- Email: [email protected]
InP ultrathin quantum wells
Physica B 301 (2001) 233}238
Optical properties of InAs/InP ultrathin quantum wells Virginie Albe*, Laurent J. Lewis De& partement de Physique et Groupe de Recherche en Physique et Technologie des Couches Minces (GCM), Universite& de Montre& al, Case Postale 6128, Succursale Centre-Ville, Montre& al, Que& bec, Canada H3C 3J7 Received 6 June 2000; received in revised form 6 December 2000
Abstract The optical properties of ultrathin InAs impurity layers embedded in bulk InP are investigated. Our calculations are based on a tight-binding description of the electronic structure, with spin-orbit interactions and strain e!ects included in a consistent manner. It is shown that the energy gap increases with decreasing number of InAs monolayers. In the limit of a single InAs monolayer, the energy gap is found to be 120 meV less than that of bulk InP. Our results are in good agreement with experimental data as far as the heavy-hole}electron transition is concerned. The energy di!erence between optical transitions is, however, in disagreement with both experiment and e!ective-mass calculations. 2001 Elsevier Science B.V. All rights reserved. PACS: 78.66.!w; 78.66.Fd; 78.20.Bh Keywords: Tight-binding model; Electronic structure; Energy transitions; InAs/InP; InAs/GaAs
1. Introduction In recent years, considerable attention has been devoted to the experimental study of ultrathin layers of impurity atoms embedded in bulk host material, i.e., quantum wells (QWs) in the case of semiconductor materials [1}7]. These structures show large radiative e$ciencies and are therefore promising candidates for highspeed and optoelectronic-device applications [8}11]. Their physical properties, however, remain largely unresolved. From a theoretical viewpoint,
* Corresponding author. V. Albe, ENFA, B.P. 87, 31326 Castanet Tolosan, France. Tel.: #33-561-753224; fax: #33-561750309. E-mail address: [email protected] (V. Albe).
QWs are usually described by macroscopic models such as the envelope-function approximation (EFA) and it is assumed that such models are valid even in the case of ultrathin layers. There has been much less e!ort in describing QWs at the microscopic level. Recent tight-binding calculations [12] indicate that the EFA is not adequate to describe the electronic properties of InAs impurity layers in GaAs [13]; because of the large lattice mismatch between InAs and GaAs (about 6.7%), QW thickness is limited to the range 0}1.6 monolayers (MLs) in this case [3,14,15]. In view of these results, the InAs/InP system, which has a much smaller mismatch (about 3.1%), can provide an interesting test of the validity of electronic structure models. Indeed, InAs layers can be grown pseudomorphically on InP with thickness up to a few MLs, thus allowing a
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comparison between experimental data and theoretical values. The photoluminescence spectra of InAs/InP ultrathin QWs generally show multiple lines which are associated with InAs layers of integer thickness [5,25,26]. The positions of the optical transitions can be deduced from the photoluminescence spectra and compared to theoretical predictions. Up to now, atomic-scale studies of ultrathin InAs/InP QWs have only been used to determine the energy gap and valence band o!set at the InAs/InP interface [16]. In this paper, we examine this system for InAs thickness varying from 1 to 6 MLs using an empirical tight-binding model to describe the electronic structure and optical transitions. We "nd that the energy gap increases with decreasing number of InAs monolayers; in the limit of a 1-ML thick InAs QW, the energy gap is 120 meV less than that of bulk InP. Our results are in good agreement with experimental data as far as the heavy-hole}electron transition is concerned. The energy di!erence between optical transitions, however, is in disagreement with e!ective-mass calculations. The heavy hole is localized in the InAs QW, while the lowest electron state is located in the host InP material, which is con"rmed by recent photoluminescence experiments.
2. Model The calculations reported here are based on the sps* tight-binding [17], with both spin}orbit interactions and strain e!ects taken into account in the electron Hamiltonian. We considered ultrathin InAs/InP QWs constructed as supercells containing between 164 and 184 atoms; periodic boundary conditions were used to eliminate surface e!ects. In-plane atoms occupy the positions of a perfect zinc-blende lattice, thus corresponding to pseudomorphic growth conditions. The distances between In and As atomic planes are, however, not those of the zinc-blende lattice; they have been calculated within the macroscopic theory of elasticity for a lattice mismatch of 3.13% between InAs and InP. Indeed, recent ab-initio calculations [18] and high-resolution X-ray di!raction measurements [19] indicate that even in the single mono-
layer limit, the macroscopic theory of elasticity does not break down. At the InAs/InP interface, we have scaled the on-site energies of the common cation (In) atom by a factor of in order to smooth out a little bit the interface boundary; this method has been used within empirical pseudopotential calculations [20] for various AlAs/GaAs systems, as well as tight-binding calculations [21] for InAs/GaAs structures, and found to be very reliable. In order to account for the e!ect of strains on the electronic structure arising from the lattice mismatch, the interatomic matrix elements were scaled according to Harrison's law [21]
d L?@ , H "H ?@ ?@ d where and are atomic orbitals, d and d are the equilibrium and distorted bond lengths, respectively, and n are parameters adjusted so as to repro?@ duce the deformation potentials. In order to perform this "t, we have set all n "2 except ?@ n "3.54, following Ref. [12]; we thus obtain the band-gap and uniaxial deformation potentials, a"!6.2 eV and b"!1.8 eV, respectively, close to the experimental values a"!6.0 eV and b"!1.8 eV [22]. For InAs, we used the parameters of Ref. [12]. To account for strain e!ects, the on-site energies of p-symmetry orbitals have been modi"ed as follows: E VW "E #b ( ! ) VV XX and E X "E !2b ( ! ), VV XX where and are the in-plane and interplane VV XX strain components, respectively. b is a constant "tted to reproduce the deformation potential b (b "0.7). If nearest-neighbour interactions only are considered, 23 parameters are needed to describe InP and pseudomorphically strained InAs; they are given in Table 1. In the tight-binding model, the valence band o!set E between the two materials is assumed to be a constant, and is added to the diagonal elements of the Hamiltonian matrix. In our case, it is the InAs on-site energies that are
V. Albe, L.J. Lewis / Physica B 301 (2001) 233}238
shifted by E (relative to bulk InP), since its valence-band edge is higher in energy when it forms an interface with InP. The value E "0.42 eV, taken from the "rst-principles calculations of Ref. [23], has been used. The electronic states*energies and wavefunctions* are then obtained by direct diagonalization; here we consider only the point.
Table 1 Tight-binding parameters (in eV) for InAs [12] and InP [31] in the sps* basis including spin}orbit interactions. a and c stand for anion and cation, respectively. and are anion and cation spin}orbit parameters, respectively
E(s, a) E(s, c) E(p, a) V W E(p, a) X E(p, c) V W E(p, c) X E(s*, a) E(s*, c) <(s, s) < VV WW < XX < VW < VX WX <(sa, pc) V W <(sa, pc) X <(sc, pa) V W <(sc, pa) X <(s*a, pc) V W <(s*a, pc) X <(s*c, pa) V W <(s*c, pa) X
InAs
InP
!9.5381 !2.7223 0.6757 0.9685 3.4858 3.7786 7.2724 6.6090 !5.9828 1.4677 2.7912 4.1215 4.7372 2.9784 3.4234 5.3143 6.1083 3.1742 3.6485 3.6715 4.2200 0.1385 0.1290
!8.5274 !1.4826 0.8285 0.8285 4.0851 4.0851 8.2129 7.0726 !5.3615 1.8801 1.8801 4.2084 4.2084 2.2227 2.2227 5.5642 5.5642 3.4081 3.4081 4.4187 4.4187 0.0244 0.1426
235
3. Results and discussion We have calculated the energy levels of ultrathin InAs/InP QWs with InAs thickness varying from 1 to 4 MLs. Excitonic e!ects have been neglected as they are expected to be small. Indeed, the freeexciton binding energy in bulk InAs is estimated to be about 1 meV [22]. For the GaAs/Al Ga As V \V system, the exciton binding energy for thin layers is possibly higher than in bulk material [24], but in InAs/GaAs, it is estimated to be about 15 meV. A tight-binding analysis including excitonic e!ects [13] indicates a binding of 12.9 meV for 1 ML and 15.7 meV for 2 MLs. These results are corroborated by the observation of high gain lasing at room temperature [11] for 1.5 ML of InAs in bulk GaAs. For the InAs/InP system, the exciton binding energy is expected to be roughly 5 meV [16]. Our results for the `"rst transition energya, i.e., the transition from the heavy-hole (hh) valence state to the "rst conduction state (CS1), are reported in Table 2. We also give, for comparison, the EFA results of Ref. [16], the photoluminescence data of Refs. [5] and [25], as well as the excited photoluminescence data of Ref. [26]. When the number of InAs MLs decreases, quantum con"nement causes the "rst transition energy to increase. For the 1-ML thick QW, the energy gap is found to be 120 meV less than that of bulk InP (1.29 vs 1.41 eV). Good overall agreement with experiment is found. The EFA results are also in good agreement with photoluminescence measurements. However, it should be noted that the thickness of the QW's, in EFA calculations, is "tted to experiment [16], in contrast to tight-binding calculations, where the only adjustable parameter*once the
Table 2 Calculated transition energies (in eV) between the hh and CS1 for InAs/InP quantum wells as a function of thickness (in MLs); also shown for comparison are the results of calculations within the envelope function approximation (EFA), as well as experimental data ML
This work
EFA [16]
Expt. [5]
Expt. [26]
Expt. [25]
1 2 3 4
1.292 1.198 1.166 1.153
1.32 1.24 1.16 1.07
1.34 1.28 1.18 1.11
1.32 1.23
1.28 1.25 1.16 1.09
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V. Albe, L.J. Lewis / Physica B 301 (2001) 233}238 Table 3 Calculated and measured (Ref. [26]) transition energies (in eV) between lh and CS1 for InAs/InP quantum wells as a function of thickness (in MLs); also shown are the energy di!erences between lh-CS1 and hh-CS1 (cf. Table 2) transitions ML
lh-CS1 This work
1 2 3 4 5
Fig. 1. Anion and cation electron densities (in arbitrary units) at the zone centre for the 1-ML thick InAs QW in InP for (a) the heavy-hole (hh) state; (b) the light-hole (lh) state; and (c) the lowest electron state (CS1). Triangles and full lines are for cations, while squares and dotted lines are for anions. The InAs ML corresponds to layer number 41 (or, equivalently, 0, since the system is periodically replicated).
model has been optimized to correctly describe bulk properties*is the valence band o!set E . The tight-binding approach is an atomistic model and no assumptions are made on the `shapea of the wavefunctions or interface matching conditions. Fig. 1(a) shows the electron density of the hh valence state for a 1-ML thick InAs QW embedded in bulk InP (40-ML thick). The hh state is strongly localized on the InAs layer and its wavefunction consists mostly of the p and V p states of the As anion. In contrast, the light-hole W (lh)valence state is localized on the InP layers, as shown in Fig. 1(b), and its wavefunction arises mainly from the p states of the P anion. These X results are a consequence of the presence of
1.318 1.1223 1.191 1.178 1.170
lh-CS1}hh-CS1 Expt.
1.37 1.32
This work 0.026 0.025 0.025 0.025 0.025
Expt.
0.05 0.09
a uniaxial strain in InAs, which causes the degenerate valence-band manifold to split into the hh state of p , p symmetry and the lh state of p symmetry. V W X Fig. 1(c) shows the electron density of the "rst conduction state for the same 1-ML thick QW; it is also found to be localized on the InP layers and is mainly composed of the s-symmetry states of the In cation. The 2-, 3-, and 4-ML thick QWs exhibit a similar behaviour of the wavefunctions. Recent photoluminescence experiments of InAs self-organized quantum dots on InP substrates [27] suggest that the recombination process involves the electrons localized in the InP substrate and the holes localized in the InAs dots. This is consistent with our calculations of the lowest transition energy, which involves the recombination of CS1 electrons in InP with hh localized in InAs. We have also calculated the optical transitions of higher energies for InAs/InP QWs with InAs thickness ranging between 1 and 6 MLs. The results for the transition from the lh valence state to the "rst conduction state are reported in Table 3. Experimental data (only the 2- and 3-ML thick QWs are available) obtained from excited photoluminescence spectra by Leonelli et al. [26] are also given for comparison. Again, here, the transition energies decrease with increasing QW thickness, characteristic of quantum con"nement. The calculated values are in reasonable agreement with experiment. We also compare in Table 3 the energy di!erence between the hh-CS1 and the lh-CS1 transitions. Our tight-binding calculations predict
V. Albe, L.J. Lewis / Physica B 301 (2001) 233}238 Table 4 Calculated and measured (Ref. [9]) transition energies (in eV) between hh and CS1 and between lh and CS1 for InAs QWs in GaAs (rather than InP); also shown are the energy di!erences between the two transitions ML hh-CS1
lh-CS1
1.428 1.318 1.276
1.463 1.506 1.385 1.471 1.371 1.448
states compares well with experimental data for the InAs/GaAs system, while for InAs/InP, our results di!er from both experimental data and EFA calculations.
lh-CS1}hh-CS1
This work Expt. This work Expt. This work Expt. 1 2 3
237
1.477 0.078 1.461 0.153 1.444 0.172
0.014 0.076 0.073
this di!erence to be essentially independent of the thickness of the QW (between 1 and 6 MLs) at about 25 meV. In contrast, excited photoluminescence experiments [26] indicate di!erences of 50 and 90 meV for the 2- and 3-ML thick InAs QWs, respectively. Using the EFA, Bastard and Marzin [28,29] "nd 89 and 136 meV. In order to ensure that the above discrepancy is not the result of some limitation of our tight-binding approach, we have calculated the hh-CS1 and lh-CS1 transition energies in the case of ultrathin InAs QWs in GaAs (rather than InP) using exactly the same procedure as above (with E "0.04 eV following Ref. [30]) The results are listed in Table 4. The calculated values are in good agreement with experiment. The energy di!erence between the hh-CS1 and the lh-CS1 transitions, further, is found to follow*at least qualitatively*the behaviour observed in experiment, thus reassuring us of the validity of our calculations in the case of InAs/InP.
4. Conclusions We have studied the electronic structure and optical properties of ultrathin InAs/InP QWs within a tight-binding model. We "nd that (i) the calculated hh-CS1 transition is in good agreement with the EFA results and experimental data. (ii) The CS1 state is located in InP while the hh valence state is localized in InAs; this is con"rmed by recent photoluminescence experiments [27]. (iii) The energy di!erence between the hh-CS1 and the lh-CS1
Acknowledgements We are grateful to R. Leonelli and P. Paki for useful discussions, and for providing a copy of Ref. [26] prior to publication. This work was supported by grants from the Natural Sciences and Engineering Research Council (NSERC) of Canada and the `Fonds pour la formation de chercheurs et l'aide a` la recherchea (FCAR) of the Province of QueH bec. We are grateful to the `Services informatiques de l'UniversiteH de MontreH ala for generous allocations of computer resources.
References [1] O. Brandt, K. Ploog, R. Bierwolf, M. Hohenstein, Phys. Rev. Lett. 68 (1992) 1349. [2] C.A. Tran, R.A. Ares, V.A. Karasyuk, S.P. Watkins, G. Letourneau, R. Leonelli, Phys. Rev. B 55 (1997) 4633. [3] A. Patane, M. Grassi Alessi, F. Intoni, M. Capizzi, F. Martelli, L. Nazi, L. Lazzarini, G. Salviati, A. Bosacchi, S. Franchi, J. Appl. Phys. 83 (1998) 5529. [4] M.V. Belousov, N.N. Ledentsov, M.V. Maximov, P.D. Wang, I.N. Yasievich, N.N. Faleev, I.A. Kozin, V.M. Ustinov, P.S. Kopev, C.M. Sottomayor Torres, Phys. Rev. B 51 (1995) 14346. [5] R. Leonelli, C.A. Tran, J.L. Brebner, J.T. Graham, R. Tabti, R.A. Masut, S. Charbonneau, Phys. Rev. B 48 (1993) 11135. [6] J.M. Sallese, J.F. Carlin, M. Gailhanou, M. Ilegems, Appl. Phys. Lett. 71 (1997) 2331. [7] C. Monier, M.F. Vilela, I. Serdiukova, A. Freundlich, Appl. Phys. Lett. 72 (1998) 1587. [8] J.H. Lee, K.Y. Hsieh, R.M. Kolbas, Phys. Rev. B 41 (1990) 7678. [9] Q.J. Xing et al., Appl. Phys. Lett. 65 (1994) 567. [10] R.A. Masut, C.A. Tran, M. Beaudoin, R. Leonelli, Proceedings of the SPIE Symposium on Optoelectronic and Micro-Optical Devices, San Jose, Vol. 2398, 1995, p.116. [11] A.R. Goni et al., Appl. Phys. Lett. 72 (1998) 1433. [12] M. Di Ventra, K.A. Mader, Phys. Rev. B 55 (1997) 13148. [13] R.C. Iotti, L.C. Andreani, M. Di Ventra, Phys. Rev. B 57 (1998) R15072. [14] O. Brandt, L. Tapfer, K. Ploog, R. Bierwolf, M. Hohenstein, F. Phillipp, H. Lage, A. Heberle, Phys. Rev. B 44 (1991) 8043.
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[15] O. Brandt, H. Lage, K. Ploog, Phys. Rev. B 45 (1992) 4217. [16] A. Bitz, C. Jordan, M. Di Ventra, K.A. Mader, L.C. Andreani, J.F. Carlin, A. Rudra, J.L. Staehli, Il Nuovo Cimento D 17 (1995) 1367. [17] P. Vogl, H.P. Hjalmarson, J.D. Dow, J. Phys. Chem. Solids 44 (1983) 365. [18] J.E. Bernard, A. Zunger, Appl. Phys. Lett. 65 (1994) 165. [19] Y. Zheng, J.C. Boulliard, B. Capelle, A. Lifchitz, S. Lagomarsino, Europhys. Lett. 41 (1998) 623. [20] K.A. Mader, A. Zunger, Phys. Rev. B 50 (1994) 17393. [21] W.A. Harrison, Electronic Structure and the Properties of Solids, Freeman, San Francisco, 1980. [22] O. Madelung (Ed.), Landolt-BoK rnstein New Series, Numerical Data and Functional Reliationships in Science and Technology, Group III, Berlin, Springer, 1982. [23] S.-H. Wei, A. Zunger, Appl. Phys. Lett. 72 (1998) 2011.
[24] L.C. Andreani, A. Pasquarello, Phys. Rev. B 42 (1990) 8928. [25] R.P. Schneider, B.W. Wessels, J. Appl. Phys. 70 (1991) 405. [26] P. Paki, R. Leonelli, L. Isnard, R.A. Masut, Appl. Phys. Lett. 74 (1999) 1445. [27] B.Z. Wang, Z. Jin, F.H. Zhao, Y.H. Peng, Z.T. Li, S.Y. Liu, Opt. Quantum Electron. 30 (1998) 187. [28] G. Bastard, Wave Mechanics Applied to Semiconductor Heterostructures, Editions de Physique, Les Ulis, Paris, 1988. [29] J.Y. Marzin, J.M. Gerard, P. Voisin, J.A. Brun, in: Strained-Layer Superlattices: Physics Semiconductors and Semimetals, Vol. 32, Academic, San Diego, 1990. [30] N. Tit, M. Peresi, S. Baroni, Phys. Rev. B 48 (1993) 17607. [31] C. Priester, G. Allan, M. Lannoo, Phys. Rev. B 37 (1988) 8519.
Optical properties of InAs/InP ultrathin quantum wells Virginie Albe*, Laurent J. Lewis De& partement de Physique et Groupe de Recherche en Physique et Technologie des Couches Minces (GCM), Universite& de Montre& al, Case Postale 6128, Succursale Centre-Ville, Montre& al, Que& bec, Canada H3C 3J7 Received 6 June 2000; received in revised form 6 December 2000
Abstract The optical properties of ultrathin InAs impurity layers embedded in bulk InP are investigated. Our calculations are based on a tight-binding description of the electronic structure, with spin-orbit interactions and strain e!ects included in a consistent manner. It is shown that the energy gap increases with decreasing number of InAs monolayers. In the limit of a single InAs monolayer, the energy gap is found to be 120 meV less than that of bulk InP. Our results are in good agreement with experimental data as far as the heavy-hole}electron transition is concerned. The energy di!erence between optical transitions is, however, in disagreement with both experiment and e!ective-mass calculations. 2001 Elsevier Science B.V. All rights reserved. PACS: 78.66.!w; 78.66.Fd; 78.20.Bh Keywords: Tight-binding model; Electronic structure; Energy transitions; InAs/InP; InAs/GaAs
1. Introduction In recent years, considerable attention has been devoted to the experimental study of ultrathin layers of impurity atoms embedded in bulk host material, i.e., quantum wells (QWs) in the case of semiconductor materials [1}7]. These structures show large radiative e$ciencies and are therefore promising candidates for highspeed and optoelectronic-device applications [8}11]. Their physical properties, however, remain largely unresolved. From a theoretical viewpoint,
* Corresponding author. V. Albe, ENFA, B.P. 87, 31326 Castanet Tolosan, France. Tel.: #33-561-753224; fax: #33-561750309. E-mail address: [email protected] (V. Albe).
QWs are usually described by macroscopic models such as the envelope-function approximation (EFA) and it is assumed that such models are valid even in the case of ultrathin layers. There has been much less e!ort in describing QWs at the microscopic level. Recent tight-binding calculations [12] indicate that the EFA is not adequate to describe the electronic properties of InAs impurity layers in GaAs [13]; because of the large lattice mismatch between InAs and GaAs (about 6.7%), QW thickness is limited to the range 0}1.6 monolayers (MLs) in this case [3,14,15]. In view of these results, the InAs/InP system, which has a much smaller mismatch (about 3.1%), can provide an interesting test of the validity of electronic structure models. Indeed, InAs layers can be grown pseudomorphically on InP with thickness up to a few MLs, thus allowing a
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comparison between experimental data and theoretical values. The photoluminescence spectra of InAs/InP ultrathin QWs generally show multiple lines which are associated with InAs layers of integer thickness [5,25,26]. The positions of the optical transitions can be deduced from the photoluminescence spectra and compared to theoretical predictions. Up to now, atomic-scale studies of ultrathin InAs/InP QWs have only been used to determine the energy gap and valence band o!set at the InAs/InP interface [16]. In this paper, we examine this system for InAs thickness varying from 1 to 6 MLs using an empirical tight-binding model to describe the electronic structure and optical transitions. We "nd that the energy gap increases with decreasing number of InAs monolayers; in the limit of a 1-ML thick InAs QW, the energy gap is 120 meV less than that of bulk InP. Our results are in good agreement with experimental data as far as the heavy-hole}electron transition is concerned. The energy di!erence between optical transitions, however, is in disagreement with e!ective-mass calculations. The heavy hole is localized in the InAs QW, while the lowest electron state is located in the host InP material, which is con"rmed by recent photoluminescence experiments.
2. Model The calculations reported here are based on the sps* tight-binding [17], with both spin}orbit interactions and strain e!ects taken into account in the electron Hamiltonian. We considered ultrathin InAs/InP QWs constructed as supercells containing between 164 and 184 atoms; periodic boundary conditions were used to eliminate surface e!ects. In-plane atoms occupy the positions of a perfect zinc-blende lattice, thus corresponding to pseudomorphic growth conditions. The distances between In and As atomic planes are, however, not those of the zinc-blende lattice; they have been calculated within the macroscopic theory of elasticity for a lattice mismatch of 3.13% between InAs and InP. Indeed, recent ab-initio calculations [18] and high-resolution X-ray di!raction measurements [19] indicate that even in the single mono-
layer limit, the macroscopic theory of elasticity does not break down. At the InAs/InP interface, we have scaled the on-site energies of the common cation (In) atom by a factor of in order to smooth out a little bit the interface boundary; this method has been used within empirical pseudopotential calculations [20] for various AlAs/GaAs systems, as well as tight-binding calculations [21] for InAs/GaAs structures, and found to be very reliable. In order to account for the e!ect of strains on the electronic structure arising from the lattice mismatch, the interatomic matrix elements were scaled according to Harrison's law [21]
d L?@ , H "H ?@ ?@ d where and are atomic orbitals, d and d are the equilibrium and distorted bond lengths, respectively, and n are parameters adjusted so as to repro?@ duce the deformation potentials. In order to perform this "t, we have set all n "2 except ?@ n "3.54, following Ref. [12]; we thus obtain the band-gap and uniaxial deformation potentials, a"!6.2 eV and b"!1.8 eV, respectively, close to the experimental values a"!6.0 eV and b"!1.8 eV [22]. For InAs, we used the parameters of Ref. [12]. To account for strain e!ects, the on-site energies of p-symmetry orbitals have been modi"ed as follows: E VW "E #b ( ! ) VV XX and E X "E !2b ( ! ), VV XX where and are the in-plane and interplane VV XX strain components, respectively. b is a constant "tted to reproduce the deformation potential b (b "0.7). If nearest-neighbour interactions only are considered, 23 parameters are needed to describe InP and pseudomorphically strained InAs; they are given in Table 1. In the tight-binding model, the valence band o!set E between the two materials is assumed to be a constant, and is added to the diagonal elements of the Hamiltonian matrix. In our case, it is the InAs on-site energies that are
V. Albe, L.J. Lewis / Physica B 301 (2001) 233}238
shifted by E (relative to bulk InP), since its valence-band edge is higher in energy when it forms an interface with InP. The value E "0.42 eV, taken from the "rst-principles calculations of Ref. [23], has been used. The electronic states*energies and wavefunctions* are then obtained by direct diagonalization; here we consider only the point.
Table 1 Tight-binding parameters (in eV) for InAs [12] and InP [31] in the sps* basis including spin}orbit interactions. a and c stand for anion and cation, respectively. and are anion and cation spin}orbit parameters, respectively
E(s, a) E(s, c) E(p, a) V W E(p, a) X E(p, c) V W E(p, c) X E(s*, a) E(s*, c) <(s, s) < VV WW < XX < VW < VX WX <(sa, pc) V W <(sa, pc) X <(sc, pa) V W <(sc, pa) X <(s*a, pc) V W <(s*a, pc) X <(s*c, pa) V W <(s*c, pa) X
InAs
InP
!9.5381 !2.7223 0.6757 0.9685 3.4858 3.7786 7.2724 6.6090 !5.9828 1.4677 2.7912 4.1215 4.7372 2.9784 3.4234 5.3143 6.1083 3.1742 3.6485 3.6715 4.2200 0.1385 0.1290
!8.5274 !1.4826 0.8285 0.8285 4.0851 4.0851 8.2129 7.0726 !5.3615 1.8801 1.8801 4.2084 4.2084 2.2227 2.2227 5.5642 5.5642 3.4081 3.4081 4.4187 4.4187 0.0244 0.1426
235
3. Results and discussion We have calculated the energy levels of ultrathin InAs/InP QWs with InAs thickness varying from 1 to 4 MLs. Excitonic e!ects have been neglected as they are expected to be small. Indeed, the freeexciton binding energy in bulk InAs is estimated to be about 1 meV [22]. For the GaAs/Al Ga As V \V system, the exciton binding energy for thin layers is possibly higher than in bulk material [24], but in InAs/GaAs, it is estimated to be about 15 meV. A tight-binding analysis including excitonic e!ects [13] indicates a binding of 12.9 meV for 1 ML and 15.7 meV for 2 MLs. These results are corroborated by the observation of high gain lasing at room temperature [11] for 1.5 ML of InAs in bulk GaAs. For the InAs/InP system, the exciton binding energy is expected to be roughly 5 meV [16]. Our results for the `"rst transition energya, i.e., the transition from the heavy-hole (hh) valence state to the "rst conduction state (CS1), are reported in Table 2. We also give, for comparison, the EFA results of Ref. [16], the photoluminescence data of Refs. [5] and [25], as well as the excited photoluminescence data of Ref. [26]. When the number of InAs MLs decreases, quantum con"nement causes the "rst transition energy to increase. For the 1-ML thick QW, the energy gap is found to be 120 meV less than that of bulk InP (1.29 vs 1.41 eV). Good overall agreement with experiment is found. The EFA results are also in good agreement with photoluminescence measurements. However, it should be noted that the thickness of the QW's, in EFA calculations, is "tted to experiment [16], in contrast to tight-binding calculations, where the only adjustable parameter*once the
Table 2 Calculated transition energies (in eV) between the hh and CS1 for InAs/InP quantum wells as a function of thickness (in MLs); also shown for comparison are the results of calculations within the envelope function approximation (EFA), as well as experimental data ML
This work
EFA [16]
Expt. [5]
Expt. [26]
Expt. [25]
1 2 3 4
1.292 1.198 1.166 1.153
1.32 1.24 1.16 1.07
1.34 1.28 1.18 1.11
1.32 1.23
1.28 1.25 1.16 1.09
236
V. Albe, L.J. Lewis / Physica B 301 (2001) 233}238 Table 3 Calculated and measured (Ref. [26]) transition energies (in eV) between lh and CS1 for InAs/InP quantum wells as a function of thickness (in MLs); also shown are the energy di!erences between lh-CS1 and hh-CS1 (cf. Table 2) transitions ML
lh-CS1 This work
1 2 3 4 5
Fig. 1. Anion and cation electron densities (in arbitrary units) at the zone centre for the 1-ML thick InAs QW in InP for (a) the heavy-hole (hh) state; (b) the light-hole (lh) state; and (c) the lowest electron state (CS1). Triangles and full lines are for cations, while squares and dotted lines are for anions. The InAs ML corresponds to layer number 41 (or, equivalently, 0, since the system is periodically replicated).
model has been optimized to correctly describe bulk properties*is the valence band o!set E . The tight-binding approach is an atomistic model and no assumptions are made on the `shapea of the wavefunctions or interface matching conditions. Fig. 1(a) shows the electron density of the hh valence state for a 1-ML thick InAs QW embedded in bulk InP (40-ML thick). The hh state is strongly localized on the InAs layer and its wavefunction consists mostly of the p and V p states of the As anion. In contrast, the light-hole W (lh)valence state is localized on the InP layers, as shown in Fig. 1(b), and its wavefunction arises mainly from the p states of the P anion. These X results are a consequence of the presence of
1.318 1.1223 1.191 1.178 1.170
lh-CS1}hh-CS1 Expt.
1.37 1.32
This work 0.026 0.025 0.025 0.025 0.025
Expt.
0.05 0.09
a uniaxial strain in InAs, which causes the degenerate valence-band manifold to split into the hh state of p , p symmetry and the lh state of p symmetry. V W X Fig. 1(c) shows the electron density of the "rst conduction state for the same 1-ML thick QW; it is also found to be localized on the InP layers and is mainly composed of the s-symmetry states of the In cation. The 2-, 3-, and 4-ML thick QWs exhibit a similar behaviour of the wavefunctions. Recent photoluminescence experiments of InAs self-organized quantum dots on InP substrates [27] suggest that the recombination process involves the electrons localized in the InP substrate and the holes localized in the InAs dots. This is consistent with our calculations of the lowest transition energy, which involves the recombination of CS1 electrons in InP with hh localized in InAs. We have also calculated the optical transitions of higher energies for InAs/InP QWs with InAs thickness ranging between 1 and 6 MLs. The results for the transition from the lh valence state to the "rst conduction state are reported in Table 3. Experimental data (only the 2- and 3-ML thick QWs are available) obtained from excited photoluminescence spectra by Leonelli et al. [26] are also given for comparison. Again, here, the transition energies decrease with increasing QW thickness, characteristic of quantum con"nement. The calculated values are in reasonable agreement with experiment. We also compare in Table 3 the energy di!erence between the hh-CS1 and the lh-CS1 transitions. Our tight-binding calculations predict
V. Albe, L.J. Lewis / Physica B 301 (2001) 233}238 Table 4 Calculated and measured (Ref. [9]) transition energies (in eV) between hh and CS1 and between lh and CS1 for InAs QWs in GaAs (rather than InP); also shown are the energy di!erences between the two transitions ML hh-CS1
lh-CS1
1.428 1.318 1.276
1.463 1.506 1.385 1.471 1.371 1.448
states compares well with experimental data for the InAs/GaAs system, while for InAs/InP, our results di!er from both experimental data and EFA calculations.
lh-CS1}hh-CS1
This work Expt. This work Expt. This work Expt. 1 2 3
237
1.477 0.078 1.461 0.153 1.444 0.172
0.014 0.076 0.073
this di!erence to be essentially independent of the thickness of the QW (between 1 and 6 MLs) at about 25 meV. In contrast, excited photoluminescence experiments [26] indicate di!erences of 50 and 90 meV for the 2- and 3-ML thick InAs QWs, respectively. Using the EFA, Bastard and Marzin [28,29] "nd 89 and 136 meV. In order to ensure that the above discrepancy is not the result of some limitation of our tight-binding approach, we have calculated the hh-CS1 and lh-CS1 transition energies in the case of ultrathin InAs QWs in GaAs (rather than InP) using exactly the same procedure as above (with E "0.04 eV following Ref. [30]) The results are listed in Table 4. The calculated values are in good agreement with experiment. The energy di!erence between the hh-CS1 and the lh-CS1 transitions, further, is found to follow*at least qualitatively*the behaviour observed in experiment, thus reassuring us of the validity of our calculations in the case of InAs/InP.
4. Conclusions We have studied the electronic structure and optical properties of ultrathin InAs/InP QWs within a tight-binding model. We "nd that (i) the calculated hh-CS1 transition is in good agreement with the EFA results and experimental data. (ii) The CS1 state is located in InP while the hh valence state is localized in InAs; this is con"rmed by recent photoluminescence experiments [27]. (iii) The energy di!erence between the hh-CS1 and the lh-CS1
Acknowledgements We are grateful to R. Leonelli and P. Paki for useful discussions, and for providing a copy of Ref. [26] prior to publication. This work was supported by grants from the Natural Sciences and Engineering Research Council (NSERC) of Canada and the `Fonds pour la formation de chercheurs et l'aide a` la recherchea (FCAR) of the Province of QueH bec. We are grateful to the `Services informatiques de l'UniversiteH de MontreH ala for generous allocations of computer resources.
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