Origin of second-harmonic generation in the incommensurate phase ofK2SeO4

Abstract

We show that a ferroelectric phase transition takes place in the incommensurate phase of the K 2 SeO 4 crystal. The ferroelectric character of the incommensurate phase explains the second-harmonic generation observed in the corresponding temperature range.

On the dielectric susceptibility calculation in the incommensurate phase of K 2 SeO 4

“…In what follows we show that the thermodynamic potential of the crystal with the domain-induced microscopic electric field can be reduced to the potential, analyzed in the paper [4], and the behavior of the system in such a case should be that described ibidem. Minimizing the potential over P (R) andP , under the condition (P (r) −P )dx = 0, which should be added to the potential equation with the Lagrange multiplier, one obtains…”

“…Spatial distributions of the IC modulation phase and polarization vector in the ferroelectric (or z-polarized) IC phase 4 are depicted in Fig.1. Below we explain the origin of the difference between the results of the two calculations, 3,4 and show that the thermodynamic potential used for calculations by Nattermann and Trimper 3 should be supplemented with a term, taking into account the local (Lorentz) microscopic electric field, induced by the dipoles (the neighboring pairs of the domains in Fig.1 can be viewed as periodically arranged dipoles). In such a case, calculation of the dielectric susceptibility by Nattermann and Trimper 3 should be revised, and it reduces to that carried out by Aslanyan.…”

“…1,2 Nattermann and Trimper 3 calculated the dielectric susceptibility χ of the crystal in the IC phase, and showed that χ diverges on approaching the lock-in transition point (i.e., a loss of stability of the IC phase with respect to the commensurate phase takes place). The dielectric susceptibility for the same IC structure was calculated also by Aslanyan, 4 and it was obtained that χ diverges not in the lock-in transition point, but at a higher temperature, giving rise to the phase transition from the IC to the ferroelectric IC structure. Spatial distributions of the IC modulation phase and polarization vector in the ferroelectric (or z-polarized) IC phase 4 are depicted in Fig.1.…”

“…In such a case, calculation of the dielectric susceptibility by Nattermann and Trimper 3 should be revised, and it reduces to that carried out by Aslanyan. 4 The starting point in both calculations 3,4 is the thermodynamic potential of the following form:…”

“…In this expression (and below as well) the notations of paper [4] are used, where η 0 and ϕ are, respectively, the amplitude and the phase of the IC modulation, k is the IC vector at the temperature of the IC transition (from the normal to the IC phase), P is the polarization vector in the z direction, and the IC modulation is given by η 0 cos(kx + ϕ). However, potential (1) should be supplemented with a…”