Skyrmionic order and magnetically induced polarization change in lacunar spinel compounds ${\mathrm{GaV}}_4{\mathrm{S}}_8$ and ${\mathrm{GaMo}}_4{\mathrm{S}}_8$: Comparative theoretical study

We show how low-energy electronic models derived from the first-principles electronic structure calculations can help to rationalize the magnetic properties of two lacunar spinel compounds $\mathrm{Ga}{M}_4{\mathrm{S}}_8$ with relatively light ($M=\mathrm{V}$) and heavy ($M=\mathrm{Mo}$) transition-metal elements, which are responsible for different spin-orbit interaction strength. In the model, each magnetic lattice point was associated with the ${\left(M_4{\mathrm{S}}_4\right)}^{5+}$ molecule, and the model itself was formulated in the basis of molecular Wannier functions constructed for three magnetic $t_2$ bands. The effects of rhombohedral distortion, spin-orbit interaction, band filling, and the screening of Coulomb interactions in the $t_2$ bands are discussed in details by stressing similarities and differences between ${\mathrm{GaV}}_4{\mathrm{S}}_8$ and ${\mathrm{GaMo}}_4{\mathrm{S}}_8$. The electronic model is further treated in the superexchange approximation, which allows us to derive an effective spin model for the energy and electric polarization ($\mathit{P}$) depending on the relative orientation of spins in the bonds, and study the properties of this model by means of classical Monte Carlo simulations with the emphasis on the possible formation of the skyrmionic phase. While isotropic exchange interactions clearly dominate in ${\mathrm{GaV}}_4{\mathrm{S}}_8$, all types of interactions (isotropic, antisymmetric, and symmetric anisotropic) are comparable in the case of ${\mathrm{GaMo}}_4{\mathrm{S}}_8$. Particularly, large uniaxial exchange anisotropy has a profound effect on the properties of ${\mathrm{GaMo}}_4{\mathrm{S}}_8$. On the one hand, it raises the Curie temperature by opening a gap in the spectrum of magnon excitations. On the other hand, it strongly affects the skyrmionic phase by playing the role of a molecular field, which facilitates the formation of skyrmions, but makes them relatively insensitive to the external magnetic field in the large part of the phase diagram. We predict reversal of the magnetic dependence of $\mathit{P}$ in the case of ${\mathrm{GaMo}}_4{\mathrm{S}}_8$ caused by the reversal of the direction of rhombohedral distortion.

Figure 1

Fragments of the crystal structure of $\mathrm{Ga}{M}_4{\mathrm{S}}_8$: (a) Network of the $M_4{\mathrm{S}}_4$ “molecules”; (b) schematic view on the network with the notation of rhombohedral parameters $a_r$ and ${\alpha }_r$; (c) $M_4$ tetrahedra interconnected by the S atoms; (d) single $M_4$ tetrahedron with the notation of two inequivalent types of $M$ atoms, the $M_1-M_2$ bond length $a_t$ and the angle ${\alpha }_t$, characterizing the distortion of this tetrahedron.

Figure 2

Electronic structure and densities of states of ${\mathrm{GaMo}}_4{\mathrm{S}}_8$ and ${\mathrm{GaV}}_4{\mathrm{S}}_8$ in the local density approximation as obtained for the high-temperature cubic $F\overline{4}3m$ phase and the low-temperature rhombohedral $R3m$ phase with and without spin-orbit interaction. The Fermi level is at zero energy (shown by dashed line).

Figure 3

Notations of $M_4$ clusters in $\mathrm{Ga}{M}_4{\mathrm{S}}_8$: (a) side view, (b) top view.

Figure 4

Energies of two-particle excitations obtained using matrix elements of the bare (left) and screened (right) Coulomb interactions and corresponding averaged Kanamori parameters of intraorbital Coulomb interaction $U$ and Hund's rule exchange interaction $J$.

Figure 5

Typical spin patterns obtained in Monte-Carlo simulation for the model (5) on the lattice $30\times 30\times 18$ (${\mu }_{\mathrm{B}}h\sim 0.15J_{\parallel }$ and $T=0.1J_{\parallel }$).

Figure 6

Schematic illustration of interpenetrating skyrmionic textures in adjacent planes $z=0$ and $z=c$ with the notation of bonds formed by neighboring spins: (a) top view and (b) side view. Owing to the stacking misalignment, each spin couples with three neighboring spins in the next plane. Therefore, some of the neighboring spins between the planes will always be noncollinear.

Figure 7

(a) Spin patterns as obtained in Monte Carlo calculations for the model (5) in an external magnetic field $h\parallel z$ at the temperature $T=0.1J_{\parallel }$. (b) Magnetization and (c) electric polarization (total and partial contributions) versus magnetic field. The meaning of the regions I, II, and III is explained in the text.

Figure 8

Temperature dependence of the specific heat as obtained in the Monte Carlo calculations for the spin model (5) with the parameters derived in the superexchange approximation for ${\mathrm{GaV}}_4{\mathrm{S}}_8$ and ${\mathrm{GaMo}}_4{\mathrm{S}}_8$.

Figure 9

Densities of states for the ferromagnetic state as obtained in the Hartree-Fock approximation for the model (1).