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a) Spin patterns as obtained in Monte-Carlo calculations for the model (4) with an applied magnetic field h z at temperature T = 0.1J for the supercell of 72?72?3 spins with periodic boundary conditions. In these notations, a "skyrmion lattice" means the lattice of well distinguished skyrmionic tubes of the same size, while a "cycloidal phase" includes large interconnected regions with the same direction of spins. The corresponding h-dependence of (b) the magnetization and (c) electric polarization: total and partial contributions, calculated from Eq. (7) relative to the FM state.
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The lacunar spinel GaV$_{4}$S$_{8}$ was recently suggested to be a prototype multiferroic material hosting skyrmion lattice states with a sizeable polarization $\boldsymbol{P}$ coupled to magnetic order. We explain this phenomenon on the microscopic level. On the basis of density functional theory, we construct an effective model describing the beh...
Contexts in source publication
Context 1
... diagram. We perform classical Monte-Carlo cal- culations for the spin model (4) with an applied mag- netic field h z by using a heat-bath algorithm com- bined with overrelaxation [22,42]. The results shown in Fig. 2a nicely reproduce the main sequence of cycloidal ? skyrmionic ? FM states in the phase diagram of GaV 4 S 8 with the increase of h [15]. Furthermore, in the cycloidal and skyrmion phases, the two-dimensional spin patterns in the xy plane tend to stack ferromagnetically along z, that is naturally explained by the FM coupling J ? . How- ever, in the GaV 4 S 8 structure, this stacking is misaligned by the rhombohedral translations, so that the adjacent skyrmionic layers experience an additional shift in the xy plane. Therefore, there will always be some noncollinear- ity of spins between the adjacent layers, which according to Eq. (7) will contribute to the spin-excess ...
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... nonmagnetic electronic band structures calculated for the high-and low-temperature phases of GaV 4 S 8 are presented in Fig. 2a. The group of bands located near the Fermi level is dominated by metal-metal bonding states of the V 4 cluster generated by the V 3d orbitals. In the high-temperature cubic phase (point group T d ), these states are split into a 1 , e and t 2 levels filled with seven electrons in such a way that the highest 3-fold degenerate t 2 level contains one unpaired electron. At 38 K GaV 4 S 8 undergoes a cubic-to-rhombohedral structural phase transition driven by the cooperative Jahn-Teller distortion [11] which lifts the degeneracy of the t 2 level and splits it into a low-lying single a 1 and double degenerate e states (point group C 3v ), as shown in Fig. 2b. Since each V 4 cluster carries a local moment S = 1/2 formed by three bands located near the Fermi level, these states can be chosen as a basis for the effective low-energy model to describe magnetic properties of GaV 4 S 8 . The corresponding Wannier functions obtained by projecting these states onto atomic-like orbitals are shown in Fig. 2c. As seen, the resulting Wannier functions are well defined and localized, but have a complicated structure representing metal-metal bonding in the V 4 ...
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... nonmagnetic electronic band structures calculated for the high-and low-temperature phases of GaV 4 S 8 are presented in Fig. 2a. The group of bands located near the Fermi level is dominated by metal-metal bonding states of the V 4 cluster generated by the V 3d orbitals. In the high-temperature cubic phase (point group T d ), these states are split into a 1 , e and t 2 levels filled with seven electrons in such a way that the highest 3-fold degenerate t 2 level contains one unpaired electron. At 38 K GaV 4 S 8 undergoes a cubic-to-rhombohedral structural phase transition driven by the cooperative Jahn-Teller distortion [11] which lifts the degeneracy of the t 2 level and splits it into a low-lying single a 1 and double degenerate e states (point group C 3v ), as shown in Fig. 2b. Since each V 4 cluster carries a local moment S = 1/2 formed by three bands located near the Fermi level, these states can be chosen as a basis for the effective low-energy model to describe magnetic properties of GaV 4 S 8 . The corresponding Wannier functions obtained by projecting these states onto atomic-like orbitals are shown in Fig. 2c. As seen, the resulting Wannier functions are well defined and localized, but have a complicated structure representing metal-metal bonding in the V 4 ...
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... nonmagnetic electronic band structures calculated for the high-and low-temperature phases of GaV 4 S 8 are presented in Fig. 2a. The group of bands located near the Fermi level is dominated by metal-metal bonding states of the V 4 cluster generated by the V 3d orbitals. In the high-temperature cubic phase (point group T d ), these states are split into a 1 , e and t 2 levels filled with seven electrons in such a way that the highest 3-fold degenerate t 2 level contains one unpaired electron. At 38 K GaV 4 S 8 undergoes a cubic-to-rhombohedral structural phase transition driven by the cooperative Jahn-Teller distortion [11] which lifts the degeneracy of the t 2 level and splits it into a low-lying single a 1 and double degenerate e states (point group C 3v ), as shown in Fig. 2b. Since each V 4 cluster carries a local moment S = 1/2 formed by three bands located near the Fermi level, these states can be chosen as a basis for the effective low-energy model to describe magnetic properties of GaV 4 S 8 . The corresponding Wannier functions obtained by projecting these states onto atomic-like orbitals are shown in Fig. 2c. As seen, the resulting Wannier functions are well defined and localized, but have a complicated structure representing metal-metal bonding in the V 4 ...
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... order to describe this effect quantitatively, we eval- uate the total and partial contributions to P z by us- ing Eq. (7) and the distribution of spins {e i } obtained in Monte-Carlo calculations. The results are summa- rized in Fig. 2c. Particularly, we note a strong com- petition of the isotropic (? e i e j ) and antisymmetric (? e i ? e j ) contributions, while the anisotropic symmet- ric part (? e i ? ? ij e j ) is negligibly small. As expected, the antisymmetric contribution decreases with the increase of h and vanishes in the collinear FM state. On the con- trary, the isotropic contribution takes its maximal value in the FM state and is further reduced by a noncollinear alignment of spins. Since the change of e i ?e j and e i e j is proportional to ? ij and ? 2 ij , respectively (with ? ij being the angle between e i and e j , which is induced by DM interactions and proportional to ? SO ), both the isotropic and antisymmetric mechanisms are of the 2nd order in ? SO , while the change of e i ? ? ij e j is only of the 3rd order. This naturally explains the hierarchy of partial contribu- tions to ?P z in Fig. 2c. Furthermore, the antisymmetric mechanism dominates when the skyrmions are large and the spin texture slowly varies in space. In this region, electric polarization decreases with h, in agreement with the experimental observation [15]. The corresponding polarization change of about 4 ?C/m 2 is also consistent with the experimental data [15]. Finally, our conclusion clearly differs from the phenomenological analysis pre- sented in [15], arguing that the antisymmetric DM inter- actions are solely needed to stabilize the cycloidal and skyrmion phases, while the corresponding polarization change is driven by the isotropic and anisotropic sym- (4) with an applied magnetic field h z at temperature T = 0.1J for the supercell of 72?72?3 spins with periodic boundary conditions. In these notations, a "skyrmion lattice" means the lattice of well distinguished skyrmionic tubes of the same size, while a "cycloidal phase" includes large interconnected regions with the same direction of spins. The corresponding h-dependence of (b) the magne- tization and (c) electric polarization: total and partial contri- butions, calculated from Eq. (7) relative to the FM state. metric terms. In fact, we also expect a small region in the phase diagram, where the magnetization is nearly saturated (Fig. 2b) and the skyrmion size is small, so the polarization change is mainly governed by the isotropic mechanism and is expected to increase with h. Over- all, the h-dependence of spin-driven polarization in the skyrmion phase depends on the skyrmion size and the way a skyrmion lattice is packed leading to different com- peting ...
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... order to describe this effect quantitatively, we eval- uate the total and partial contributions to P z by us- ing Eq. (7) and the distribution of spins {e i } obtained in Monte-Carlo calculations. The results are summa- rized in Fig. 2c. Particularly, we note a strong com- petition of the isotropic (? e i e j ) and antisymmetric (? e i ? e j ) contributions, while the anisotropic symmet- ric part (? e i ? ? ij e j ) is negligibly small. As expected, the antisymmetric contribution decreases with the increase of h and vanishes in the collinear FM state. On the con- trary, the isotropic contribution takes its maximal value in the FM state and is further reduced by a noncollinear alignment of spins. Since the change of e i ?e j and e i e j is proportional to ? ij and ? 2 ij , respectively (with ? ij being the angle between e i and e j , which is induced by DM interactions and proportional to ? SO ), both the isotropic and antisymmetric mechanisms are of the 2nd order in ? SO , while the change of e i ? ? ij e j is only of the 3rd order. This naturally explains the hierarchy of partial contribu- tions to ?P z in Fig. 2c. Furthermore, the antisymmetric mechanism dominates when the skyrmions are large and the spin texture slowly varies in space. In this region, electric polarization decreases with h, in agreement with the experimental observation [15]. The corresponding polarization change of about 4 ?C/m 2 is also consistent with the experimental data [15]. Finally, our conclusion clearly differs from the phenomenological analysis pre- sented in [15], arguing that the antisymmetric DM inter- actions are solely needed to stabilize the cycloidal and skyrmion phases, while the corresponding polarization change is driven by the isotropic and anisotropic sym- (4) with an applied magnetic field h z at temperature T = 0.1J for the supercell of 72?72?3 spins with periodic boundary conditions. In these notations, a "skyrmion lattice" means the lattice of well distinguished skyrmionic tubes of the same size, while a "cycloidal phase" includes large interconnected regions with the same direction of spins. The corresponding h-dependence of (b) the magne- tization and (c) electric polarization: total and partial contri- butions, calculated from Eq. (7) relative to the FM state. metric terms. In fact, we also expect a small region in the phase diagram, where the magnetization is nearly saturated (Fig. 2b) and the skyrmion size is small, so the polarization change is mainly governed by the isotropic mechanism and is expected to increase with h. Over- all, the h-dependence of spin-driven polarization in the skyrmion phase depends on the skyrmion size and the way a skyrmion lattice is packed leading to different com- peting ...
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... order to describe this effect quantitatively, we eval- uate the total and partial contributions to P z by us- ing Eq. (7) and the distribution of spins {e i } obtained in Monte-Carlo calculations. The results are summa- rized in Fig. 2c. Particularly, we note a strong com- petition of the isotropic (? e i e j ) and antisymmetric (? e i ? e j ) contributions, while the anisotropic symmet- ric part (? e i ? ? ij e j ) is negligibly small. As expected, the antisymmetric contribution decreases with the increase of h and vanishes in the collinear FM state. On the con- trary, the isotropic contribution takes its maximal value in the FM state and is further reduced by a noncollinear alignment of spins. Since the change of e i ?e j and e i e j is proportional to ? ij and ? 2 ij , respectively (with ? ij being the angle between e i and e j , which is induced by DM interactions and proportional to ? SO ), both the isotropic and antisymmetric mechanisms are of the 2nd order in ? SO , while the change of e i ? ? ij e j is only of the 3rd order. This naturally explains the hierarchy of partial contribu- tions to ?P z in Fig. 2c. Furthermore, the antisymmetric mechanism dominates when the skyrmions are large and the spin texture slowly varies in space. In this region, electric polarization decreases with h, in agreement with the experimental observation [15]. The corresponding polarization change of about 4 ?C/m 2 is also consistent with the experimental data [15]. Finally, our conclusion clearly differs from the phenomenological analysis pre- sented in [15], arguing that the antisymmetric DM inter- actions are solely needed to stabilize the cycloidal and skyrmion phases, while the corresponding polarization change is driven by the isotropic and anisotropic sym- (4) with an applied magnetic field h z at temperature T = 0.1J for the supercell of 72?72?3 spins with periodic boundary conditions. In these notations, a "skyrmion lattice" means the lattice of well distinguished skyrmionic tubes of the same size, while a "cycloidal phase" includes large interconnected regions with the same direction of spins. The corresponding h-dependence of (b) the magne- tization and (c) electric polarization: total and partial contri- butions, calculated from Eq. (7) relative to the FM state. metric terms. In fact, we also expect a small region in the phase diagram, where the magnetization is nearly saturated (Fig. 2b) and the skyrmion size is small, so the polarization change is mainly governed by the isotropic mechanism and is expected to increase with h. Over- all, the h-dependence of spin-driven polarization in the skyrmion phase depends on the skyrmion size and the way a skyrmion lattice is packed leading to different com- peting ...
Context 8
... diagram. We perform classical Monte-Carlo cal- culations for the spin model (4) with an applied mag- netic field h z by using a heat-bath algorithm com- bined with overrelaxation [22,42]. The results shown in Fig. 2a nicely reproduce the main sequence of cycloidal ? skyrmionic ? FM states in the phase diagram of GaV 4 S 8 with the increase of h [15]. Furthermore, in the cycloidal and skyrmion phases, the two-dimensional spin patterns in the xy plane tend to stack ferromagnetically along z, that is naturally explained by the FM coupling J ? . How- ever, in the GaV 4 S 8 structure, this stacking is misaligned by the rhombohedral translations, so that the adjacent skyrmionic layers experience an additional shift in the xy plane. Therefore, there will always be some noncollinear- ity of spins between the adjacent layers, which according to Eq. (7) will contribute to the spin-excess ...
Context 9
... order to describe this effect quantitatively, we eval- uate the total and partial contributions to P z by us- ing Eq. (7) and the distribution of spins {e i } obtained in Monte-Carlo calculations. The results are summa- rized in Fig. 2c. Particularly, we note a strong com- petition of the isotropic (? e i e j ) and antisymmetric (? e i ? e j ) contributions, while the anisotropic symmet- ric part (? e i ? ? ij e j ) is negligibly small. As expected, the antisymmetric contribution decreases with the increase of h and vanishes in the collinear FM state. On the con- trary, the isotropic contribution takes its maximal value in the FM state and is further reduced by a noncollinear alignment of spins. Since the change of e i ?e j and e i e j is proportional to ? ij and ? 2 ij , respectively (with ? ij being the angle between e i and e j , which is induced by DM interactions and proportional to ? SO ), both the isotropic and antisymmetric mechanisms are of the 2nd order in ? SO , while the change of e i ? ? ij e j is only of the 3rd order. This naturally explains the hierarchy of partial contribu- tions to ?P z in Fig. 2c. Furthermore, the antisymmetric mechanism dominates when the skyrmions are large and the spin texture slowly varies in space. In this region, electric polarization decreases with h, in agreement with the experimental observation [15]. The corresponding polarization change of about 4 ?C/m 2 is also consistent with the experimental data [15]. Finally, our conclusion clearly differs from the phenomenological analysis pre- sented in [15], arguing that the antisymmetric DM inter- actions are solely needed to stabilize the cycloidal and skyrmion phases, while the corresponding polarization change is driven by the isotropic and anisotropic sym- (4) with an applied magnetic field h z at temperature T = 0.1J for the supercell of 72?72?3 spins with periodic boundary conditions. In these notations, a "skyrmion lattice" means the lattice of well distinguished skyrmionic tubes of the same size, while a "cycloidal phase" includes large interconnected regions with the same direction of spins. The corresponding h-dependence of (b) the magne- tization and (c) electric polarization: total and partial contri- butions, calculated from Eq. (7) relative to the FM state. metric terms. In fact, we also expect a small region in the phase diagram, where the magnetization is nearly saturated (Fig. 2b) and the skyrmion size is small, so the polarization change is mainly governed by the isotropic mechanism and is expected to increase with h. Over- all, the h-dependence of spin-driven polarization in the skyrmion phase depends on the skyrmion size and the way a skyrmion lattice is packed leading to different com- peting ...
Context 10
... order to describe this effect quantitatively, we eval- uate the total and partial contributions to P z by us- ing Eq. (7) and the distribution of spins {e i } obtained in Monte-Carlo calculations. The results are summa- rized in Fig. 2c. Particularly, we note a strong com- petition of the isotropic (? e i e j ) and antisymmetric (? e i ? e j ) contributions, while the anisotropic symmet- ric part (? e i ? ? ij e j ) is negligibly small. As expected, the antisymmetric contribution decreases with the increase of h and vanishes in the collinear FM state. On the con- trary, the isotropic contribution takes its maximal value in the FM state and is further reduced by a noncollinear alignment of spins. Since the change of e i ?e j and e i e j is proportional to ? ij and ? 2 ij , respectively (with ? ij being the angle between e i and e j , which is induced by DM interactions and proportional to ? SO ), both the isotropic and antisymmetric mechanisms are of the 2nd order in ? SO , while the change of e i ? ? ij e j is only of the 3rd order. This naturally explains the hierarchy of partial contribu- tions to ?P z in Fig. 2c. Furthermore, the antisymmetric mechanism dominates when the skyrmions are large and the spin texture slowly varies in space. In this region, electric polarization decreases with h, in agreement with the experimental observation [15]. The corresponding polarization change of about 4 ?C/m 2 is also consistent with the experimental data [15]. Finally, our conclusion clearly differs from the phenomenological analysis pre- sented in [15], arguing that the antisymmetric DM inter- actions are solely needed to stabilize the cycloidal and skyrmion phases, while the corresponding polarization change is driven by the isotropic and anisotropic sym- (4) with an applied magnetic field h z at temperature T = 0.1J for the supercell of 72?72?3 spins with periodic boundary conditions. In these notations, a "skyrmion lattice" means the lattice of well distinguished skyrmionic tubes of the same size, while a "cycloidal phase" includes large interconnected regions with the same direction of spins. The corresponding h-dependence of (b) the magne- tization and (c) electric polarization: total and partial contri- butions, calculated from Eq. (7) relative to the FM state. metric terms. In fact, we also expect a small region in the phase diagram, where the magnetization is nearly saturated (Fig. 2b) and the skyrmion size is small, so the polarization change is mainly governed by the isotropic mechanism and is expected to increase with h. Over- all, the h-dependence of spin-driven polarization in the skyrmion phase depends on the skyrmion size and the way a skyrmion lattice is packed leading to different com- peting ...
Context 11
... order to describe this effect quantitatively, we eval- uate the total and partial contributions to P z by us- ing Eq. (7) and the distribution of spins {e i } obtained in Monte-Carlo calculations. The results are summa- rized in Fig. 2c. Particularly, we note a strong com- petition of the isotropic (? e i e j ) and antisymmetric (? e i ? e j ) contributions, while the anisotropic symmet- ric part (? e i ? ? ij e j ) is negligibly small. As expected, the antisymmetric contribution decreases with the increase of h and vanishes in the collinear FM state. On the con- trary, the isotropic contribution takes its maximal value in the FM state and is further reduced by a noncollinear alignment of spins. Since the change of e i ?e j and e i e j is proportional to ? ij and ? 2 ij , respectively (with ? ij being the angle between e i and e j , which is induced by DM interactions and proportional to ? SO ), both the isotropic and antisymmetric mechanisms are of the 2nd order in ? SO , while the change of e i ? ? ij e j is only of the 3rd order. This naturally explains the hierarchy of partial contribu- tions to ?P z in Fig. 2c. Furthermore, the antisymmetric mechanism dominates when the skyrmions are large and the spin texture slowly varies in space. In this region, electric polarization decreases with h, in agreement with the experimental observation [15]. The corresponding polarization change of about 4 ?C/m 2 is also consistent with the experimental data [15]. Finally, our conclusion clearly differs from the phenomenological analysis pre- sented in [15], arguing that the antisymmetric DM inter- actions are solely needed to stabilize the cycloidal and skyrmion phases, while the corresponding polarization change is driven by the isotropic and anisotropic sym- (4) with an applied magnetic field h z at temperature T = 0.1J for the supercell of 72?72?3 spins with periodic boundary conditions. In these notations, a "skyrmion lattice" means the lattice of well distinguished skyrmionic tubes of the same size, while a "cycloidal phase" includes large interconnected regions with the same direction of spins. The corresponding h-dependence of (b) the magne- tization and (c) electric polarization: total and partial contri- butions, calculated from Eq. (7) relative to the FM state. metric terms. In fact, we also expect a small region in the phase diagram, where the magnetization is nearly saturated (Fig. 2b) and the skyrmion size is small, so the polarization change is mainly governed by the isotropic mechanism and is expected to increase with h. Over- all, the h-dependence of spin-driven polarization in the skyrmion phase depends on the skyrmion size and the way a skyrmion lattice is packed leading to different com- peting ...
Context 12
... nonmagnetic electronic band structures calculated for the high-and low-temperature phases of GaV 4 S 8 are presented in Fig. 2a. The group of bands located near the Fermi level is dominated by metal-metal bonding states of the V 4 cluster generated by the V 3d orbitals. In the high-temperature cubic phase (point group T d ), these states are split into a 1 , e and t 2 levels filled with seven electrons in such a way that the highest 3-fold degenerate t 2 level contains one unpaired electron. At 38 K GaV 4 S 8 undergoes a cubic-to-rhombohedral structural phase transition driven by the cooperative Jahn-Teller distortion [11] which lifts the degeneracy of the t 2 level and splits it into a low-lying single a 1 and double degenerate e states (point group C 3v ), as shown in Fig. 2b. Since each V 4 cluster carries a local moment S = 1/2 formed by three bands located near the Fermi level, these states can be chosen as a basis for the effective low-energy model to describe magnetic properties of GaV 4 S 8 . The corresponding Wannier functions obtained by projecting these states onto atomic-like orbitals are shown in Fig. 2c. As seen, the resulting Wannier functions are well defined and localized, but have a complicated structure representing metal-metal bonding in the V 4 ...
Context 13
... nonmagnetic electronic band structures calculated for the high-and low-temperature phases of GaV 4 S 8 are presented in Fig. 2a. The group of bands located near the Fermi level is dominated by metal-metal bonding states of the V 4 cluster generated by the V 3d orbitals. In the high-temperature cubic phase (point group T d ), these states are split into a 1 , e and t 2 levels filled with seven electrons in such a way that the highest 3-fold degenerate t 2 level contains one unpaired electron. At 38 K GaV 4 S 8 undergoes a cubic-to-rhombohedral structural phase transition driven by the cooperative Jahn-Teller distortion [11] which lifts the degeneracy of the t 2 level and splits it into a low-lying single a 1 and double degenerate e states (point group C 3v ), as shown in Fig. 2b. Since each V 4 cluster carries a local moment S = 1/2 formed by three bands located near the Fermi level, these states can be chosen as a basis for the effective low-energy model to describe magnetic properties of GaV 4 S 8 . The corresponding Wannier functions obtained by projecting these states onto atomic-like orbitals are shown in Fig. 2c. As seen, the resulting Wannier functions are well defined and localized, but have a complicated structure representing metal-metal bonding in the V 4 ...
Context 14
... nonmagnetic electronic band structures calculated for the high-and low-temperature phases of GaV 4 S 8 are presented in Fig. 2a. The group of bands located near the Fermi level is dominated by metal-metal bonding states of the V 4 cluster generated by the V 3d orbitals. In the high-temperature cubic phase (point group T d ), these states are split into a 1 , e and t 2 levels filled with seven electrons in such a way that the highest 3-fold degenerate t 2 level contains one unpaired electron. At 38 K GaV 4 S 8 undergoes a cubic-to-rhombohedral structural phase transition driven by the cooperative Jahn-Teller distortion [11] which lifts the degeneracy of the t 2 level and splits it into a low-lying single a 1 and double degenerate e states (point group C 3v ), as shown in Fig. 2b. Since each V 4 cluster carries a local moment S = 1/2 formed by three bands located near the Fermi level, these states can be chosen as a basis for the effective low-energy model to describe magnetic properties of GaV 4 S 8 . The corresponding Wannier functions obtained by projecting these states onto atomic-like orbitals are shown in Fig. 2c. As seen, the resulting Wannier functions are well defined and localized, but have a complicated structure representing metal-metal bonding in the V 4 ...
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