Theory of central peak and acoustic anomaly in cubic BaTiO3 close to ferroelectric transition

Abstract

We present a Ginzburg-Landau theory on statics and dynamics of BaTiO$_3$-type ferroelectrics in the paraelectric phase with the cubic structure, where the order parameter is the polarization $\bi p$. Unique effects are caused by the electrostrictive (ES) coupling between ${\bi p}$ and the elastic displacement $\bi u$. We show that the ES coupling gives rise to a central peak in the Fourier-Laplace transform of the displacement time-correlation function at small wave numbers. It emerges and grows with a narrow width as the transition is approached. Such central peaks have long been observed in a number of scattering experiments in various ferroelectrics, but their origin has not been well understood. From the acoustic part of the displacement dynamic correlation we obtain the frequency-dependent elastic moduli $C_{11}^*(ω)$, $C_{12}^*(ω)$, and $C_{44}^*(ω)$, whose singular parts arise from the ES coupling, We then calculate the singular sound velocity and attenuation. In the central peak and the elastic moduli, the frequency $ω$ appears in the scaled form $ωτ_D$, where $τ_D$ is the Debye relaxation time in the frequency-dependent dielectric constant. {Keywords}: ferroelectric transition, central peak, acoustic anomaly, electrostrictive coupling

Figures (4)

• Figure 1: Anisotropy factor $D({\hat{\hbox{\boldmath$q$}}})$ in Eqs.(42) and (43) in the density correlation for cubic BaTiO$_3$ as a function of $(\theta,\varphi)$ with ${\hat{\hbox{\boldmath$q$}}}= (\sin\theta\cos\varphi, \sin\theta\sin\varphi, \cos\theta)$. Its maximum is 2.8, which is attained at $\theta=0$ ($[001]$), $(\theta,\varphi)=(\pi/2,0)$ ($[100]$), and $(\theta, \varphi)=(\pi/2,\pi/2)$ ($[010]$). Its minimum is 0.44 for $(\theta,\varphi)=(\pi/4,\pi/4)$ ($[111]$). Use is made of $c_{ij}$ and $M_{ij}$ in Table 1.
• Figure 2: (a) Scaling function $K_c(s)$ (blue line) in Eq.(68) with $s=2t/\tau_D$, which has a cusp at $s=0$ and decays faster than $e^{-s}$ (red line). (b) Real part and imaginary parts of the scaling function $F_c(i\Omega)$ in Eq.(69) with $\Omega=\omega \tau_D/2$, which decay slowly for $\Omega\gtrsim 4$. (c) Real part and imaginary parts of the scaling function $G_c(i\Omega)$ in Eq.(77), which also decay slowly for $\Omega\gtrsim 4$.
• Figure 3: (a) $F^{\rm R}(\omega, A)$ in Eq.(70) as a function of $\omega\tau_0/2$ and $A$, which represents the near-critical strong growth of the central peak and the sound adsorption (see Eqs.(85) and (89)). (b) $G^{\rm R}(\omega, A)$ in Eq.(86) vs $\omega\tau_0/2$ for three values of $A$, which gives the near-critical mild growth of the longitudinal sound velocity $v_{11}$ along $[001]$ in Eq.(84).
• Figure 4: Normalized dynamic structure factor $(c_{11}/k_BT\tau_0)I_D(q,\omega)$ in Eq.(89) vs scaled frequency $\omega/qv_{11}^0$ along $[001]$ for $q=0.08~{\rm \AA}$ and $v_{11}^0=5.87\times 10^5$ cm$/s$. (a) The central peak is shown for $A=0.001$, $0.002$, and $0.004$ in the range $0<\omega/qv_{11}^0<1.2$, where $qv_{11}^0\tau_D= 0.052/A\gg 1$ for these $A$. (b) The acoustic peak of $(c_{11}/k_BT\tau_0)I_D(q,\omega)$ is shown in the range $0.95<\omega/qv_{11}^0<1.01$.