Many-electron and many-body force effects in the lattice dynamics of metals and semiconductors

Abstract

Using a response function formalism developed earlier, a study is made of the basic form of the dynamical matrix when many-electron and many-body force effects are included. Attention is focussed separately on (l) a one-body response function F generated from that one-body potential V(r) which reproduces the exact ground-state charge density in the perfect lattice, and which tells us the density change Δρ when V → V + ΔV and (2) an interaction function U(rr′) which is related to a functional derivative with respect to density of the exchange and correlation potential. With a metal like Al in mind, the form of F is given to first order in a weak model potential. In the same nearly free electron regime, U is discussed by starting from the Dirac expression for the exchange energy. Contact is also established, by an alternative k space calculation, with the static calculation of elastic constants of Brovman and Kagan. In a very different situation corresponding to the valence bands of semiconducting Si or Ge, a tight-binding approximation appears to be more appropriate. The response function F, treated as diagonal in zeroth order for nearly free electrons, now has essential off-diagonal terms. The factorization of F is used to enable one to obtain an explicit expression for the “tensor” charge density in terms of quantities characteristic of the shell model of lattice dynamics. The shell model is thereby shown to contain many-body forces to all orders, whereas nearly free electron theory develops lattice dynamics as a sequence in two-, three-body forces etc. It is finally stressed that, for a given potential V, the calculation of F can be carried out, in principle exactly, from energy band theory, whereas of course U is only to be obtained approximately from many-electron theory. Nevertheless, a number of properties of U can be established precisely, in the long wavelength limit, and these are also presented.