Electric-field-induced magnetic toroidal moment and nonlinear magnetoelectric effect in antiferromagnetic olivines

TL;DR

The paper investigates the microscopic origin of electric-field-induced magnetic toroidal moments in antiferromagnetic olivines, prompted by directional dichroism in Co$_2$SiO$_4$. It introduces a minimal spin model with $d$-$p$ hybridization–based magnetoelectric coupling, solved at the mean-field level, that yields a finite magnetic toroidal monopole $T_0$ in the AFM state and reproduces qualitative dielectric anomalies. It demonstrates a linear magnetoelectric response linking $\mathbf{t}$ and $\mathbf{E}$ that is highly anisotropic, and predicts a second-order, antisymmetric magnetoelectric effect producing a transverse magnetization. It shows similar toroidal-type magnetoelectric responses in related olivine compounds, suggesting the mechanism is robust across this material family and offering pathways to domain control and toroidal-based devices.

Abstract

Beyond conventional electric and magnetic monopoles, electric and magnetic toroidal monopoles, which are rank-0 multipoles distinguished by opposite parities under spatial inversion and time reversal, can exist in nature. The recent observation of electric-field-induced directional dichroism in antiferromagnetic olivine Co$_2$SiO$_4$ has provided the first concrete example of a magnetic toroidal monopole; however, its microscopic origin remains elusive. Here, we propose a minimal spin model that incorporates magnetoelectric coupling via the $d$-$p$ hybridization mechanism and analyze it within the mean-field approximation. The model qualitatively reproduces the experimentally observed temperature dependence of the dielectric constant and its pronounced sensitivity to the direction of the applied electric field. Furthermore, it elucidates the temperature evolution of the magnetic toroidal monopole and the strong electric-field-direction dependence of the magnetic toroidal moment. Our calculations also predict a second-order nonlinear magnetoelectric response, consistent with the symmetry classification of Co$_2$SiO$_4$ as an altermagnet. Additionally, we demonstrate that the same framework is applicable to other antiferromagnetic olivines with analogous magnetic order, indicating the robustness and generality of the toroidal-type magnetoelectric response in this material family.

Figures (8)

• Figure 1: Crystal structure of Co$_2$SiO$_4$Sazonov2008. For clarity, Si atoms are omitted, showing only Co and O. (a) The unit cell (thin black lines) contains two crystallographically distinct Co$^{2+}$ sites, denoted as Co1 (light blue) and Co2 (dark blue). Each Co$^{2+}$ ion is octahedrally coordinated by six O$^{2-}$ ligands (red). In this study, we focus on the Co2 sites and assign sublattice indices A, B, C, and D within the unit cell. (b) The Co2 sublattices A and B, and C and D, respectively, form square-lattice-like networks in the $bc$ plane through corner sharing of octahedra. Arrows indicate the spin directions (parallel or antiparallel to the $b$ axis) in the ordered phase. (c) Octahedron formed by a Co2 ion of sublattice A and its surrounding ligands. $\mathbf{e}_{\mathrm{A}1}$--$\mathbf{e}_{\mathrm{A}6}$ represent the relative position vectors from the Co2 site to the ligands. Crystal structures are visualized using VESTA Momma2011.
• Figure 2: Temperature dependence of the dielectric constant from theory and experiment. Panels (a)--(c) show the theoretical results of (a) $\varepsilon^{aa}$, (b) $\varepsilon^{bb}$, and (c) $\varepsilon^{cc}$. Panels (d)--(f) show the corresponding experimental data of (d) $\varepsilon^{aa}$, (e) $\varepsilon^{bb}$, and (f) $\varepsilon^{cc}$.
• Figure 3: Antiferromagnetic order and magnetic toroidal monopole. (a) Real-space distributions of the local magnetization $\mathbf{m}_\alpha$, the local electric polarization $\mathbf{p}_\alpha$, and the local magnetic toroidal moment $\mathbf{t}_\alpha$ in the low-temperature antiferromagnetically ordered phase. (b) Temperature dependence of the antiferromagnetic order parameter $m_{\mathrm{AF}}^b$. (c) Temperature dependence of the magnetic toroidal monopole $T_0$.
• Figure 4: Electric-field-induced magnetic toroidal moment. (a)--(c) Field-angle dependence of the induced components: (a) $t^a$ and $t^b$ for $\mathbf{E} \perp c$, (b) $t^b$ and $t^c$ for $\mathbf{E} \perp a$, and (c) $t^a$ and $t^c$ for $\mathbf{E} \perp b$. Solid lines are the numerical results obtained within the mean-field approximation, and dashed lines are fits to either $\cos\theta$ or $\sin\theta$ functions. (d) Distribution of the induced toroidal-moment response with respect to the electric-field direction, visualized on a sphere. The color of each arrow indicates the magnitude of the magnetic toroidal moment $|\mathbf{t}| = \sqrt{\sum_{\mu} (X^{tE}_{\mu\mu} E^{\mu})^2}$, where $X^{tE}_{\mu\mu}$ is the magnetoelectric coefficient estimated from the fitting, for an electric field with normalized amplitude $|\mathbf{E}|=1$.
• Figure 5: Electric-field angle dependence of the second-order magnetoelectric effect. A uniform magnetization $m^{\mu}$ along the $\mu$ axis is induced in the perpendicular direction to an applied electric field $\mathbf{E}$. Solid lines are the numerical results obtained within the mean-field approximation, and dashed lines are fits to $X^{mEE}E^{2}\sin\theta\cos\theta$, where $X^{mEE}$ is the coefficient of the second-order magnetoelectric response and $\theta$ is the rotation angle of $\mathbf{E}$ in the measurement plane. The horizontal axis is $\theta_{\nu\gamma}$ with $\{\mu\nu\gamma\}=\{abc\}$, $\{bac\}$, or $\{cab\}$, corresponding respectively to the field rotations in the $ab$, $bc$, and $ac$ planes. For visibility, the data for $\mu=b$ are multiplied by $10$.