Superlattices and Microstructures, Vol. 23, No. 2, 1998

Electronic structure of (311)-InAs monolayers embedded in GaAs

Toshio SaitoInstitute of Industrial Science, University of Tokyo, 7-22-1 Roppongi, Minato-ku, Tokyo 106, Japan

(Received 15 July 1996)

We have theoretically studied the electronic structure of a (311)-oriented InAs mono-layer embedded in GaAs by using the semi-empiricalsp3s∗ tight-binding method;(InAs)1/(GaAs)n [311] superlattices withn = 10 ∼ 26 are used for the calculations.The (311)-InAs monolayer induces an electron level (near the conduction-band edge) anda hole level (near the valence-band edge) in the GaAs band gap. The electron–hole energyseparation is smaller than the GaAs band-gap energy by 0.06 eV (n = 10) and 0.03 eV(n = 26). The charge densities of the electron state and the hole state are weakly localizednear the (311)-InAs monolayer.

c© 1998 Academic Press LimitedKey words: GaAs, InAs monolayer, (311) orientation, tight-binding method.

1. Introduction

Ultrathin InAs layers grown pseudomorphically on GaAs substrates with different orientations have at-tracted much attention both from fundamental and application points of view. In particular, the structuraland optical properties of ultrathin InAs layers on (311) GaAs substrates have been experimentally studied indetail in order to investigate the electronic structure of (311)-InAs monolayers and submonolayers [1–5]. It isfound that the energies of the InAs-related optical transitions depend on orientation [3]. The transition energycalculation has been performed by using eight-bandk · p-typeeffective-masstheory [3].

In contrast with the detailed experimental studies, theoretical investigations of the electronic structure ofultrathin (311)-InAs layers based onatomic-scalecalculations, such as a tight-binding theory, have not beenperformed. For (100)-InAs monolayers in GaAs,atomic-scalecalculations using a tight-binding method [6]and a self-consistent pseudopotential method [7] have been performed in order to investigate the electronicstructures.

In this study, we theoretically investigate the electronic structure of a (311)-InAs monolayer embedded inGaAs, giving attention to (311)-InAs-induced electron and hole levels in the GaAs band gap. The (311)-InAs-induced levels relate to the optical transitions observed in experiments [3]. Figure 1 schematically shows theatomic arrangement of the (311)-InAs monolayer in GaAs. The theoretical calculations are performed by usinga semi-empiricalsp3s∗ tight-binding method [8] on (InAs)1/(GaAs)n [311] superlattices withn = 10∼ 26,to approximate a single (311)-InAs monolayer embedded in bulk GaAs; heren is the number of GaAs layers.We calculate the energies and the charge densities|9|2 of the electron and hole states.

2. Method of calculation

The theoretical calculations are performed by using the semi-empirical tight-binding method with thesp3s∗

0749–6036/98/020219 + 05 $25.00/0 sm960218 c© 1998 Academic Press Limited

220 Superlattices and Microstructures, Vol. 23, No. 2, 1998

[311]

In

In

Ga

As

Fig. 1. Atomic arrangement of a (311)-InAs monolayer embedded in GaAs (viewed along the [011] direction)

basis [8]. We adopt the tight-binding parameters of bulk GaAs and InAs from the values of Voglet al. [8],which give the band-gap energiesEg = 1.55 eV (GaAs) and 0.43 eV (InAs). In our study, spin–orbit couplingis not included.

The length of the strained In–As bonds,dIn–As, is estimated from the lattice deformation of the pseudo-morphic InAs layer on a (311) GaAs substrate. Using the elastic strain tensorε′i j of the pseudomorphic layer,

which is given by De Caro and Tapfer [9], we obtaineddIn–As = 2.55A.The bond-length dependence of the InAs tight-binding parameters (the hydrostatic strain) is included by the

formula; the interatomic parameters are proportional to(dIn–As)−ηα,β , whereα andβ (= s, px, py, pz,ands∗)

indicate the adjacent orbitals. We adopt the exponentsηα,β from the values of Honget al.[10] On the other hand,the bond-angle dependence (the uniaxial strain) is not included. Due to the omission of spin–orbit couplingand uniaxial strain in the calculation for InAs, the hole levels near the valence-band edge are calculated lessaccurately in the present study. A similar treatment of strain effects on tight-binding parameters was done byWilke and Hennig [6] for the calculation of the (100)-InAs monolayer in GaAs.

(InAs)1/(GaAs)n [311] superlattices withn = 10 ∼ 26 are used to model the (311)-InAs monolayerembedded in GaAs. In the superlattices, the unit cell ((InAs)1(GaAs)n layers) is stacked repeatedly along the[311] direction as follows;

−→[311]· · · (InAs)1(GaAs)n(InAs)1(GaAs)n(InAs)1(GaAs)n · · · .

The (311)-InAs monolayer is surrounded by the (GaAs)n layers.

3. Results and discussions

First, we show the result for the (InAs)1/(GaAs)10 [311] superlattice. Figure 2 shows the energies of theelectron level and the hole level in the GaAs band-gapat the0 point, which are induced by the presenceof a (311)-InAs monolayer. The (311)-InAs monolayer induces electron states (near the conduction-bandedge) and hole states (near the valence-band edge) which have energy dispersion in the reciprocal space. Theelectron and hole levels shown in Fig. 2 are the minimum energies of an electron and a hole, respectively, atthe0 point. The conduction-band offset (1Ec) between GaAs and strained InAs is varied within the range

Superlattices and Microstructures, Vol. 23, No. 2, 1998 221

1.8

1.6

1.4

1.2

1.0

0.8

Ene

rgy

(eV

)

0.6

0.4

0.2

0

0.3 0.4 0.5∆Ec (eV)

(InAs)1/(GaAs)10 [311] superlattice

Ec(GaAs)

Ev(GaAs)

Ee–h=1.49 eV 1.49 eV 1.49 eV

0.6–0.2

e

h

Fig. 2. Energies of the electron level (e) and the hole level (h) in the GaAs band-gap at the0 point, which are induced by a (311)-InAsmonolayer in GaAs, dependent on the conduction-band offset1Ec(GaAs-InAs). An (InAs)1/(GaAs)10 [311] superlattice is used for thecalculation.Ee−h is the energy separation between the electron level and the hole level.

1Ec = 0.34∼ 0.54 eV. (Alonsoet al. [3] deduced1Ec = 0.43 eV from optical measurements.) The energyseparation between the electron and hole levels (Ee−h), which corresponds to the optical transition energy, iscalculated to be 1.49 eV, and is found to be independent of1Ec. The valueEe−h = 1.49 eV is smaller by0.06 eV than the GaAs band-gap energy ofEg = 1.55 eV.

Figure 3 shows the dependence ofEe−h on the number of GaAs layersn in the (InAs)1/(GaAs)n [311]superlattice. (We assume1Ec = 0.34 eV in this figure, but we have checked that the dependence ofEe−h

on1Ec is negligible; this is clear in Fig. 2.) As shown in the figure,Ee−h increases with increasingn. Forn = 26, we obtainEe−h = 1.52 eV, which is smaller by 0.03 eV thanEg = 1.55 eV. In the figure,Ee−h showsa tendency to saturate at a certain value belowEg(GaAs)= 1.55 eV for largen. The saturated value ofEe−h

for largen corresponds to the optical transition energy measured in experiments, where an isolated (311)-InAsmonolayer is embedded in GaAs. A further calculation for largern is required to obtain the saturated value.

In experiments, Alonsoet al. [3] observed a photoluminescence peak atEPL = 1.482 eV (at 8 K) fora sample grown by molecular beam epitaxy with the InAs thickness 1.8A(1 monolayer) on a (311) GaAssubstrate. The valueEPL = 1.482 eV is smaller by 0.04 eV thanEg(experiment)= 1.52 eV. The theoreticaltrend in Fig. 3 roughly accounts for the above experiment, although an improvement of accuracy by includingspin–orbit coupling and uniaxial strain effect is required for more quantitative discussions.

222 Superlattices and Microstructures, Vol. 23, No. 2, 1998

1.60

1.55

1.50

Ee–

h (e

V)

1.45

1.400 5 10 15 20 25

Eg (GaAs)

30

(InAs)1/(GaAs)n [311] superlattice ∆Ec = 0.34 eV

Number of GaAs layers n (molecular layer)

Fig. 3. Energy separation between the electron level and the hole level in the GaAs band-gap at the0 point (Ee−h), which areinduced by a (311)-InAs monolayer in GaAs, dependent on the number of GaAs layersn in the (InAs)1/(GaAs)n [311] superlattice. Theconduction-band offset1Ec(GaAs-InAs) is assumed to be 0.34 eV.

Figure 4 shows the charge densities|9|2 of the electron and hole states induced by the (311)-InAs monolayerin an (InAs)1/(GaAs)26 [311] superlattice. The charge densities of the electron and hole states are weaklylocalized near the (311)-InAs monolayer. The degree of localization of the electron increases with increasing1Ec (1Ec = 0.34 eV in Fig. 4A, 0.44 eV in B). Such weakly localized electron and hole states were foundaround the (100)-InAs monolayer in GaAs by Shiraishi and Yamaguchi [7].

4. Conclusions

We have theoretically studied the electronic structure of a (311)-InAs monolayer embedded in GaAs byusing the semi-empiricalsp3s∗ tight-binding method. (InAs)1/(GaAs)n [311] superlattices withn = 10∼ 26are used for the calculations. The results are summarized as follows.

1. The (311)-InAs monolayer induces an electron level near the conduction-band edge and a hole levelnear the valence-band edge in the GaAs band gap.

2. The electron–hole energy separation is smaller than the GaAs band-gap energy by 0.06 eV (n = 10)and 0.03 eV (n = 26).

3. The charge densities of the electron and hole states are weakly localized near the (311)-InAs monolayer.

Acknowledgements—We thank Professor K. Hirakawa for valuable discussions. This work was partly sup-ported by the Grant in Aid from the Ministry of Education, Science and Culture, Japan, for scientific researchon priority areas, ‘Quantum Coherent Electronics’, and Scientific Research (C).

References

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Superlattices and Microstructures, Vol. 23, No. 2, 1998 223

0.05

0.04

0.03

0.02

|Ψ|2

0.01

0(GaAs)13 (GaAs)13

[311]

(InAs)1/(GaAs)26 [311] superlattice(311)-InAs induced electron state∆Ec = 0.34 eV In

A

(InAs)1

0.05

0.04

0.03

0.02

|Ψ|2

0.01

0(GaAs)13 (GaAs)13

[311]

(InAs)1/(GaAs)26 [311] superlattice(311)-InAs induced electron state∆Ec = 0.44 eV

B

(InAs)1

In

0.05

0.04

0.03

0.02

|Ψ|2

0.01

0(GaAs)13 (GaAs)13

[311]

C

(InAs)1

In

(InAs)1/(GaAs)26 [311] superlattice(311)-InAs induced hole state∆Ec = 0.34 eV

Fig. 4. Charge densities|9|2 of the states induced by a (311)-InAs monolayer: A, The electron state (1Ec = 0.34 eV); B, the electronstate (1Ec = 0.44 eV); C, the hole state (1Ec = 0.34 eV). An (InAs)1/(GaAs)26 [311] superlattice is used for the calculations.

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