Dynamical coupling in $\text{Pb(Zr,Ti)}{\mathrm{O}}_3$ solid solutions from first principles

A first-principles-based approach is developed to model dynamical coupling between different degrees of freedom in the terahertz frequency range at any temperature. Such an approach is applied to $\text{Pb(Zr,Ti)}{\text{O}}_3$ alloys, which exhibit both ferroelectric (FE) and antiferrodistortive (AFD) motions. It reproduces very well the existence of two modes (rather than the single $E(1\text{TO})$ soft mode) in the 50–75 cm${}^{-1}$ range for temperatures smaller than $\simeq 200$ K. It also provides an insight into the existence of these two modes, which originate from the coupling between long-range-ordered FE and AFD motions. The proposed scheme is further used to reveal a field-induced anticrossing involving FE and AFD degrees of freedom.

Figure 1

Temperature dependence of some dynamical characteristics in the $\mathrm{Pb}({\mathrm{Zr}}_{0.55},{\mathrm{Ti}}_{0.45}){\mathrm{O}}_3$ solid solution. (a) The ${\nu }_r$ resonant frequency of the lowest-in-frequency dielectric peaks for any temperature as well as the resonant frequency of ${\varepsilon }_{x,x}^{\mathrm{AFD}}(\nu )$ for temperatures above 200 K (denoted AFD mode) found in our simulations. The solid lines represent fittings by square-root laws (i.e., ${\nu }_r~|T-T_c|{}^{1/2}$) of the resonant frequency of the $E(1\mathrm{TO})$ mode and of the soft mode above $T_C$. (b) The electric dipole spectral weight (extracted from the dielectric response) of different modes (see text).

Figure 2

Complex responses of the $\mathrm{Pb}({\mathrm{Zr}}_{0.55},{\mathrm{Ti}}_{0.45}){\mathrm{O}}_3$ solid solution in the 20–100 cm${}^{-1}$ frequency range at different temperatures. The (a) real and (b) imaginary parts of the ${\varepsilon }_{x,x}(\nu )$ dielectric response. (c) The imaginary part of the AFD-related ${\varepsilon }_{x,x}^{\mathrm{AFD}}(\nu )$ function of Eq. (3). The displayed data correspond to a fit of the raw data by two classical damped harmonic oscillators (except at and above $T=200\hspace{0.5em}\mathrm{K}$, where we use a single oscillator because of the quality of the fit).[27, 28] The data for 50, 75, 100, 150, 200, and 300 K have been vertically shifted by 2500, 5000, 7500, $10\hspace{0.5em}000$, $12\hspace{0.5em}500$, and $15\hspace{0.5em}000$, respectively, in order to distinguish them from the 10 K data.

Figure 3

(a) Resonant frequency and (b) relative square of the strength of the oscillator of the $E^{(1)}$ and $E^{(2)}$ modes (as extracted from their dielectric peaks). Solid lines in (a) and (b) represent fittings by the eigenvalues and eigenvectors of a $2\times 2$ matrix, respectively. In this matrix, the two diagonal terms [indicated by dashed lines in (a)] are frequencies that linearly depend on the magnitude of the electric field, while the off-diagonal terms are constant frequencies associated with dynamical coupling.