Dynamical Lattice Instabilities in Alloy Phase Diagrams

Abstract

Ab initio electron structure calculations can accurately give total energies of solids in assumed atomic configurations for which there are no experimental data. As an example, one may calculate the total energy of silicon and germanium not only in the observed and stable diamond-type lattice structure but also in body centred cubic, face centred cubic, hexagonal close packed and other structures.1 Similarly the difference in cohesive energy between bcc, fcc and hcp lattice structures can be obtained across a transition-metal row in the Periodic Table.2 As another example, one may find the vacancy formation energy in 3d-, 4d- and 5d-transition metals when they are assumed to have a bcc structure and compare that with the results in an assumed fcc structure.3 In all such calculations, the atomic positions are kept fixed in a certain lattice structure, i.e. bcc, fcc, hcp etc. However many of these structures, for a given chemical composition, are dynamically unstable. The well-known conditions for elastic stability under shear, in a lattice of cubic symmetry, are4

$$ {{c}_{{44}}} > 0;\quad C' = ({{c}_{{11}}} - {{c}_{{12}}})/2 > 0 $$

(1)

where c _ij are single-crystal elastic constants. Even if these inequalities referring to long-wavelength deformations are fulfilled, there may be instabilities under a lattice modulation of short wavelength. To ensure stability of a lattice for any small displacement of the atoms from their assumed equilibrium positions, all phonon frequencies ω(q,s) of wavevectors q and mode indices s must be real, i.e.,

$$ {{\omega }^{2}}(q,s) > 0 $$

(2)