Topological staggered field electric effect with bipartite magnets

We study the interface physics of bipartite magnetic materials deposited on a topological insulator. Our study comprises antiferromagnets as well as ferrimagnets and ferromagnets with multiple magnetic moments per unit cell. If an energy gap is induced in the Dirac states on the topological surface, a topological magnetoelectric effect has been predicted. Here, we show that this effect can act in opposite directions on the two components of the magnet in a certain parameter region. Consequently, an electric field will mainly generate a staggered field rather than a net magnetization in the plane. This result is relevant for current attempts to detect the magnetoelectric effect experimentally, as well as for possible applications. We take a field-theoretic approach that includes the quantum fluctuations of both the Dirac fermions on the topological surface and the fermions in the surface layer of the magnet in an analytically solvable model. The effective Lagrangian and the Landau-Lifshitz equation describing the interfacial magnetization dynamics are derived.

Figure 1

The model system: (a) Bilayer heterostructure consisting of a bipartite magnetic insulator (BMI) deposited on a topological insulator (TI). (b) By means of the parameter $\mu ={\overline{m}}_2/{\overline{m}}_1$, the magnet can be tuned to be in an antiferromagnetic (AFM), ferrimagnetic (FiM), or ferromagnetic (FM) configuration at mean field. (c) The model involves fermionic fields $\mathrm{\Psi }$ and ${\chi }_{1,2}$ on the surfaces of both the TI (blue plane) and the BMI (gray plane), respectively, which are coupled by the amplitudes $h$ (hopping across the interface) and $t$ (local coupling of the two sublattices).

Figure 2

The anisotropic fluctuation-induced antiferromagnetic exchange coupling of ${\mathbf{m}}_1$ and ${\mathbf{m}}_2$ at the surface, which renormalizes the exchange coupling induced from the bulk. (a) In the component along the mean-field direction, the coupling constant is given by $T^{00}+T^{zz}$ (see main text) and shows a finite discontinuity at $\mu =\tau$ (dashed line). (b) In the component orthogonal to the mean-field direction, the AFM coupling $T^{00}-T^{zz}$ diverges at the discontinuity. The color scale is identical in both plots. (c) The quantities $T^{00}$ (thin solid blue line), $T^{zz}$ (dash-dotted blue line), $T^{00}+T^{zz}$ (bold solid red line), and $T^{00}-T^{zz}$ (dashed red line) as a function of $\mu$ for a specific value of $\tau$ $\left(\tau =0.45\right)$, which is indicated by the thin white dotted lines in (a) and (b). (d) The anisotropy terms $D_1^{zz}$ (thin solid blue line), $D_2^{zz}$ (dashed blue line), and $D_i^{00}+D_i^{zz}$ (bold solid red line, identical for $i=1,2)$ behave similarly, showing a discontinuity at $\tau =\mu$. The vicinity of this line is excluded from the further analysis.

Figure 3

Left: parameter regions of the bipartite magnet where the topological response to an electric field has the same (white area) or opposite (gray area) direction on the two sublattices, corresponding to a predominantly magnetoelectric (TME) or staggered field electric (TSE) effect, respectively. Here, $\tau$ is the dimensionless amplitude of the coupling of the fermions on the two sublattices, and $\mu$ is the ratio of the mean-field values of the magnetizations on the sublattices. Close to the dashed line at $\mu =\tau$, our results may not be applicable. Right: illustration of the topological effects for a FiM. (a) Both magnetizations ${\mathbf{m}}_1$ and ${\mathbf{m}}_2$ (black) pointing in their mean-field directions. (b) TME effect: if the topological response to the electric field $\mathbf{E}$ (red) has the same sign on both sublattices, an in-plane net magnetization ${\mathbf{m}}_{\parallel }$ (blue) is generated, while the induced in-plane staggered field ${\mathbf{l}}_{\parallel }$ (orange) is small. (c) TSE effect: if the topological response to $\mathbf{e}$ has opposite signs for ${\mathbf{m}}_1$ and ${\mathbf{m}}_2$, an in-plane staggered field is generated, while ${\mathbf{m}}_{\parallel }$ is small. The overall sign of these effects depends on the sign of the mass term $m_{\mathrm{\Psi }}$.