Self-consistent calculation of the dielectric function of a periodic lattice

19 June 1972

PHYSICS LETTERS

Volume 40A, number 1

SELF-CONSISTENT CALCULATION OF THE DIELECTRIC FUNCTION OF A PERIODIC LATTICE T. LUKE& D.J. MORGAN and B. NIX Department

of Applied Mathematics and Mathematical Physics, University College, Cardiff, UK

Received 4 May 1972

It is shown that the method used by Lukes and Morgan to calculate the resolvent operator of a perfect lattice can be used to obtain the one electron Green function and hence the dielectric constant in random phase approximation provided the matrix elements of the pseudopotential are known. The method can be genera&d to give a self-consistent determination of the dielectric function.

A convenient definition of the dielectric function ~(4, w) is provided by the ratio of the effective interaction, V(q, w) between electrons in a medium and the bare Coulomb interaction V,(q). In general E is a matrix and eY= V,. The equation Y= V, t V,nV, where 1 is the irreducible polarisation, enables E to be calculated if I is known. In real space the approximation +,x1)

= -iG,(x,xI)

G,(xl,x)

(1)

where G(x,xI) is the one-electron Green function and G,(x,xI) is the corG(x,x,) = -iVu(x)u+(xI)) responding function evaluated without electron-electron interactions, can be shown to be equivalent to the random phase approximation for a free electron gas [ 11. For a periodic lattice the only method used hitherto has been the self-consistentapproach of Ehrenreich and Cohen [2] which expresses the dielectric function as an infinite sum involving the energies and matrix elements of one-electron wavefunctions. It may be shown quite generally that this approach is exactly equivalent to the approximation (1) [4]. The purpose of the present note is to point out, firstly, that e may be calculated for a periodic lattice directly from eq. (l), and, secondly, that such a calculation may be made self-consistent. We introduce the four-dimensional Fourier trans-

(2)

s

* Go(x1’x2) = &4

*

(-&,)‘t

where G,(I, m) is given by

exp {i [fixI - m*x2])Go(l, m)

G,(l,m)=(Ile(H-EF)+ WG-Jo E-H-ie

E-H+ie

Irn)

(3)

3

and H = Hot VP, where VPis the periodic lattice potential. Defining go=B[f(EF--H,)]/(E-H,+ie),

(4)

G,’ obeys the integral equation Go = g,‘V Gof and is therefore given by Go = (6’ - VP)-‘. &is can be calculated by the methods of Lukes and Morgan [ 1] by direct inversion: E-(k+K#fie

QK2

. .. . . .. . .

-1

f-,WW G; = . . . .. . . . .. . .. . . . .. . . .. . . . .. . . . .. .. . .. . . . .. . . . .. . . . E- (ktK,J -+ic .................... ... fo WK,J

(5)

where the VK’s are pseudopotential ma; tr ix elements of the total effective potential and f. is the Fermi factor at T = 0. Hence E = I- V, I where n(qtK, q+K’) =

2 K”,K”‘B.Z.

j-

d31dE’ (2n)4

X Go<1tK”,ItK”‘,E’)Go(l-qtK”-KJ-qtK”-K). If the empirical pseudopotential matrix elements are used, this enables E to be evaluated directly from the Green function. The method can, however, be made self-consistent by using ionic potentials Y instead of effective potentials in (5). In this case each off diagonal element becomes typically e-1 I$ (or to a lower order of approximation E(K)-l q(K), and the method becomes a self-consistent one for determining the matrix elements of the dielectric tensor. 91

Volume 40A, number 1

PHYSICS LETTERS

References [l] A.L. Fetter and J.D. Walecka, Quantum theory of many particle systems (McGraw Hill, New York, 1971). (21 H. Ehrcnreich and M.H. Cohen, Phys. Rev. 115 (1954) 786.

19 June 1972

[ 31 T. Lukes and D.J. Morean, J. Phys. C: Solid St. Phys. 4 (1971) 2623. [4] T. Lukes and B. Nix, to be published.