Solitary Wave Solutions in a Diatomic Lattice

Abstract

Recently, it has been shown that the non-linear and anisotropic oxygen polarizability explains successfully the unusual dynamical properties of oxidic perovskites (1). In particular the temperature dependence of the ferroelectric (transverse optical) soft mode and its coupling to the transverse acoustical modes in incipient ferroelectrics (SrTiO₃ and KTaO₃) has been described by a single quartic electron-phonon coupling parameter (localised at the oxygen lattice site) in the renormalised harmonic approximation (RHA (2)). This coupling leads, using the adiabatic condition, to a long-range anharmonic interionic potential along the (100)-direction where we have diatomic chains of alternating transition metal and oxygen ions. In view of the well-known limits of the RHA (3) we are interested in more general solutions of the problem. The structure of the Hamiltonian suggests a simplified version which allows analytical solutions while keeping the essential features of the underlying physics. The simplest model consists of a quasi-one dimensional diatomic lattice with harmonic and quartic nea-rest-neighbour interactions. It is shown that this lattice has at least two kinds of solitary wave solutions differing in their acoustical or optical-mode character. The low-energy acoustical excitation obeys a modified Korteweg-de Vries equation similar to the soliton in a monoatomic chain (4). The optical mode is the solution of a ⌽⁴-type wave equation and represents a ‘kink-soliton’; it exhibits a close relation to the Krumhansl-Schrieffer model (5) for structural phase transitions.