Theory of phase transitions and modulated structures in ferroelectrics

Abstract

For ferroelectric systems described by an n>1 polar vector order parameter p, a new term in the free-energy density, β${[\mathrm{p}}^2$∇⋅p-p⋅∇${(\mathrm{p}}^2$)], is allowed. Considering an isotropic, centrosymmetric model and using Landau theory, we show that as a result the ferroelectric phase becomes locally unstable when ‖β‖ is sufficiently large. For ‖β‖ values above the critical one, the ordered phase has a modulated antiferroelectric structure. Several alternative structures are considered, and it is argued that for n=3 the most likely phase is a cubic one (space group $O_h^9$). Immediately above the critical value of ‖β‖, the transition to the ordered phase is of second order. When ‖β‖ is further increased, a tricritical point is reached above which the transition is first order. The effect of fluctuations on the above results is analyzed by using renormalization-group theory and expanding to O(ε) in 4-ε dimensions. The new term is found to be relevant when n$n_0$=4-11ε/4+O(${\epsilon }^2$), in which case no stable fixed points exist for 1.86≲n$n_0$. Thus ferroelectrics and ferromagnets are not necessarily in the same universality class. An examination of the renormalization-group flows, together with the results of the classical Landau analysis, indicates that modulated structures are possible immediately below the order-disorder transition regardless of the magnitude of β(≠0). Some experimental data which appear to agree with the theoretical results are discussed.