J. Phys. Chum. Solids Vol. 51. No. 2. pp. 93-100. 1990
Printed in Great Britain.
@x2-3697190 13.00 + 0.00
0 1990 Pcrpmoa Press pk
SYLVES~ N. EKPENUMA and CHARLES
W. MYLE~
Department of Physics and Engineering Physics, Texas Tech University, Lubbock, TX 79409-1051, U.S.A.
(Received I May 1989; accepted in revisedform 13
1989)
Al&art-We have used a modified version of the sp%+ semi-empirical nearest-neighbor tightbinding formalism to compute the bandstructures of the II-VI zincblende compounds ZnTe, ZnSc. ZnS, and HgSc.
Our scheme includes the effects of spin-orbit coupling and, in contrast to previous sp?F* schemes which fit differences in diagonal tightbinding matrix elements to atomic energy differences, it fits every tightbinding parameter to available band structure data. The resulting parameters show that the scaling factors for the diagonal matrix elements used in previous sp’s* calculations are approximately valid for the compounds considered. The resulting bandstructures are in reasonable agreement with published pseudopotential calculations.
Keywor&: Electronic structure, xincblende compounds, bandstructure, Brillouin zone, tightbinding.
1.
INTRODUCTION
A general feature of II-VI semiconducting com- pounds is the direct energy gap. This feature makes them ideal candidates for device applications requir- ing high absorption or emission of radiation in the energy gap region [ 11. For most of these compounds, especially those containing heavier atoms, relativistic effects can be important. One way of accounting for such effects on the material electronic properties is to include the spin-orbit splitting in the bandstructure parameters.
The purpose of the present paper is to outline a semi-empirical tightbinding bandstructure scheme for
ZnTe, ZnSe, ZnS, and HgSe which includes spin-orbit coupling effects and to present the result- ing bandstructures for these materials. Our method is a slightly modified version of the sp%* nearest- neighbor scheme [2] which has been successfully used to describe the bandstructures of several other materials [2,3]. The addition of the excited s*-state to the sp’ basis on each atom enables one to better simulate the conduction bands, which cannot be properly done within the sp’ model alone [4]. Previ- ous calculations using the sp3s* model without spin-orbit coupling for mostly III-V and group IV semiconductors [2] and with spin-orbit coupling for
CdTe and HgTe [3] were carried out primarily for use in studies of chemical trends in defect energy levels
[5], although several applications of these band- structures to the studies of the variation in the semiconductor alloy electronic properties with alloy- composition have also been made [6]. In these previ- ous sp3s* schemes, the differences in the s or p diagonal matrix elements (E, - E,.,) and (E,,, - E,,) of the tightbinding Hamiltonian were empirically fit to the corresponding free atomic energy differences.
Here the subscripts s, a(c) and p, a(c) refer to the s- and p-states on the anion (cation), respectively. For applications to defect energy levels, this enabled a natural description of the defect potential in terms of such differences. Furthermore, it utilized the reason- able assumption that the cation or anion of a partic- ular zincblende compound is chemically similar to the cation or anion of another member of the group.
This approximation has been shown to produce very reasonable chemical trends in deep levels for several materials [5] and in the variation of electronic properties with alloy composition in semiconductor alloys [6].
In the present work, we are less interested in predicting chemical trends than in obtaining band- structures that are as accurate as possible and, at the same time, are computationally easy to use in con- junction with electronic structure calculations for particular semiconductor alloys in which we are interested [7]. We have therefore chosen to fit the sp3s* tightbinding parameters for the compounds of interest exclusively to available experimental and theoretical bandstructure data. In this regard, we have used electroreflectance [g, 91 and photoemission
[lo] experimental data and empirical pseudopotential
[l
11, the Green’s Function method [Korringa,
Kohn and Rostoker (KKR)] and adjusted orthogo- nalized plane wave [12] theoretical results for the materials we consider. We note in passing that since most measurements of the spin-orbit splittings indi- cate the sign of the dipole [13], bandstructures with spin-orbit coupling effects may be construed as in- cluding partial corrections for the ionic character of the compounds. (Such ionic effects are more impor- tant for the II-VI compounds than for the III-V

94
SYLMSTER
N. EKPENUMA and CHARLES W. MYLES materials treated in Ref. 2.) We have thus used, where available, such experimentally-determined values of the spin-orbit splitting of the bands at the I&) and the L(A,) symmetry points of the Brillouin zone, and have fitted these to their band representations for zincblende materials in a manner which we describe below.
In the actual fitting procedure, sometimes an average of eqns (2) and (3) is needed to obtain a good value of (E,, - E,,)(i, - j.,).
We further use the definitions of the valence band r-point spin-orbit splitting A,,(Te - r,), and the interband transitions, E; from E(Ti) to E(I$), and
E,, from E(T;) to E(r5) (see Appendix A). Using these definitions, we obtain
2. PROCEDURE FOR OBTAINING
BANDSTRUCTURE PARAMETERS
We begin with a semi-empirical nearest-neighbor tightbinding Hamiltonian with spin-orbit coupling of the form discussed in Ref. 3, which we write as
(A., - j.,) = [$(Ei’ - Es.) - i(E,, - Ep,c)(i., - A,)]’ *.
(4)
This expression for (& - 3.,) can also be written in terms of the A,, spin-orbit splitting as
The spin-orbit part of the Hamiltonian H,, couples the different spin states of the p orbitals on the same atom. There are 10 orbital states per atom, the four sp’ states plus the excited s* state, with spin up and spin down states associated with each. The Hamil- tonian is thus a 20 x 20 matrix with spin-orbit coupling parameters for the anion and cation states respectively, of 1, and i.,, with 13 other parameters
(for a total of IS) which are related to the two-center
Slater-Koster parameters [14]. These parameters are fit to experimental and theoretical bandstructure data.
By utilizing the expressions for the band energies at the r, L and X symmetry points of the Brillouin zone in terms of the tightbinding parameters, and by inputting known experimental or theoretical values for these energies, the 15 tightbinding’parameters can be determined. (See Appendix A for more details.)
Specifically, very straightforward manipulations of the expressions for the lYs, r, and r6 band energies
[3, IS] yield simple expressions for the sums of the diagonal (on-site) tightbinding matrix elements
(E,,+ E,,), (EPe + E,,), and for the sum of the spin-orbit splitting parameters of the anion and cation (A,, + ;I,). As mentioned above, in previous
Sp’S* models [2,3], the differences in the on-site matrix elements E,dC, and Epdc, were fitted empiri- cally to the s- and p-atomic energy differences. In our case, we employ the definitions of the valence band
L-point spin-orbit splitting term A, (.& - L6), and an interband “transition” EL from E(L:,) to
E(f&) [15]. We also define another “transition;’ EL6, from E(L$*)) to E(L.2’)). Expressions for EL and ELs may be found in Appendix A. With the aid of these, we obtain the simplified expressions: and
(E,, - E,,)(i,
H = H,+.H,_,. x [EL + A, - (j., + i,)]
(1)
(2)
- &) = $5 - EL&% + EL,). (3)
(A, - j.,) = [f(2 A0 - 3i., - 31,)(2 A0 + 2Ei - 3J.,
- 3i.,) + 2(E,, - E,,)(E., - I.,)]“‘. (5)
The explicit inclusion of the A0 and A, spin-orbit splitting terms in the above equations enables the use of their experimentally-determined values in our cal- culations. Also, it is clear from eqns (l)--(4) that the product (E,, - E,,,.)(i., - E.,) and hence (,I, - E.,), is not uniquely determined.
“reasonable” values for (Ep,a - E,,)(i., - &) and
(2. - &) such that spin-orbit splitting parameters i., and 1, are positive quantities. We also require that the correct sign of (E,,,a - Ep,c)(i., - A.,) be obtained. This is negative for three of the II-VI compound semi- conductors studied, being positive only in HgSe [ 161.
Even though we have not fitted our parameters to atomic spin-orbit splitting values, we have compared our I values to A, and A, [17], which are the atomic contributions of the anion and cation, respectively, to the spin-orbit splitting of the compound. This enables us to tell if we have obtained the correct trend.
The s-orbital splitting [2,4] (E,, - related to the p-orbital the band splitting at the X symmetry point of the
Brillouin zone. This relation, which is usually given in terms of the single group notation rewritten (approximately)
Es, - ~5, = Ep, - Ep.o
Our aim is to obtain splitting as
E,,a) can be
(Ep,c - E,,,) and
[2], can be
- EGG;) + E(X;) - E(X;), (6) where the double group notation has been used.
Thus by inputting the energies at the r and X points, and the spin-orbit splittings A,, and A,, we are able to determine the diagonal tightbinding matrix elements ES,, E,,, Epa and E,,, the spin-orbit split- ting parameters x,, and i., , and the off-diagonal matrix elements WI, and tJC,, For the latter, we have adopted the notation (4Vij)? = Ut of Ref. 3, where i,j = S, x, y, z. Our procedure for fitting the parame-
, CJ,, and the parameters associated with the higher-lying s* states (ES,,, E,,,, U,., and Up,,,.)

II-VI zincblende compounds 95 to the E(X,) and E(X,) band energies follows in the manner of Ref. 2 for the conduction band and Ref. 4 for the valence band. We use the lower values of
E(&) and E(X,) of the valence band to fit to the parameters W, and UP,. The upper values of E(&) and E(X,) of the valence band which result from the splitting of E(X$) (single group notation) are fitted to
U,, and U,#, and U,, and UPI,, respectively. The form of the equation for E(&) is f(E,, + b - ~KME,, + & - E(G)] - %,j.:,l x [{E,, - &G)]{E,, + 1, - EM,)] -
A similar equation holds for E&X’,), but it involves
U,,J and U,,. The approach we take in solving these equations is to use the values of U,$ and U,,, obtained from the lower valence band E(&) and E(X,) as initial values and to proceed iteratively [3f. This then leaves four parameters to be determined, the diagonal and off-diagonal parameters associated with the ex- cited P-states. To find these, we fit to the E(Xi) and the E(X$) energies for which we obtain an expression of the form
[Es., - EG’;)l[fE, determined.
(7)
- E@‘C,))fE,, - 2i, - GGN
- 2Uj ] - 4{ E,, - E(X;)} U;., = 0. (8)
A similar equation hofds for E(X$) with interchange of anion and cation [3]. The values of the excited state energies are chosen to be slightly above the energies
E(X$) and
E(X:).
They are comparable to the ener- gies E(.L.;) and E(.L$*). With these values, we then solve for U,., and U,,., and our 15 parameters are
3. RESULTS AND SUMMARY
The parameters as determined above are shown for the four compounds HgSe, ZnS, ZnSe and ZnTe in
Table 1. The off-diagonal matrix elements scale ap proximately as the inverse bond length squared, in agreement with Harrison’s bond orbital model [18].
Also, we have investigated the validity of empirically fitting the differences in the on-site matrix elements
(E,, and
EpAE)) atomic values, a procedure that was followed in previous sp’s * calculations [2,3]. Interestingly, we find that the differences in these on-site parameters which result from our fitting scheme still scale ap- proximately as the difference in the corresponding atomic energies, as they would had we fit them to such differences. However, the scaling factors (8,, fl,) we find for the II-VIs need to be slightly higher than those obtained in Ref. 3. We obtain average values of
#?, = 0.80 and &, = 0.70, compared with values of 0.70 and 0.50, respectively, obtained in Ref. 3. Our values are thus close to those found for the III-V and group
IV semiconductors in Ref. 2. This dependence on atomic energy differences shows that the chemistry of the materials is correctly accounted for by our method.
For the four materials we have considered, we list in Table 2 the values of the bandgap,
E,, the spin-orbit splittings d, and A,, and the interband transitions
E;, EL, E,,, and
Et, which were defined earlier. We list both the values obtained from our bandstructure calculations (first column for a given material) and the values which we have used as input into the determination of our tightbinding par- ameters, as discussed above (second column for a given material). Comparison of the two columns for each material gives an indication of how well the resulting bandstructures reproduce the input data. As can be seen from the table, the calculated values compare quite favorably with the input data for energies at the r point of the Brillouin zone, while larger discrepancies occur for energies at the L point.
Thus, our calculated bandstructures are less accurate at the L point than they are at the r point. This genera1 trend has been noted earlier for sp%* band- structures in other materials [19].
Table 1. Nearest-neighbor tightbinding Hamiltonian matrix elements for HgSe,
ZnS, &Se, and ZnTe. All energies are in electron volts
Matrix elements HgSe ZnS ZnSe ZnTe
- 12.2751
1.0876
-0.1251
5.4874
- 1.2127
2.6355
5.0586
3.2701
- 2.6476
~~
0:8799
-0.5524
0.0485
0.6265
- 12.2510
0.4490
3.3510
7.9510
-2.2846
2.0949
5.0865
2.8957
-4.5769
8.2500
8.5000
1.5180
-2.5210
0.0480
0.0220
- 10.8447
0.7153
1.3547
6.5247
-4.3724
2.4022
6.0245
3.3316
- 5.3090
7.5000
8.0000
I .5867
- 2.6580
0.1674
0.0126
- to.3990
0.21 to
1.3990
5.2890
-3.5115
1.6979
4.3154
1.0889
-4.9893
6.5000
7.7500
1.3235
-4.2076
0.3311
0.0289

96 SYLVESTER EKPENUMA and CHARLES W. MYLES
Table 2. Comparison of various energies obtained from bandstructure calculations (first column for each material) with values used to fit tightbinding parameters (second column for each material). All values are in electron volts
ZnSe Parameter
HgSe
ZnS ZnTe
E;
;
E;
Er,
Eh
6.12
8.42
0.31
0.22
-0.15
6.31
7.09
I.257
8.3011
0.31tt
0.30tt
-0.25t
6.OOt
7.907
8.50 8.57% 5.86
9.83
0.14
0.05
9.09%
0.07$$
1.68
0.91
0.05$% 0.32
2.37 3.80
8.39
3.80%
8.50%
10.40 9.06%
6.66
7.81 t Bloom S. and Bergstreser T. K., Phys. S~arus Solidi 42, 191 (1970).
$ Herman F. er al., see Ref. 12.
8 Walter J. P. et al., Phys. Rev. Bl, 2661 (1970).
‘;Chelikowsky J. R. and Cohen M. L.,
Phys. Rev.
B14, 556 (1976).
11 Rev.
133,A 736 (1961). tt Cardona M. et al.. Phys. Rev. 154, 690 (1967).
$1 Eckelt P.,
Solid Stare Commun.
6, 489 (1968).
# Cardona M. and Harbeke G., J. appl.
Phys. 34, 813 (1963).
5.86
7.1@
0.90$
0.585
2.37#
6.628
7.618
7.42
9.15
0.45
0.13
2.76
7.81
8.54
7.42?
8.481
0.457
0.287
2.767
7.787
8.72:
In Figs l-4, we show the bandstructure diagrams For the case of HgSe (Fig. I), however, and to a obtained (solid curves) using our parameters for the lesser extent for the ZnS bandstructures (Fig. 4). the compounds HgSe, ZnSe, ZnTe, and ZnS, respec- lowest valence band (not shown for HgSe) is not tively. These bandstructures compare favorably with accurately represented by our results. This may be pseudopotential [9, lo] and KKR [ 121 energy bands due to the fact that while a wealth of information as indicated by the dashed and dotted curves respec- relating to the bandstructure data exists for the tively in the figures. The ZnSe and ZnTe bands are ZnSe and ZnTe zincblende systems, very little is well reproduced over a wide range of energies in both known about the ZnS and HgSe bandstructures. the conduction and the valence-bands. Thus. there is less available data with which to fit our
12.0
HgSe
8.0
F
3
& ti
TI
4.0
0.0
-4.0 r X U,K
WAVEVECTOR k
Fig. 1. HgSe bandstructure obtained with our procedure (solid curves) compared with pseudopotential
(dashed curves) bandstructure of Bloom and Bergstresser, Ref. 1 I.

II-VI zincblendc compounds 97
0 cd
0 i
9
0
(K’) AOtElN3
9 f
9
9)

II-VI xincblende compounds 99
L r X U,K r
WAVEVECTOR k
Fig. 6. Bandstructure for the alloy ZnTe, _,S,Se, at x = 0.25, along the principal symmetry directions of the Brillouin zone. bandstructures for these two materials than for ZnTe bandstructures are shown for compositions x = 0.50 and ZnSe.
For ZnS for instance, previous bandstructure and 0.25 in Figs 5 and 6, respectively. As can be seen, the correct shape for all the bands is preserved for the calculations have given results without the inclusion alloys. Also, the trends in the bands and energy gaps of the spin-orbit splittings (single space group with composition x appear to be reasonable. So, notation). Using the information from the single despite the discrepancy in the dispersion of the lowest group notation bandstructure [12] and a knowledge valence band in the ZnS bandstructure, the proper of the size of the splittings at the various band chemical trends are still found for the alloys with our symmetry points in the Brillouin zone [8, 121, we have fitting procedure. obtained the spin-orbit split levels for this material by forcing the band energies at these points in the Acknowlecfgemenrs-We are grateful to Texas Instruments, absence of spin-orbit coupling to be the average
Inc., for a grant which partially supported this work. value of the split levels involved. Even with this, the conduction band is still reasonably reproduced. The theoretical bandstructure result we have used for
HgSe [ll] does not contain information about the valence band below about -4.0 eV, so we have used photoemission data [lo] values for the r; and the lower Xfi bands, deeper in the valence band. As with the ZnS structure, we obtain a reasonable conduction bandstructure for HgSe. In particular, the inverted gap is well reproduced.
As a further test that the correct chemistry of the sp’ bonding of the materials is preserved by our procedure, we have computed the bandstructures of the alloys ZnS,Se,_.r and ZnTe,_&Se, using the virtual crystal approximation to model the variation in properties with alloy composition. The resulting
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100 SYLWR N. EKPENUW and
CHARLES
W. MYLE~
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W,) = &., + E,,) - (4 + 4) and
In the above equations, we have adopted the definition
(4l’,,)* = ui of Ref. 3. where the are the transfer
(off-diagonal) matrix elements of the tightbinding Hamil- tonian. The valence band spin-orbit written as: splitting A0 can be
4=E(T,)-E(T,)=~I,+I,)-iE;
+ f[{E,, - E,, - 2(& - A,,j’ + 4C’?,,]“‘, (A4) where the twice the square root term of eqn (A2). Also, the approxi- mate expressions for the L,,, and Ib band energies are given by 1151
QLIJ) = #,,a + E,,<) + 31, + A,) St: f[{(E,., - E,,) and k f[{W,, -
E,,<) - 2(1, - ,$)I? + ~LJ’:_~]“’ (A2)
E(T,) = #,,
E;
+ E,,) + !(A. + &) + #(E,.,~ - E,,) transition [from
+ (i -I
E(TC,) c
Vl,s to E(rl)] is given by
)}r + (LI
E(L,) = f(E,., + E,,) - !(A. + j.,) f f[{(E,., - E,,,)
(AS)
The valence band L-point spin-orbit splitting can thus be written as
A, = E(L,,) - E(L,) = (j., + A,) - f& + %(E,,, - E,,.,)
-@ a c r r X’J 9 (A7)
APPENDIX A
The expressions for the Fe, F, and Ts band energies are [3, 151:
E(T,) = #E,, + E,,) + $(&,a - E,.,)? + 4L1:,]“‘, (Al) where EL is defined similarly to E;, and is given by twice the square root term of eqn (AS). The above equations and the transitions which have been defined, enable us to obtain the necessary equations [eqns (2)-(8) of the text] for fitting the band parameters.