Ferrotoroidic polarons in antiferrodistortive $\mathrm{SrTi}{\mathrm{O}}_3$

Ultrasmall ferroelectrics with nontrivial topological field textures such as polar vortices, skyrmions, and merons hold promise in technological paradigms. Such nontrivial ferroic orders and their functionalities, however, inevitably disappear below a critical size of several nanometers. Here, we propose a strategy to overcome this limitation and design atomically small ferroelectrics with topological polarization vortices by engineering excess-electron polarons. Our first-principles calculations demonstrate that excess-electron polarons formed in antiferrodistortive $\mathrm{SrTi}{\mathrm{O}}_3$ induce localized ferroelectric polarization with a topological vortex form due to local symmetry breaking and demonstrate the possibility of an atomic-scale “ferrotoroidic” materials. We further show that the electron polaron carries a magnetic moment coupled with ferrotoroidicity, i.e., the magnetoelectric effect. We also discuss possible methods to switch the toroidal moment via the magnetoelectric effect. Our result, thus, provides insight into the ultimate miniaturization of ferrotoroic materials and a class of functional polaron families.

Figure 1

(a) Electronic density of states (DOS) of the excess-electron polaron in $\mathrm{SrTi}{\mathrm{O}}_3$. The red and blue areas (lines) indicate the occupied (unoccupied) states of up- and down-spins, respectively. The mid-gap state is the Ti $d_{xz}$ state occupied by the one excess electron. (b) Charge density distribution of the polaronic state in $\mathrm{SrTi}{\mathrm{O}}_3$. The orange area represents the isosurface of charge density of $0.01e/{\AA }^3$. White arrows indicate the displacements of O atoms around the polaron. (c) Atomic displacement $d$ (white arrows) around the polaron in $\mathrm{SrTi}{\mathrm{O}}_3$.

Figure 2

The distribution of polarization $P$ in $\mathrm{SrTi}{\mathrm{O}}_3$ (a) without polaron and (b) with polaron. The yellow arrows indicate local polarization. The orange area represents the iso-surface of polaron charge densities of $0.01e/{\AA }^3$. The local polarization of $11.7\mu \mathrm{C}/\mathrm{c}{\mathrm{m}}^2$ in undoped AFD $\mathrm{SrTi}{\mathrm{O}}_3$ is shown as reference in the left panel.

Figure 3

The polarization induced by the formation of excess-electron polaron in $\mathrm{SrTi}{\mathrm{O}}_3$. The yellow arrows indicate the difference of local polarization Δ$P$ between $\mathrm{SrTi}{\mathrm{O}}_3$ with and without the polaron. The orange area represents the isosurface of polaron charge densities of $0.01e/{\AA }^3$. (a) Overview, (b) top view, (c) top view (zoom), (d) radial component, and (e) azimuthal component.

Figure 4

Schematic illustration of polarization vortices in AFD $\mathrm{SrTi}{\mathrm{O}}_3$ (a) without polaron and (b) with polaron. The blue and red arrows indicate local polarization. The black and red circles indicate the direction of vortices.

Figure 5

(a) Schematic illustration of toroidal moment $G$ induced by polaron in $\mathrm{SrTi}{\mathrm{O}}_3$. The direction of $G$ corresponds to the direction of local AFD rotation at the site where the polaron is formed. (b) Energy barrier for polaronic migration in $\mathrm{SrTi}{\mathrm{O}}_3$ from a site with clockwise AFD to the neighboring site with counterclockwise AFD rotation.

Figure 6

(a) Magnetic spin-density distribution of the excess-electron polaron in $\mathrm{SrTi}{\mathrm{O}}_3$. The yellow area represents the iso-surface of spin-densities of $0.05{\mu }_{\mathrm{B}}/{\AA }^3$. (b) Magnetocrystalline anisotropy energy (MAE) surface of excess-electron polaron. The white lines represent the contour lines at every 0.25 μeV. (c) MAE of around the [100] direction. (d) The direction of magnetic moment of the excess-electron polaron.

Figure 7

Schematic illustration of the possible magnetic moment directions of the polaron with positive and negative toroidal moments.