Finite-temperature properties of $(\mathrm{Ba},\mathrm{Sr})\mathrm{Ti}{\mathrm{O}}_3$ systems from atomistic simulations

An effective Hamiltonian approach is developed to mimic finite-temperature properties of $\left({\mathrm{Ba}}_x{\mathrm{Sr}}_{1-x}\right)\mathrm{Ti}{\mathrm{O}}_3$ perovskite systems. It is found that this atomistic approach is overall quite accurate to qualitatively and quantitatively reproduce the experimental composition-temperature phase diagram of the disordered solid solutions (especially, for compositions $x$ larger than 25%), when allowing one single parameter to deviate from its first-principles-derived value. Interestingly, such approach also yields predictions that are in good agreement with direct first-principles calculations for $\mathrm{Ba}\mathrm{Ti}{\mathrm{O}}_3∕\mathrm{Sr}\mathrm{Ti}{\mathrm{O}}_3$ superlattices. The proposed approach is thus promising to investigate phenomena in any perovskite system made of Ba, Sr, Ti, and O atoms, independently of the overall composition and atomic arrangement.

Figure 1

Largest, middle, and smallest average Cartesian coordinates $u_1$, $u_2$, and $u_3$ of the local-mode vector as a function of temperature in disordered single crystals of $\left({\mathrm{Ba}}_{0.5}{\mathrm{Sr}}_{0.5}\right)\mathrm{Ti}{\mathrm{O}}_3$, using the numerical approach described in Eqs. (1, 2, 3) with ${\kappa }_2$ having been reduced by 20% from its LDA-derived value.

Figure 2

Phase diagram of disordered $\left({\mathrm{Ba}}_x{\mathrm{Sr}}_{1-x}\right)\mathrm{Ti}{\mathrm{O}}_3$ solid solutions. Filled symbols show our predictions using the numerical approach described in Eqs. (1, 2, 3) with ${\kappa }_2$ having been reduced by 20% from its LDA-derived value. Open dots and squares symbols represent the experimental data of Refs. 17, 18, respectively. Lines are guides for the eyes. The uncertainty on the theoretical transition temperatures is estimated to be around $5\mathrm{K}$.

Figure 3

Composition-dependency of the ${\kappa }_2$ parameter leading to theoretical Curie temperature matching the experimental values for the whole composition range.

Figure 4

Dependency of the polarization along the [001] direction on the ratio of the number of $\mathrm{Sr}\mathrm{Ti}{\mathrm{O}}_3$ layers over the number of $\mathrm{Ba}\mathrm{Ti}{\mathrm{O}}_3$ layers in epitaxially-strained $\mathrm{Ba}\mathrm{Ti}{\mathrm{O}}_3∕\mathrm{Sr}\mathrm{Ti}{\mathrm{O}}_3$ superlattices at $5\mathrm{K}$. The filled symbols represent our predictions using the numerical approach described in Eqs. (1, 2, 3) with ${\kappa }_2$ having been reduced by 20% from its LDA-derived value. The dotted line displays the best fit of our results to Eq. (4), which is derived from simple macroscopic electrostatics.