Crossover from Linear to Quadratic Electro-optic Behavior in ${\mathrm{BaTiO}}_3$ and $(\mathrm{Ba},\mathrm{Sr}){\mathrm{TiO}}_3$ Solid Solution

We derive a numerical method based on coupled density functional theory and effective Hamiltonian schemes to calculate the linear and quadratic electro-optic response of ferroelectrics at finite temperature and in different frequency ranges. By applying the developed method to ${\mathrm{BaTiO}}_3$, we successfully resolve apparent discrepancies in the experimental literature that reported a linear or quadratic electro-optic response when visible or terahertz radiation was employed to measure the optical index, respectively. We further demonstrate that (and explain why), in the case of the ${\mathrm{Ba}}_{1-\mathrm{x}}{\mathrm{Sr}}_{\mathrm{x}}{\mathrm{TiO}}_3$ disordered solid solutions, structural phase transitions not only lead to larger linear electro-optic constants, as previously demonstrated in the literature, but also significantly enhance the quadratic electro-optic constants.

Figure 1

(a) Contributions to the linear EO constant $r_{33}^{\sigma }$ in BTO at $\hslash \omega =4\mathrm{meV}$ and 1.55 eV. (b) The change of the refractive index $\mathrm{\Delta }n$ for BTO at $\hslash \omega =4\mathrm{meV}$ (in blue) and $\hslash \omega =1.55\mathrm{eV}$ (in orange).

Figure 2

(a) Calculated linear EO constants versus composition $x$ in bulk BST; open symbols denote experimental values [4]. (b) Computed quadratic EO constant $R_{333}$ versus composition $x$. (c) Expected change of refractive index for electric fields along the polar direction. (d) Crossover field at which the linear and quadratic contributions to the change of refractive index are equal. All data were calculated at 300 K and for $\hslash \omega =1.55\mathrm{eV}$.