination of the molecular beam epitaxy and lithography.21 Also, the hope of achieving very high m o b i l i t i e ~ by ~ ~confining ~ - ~ ~ ~even more the carriers compared with the quasi-2D heterojunctions has been one of the motivations in the search for quasi-1D electron gases. We consider the case of a rectangular GaAs wire embedded in a matrix of Gal -.Al,As. The donors are assumed to be placed on a sheet separated by a distance d from the closer interface. We consider a uniform doping along the x and y directions; the donors are depleted along a distance 1, (see Fig. 98). The Fermi energy has to be the same through the entire structure to ensure

Id

;H++++++++;

4

FIG.98. Cross section of a one side, n-spike, modulation-doped quantum wire. /x is the donor depletion length, and d is the distance separating the donor plane from the x side of the wire.

ELECTRONIC STATES IN SEMICONDUCTOR HETEROSTRUCTURES

389

thermodynamic equilibrium. Some donors will ionize, transferring their electron to the GaAs region. This will create a band bending that will lower the Ga,-,Al,As conduction band edge and, thus, the Fermi level in the donor region. On the other hand, the GaAs levels will move up because of the electron-electron interaction, treated here in the Hartree approximation. Being a repulsive potential, it will spread the electrons in the well and push up the bound levels as well as the Fermi level in the wire region. The transfer will proceed up to the equilibrium. The electrostatic potential created by the charge transfer, - ecpsc(x,z), has to be added to the Schrodinger equation. This potential is obtained by solving the Poisson equation. Both equations are then solved self-consistently. The ionization of the donors in the barrier creates a positive density of charges given by

+ d + LZ/2)Y(1:/4 - x2),

p+(r) = eN,d(z

(17.1)

and the electrons that occupy the quantum wire states create a negative density of charges equal to

where f D stands for the Fermi-Dirac consistent potential can be written as

1

+m

-

X:(z)exPc-4q,lz

-

distribution function. The self-

z'lldz'

cli(x')cos qxx'dx',

(17.3)

-OD

where IC is the dielectric permittivity of the heterostructure and the an(x))sare the solutions of

{&:+'1 =(&

dz -

XI(.)

El

[

VbY (x2 -

-G

cl,(x).

$) Y ($

-

z2)

- eqsc(x,]) z

cl,(x)

(17.4)

We have solved the above equations by projecting the entire hamiltonian onto the basis generated by the eigenfunctions, m:(x), the solutions of Eq. (17.4) under flat-band conditions (q,,= 0). The eigenvalues were obtained by

390

G. BASTARD et al.

I

300-

I

Lz-1008

d-1008

-

iterating to selfconsistency. The neutrality of charge has been used in Eq. (17.1): 2

Nd1, = -

1

7111

(17.5a)

kF,,

kFn = J(2m*/h2)(&,

-

El

- 8.).

(17.5b)

The Fermi energy, cF, being constant throughout the heterostructure, should take a value in the well that coincides with the value in the donor part: E~ =

El

h2kin + += V, - R* 2m* E,

-

(i, -2).

eqsc

-d

(17.6)

The charge transfer will depend on sevefal parameters: the barrier height, the x and z confinement lengths, the separation between the donors and the wire, and the donor concentration. We performed the calculations for V, = 100meV, m* = 0.07m0, R* (donor binding energy) = lOmeV, L, = lWA, and a sheet donor concentration nd = 101Zcm-2. Figure 99 shows the depletion length I, as a function of the lateral width L,. One notices that the transfer is never complete, since ,Z < L,. The equality between ,Z and L, is probably recovered at very large L,, when many x-related subbands are occupied and the wire practically becomes a 2D system. The curve exhibits kinks any time a new x-related subband is occupied. These weak singularities reflect the square-root-like behavior of the occupancy of a given one-dimensional subband versus the Fermi energy. In Fig. 100a, we show the self-consistent potential for L, = 150& 250A. As we can see from Fig. 99,

200

-

50 N

-200

-200

0

200

FIG. 100a. Contour plots of the electrostatic potential energy in a GaAs-Ga,,,AI,,,As modulation-doped quantum wire. L,= lOOA. N d =10L2cm-Z.d = lOOA and three different values of L,:L,= l 5 0 & 250& 350A.

392

G. BASTARD et al.

1(

L I Z ,5108 -Lz.lOOA

-

h

4

N

1 Oo

-1 (

I -100

I

I

0

100

X ( h

FIG. 100b. Contour plot for the charge distribution in a GaAs-GaO,,A1,,,As modulationdoped quantum wire. L, = loo& L, = lSOA, N , = 10'2cm-2.

these L, values correspond to the occupancy of one, two, and three 1D subbands, respectively. In general - e(psc(x,z ) shows an asymmetric dipole shape. This is to be expected since the positive charges are delta functions distributed along the z axis, whereas the negative charges are spread out on the wire cross section, as we can see in Fig. 100b. In Fig. 101 we plot the energy levels and the Fermi level as a function of L,. The origin of energies is chosen at the corners of the rectangle closer to the sheet of donors ( z = - L,/2, x = fL,/2). As more levels get occupied, the interband separation decreases. The reason is that L, increases. The Fermi level shows small accidents when a new subband is occupied but remains essentially constant, at about 85 meV. The modulation-doped structure we have discussed is quite idealized. In practice, we should expect fluctuations in the doping densities. Also, constrictions along the wire axis from the growth process are expected. For the actual cases of interest such fluctuations and their influence on the energy levels, charge transfer, and transport properties have to be considered. Finally, note that for the typical parameters used here, the excited subbands in the quasi- 1D heterostructures are easily occupied; that is, the electric quantum limit is not reached. This implies that intersubband ~ c a t t e r i n g ~ ~will ' - ~ be ~ ~ operative and that the high mobility figures 230H.Sakaki, Jpn. J . Appl. Phys. 19, L735 (1980); J . Vuc. Sci. Technol. 19, 148 (1981). 231J. Lee and H. N. Spector, J . Appl. Phys. 54, 3921 (1983). 232G.Fishman, Phys. Rev. B 34, 2394 (1986).

ELECTRONIC STATES IN SEMICONDUCTOR HETEROSTRUCTURES

150

-

2E

d.lOOi

Nd=1012d2)

I

v

W

Lz-lObh

393

I

100

n=l

1

200

Lx

(A,

300

400

FIG. 101. Calculated eigenenergies versus L, in a GaAs-Gao,,Al0,,As quantum wire. The parameters are the same as in Fig. 100a.

associated with the quenching of the available final states for elastic transitions will not easily be reached. 18. MAGNETOELECTRIC SUBBANDS IN A QUASI-1DELECTRON GAS As discussed earlier, the application of external fields is a very important tool for studying the electronic structure and for looking for new effects and applications. Here, we follow the same path as in the study in quasi-2D electron gases. In the latter systems, the application of a tilted magnetic field with respect to the growth axis was decisive in establishing the existence of a 2D electron gas through the anisotropy of the magnetic-field-dependent spectrum.’ The energy spectrum of a 3D electron gas subjected to a constant magnetic field B is organized into Landau levels. Since for a bulk system all the directions are approximately equivalent (for the spherical parabolic dispersions we are considering here they are exactly equivalent) the effect is independent of the direction of the magnetic field. Surface effects will only be present at very weak fields, when the finite dimension of the crystal plays a role. The 2D electron gas instead has a cylindrical symmetry, arising from the 233F.F. Fang and P. J. Stiles, Phys. Reo. 174, 823 (1968).

394

G. BASTARD et a1

existence of a preferential axis (the growth axis). The magnetosubbands show a completely different behavior, depending on the direction of the magnetic field. If B is along the growth direction, the Landau levels are again independent of interface effects. If B has a nonzero component in the 2D electron gas plane, this component gives rise to a diamagnetic Zeeman shift, and only the component parallel to z enters in the Landau level splitting. For the 1D electron gas the situation is even more drastic: the carriers are confined in two dimensions, and one cannot expect any interface-free Landau levels. Again, the magnetosubbands reflect the symmetry of the heterostructure. However, in the 1D case, it is the difference in the confinement strengths that will give rise to some interesting effects. The electric field plays a different role in the low-dimensional heterostructures. Instead of increasing the confinement by the cyclotron motion like the magnetic field does, the electric field, when applied along a direction of confinement, weakens the confinement. For quasi-independent confined directions, the electric field effects are not expected to differ significantly from those found in the 2D electron gas: the electric field leads to a carrier accumulation toward one edge of the quantum well. As discussed in Section V, the levels, although formally resonances (due to the field-induced tunneling across the finite barrier), can be considered as discrete to a good accuracy if the field is weak enough. The effects of the external fields are strongly dependent upon their direction. If they are applied along the directions of confinement and if they show different strengths (as in epitaxial and lithographic confinements), the effects would be quite sensitive to the difference of scale in the confinements. On the other hand, if the fields are applied along the remaining free direction, a qualitatively different behavior is expected. For instance, let us consider a magnetic field applied on a quantum wire with anisotropic confinement: L, << L,. If B is applied along x(z), the carriers tend to develop cyclotron orbits in the y,z (x,y) plane. Thus there is no longer a free direction. The energy levels will depend on the relative magnitude of the magnetic field effect (the cyclotron energy ho,), and the size quantization along x or z. If, on the other hand, B is applied along the free direction, the magnetic field couples both confined directions, and the carriers remain fundamentally onedimensional. This 1D spatial anisotropy has been successfully exploited by Smith and c o - w o r k e r ~to~ ~evidence ~ the quasi-1D behavior of heterostructures grown by epitaxial and lithographic techniques. We focus on a particular case, which corresponds to B 11 z (narrow-well width), while the electric field F is along the x-direction (large-well width). In this configuration, the magnetic and electric fields effects are coupled as well as the confinements. The other configurations can be studied in a similar way but will not be discussed here.

ELECTRONIC STATES IN SEMICONDUCTOR HETEROSTRUCTURES

395

In general, the hamiltonian in the presence of the electric and magnetic fields can be written in the effective mass approximation as

H

1 2m*

= - {p - eA/c}’

+ V,(z) + Aw(x, z) + eFx.

(18.1)

The spin contribution can be included by adding g*,uBa’B to the eigenenergies of Eq. (18.1) (where g* is the effective Lande g factor, pB is the Bohr magneton, and the operator a has the eigenvalues f 1/2). In the decoupled approximation, the z-motion is not altered by the external fields. We have to study the x- and y-dependent hamiltonian for the states belonging to the ground z-related subband. By defining the cyclotron frequency o, (where w, = eB/m*c) and the magnetic length I , (where A: = hc/eB), and using the Landau gauge {A = (0, Bx, 0)}, the hamiltonian can be written as

+

PZ

+

HI = El 7 v$i(x) 2m

1 + -m*w,(x 2

- x0)’ - -

2

(1 8.2) and,

xo = - I: ky - eF/m*oz,

(18.3)

where xo gives the position of the center of the classical orbit, and Vk:i(x) is given by Eq. (16.8). Since p , commutes with the hamiltonian, we can search for solutions having the form

Y(x, y) = (L,) - ‘1’ eikyyq(x).

(1 8.4)

We have calculated the subbands of HI by projecting it on the basis spanned by the bound levels obtained at the zeroth order of approximation with both F and B equal to zero.235We have also calculated the density of states for this problem, assuming that the eigenstates are phenomenologically broadened, namely by replacing the delta functions by gaussian ones:

(18.5) 234T.P. Smith, 111, J. A. Brum, J. M. Hong, C. M. Knoedler, H. Arnot, and L. Esaki, Phys. Rev. Lett. 61, 585 (1988). 235J. A. Brum and G. Bastard, Superlattices and Microstructures 4, 443 (1988).

396

G. BASTARD et af.

where

r, = TI + rg,,

(18.6)

where P , is the integrated probability of finding the carrier in the barriers of the x-dependent quantum well for the nth x-related subband, and f, and are two energies assumed to be equal to 2 and lOmeV, respectively. This model intends to simulate the defects and/or segregated impurities accumulated in the interface produced by lithographic techniques. We plot in Fig. 102a, b the density of states at B = 0 for different electric field strengths and for L, = 200 A and 600 A, respectively. The density of states shows the singularities typical of one-dimensional behavior, similar to that discussed at the beginning of this section. The peak positions shift with the electric field according to the expected Stark shift (e.g., a red shift for the ground state). In the presence of a magnetic field, but with F = 0, there is a competition between the barrier confinement and the magnetic confinement. The energy

r,

UY

0

D

0.000 20

--- F=40kV/crn

I 1

1

I

80 E (meV)

60

40

I

I

100

120

0.0 2

o.ool..: 10

:

i

-' J

1

30

I

50

E (mcV 1 FIG. 102. (a) Calculated density of states for a GaAs-Gao.,A1o.,As quantum wire with L, = 100 ! Iand L, = 200 !Ifor an electric field of 10 kV/cm (solid line) and 40 kV/cm (dashed line). (b) Same as in Fig. 102a, but L x = 6 O 0 ~ .F = O solid line, F = 5 kV/cm: dashed line and F = 10 kVicm: dotted line.

ELECTRONIC STATES IN SEMICONDUCTOR HETEROSTRUCTURES

397

levels are governed by the position of the center of the classical cyclotron orbit, x,,, and the relative values for the barrier potential, the quantum-well width, and the magnetic field. For 1, >> L, the quantum-well confinement dominates. The dispersion is quasi-quadratic versus the position of the center of the orbit. The density of states recalls a quasi-1D one, and the magnetic field can be considered as a small perturbation. This is illustrated in Fig. 103 (upper panel), where we show the energy levels for L, = 200A and B = 10T as a function of k, (for the k, units used in this figure k, = & 1 implies xo = k L J 2 ) and in Fig. 104 (upper panel) where the respective density of states is plotted. As we consider larger values for L, (keeping the same magnetic field), the magnetic confinement increases. The energy levels become Landau-level-like. When the center of the orbit is close to the interfaces, the levels show a strong dispersion in k,, corresponding to the classical reflections by the barrier. These edge effects are similar to those found in Landau levels in quasi-2D systems.236The edge effects start to play some part whenever the distance of the center of orbit to the interface is equal

200

150 100 50

G.

103. Energy levels as a

units of L,/21; where 1, is the magnetic 1ength)in a L , ~ , o o oL2;100i ~ F,o B,lOT

398

G. BASTARD et a/.

0.1

0

01 0

0.3 0.2

0.1 0 20

40

E

100 FIG.104. Density of states for the same structures

60 80 (meV)

and parameters as used in Fig. 103.

to the cyclotron radius (=A,,/-, where n is the usual Landau level). Thus we obtain the criterion Ik,J < k,, where

22: ~

LX

k,

=

d G ,

1 -1, LX

(18.7)

to obtain dispersionless Landau levels. We observe the evolution of this behavior in Fig. 103, (middle and lower panels) and Figs. 104 (middle and lower panels), where we plot the energy levels as a function of k, for L, = 400A and lOOOA, respectively, and the corresponding densities of states. The dispersionless region increases with Lx, and the density of states gets closer to that of a 2D system (a series of delta functions), with a gaussian broadening, and an increasing quasi-Landau-level degeneracy. In the presence of crossed electric and magnetic fields, the effects are mixed together. The electric field pushes the center of orbit toward one of the interfaces. The dispersionless region is replaced by a k,-dependent dispersion that is linear in k, when Ik,l < k,(F), where (18.8)

ELECTRONIC STATES IN SEMICONDUCTOR HETEROSTRUCTURES

399

150 100

50 0

-,,5

-l,o

-o,5

k y (L,/2L;

o,5

1

1D 15

FIG. 105. Energy levels as a function of k, (in units of Lx/21i,where I , is the magnetic length) for the same structure as in Fig. 103 in the presence of crossed electric and magnetic fields (10 kV/cm and 10 T, respectively).

Essentially, when edge effects are unimportant, the dispersion shown by the center of the orbit reveals the quantum-well shape: rectangular when F = 0, and triangular for F # 0. When edge effects become important, the dispersion as a function of k , becomes strongly nonlinear, showing the effect of the barriers on the carrier motion. Figures 105 and 106 show the energy dispersion upon k, and the density of states, respectively, for the same structures as in Fig. 103, for F = 10kV/cm and B = 10T. Again, we see the evolution from a configuration where the size quantization dominates and where the external fields have little effect on the carrier motion (narrow well) to a situation where the magnetic field is dominant and the carriers are pushed toward the interface x = - L J 2 (wide wells). The density of states displays a completely different behavior, as shown in Fig. 106, reverting its shape to that of a 1D structure. The small differences with the purely parabolic dispersion DOS (Fig. 102) are smeared out by the broadening. 19. VALENCEBANDSTATES AND DISPERSION RELATIONS In previous sections we studied the energy levels of electrons confined by a two-dimensional potential. We considered a simple parabolic dispersion for the conduction band spectrum of the host. In reality, the conduction band

400

G. BASTARD et a/.

0.3 L1:200A Lz;lOO;

o.2 L,.lOOO

0

20

L,:lOOA

40

F=lOkV/mB.1OTI

F=iOWcmM O T

60

E (meV)

80

100 FIG.106. Density of states for the same structure and parameters as used in Fig. 105.

shows nonparabolic effects that have the same origin (k* p coupling between r6,r7,and r8),as discussed in the 2D gas (see Sections I11 and IV). Since the additional confinement is in general not as strong as the epitaxial one, the parabolic approximation is quite good for the first 1D subbands. This simplification is impossible for the valence band energy levels of quantum wires. As analyzed in Section 111, it is always possible to regard the valence states as originating from two decoupled hamiltonians (the heavy and light hole ones) for the envelope functions (at k , = 0) in quasi-bidimensional structures. These k , = 0 states are coupled among them and with the r, (split-off)band whenever we consider in-plane motion. When we add a lateral confinement, we localize the holes along a new direction (again, we call it the x-direction). The localized states for the x motion are composed of combinations of real (when we are in the well part of the potential) and imaginary (in the barrier part of the potential) parts of the x component of the hole wave vector, and, thus, by confining laterally the carriers, we are necessarily coupling the heavy and light hole Ts states and the r7states even at k, = 0. The situation becomes even more complicated when we consider the dispersion along the wire axis (k, dispersion).

ELECTRONIC STATES IN SEMICONDUCTOR HETEROSTRUCTURES

401

For states close to the top of the valence band, we expect that the coupling between r8 and r7will play a minor part. For the quantum wells (see Section IV), at k, = 0 the split-off bands are coupled to the light holes bands, leaving the heavy holes uncoupled. For the quantum wires, the additional lateral confinement couples all the bands together. Citrin and Chang237calculated the effect of the split-off bands for square-shape quantum wires and compared it with the same effect in quantum wells. They found small energy shifts for all subbands, with a larger effect for the deeper subbands. Here, we shall only take into account the heavy and light hole coupling in the 1D subbands. Following the effective mass approximation, the 4 x 4 Ts bands are described by the Luttinger hamiltonian. In the envelope function approach, the k, and k, wave vectors, in the confined directions, are replaced by the operators - i q a z and - ia/ax, respectively. The Schrodinger equation to be solved is238 CHrs

where -

+ (vb(z) + w(x, z))i]$ky

rc*

H,,

= &Qky,

(19.1)

0

and

(19.3~) b(k,,

$ m0

X , Z) = -( P , -

ihk,)y3pz.

(19.3d)

The wave function Ykyis a 4 x 1 column vector to be determined, as well as the eigenenergies. For the barrier potential we follow the decoupling procedure introduced for electrons [see Eq. (16.6)]. For different directions in 237D.S. Citrin and Y. C. Chang. Phys. Rev. B 40,5507 (1989).

238J.A. Brum, G. Bastard, L. L. Chang, and L. Esaki, Superlattices and Microstructures 3, 47 (1987);J. A. Brum and G. Bastard, Superlattices and Microstructures 4, 443 (1988).

402

G. BASTARD et al.

the x, y plane we have slightly different dispersions due to the warping of the host valence bands. Rigorously, we need to know the exact direction of the lateral confinement. For actual samples, so far, that is not an easy task, due to the imperfect control of the etching and lithography processes. Also, the warping is a small effect for the semiconductors considered. Thus, we have chosen to average the Luttinger parameters responsible for the x, y dispersions according to the spherical approximation used in the bulk materials (we just replace y z and y 3 by y = (2y, + 3 y J 5 in the c and b terms of Eqs. (19.3). With these approximations both x and y directions are equivalent, and the question about the lateral confinement direction becomes irrelevant. We have also neglected the eventual mismatches between the Luttinger parameters of the host materials. We look for the solutions of this problem, using the same procedure as followed in Section IV. We first generate a basis for a fixed value of k, (for convenience, we chose ky = 0, the zone center) and project the whole hamiltonian onto this basis. Since the basis only includes the bound z- and xrelated quantum-well states, the dispersion is more accurate for (k,) values that are closer to the zone center. Equations (19.3) show that even at k, = 0 the heavy and light hole subbands are coupled, as expected. The solutions of the k, = 0 problem are no more straightforward. We can still generate a basis by solving the diagonal part of the hamiltonian. In the decoupled approximation for this diagonal part, a basis is formed by the z-related subbands and their respective family of x-related subbands. These states have even and odd parity in x and z, according to the rectangular symmetry of the barrier potential. The motion in the y-direction is described by a plane wave. To obtain the solutions for any k, (including ky = 0), it is necessary to diagonalize the hamiltonian projected onto this basis. We can see that the term c in the Luttinger hamiltonian couples the heavy hole subbands with light hole subbands with the same parity in both z- and x-related states. On the other hand, the term b couples the heavy and light hole subbands with opposite parities. In this approximation, the hole wavefunction is written as

(19.4)

where xi(z) (qj(z)) are the heavy (light) hole eigenfunctions at k, = 0 for the z decoupled motion confined by the potential V,(z). &,,(i)(x)and CnG)(x) are the

ELECTRONIC STATES IN SEMICONDUCTOR HETEROSTRUCTURES

403

solutions of the zeroth-order diagonal part of Eqs. (19.2)-(19.3) for the x motion at k, = 0 for the heavy and light holes, respectively. The eigenvalues are obtained by projecting the whole hamiltonian onto this basis followed by a numerical diagonalization. First, let us consider the solutions of the quantum wire bound states, that is, the states at k , = 0. Since the coupling between the heavy and light hole states originates from the lateral confinement, we expect the coupling will decrease when the x confinement becomes weaker. This is exactly what we observe. In Fig. 107a we plot the first five hole states for a GaAs rectangular quantum wire with L, = 50A and barriers of Ga(A1)As with 20% of A1 as a function of the lateral confinement dimension (LJ.The dashed lines represent the z-related quantum-well states. In Fig. 107b we plot the corresponding values of which gives us a measure of the coupling between the heavy and light hole components. At small values of L, the coupling is more effective,and there is no state with a clear heavy or light hole nature except,

m,

0

-5.0

X=0.2

GaAr-Ga(A1)As

80

I

I

180

~

L,(A)

280

(a) I

380

FIG.107. (a) The energies of the first five hole states (at k, = 0) are plotted versus L, for a GaAsGa,,,AI,,,As rectangular quantum wire. L,= SO.& (b) The quantity is plotted versus L, for the first five hole states (at k,=O) of Fig. 107a.

404

G. BASTARD et al.

eventually, the ground state. As L, gets larger, which leads to a weaker x confinement, the 1D ground state evolves to the z-related heavy hole subband. Its character is increasingly heavy-hole-like. For the excited 1D subbands the picture is more complex. In the absence of lateral confinement, the ground light hole subband and the other excited subbands lie in the continuum of the ground heavy hole subband (HH,). It is this continuum that is broken down to form the x-related subband while being coupled to the light hole related states. Moreover, for the parameters of Figs. 107, the second z-related heavy hole subband (HH,) is close in energy to the ground z-related light hole subband (LH J, increasing the intensity of the coupling among the 1D states. All these aspects result in a rather complex picture. For values of L, close to L, the higher 1D subbands have a mixed character, most of them being predominantly heavy-hole-like (since they are built mostly out of HH, and HH,). One at least displays a prevalent light hole character. As L, increases, the states that were, at small L,, mostly HH,- and HH,-like, start to display a light hole character as they “cross” the energy region close to LH Subsequently, they recover a heavy hole character, now mainly related to HH,. This is indicated by the “hills” and “valleys” exhibited by m a s a function of L,. At very large values of L, they would constitute a quasicontinuum for the ground 2D heavy hole subband. Since the ground z-related light hole subband is degenerate with the continuum of HH,, even a small coupling is enough to mix their heavy and light hole characters. For more excited subbands the picture is even more involved. This complex mixing between the heavy and light hole characters should certainly influence the optical properties. Since the kind of mixing depends on the geometry of the confinement, detailed polarization spectroscopy can be a useful tool to study the lateral confinement (for the study of the 1D k, dispersion with a circular confinement, see Ref. 239). Notice that we have found a situation formally similar to the wire subbands in the study of the quantum-well excitonic transitions, taking into consideration the heavy and light hole coupling (Section VII). In that problem, the coulomb interaction plays the part of the additional confinement, although acting in both in-plane directions. In both the quasi-2D exciton and the quantum wire cases, the degeneracy of the ground light hole state with the ground heavy hole continuum complicates the interpretation of the results. The study of such effects requires very detailed spectroscopy combined with a very complete theory. As should be expected, one also needs to include the excitonic effects to study the optical transitions in the quasi-ID gas. To calculate the excitonic effects, the coupling between the heavy and light hole states has to be considered in the same way as above and as for the

,.

Z39M. Sweeny, J. Xu,and M. Shur, Superlattices and Microstructures 4, 623 (1988).

ELECTRONICSTATES IN SEMICONDUCTOR HETEROSTRUCTURES

405

quasi-2D exciton states (see Section VII). As far as we know a complete theory of the excitonic transitions in quantum wires is not yet available. Let us now focus on the k, dispersion of the quasi-1D hole subbands. In Figs. 108a-d we plot the k, dispersion for GaAs-Ga,,,Al,.,As quantum wires with L, = 50A and L, = (a) lOOA, (b) 150A, (c) 200& and (d) L, = l O O A and L, = 200A. In Figs. 109a-d we plot the respective values of as a function of k,. Finally, in Figs. 110a-d we plot their respective density of states, assuming a phenomenological gaussian broadening of r = 1 meV. As we noticed before, the first subbands try to follow the pattern of their z-related host subband. For instance, for L, = lOOA, the first 1D subband follows a k, dispersion that is nearly parallel to the HH, 2D subband. A more complex picture is obtained for the excited subbands. As we consider larger L, values, a complicated crossing and anticrossing picture with k, is of interest. As k, appears. The behavior of the variation of increases, the mixing of the heavy and light hole characters increases, as expected. However, for all values of 1D confinement considered here, the values of tend to converge to a common value for all subbands. This value is such that any heavy or light hole character is lost, and, as far as is concerned, the subbands are quasi-indistinguishable. For large values of L,, the 1D subbands form a quasi-2D continuum (see Fig. 108d). This complex pattern, with its anticrossing behaviors, gives rise to some subbands displaying a very heavy mass for their y motion and, thus, a large density of states. This is shown in Fig. 110,where one notices quite an uneven behavior in the DOS for several quantum wires with respect to the contribution of the different subbands. In contrast to the parabolic conduction band, where the DOS evolves smoothly from a typical 1D DOS to a quasi-2D DOS as L, increases, there exists in the valence DOS sharp peaks, which correspond to the subbands with a heavy mass or even a camelback shape for their k, dispersions. The existence and energy location of such peaks depend sensitively on the quantum wire parameters. Their relative intensities weaken when we consider a larger intrinsic broadening. Notice also that for large values of k,, the anticrossing region is largely absent and the dispersion regains a nearly parabolic behavior upon k,. Consequently, the density of states recovers a behavior similar to that of a wire with spherical, parabolic, and nondegenerate host bands. Although the above characteristics should have some influence on the transport and optical properties that include the valence states, the intrinsic broadening might mask some of their consequences. More conclusive observations need more detailed transport or optical studies. It is clear, however, that the 1D valence subbands are quite complex and present an intricate behavior. Nevertheless, these aspects once properly recognized

406

G. BASTARD et al.

1

0

c)

Lz:5Oi

L1=200i

0

0

ky (SI. I lo6 crn-‘ 1

1

ELECTRONIC STATES IN SEMICONDUCTOR HETEROSTRUCTURES

407

1.0 -

0.5

L,=50B 0

I

0.2

L,=100A I

0.4

X,0.2

I

I

0.6

0.8

1.0

0.5 0

L,,’iOA

L,=150d I

0.4

0.2

ky

k y ( K x106cm-’)

I

0.6

X,0.2 I

0.8

1.0

(Kx1O6 ern-’)

1.5

1.0

0.5

I

I

I

0.5

I

should stimulate further studies, in the search of a better understanding of the interesting phenomena present in the quasi-1D valence electronic structures. 20. QUANTUM WIRES

WITH

LATERALDEFECTS

The electronic properties we have studied considered the situation of an electron, in the conduction or valence band, in the presence of a twodimensional confining barrier. So far this barrier has been considered as ideal, and any effect due to fluctuations in the lateral confinement as well as to other interface defects, such as impurities, was neglected. The actual situation is far from being that ideal. The techniques developed to build a lateral confinement in a 2D epitaxial electron gas are still under development. Lateral

408

G. BASTARD et al.

In .-"

=

I=

n L tu v

0.01

-

-

m D

0

0

-100

-80

~ ~ . 5 o iLl:150i i

-60

-40

x:0.2

-20

(b)

0 -100

-80

-60

-40

-20

E (meV FIG. 110. (a) Energy dependence of the valence band density of states for a GaAs-Ga,,,A1,.,As rectangular quantum wire with L, = 50 8, and L, = 100A. The energy zero is taken at the top of the bulk GaAs valence band edge. (b) Same as in Fig. IlOa, but L,= 1508,. (c) Same as in Fig. llOa, but L,= 200 A. (d) Same as in Fig. llOa, but L, = 100A, L, = 200 8,.

fluctuations are quite important. Also, the segregation of impurities at the lateral interfaces is a common phenomenon. For lateral confinement by lithographic techniques, the damage of the first layers close to the interfaces is still an unsolved problem. All these aspects severely jeopardize the observation of some of the expected 1D behavior in available samples. Although we do expect that the 1D samples will improve in the next years, minimizing some of these problems, it is important to have a better insight of what happens at the surfaces and in which way the mentioned defects modify the electronic structure of the quantum wires. On the other hand, increasing improvement in the techniques of building quasi-1D structures has led to the engineering of new kinds of structures, by introducing lateral modulations in quantum wires. The short lengths of some nanostructures allow the study of ballistic transport of electrons in a system of reduced dimensionality. Interesting phenomena have been studied, such as the physics of point contact^^'^ and Aharonov-Bohm rings.240Interference 240G.Timp, A. M. Chang, J. E. Cunningham, T. Y. Chang, P. Mankiewich, R. Behringer, and R. E. Howard, Phys. Reu. Lett. 58, 2814 (1987).

0

ELECTRONIC STATES IN SEMICONDUCTOR HETEROSTRUCTURES

409

FIG.1 1 1. Sketch of a (x, y ) cross section of a rectangular (Lz,Lx)quantum wire presenting a constriction ( L y ,lJ. An incident wave in the ith x mode is partially reflected in the jth mode (rij) and partially transmitted (tij)in the jth mode by the constriction. (S) and (D) respectively denote the (remote) contacts.

phenomena should be expected in quantum wires with lateral indentations or

constriction^.^^^

A first step for the understanding of such structures, quantum wires presenting lateral indentations or constrictions-intentional or accidentalis the study of their electronic properties. We discuss a simple but powerful method to calculate the electronic spectrum of such structures.242 To exemplify the method, we discuss the transmission coefficients of a rectangular quantum wire containing a lateral repulsive defect (see Fig. 111). The electrons are assumed to be in a regime of ballistic transport and to suffer only elastic scattering at the constriction. In the framework of the theory of Buttiker et ul.,243 we discuss their transport properties. Finally, we extend it to the discussion of the experimental results obtained by van Wees and coworkers4 and Wharam and c o - ~ o r k e r s This . ~ problem has been extensively and successfully discussed by several a ~ t h o r s . ~ ~ ~ , ~ ~ ~ In the framework of the envelope approximation and the decoupled approximation, the potential barrier for the quantum wire showing a lateral construction can be written as

( L;I) [ ( Lq:)

V(X, y , z ) = V,Y x

z2 - -

Y

x2--

(Y ) ( 5)

+ Y --x2

Y x2--

1

Y(-y(y-LJ).

(20.1)

241F.Sols, M. Macucci, U. Ravaioli, and K. Hess, Appf. Phys. Lett. 54, 350 (1989). 242J. A. Brum and G. Bastard, to appear in “Science and Engineering of 1- and 0-Dimensional Semiconductors” (S. P. Beaumont and C. M. Sotomayor Torres, eds.). 243M.Buttiker, Y.Imry, R. Landauer, and S. Pinhas, Phys. Rev. B 31, 6207 (1985). 244A. Szafer and A. D. Stone, Phys. Rev. Lett. 62, 300 (1989). 245G.Kirczenow, Solid State Comm. 68, 715 (1988).

410

G . BASTARD et al.

Again, we consider the case of a much stronger z confinement. We restrict our analysis to the states originating from the ground z-related subband. The z-motion is then locked in a tightly bound state and will not interfere with the motion in the x,y plane. The zero of energy is chosen at the edge of the ground z-related subband. The origin of the y-coordinate is placed on the constriction interface on the left. The solution for the electronic state is

$(r) For y < 0 and y > L,,

E,

= X1(z)a,(x)eiqyy.

(20.2)

is the eigenvalue of (20.3)

and E =

+ E, + h2q,2/2m*.

El

(20.4)

Let us consider an incident mode at y < 0, associated with the nth x related subband. The effect of the constriction is to scatter the incoming wave function into different modes, which are then partially reflected and transmitted. For y < 0, the general form of the wave function is

(20.5) where we are limiting the problem to the first N x-related subbands. . the Notice that some solutions might be evanescent [i.e., q;,) = i ~ 3For region y > L, we can write N

(20.6) The solutions in the constricted region are obtained by projecting the hamiltonian onto the basis generated by the am’s.The solution we look for is of the form

and we obtain the coefficients am’sby diagonalizing the determinant formed from the set of equations:

(20.8)

ELECTRONIC STATES IN SEMICONDUCTOR HETEROSTRUCTURES

41 1

Here, the energy E is fixed (the energy of the impinging electron on the constriction), and k, are the unknown variables to be found. Again, some of the k, values can be imaginary. We can then write the general solution in the constricted part as

c (djeikjy+ e j e - i k J y )c a,(kf)a,(x), N

f(x,y)

=

j= 1

N

n=

1

(20.9)

where the kf’s (j= 1 , 2, . . . , N) are the solutions of Eq. (20.8), and were we used the property that a, depends on k;. It is now necessary to determine the 4N unknown coefficients: r,, t,, d j , and e j . We obtain the required relations by demanding the continuity off(x, y ) and d f ( x , y ) / d y at the interfaces y = 0 and y = L, and by exploiting the orthogonality of the a’s: (20.10a) iqm(6,,

-

r,) =

1ikj(dj - ej)am(k;),

j= 1

(20.10b) (20.10c)

(20.1Od) where qm stands for q:. We obtain then a linear system of 4 N x 4 N nonhomogeneous equations. By combining the equations, we can reduce them to a 2N x 2N system. For the case we are considering, a symmetric constriction, the x modes interact only with modes of the same symmetry. We can separate the problem into two blocks, with odd and even x-related solutions. We finally obtain the solutions by solving the system numerically. In what follows we have retained N = 40-80 x-related subbands in the analysis, depending on L,. In Figure 112 we show the transmission coefficients as a function of the incident energy for the three first modes for an incident mode associated with the first x-related subband. The parameters used in the calculations are V, = 600 meV, 1, = 600 A, L, = 500 A, L, = 1000 A, and m* = 0.067m0. Here, Tli is the normalized transmission coefficient ( = q i / q l l t J ) , so as to conserve the probability current246Ci(Tli+ R l i ) = 1. Each time a new mode becomes propagating, we observe strong resonances. At higher energies, we 246D.S. Fisher and P. A. Lee, Phys. Reo. B 23, 6851 (1981). 247R.Landauer, In “Nanostructure Physics and Fabrication,”(M. A. Reed and W. P. Kirk, eds.), p. 17. Academic Press, New York 1989. 248E. Deleporte, J. M. Berroir, G. Bastard, C. Delalande, J. M. Hong, and L. L. Chang, Phys. Rev. B 42, 5891 (1990).

;;k]

412

G. BASTARD et al.

0.8

--

I-

04 02 0 0

10

20

30 40 E (meV)

50

FIG. 112. Energy dependence of the Tlz coefficients ( I =1,2,3) for a constricted wire. L,= l0ClO.k l,=600A, L, = 500 A, V, = 0.6 eV.

observe a scattering among the different modes. Although the first mode dominates the transmission, the other modes have a nonnegligible contribution. At the edge of the x-related subband of a mode, the transverse kinetic energy is very small, which explains the strong resonances. It is, however, very difficult to probe experimentally the different modes. For example, transport experiments are unselective of the x mode but rather measure the total transmission through the constriction. To calculate it, we follow the theory developed by Buttiker and c o - w o r k e r ~where ~ ~ ~ they studied the conductance of a sample containing many channels coupled by elastic scatterings. In our case, the channels are the multiple subbands in the constricted region. In the wider regions we consider the electrons in a regime of ballistic transport. In our utilization of Buttiker et aZ.243theory we have to be careful about what is being measured. In their work they derived two different expressions for the multichannel conductance, according to where the contacts are introduced to measure it. We will not discuss the details of their work, but will just mention the results obtained (which we want to use) and the conditions of applicability. If we consider the conductance between regions S and D (see Fig. 1 1 1) and we are considering the transition from S to region 1 and from region 2 to D as being smooth, that is, without any scattering, then the conductance at T = OK is given by (20.1la) (20.11 b) j

If we consider the conductance as measured between regions 1 and 2, the 249X.Liu, A. Petrou, J. Warnock, B. T. Jonker, G. A. Prinz, and J. J. Kretz, Phys. Rev. Lett. 63, 2280 (1989). "OM. M. Dignam and J. E. Sipe, Phys. Rev. Lett. 64, 1797 (1990).

ELECTRONIC STATES IN SEMICONDUCTOR HETEROSTRUCTURES

41 3

electronic structure in the quantum wires of the wider region make an important contribution. The expression for the conductance is then

where Ri =

lrijI2

(20.13a)

j

and (20.13b)

,x( G N

gr =

0

1

E (mtV)

2

1=1

l,

(20.13c)

FIG. 113. Energy dependence of the conductances G , , G , (in units of 2ez/h) for a rectangular quantum wire containing a 2000 A x 2000 A constriction. Vb = 0.6 eV. (a) L, = 4000 A; (b) L, = 8000 A; (c) L, = 20000 A.

414

G. BASTARD et al.

where uiis the velocity of the electrons in channel i and 1 and r refer to the lefthand side and right-hand side of the unperturbed wire, respectively. In Fig. 113 we show the conductance in units of 2ez/h as a function of the incident energy, in both cases, for a quantum wire 2000 A wide and 2000 A long. The barrier potential is 600meV. The wide quantum wire has L, = (a) 4000 A, (b) 8000 A, and (c) 20,000 A. We observe striking differences between the two conductances. While the conductance measured from S to D ( G , ) shows nearly a steplike behavior, the conductance as measured from 1 to 2 (C,) shows very strong structures between two successive steps. The results obtained for C, are clear: each time a new channel (here, a new subband) becomes propagating, the conductance increases by an amount equal to 2ez/h. From the experimental point of view, each time the energy of the impinging carrier crosses a new one-dimensional subband edge in the constricted region, the electron has a new path to follow. The conductance is given by 2ne2/h,where n is the number of 1D subbands in the narrow region that are below the incident energy. We observe that this behavior is independent of the width of the wide region. When we consider the conductance between 1 and 2, we observe strong resonances. These structures are directly related to the quantization of regions 1 and 2. The conductance shows very sharp features and a very complex behavior. As we consider larger values for the wide part of the wire, the resonances decrease in intensity and get smoother. G, gets close to GI.The experiments of van Wees et d4and Wahram et aL5 show a quantization of the sample conductance at low temperatures. They measured the conductance from a 2D electron gas through a constriction (point contact) toward another 2D region. Their results were explained in Refs. 244 and 245 by using the expression for G , . However, as pointed out by L a n d a ~ e r , ~the ~ ’experiments are more closely related to a situation like the one described by G,, where the conductance is measured between the two 2D regions, and not from the contacts (S and D). Landauer also pointed out that the difference between both expressions is practically irrelevant if we have real 2D regions (i.e., large L,). The difference between both measurements should be a small correction to G I , and the steps of the conductance should be a little larger than 2e2/h. In our case, we get closer to the experiments described in Refs. 4 and 5 as we increase L,. As we observe in Fig. 113c, the structures become smooth and both conductances are very close in magnitude. It is expected that as we keep increasing L,, to simulate a 2D region, G , will be even smoother (no 1D structure should be present for a 2D region) and its value will converge to G , . Besides, broadening mechanisms in the wide regions will unavoidably come into play, and for L, > Lo, where Lo is such that the size quantization in the wide region becomes comparable to the damping h/z, where z is the level lifetime due to elastic and/or inelastic scattering, the very notion of x quantization in the

ELECTRONIC STATES IN SEMICONDUCTOR HETEROSTRUCTURES

-

415

4-

FIG. 114. Energy dependence of the conductances G , , G 2 (in units of 2e2/h) for a 0

0

I

I

2

1

E (rneV)

3

rectangular quantum wire containing a constriction. V, = 0.6 eV, I, = 2000 A, L, = 20000 A, L, = 3000 A.

wide region will become irrelevant. Then, of course, the assumption of ballistic transport in the wide regions will become incorrect. One final word about the interference we observe in both conductances superimposed on the steps. Clearly, they are associated with the Fabry-Perot effects in the constriction, the electrons being reflected by the two interfaces y = 0 and y = L, and oscillating in the constriction before being reflected toward the region 1 or transmitted toward the region 2. For a longer constriction this interference pattern is even stronger. In Fig. 114 we plot the energy dependence of the conductances for L, = 3000A while keeping the other parameters equal to those of Fig. 113c. The spikes superimposed on the plateaux are clearly larger for L, = 3000A than for L, = 2000A. However, these structures have not shown up in the experiment^.^.^ The main reason is that the point contacts are not characterized by abrupt interfaces but by smoother ones. The transmission should then occur adiabatically and should not exhibit any interference pattern.

ACKNOWLEDGMENTS We are pleased to thank the CAPES (Brazil) and CNRS (France) for financial support. We have enjoyed illuminating discussions with many colleagues, notably L. L. Chang, A. Chomette, C. Delalande, B. Deveaud, L. Esaki, Y. Guldner, J. Y.Marzin, E. E. Mendez, P. Voisin, and M. Voos. We are grateful to K. Moore, G. Duggan, J. Y. Marzin, J. M. Gerard, and E. E. Mendez for supplying us with their drawings.