Quantized topological magnetoelectric effect of the zero-plateau quantum anomalous Hall state

The topological magnetoelectric effect in a three-dimensional topological insulator is a novel phenomenon, where an electric field induces a magnetic field in the same direction, with a universal coefficient of proportionality quantized in units of $e^2/2h$. Here, we propose that the topological magnetoelectric effect can be realized in the zero-plateau quantum anomalous Hall state of magnetic topological insulators or a ferromagnet-topological insulator heterostructure. The finite-size effect is also studied numerically, where the magnetoelectric coefficient is shown to converge to a quantized value when the thickness of the topological insulator film increases. We further propose a device setup to eliminate nontopological contributions from the side surface.

Figure 1

Zero-plateau QAH state and magnetic field dependence of ${\sigma }_{xy}$. (a) Magnetic TI in an external field $H$, and sketch of ${\sigma }_{xy}$ as a function of $H$. The ${\sigma }_{xy}=0$ plateau occurs at the coercivity. The arrow indicates the magnetization direction. (b) FM-TI-FM heterostructure, and ${\sigma }_{xy}$ vs $H$. The zero plateau appears in the hysteresis loop due to the different $H_1^c$ and $H_2^c$ of FM A and B. The blue and red arrows indicate the magnetization directions of FM A and B, respectively.

Figure 2

TME effect. (a) Illustration of the ac electric current induced by an ac magnetic field $B_x$; the current density $j_x^{\text{3D}}\left(z\right)$ is defined in Eq. (5). (b) Magnetic field $B_y$ induced by applying an electric field $E_y$ through a capacitor. $E_y$ (with direction into the paper) will induce Hall currents ${\mathbf{J}}^t$ and ${\mathbf{J}}^b$ for antiparallel magnetization. (c) The functions $\kappa \left(z\right)$ and $\eta \left(z\right)$ for different thicknesses 6, 10, and 20 QL. Each QL is about 1 nm thick.

Figure 3

Finite-size effect of TME. (a) The $\gamma$ and $\kappa (z=0)$ as a function of $d$. The inset shows $\gamma$ plotted vs the inverse of thickness $1/d$. (b) The current amplitude ${\mathcal{J}}_0$ scales linearly with $d$. Here, $\theta /\pi =\gamma$.

Figure 4

Schematic of the FM-TI-FM heterostructure to observe the quantized TME, where the side surface of the TI is gapped by a FM proximity in (a). Au is the electrode. Such a geometry of the TI can be made by lithography. In reality, such a configuration in experiments may resemble that in (b), where the side surface is uneven and steplike. In this case, the side surface is gapped by both FM ordering and quantum confinement.