Layer dipole magnetoelectric polarizability of antiferromagnetic bilayers

Abstract

In this paper we study magnetoelectric effects in two-dimensional magnetic bilayers and introduce the notion of a layer dipole magnetoelectric polarizability. This magnetoelectric polarizability describes the magnetization response to an applied electric field perpendicular to the bilayer. As such, it represents the electric analog of the spin magnetoelectric polarizability, governing the charge polarization response to an applied Zeeman field. Starting from the orbital magnetization produced by a perpendicular displacement field, we derive a microscopic expression for the layer dipole magnetoelectric polarizability and apply it to two minimal models for bilayer magnets, i.e., a buckled square lattice model and a magnetic topological insulator model. In the case of the buckled square lattice model we show that the layer dipole magnetoelectric polarizability has a (quasi-)topological contribution, revealing a topological magnetoelectric response of two-dimensional antiferromagnets associated with the layer pseudospin degree of freedom.

Figures (4)

• Figure 1: (a) Illustration of the layer dipole magnetoeletric effect in bilayer antiferromagnets. An orbital magnetization ($M_z$) develops in response to a perpendicular displacement field $D_z$. The 2D antiferromagnet is formed from two oppositely aligned easy-axis ferromagnetic layers (shown on the right). (b) The buckled square lattice with two sublattices (white and black sites) displaced in the negative and positive $z$ direction (i.e., out-of-plane direction), respectively. The inversion center at the origin of the unit cell is marked as a blue dot. (c) Energy spectrum of the buckled square lattice model in the presence of displacement field, as calculated from Eqs. \ref{['eq:H_k']} and \ref{['eq:buckled-square']}. Here we have set $\lambda=0.6W$, $N_z=0.4W$, and $edD_z=0.2W$, with $W\equiv \hbar^2/2m_ea^2$ (we always set $t_1 =W$). The inset shows the Brillouin zone.
• Figure 2: (a) Orbital magnetization as a function of $D_z$, calculated directly from Eq. \ref{['eq:M_z']} (blue dots) and from a linear approximation using Eq. \ref{['eq:alpha_zz']} (red lines). The two sets of curves correspond to $N_z/W=0.2,0.4$. Note that the shown values of $M_z$ have been enlarged by a factor $10^2$. (b) Polarizability $\alpha_{zz}$ [Eq. \ref{['eq:alpha_zz']}] as a function of $N_z$. Different curves correspond to $\lambda/W=0.6,0.8,1.0,1.2$ (bottom to top for $N_z>0$; top to bottom for $N_z<0$). (c) Sketch of a fourfold Dirac fermion in two dimensions. (d) A perpendicular displacement field $D_z$ shifts the twofold Dirac points in energy by an amount $\hbar b_0 = edD_z$. (e) Energy bands of the buckled square lattice model in the presence of a displacement field $\Delta=edD_z = 0.2W$ but with $N_z=0$. The resulting energy shift of the Dirac crossings at $M$ is indicated.
• Figure 3: (a) Current density $j_y(x)$ for the buckled square lattice model with open boundary in the $x$ direction ($N_x=100$ sites; $\lambda =0.8W$; $N_z=0.25W$; $\Delta=0.05W$). (b) Integrated current of the right edge as a function of $\Delta$, divided by $\Delta \zeta$ and calculated for $N_z=0.4W$ (red), $N_z=0.2W$ (blue), $N_z=0.15W$ (orange).
• Figure 4: (a) Surface state model of a 2D (thin film) antiferromagnetic TI. Top and bottom surfaces are coupled to surface-normal ferromagnetic order. (b) Schematic energy spectrum of surface state with (left) and without (right) displacement field $D_z$. The chemical potential $\mu$ lies in the gap.