Strain and optical transitions in InAs quantum dots on (001) GaAs

Superlattices and Microstructures, Vol. 30, No. 4, 2001 doi:10.1006/spmi.2001.1009 Available online at http://www.idealibrary.com on

Strain and optical transitions in InAs quantum dots on (001) GaAs Y. F U , Q. X. Z HAO Physical Electronics and Photonics, Microtechnology Center at Chalmers, Department of Physics, Fysikgränd 3, Chalmers University of Technology and Gothenburg University, S-412 96 Göteborg, Sweden F. F ERDOS , M. S ADEGHI , S. M. WANG , A. L ARSSON Photonics Laboratory, Microtechnology Center at Chalmers, Department of Microelectronics ED, Chalmers University of Technology, S-412 96 Göteborg, Sweden (Received 9 October 2001)

To investigate the strain characteristics of InAs quantum dots grown on (001) GaAs by solid source molecular beam epitaxy we have compared calculated transition energies with those obtained from photoluminescence measurements. Atomic force microscopy shows the typical lateral size of the quantum dots as 20–22 nm with a height of 10–12 nm, and photoluminescence spectra show strong emission at 1.26 µm when the sample is capped with a GaAs layer. The luminescence peak wavelength is red-shifted to 1.33 µm when the dots are capped by an In0.4 Ga0.6 As layer. Excluding the strain it is shown that the theoretical expectation of the ground-state optical transition energy is only 0.566 eV (2.19 µm), whereas a model with three-dimensionally-distributed strain results in a transition energy of 0.989 eV (1.25 µm). It has thus been concluded that the InAs quantum dot is spatially strained. The InGaAs capping layer reduces the effective barrier height so that the transition energy becomes red-shifted. c 2001 Academic Press

Key words: InAs quantum dots, photoluminescene, strain distribution, capping layers.

1. Introduction Optical gain materials that enable long wavelength (1.3 and 1.55 µm) emission and amplification on GaAs substrates are attractive for the realization of temperature insensitive lasers and vertical cavity surface emitting lasers intended for high bit rate communication on single mode fibres. Among the promising materials are GaInNAs quantum wells [1, 2] and In(Ga)As quantum dots (QDs) [3, 4]. Gain materials based on In(Ga)As QDs could benefit from extra advantages related to the three-dimensional confinement of carriers injected into the ground states of the QDs. Of importance for the gain characteristics is the density of states in the QDs which is determined by the composition and shape of the dots, the composition of the surrounding material, and the strain distribution in the QDs [5, 6]. It is the purpose of this work to investigate the strain characteristics of InAs QDs grown on GaAs by solid source molecular beam epitaxy and the effect of capping the QDs with InGaAs. This is done by comparing results from photoluminescence (PL) measurements with numerical calculations of the density of state functions and emission spectra under assumptions of various degrees of strain. We conclude 0749–6036/01/100205 + 09

$35.00/0

c 2001 Academic Press

206

Superlattices and Microstructures, Vol. 30, No. 4, 2001

Fig. 1. PL spectra of InAs QDs covered with five, seven and 10 monolayers of In0.4 Ga0.6 As. PL from GaAs capped InAs QDs is presented as the dotted line.

that InAs QDs are three-dimensionally strained and the wavefunction penetration into the InGaAs capping layer reduces the optical transition energy. In Section 2 we present the fabrication and PL characterization of the InAs QDs. Experimental results are analysed in Section 3 theoretically. A brief summary and discussions are made in Section 4.

2. Experiments The InAs QD samples were grown on semi-insulating (001) GaAs substrate. First a 50 nm GaAs buffer layer was grown by solid source molecular beam epitaxy. The InAs QDs were formed under a repeated growth sequence of 0.1 monolayer InAs followed by a 2 s growth interruption while having a continuous flux of As2 . The nominal InAs layer thickness is 3.7 monolayers. For some samples a thin In0.4 Ga0.6 As layer with a nominal thickness of 5 ∼ 10 monolayers was deposited on the InAs QDs. Finally, a 200 nmthick GaAs cap layer was grown for PL studies. Typical lateral size of the InAs QDs is 20 ∼ 22 nm with a height of 10 ∼ 12 nm, and the dot density is about 1 ∼ 2 × 1010 cm−2 . Surface studies using atomic force microscopy (AFM) indicate that the surface remains three-dimensional when capping the InAs dots with a higher In-content InGaAs layer, thus creating a ‘dots in dots’ structure. Details of the sample fabrication and surface quality characterization are published elsewhere [7]. An argon–ion laser (514.5 nm line) was used as the excitation source for room temperature PL measurements. The excitation density was typically about 8 W cm−2 . At growth conditions for maximum PL intensity, a strong emission at 1.21 µm in the PL spectrum is believed to originate from the transition of the ground states confined in InAs QDs, while a weak emission at 1.13 µm is from the InAs wetting layer or transitions related to the excited states in InAs QDs. The full-width at half-maximum (FWHM) of the InAs QD emission is 28.7 meV. It decreases slightly when the sample is capped by an Inx Ga1−x As layer. For InAs QDs capped with pure GaAs, the longest achieved emission wavelength is 1.26 µm, whereas covering the QDs by Inx Ga1−x As with different In mole fraction and nominal thickness, the luminescence wavelength becomes red-shifted and reaches 1.33 µm when x = 0.4 and a thickness of 10 monolayers. Typical PL spectra are presented in Fig. 1.

Superlattices and Microstructures, Vol. 30, No. 4, 2001

207

3. Theoretical analysis of strain and optical transitions By defining the sample growth direction as the z-axis and the plane perpendicular to this axis as the x y-plane, we commence the theoretical analysis by solving the Schrödinger equation for the energy band structure. The Coulomb potential is neglected since the samples are not doped. The total wavefunction of sublevel i in band n (we discuss both the conduction band, n = c, and the valence band, n = v) is denoted as 9ni (r) = ψni (r )u n (r) E ni = E n + E i

(1)

where E n is the band edge of band n and u n is the Bloch function. E c − E v = E g is the bandgap. We consider inter-band optical recombination between conduction-band and valence-band sublevels (luminescence). The total transition rate per unit time is 1 X W (h¯ ω) = wi j (h¯ ω)[1 − f (E v j )] f (E ci ) (2) h¯ ω ij where h¯ ω is the photon energy. The matrix element of an inter-band transition between valence-band sublevel (9vi , E vi ) and conduction-band sublevel (9cj , E cj ) is pi j = Pcv hψci (r)|ψv j (r)i ωi j (h¯ ω) =

2πpi2j

δ(E ci − E v j − h¯ ω) (3) h¯ 2 f (E v j ) and f (E ci ) are occupations of the two sublevels. pcv = hu c |∇|u v i is the dipole moment. We consider weak and constant optical excitation and assume a constant dipole moment so that the total transition rate becomes 2π|pcv |2 X |hψci (r)|ψv j (r)i|2 0 w(h¯ ω) = [1 − f (E v j )] f (E ci ). (4) h¯ ω (E ci − E v j − h¯ ω)2 + 0 2 ij

The δ function of the energy conservation is replaced by a Lorentzian function by introducing a phenomenological relaxation energy 0. An extra broadening effect due to the inhomogeneous dot sizes should be included, whereas statistics shows experimentally that about 75% of QDs have a typical lateral size of 20– 22 nm and a vertical size of 10–12 nm. Less than 5% of dots have sizes smaller than 15 nm/5 nm (lateral/vertical), or larger than 25 nm/15 nm. Thus we neglect this inhomogeneous effect in the current work. 0 is obtained from the PL spectra as FWHM. Based on the AFM we assume that the InAs QD has a geometric structure of y2 z2 x2 + + ≤ 1.0 (5) a2 a2 c2 for z > 0. Elsewhere it is GaAs. We try to find the relationship between the E ni , hψci (r)|ψv j (r)i and geometric parameter a and c, while keeping the volume of the QD constant, i.e. a 2 c = r03 and r0 = 10 nm. It is expected that the InAs QDs and the Inx Ga1−x As cap layer grown on the (001) GaAs substrate are spatially strained. The descriptions of the strain given by [8] and [9] are based on the deformation potential of Pikus and Bir [10]. Such a deformation potential has been applied to spatially distributed strains, e.g. the strain in lattice-mismatched InAsx P1−x /InP quantum wires [11]. However, physical parameters of this deformation potential are limited. We adopt a sp3 s∗ tight-binding model [12] with the universal a −2 scaling rule [13, 14] which has been developed for almost all commonly used semiconductor materials, where a is v , is referred to as the zero the distance between interacting atoms. In [12], the top of the valence band, 015

208

Superlattices and Microstructures, Vol. 30, No. 4, 2001 Table 1: Conduction-band and valence-band edges (E c and E v in units of eV) of relaxed and strained Inx Ga1−x As grown on (001) GaAs.

GaAs

In0.2 Ga0.8 As

In0.4 Ga0.6 As

InAs

Strained

Ec Ev

1.5512 0.0

1.5458 0.0363

1.5419 0.0728

1.5394 0.1823

Relaxed

Ec Ev

1.5512 0.0

1.3853 0.0814

1.2309 0.1622

0.8337 0.3999

Fig. 2. Band edge profiles [eV] of the InAs/In0.4 Ga0.6 As system embedded in GaAs.

reference energy for every individual elemental or compound semiconductor material. To calculate the energy band structures of InAs/Inx Ga1−x As/GaAs heteromaterials, we add a valence-band offset of 0.4 eV to the energies of orbitals in InAs with respect to GaAs so that the calculated conduction-band offset agrees with the general rule of 0.65[E g (A) − E g (B)] when the III–V composite materials A and B are lattice-relaxed, where E g (InAs) = 0.354 and E g (GaAs) = 1.424 eV are energy band gaps at room temperature. We consider three extreme strain situations in this work: (i) the InAs/Inx Ga1−x As region is completely strained having the lattice constant of the GaAs substrate, aGaAs ; (ii) the InAs/Inx Ga1−x As region is completely relaxed with a lattice constant aInx Ga1−x As = xaInAs + (1 − x)aGaAs

(6)

The lattice constant of InAs and GaAs bulk materials, aInAs and aGaAs are 6.0583 and 5.6611 Å [15], respectively. Knowing the lattice constants, the interaction matrix elements among orbital states in the tight-binding model are determined by the a −2 scaling rule. By the tight-binding model, the energy band structures of the heterostructure InAs/Inx Ga1−x As system are presented in Table 1 corresponding to the two extreme strain situations. It is noticed that the parameters in the sp3 s∗ tight-binding model of Vogl et al. [12] are derived at low temperature so that the bandgap of GaAs is 1.551 eV. In the current investigation at room temperature, we simply shift down the conduction-band edge by 0.127 eV. The method has been applied to strain fields in Si–C [16] and InGaAs quantum wire [17]. As observed from Table 1, the ground transition energy in a completely strained InAs QD must be more than 1.3571 eV at low temperature and 1.2301 eV at room temperature (wavelength shorter than 1.008 µm), while for a completely relaxed InAs QD, it is only 0.4338/0.3068 eV (2.858/4.042 µm) at low/room temperature. In the following analysis we assume the third model for the strain distribution. (iii) The point of x = y = z = 0 is completely strained, while the surface of x 2 /a 2 + y 2 /a 2 + z 2 /c2 = 1 is lattice relaxed. In between the strain intensity varies linearly. We further assume that the In0.4 Ga0.6 As capping layer is lattice relaxed. The energy band structure of the InAs/In0.4 Ga0.6 As QD is schematically depicted in Fig. 2.

Superlattices and Microstructures, Vol. 30, No. 4, 2001

209

A

B

Fig. 3. A, LDOS of conduction-band electrons along the z-axis (x = y = 0); B, squared wavefunction of the ground sublevel in the conduction band. ‘+’ marks the peak position of the wavefunction. Note that the ground-sublevel wavefunction is rotationally symmetric along the z-axis. a = c = 10 nm and the mole fraction of In in the capping layer is 0.4.

With the energy band structure thus determined, we apply the effective mass approximation to study the optical properties of our InAs/Inx Ga1−x As heterostructure using the energy band structures of Fig. 2 and the ∗ = 0.0239 and 0.063, and of heavy holes, m ∗ = 0.50 and 0.35 in InAs and effective masses of electrons, m C hh GaAs, respectively, in units of free electron mass [15]. It is interesting to note that the energy band structure of completely strained InAs bulk material having a lattice constant of GaAs is very similar to the one of GaAs, as observed by its much increased bandgap (Table 1). This is, of course, the physical reason for the universal a −2 scaling rule. A similar effect is expected concerning the values of the effective masses of electrons and holes as functions of the strain, whereas we have assumed uniform effective masses in the InAs QD region. Numerically we expect that the modification of the effective masses by the strain does not significantly affect the ground sublevels, since the ground sublevels are largely confined in the low-band edge relaxed part of the InAs dot. Along the z-axis (x = y = 0), the local densities of states (LDOS), νc (r, E), of conduction-band electrons are presented in Fig. 3A, the x z-contour plot of the LDOS at the ground sublevel energy is shown in Fig. 3B. It is observed that there are two localized sublevels confined in the InAs QD below the conduction-band edge of GaAs (here the GaAs conduction-band edge is referred to as the energy reference level), and the excited sublevel is very close to the GaAs conduction-band edge. Detailed numerical analysis of the local density of states X νn (r, E) = |ψni (r)|2 δ E,E ni (7) i

by the local Green’s function theory [18] shows that the ground sublevel is not degenerate so that the LDOS presented in Fig. 3B directly corresponds to the squared wavefunction of the ground sublevel. The excited state extends in the InAs QD, whereas the wavefunction of the ground sublevel is largely confined in the relaxed InAs region where the potential energy is low (see Fig. 2). We have calculated the local densities of heavy-hole states and it is noticed that there are many sublevels confined in the InAs QD, and the spatial distributions of these sublevels are rather complicated due to the large effective mass.

210

Superlattices and Microstructures, Vol. 30, No. 4, 2001

A

B

√ Fig. 4. Local density of states conduction-band electrons at (0, 0, 5) nm as a function of A, s and B, s1 . a = sr0 , c = r0 /s, a1 = s1 a, c1 = s1 c, and r0 = 10 nm. The LDOS spectra are shifted vertically for clear presentation. The mole fraction of In in the capping layer is 0.4.

We now study the relationship between the energy sublevels and the geometric structure of the InAs QD. The LDOS at x = y = 0 and √ z = 5 nm are presented in Fig. 4A, where the InAs QD is described by a parameter s defined by a = sr0 , c = r0 /s, and r0 = 10 nm. As demonstrated by Fig. 4A, the ground sublevel is at its lowest energy when s = 0.75. Increasing or decreasing s from s = 0.75 shifts the ground sublevel upwards in the energy scale. The energy sublevel is determined largely by the size of the quantum confinement space, i.e. the sizes of InAs QDs in the x y-plane and z-direction. Reduced (enhanced) quantum confinement size indicates an up-shifted (down-shifted) sublevel. Increasing s results in a reduced quantum confinement in the x y-plane and an enhanced quantum confinement along the z-axis. A large value of s indicates a disk-like system where the quantum confinement in the x y-plane can be neglected, so that the system becomes effectively a quantum well whose energy sublevel is determined by the width of the quantum well (i.e. c). A small c value (large s) results in a high-energy sublevel. On the other hand, when s is very small, the z-enhancement (c = r0 /s) can be neglected so that InAs system effectively becomes a quantum wire whose energy sublevels are determined by a, and small a (small s) results in high-energy sublevels. Thus a minimal sublevel is expected as a function of s, which is observed in Fig. 4A. The In0.4 Ga0.6 As capping effect (‘dots in dots’) is considered by modelling an In0.4 Ga0.6 As layer as y2 z2 x2 + + ≥ 1.0, a2 a2 c2

x2 y2 z2 + + ≤ 1.0 a12 a12 c12

(8)

for z ≥ 0, where a1 = s1 a and c1 = s1 c. The corresponding LDOS at (0, 0, 5) nm is presented in Fig. 4B, where a = c = r0 = 10 nm (s = 1.0). The observed downward-shift of the sublevel energy is expected by considering the lowering of the barrier height (Fig. 2) between InAs and Inx Ga1−x As as compared with the InAs/GaAs interface, and therefore the increased wavefunction penetration into the barrier. The saturation of the down-shift is also expected in the same wavefunction-penetration context. The extension of the wavefunction (penetration into the barrier region) is further reflected in the reduced LDOS intensity (the wavefunction is normalized so that the increase of the distribution extension results in a reduced wavefunction amplitude).

Superlattices and Microstructures, Vol. 30, No. 4, 2001

211

Fig. 5. Calculated PL spectra of InAs/In0.4 Ga0.6 As systems. s = 10 and T = 300 K.

Similar results are obtained for valence-band heavy holes (light holes are not directly relevant when discussing the PL study). The transition energy, obtained for s = 1.0, between the two ground sublevels (conduction-band electron and valence-band heavy hole) of the strained system is 0.989 eV (1.25 µm) which agrees rather well with experimental result of 1.26 µm emission from GaAs-capped InAs QDs. The theoretical expectation of the transition energy becomes 0.8839 eV (1.40 µm) when s1 = 1.4, while experimentally a 1.33 µm luminescence is observed when capping the InAs dot with a thick In0.4 Ga0.6 As layer. Discrepancy can be explained by the facts that (a) the parameters in the sp3 s∗ model are obtained at low temperature, (b) the simple effective mass approximation model is applied and (c) the strain in the InAs/Inx Ga1−x As should be treated as distributed, as well as the exciton effect which is expected to be more than 30 meV [19, 20]. Furthermore, fine tuning the geometric structure of the InAs QD can result in a better agreement between theory and experiment. Taking the relaxation energy 0 = 28.7 meV as determined by the FWHM of InAs QD emission and including only the optical transition between ground sublevels of conduction-band electron and valenceband heavy hole, the calculated PL spectra are presented in Fig. 5. We observe an increased PL intensity when the InAs QD is capped by an In0.4 Ga0.6 As layer, which is numerically contributed by the h¯ ω factor in eqn (4). A decreased excitation (decreased photon energy) indicates an increased PL intensity. However, we would expect further PL intensity enhancement. In calculating Fig. 5 we have assumed constant electron and hole concentrations occupying the ground sublevels. Under the same optical excitation power, however, the carrier concentrations are expected to increase when the InAs QDs are capped by an Inx Ga1−x As layer, since the wavefunctions of the carriers confined in the dot become more extended spatially so that their overlappings with the continuum states (above the GaAs band edges) are increased. With increased carrier concentrations, the luminescence intensity increases. However, detailed theoretical investigation about the coupling between the continuum states and the localized sublevel is much more involved [21]. On the other hand, the effect of the exciton is also expected to be important due to the forced confinements of the electron and hole in the same small InAs QD region. Experimental spectra in Fig. 1 show a much decreased luminescence intensity when the InAs QDs are capped by an Inx Ga1−x As layer as compared with the GaAs capping, while the PL intensity increases with the thickness of the In0.4 Ga0.6 As capping layer. The result is explained by the large difference in the material quality when samples are capped differently (i.e. either directly by a GaAs layer or by an Inx Ga1−x As layer). The sample qualities of In0.4 Ga0.6 As layer capping are more or less the same, so that we should compare the PL spectra of three different Inx Ga1−x As layer cappings in Fig. 1 with our theoretical expectation of Fig. 5.

212

Superlattices and Microstructures, Vol. 30, No. 4, 2001

It has been numerically obtained that the overlapping between the ground sublevels of conduction-band electron and valence-band heavy hole, hψc0 (r)|ψv0 (r)i, depends slightly on the model geometric shape of the InAs QD under investigation. The effect of the Inx Ga1−x As capping layer is more evident. For s = 1.0, hψc0 (r)|ψv0 (r)i = 0.943 when s1 = 1.0 and hψc0 (r)|ψv0 (r)i = 0.921 when s1 = 1.4. The reduction of the overlapping is due to the increased penetration of the conduction-band electron wavefunction into the Inx Ga1−x As capping layer (the penetration of the valence-band heavy hole is negligible due to the large effective mass). In the extreme case of an infinitely high barrier approximation, ψc0 = ψv0 so that hψc0 |ψv0 i = 1.0.

4. Conclusions and discussions We have studied the optical properties of InAs QDs grown on GaAs by solid source molecular beam epitaxy, both experimentally and theoretically. AFM shows a typical lateral size of the QDs as 20–22 nm and a height of 10–12 nm, and PL spectra indicate strong emission at 1.26 µm when the sample is GaAslayer capped. Capping the InAs QDs by a thin layer of In0.4 Ga0.6 As results in a red-shift in luminescence wavelength. By the sp3 s∗ tight-binding model to account for the strain in the InAs/Inx Ga1−x As system and the effectivemass-approximation calculation for the energy sublevels of conduction-band electrons and valence-band heavy holes, it has been concluded that the InAs/Inx Ga1−x As heterosystem embedded in GaAs is threedimensionally strained. The extra Inx Ga1−x As capping layer enhances the wavefunction penetration and, therefore, a more extended distribution of the wavefunction so that the transition energy is red-shifted. We have assumed that the In0.4 Ga0.6 As capping layer is lattice relaxed. In reality the strain field in this capping layer is distributed as well. When assuming a completely strained In0.4 Ga0.6 As capping layer, the band offsets between this layer and the GaAs layer become smaller, whereas the band offsets to the InAs QD increase. As observed from the numerical energy sublevel calculations, the ground sublevel in the QD is already far below the conduction-band edge of the relaxed In0.4 Ga0.6 As capping layer; it is thus expected that the detailed strain distribution in the In0.4 Ga0.6 As capping layer does not affect the ground sublevels in the QD. In addition, the tight-binding model has been applied to find the energy band edges of either relaxed or strained bulk InAs material. Further work is underway to study the spatially oriented bond between two interacting atoms which is not in the form of a diamond structure, by the two-centre approximation of Slater and Koster [13], to include the spatial distribution effect of the strain in the QD. Despite many approximations made in the current work, the agreement between theory and experiment is quite good. Such results are due to the facts that the experimental data only reveal information about the ground sublevels, and theoretically, the ground sublevels are largely confined to the relaxed part of the InAs QD, and the energy band structure in this region does not significantly depend on the overall strain distribution. Acknowledgement—The Swedish Foundation for Strategic Research is acknowledged for supporting this work.

References

[1] B. Borchert, A. Y. Egorov, S. Illek, and H. Riechert, Static and dynamic characteristics of 1.29 µm GaInNAs ridge-waveguide laser diodes, IEEE Photonics Technol. Lett. 12, 597 (2000). [2] K. D. Choquette et al., Room temperature continuous wave InGaAsN quantum well vertical-cavity lasers emitting at 1.3 µm, Electron. Lett. 36, 1388 (2000). [3] D. L. Huffaker, G. Park, Z. Zou, O. B. Shchekin, and D. G. Deppe, 1.3 µm room-temperature GaAsbased quantum-dot laser, Appl. Phys. Lett. 73, 2564 (1998).

Superlattices and Microstructures, Vol. 30, No. 4, 2001

213

[4] K. Mukai, Y. Nakata, K. Otsubo, M. Sugawara, N. Yokoyama, and H. Ishikawa, 1.3 µm CW lasing characteristics of self-assembled InGaAs–GaAs quantum dots, IEEE J. Quantum Electron. 36, 472 (2000). [5] M. Sugawara (ed.), Self-assembled InGaAs/GaAs Quantum Dots, Semiconductors and Semimetals (Academic Press, San Diego, 1999) Vol. 60. [6] B. Gil, Stress effects on optical properties, in Gallium Nitride (GaN) II, Semiconductors and Semimetals, edited by J. I. Pankove and T. D. Moustakas (Academic Press, San Diego, 1999) Vol. 57, Chap. 6, pp. 209–274. [7] F. Ferdos, M. Sadeghi, Q. X. Zhao, S. M. Wang, and A. Larsson, Optimisation of MBE growth conditions for InAs quantum dots on (001) GaAs for 1.3 µm luminescence, J. Cryst. Growth 227–228, 1140 (2001). [8] T. G. Andersson, Z. G. Chen, V. D. Kulakovskii, A. Uddin, and J. T. Vallin, Photoluminescence and photoconductivity measurements on band-edge offsets in strained molecular-beam-epitaxygrown Inx Ga1−x As/GaAs quantum well, Phys. Rev. B37, 4032 (1988). [9] M. Beaudoin, A. Bensaada, R. Leonelli, P. Desjardins, R. A. Masut, L. Isnard, A. Chennouf, and G. L’Esperance, Self-consistent determination of the band offsets in InAsx P1−x /InP strained-layer quantum wells and the bowing parameter of bulk InAsx P1−x , Phys. Rev. B53, 1990 (1996). [10] G. E. Pikus and G. L. Bir, Effect of deformation on the hole energy spectrum of germanium and silicon, Sov. Phys. Solid State 1, 1502 (1959). [11] M. Notomi, J. Hammersberg, H. Weman, S. Nojima, H. Sugiura, M. Okamoto, T. Tamamura, and M. Potemski, Dimensionality effects on strain and quantum confinement in lattice-mismatched InAsx P1−x /InP quantum wires, Phys. Rev. B52, 11147 (1995). [12] P. Vogl, H. P. Hjalmarson, and J. D. Dow, A semi-empirical tight-binding theory of the electronic structure of a semiconductor, J. Phys. Chem. Solids 44, 365 (1983). [13] J. C. Slater and G. F. Koster, Simplified LCAO method for the periodic potential problem, Phys. Rev. 94, 1498 (1954). [14] W. A. Harrison, Electronic Structure and the Properties of Solids (Freeman, San Francisco, 1980). [15] O. Madelung (ed.), Semiconductors Group IV Elements and III–V Compounds (Springer, Berlin, 1991). [16] Y. Fu, M. Willander, P. Han, T. Matsuura, and J. Murota, Si–C atomic bond and electronic band structure of a cubic Si1−y C y alloy, Phys. Rev. B58, 7717 (1998). [17] Y. Fu, M. Willander, W. Lu, X. Q. Liu, S. C. Shen, C. Jagadish, M. Gal, J. Zou, and D. J. H. Cockayne, Strain effect in a GaAs-In0.25 Ga0.75 As-Al0.5 Ga0.5 As asymmetric quantum wire, Phys. Rev. B61, 8306 (2000). [18] E. N. Economons, Green’s Function in Quantum Physics, edited by M. Cardona (Springer, Berlin, 1979). [19] W. Xie, Exciton states in a disk-like quantum dot, Physica B279, 253 (1999). [20] K. Chang, Quantum-confinement-effect-driven type I–type-II transition in inhomogeneous quantum dot structures, Phys. Rev. B61, 4743 (2000). [21] Y. Fu, M. Willander, X.-Q. Liu, W. Lu, S. C. Shen, H. H. Tan, C. Jagadish, J. Zou, and D. J. H. Cockayne, Optical transition in infrared photodetector based on V-groove Al0.5 Ga0.5 As/GaAs multiple quantum wire, J. Appl. Phys. 89, 2351 (2001).