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Optical study of probability densities in quantum well eigenstates
Surface Science 229 ( 1990) 433-438 North-Holland
OPTICAL Jean-Michel
STUDY
OF PROBABILITY
GERARD
433
DENSITIES
IN QUANTUM
WELL EIGENSTATES
* and Jean-Yves MARZIN **, 196, Av. Henri Ravera. 92220 Bagneux, France
C’NET-Laboratoire de Bagneux Received
I 1 July 1989; accepted for publication
14 September
1989
A novel technique is applied to the study of the local electronic properties of heterostructures. We insert a planar isoelectronic probe in the structure, and extract information on the probability densities in the bound states from an optical study of the effect of this local perturbation. This technique is used to reveal directly type II configurations in semiconductor superlattices. The spatial variation of the probability densities in the first bound electron states was also measured for the first time in the case or a GaAs/GaAlAs quantum well.
1. Introduction Since the earlier studies of semiconductor heterostructures, it has been a challenge to obtain experimental information on the spatial distribution of the probability densities in the confined electronic states. Many properties attest the validity of envelope function calculations, which predict particle-in-a-box type probability densities, but very few experiments allow one to extract directly information on these quantities. For some superlattices, it is even difficult to known in which material the confined particles are localized. We have performed an isoelectronic planar substitution (IPS) in these layered structures to obtain such a kind of information. A powerful approach to study the structural and electronic properties of solids is indeed to incorporate impurities in the material and use them as built-in probes. Isoelectronic [ 1,2], deep-level [ 3 ] or charged [ 4 ] impurities have been used for a wide range of applications. However, in layered structures, the IPS is particularly attractive. Unlike charged impurities (which create Coulomb potentials) or isolated isoelectronic impurities (which generally distort the host matrix), an IPS introduces a short range perturbation in the structure.
* Member
of the Direction des Recherches niques, French Ministry of Defense. ** Unite associee No. 250 du Centre National Scientifique. 0039-6028/90/$03.50 (North-Holland )
Etudes
et Tech-
de la Recherche
0 Elsevier Science Publishers
B.V.
In this paper, we first show that the probability densities in the unperturbed quantum well (QW) eigenstates at the location of the planar probe can be extracted from the energy shifts of the QW levels. We first apply this technique to test the occurrence of type II band configurations in some multi-QW (MQW) structures: an IPS of the sublayers of larger bandgap probes the localization of QW eigenstates in these layers. In a second part, the location of the probe is varied from sample to sample so as to determine experimentally the spatial dependence of the probability densities in the bound electron levels of a GaAs/GaAlAs QW. For both studies, the MQW samples are grown by molecular beam epitaxy. which allows a precise control, and an excellent reproducibility of the MQW parameters from sample to sample. The shifts of the energy levels due to the perturbation are extracted from a study of the optical transitions in the perturbed samples and in a reference sample without perturbation.
2. Localized perturbation of a quantum well
The electronic properties of the QW are well described by effective mass models [ 51. In this framework, a series of energy levels for each band of interest are found as eigenenergies of the effective Hamiltonian:
where A is a high symmetry point of the Brillouin zone, m* the effective mass associated to the band n near A, and V,,, the square potential which describes the variation of the energy position of the nth band extremum at A across the heterostructure. The eig~nfunctions of this effective Hamiltonian are the enveloppe functions FFA( Z) of the QW levels at the Brillouin zone center. An IPS affects this series of energy levels only via a change Ali,,* of V,,, on the substituted area. If the perturbation localized at z0 has a width AzO, the firstorder shift of the energy level E, is equal to AZ, AV,,4 iF~4(zo) 12. This approach is valid for a small strength IV= AZ, AV!,, of the pe~urbation, and more precisely when the shift of the levels are small with respect to the level spacing. For the first level and a repulsive probe, the maximum first-order shift 2 W/ d should for example be compared to the spacing Es--E, which is typically 3 h2/8m*d2. A stronger perturbation couples efficiently the Fi levels, for given n,A. Now, for a delta perturbation localized at z,, two sets of eigenenergies E: are found for the perturbed problem: - the E, for which F,(q) =O, - the roots of the equation: G,,(E)=
2 I
‘;y2 =:+,.
(2)
I
F/(ZO)#O
The function GZ, decreases from +a to --co between two adjacent values of E, corresponding to states with F,(z,) # 0. For a repulsive potential, I/ W is positive, and the nth eigenenergy EL of the perturbed Hamiltonian is found in the segment [E,,,En_+., 1. If IV is negative, E’, is lower than the fundamental E, of Ho, and El, for n # 0 is in the segment [E,_,,E,]. These general results are illustrated in fig. 1; the energies of the bound electron levels of a GaAsf Ga0.7A10.XAs quantum well are calculated in an effective mass model, and plotted as a function of the location of an attractive (a) or repulsive (b) perturbation, for various strengths W. Several approaches are then possible to get a local information on the probability densities in the un-
et a
b
1 ‘..__
i
____--’,‘I’~
e,
Fig, 1.The calculated bound electron levels of a 200 8, thick QW, are plotted as a function of the position of a perturbing repulsive (a) or attractive (b) delta potential. The strength of the potential is 0.25.0.5 and 50 in units of h*/8m*d for the dashed, dashdotted, and full lines.
perturbed QW (i.e. on the IF, ( zo) I ‘) from the level shifts due to the perturbation We: - For small 1W( , the dependence of the level shifts already reflect the shape of the probability densities in the bound levels of the unpe~urbed QW. Larger / WI are however generally required experimentally to measure the shifts E; -En with enough precision to extract the I Fi( zo) / 2. - The resolution of system (2 ), truncated to the finite set of the lower energy levels which are accessible experimentally, allows to extract approximate values for the ]F,(z,) I2 from the measure of the E: and E,. - The pe~urbations of the well and barrier layers have drastically different effects for a confined level: the study of the energy shift E; -E, gives therefore a binary answer concerning the localization of the considered particle (band n, band extrema A) in the perturbed layer. The two latter approaches are implemented in the following parts.
3. Direct identi~~ation
of type II superlatti~es
In a semiconductor superlattice (SL), the electrons and the holes can be both confined in the same sublayers of the material of smaller bandgap (type I
XM. Gerard, J. Y. MarzinlOpticaistudy
ofprobability densities in quantum well eigenstates
SL), or spatially separated and localized in different sublayers (type II SL). A IPS of the layers of larger bandgap can be used to test directly the occurrence of a type II band con~guration, since large shifts of an optical transition of the SL will indicate that one of the involved electron or hole levels is localized in the perturbed sublayer. Either a third material or the SL’s constituent of lower bandgap can be used to perform the IPS. The latter case is of particular interest: the pe~urbation is then attractive (respectively repulsive) for a particle confined in the sublayer of smaller (larger) bandgap. For a type I system, a (generally small) reduction of the SL’s optical transition energies is always expected. If detected, a shift toward higher energies of these transitions reveals then unambiguously a type II band configuration for the studied SL. This approach has been applied to two systems which displays non-trivial electronic properties: GaAs/AlAs indirect SL’s and InO.,,GaO,,SAs/GaAs strained SL’s. The band co~~guration for a GaAs/ AlAs SL is shown in fig. 2a [ 6,7]. Since the X conduction extrema lies in AlAs at a lower energy than in GaAs, the AlAs layers act as wells for the X valley conduction states of the SL. For a constant width of AlAs and the thinner GaAs layers, the SL’s bandgap is related to the transition Xi-HH, involving the first X conduction state and the first heavy hole state, which are localized in different sublayers. On the other hand, two different band configurations have been proposed for In,YGa,_,As/GaAs strained SL’s, of mixed type for x= 0.15 [ 8,9] (type I for heavy holes and type II for light holes as in fig. 2c), and of type I for heavy and light holes for x=0.05 [9,10]
(fig. 2b). In both cases the band discontinuity I$, for the light holes is however rather small ( -20 meV). All these results have been obtained via a determination of the band discontinuities which is based on a fit of the optical transitions. It is thus interesting to get a direct and reliable identification of type II conligurations. This is particularly the case for InGaAs/GaAs SL’s, since the sample parameters and even the alloy bandgap and deformation potentiafs are not known with enough precision to allow a perfect confidence in the extracted value of l$,. We present below some results obtained performing a IPS in both kinds of SLs. To validate this approach, we first studied an indirect GaAs/AlAs SL (SL A, GaAs: 32.5 A, AlAs: 110 A) and two perturbed SL’s obtained by a IPS of thin GaAs layers at the middle (SL Al ), or at the first and second thirds (SL A2) of the AlAs sublayers. Their low temperature photoluminescence (PL) spectra are shown in fig. 3. The two sets of peaks seen for SL A correspond to the Xi-HH, indirect transition and its phonon replica, and to the higher energy F,-HH, transition. This direct I,-HH, transition is also detected in PL for SL Al and in PL excitation in SL A2; it is clearly insensitive to an IPS in the AlAs layer since the conduction and valence
AIAsr (5
-__ 4__J-GaAs
bi
-___ X
-6
GaAs
GaAs --L-“-Jr
-3i51\
InGaAs
-1lOi
_I$/LH L-----J
““Yl---L-r ai
435
bl
cl
1 IO
1.74 ENERGY feYI
1.10
GaAs AlAs
1
Fig. 3. 10 K PL f-) and PL excitation (- -) spectra for a GaAs/ Fig. 2. (a) Energy band configuration in a GaAs/AIAs SL for the conduction band (at T and X) and valence band extrema. (b) and (c): Two possible configurations for the T conduction and valence band extrema in a GaInAs/GaAs SL.
AlAs SL, unperturbed (a) and perturbed (b) and (c). The com-
position of one period is shown for each SL. Open (full) arrows mark the calculated energyofthe XI-HH, (f,-HH, ) transitions (Exciton binding energies are not included.)
J.M. Gerurd, J. Y. .~arz~~/O~~lral study o~probabill~.vdensities in q~ani~~ well eigenstntes
436
band extrema at F display a type I configuration. On the other hand, the large increase of the X,-HI-I, transition energy (in the 15 and 30 meV range for SL Al and A2 respectively) confirms unambiguously that the fundamental X, electron level is localized in the ALAS sublayers of the SL. This technique has also been successfully applied to the In,,,Ga,,8,As/GaAs SL 3 (GaInAs: 89 A, GaAs: 200 A). The 8 K transmission spectra of SL B and of the perturbed SL obtained for two different kinds of probes are viewed in fig. 4. The three transitions observed for SL B have been assigned to ElHH,, E,-LH, and E2-HH2 transitions for a similar sample [ 8 7. They are also observed in the perturbed SL’s and have the same nature since their energy shifts are small from sample to sample. An overall increase of the transition energies is observed when a 2 8, thick AlAs layer is inserted in the GaAs layer (SL I3 I), since this probe is repulsive for both electrons and holes. This effect is larger for ElLH, (+5 meV) than for E,-HH, (+ 1 meV) which indicates that the light holes have a large probability density (PD) in the GaAs layers. We show below that the shift of LHi (at least i-4 meV) reveals in fact that they are mainly localized in these layers. For a type I system, the PD of the particle is smaller than the average value of the PD over the SL’s period (in our case l/289 = 3.4 x 1Om3A- ’ ) in the barrier layer. The strength W of the perturbation is for the holes
-89i -2OOi
ENERGY
GaInAs GaAs
4. Experimental determination of probability densities in quantum well eigenstates
IeV)
Fig. 4. 10 K transmission spectra of a Gao.ssIno.,,As/GaAs unperturbed (a) and perturbed (b) and (c). Thecomposition one period is also shown for each SL.
Az,AV= I .06 A eV, where AV is the valence band offset between GaAs and AlAs. Since the shift of the first level LH, due to the perturbation is majored by its first-order estimate, it cannot exceed 3.5 meV for a type I system. This first evidence of a type II configuration for the electrons and light holes relies however on the precise knowledge of I@. We therefore use in a second step, as for GaAs/ AlAs SLs above, an IPS using the SL’s material of smallest band gap: here, two 10 A thick Ga0.8sIn0,,,As layers are inserted at the first and second thirds of the GaAs sublayers of the SL B2. The expected effect is observed in fig. 4c: whereas Er-HH, ( - 1 meV) and E,-HHZ ( - 8 meV ) transition energies are reduced, an increase is detected for Ei-LH, ( + 2 meV). These effects confirm that the second transition does not belong to the I$-HH, family (and is thus ElLH, ), and evidences a type II band alignment for the conduction and light hole band extrema. Finally, two probes have been used so as to relax the condition LH,
SL, of
As mentioned above, the PD for a discrete set of levels can be probed locally, and extracted from the
J.M. Gerard, J. Y. Marzin/Optical study ofprobability densities in quantum well eigenstates
-GaAs
.-*
0
well-
..
as functions of the location of the probe. The energy shifts of E,--HH, or E,-LH, transitions already reflect the shapes of the PD in the nth electron or hole level (since these are roughly similar for a type I system), and are as expected of opposite signs for attractive and repulsive probes. Finally, they are large compared to the experimental uncertainties (1 meV) but also of the order of the spacing between the transitions so that the coupling of the QW levels induced by the perturbation has to be taken into account. In this experiment, we have obtained the PDs for the three first bound electron levels. For that purpose, one has first to extract for each location z. of the probe the energy differences E; -E, (n,m 53) which appear in (2 ) from the E,-HH, (n 5 3 ), E,LH, (n I2 ), E ,-HHX and E2-LHI optical transition energies in the corresponding sample, and in a reference (unperturbed) QW. The detail of this procedure has been extensively described elsewhere [ 11,121. Now, (2) is restricted to the set of the three first levels under the form:
well-
l
160
80
Indium plane
--GaAs
0
position (A)
Aluminum
la)
80 plane
160 position
431
(it
Ib)
Fig. 5. Experimental transition energies at 10 K, as functions of the position of the isoelectronic probe (a) In plane; (b) Al plane. The dots plotted at -40 8, and 200 Bi are those of the reference sample. (m) E,-HH,; (0) E,-LH,; (*) E,-HH,; (0) E2-LH2, (0 ) ES-HH3; (A ) E4-HH+ For clarity, the E,-HH, transitions are not indicated.
shifts of the levels due to a localized perturbation. When the position of the probe is varied, the spatial dependence of the PDs is mapped. This technique has been successfully applied to the case of a GaAs/ GaO,,A1,,jAs QW [ 111 (GaAs: 200 A; GaAlAs: 85 A); 1 monolayer thick attractive (InAs) and repulsive (AlAs) probes have been used in two independent series of experiments. For each kind of probe, a series of 6 period multi-QW samples has been grown by MBE (one for each probe position) and studied by PL excitation and wavelength-modulated reflectivity at low temperature (10 K). The experimental transition energies of the QWs are plotted in fig. 5
$ IFt(zo)lZ= L I=,
E:-E,
i= 1,3 )
W,’
(3)
where W, includes a first-order correction (taken independent of zo) on the potential W, due to the coupling with the remote levels. We use for each (F, (’ the normalization condition on one period of the multi-QW structure to eliminate the W,‘s. This constitutes a major interest of this technique; no detailed knowledge of the perturbing potential is required to extract the PD’s. The reproducibility of the IPS from sample to sample (at a submonolayer scale) is far more important in this experiment than the
b)
(al 0.15 -7
0.10
“5 O’ 0.05 0.00 j 80
Fig. 6. Experimental
PD’sp,=
160
0
80
160
0
80
160
IF,I’ for the thiee first bound electron levels, compared to the theoretical effective-mass triangles indicate the data obtained using the In and Al probes, respectively.
result. Dots and
438
J.M. Gerard, J. Y. Marzin/Optical study ofprobability densities in quantum well eigenstates
precise strength of the perturbing potential. The “experimental” PD’s which result of this procedure are in good agreement with the theoretical effective mass estimate, for both kinds of probes, as shown in fig. 6. The error bars on the PD’s are plotted for three standard deviations; they include the effect of some simplifying assumptions in the treatment of the data and the errors on the determination of the transition energies. The lack of precision on the higher transition energies is mainly responsible for the increasing error bars on 1F, ) *, 1F212 and IF312. The fundamental limitations of this technique are related to the cut-off in the spatial frequencies of the experimental PDs which are introduced by the sampling of z, ( lo- 15 A between successive values), and the fact that a finite set of levels is considered in our treatment of the data. The latter point introduces an error on the experimental PD’s of the same typical variation length as the first neglected level, F4 (d/ (4x) = 12 A), and whose amplitude is smaller for the lower energy levels. The effect of the finite width (3 A) of the probe is negligible, when compared to those limiting factors.
5. Conclusion The insertion of built-in isoelectronic planar probes is a powerful and elegant approach to study locally the electronic properties of heterostructures. This
technique can also be used to tailor the electronic properties of a given QW (transition energies, shape of the probability densities . ..). Due to its very general character, it should find numerous novel applications in the future.
References [ 1 ] H. Mariette, J. Chevallier and P. Leroux-Hugon, Phys. Rev. B21 (1980) 5706. [2] T.C. Hsieh, T. Miller and T.C. Chiang, Phys. Rev. Lett. 55 (1985) 2483. [ 31 M.J. Caldas, A. Fazzio and A. Zunger, Appl. Phys. Lett. 45 (1984) 671. [ 41 G. Abstreiter, H. Brugger, T. Wolf, H. Jorke and J.H. Herzog, Surf. Sci. 174 (1986) 640. [5] Seee.g., G. Bastard, Phys. Rev. B 24 (1981) 5693. [ 61 G. Danan, B. Etienne, F. Mollot, R. Planel, A.M. Jean-Louis, F. Alexandre, B. Jusserand, G. Le Roux, J.Y. Marzin, H. Savary and B. Sermage, Phys. Rev. B 35 (1987) 6207. [ 7 ] K.J. Moore, G. Duggan, P. Dawson and C.T. Foxon, Phys. Rev. B 38 (1988) 3368. [8] J.Y. Marzin, M.N. Charasse and B. Sermage, Phys. Rev. B 31 (1985) 8298. [9] M.J. Joyce, M.J. Johnson, M. Gal and B.F. Usher, Phys. Rev. B 38 (1988) 10978. [ lo] J. Menendez, A. Pinczuk, D.J. Werder, S.K. Sputz, R.C. Miller, D.L. Sivco and A.Y. Cho, Phys. Rev. B 36 (1987) 8165. [ 111 J.Y. Marzin and J.M. Gerard, Phys. Rev. Lett. 62 (1989) 2172. [ 121 J.Y. Marzin and J.M. Gerard, in: Proc NATO Advanced Research Workshop on Spectroscopy on Semiconductor Microstructures, Venice, May 1989 (Plenum, New York), to be published.
OPTICAL Jean-Michel
STUDY
OF PROBABILITY
GERARD
433
DENSITIES
IN QUANTUM
WELL EIGENSTATES
* and Jean-Yves MARZIN **, 196, Av. Henri Ravera. 92220 Bagneux, France
C’NET-Laboratoire de Bagneux Received
I 1 July 1989; accepted for publication
14 September
1989
A novel technique is applied to the study of the local electronic properties of heterostructures. We insert a planar isoelectronic probe in the structure, and extract information on the probability densities in the bound states from an optical study of the effect of this local perturbation. This technique is used to reveal directly type II configurations in semiconductor superlattices. The spatial variation of the probability densities in the first bound electron states was also measured for the first time in the case or a GaAs/GaAlAs quantum well.
1. Introduction Since the earlier studies of semiconductor heterostructures, it has been a challenge to obtain experimental information on the spatial distribution of the probability densities in the confined electronic states. Many properties attest the validity of envelope function calculations, which predict particle-in-a-box type probability densities, but very few experiments allow one to extract directly information on these quantities. For some superlattices, it is even difficult to known in which material the confined particles are localized. We have performed an isoelectronic planar substitution (IPS) in these layered structures to obtain such a kind of information. A powerful approach to study the structural and electronic properties of solids is indeed to incorporate impurities in the material and use them as built-in probes. Isoelectronic [ 1,2], deep-level [ 3 ] or charged [ 4 ] impurities have been used for a wide range of applications. However, in layered structures, the IPS is particularly attractive. Unlike charged impurities (which create Coulomb potentials) or isolated isoelectronic impurities (which generally distort the host matrix), an IPS introduces a short range perturbation in the structure.
* Member
of the Direction des Recherches niques, French Ministry of Defense. ** Unite associee No. 250 du Centre National Scientifique. 0039-6028/90/$03.50 (North-Holland )
Etudes
et Tech-
de la Recherche
0 Elsevier Science Publishers
B.V.
In this paper, we first show that the probability densities in the unperturbed quantum well (QW) eigenstates at the location of the planar probe can be extracted from the energy shifts of the QW levels. We first apply this technique to test the occurrence of type II band configurations in some multi-QW (MQW) structures: an IPS of the sublayers of larger bandgap probes the localization of QW eigenstates in these layers. In a second part, the location of the probe is varied from sample to sample so as to determine experimentally the spatial dependence of the probability densities in the bound electron levels of a GaAs/GaAlAs QW. For both studies, the MQW samples are grown by molecular beam epitaxy. which allows a precise control, and an excellent reproducibility of the MQW parameters from sample to sample. The shifts of the energy levels due to the perturbation are extracted from a study of the optical transitions in the perturbed samples and in a reference sample without perturbation.
2. Localized perturbation of a quantum well
The electronic properties of the QW are well described by effective mass models [ 51. In this framework, a series of energy levels for each band of interest are found as eigenenergies of the effective Hamiltonian:
where A is a high symmetry point of the Brillouin zone, m* the effective mass associated to the band n near A, and V,,, the square potential which describes the variation of the energy position of the nth band extremum at A across the heterostructure. The eig~nfunctions of this effective Hamiltonian are the enveloppe functions FFA( Z) of the QW levels at the Brillouin zone center. An IPS affects this series of energy levels only via a change Ali,,* of V,,, on the substituted area. If the perturbation localized at z0 has a width AzO, the firstorder shift of the energy level E, is equal to AZ, AV,,4 iF~4(zo) 12. This approach is valid for a small strength IV= AZ, AV!,, of the pe~urbation, and more precisely when the shift of the levels are small with respect to the level spacing. For the first level and a repulsive probe, the maximum first-order shift 2 W/ d should for example be compared to the spacing Es--E, which is typically 3 h2/8m*d2. A stronger perturbation couples efficiently the Fi levels, for given n,A. Now, for a delta perturbation localized at z,, two sets of eigenenergies E: are found for the perturbed problem: - the E, for which F,(q) =O, - the roots of the equation: G,,(E)=
2 I
‘;y2 =:+,.
(2)
I
F/(ZO)#O
The function GZ, decreases from +a to --co between two adjacent values of E, corresponding to states with F,(z,) # 0. For a repulsive potential, I/ W is positive, and the nth eigenenergy EL of the perturbed Hamiltonian is found in the segment [E,,,En_+., 1. If IV is negative, E’, is lower than the fundamental E, of Ho, and El, for n # 0 is in the segment [E,_,,E,]. These general results are illustrated in fig. 1; the energies of the bound electron levels of a GaAsf Ga0.7A10.XAs quantum well are calculated in an effective mass model, and plotted as a function of the location of an attractive (a) or repulsive (b) perturbation, for various strengths W. Several approaches are then possible to get a local information on the probability densities in the un-
et a
b
1 ‘..__
i
____--’,‘I’~
e,
Fig, 1.The calculated bound electron levels of a 200 8, thick QW, are plotted as a function of the position of a perturbing repulsive (a) or attractive (b) delta potential. The strength of the potential is 0.25.0.5 and 50 in units of h*/8m*d for the dashed, dashdotted, and full lines.
perturbed QW (i.e. on the IF, ( zo) I ‘) from the level shifts due to the perturbation We: - For small 1W( , the dependence of the level shifts already reflect the shape of the probability densities in the bound levels of the unpe~urbed QW. Larger / WI are however generally required experimentally to measure the shifts E; -En with enough precision to extract the I Fi( zo) / 2. - The resolution of system (2 ), truncated to the finite set of the lower energy levels which are accessible experimentally, allows to extract approximate values for the ]F,(z,) I2 from the measure of the E: and E,. - The pe~urbations of the well and barrier layers have drastically different effects for a confined level: the study of the energy shift E; -E, gives therefore a binary answer concerning the localization of the considered particle (band n, band extrema A) in the perturbed layer. The two latter approaches are implemented in the following parts.
3. Direct identi~~ation
of type II superlatti~es
In a semiconductor superlattice (SL), the electrons and the holes can be both confined in the same sublayers of the material of smaller bandgap (type I
XM. Gerard, J. Y. MarzinlOpticaistudy
ofprobability densities in quantum well eigenstates
SL), or spatially separated and localized in different sublayers (type II SL). A IPS of the layers of larger bandgap can be used to test directly the occurrence of a type II band con~guration, since large shifts of an optical transition of the SL will indicate that one of the involved electron or hole levels is localized in the perturbed sublayer. Either a third material or the SL’s constituent of lower bandgap can be used to perform the IPS. The latter case is of particular interest: the pe~urbation is then attractive (respectively repulsive) for a particle confined in the sublayer of smaller (larger) bandgap. For a type I system, a (generally small) reduction of the SL’s optical transition energies is always expected. If detected, a shift toward higher energies of these transitions reveals then unambiguously a type II band configuration for the studied SL. This approach has been applied to two systems which displays non-trivial electronic properties: GaAs/AlAs indirect SL’s and InO.,,GaO,,SAs/GaAs strained SL’s. The band co~~guration for a GaAs/ AlAs SL is shown in fig. 2a [ 6,7]. Since the X conduction extrema lies in AlAs at a lower energy than in GaAs, the AlAs layers act as wells for the X valley conduction states of the SL. For a constant width of AlAs and the thinner GaAs layers, the SL’s bandgap is related to the transition Xi-HH, involving the first X conduction state and the first heavy hole state, which are localized in different sublayers. On the other hand, two different band configurations have been proposed for In,YGa,_,As/GaAs strained SL’s, of mixed type for x= 0.15 [ 8,9] (type I for heavy holes and type II for light holes as in fig. 2c), and of type I for heavy and light holes for x=0.05 [9,10]
(fig. 2b). In both cases the band discontinuity I$, for the light holes is however rather small ( -20 meV). All these results have been obtained via a determination of the band discontinuities which is based on a fit of the optical transitions. It is thus interesting to get a direct and reliable identification of type II conligurations. This is particularly the case for InGaAs/GaAs SL’s, since the sample parameters and even the alloy bandgap and deformation potentiafs are not known with enough precision to allow a perfect confidence in the extracted value of l$,. We present below some results obtained performing a IPS in both kinds of SLs. To validate this approach, we first studied an indirect GaAs/AlAs SL (SL A, GaAs: 32.5 A, AlAs: 110 A) and two perturbed SL’s obtained by a IPS of thin GaAs layers at the middle (SL Al ), or at the first and second thirds (SL A2) of the AlAs sublayers. Their low temperature photoluminescence (PL) spectra are shown in fig. 3. The two sets of peaks seen for SL A correspond to the Xi-HH, indirect transition and its phonon replica, and to the higher energy F,-HH, transition. This direct I,-HH, transition is also detected in PL for SL Al and in PL excitation in SL A2; it is clearly insensitive to an IPS in the AlAs layer since the conduction and valence
AIAsr (5
-__ 4__J-GaAs
bi
-___ X
-6
GaAs
GaAs --L-“-Jr
-3i51\
InGaAs
-1lOi
_I$/LH L-----J
““Yl---L-r ai
435
bl
cl
1 IO
1.74 ENERGY feYI
1.10
GaAs AlAs
1
Fig. 3. 10 K PL f-) and PL excitation (- -) spectra for a GaAs/ Fig. 2. (a) Energy band configuration in a GaAs/AIAs SL for the conduction band (at T and X) and valence band extrema. (b) and (c): Two possible configurations for the T conduction and valence band extrema in a GaInAs/GaAs SL.
AlAs SL, unperturbed (a) and perturbed (b) and (c). The com-
position of one period is shown for each SL. Open (full) arrows mark the calculated energyofthe XI-HH, (f,-HH, ) transitions (Exciton binding energies are not included.)
J.M. Gerurd, J. Y. .~arz~~/O~~lral study o~probabill~.vdensities in q~ani~~ well eigenstntes
436
band extrema at F display a type I configuration. On the other hand, the large increase of the X,-HI-I, transition energy (in the 15 and 30 meV range for SL Al and A2 respectively) confirms unambiguously that the fundamental X, electron level is localized in the ALAS sublayers of the SL. This technique has also been successfully applied to the In,,,Ga,,8,As/GaAs SL 3 (GaInAs: 89 A, GaAs: 200 A). The 8 K transmission spectra of SL B and of the perturbed SL obtained for two different kinds of probes are viewed in fig. 4. The three transitions observed for SL B have been assigned to ElHH,, E,-LH, and E2-HH2 transitions for a similar sample [ 8 7. They are also observed in the perturbed SL’s and have the same nature since their energy shifts are small from sample to sample. An overall increase of the transition energies is observed when a 2 8, thick AlAs layer is inserted in the GaAs layer (SL I3 I), since this probe is repulsive for both electrons and holes. This effect is larger for ElLH, (+5 meV) than for E,-HH, (+ 1 meV) which indicates that the light holes have a large probability density (PD) in the GaAs layers. We show below that the shift of LHi (at least i-4 meV) reveals in fact that they are mainly localized in these layers. For a type I system, the PD of the particle is smaller than the average value of the PD over the SL’s period (in our case l/289 = 3.4 x 1Om3A- ’ ) in the barrier layer. The strength W of the perturbation is for the holes
-89i -2OOi
ENERGY
GaInAs GaAs
4. Experimental determination of probability densities in quantum well eigenstates
IeV)
Fig. 4. 10 K transmission spectra of a Gao.ssIno.,,As/GaAs unperturbed (a) and perturbed (b) and (c). Thecomposition one period is also shown for each SL.
Az,AV= I .06 A eV, where AV is the valence band offset between GaAs and AlAs. Since the shift of the first level LH, due to the perturbation is majored by its first-order estimate, it cannot exceed 3.5 meV for a type I system. This first evidence of a type II configuration for the electrons and light holes relies however on the precise knowledge of I@. We therefore use in a second step, as for GaAs/ AlAs SLs above, an IPS using the SL’s material of smallest band gap: here, two 10 A thick Ga0.8sIn0,,,As layers are inserted at the first and second thirds of the GaAs sublayers of the SL B2. The expected effect is observed in fig. 4c: whereas Er-HH, ( - 1 meV) and E,-HHZ ( - 8 meV ) transition energies are reduced, an increase is detected for Ei-LH, ( + 2 meV). These effects confirm that the second transition does not belong to the I$-HH, family (and is thus ElLH, ), and evidences a type II band alignment for the conduction and light hole band extrema. Finally, two probes have been used so as to relax the condition LH,
SL, of
As mentioned above, the PD for a discrete set of levels can be probed locally, and extracted from the
J.M. Gerard, J. Y. Marzin/Optical study ofprobability densities in quantum well eigenstates
-GaAs
.-*
0
well-
..
as functions of the location of the probe. The energy shifts of E,--HH, or E,-LH, transitions already reflect the shapes of the PD in the nth electron or hole level (since these are roughly similar for a type I system), and are as expected of opposite signs for attractive and repulsive probes. Finally, they are large compared to the experimental uncertainties (1 meV) but also of the order of the spacing between the transitions so that the coupling of the QW levels induced by the perturbation has to be taken into account. In this experiment, we have obtained the PDs for the three first bound electron levels. For that purpose, one has first to extract for each location z. of the probe the energy differences E; -E, (n,m 53) which appear in (2 ) from the E,-HH, (n 5 3 ), E,LH, (n I2 ), E ,-HHX and E2-LHI optical transition energies in the corresponding sample, and in a reference (unperturbed) QW. The detail of this procedure has been extensively described elsewhere [ 11,121. Now, (2) is restricted to the set of the three first levels under the form:
well-
l
160
80
Indium plane
--GaAs
0
position (A)
Aluminum
la)
80 plane
160 position
431
(it
Ib)
Fig. 5. Experimental transition energies at 10 K, as functions of the position of the isoelectronic probe (a) In plane; (b) Al plane. The dots plotted at -40 8, and 200 Bi are those of the reference sample. (m) E,-HH,; (0) E,-LH,; (*) E,-HH,; (0) E2-LH2, (0 ) ES-HH3; (A ) E4-HH+ For clarity, the E,-HH, transitions are not indicated.
shifts of the levels due to a localized perturbation. When the position of the probe is varied, the spatial dependence of the PDs is mapped. This technique has been successfully applied to the case of a GaAs/ GaO,,A1,,jAs QW [ 111 (GaAs: 200 A; GaAlAs: 85 A); 1 monolayer thick attractive (InAs) and repulsive (AlAs) probes have been used in two independent series of experiments. For each kind of probe, a series of 6 period multi-QW samples has been grown by MBE (one for each probe position) and studied by PL excitation and wavelength-modulated reflectivity at low temperature (10 K). The experimental transition energies of the QWs are plotted in fig. 5
$ IFt(zo)lZ= L I=,
E:-E,
i= 1,3 )
W,’
(3)
where W, includes a first-order correction (taken independent of zo) on the potential W, due to the coupling with the remote levels. We use for each (F, (’ the normalization condition on one period of the multi-QW structure to eliminate the W,‘s. This constitutes a major interest of this technique; no detailed knowledge of the perturbing potential is required to extract the PD’s. The reproducibility of the IPS from sample to sample (at a submonolayer scale) is far more important in this experiment than the
b)
(al 0.15 -7
0.10
“5 O’ 0.05 0.00 j 80
Fig. 6. Experimental
PD’sp,=
160
0
80
160
0
80
160
IF,I’ for the thiee first bound electron levels, compared to the theoretical effective-mass triangles indicate the data obtained using the In and Al probes, respectively.
result. Dots and
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J.M. Gerard, J. Y. Marzin/Optical study ofprobability densities in quantum well eigenstates
precise strength of the perturbing potential. The “experimental” PD’s which result of this procedure are in good agreement with the theoretical effective mass estimate, for both kinds of probes, as shown in fig. 6. The error bars on the PD’s are plotted for three standard deviations; they include the effect of some simplifying assumptions in the treatment of the data and the errors on the determination of the transition energies. The lack of precision on the higher transition energies is mainly responsible for the increasing error bars on 1F, ) *, 1F212 and IF312. The fundamental limitations of this technique are related to the cut-off in the spatial frequencies of the experimental PDs which are introduced by the sampling of z, ( lo- 15 A between successive values), and the fact that a finite set of levels is considered in our treatment of the data. The latter point introduces an error on the experimental PD’s of the same typical variation length as the first neglected level, F4 (d/ (4x) = 12 A), and whose amplitude is smaller for the lower energy levels. The effect of the finite width (3 A) of the probe is negligible, when compared to those limiting factors.
5. Conclusion The insertion of built-in isoelectronic planar probes is a powerful and elegant approach to study locally the electronic properties of heterostructures. This
technique can also be used to tailor the electronic properties of a given QW (transition energies, shape of the probability densities . ..). Due to its very general character, it should find numerous novel applications in the future.
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