band maximum at point and conduction band minimum at X point, while those for other AIIIBV compounds 3. Results are direct. As expected, band gaps for AlP, AlAs, AlSb, GaP, GaAs, GaSb calculated using both of the lattice 3.1. Lattice constants constants are underestimated, while those for InP, InAs and InSb are overestimated for calculated lattice constant It will become clear that the band structure depends and underestimated for experimentally determined lattice critically upon such parameters as the lattice constant, constant. The overestimate can be related to the small among others. By a search of the totalenergy minimum, gap and the neglect of the spinorbit coupling; similar lattice constants have been found (Table 1) which for results were obtained by Huang and Ching [14]. One the AIIIBV compounds and AIIBVI semiconductors with can also see that LDA calculations of different authors delectrons included into the valence complex differ from differ significantly. For example, the direct band gap of those determined experimentally by < 3 and < 1.5%, GaAs, GaSb, InAs and InSb calculated by Geller et al. [10] respectively. However, if the delectrons of groupII within LDA by the fullpotential linearized augmented atoms are included into the core, then the calculated planewave (FLAPW) method is 0.25, 0.3, 0.53, lattice constants for the AIIBVI compounds differ from experimentally determined ones significantly (> 7%). It 0.57 eV respectively, while that of AlAs, GaAs and GaSb calculated by Wei and Zunger [41] within LDA are 1.84, should be noted that experimentally determined lattice constants for AIIBVI compounds given in handbooks 0.15 and 0.165 eV, respectively. Our calculated band gaps differ from each other. For example, in Ref. [37], the of AIIIBV compounds are similar to those of Ref. [8,9,15]. lattice constant for CdSe is a = 6.052 while in another It should also be noted that the direct band gaps handbook, Ref. [38], a = 6.084. Our calculated lattice calculated in this work using the calculated lattice constant 1 Физика и техника полупроводников, 2005, том 39, вып. 180 S.Zh. Karazhanov, L.C. Lew Yan Voon Table 2. Eigenvalues (eV) at, X, and L for the zincblende AIIIBV and AIIBVI compounds calculated using the theoretical (a0, ad0) and experimental (ae, ade) lattice constants including the delectrons of groupII atoms into the core (a0, ae) and into valence complex (ad0, ade). The results have been compared to experimental data [39,40] and theoretical calculations within k · p theory [2], LDA and GW [25] AlP AlAs AlSb GaP GaAs GaSb InP InAs InSb c a0 3.3457 2.1517 1.6488 2.5982 1.1885 0.5885 1.3831 0.4131 0.ae 3.0929 1.9979 1.4354 1.7736 0.5337 0.0000 0.7205 0.0000 0.c X1 a0 1.4194 1.3127 1.1291 1.4058 1.2861 0.8275 1.5522 1.3821 1.ae 1.4739 1.3437 1.1766 1.5872 1.4245 0.9086 1.7198 1.5340 1.Lc a0 2.7809 2.1399 1.3611 1.9424 1.2413 0.5738 1.8019 1.2331 0.ae 2.6692 2.0680 1.2806 1.6262 0.9888 0.3868 1.4918 0.9624 0.Experiment [39] 3.6200 3.1400 2.2190 2.7800 1.4240 0.7500 1.3400 0.4180 0.k · p [2] 5.1200 3.0600 2.8700 1.5200 0.8100 1.4200 0.4200 0.ZnS ZnSe ZnTe CdS CdSe CdTe c a0 5.325 3.908 2.971 3.646 2.780 2.ae 2.933 1.968 1.587 2.155 1.513 1.ad0 2.004 1.180 1.077 0.913 0.394 0.ade 1.868 1.076 0.901 0.901 0.371 0.c X1 a0 2.953 2.594 1.971 3.379 2.932 2.ae 3.618 3.171 2.451 3.757 3.322 1.ad0 3.132 2.764 2.125 3.301 2.923 2.ade 3.242 2.820 2.167 3.311 2.943 2.Lc a0 5.159 4.073 2.621 0.487 4.003 2.ae 3.985 3.107 2.117 3.908 3.186 0.ad0 3.128 2.395 1.610 2.770 2.193 1.ade 3.084 2.351 1.563 2.761 2.179 1.Experiment [40] 3.680 2.700 2.280 2.500 1.900 1.k · p [2] 3.800 2.820 2.390 2.560 1.840 1.LDA [25] 2.370 1.450 1.330 1.370 0.760 0.GW [25] 3.980 2.840 2.570 2.830 2.010 1.is bigger than those found using the experimentally It is well known that the inclusion of delectrons of determined lattice constants. This can be attributed, in part, atoms of groupII into the valence complex causes strong to the direct gap increase with compression (equivalent pd coupling of the upper valence band with the dstates, to small a). In the latter case, the band gaps are even which results in the repulsion of the former upward and zero for GaSb, InAs and InSb. Also, the band gaps for reduces the band gap [24–26,29]. Our results in Table AlP, GaP, GaAs, GaSb, InP, InAs and InSb calculated and Fig. 2 are consistent with this statement. Furthermore, d d using the calculated lattice constants are much closer to the energy levels of the delectrons ( and ) fall in between 12 experimental band gaps. the s and p valence bands, which are in good agreement For the AIIBVI compounds, the calculated eigenenergies with calculations of Wang and Klein [13] for ZnS and are in general agreement with previous calculations [24–26]. ZnSe using the linear combination of Gaussian orbitals If the delectrons of the atoms of groupII are included into method with LDA. However, energies of the bands are much the core with nonlinear core corrections and theoretical higher than the experimentally determined ones [30], which lattice constants are used in bandstructure calculations, indicates overestimation of the pd coupling by LDA in then the band gaps of all the AIIBVI compounds (except agreement with calculations in Ref. [13,24–26,29]. The d d CdSe and CdTe) become indirect with the valenceband and bands are almost dispersionless with very narrow maximum at the point and the conductionband minimum widths. It indicates that these states are well localized. at the X point (see Table 2 and Fig. 1), which contradicts So, the abovementioned overestimation is consistent with the experimental data [37,38,40]. This error is because statement of Wang and Klen [13]: „... the more these of the big difference of the LDA lattice constants from lower states are localized (narrower bands), the larger is the experimentally determined ones (see Table 1). If the the disagreement...“. Comparing Fig. 1 with Fig. 2, one experimental lattice constants are used in the bandstructure can see that, due to involvement of the delectrons into calculations, then eigenenergies (Table 2 and Fig. 1) agree the valence shell, band gaps are reduced significantly for much better with LDA [25,26] and GW [25] calculations. the AIIBVI compounds considered. Note that the difference Физика и техника полупроводников, 2005, том 39, вып. Ab initio studies of band parameters of AIIIBV and AIIBVI zincblende semiconductors Figure 1. Results of the bandstructure calculations. Solid curves are calculated with theoretical lattice constants (a0) and dashed curves with experimental ones (ae). Figure 2. Results of the bandstructure calculations with the inclusion of delectrons of atoms of groupII into the valence complex. Solid curves are calculated with theoretical lattice constants (ad0) and dashed curves with experimental ones (ade). between eigenenergies calculated using the theoretical and Analysis of Table 2 (also Fig. 1 and Fig. 2) shows that the the experimental lattice constants in Fig. 2 are smaller band gaps calculated in this work using the experimental compared to that in Fig. 1. Such small difference for lattice constant and including the delectrons into the core the former is because the LDA and experimental lattice are much bigger than those of previous ones calculated constants are close to each other. by LDA [25] and smaller than those determined by the Физика и техника полупроводников, 2005, том 39, вып. 182 S.Zh. Karazhanov, L.C. Lew Yan Voon Table 3. Momentum matrix elements (eV) and Luttinger parameters for zincblende AIIIBV compounds using the theoretical (a0) and the experimental (ae) lattice constants L L L Ep Ep 1 2 AlP a0 17.183 0.015 3.696 0.826 1.ae 16.807 0.036 3.750 0.893 1.k · p [2] 17.700 3.470 0.060 1.AlAs a0 15.678 0.144 5.203 1.526 2.ae 15.437 0.172 5.355 1.628 2.[39] 21.100 3.45,3.25 0.68,0.64 1.29,1.k · p [2] 21.100 4.040 0.780 1.AlSb a0 14.548 0.592 5.810 1.659 2.ae 14.206 0.634 6.180 1.898 2.k · p [2] 18.700 4.150 1.010 1.GaP a0 18.758 0.150 5.729 1.222 2.ae 17.496 0.366 6.517 1.872 2.[39] 22.200 4.050 0.490 1.k · p [2] 22.200 4.200 0.980 1.GaAs a0 17.189 0.440 10.336 3.526 4.ae 16.136 0.634 18.198 7.674 8.[39] 25.700 6.807.20 2.102.50 1.002.k · p [2] 25.700 7.650 2.410 3.k · p, TB[45] 22.500 0.025 7.070 2.400 3.Experiment [17] 22.5328.9 3.556.0 5.647.21 1.352.49 2.493.Experiment [18] 27.860 2.36 4.808.56 1.222.90 1.853.LDA [15] 15.820 2.27 9.270 3.120 4.GaSb a0 15.726 0.997 16.291 6.305 7.ae 0.000 8.540 115.200 56.033 56.k · p [2] 22.400 11.800 4.030 5.[39] 25.00,26.1 13.10,13.30 4.40,4.50 5.70,6.InP a0 15.456 0.048 6.638 3.033 2.ae 14.314 0.205 9.768 3.836 4.[39] 17.000 4.955.15 0.941.65 1.622.k · p [2] 20.400 6.280 2.080 2.InAs a0 14.427 0.169 19.481 8.428 9.ae 0.000 7.132 33.807 15.819 16.k · p [2] 22.200 19.670 8.370 9.InSb a0 13.866 0.454 24.259 10.681 11.ae 0.000 7.152 23.493 10.545 11.[39] 21.200 33.50,40.10 14.50,18.10 15.70,19.k · p, TB[45] 23.100 35.080 15.640 16.GW [25] approach and using experimental lattic constants. atoms of groupII, III, V and VI, and different values of Our calculated band gap for ZnS is also smaller than the Ecut and lattice constant values. 3.55 eV [26]. However, our calculated band gaps obtained The other important conclusions which can be derived with the delectrons included in the valence complex are from the comparative analysis is that the band gaps (Eg) in good agreement with 1.02 eV for ZnTe and 0.47 eV for are underestimated and the energies of the semicore states CdTe [24] and 1.84 eV for ZnS [26], which are too small are overestimated systematically. However, this error, as compared to the experimental data. discussed in Section 1, in not an artifact of numerical bandThe eigenvalues at some special kpoints for some of structure approximations, but rather is a failure of DFT. the compounds considered is in general agreement with other LDA calculations (e. g. with those by Huang and Ching [14] using a minimal basis semiab initio approach 3.3. Momentum matrix elements and Luttinger and by Wang and Klein [13] using the LDA). However, parameters the agreement is not systematic. Note that the discrepancies between the theoretical band gaps of different authors can Momentum matrix elements, are the key parameters for be related to the accuracy of the LDA calculations, different discussing the optical properties of semiconductors. So, to parameters used to generate the pseudopotentials of the study the problem using ab initio band structure calculations Физика и техника полупроводников, 2005, том 39, вып. Ab initio studies of band parameters of AIIIBV and AIIBVI zincblende semiconductors Table 4. Momentum matrix elements (eV) and Luttinger parameters for AIIBVI semiconductors for delectorns of groupII atoms included into the core (a0, ae) and into the valence complex (ad0, ade) using the theoretical (a0, ad0) and the experimental (ae, ade) lattice constants. The results have been compared to calculations within other methods such as TBLMTO [5] and k · p [2,23] L L L Ep Ep 1 2 ZnS a0 19.649 0.008 3.158 0.746 1.ae 14.735 0.279 2.981 0.973 1.ad0 12.262 0.232 4.113 1.160 1.ade 12.013 0.301 4.143 1.241 1.TBLMTO [5] 24.882 20.998 2.120 0.510 1.TBLMTO [5] 15.295 6.708 1.280 0.090 1.k · p [2] 20.400 2.540 0.750 1.ZnSe a0 16.961 0.048 3.954 1.079 1.ae 12.843 0.545 4.238 1.542 1.ad0 11.358 0.484 6.465 2.371 2.ade 11.146 0.544 6.817 2.530 3.[37] 4.300 1.140 1.k · p [23] 23.TBLMTO [5] 16.230 9.835 3.210 0.750 2.k · p [2] 24.200 3.770 1.240 1.ZnTe a0 16.075 0.507 4.327 1.132 1.ae 12.758 1.062 4.835 1.721 2.ad0 12.272 0.907 7.215 2.511 3.ade 11.820 1.020 7.790 2.897 3.TBLMTO [5] 19.667 0.837 3.440 0.590 2.[37] 3.900 0.600 0.[37] 3.900 0.830 1.[37] 4.000 0.830 1.[37] 0.800 1.k · p [2] 19.100 3.740 1.070 1.CdS a0 15.356 0.000 2.721 0.841 1.ae 11.631 0.195 2.647 0.975 1.ad0 9.301 0.219 5.101 1.880 2.ade 9.273 0.228 5.121 1.897 2.TBLMTO [5] 3.440 0.970 2.TBLMTO [5] 2.200 0.350 1.k · p [23] 21.CdSe a0 13.563 0.036 3.265 1.162 1.ae 10.125 0.374 3.806 1.515 1.ad0 8.670 0.384 11.694 5.144 5.ade 8.600 0.405 12.160 5.394 5.TBLMTO [5] 4.400 1.600 2.k · p [23] 20.CdTe a0 13.600 0.256 3.862 1.258 1.ae 10.732 0.663 4.353 1.709 1.ad0 9.735 0.628 10.072 4.207 4.ade 9.488 0.697 11.208 4.831 5.TBLMTO [5] 21.066 5.098 4.340 1.600 2.[37] 5.300 1.700 2.[37] 4.110 1.080 1.k · p [23] 18.k · p [2] 20.700 5.290 1.890 2.is an important problem. Our calculated results for Ep, Ep smaller than those found from k · p theory for all the L L L and Luttinger parameters 1, 2, 3 are given in Table 3 and semiconductors considered except AlP, ZnS and ZnSe for Table 4 for the AIIIBV and AIIBVI compounds, respectively. which good agreement was achieved. The small values of Ep Analysis of Table 3 and Table 4 shows that momentum can be related to the underestimation of the coupling of matrix elements calculated in this work are much the valenceband maximum [20]. For InP our result agrees Физика и техника полупроводников, 2005, том 39, вып. 184 S.Zh. Karazhanov, L.C. Lew Yan Voon Table 5. Effective masses (in units of the freeelectron mass m0) for zincblende AIIIBV compounds calculated using the theoretical (a0) and experimentally determined (ae) lattice constants.
