Optical Properties of Materials using the Empirical Tight-Binding Method
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L. C. LEW YAN VOON Department of Physics, Worcester Polytechnic Institute, 100 Institute Road, Worcester, Massachusetts 01609 ABSTRACT Procedures for calculating the optical properties of periodic systems using the empirical tight-binding method are compared. Results of the linear and nonlinear susceptibilities are presented using a recently developed exact formalism [Lew Yan Voon and Ram-Mohan, Phys. Rev. B47, 15500 (1993)]. INTRODUCTION The multiband empirical tight-binding method was developed by Slater and Koster [1] in 1954 in order to provide an efficient full Brillouin zone parametrization of the electronic band structures of periodic systems. The Hamiltonian was formally represented in terms of a finite number of localized atomiclike basis functions satisfying the Bloch condition, the functions themselves never being explicitly given. The unknown parameters (the tightbinding [TB] parameters) were then obtained by fitting the eigenvalues of the Hamiltonian matrix (i.e., energies) to experimental data. For the next 30 years or so, the TB model was used extensively to solve a wide variety of problems: point defects levels, surface and heterostructure band structures, bulk and surface structures, transport coefficients. Missing was the knowledge of how to calculate optical coefficients (e.g., absorption coefficient)
exactly and completely. The argument was that the calculation of the optical coefficients requires first calculating momentum matrix elements and the latter could not be done because the actual explicit wave functions were never known. Approximate schemes were introduced which relied mainly on fitting the momentum operator in a fashion analogous to the Hamiltonian [2, 3]. In 1993, we managed to prove explicitly that the momentum operator could be defined exactly in terms of the Hamiltonian operator [4]. Our aim here is to compare the fitting methods to our exact expression. We also present results of the linear and nonlinear optical susceptibilities obtained using our procedure to illustrate the difference between different TB parameter sets for optical properties. THEORY The long-wavelength optics we are interested in here is obtained from coefficients characteristic of the material. For linear optics, we will refer to the imaginary part of the dielectric function, 2 7-e 2 E2(w) -E €omw 2v f"(E") [1 - fc(Ec)] I(cIpIv)126(E., - hw), (1) f00
cvk
while, for second-order nonlinear optics, we will compute the coefficient of second-harmonic 377 Mat. Res. Soc. Symp. Proc. Vol. 491 ©1998 Materials Research Society
generation [5]:
Zk
3 ieh3(2E)
X2jk(2E)E-0 mE E3 ×
i (Ec,v + 2E)(Eoo + E) +
vccldk
(2)
For the above, one needs the band structure Enk and the momentum matrix element p,", (k). In computing the momentum matrix element within the TB formalism, a number of procedures had been adopted in the past. The most appealing was to represent the momentum operator in the same basis set as the Hamiltonian: Pnm(k) =
(nkIPImk) = E
Cb(nk)Cb',(mk)
afibb'
Z
eikRtbb'
I
(3)
x fdr k*b(r) POpb,(r - R
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