Magnetic properties and spin-driven ferroelectricity in multiferroic skyrmion host ${\mathbf{GaV}}_4{\mathbf{S}}_8$

The electronic structure, magnetic exchange interactions, and spin-driven ferroelectricity in skyrmion host ${\mathrm{GaV}}_4{\mathrm{S}}_8$ have been studied on the basis of first-principles calculations and symmetry analysis. We find the reconstruction of band structure with the increasing value of Hubbard parameter U. The ferromagnetic ground state is determined by the symmetric exchange interaction along interplane bonds. The antisymmetric Dzyaloshinskii-Moriya interaction along intraplane bonds plays a major role in the origins of cycloidal and skyrmion phases. The exchange-striction mechanism is identified as the main origin of spin-driven polarization. Moreover, two spin-current mechanisms expressed by ${\mathbf{P}}_1\propto {\mathbf{r}}_{ab}\times ({\mathbf{S}}_a\times {\mathbf{S}}_b)$ and ${\mathbf{P}}_2\propto {\mathbf{S}}_a\times {\mathbf{S}}_b$ appear in the cycloid, while only the former mechanism appears in the skyrmion.

Figure 1

(a) Crystal structure of ${\mathrm{GaV}}_4{\mathrm{S}}_8$ in high-temperature cubic phase. (b) Crystal and FM magnetic structure in low-temperature hexagonal $R3m$ phase. The V atoms occupy different Wyckoff positions, marked as V1 and V2. $J_1$ and $J_2$ denote the exchange couplings between neighbor ${\mathrm{V}}_4$ clusters along interplane V1-V2 and intraplane V2-V2 bonds, respectively.

Figure 2

Spin polarized and projected band structures for (a) up-spin and (b) down-spin states. (c) Molecular orbital scheme of ${\mathrm{V}}_4$ cluster. Only the splitting of half-filled $a_1$ orbital is shown. (d) Partial DOS of $3d_{z^2}$ orbital for V1 and V2 atoms. The Fermi energy is set to zero.

Figure 3

(a) Symmetric exchange interactions and (b) magnetic moments located at V1 and V2 atoms with different $U_{\mathrm{eff}}$ values.

Figure 4

(a) Orientations of DM vectors for interplane and intraplane bonds, denoted as ${\mathbf{D}}_1$ (solid arrows) and ${\mathbf{D}}_2$ (dashed arrows), respectively. According to the symmetry analysis, the DM vector of the specified interplane bond in the mirror plane ($yz$ plane) is along the $x$-axis direction, and the DM vector of specified intraplane bond (along the $x$ axis) lies within the mirror. (b) The $2\times 1\times 1$ hexagonal supercell and spin configuration used in the calculations of ${\mathbf{D}}_1$ vector. The arrows denote the spins of ${\mathrm{V}}_4$ clusters. (c) The orthorhombic supercell and spin configuration used in the calculations of ${\mathbf{D}}_2$ vector. The in-plane basis vectors of orthorhombic supercell a and b are along the [1-10] and [110] directions of hexagonal representation (in-plane basis vectors ${\mathbf{a}}_h$ and ${\mathbf{b}}_h$). (d) The electron density difference of ${\mathrm{V}}_4$ cluster between including and excluding SOC.

Figure 5

Schematic diagrams of magnetic structure, orientations of ferroelectric polarization, and electric dipole moment of bonds in (a) FM, (b) cycloidal, and (c) skyrmion phases. The components of electric dipole moment are mapped according to the local coordinate system of each bond. In cycloidal and skyrmion phases, they mean the average of electric dipole moment through the whole period; i.e., the oscillating part is ignored.