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Electronic states in near-surface quantum wells
Surface Science 418 (1998) 536–542
Electronic states in near-surface quantum wells S.J. Vlaev a,b, M.R. Muro-Ortega a, D.A. Contreras-Solorio a, V.R. Velasco c,* a Escuela de Fı´sica, Universidad Auto´noma de Zacatecas, Apartado Postal C-580, Zacatecas, 98068 Zac., Mexico b Institute of General and Inorganic Chemistry, Bulgarian Academy of Sciences, Sofia 1113, Bulgaria c Instituto de Ciencia de Materiales, CSIC, Cantoblanco, 28049 Madrid, Spain Received 28 May 1998; accepted for publication 19 September 1998
Abstract We have calculated the energies and the spatial distributions of the electronic bound states of AlGaAs/GaAs near-surface quantum wells. We have employed a semi-empirical sp3s* tight-binding Hamiltonian including spin–orbit coupling, a Green function technique and the slab approximation. We have found blue shifts for the transition energies associated with the main optical transitions when the thickness of the AlGaAs top barrier decreases. The sign and the magnitude of these shifts agree quite well with the experimental data obtained recently. Both types of surface termination, cation and anion, are considered. In the case of As termination we have found a red shift for the transition energy of the first excited optical transition when the top barrier thickness decreases. An interpretation of these results in terms of the spatial distributions and orbital components of the electronic bound states is discussed. © 1998 Elsevier Science B.V. All rights reserved. Keywords: Semiconductor–semiconductor heterostructures; Semi-empirical models and model calculations
1. Introduction Interest in the study of the electronic structure of near-surface quantum wells (NSQWs) and semiinfinite superlattices has been growing in recent years [1–20]. The problem of the interaction between the bound states in a NSQW and the surface states is relevant to both the fundamental and practical aspects of materials science [1–20]. In experimental works [1–4], near-surface quantum wells of Al Ga As/GaAs [1–3] and 0.3 0.7 GaAs/Ga In As [4] have been studied when the 0.8 0.2 top barrier thickness decreases. The possible influence of the (001) GaAs reconstructed surface on * Corresponding author. Fax: +34 91 3720623; e-mail: [email protected]
the photoluminescence spectra of near-surface quantum wells is shown in Ref. [5]. Due to the surface states supporting the nonradiative recombination channel, the intensity of the optical transitions in a NSQW reduces drastically when the well is near the free surface [5–13]. There are several techniques for surface passivation [9–13] which recover the intensity of the radiative transitions. Modulated reflectivity spectroscopy has been used in Ref. [14] to measure optical transitions between above barrier quasi-bound electronic states in surface quantum wells. Photomodulated reflection spectra measurements are carried out in Ref. [1] for a NSQW of Al Ga As/GaAs. A blue shift of the transition 0.3 0.7 energies E(C1–HH1) and E(C1–LH1) is found when the NSQW approaches the free surface [1].
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This result is qualitatively different from the photoluminescence measurements in Refs. [2,3], where the authors have found a red shift for the transition (C1–HH1). When the thickness of the top barrier decreases, the effective barrier height increases, the bound states move apart from the quantum well bottom and a blue shift appears. This kind of blue shift is measured in Ref. [4] for the Ga In As/GaAs quantum well. The red shift in 0.8 0.2 Ref. [2] can be explained as a result of the c(4×4) AlGaAs surface reconstruction. Although simple one-band effective mass models have been used to explain the main features of the experiments, it has been noted [21] that it is difficult to accommodate in these models various critical factors such as the multiband and bandmixing effects, indirect energy bandgap, intervalley transfer and band non-parabolicity. On the other hand, these effects can readily be included in tightbinding models. Because of this, in the present paper we perform realistic tight-binding calculations for the energies and the spatial distributions of the bound electronic states in a near-surface quantum well. We have considered also the effects introduced by the surface layer being an anion or cation atomic layer. This effect cannot easily be included in the former models. The studied NSQW is the structure experimentally grown and measured in Ref. [1].
2. Model and method We consider a rectangular quantum well, near the free surface of its top barrier, the so-called NSQW. A GaAs quantum well with Al Ga As barri0.3 0.7 ers is schematically shown in Fig. 1. The bulk QW is presented in Fig. 1a, the NSQW is sketched in Fig. 1b, and a quantum well embedded in a finite slab can be seen in Fig. 1c. The interface vacuum/Al Ga As (see the left interfaces in 0.3 0.7 Fig. 1b and Fig. 1c) is a real one, while the interface Al Ga As/vacuum (see the right interface 0.3 0.7 in Fig. 1c) does not exist in the real structure and is artificially introduced within the slab approximation. We have varied the distance NL between the QW and the free surface from 0 to 20 monolayers
Fig. 1. Sketch of the Al Ga As/GaAs quantum well. The well 0.3 0.7 width is NM=14, 18 or 25 ML. (a) Bulk quantum well; (b) near-surface quantum well. The top barrier thickness is NL= 0, 1, …, 20 ML. (c) Near-surface quantum well embedded in a finite slab; (NL+NM+NR)=300 ML.
(ML) during the calculations. The size of the whole slab was always 300 ML to ensure a good convergence for the energies of the bound states with a precision of 1 meV. NSQWs of width 14, 18 and 25 ML correspond to those experimentally studied in Refs. [1,2]. The crystal growth direction
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is (001) and the central point (K =0, K =0) of x y the two-dimensional Brillouin zone is studied. We assume an ideal surface in the calculations, which corresponds to the case investigated in Ref. [1], and a possible surface reconstruction as in Ref. [2] is not taken into account. Anion and cation surface atomic layers have been considered in the calculations. We employ a one-electron semi-empirical tight-binding model with sp3s* spin-dependent basis [22,23]. The Schro¨dinger equation in our discrete model in terms of the Green functions is: (EI−H )G=I.
(1)
In Eq. (1), I is the unit matrix, H is the Hamiltonian matrix, G is the Green function matrix, and E is the energy of the system. In a mixed representation (n, m; K , K ) within the slab x y approximation we find: G (K , K ; E )=[EI−H(K , K )]−1 , (2) n,m x y x y n,m where n and m are discrete real space coordinates in the (001) direction. The local density of states in an arbitrary layer of the slab with layer index n can be calculated from the expression [24,25]: −
1 p
Im Tr G (K , K ; E). n,n x y
(3)
In the computation we have added a finite imaginary part between 0.01 eV and 0.0001 eV to the energy to broaden in the computer the d peaks associated with the eigenstates. The ‘‘band-offset’’ 66/34 is taken from Ref. [26 ].
3. Results and discussion Fig. 2 presents the dependency of the bound state energies of the NSQW on the distance NL between the free surface and the QW. Fig. 2a concerns the electrons (C1) and Fig. 2b,c is for the heavy holes (HH1) and light holes (LH1), respectively. All the states are shifted upwards from the bottom of the NSQW when the well approaches the surface, except the LH1 states for the anion terminated surface which shift downwards to the bottom of the well. We have considered three
Fig. 2. The bound state energies of electrons (a), heavy holes (b) and light holes (c) versus the distance NL between the surface and the quantum well. The origin of the energy scale is taken at the top of the AlGaAs valence band. The notation a-c (c-a) means that the surface is terminated in an anion (cation) atomic layer.
different well widths, 14, 18 and 25 ML (cation termination), corresponding to the thicknesses of the samples studied in Refs. [1,2]. The influence of the vacuum/Al Ga As interface is more pro0.3 0.7 nounced for thinner wells. For instance, the energies of the ground electron states shift upwards 42, 25 and 12 meV, respectively for well widths of 14, 18 and 25 ML when the QW moves from the bulk to the surface. The energies of the bound states with smaller effective masses have a larger shift for a fixed well width. In the surface well of 18 ML (cation termination), the HH1 state shifts only 6 meV with respect to the bulk one, while the shift of the LH1 state is 28 meV. The same effect
S.J. Vlaev et al. / Surface Science 418 (1998) 536–542
is present in the 14 and 25 ML quantum wells. The above mentioned results, presented in Fig. 2, explain the dependencies found in our calculations for the transition energies. Fig. 3a and Fig. 3b shows the transition energies E(C1–HH1) and E(C1–LH1) in quantum wells of 14, 18 and 25 ML as a function of the top barrier thickness NL. Both transitions in cation terminated surface, for all well widths, reveal blue shifts when the QW approaches the surface. The narrower the well the larger the blue shift it has. For instance, the bulk/surface shift of the (C1–HH1) transition is 31 meV (15 meV ) for the well of 18 ML (25 ML). For a fixed well width, the (C1–LH1) transition shows a larger shift due to the different
Fig. 3. The energies of: (a) the (C1–HH1) and (b) the (C1–LH1) transitions versus the distance NL between the surface and the quantum well. The notation a-c (c-a) means that the surface is terminated in an anion (cation) atomic layer.
539
effective masses of HH1 and LH1 states. The transition energy E(C1–LH1) increases 53 meV, while the transition energy E(C1–HH1) increases 31 meV in the surface well of 18 ML, cation surface termination. The optical transition (C1–LH1) for anion surface termination reveals a qualitatively different behaviour when the top barrier thickness decreases. The transition energy decreases, and thus we have a red shift. Fig. 4 represents the spatial distributions of the bound states C1, HH1 and LH1 in bulk QW ( Fig. 4a) and surface QW (Fig. 4b,c), both with 18 ML widths. Squared optical matrix elements for GaAs–Ga Al As superlattices [27] have 0.7 0.3 been calculated. As the AlGaAs slab is 20 ML thick, the situation can be considered quite similar to that of a quantum well. Similar calculations for a quantum well considered as a superlattice with wide barriers [28] give similar conclusions. By considering the results of Ref. [27] for a GaAs slab 18 ML thick, it can be seen that the (C1–LH1) transition is less intense than the (C1–HH1) transition. One can reach the same conclusion for the quantum well by using the simple explanation associated with the spatial overlaps for electron and hole states. We can then more confidently use the same simple argument to justify that the intensities of the optical transitions will decrease in NSQW, due to the smaller overlap between the electron and hole wavefunctions, as shown in Fig. 4b and Fig. 4c. The experimental study of the dependence of the transition intensities on the top barrier thickness NL in Refs. [1,2] gives results in qualitative agreement with our theoretical results. The spatial distributions of LH1 states in Fig. 4b,c are very different. When the first atomic layer on the surface is an anion layer, the probability amplitude has a maximum at this layer, see Fig. 4c. The spatial overlap of the LH1 state with the C1 state is sufficient to provide a non-zero intensity of the (C1–LH1) transition. If the surface terminates with a cation atomic layer, only a weak local maximum appears, see Fig. 4b. The curve of the C1 state in Fig. 4b has a similar local maximum at the surface layer. In both types of surface termination, the HH1 state does not show such a weak maximum. We think that the presence of surface potential could explain the above men-
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Fig. 4. Spectral strength of the C1, HH1 and LH1 bound states in (a) bulk, (b) near-surface quantum well terminated in a cation atomic layer and (c) near-surface quantum well terminated in an anion atomic layer versus the layer index. The well width is 18 ML. The vertical dotted lines indicate the position of the well.
tioned differences between the c-a (cation terminated surface) and the a-c (anion terminated surface) cases. This potential has different signs for the a-c and c-a surfaces. The influence of the surface potential will be stronger for more delocalized states at higher energy. The LH1 state contains intense p components and is more easily delocalz ized than the HH1 and C1 states. For this reason the LH1 state feels strongly the surface potential and shifts in opposite directions when the well approaches the a-c or c-a surface, see Fig. 2c. The energy of the C1 state changes slightly and keeps the same sign in agreement with the occurrence of the weak local maxima. The most important orbital component of the C1 state is the s component, which is slightly delocalized. The energy of the HH1 state practically does not change (see Fig. 2b) due to the more localized character of the p and p orbitals. Local maxima do not x y appear in the spatial distributions of the HH1 states. Let us now compare our theoretical results for the cation terminated surface (c-a case) with the experimental data available for the same quantum well structure [1]. In Ref. [1] the authors measured the photomodulated reflection spectra of GaAs quantum wells where the top barrier is confined by thin Al Ga As layers. They observed a sig0.3 0.7 nificant blue shift of the transitions (C1–HH1) and (C1–LH1) as compared with the transitions of a quantum well with a thick barrier. The NSQW width is 18 ML and the surface reconstruction of the (001) face is absent. The thickness of the Al Ga As barrier between the vacuum and the 0.3 0.7 well varies in a wide interval, from about 1000 ML to practically zero. The transition energies decrease slowly in the interval 1000–200 ML. This red shift is small, about 7 meV for the transition (C1–HH1) and about 2 meV for the transition (C1–LH1). After that, with the decreasing of the top barrier thickness, the energies increase and a blue shift of about 10 meV for both transitions is measured for the last 50 ML. Our calculations also show blue shifts for the transitions (C1–HH1) and (C1–LH1) in NSQW of 18 ML. The dependence of these shifts on the distance NL between the vacuum and the well can be seen in Table 1, column c-a. There are several possible reasons for the quanti-
S.J. Vlaev et al. / Surface Science 418 (1998) 536–542 Table 1 Energies (meV ) of the transitions C1–HH1 and C1–LH1 of the NSQW of 18 ML as a function of the distance NL between the vacuum and the well. The notation a-c (c-a) means that the surface terminates with an anion (cation) atomic layer NL (ML)
0 1 2 3 4 5 6 7 8 9 10 11 12
DE(C1–HH1) (meV )
DE(C1–LH1) (meV )
a-c
c-a
a-c
c-a
20 14 7 4 3 1 1 1 0 0 0 0 0
31 21 14 9 5 3 2 1 1 0 0 0 0
−32 −15 −9 −8 −6 −4 −2 −1 −1 −1 −1 −1 0
53 40 28 20 13 9 6 4 3 1 1 0 0
tative differences between the experimentally measured and theoretically calculated shifts. We do not take into account in the calculations the so-called ‘‘built-in’’ electric field. This field gives a small red shift when the top barrier thickness NL decreases, so it could decrease the theoretical values of the blue shifts, thus improving the agreement between the theory and experiment. Although quite recently the use of tight-binding models with electromagnetic fields has been put on formal grounds [29] and it has been noted that the built-in electric field can give a contribution to the energy of bound states [15], we have not included this effect in our calculations due to the fact that in some of the experiments the authors stress that they took efforts to minimize this effect. Another important reason can be the precision of the chemical etching and the thickness determination. Our numerical calculations demonstrate a great sensibility of the transition energies to the top barrier size NL for the last 5–6 ML (see Table 1). If there is no Al Ga As layer at the 0.3 0.7 end of the chemical etching process, NL=0, the bulk-to-surface blue shifts of the (C1–HH1) and (C1–LH1) transitions are 31 meV and 53 meV, respectively. But if the chemical etching finishes with a barrier thickness of 3–4 ML, the above
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mentioned blue shifts are about 9–13 meV, in good agreement with the experimental value of 10 meV [1]. We have not included temperature and excitonic corrections in the numerical calculations, but presumably these corrections are not important for the energy differences entering the transition energies. In fact, we find an energy difference of 23 meV between the bulk QW transitions (C1–HH1) and (C1–LH1), while the experimental value is 22 meV, which is in excellent agreement. The bulk-to-surface transition energy shift for (C1–LH1) optical transition is negative if the surface terminates with anions, see a-c column of Table 1. We would like to emphasize that the first hole state for NL=0, 1, 2 is a light-hole state and not a heavy-hole one. This change occurs only in the a-c case. If the (C1–LH1) transition is sufficiently intense for very small top barrier thickness, then maybe it would be possible to observe this effect experimentally. In Ref. [2] the authors measured a red shift when the barrier thickness NL decreases in Al Ga As/GaAs quantum wells of 14 and 0.3 0.7 25 ML. In our opinion this qualitative difference in the sign of the shift follows from the surface reconstruction of the sample [2]. Probably, as indicated by the authors of Ref. [2], the reconstruction shifts the surface states from the middle of the gap to the band edges above the bound states of the well. Then the spatial overlap between the surface states and the NSQW states can give a red shift for the well transitions. If the surface is ideal, without reconstruction, as in our case, the surface states appear at the center of the gap [30,31] and their interaction with the NSQW states is impossible even for a top barrier of 0–2 ML. In our calculations, the electronic surface states of GaAs appear around the energy values of 1.28 and 1.49 eV for a cation or anion surface termination, respectively. The Al Ga As surface states (x= x 1−x 0.3) have energies of 1.30 and 1.33 eV for a cation and anion surface termination, respectively. These results agree with the published results for the (001) unreconstructed surface of GaAs, for instance see Ref. [32] and references cited therein. The zero of the energy scale in our calculations is taken at the top of the AlGaAs valence band.
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4. Conclusions We have studied the electronic states of AlGaAs/GaAs quantum wells near (001) free surface without reconstruction, considering both cation and anion terminations. The energies of the main optical transitions (C1–HH1) and (C1–LH1) in near-surface quantum wells increase when the top barrier thickness decreases, for the cation surface termination. This blue shift is larger for narrower wells and for smaller effective masses. A good agreement between theory and experiment is found. When the surface terminates with anions a red shift for the (C1–LH1) transition takes place if the top barrier thickness decreases.
Acknowledgements The authors thank Dr. L.M. Gaggero-Sager for helpful discussions and comments and the referees for critical comments and useful suggestions. This work was partially supported by CONACYT (Mexico) through Grants No. 25031E and 1852-P, by the Direccio´n General de Ensen˜anza Superior (Spain) through Grant No. PB96-0916 and by the Cooperation Agreement between the Bulgarian Academy of Sciences and the CSIC (Spain).
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Electronic states in near-surface quantum wells S.J. Vlaev a,b, M.R. Muro-Ortega a, D.A. Contreras-Solorio a, V.R. Velasco c,* a Escuela de Fı´sica, Universidad Auto´noma de Zacatecas, Apartado Postal C-580, Zacatecas, 98068 Zac., Mexico b Institute of General and Inorganic Chemistry, Bulgarian Academy of Sciences, Sofia 1113, Bulgaria c Instituto de Ciencia de Materiales, CSIC, Cantoblanco, 28049 Madrid, Spain Received 28 May 1998; accepted for publication 19 September 1998
Abstract We have calculated the energies and the spatial distributions of the electronic bound states of AlGaAs/GaAs near-surface quantum wells. We have employed a semi-empirical sp3s* tight-binding Hamiltonian including spin–orbit coupling, a Green function technique and the slab approximation. We have found blue shifts for the transition energies associated with the main optical transitions when the thickness of the AlGaAs top barrier decreases. The sign and the magnitude of these shifts agree quite well with the experimental data obtained recently. Both types of surface termination, cation and anion, are considered. In the case of As termination we have found a red shift for the transition energy of the first excited optical transition when the top barrier thickness decreases. An interpretation of these results in terms of the spatial distributions and orbital components of the electronic bound states is discussed. © 1998 Elsevier Science B.V. All rights reserved. Keywords: Semiconductor–semiconductor heterostructures; Semi-empirical models and model calculations
1. Introduction Interest in the study of the electronic structure of near-surface quantum wells (NSQWs) and semiinfinite superlattices has been growing in recent years [1–20]. The problem of the interaction between the bound states in a NSQW and the surface states is relevant to both the fundamental and practical aspects of materials science [1–20]. In experimental works [1–4], near-surface quantum wells of Al Ga As/GaAs [1–3] and 0.3 0.7 GaAs/Ga In As [4] have been studied when the 0.8 0.2 top barrier thickness decreases. The possible influence of the (001) GaAs reconstructed surface on * Corresponding author. Fax: +34 91 3720623; e-mail: [email protected]
the photoluminescence spectra of near-surface quantum wells is shown in Ref. [5]. Due to the surface states supporting the nonradiative recombination channel, the intensity of the optical transitions in a NSQW reduces drastically when the well is near the free surface [5–13]. There are several techniques for surface passivation [9–13] which recover the intensity of the radiative transitions. Modulated reflectivity spectroscopy has been used in Ref. [14] to measure optical transitions between above barrier quasi-bound electronic states in surface quantum wells. Photomodulated reflection spectra measurements are carried out in Ref. [1] for a NSQW of Al Ga As/GaAs. A blue shift of the transition 0.3 0.7 energies E(C1–HH1) and E(C1–LH1) is found when the NSQW approaches the free surface [1].
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This result is qualitatively different from the photoluminescence measurements in Refs. [2,3], where the authors have found a red shift for the transition (C1–HH1). When the thickness of the top barrier decreases, the effective barrier height increases, the bound states move apart from the quantum well bottom and a blue shift appears. This kind of blue shift is measured in Ref. [4] for the Ga In As/GaAs quantum well. The red shift in 0.8 0.2 Ref. [2] can be explained as a result of the c(4×4) AlGaAs surface reconstruction. Although simple one-band effective mass models have been used to explain the main features of the experiments, it has been noted [21] that it is difficult to accommodate in these models various critical factors such as the multiband and bandmixing effects, indirect energy bandgap, intervalley transfer and band non-parabolicity. On the other hand, these effects can readily be included in tightbinding models. Because of this, in the present paper we perform realistic tight-binding calculations for the energies and the spatial distributions of the bound electronic states in a near-surface quantum well. We have considered also the effects introduced by the surface layer being an anion or cation atomic layer. This effect cannot easily be included in the former models. The studied NSQW is the structure experimentally grown and measured in Ref. [1].
2. Model and method We consider a rectangular quantum well, near the free surface of its top barrier, the so-called NSQW. A GaAs quantum well with Al Ga As barri0.3 0.7 ers is schematically shown in Fig. 1. The bulk QW is presented in Fig. 1a, the NSQW is sketched in Fig. 1b, and a quantum well embedded in a finite slab can be seen in Fig. 1c. The interface vacuum/Al Ga As (see the left interfaces in 0.3 0.7 Fig. 1b and Fig. 1c) is a real one, while the interface Al Ga As/vacuum (see the right interface 0.3 0.7 in Fig. 1c) does not exist in the real structure and is artificially introduced within the slab approximation. We have varied the distance NL between the QW and the free surface from 0 to 20 monolayers
Fig. 1. Sketch of the Al Ga As/GaAs quantum well. The well 0.3 0.7 width is NM=14, 18 or 25 ML. (a) Bulk quantum well; (b) near-surface quantum well. The top barrier thickness is NL= 0, 1, …, 20 ML. (c) Near-surface quantum well embedded in a finite slab; (NL+NM+NR)=300 ML.
(ML) during the calculations. The size of the whole slab was always 300 ML to ensure a good convergence for the energies of the bound states with a precision of 1 meV. NSQWs of width 14, 18 and 25 ML correspond to those experimentally studied in Refs. [1,2]. The crystal growth direction
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is (001) and the central point (K =0, K =0) of x y the two-dimensional Brillouin zone is studied. We assume an ideal surface in the calculations, which corresponds to the case investigated in Ref. [1], and a possible surface reconstruction as in Ref. [2] is not taken into account. Anion and cation surface atomic layers have been considered in the calculations. We employ a one-electron semi-empirical tight-binding model with sp3s* spin-dependent basis [22,23]. The Schro¨dinger equation in our discrete model in terms of the Green functions is: (EI−H )G=I.
(1)
In Eq. (1), I is the unit matrix, H is the Hamiltonian matrix, G is the Green function matrix, and E is the energy of the system. In a mixed representation (n, m; K , K ) within the slab x y approximation we find: G (K , K ; E )=[EI−H(K , K )]−1 , (2) n,m x y x y n,m where n and m are discrete real space coordinates in the (001) direction. The local density of states in an arbitrary layer of the slab with layer index n can be calculated from the expression [24,25]: −
1 p
Im Tr G (K , K ; E). n,n x y
(3)
In the computation we have added a finite imaginary part between 0.01 eV and 0.0001 eV to the energy to broaden in the computer the d peaks associated with the eigenstates. The ‘‘band-offset’’ 66/34 is taken from Ref. [26 ].
3. Results and discussion Fig. 2 presents the dependency of the bound state energies of the NSQW on the distance NL between the free surface and the QW. Fig. 2a concerns the electrons (C1) and Fig. 2b,c is for the heavy holes (HH1) and light holes (LH1), respectively. All the states are shifted upwards from the bottom of the NSQW when the well approaches the surface, except the LH1 states for the anion terminated surface which shift downwards to the bottom of the well. We have considered three
Fig. 2. The bound state energies of electrons (a), heavy holes (b) and light holes (c) versus the distance NL between the surface and the quantum well. The origin of the energy scale is taken at the top of the AlGaAs valence band. The notation a-c (c-a) means that the surface is terminated in an anion (cation) atomic layer.
different well widths, 14, 18 and 25 ML (cation termination), corresponding to the thicknesses of the samples studied in Refs. [1,2]. The influence of the vacuum/Al Ga As interface is more pro0.3 0.7 nounced for thinner wells. For instance, the energies of the ground electron states shift upwards 42, 25 and 12 meV, respectively for well widths of 14, 18 and 25 ML when the QW moves from the bulk to the surface. The energies of the bound states with smaller effective masses have a larger shift for a fixed well width. In the surface well of 18 ML (cation termination), the HH1 state shifts only 6 meV with respect to the bulk one, while the shift of the LH1 state is 28 meV. The same effect
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is present in the 14 and 25 ML quantum wells. The above mentioned results, presented in Fig. 2, explain the dependencies found in our calculations for the transition energies. Fig. 3a and Fig. 3b shows the transition energies E(C1–HH1) and E(C1–LH1) in quantum wells of 14, 18 and 25 ML as a function of the top barrier thickness NL. Both transitions in cation terminated surface, for all well widths, reveal blue shifts when the QW approaches the surface. The narrower the well the larger the blue shift it has. For instance, the bulk/surface shift of the (C1–HH1) transition is 31 meV (15 meV ) for the well of 18 ML (25 ML). For a fixed well width, the (C1–LH1) transition shows a larger shift due to the different
Fig. 3. The energies of: (a) the (C1–HH1) and (b) the (C1–LH1) transitions versus the distance NL between the surface and the quantum well. The notation a-c (c-a) means that the surface is terminated in an anion (cation) atomic layer.
539
effective masses of HH1 and LH1 states. The transition energy E(C1–LH1) increases 53 meV, while the transition energy E(C1–HH1) increases 31 meV in the surface well of 18 ML, cation surface termination. The optical transition (C1–LH1) for anion surface termination reveals a qualitatively different behaviour when the top barrier thickness decreases. The transition energy decreases, and thus we have a red shift. Fig. 4 represents the spatial distributions of the bound states C1, HH1 and LH1 in bulk QW ( Fig. 4a) and surface QW (Fig. 4b,c), both with 18 ML widths. Squared optical matrix elements for GaAs–Ga Al As superlattices [27] have 0.7 0.3 been calculated. As the AlGaAs slab is 20 ML thick, the situation can be considered quite similar to that of a quantum well. Similar calculations for a quantum well considered as a superlattice with wide barriers [28] give similar conclusions. By considering the results of Ref. [27] for a GaAs slab 18 ML thick, it can be seen that the (C1–LH1) transition is less intense than the (C1–HH1) transition. One can reach the same conclusion for the quantum well by using the simple explanation associated with the spatial overlaps for electron and hole states. We can then more confidently use the same simple argument to justify that the intensities of the optical transitions will decrease in NSQW, due to the smaller overlap between the electron and hole wavefunctions, as shown in Fig. 4b and Fig. 4c. The experimental study of the dependence of the transition intensities on the top barrier thickness NL in Refs. [1,2] gives results in qualitative agreement with our theoretical results. The spatial distributions of LH1 states in Fig. 4b,c are very different. When the first atomic layer on the surface is an anion layer, the probability amplitude has a maximum at this layer, see Fig. 4c. The spatial overlap of the LH1 state with the C1 state is sufficient to provide a non-zero intensity of the (C1–LH1) transition. If the surface terminates with a cation atomic layer, only a weak local maximum appears, see Fig. 4b. The curve of the C1 state in Fig. 4b has a similar local maximum at the surface layer. In both types of surface termination, the HH1 state does not show such a weak maximum. We think that the presence of surface potential could explain the above men-
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Fig. 4. Spectral strength of the C1, HH1 and LH1 bound states in (a) bulk, (b) near-surface quantum well terminated in a cation atomic layer and (c) near-surface quantum well terminated in an anion atomic layer versus the layer index. The well width is 18 ML. The vertical dotted lines indicate the position of the well.
tioned differences between the c-a (cation terminated surface) and the a-c (anion terminated surface) cases. This potential has different signs for the a-c and c-a surfaces. The influence of the surface potential will be stronger for more delocalized states at higher energy. The LH1 state contains intense p components and is more easily delocalz ized than the HH1 and C1 states. For this reason the LH1 state feels strongly the surface potential and shifts in opposite directions when the well approaches the a-c or c-a surface, see Fig. 2c. The energy of the C1 state changes slightly and keeps the same sign in agreement with the occurrence of the weak local maxima. The most important orbital component of the C1 state is the s component, which is slightly delocalized. The energy of the HH1 state practically does not change (see Fig. 2b) due to the more localized character of the p and p orbitals. Local maxima do not x y appear in the spatial distributions of the HH1 states. Let us now compare our theoretical results for the cation terminated surface (c-a case) with the experimental data available for the same quantum well structure [1]. In Ref. [1] the authors measured the photomodulated reflection spectra of GaAs quantum wells where the top barrier is confined by thin Al Ga As layers. They observed a sig0.3 0.7 nificant blue shift of the transitions (C1–HH1) and (C1–LH1) as compared with the transitions of a quantum well with a thick barrier. The NSQW width is 18 ML and the surface reconstruction of the (001) face is absent. The thickness of the Al Ga As barrier between the vacuum and the 0.3 0.7 well varies in a wide interval, from about 1000 ML to practically zero. The transition energies decrease slowly in the interval 1000–200 ML. This red shift is small, about 7 meV for the transition (C1–HH1) and about 2 meV for the transition (C1–LH1). After that, with the decreasing of the top barrier thickness, the energies increase and a blue shift of about 10 meV for both transitions is measured for the last 50 ML. Our calculations also show blue shifts for the transitions (C1–HH1) and (C1–LH1) in NSQW of 18 ML. The dependence of these shifts on the distance NL between the vacuum and the well can be seen in Table 1, column c-a. There are several possible reasons for the quanti-
S.J. Vlaev et al. / Surface Science 418 (1998) 536–542 Table 1 Energies (meV ) of the transitions C1–HH1 and C1–LH1 of the NSQW of 18 ML as a function of the distance NL between the vacuum and the well. The notation a-c (c-a) means that the surface terminates with an anion (cation) atomic layer NL (ML)
0 1 2 3 4 5 6 7 8 9 10 11 12
DE(C1–HH1) (meV )
DE(C1–LH1) (meV )
a-c
c-a
a-c
c-a
20 14 7 4 3 1 1 1 0 0 0 0 0
31 21 14 9 5 3 2 1 1 0 0 0 0
−32 −15 −9 −8 −6 −4 −2 −1 −1 −1 −1 −1 0
53 40 28 20 13 9 6 4 3 1 1 0 0
tative differences between the experimentally measured and theoretically calculated shifts. We do not take into account in the calculations the so-called ‘‘built-in’’ electric field. This field gives a small red shift when the top barrier thickness NL decreases, so it could decrease the theoretical values of the blue shifts, thus improving the agreement between the theory and experiment. Although quite recently the use of tight-binding models with electromagnetic fields has been put on formal grounds [29] and it has been noted that the built-in electric field can give a contribution to the energy of bound states [15], we have not included this effect in our calculations due to the fact that in some of the experiments the authors stress that they took efforts to minimize this effect. Another important reason can be the precision of the chemical etching and the thickness determination. Our numerical calculations demonstrate a great sensibility of the transition energies to the top barrier size NL for the last 5–6 ML (see Table 1). If there is no Al Ga As layer at the 0.3 0.7 end of the chemical etching process, NL=0, the bulk-to-surface blue shifts of the (C1–HH1) and (C1–LH1) transitions are 31 meV and 53 meV, respectively. But if the chemical etching finishes with a barrier thickness of 3–4 ML, the above
541
mentioned blue shifts are about 9–13 meV, in good agreement with the experimental value of 10 meV [1]. We have not included temperature and excitonic corrections in the numerical calculations, but presumably these corrections are not important for the energy differences entering the transition energies. In fact, we find an energy difference of 23 meV between the bulk QW transitions (C1–HH1) and (C1–LH1), while the experimental value is 22 meV, which is in excellent agreement. The bulk-to-surface transition energy shift for (C1–LH1) optical transition is negative if the surface terminates with anions, see a-c column of Table 1. We would like to emphasize that the first hole state for NL=0, 1, 2 is a light-hole state and not a heavy-hole one. This change occurs only in the a-c case. If the (C1–LH1) transition is sufficiently intense for very small top barrier thickness, then maybe it would be possible to observe this effect experimentally. In Ref. [2] the authors measured a red shift when the barrier thickness NL decreases in Al Ga As/GaAs quantum wells of 14 and 0.3 0.7 25 ML. In our opinion this qualitative difference in the sign of the shift follows from the surface reconstruction of the sample [2]. Probably, as indicated by the authors of Ref. [2], the reconstruction shifts the surface states from the middle of the gap to the band edges above the bound states of the well. Then the spatial overlap between the surface states and the NSQW states can give a red shift for the well transitions. If the surface is ideal, without reconstruction, as in our case, the surface states appear at the center of the gap [30,31] and their interaction with the NSQW states is impossible even for a top barrier of 0–2 ML. In our calculations, the electronic surface states of GaAs appear around the energy values of 1.28 and 1.49 eV for a cation or anion surface termination, respectively. The Al Ga As surface states (x= x 1−x 0.3) have energies of 1.30 and 1.33 eV for a cation and anion surface termination, respectively. These results agree with the published results for the (001) unreconstructed surface of GaAs, for instance see Ref. [32] and references cited therein. The zero of the energy scale in our calculations is taken at the top of the AlGaAs valence band.
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4. Conclusions We have studied the electronic states of AlGaAs/GaAs quantum wells near (001) free surface without reconstruction, considering both cation and anion terminations. The energies of the main optical transitions (C1–HH1) and (C1–LH1) in near-surface quantum wells increase when the top barrier thickness decreases, for the cation surface termination. This blue shift is larger for narrower wells and for smaller effective masses. A good agreement between theory and experiment is found. When the surface terminates with anions a red shift for the (C1–LH1) transition takes place if the top barrier thickness decreases.
Acknowledgements The authors thank Dr. L.M. Gaggero-Sager for helpful discussions and comments and the referees for critical comments and useful suggestions. This work was partially supported by CONACYT (Mexico) through Grants No. 25031E and 1852-P, by the Direccio´n General de Ensen˜anza Superior (Spain) through Grant No. PB96-0916 and by the Cooperation Agreement between the Bulgarian Academy of Sciences and the CSIC (Spain).
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