First-principles demonstration of Roman-surface topological multiferroicity

The concept of topology has been widely applied to condensed matter, going beyond the band crossover in reciprocal spaces. A recent breakthrough suggested unconventional topological physics in a quadruple perovskite ${\mathrm{TbMn}}_3{\mathrm{Cr}}_4{\mathrm{O}}_{12}$, whose magnetism-induced polarization manifests a unique Roman surface topology [G. Liu et al., Nat. Commun. 13, 2373 (2022)]. However, the available experimental evidence based on tiny polarizations of polycrystalline samples could be strengthened. Here, this topological multiferroicity is demonstrated by using density functional theory calculations, which ideally confirms the Roman surface trajectory of magnetism-induced polarization. In addition, an alternative material in this category is proposed to systematically enhance the performance, by promoting its magnetism-induced polarization to an easily detectable level.

Figure 1

(a) Schematic of a Roman surface, which is a quartic nonorientable surface and obtained by sewing a Möbius strip to the edge of a disk [9]. (b) Crystal structure of ${\mathrm{TbMn}}_3{\mathrm{Cr}}_4{\mathrm{O}}_{12}$ with the space group of $Im\overline{3}$. (c and d) Collinear G-type antiferromagnetic structures of Cr (${\mathbf{S}}_{\mathrm{Cr}}$) and Mn sublattices (${\mathbf{S}}_{\mathrm{Mn}}$), respectively. In the ground state, all spins are pointing along the [111] direction.

Figure 2

Selected closed loops of spin rotation pathes and the induced dipoles. Sampling points in these closed loops are denoted by M, N, Q, and R, respectively. To exclude possible inaccuracy from structural relaxation, here all these spin rotations are done with the strict highly symmetric structure. (a–d) Four typical closed loops of spin rotation. (a) Longitudinal rotation (polar angle $\theta \in [0^{\circ },{180}^{\circ }]$) with fixed azimuthal angles $\phi ={45}^{\circ }$ (and ${225}^{\circ }$). (b and c) Latitudinal rotation (azimuthal angle $\varphi \in [0^{\circ },{360}^{\circ }]$) with $\theta =54.7^{\circ }$ and $\theta ={90}^{\circ }$, respectively. (d) Rotation in the ($1\overline{1}1$) plane spanned by three corner points. (e–h) The evolutions of $x$, $y$, and $z$ components of the dipole in the corresponding rotation loops of the spin axis. (i–l) The corresponding dipole trajectories of sampling points in the three-dimensional space and their projections to the $xy$ plane (red dots). In (i) and (k), for the trajectories of M and Q sampling points, the two half-periods give rise to the identical trajectory. (i) The elliptic trajectory in three dimensions and the linear projection onto the $xy$ plane. (j) The saddle-like trajectory and the circular projection. (k) The vertical line trajectory and the projection at the original point. (l) The trefoil trajectory in the (111) plane. The projection to the $xy$ plane is also a trefoil one. All these trajectories just fall onto the topological Roman surface.

Figure 3

(a) Spin rotation dependence of the individual sublattice. Here ${\mathbf{S}}_{\mathrm{Cr}}$ is fixed along the diagonal one and ${\mathbf{S}}_{\mathrm{Mn}}$ is rotated, as indicated in the inset. The energy barrier curves with and without SOC almost overlap, implying the dominant contribution is not from SOC. Since the angle ${\mathrm{\Phi }}^{\prime }$ with 0 (180) degree corresponds to the ferroelectric $+P$ ($-P$) state, this energy barrier also can be considered as the theoretical upper limit of the ferroelectric switching barrier. (b) Synchronous rotation dependence of two spin sublattices. The rotation path is indicated in the inset. (c) The linear relationship between the square root of induced polarization and the magnitude of SOC.

Figure 4

Atomic-orbital projected density of states. (a) ${\mathrm{Mn}}^{3+}$'s $3d$ and (b) ${\mathrm{Cr}}^{3+}$'s $3d$ orbitals of ${\mathrm{TbMn}}_3{\mathrm{Cr}}_4{\mathrm{O}}_{12}$. (c) ${\mathrm{Mn}}^{2+}$'s $3d$ and (d) ${\mathrm{Re}}^{4+}$'s $5d$ orbitals of ${\mathrm{BaMn}}_3{\mathrm{Re}}_4{\mathrm{O}}_{12}$. The more itinerant Re's $5d$ electrons give rise to a larger bandwidth and smaller energy gap. Right insets: the corresponding crystal field splittings and the orbital fillings. The electronic configuration in (c) and (d) can lead to much stronger SOC effects then that in (a) and (b).