Hinge Spin Polarization in Magnetic Topological Insulators Revealed by Resistance Switch

We report on the possibility of detecting hinge spin polarization in magnetic topological insulators by resistance measurements. By implementing a three-dimensional model of magnetic topological insulators into a multiterminal device with ferromagnetic contacts near the top surface, local spin features of the chiral edge modes are unveiled. We find local spin polarization at the hinges that inverts the sign between the top and bottom surfaces. At the opposite edge, the topological state with inverted spin polarization propagates in the reverse direction. A large resistance switch between forward and backward propagating states is obtained, driven by the matching between the spin polarized hinges and the ferromagnetic contacts. This feature is general to the ferromagnetic, antiferromagnetic, and canted antiferromagnetic phases, and enables the design of spin-sensitive devices, with the possibility of reversing the hinge spin polarization of the currents.

Figure 1

Magnetic TI in the FM phase, $\mathit{m}=0.05t\widehat{z}$. (a) Dispersion relation of a slab geometry infinite in the $y$ direction. The left (right) inset depicts the local spin density of states $⟨s_x⟩$ of the edge state at $k_y=-0.1\pi /a$ ($k_y=0.1\pi /a$). The edge state covers the sidewall of the slab and propagates to the right (left). (b) Local density of states of a finite square slab. The edge state circulates around the sample, covers the side surfaces perpendicular to $\widehat{x}$, and propagates along the top or bottom hinges of the side surfaces perpendicular to $\widehat{y}$. (c) Side view of transport setup geometry: metallic leads connect to the whole walls at both ends of the slab (golden color), and ferromagnetic leads connect to the sidewalls only near top hinge (red color). (d) Top view and reference numbering of the leads on the transport setup.

Figure 2

Transport simulations of a FM slab ($\mathit{m}=0.02t\widehat{z}$) between metallic leads and ferromagnetic leads with spin down $(s_x,\downarrow )$ polarization. The shaded regions depict the standard error of considering ten Anderson disorder realizations of strength $d=0.04t$. A four-terminal device allows us to measure the distinct resistance profiles. The two-terminal resistance setup between leads $L_0$ and $L_1$ is depicted in the left inset and the blue curve is the resistance profile. The right inset shows the resistance setup between leads $L_1$ and $L_2$, with resistance profile in red. In the former case, the ferromagnetic lead polarization matches the top HSP, and in the latter case, the local spin polarization is opposite.

Figure 3

2T resistance at $e_F=0$ of a magnetic TI slab in the FM phase, with magnetization $|\mathit{m}|=0.02t$ (see right inset). In (a), the two-terminal resistance is measured between leads $L_1$ and $L_2$; in (b), the resistance is measured between leads $L_0$ and $L_1$. The plots in (a) and (b) are almost identical by reflecting around $\varphi =90°$, where the magnetization component along the $z$ axis changes sign. The mismatching configurations $\varphi =0°$ in (a), and $\varphi =180°$ in (b) show the largest resistance that slowly decreases when rotating the magnetization angle.

Figure 4

Magnetic TI slab with the same parameters as in Fig. 3. HSP of the edge state at $k_y=-0.02\pi /a$ and positive energy (propagating in the $\widehat{y}$ direction) for a wide range of magnetization angles $(\theta ,\varphi )$. The arrows show the components of $⟨s⟩$ projected on the top-left (-right) hinges of an infinite slab along the $y$ axis. The direction of the arrows gives the component in the $(⟨s_x⟩,⟨s_z⟩)$ plane, and the color of the arrows is the component in $⟨s_y⟩$. (a) Propagating edge states on the top-left hinge are only present when the magnetization has a positive projection along $z$, thus, $\varphi \lesssim 90°$. (b) Propagating edge states on the top-right hinge are only present when the magnetization has a negative projection along $z$, thus, $\varphi \gtrsim 90°$.