Physica 117B & 118B (1983) 747749 NorthHolland Publishing Company
747
ELECTRONIC STRUCTURE OF SEMICONDUCTOR SUPERLATTICES
M. Altarelli
MaxPlanckInstitut f~r Festk6rperforschung 7000 Stuttgart 80, Federal Republic of Germany
A new approach to the calculation of the electronic structure of semiconductor superlattices in the envelopefunction approximation is presented and applied to the type II InAsGaSb system. The method allows a realistic description of the band edges of the constituent materials, and yields, therefore, accurate results for the subband dispersion in the direction parallel to the layers. Some novel aspects of the semiconductorsemimetal transition in InAsGaSb are pointed out.
The calculation of the electronic structure of superlattices, especially type II systems, e.g. InAs/GaSb[l], poses an interesting challenge: In fact, (a) the narrowgap nature of the constituents and the mixing of conduction and valence band states at the interfaces requires a manyband description~ (b) the investigation of transverse dynamics (k parallel to the superlattice layers) requires a realistic description of the constituents bandedges, in particular of the degenerate valence bands; and (c) the occurence of a semimetal for superlattice periods sufficiently large (>160~ in InAsGaSb) requires that charge transfers at the interfaces be taken into account self'consistently.
and appropriate boundary conditions at the interfaces. In the new method discussed here, the matching conditions are built into a variational principle thus reducing the numerical work to a standard secular matrix diagonalization. Let x denote the superlattice axis, and y, z two normal directions in the x = O plane. Following Ref. [3], we choose z as quantization direction, and consider kvectors in the (x,y) plane. The 6 states of the conduction and upper spinorbitsplit valence band decouple then in two equivalent systems. Our basis is: u I = Is+>, u 2 = 13/2,3/2>, u. = 13/2,i/2>, and the analogous u4, u5, u 6 fgr the other set. We then write:
The size of the superlattice unit cell rules out a full microscopic calculation, and the requirement (c) makes tightbinding calculations unsuitable. Here the envelopefunction approximation [2,3] is adopted: The wavefunction is a slowly varying envelope times the periodic part u of the Bloch function of each constituent. The envelopes obey effectivemasslike equations
E c + ~1k 2 + V ( x )
= elkyy Zj=l,3 F.3 (x) u.3(r) and F. satisfies: 3 Ej, Hjj, Fj,= EFj ,where
i ( k x  i k y ) P / / 6
i (kx+iky) P//2 TI+¥
H , •,
= [i ( k x  i k y ) P / / 2
Ev
2
k 2 + V (x)
3J Li ( k x + i k y ) P / / 6
/3~ ( k x + i k ) 2 / 2 Y
In Eq (i), P is Kane's matrix element, 7 and 1 are valence band parameters in the spherical ( Y 2 = Y3 = 7) approximation, and are known from magnetooptical experiments [4,5]. V(x) is the potential arising from charge redistribution in the system. With good approximation, the u functions are the same in both materials [2], which implies P = P for the momentum matrix elements. A B The continuity condition reduces then to the continuity of the envelopes, F• 3
continuous,
j = 1,2,3
(2)
0 3784363/83/00000000/$03.00 © 1983 NorthHolland
/3~ (kxik) Y Ev
YIY 2
"]
2/2 (1)
k 2+V (x)
The other three boundary conditions are derived by imposing a constant probability current across all planes perpendicular to x, in particular across the interfaces. Averaging over one unit cell of each material, we find for the expectation value of the current operator J : x =
j,j F[cDxY +DYX )k +px I Fj L
"
~
i~jj, (FjDjj,3~x
jj
jJ
y
Fj Fj xx ' ,Djj,3~x F 3)
33'I
' (3)
]14. Altarelli
748
/ Electronic structure o f semiconductor superlattices
after expressing Eq.(1), with selfevident pact notation, as: Hjj,
= ~ ,S=x,y(Djj,k
Using Eq.(2) in Eq.(3) of the P's, one gets:
k~+Pjj,ke)
com
+ V(x)~jj,
(4)
together with the equality
rl Xy + yx . xx 3 ~j,[~(Djj, D j j , ) k y F j ,  1 D j j , ~ x F j , ]
continous,
j = 1,2,3
Eq.(2) and (5) contain, as particular cases, conditions derived in Ref. [2,3] and [6].
(5)
the
rather than a semimetal, results, although the gap is extremely small throughout the Brillouin zone. For slightly thicker superlattices, the conductionlike subband moves further downwards, until all of it lies below the topmost heavyhole subband. One has then a very small gap s e m i conductor, with gaps
We proceed to solve Eq. (1) with boundary condition (2)and (5)by a generalization of a procedure adopted in band theory [7 ]. Let us denote by ~ = (~A,~B) a wavefunction on both sides of an AB inter£ace. The Hamiltonian is a diagonal operator in this 2component basis. We can introduce an "interface operator" F, which is non diagonal, and has the following property: If Y is an ein genfunction of H, H~ = E ~ , satisfying the c o r • n n n rect boundary cond~tlons at the AB interface, then ~ is a stationary function for the functionaln<~IH+F ~>, and furthermore <~ H+FI~ > = En<~ n ~n >. Thus the addition of the nlnterfane operator F to the Hamiltonian automatically selects eigenfunctions of H with the correct boundary conditions. Proceeding along lines similar to those of Ref. [7], one can show that [8]
200
~ /
(a)
100
20o 100
i
= m(<~AIJx(o) l~A>  <%ISx(o) l%>
n;/d kx 0
÷ <%lSx(o) l~h>  <~A13x(o) l%>) + II~A(O)

~B(O+)I 2
(6)
where J (o) is the current operator in Eq.(3) at the interface x = o, and I is an arbitrary real number (F is far from being unique), which can thus be varied to optimize convergence and stability. By choosing a set of functions in which to expand the components of ~, we obtain a straightforward matrix diagonalization problem. For the InAsGaSb superlattice, the set consists of 12 trigonometric functions in each layer. In Fig. i we show the resulting band structures for two thicknesses, a = b = 60 ~, and a = b = 80 ~, respectively, in the non selfconsistent flatband approximation V(x) = O. In (b), the lowest conductionlike subband has dropped below the highest valencelike band. Notice however that band crossing only occurs for k = O, while for k # O bands strongly hybridize a~d repel each other. The fermi surface therefore consists of only one point along k x and a zero gap semiconductor,
Fig.
i.
ky
Band structure of InAsGaSb superlattices with (a) 2 x 60 ~ and (b) 2 x 80 ~ period. Energies are in meV and the zero is at the bottom of the InAs conduction band.
M. Altarelli / Electronic structure of semiconductor superlattices
References:
[i]
SaiHalasz G.A., Tsu R. and Esaki L., Appl. Phys. Lett. 30 (1977) 651653; SaiHalasz G.A., Esaki L. and Harrison W.A., Phys. Rev. B 18 (1978) 28122818
[2]
White S.R. and Sham L.J., Phys. Rev. Lett. 47 (1981) 879882
[3]
Marques G.E. and Sham L.J., Surface Sci. 113 (1982) 131133
[4]
Pidgeon C.R., Mitchell D.L. and Brown R.N., Phys. Rev. 154 (1967) 737745
[5]
Suzuki K. and Miura N., J. Phys. Soc. Japan 39 (1975) 148154
[6]
Bastard G., Phys. Rev. B 24 (1981) 56935697
[7]
Schlosser H. and Marcus P., Phys. Rev 131 (1963) 25292546
[8]
Altarelli M., to be published
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