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Semiconductor effective charges from tight-binding theory J. Bennetto and David Vanderbilt Department of Physics and Astronomy, Rutgers University, Piscataway, New Jersey 08855-0849 (February 8, 2008) We calculate the transverse effective charges of zincblende compound semiconductors using Har- rison’s tight-binding model to describe the electronic structure. Our results, which are essentially exact within the model, are found to be in much better agreement with experiment than previous perturbation-theory estimates. Efforts to improve the results by using more sophisticated variants of the tight-binding model were actually less successful. The results underline the importance of includingquantitiesthataresensitivetotheelectronic wavefunctions,suchastheeffectivecharges, in thefitting of tight-bindingmodels. 77.22.Ej, 77.22.-d, 71.15.Fv 6 9 9 1 ∗ TheBorneffectivechargese ,alsoknownastransverse this question does not appear to have been answered T n or dynamic effective charges, are the fundamental quan- previously. The only previous work of which we are a tities which specify the leading coupling between lattice aware made use of a two-center perturbation approxi- J displacements and electrostatic fields in insulators.1 In mation to obtain estimates of the effective charges.6,7 5 general the effective charges are site-dependent tensors, This approach used an expedient division of the effec- 1 tive charge into “static” and “transfer” charge contribu- P = e∗(l)u(l) + O(u2) , (1) tions, with the interpretation of the latter being open to v1 i Xl,j Tij j somequestion.8 The purposeofthis Reportis to present 2 essentially exact calculations of the transverse effective 0 where Pi is the polarization in cartesian direction i, and charges computed for zincblende II-VI, III-V, and IV- 0 u(l) is the displacement of sublattice l in cartesian di- IV semiconductor compounds using the Harrison tight- j 1 rectionj. However,forcompound semiconductorsofthe binding parametrization. While the results could have 0 zincblendestructure,whicharethefocushere,itiseasily beenobtainedusinglinear-responsetechniques,wefound 6 ∗ shownthattheeffectivechargesarescalars,andareequal it simpler to to compute the e ’s instead from finite dif- 9 T / and opposite for cation and anion; it is conventional to ferences, calculating directly the change in bulk polar- h use the positive cation effective charge to characterize a izationfromasmalldisplacementofonesublattice using t - given compound. The effective charges for a variety of the formulationofKing-SmithandVanderbilt.9 Ourcal- rl zincblendesemiconductorshavebeencomputedusingab- culationsareboth closerto the experimentalvalues,and mt initio density-functional linear-responsetheory and have more strongly correlated with them, than the previous been found to agree very well with experiment.2,3 How- reults. However,theyarestillsignificantlylowerthanex- : v ever, it is interesting to inquire whether more approxi- periment, and the correlation is still not very good. We i mate schemescangivea goodaccountingofthe effective alsotriedincludingoff-diagonalpositionmatrixelements, X charges in compound semiconductors. If so, additional and considered a modified universaltight-binding model r a insight into the chemical and physical factors that affect that was proposed to incorporate non-orthogonality of the e∗ might be obtained. the basis functions.10 Unfortunately, both modifications T One particularly attractive and well-known approx- were found to worsen the results. imate scheme is the universal tight-binding model of The details of our theoretical approach are as follows. Harrison.4,5Itprovidesastraitforwardandcomputation- We consider each zincblend compound at its experimen- allyefficientapproachtocalculatingelectronicproperties tal lattice constant, with and without displacements of of solids using a minimal orthogonal sp3 basis set, with one sublattice along the zˆ direction by ±0.0001˚A. The theHamiltonianlimitedtotheon-siteandnearestneigh- Bloch functions are computed in the tight-binding rep- borterms. Theon-siteelementsǫ andǫ aretakenfrom resentationusing standard direct matrix diagonalization s p calculated free-atom term values, while the interatomic onameshofk-points. AccordingtothetheoryofRef.9, elements (V , V , V and V ) are taken to be the electronic contribution to the polarization takes the ssσ spσ ppσ ppπ species-independent “universal” constants times the in- form verse square of the distance. Given its simplicity, the M modelisimpressivelysuccessfulinestimatingmanyelec- ie tronic properties of a wide variety of materials.4,5 Pe =−(2π)3 Z dkhunk|∇k|unki , (2) It is thus natural to ask what the Harrison tight- nX=1 BZ binding model would predict for the effective charges wherethesumrunsoveroccupiedbandsandtheunk are of the zincblende compound semiconductors. Oddly, the periodic parts of the Bloch wavefuctions, 1 TABLE I. The transverse charge e∗T for zincblende semi- 3.0 conductors calculated at the experimental lattice spacing d, O-D compared with perturbationestimates of Kitamuraand Har- rison (KH) and experimental values. O– and NO– indicate ge 2.5 O-D ar KH orthogonal and non-orthogonal tight-binding models respec- h c tively, while –O and –OD refer to diagonal and off-diagonal e 2.0 representations of theposition operator. v ti c d(˚A) O–D O–OD NO–D KHa Expt.b fe 1.5 f SiC 1.88 1.97 2.20 1.84 2.57 e d BN 1.57 1.24 0.96 1.01 2.47 e t 1.0 BP 1.97 -0.09 -0.18 -0.23 ula BAs 2.07 -0.39 -0.42 -0.54 c AlP 2.36 1.92 1.61 1.64 2.28 Cal 0.5 AlAs 2.43 1.75 1.50 1.50 2.30 AlSb 2.66 1.48 1.22 1.32 1.93 0.0 GaP 2.36 1.88 1.57 1.62 0.89 2.04 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Experimental effective charge GaAs 2.45 1.73 1.47 1.51 0.71 2.16 GaSb 2.65 1.41 1.12 1.29 0.40 2.15 FIG.1. Comparisonoftheoreticale∗T valuesfromthiswork InP 2.54 2.26 1.94 1.99 1.26 2.55 (filledsymbols)andfromperturbationestimatesofKH(open InAs 2.61 2.11 1.85 1.86 1.07 2.53 symbols, from Ref. 6), plotted against experimental values. InSb 2.81 1.86 2.14 0.75 2.42 Filled squares indicate results for compounds not considered BeS 2.10 1.61 1.08 0.71 byKH. BeSe 2.20 1.56 1.04 0.71 BeTe 2.40 1.51 0.96 0.57 ZnS 2.34 1.89 1.46 0.53 1.25 2.15 ofthepositionoperatorarespecifiedinthetight-binding ZnSe 2.45 1.86 1.47 0.50 1.15 2.03 basis. In the context of the above formulation, these po- ZnTe 2.64 2.05 2.50 0.98 2.00 sition matrix elements are needed for the conversion (3) CdS 2.53 1.98 1.61 1.10 2.77 between the unk and ψnk. The simplest ansatz is to as- CdTe 2.81 1.92 1.53 0.41 1.24 2.35 sume that the position operator is diagonal in the tight- aRef. 6. bindingrepresentation,withelementsreflectingthecoor- bRef. 7. dinates of the atoms. However, such an ansatz is rather unphysical; it would imply that the center of charge of an sp hybrid on an atom would lie exactly at the center unk(r)=e−ik·rψnk(r) . (3) of that atom, whereas in reality it would be displaced toward the principal lobe of the hybrid. We report our We are only interested in the z components of P for the results first for the simple “diagonal” ansatz. Later, we distortions considered. After discretization in k-space, discuss the effects of trying to improve upon this ansatz, these are given9 as aswellasthe effectofincludingthe non-orthogonalityin the model of Ref. 10. 2e(∆k)2 Pz =− (2π)3 Xk φ(k⊥). (4) theThdeiagreosnualltsrefporretshenetoarttiohnogoofnralaHreagrriviseonninmothdeelc4oulusminng ⊥ labeled “O–D” (orthogonal, diagonal) for a variety of where k⊥ = (kx,ky) is discretized on a mesh of spacing zincblende structures in Table I.12 The last two columns ∆k, and the contribution from a string of J k -points givethevaluesoftheperturbationestimatesofKitamura z takes the Berry-phase form11 and Harrison (KH),6 and the experimental values, for comparison. The data is also represented graphically in J−1 Fig. 1. The filled symbols are our results; the open ones φ(k⊥)=Imln det um,k⊥,kj|un,k⊥,kj+1 . (5) are those given in Kitamura and Harrison.6 (The filled jY=0 (cid:10) (cid:11) squaresrepresentcompoundsnotstudiedinRef.6.) Our calculationsshowsaclearimprovement,althoughwestill Here the argument of the determinant is a 4×4 matrix systematically underestimate the experimental values of correspondingtothe factthatmandnrunoverthe four ∗ e . The correlationbetween our calculations and exper- occupied bands. We typically use a discretization onto T iment is not very good,althoughit shouldbe noted that a 16×16 mesh in k⊥ space, and extrapolate to J = ∞ the lowest filled point is BN, a first row compound for usingstringsofJ =32andJ =64k points. Thetrivial z which the model is less accurate. ∗ ionic contribution to P is added, and the value of e z T Whilethepresentresultsarecertainlyanimprovement deduced by simple finite differences. over the perturbation estimates of KH, there is clearly Strictly speaking, the polarization P and effective roomforimprovement. Wethusinvestigatedtwopossible ∗ charge e are not well-defined until the matrix elements T 2 3.0 (11) of Ref. 10. Some care is required in the appli- O-D cation of the theory of Ref. 9 to this case: the inner ge 2.5 O-OD product appearing in Eq. (5) has to be generalized to har NO-D takea formlike hφm,k⊥,kj|Sk⊥,k¯|φn,k⊥,kj+1i, where|φnki c is the vector of tight-binding coefficients corresponding ve 2.0 to |unki, Sk is the overlap matrix at wavevector k, and ti k¯ =(k +k )/2. The results are shown in the column c j j+1 fe 1.5 labeled “NO–D” (non-orthogonal, diagonal) in Table I, f e d andasthefilledtrianglesinFig.2. Onceagain,this“cor- e rection” is seen to act in the wrong direction, worsening t 1.0 a ul the agreement with experiment. alc 0.5 The failures of the abovetwo attempts to improve the C tight-binding model are disappointing, but perhaps in hindsight they are not surprising. For the case of the 0.0 non-orthogonal model, a partial explanation may lie in 0.0 0.5 1.0 1.5 2.0 2.5 3.0 the fact that the non-orthogonality was added in large Experimental effective charge part to improve the fit for structures that were not four- FIG. 2. Comparison of theoretical e∗T values using differ- fold coordinated, which is not relevant here. But more ent variants of the tight-binding model, plotted against ex- fundamentally, we note that the model Hamiltonians we perimental values. The notation is the same as in Table I. tested were developed by fitting to energy bands; thus, the fit included only information about energy eigenval- ues, and not the wavefunctions per se. However, the modifications of the tight-binding model to see whether electric polarization is a quantity which depends sensi- theywouldbringthetheoreticalresultsintobetteragree- tively on the electronic wavefunctions themselves. Thus, ment with experiment. First, we tried going beyond the arealimprovementin the tight-binding modelcanprob- artificial diagonal ansatz for the tight-binding represen- ably best be accomplished by including quantities that ∗ tation of the position operator by including some off- are sensitive to the wavefunctions, such as e values, in T diagonal terms. Specifically, we included on-site matrix the fitting procedure itself. elements between s and p orbitals, e.g., hs|z|p i. The In summary, we have carried out essentially exact cal- z ∗ valuesofthesematrixelementswereobtainedfromsepa- culations of the transverse effective charge e in com- T rateLDAcalculationsonfree(neutral,spin-unpolarized) poundsemiconductionswithinHarrison’suniversaltight- atoms. By symmetry, off-diagonalp−p matrix elements binding scheme. We find a significantly improved agree- of r are zero, and we assumed all off-diagonal intersite ment with experiment, compared with previous pertur- elements to be zero as well. The contribution of these bation estimates. However, the theoretical results still extra off-diagonal terms to the polarization P was cal- show a systematic underestimate relative to experiment, culatedas a simple expectationvalue, using the already- by an average of 20%. Attempts to improve the agree- calculated wave functions (the Berry-phase approach is mentby including off-diagonalpositionmatrix elements, not needed). The results are given in the column la- or non-orthogonalityof the basis,were actually found to beled “O–OD”(orthogonal,off-diagonal)in Table I, and lead to a worsening agreement with experiment. Based are compared with the previous results (open vs. closed onthisexperience,wesuggestthatitmightbehelpfulto circles) in Fig. 2. Unfortunately, the correction appears use the effective charge as a fitting parameter in future to be in the wrong direction, and there is no apparent tight-binding models. Such an approach might lead to improvement in the correlation between theoretical and a more accurate description of the electronic properties experimental values. of semiconductors within this class of simple, but very Second, we attempted to improve the results by using useful, models. a tight-binding model thatincludes non-orthogonalityof ThisworkwassupportedbyONRGrantN00014-91-J- the basis, as proposed by van Schilfgaarde and Harri- 1184. J.B.acknowledgessupportofONRGrantN00014- son.10 They used extended Hu¨ckel theory to derive the 93-I-1097. overlap elements 2Vll′m Sll′m = , (6) K(ǫl+ǫl′) from the original model, ǫl and ǫl′ are the onsite ener- gies from the same model, and K is a parameter de- 1R.M. Pick, M.H. Cohen, and R.M. Martin, Phys. Rev. B pending on row of the Periodic Table, chosen to fit the 1, 910 (1970). equilibrium spacings of the IV–IV crystals. The Hamil- 2S. de Gironcoli, S. Baroni, and R. Resta, Phys. Rev. Lett. tonian parameters were also renormalized following Eq. 62, 2853 (1989). 3 3K. Karch et al.,Phys. Rev.B. 50, 17054 (1994). 4W.A. Harrison, Phys. Rev.B 24, 5835 (1981). 5W.A. Harrison, Electronic Structure and the Properties of Solids(Freeman,SanFrancisco,1980)(reprintedbyDover, New York,1988). 6M. Kitamura and W.A. Harrison, Phys. Rev. B 44 7941 (1991). 7M. Kitamura, S. Muramatsu, and W.A. Harrison, Phys. Rev.B 46 1351 (1992). 8In Refs. 6 and 7, the polarization change is taken to have contributions from a “static charge” and a “transfer charge.” The former contribution is just the static charge residingon anatom timesthedistanceitmoves,while the latteristakenastheinducedchangeinstaticchargetimes the interatomic separation. This last approximation is es- pecially questionable; even within the stated assumption that electrons transfer only to neighboring atoms, it ne- glects the possibility of charge transfer along bonds other than the one on which the perturbation analysis is being performed. 9R.D.King-SmithandD.Vanderbilt,Phys.Rev.B471651 (1993). 10M. van Schilfgaarde and W. Harrison, Phys. Rev. B 33 2653 (1986). 11M.V. Berry, Proc. Roy.Soc. London A 392 451 (1984). 12Wealsocalculatedthee∗T tensorforthesamecompondsin the ideal wurtzite stucture, and found that neither of the two independent e∗T components typically differs by more than ∼2% from the zincblendevalues. 4