Journal of Luminescence 30 (I953) 455 50 North—Holland, Amsterdam
COULOMBIC BOUND STATES IN SEMICONDUCTOR QUANTUM WELLS U. BASTARD i/c P/irsiq uc i/eu So/ise.s i/C 1’ Leo/c Norniu/e Siiperieurc, 24 rue Llionio,ii/. 7)231 Purrs (~ei/ex05, France
Recent theoretical works on ~_oulornbic hound states in semiconductor quantum wells Q.W.) are reviewed. Due to carrier confinement along the growth axis the hound impurity or exciton states display enhanced binding energies over the hulk values. The presence offree carriers in modulation-doped quantum wells decreases the impurity binding energies. However the quasi-bidimensionality of the carrier motion prevents a complete vanishing of impurity hound States. The photoluminescence line of high-quality quantum well is often Stokes-shifted with respect to the absorption or excitation spectra. This Stokes shift can he correlated with interface defects in a qualitative fashion.
1. Introduction The modern growth techniques such as Molecular-Beam-Epitaxy or MetalOrganic-Chemical-Vapor- Deposition have made possible the realization of high quality semiconductor heterojunctions. e.g.. quantum wells (Q.W.) or superlattices (S.L.). In these systems quantum size effects become a key feature [1,2]. For instance, GaAs Ga~ 5AIxM Q.W.’s have an effective bandgap which can be blue-shifted by as much as 445 meV from bulk GaAs bandgap  due to electron and hole confinements in GaAs slabs. In a defectless Q.W. the carriers move freely in the layer planes but are bound along the growth axis. This leads to the formation of quasi-two-dimensional (2D) subbands and to a step-like density of states (D.O.S.). Despite their high quality the Q.W. structures contain residual charged impurities. Also a Q.W. can be intentionally doped and the dopant can be placed either in the barrier (modulation doping) or in the well (anti-modulation doping). As a result of doping the Q.W. structure may contain free carriers which will screen the impurity potential Since the Q.W. D.O.S. strongly differs from the bulk one, one may expect modifications of thc Coulombic bound states (unscreened or screened) in such structures. The absorption and photoluminescence spectra of high quality single and multiple GaAs Ga~.xAlrAs Q.W. structures are dominated by excitonic 0022-231 3/85~S03.30 © El sevier Science Publishers B. V. (North-Holland Physics Publishing Division)
G. Bastard / Coulombic bound states in semiconductor quantum wells
effects . In fact the room temperature absorption spectra of GaAs Q.W.’s display very strong excitonic lines  whereas bulk GaAs hardly shows an exciton-related shoulder under the same conditions. This is a clear indication that the exciton binding energy, another Coulombic problem, is markedly increased by the electron and hole confinement. In this paper we present a review of recent theoretical results obtained on the problem of Coulombic binding in quasi-2D semiconductor heterostructures. Section 2 is devoted to Coulombic impurities (unscreened or screened) in single Q.W.’s whereas section 3 will deal with the exciton problem. The exciton trapping on a model interface defects is discussed in section 4.
2. Coulombic impurities in single Q.W. structures [6—161 Consider an idealized Q.W. and assume that (i) a single spherical parabolic band is involved in the carrier kinematics in each layer; (ii) the carrier effective mass m* and the relative dielectric constant ic are constant throughout the whole structure. These assumptions are rather well justified in the case of the bound states supported by Coulombic donors in GaAs~Ga1_~Al~As Q.W. Thus the effective mass Hamiltonian we have to investigate is 2/2rn*+ Vb(z)—e2/1~[p2+(z—zI)2]’~2,
In eqs. (1) and (2) the z origin is taken at the center of the Q.W.; Y(x) is the step function which is equal to I ifx >0 and vanishes if x <0; L is the Q.W. thickness; Vb the barrier height; p2 =x2 +y2 and z~is the position of the impurity along the growth (z) axis. The characteristic length associated with the sole Coulombic potential (Vb=O) is the three-dimensional (3D) Bohr radius a~(a~=,dI2/m*e2,..~ 100 A for donors in bulk GaAs) and the characteristic energy is the 3D Rydberg R~(R~= m*e4/2ic2112 5 meV for donors in bulk GaAs). Two dimensionless parameters govern the behaviour of the Coulombic bound states in Q.W.: L/a~ and ztI/a~.The first parameter gives a measure of the dimensionality of the motion: if Vb is finite and L/a~is ~s 1 or ~ 1 the problem is almost three dimensional. If L/a~y~1 the size quantization becomes negligible compared to R~, lYit2/2m*L2 = it2 R~(a~/L)2 ~ 1: many quasi-2D subbands are admixed by the Coulombic potential and participate in the building of the impurity bound state. For on-center impurity the binding energy extrapolates to R~when L —p fJ and the shape of the ground bound state wavefunction approaches that of a Is hydrogenic wavefunction. If on the other hand L/a~~ 1 the ground bound level of the Q.W. E~raises up to 1’~and its associated wavefunction ~ 1(z)
increasingly leaks in the barrier material. ~1(z) is more and more coupled to the Q.W. continuum states by the Coulombic potential. In turn the Q.W. continuum
( oii/onihii hound stoics in semiconductor i/iiant urn sic//s
increasingly contributes to the impurity wavefunction. For L/a~~ 1, the Q.W. becomes a perturbation and at L=0 the impurity binding energy is R~(since the well and the barrier materials are assumed to have identical effective mass and dielectric constant( [7 9]. For finite l~and intermediate thickness (L,.a~ I) the binding energy is larger than that found at both L = 0 and L = The = motion becomes forced by the Q.W. effect and the impurity binding energy is larger than in 3D situations because the electron is held close to the attractive center by the Q.W. walls. The positive energy associated with this extralocalization is taken up by E~(the onset energy of the Coulombic continuum(. Summarizing the previous hand-waving arguments we note that the Coulombic binding energy first increases from R~by decreasing L. then reaches a maximum value and then decreases to R~at L=0. If however I~,is infinite E1 is liable to diverge and is increasingly separated from the excited states when L decreases. The situation is qualitatively opposite at infinite T~to that found at finite ~: decreasing L to zero reinforces the twodimensional character if l~j~ = ~ Consequently the impurity binding energy monotically increases reaching 4R~at L = 0 which is the binding energy of a two-dimensional hydrogenic atom. The second parameter relevant to the discussion of impurity binding energy in heterostructures is the position of the impurity along the growth axis. In a bulk material the ground bound state is degenerate with respect to the impurity position: two impurity sites separated by a lattice vector are equivalent for impurity binding energy. That is no longer true in Q.W. structures which lack for translational invariance along Consider for instance a thick well. The on-center impurity binding energy is close to R~.If on the other hand = the Q.W. wall prevents a quasi-IS state to be an acceptable solution of the onedge impurity problem. Instead a quasi-2P state which nearly vanishes at the impurity site will he an acceptable solution. Thus for thick wells the on-edge impurity binding energy should approach R~/4.A dispersion of the binding energy upon =~is a characteristic feature of the impurity problem in Q.W. . If the impurities are uniformly distributed in the structure the on-center impurities are statistically dominant. An impurity placed in the barrier material (~zJ> L/2) can still bind a state (i.e., a state whose energy is lower than the onset of the ground Q.W. subband E1). At large its binding energy approaches zero [6,9]. There exists a second impurity state attached to the edge of the barrier material (energy V, R~).That state is not a true bound state since it interferes with the continuum associated with the 2D subbands E~,E, However its lifetime T roughly given by 1 1 1 4itm*a~ 2 R~ 12 .
can be very long for remote impurities. Various variational solutions of(I)have been proposed. For instance [6,9. 10],
6. Bastard Coulombic bound states in semiconductor quantum wells
is exact in both L=0 and L= c~limits. For Q.W. of current interest (50A~L~25OA)i~’~ is as good as tji~for on-center impurities and allows for simpler calculations when the Q.W. is perturbed (e.g., by an electric field applied parallel to z; when one wants to discuss the effect of effective mass mismatch at the interfaces or when the impurity is screened by free carriers). In fig. I we present the L dependence of on-edge donor binding energy in GaAs -Gai AlIAs Q.W. calculated with i/i and in fig. 2 the z1 dependence of the binding energy of the bound level attached to E1 and created by a donor ion placed in the barrier material (1’~,= cfD). Note that Tanaka et al.  have shown that finite Vb sensitively increases the binding energy of the in-barrier impurities compared to the results shown on fig. 2. Suppose that the Q.W. barriers are selectively doped with donors. At thermal equilibrium some of these donors have released their electrons (fig. 3) which are transferred in the Q.W. leaving fixed positive charges in the barrier. This spatial separation between electrons and their parent donors creates band bending on the one hand and strongly decreases the ionic scattering suffered by the quasi-2D gas on the other hand. We are interested in calculating the bound states supported by a single residual impurity, which we assume to be placed ~
CD uJ w CD
I (A) Fig. 1. Thickness dependence of the on-edge donor binding energy in GaAs Ga~_~Al~As Q.W. The curves labelled 1,2,3,4 correspond to Li,, = 212 meV. 318 meV, 424 meV and Vb = ~x.,respectively.
(ou/o,nhic hound states in ,seniiconductor i/uaflturn we//.s
1-mg. 2. Position dependence ol the donor binding energy (in dimensionless units( in a Q.W. of infinite height. The donor is assumed to be placed in the barrier. 1. = a~.
at the center of the well (=~=O)[II]. Due to the presence of free carriers the electrostatic interaction between the electron and the on-center impurity is screened. In the random-phase approximation, if a single Q.W. subband is occupied and if T=0, this quasi-2D screening is expressed by  (yJHq.
:~=0)~/i)=~(Xt~_ 2ite2 exp (—qH)~Xi~.
where q is a 2D (in-plane) wavevector; V~q, :~=0)is the coefilcient of the 2D Fourier expansion of the screened Coulombic potential and t~(q)the dielectric constant: .-,
exp (—q~— ‘~)d d:. (7( 2]’2 (8) g(q( = 1 Y(q 2k1.)[l (2k~,q) In eqs. (6) (8) qT~=2/u~is the 2D Thomas Fermi wavevector and kf is the Fermi wavevector related to the 2D concentration n~by k t2. The 1. =(2itne) quasi-2D screening differs from the bulk one by two important factors. (i) Only the averages over Zi of the 2D Fourier coefficients of the screened potential are easily evaluated. In the bulk instead, one knows the 3D Fourier coefficients I~(q)exactly. Thus k~(r)is easily calculated. In the Thomas Fermi approximation, for instance, one gets ~(r)= —(e2/kr)exp(-—QTFr). —
G. Bastard / Coulomnbic bound states in semiconductor quantum wells
Fig. 3. Conduction band profile ofa modulation-doped Q.W. The donors are placed in the barrier and some of them have released their electrons which accumulate in the well. E? labels the Q.W. ground state in the absence of charge transfer and E~is the Q.W. ground state taking into account the band bending effects.
~e (cm 1010
2krao Fig. 4. Binding energy of a on-center screened donor in GaAs—Ga 1 _~Al~As Q.W. (l~,=0.2eV) plotted versus the 2D electron concentration n, for three Q.W. thicknesses L.
(ou/isnhii hound stat is in scm uoni/uitor q stilt urn sec//u
(ii) The 2D Thomas Fermi wavevector 9mm is constant whereas the bulk quantity Qi is concentration-dependent. The latter feature implies that a screened Coulombic potential may have no bound state if the 3D-concentration N~is larger than a critical value N~ such that Q1 ia~ I. In quasi-2D ~.
systems the constant 9-i-i implies that there always exists a bound state: the binding energy saturates to a finite value at large n~.The binding energy of a screened on-center donor states has been recently calculated by Brum ci al. [II]. The results of these calculations are presented on fig. 4 for GaAs Ga1 .5Al5As Q.W. (l~=0.2eV).A sizeable binding energy remains at large mi, which should allow the observation of impurity-related features~’~t low temperature in modulation-doped Q.W.’s. Various impurity features have been reported [1 2,13.16] in photoluminescence spectra of GaAs Ga~ 5Al~AsQ.W. structures. The impurity lines are much weaker than in bulk GaAs and have been associated either to donors or to acceptors. The observed donor binding energies reported by Shanabrook ci al.  are well described by Mailhiot et al.’s or Greene and Bajaj’s calculations [7.8]. The acceptor binding energies reported by Miller et al.  have been quantitatively interpreted by Masselink et al. . Note that acceptor calculations are more complicated than donor calculations due to the coupling between light and heavy hole subbands. 3. Exciton binding energy in single quantum well [I7—241 When a Q.W. structure is shined by near- (effective( bandgap light the lowest lying electron hole state corresponds to a bound state for the reduced electron hole motion: the exciton. To its energy one should add the kinetic energy of the center-of-mass (C.M.). Since the electron and hole = motions are confined by Q.W. walls one deals with quasi-2D excitons and the CM. motion occurs only in the layer plane. Thus, to a great deal, the calculations of exciton binding energy resemble those of Coulombic impurities. There are however two important new features brought by the mobile nature of the positive charge. (i) The electron and the hole can he mostly localized within the same slab (type I Q.W., e.g., GaAs Gat ~Al5As)or within adjacent layers (type II Q.W.. e.g., InAs GaSb) (fig. 5). Clearly the exciton binding should be larger in type I Q.W. than in type 11 Q.W. , all other parameters being identical. This difference is reminiscent of the much larger binding energies found for in-well impurities compared to those of in-barrier impurities. (ii( The in-plane hole kinematics is complicated because of the coupling between light and heavy hole subbands at finite wavevector k in the layer plane [19 21]. In bulk materials the F8 valence Hamiltonian does not reduce to a scalar F6 conduction Hamiltonian. To the the growth scalar part add 2 like termthewhere J=~.Quantizing J along axisone andshould adding thea (kJ)
6. Bastard / Coulombic bound states in semiconductor quantum wells
lYp~ I EC (z)
TYPt ~ Fig. 5. Schematic representation of electron (Es) and hole wavefunction in type 1 and type II (GaSb InAs- Sb, Li,,A. ~ 100 A) quantum wells, E,(z( and E~(z)denote the position-dependent conduction and valence band edges respectively.
valence barrier term V~(z)leads at k~=Oto decoupled light (LH~)and heavy (HH5) subbands whose confinement energies involve the bulk heavy and light hole effective masses. At k ~ 0 these subbands are admixed and for instance anti-crossings take place [19---21].The in-plane effective mass of heavy hole and light hole subbands have no obvious relationships with the bulk hole masses. Experimentally the existence of two kinds of excitons formed between electron and heavy hole on the one hand and electron and light hole on the other hand is very well documented in GaAs—Ga~5Al~As Q.W.’s [2,4,5,17]. That is due to a sizeable difference between HH~and LH~confinement energies when L~150A. No direct information on the exciton binding energy itself are available (except in  where a small hump adjacent to the main E1 I-1H1 exciton line has been interpreted as the “2S” state of the E~—HH~ exciton). In principle the exciton binding energy is obtained by finding the eigenenergies of a 4 x 4 matrix which represents the reduced motion of a F6 electron and f’5 hole. In bulk materials Baldereschi and Lipari  have shown that the ,
hound ,statc.s in semiconductor quaniton ire//s
isotropic parts of the diagonal terms of this matrix are much larger than the other terms. Thus the exeiton problem reduces to a donor-like problem if only the isotropic terms are retained and if the CM. wavevector is assumed to be zero. In Q.W. structures model exeiton calculations have been proposed [18.22]. If one neglects the valence band intricacies and thus deals with idealized isotropic valence bands the exciton binding energies vary with the Q.W. thickness in a very similar way as the donor binding energy does . Greene et al.  and Miller eta!.  have used part of the bulk procedure to solve the Q.W. exciton problem, i.e.. they have neglected the off-diagonal terms of the exciton matrix. The diagonal terms involve reduced masses which are different for the and the in-plane motions and also different for electron light hole. E,, LEt,,. and electron heavy hole, F,, H H,,, exeitons. Greene et al.  show-ed that finite barrier effects are essential in exciton calculations. They found that the binding energies of E~ H l~I~ and E~ LH~excitons are influenced by two effects. (i) The electron and hole leakages in the Ga~_.~Al~As barrier. The LH~state is less localized in GaAs than the HH state. Thus the heavy hole exciton is more two dimensional, i.e., more tightly bound than the light hole exeiton for a given L. (ii) However the in-plane light hole exeiton effective mass is larger than the one of the heavy hole exciton. Thus the LH F1 exciton has a tendency to a larger binding than the HI-I1 F1 exciton. The mass effect prevails at large L whereas at small L the E1 HH1 exciton is more bound than the E~ LH1 exciton. It would be desirable to calculate the effects of the off-diagonal terms
0.5 L Ia ~ 6. Thickness dependence of the exciton binding s~lOOA(.
clserg\ in GaSh
litAs GaSh quantum ssclls
6. Bastard / Coulombic hound states in semiconductor quantum ire/Is
of the exciton matrix and also of the effective mass mismatch between GaAs Ga1 _5Al~Asbefore drawing definite conclusions on the Q.W. exciton problem. As far as excitons in type II Q.W.’s are concerned, only a model calculation has been proposed . As expected, the spatial separation between electron and hole drastically reduces the exeiton binding energy (fig. 6). -
4. Low temperature exciton trapping on interface defects [25,261 The interfaces between two semiconductors, even lattice-matched, are not perfect. In Si02- Si metal-oxide-semiconductor structures the Si Si02 interface roughness is known to contribute to a limitation of channel mobility . In GaAs -Ga1 _~Al~Asmodulation-doped heterojunctions whose interface quality is far superior to that found in Si—SiO2 the interface roughness has been calculated to be a scattering mechanism of secondary importance . In GaAs— Ga1 ~Al~As Q.W.’s the optical probes (luminescence, excitation spectroscopy) may provide a finer access to the interface roughness than transport measurements. This is due to the influence of interface defects on the optical properties of Q.W.’s at low temperature. Suppose an exciton has acquired by some means a finite in-plane C.M. wavevector K and assume that Coulombic impurities are too scarce to play a significant part. The interfaces being imperfect deviate locally from a plane (here as in previous sections we are only concerned with envelope functions which makes possible the definition of planar interfaces). When this deviation corresponds to an attractive potential for both electron and hole, i.e., when extra GaAs protrudes in the Gai .~Al~As barriers, the moving exciton is scattered and even get trapped in this quasi-2D potential well. Little is known on ‘~~d~f the potential energy associated with interface defects and to a great deal we have to define it. In an empirical approach an interface defect can be characterized by a depth b (along z) and a lateral (inplane) extension a. Many .)t’d~f fulfill these requirements. We took  a semi-Gaussian model 2)exp( z~/2b2) Iz~exp( p~/2a Y[ Ze] E~exp( p~/2a2)exp( —
z~/2b2)Y[ zh] —
for a defect protruding in the left-hand side barrier. In eq. (9) V~(JK~)is the height of the electron (hole) confining barriers; the subscripts e (h) refers to the electron (hole) and L is the average Q.W. thickness. High quality Q.W.’s are characterized by intense, narrow (few meV) photoluminescence lines which are little Stokes-shifted (0 to 5 meV for L ~ 100 A) from the maximum of the E 1—HH1 free exciton line seen in absorption or cxci-
Cou/omhii- hound stilt c.s in scmconduit or 9 san turn icc//s
I ig. 7. Photol uminescence I solid I ule( and Ga, ALAs single Q.W. (x =/).52f
(dashed I inc I spectra of a 70 A thick GaAs
tation spectra (X.S.( (fig. 7). Qualitatively one may define two limiting regimes characterizing the optical properties of the excitons interacting with interface defects. (i) Either a is very large compared to a~ (say a 10 A). Then each of the defect acts like a micro-sample having its ow-i-i thickness L. Inside such an island the exciton is free, the CM. wavevector is a good quantum number. The absorption or excitation spectrum is essentially governed by the states of high D.O.S. Thus it will reflect the existence of many micro-samples, each of different F and thus of different effective bandgap H ~(L’) + E (L’) + t:g R~,,,(L’). Suppose one deals with a set of samples of varying average thickness L and assume that. irrespective of L. the lateral island sizes are very large (a—÷ii) and the defect depth his constant. The width of the E1—~HH1exciton peak seen in X.S. will —
increase like h/L. This behaviour was observed in high quality MBE-grown samples by Weisbuch et al. . The width of the P.L. line and its (eventual) Stokes-shift with respect to X.S. peak can hardly be correlated with the X.S. linewidth. This is because P.L. involves excited carriers which, during their lifetime before recombination, may relax or not through all the quantum states of low energy which exist in the excited band. The P.L. emphasizes the quantum state of low energy (which may have a low D.O.S.) provided the capture processes in these low energy states are faster than the radiative lifetime. (ii) The other limiting regime occurs for a a~. A single defect has few hound states (often a single one) because of the in-plane confinement. The CM. -~
wavevector is no longer a good quantum number. The exciton states consist of a
2D continuum associated with moving excitons scattered by attractive or repulsive interface defects. At lower energy bound states of exciton trapped on attractive interface defects occur. Defects of various (a. 6) give rise to different binding energies (fig. 8). Thus a whole band of trapped excitons is generated in
Coulombic bound states in semiconductor quantum wells
=052 L = 70 A
Fig. 8. Trapped exciton binding energy versus the lateral size a of attractive interface defects for different defect depth b in GaAs-- Ga 1 ~,Al,Asquantum well (x=0.52). The average quantum well thickness is L=70A.
the limit of small defects. The X.S. probes the exciton continuum near K = 0. The P.L. line is more sensitive to trapped excitons and must be structureless unless a definite pair (a, b) is statistically dominant, which seems unlikely. As far as thermalization of trapped excitons is concerned it appears difficult in a given site since most of the defects have a single bound level. It may occur through phonon-assisted hopping from site to site. To be efficient, that mechanism requires the interface defects to be numerous since they have small radii. In addition to optical experiments there have been interface characterizations  which provide independent evidence that in high quality samples the mean defect depth b is one monolayer (2.83 A in GaAs) and its lateral extension is 200—300 A. If we take these numbers into consideration model (ii) seems preferable to model (i). The optical experiments performed by the E.N.S. group on MOCVD-grown GaAs—Ga~_~Al~As single Q.W.’s (L=70A, xz=0.52) reveal that the P.L. line of a representative high quality sample was 4.4 meV Stokes-shifted from the E1-HH1 exciton peak seen in the X.S. The P.L. line was structureless and did not change shape when the laser intensity varied from 1 mW to 1 W: neither carrier heating nor saturation effects were detectable. We have qualitatively evaluated the characteristics of the trapped exciton band, (i) by calculating the exciton binding energy on the semi-Gaussian interface defects (eq. (9)) versus defect sizes a and b, (ii) by assuming a Gaussian distribution of defect sizes. ‘=.~
(ou/omhi,- hound states in sc,nd-ondui-t or quantimi lvc//.s
BINDING ENERGY (meV) 9. l rapped exciton density-of-states versus binding energy. ihe structures are associated to discrete h whereas the smooth curve sxould correspond to a continuous variation of 6 in a GaAs ~Al,AsIs =0.52/ Q.W. of average thickness 70 A. FIg.
The trapped exciton D.O.S. is shown on fig. 9. It peaks —4 meV below the edge of the unperturbed C.O.M. continuum and is 2.5 meV broad at half maximum. We did not attempt to optimize the parameters of the defect sizes distribution. To ideiitify the D.O.S. to the P.L. line one needs to assume that the optical matrix element governing recombination is constant and that there is no thermalization inside the trapped exciton band. At low temperature (T= 1.7 K) once an exciton is trapped in a given defect it can only emit acoustical phonons to tunnel to another defect. We showed that for reasonable defect concentrations that process lasts longer than the measured luminesceiice decay time , supporting the contention that trapped excitons are not thermalized. 5. Conclusion Coulombie problems in semiconductor Q.W. have rapidly diversified. The gross features of basic problems (such as the determination of the binding energy of a Coulombic impurity placed somewhere in a Q.W.( are now understood. The calculations have now to address to more difficult situations (perturbed Q.W.’s. interplay between interfaces and impurities. etc.) to match the challenge of present experiments. Acknowledgements I am much indebted to Drs J.A. Brum, C. Delalande, C. Guillemot, M.H.
6. Bastard / Coulo,nhic hound states
semiconductor quantum wells
Meynadier, P. Voisin, M. Voos at the E.N.S. and Drs L.L. Chang, L. Esaki, E.E. Mendez at IBM for their active participation in the works described in this paper. The Groupe de Physique de l’E.N.S. is “Laboratoire associê au C.N.R.S.”. The work at the E.N.S. has been partly supported by C.N.E.T.
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