Hard antiphase domain boundaries in strontium titanate unravelled using machine-learned force fields

We investigate the properties of hard antiphase boundaries in ${\mathrm{SrTiO}}_3$ using machine-learned force fields. In contrast to earlier findings based on standard ab initio methods, for all pressures up to $120\mathrm{kbar}$ the observed domain wall pattern maintains an almost perfect Néel character in quantitative agreement with Landau-Ginzburg-Devonshire theory, and the in-plane polarization $P_3$ shows no tendency to decay to zero. Together with the switching properties of $P_3$ under reversal of the Néel order parameter component, this provides hard evidence for the presence of rotopolar couplings. The present approach overcomes the severe limitations of ab initio simulations of wide domain walls and opens avenues toward concise atomistic predictions of domain-wall properties even at finite temperatures.

Figure 1

Comparison of calculated phonon dispersions in the tetragonal I4/mcm structure as determined directly from the PBEsol exchange-correlation functional (dotted lines) and using the MLFF (full lines).

Figure 2

Energy double well for the mode amplitude ${\mathrm{\Gamma }}_3$ at $p=0$ along the displacement pattern ${\mathit{E}}_3$ in the bulk $\mathit{I}4/\mathit{mcm}$ phase calculated directly from DFT using the PBEsol exchange-correlation functional and from the MLFF. We find that using our MLFF, the FE instability disappears at $p^{\left(\mathrm{MLFF}\right)}=10\mathrm{kbar}$ (blue line in the plot), while in DFT stabilization sets in at $p^{\left(\mathrm{DFT}\right)}=13\mathrm{kbar}$.

Figure 3

(a) Pressure dependence of the DW width $\delta \left(p\right)$ measured in units of the pseudocubic lattice constant $a_0\left(p\right)$. (b) Comparison of rescaled OP profiles ${\psi }_1\left(x\right)$ and ${\psi }_3\left(x\right)$ as obtained from our MLFF simulations at pressures $p=0,10,20,40,60,80,100$, and $120\mathrm{kbar}$ (colored) and numerical solutions of LGD theory (gray). As explained in the text, distances $x_1$ are measured in units of the DW width $\delta \left(p\right)$, i.e., $x_1\equiv x·\delta \left(p\right)$, and OP components ${\varphi }_i$ are normalized with respect to the bulk OP modulus $\overline{\varphi }\left(p\right)$, i.e., ${\psi }_i\left(x\right)={\varphi }_i\left(x\delta \left(p\right)\right)/\overline{\varphi }\left(p\right)$. (c), (d) Polarization profiles $P_1\left(x\right)$ and $P_3\left(x\right)$, respectively, for pressures $p=20,40,60,80,100$, and $120\mathrm{kbar}$. Polarization profiles for unstable configurations at $p=0\mathrm{kbar}$ and $p=10\mathrm{kbar}$ were found to be completely out of scale and were thus deliberately omitted from the plot.

Figure 4

Pressure evolution of the extrema of the polarization profile $P_3\left(x\right)$. Inset: Polarization profiles $P_1\left(x_1\right)$ and $P_3\left(x_1\right)$ for ${\varphi }_1\left(x\right){\varphi }_3^{\prime }\left(x\right)<0$ (label $+$) and ${\varphi }_1\left(x\right){\varphi }_3^{\prime }\left(x\right)>0$ (label –) at pressure $p=40\mathrm{kbar}$.

Figure 5

Polarization profiles $P_1\left(x\right)$ (red), $P_2\left(x\right)$ (green), and $P_3\left(x\right)$ (blue) for (a) $T=30\mathrm{K}$, (b) $T=50\mathrm{K}$, and (c) $T=80\mathrm{K}$; (d) temperature evolution of maxima of polarization component $P_3\left(x\right)$.