Out-of-plane polarization and topological magnetic vortices in multiferroic ${\mathrm{CrPSe}}_3$

Two-dimensional multiferroic materials are ideal systems for exploring new coupling mechanisms between different ferroic orders and producing novel quantum phenomena with potential applications. We employ first-principles density functional theory calculations to discover intrinsic ferroelectric and antiferroelectric phases of $\mathrm{Cr}\mathrm{P}{\mathrm{Se}}_3$, which show ferromagnetic order and compete with the centrosymmetric phase with an antiferromagnetic order. Our analysis shows that the electrical dipoles of such type-I multiferroic phases come from the out-of-plane displacements of phosphorus ions due to the stereochemically active lone pairs. The coupling between polar and magnetic orders creates the opportunity for tuning the magnetic ground state by switching from the centrosymmetric to the ferroelectric phase using an out-of-plane electric field. In ferroelectric and antiferroelectric phases, the combination of easy-plane anisotropy and Dzyaloshinskii-Moriya interactions indicates that they can host topological magnetic vortices such as meron pairs.

Figure 1

Atomic models for (a) the centrosymmetric phase and (b) the polar phase [35]. (c) Comparison of the total energies between different multiferroic configurations when calculated with different $U_{\mathrm{eff}}$.

Figure 2

(a) Phonon dispersion of the FE-FM phase. (b) Energy barriers for switching the electric polarization of the ferroelectric (i.e., polar) phase. (c) The energy profile for a structural transition from the FE-FM phase to the AFE-FM phase calculated with $\mathrm{PBEsol}+U$ (with $U_{\mathrm{eff}}=2.8$ eV). (d) Isosurfaces of the electron localization function in the polar phase, showing the lone-pair electrons near phosphorus atoms.

Figure 3

(a) A schematic plot showing the nearby atoms and the direction of DMI vector ${\mathbf{D}}_1$ between nearest-neighbor Cr ions. (b) A snapshot of a Monte Carlo simulation for the FE-FM phase ($T=2$ K) which shows a pair of antimerons (topological charge of $\frac{1}{2}$) with winding numbers $-1$ and 1, respectively. The local topological charge density is plotted with contours on the background. Spins are colored in green or blue according to the out-of-plane components. (c) A screenshot for six long-lived merons of the AFE-FM phase. Spins are colored according to the direction of their in-plane components.