Tunneling Anomalous and Spin Hall Effects

We predict, theoretically, the existence of the anomalous Hall effect when a tunneling current flows through a tunnel junction in which only one of the electrodes is magnetic. The interfacial spin-orbit coupling present in the barrier region induces a spin-dependent momentum filtering in the directions perpendicular to the tunneling current, resulting in a skew tunneling even in the absence of impurities. This produces an anomalous Hall conductance and spin Hall currents in the nonmagnetic electrode when a bias voltage is applied across the tunneling heterojunction. If the barrier is composed of a noncentrosymmetric material, the anomalous Hall conductance and spin Hall currents become anisotropic with respect to both the magnetization and crystallographic directions, allowing us to separate this interfacial phenomenon from the bulk anomalous and spin Hall contributions. The proposed effect should be useful for proving and quantifying the interfacial spin-orbit fields in metallic and metal-semiconductor systems.

Figure 1

(a) Schematic of a ferromagnet-semiconductor-normal metal tunnel junction. The tunneling current flowing in the $z$ direction generates the anomalous Hall voltage ($V_{\mathit{H}}$) in the nonmagnetic electrode. (b) Side view of (a). Taking the [110] axis as a reference, the magnetization direction ($\mathbf{m}$) and the direction along which the Hall voltage is measured ($\mathbf{t}$) are determined by the angles $\phi$ and $\varphi$, respectively.

Figure 2

Dependence of the TAHE conductance ratios, $G_{xz}/G_{zz}$ [(a), (c)] and $G_{yz}/G_{zz}$ [(b), (d)], on the magnetization direction for different values of the BR parameter ($\alpha$) and barrier thickness ($d$).

Figure 3

Spin-dependent momentum filtering resulting from tunneling through a barrier with BR SOC characterized by the parameter $\alpha >0$. Particles propagate from the ferromagnetic (left) electrode into the nonmagnetic (right) region. For $k_y=0$ and $\mathbf{m}\parallel \mathbf{y}$ the height of the effective, spin-dependent potential barrier is $h=V_{0}d-\sigma \alpha k_x$ [see Eq. (8)] with $\sigma =1$ ($\sigma =-1$) for an incident particle with spin parallel (antiparallel) to the magnetization. Therefore, for the majority channel ($\sigma =1$) the transmission of particles with $k_x>0$ is favored, as schematically shown in (a). The momentum imbalance in the transmitted states generates a transverse current. For the minority channel (b) the situation is opposite. However, due to the presence of spin polarization, the contribution of the majority channel dominates. The result is a finite TAHE current in the $x$ direction, $J_x<0$ (and a finite TAHE conductance, $G_{xz}=J_x{A}_x/V_z$). The situation for $\alpha <0$ is analogue but the TAHE current flow is reversed, i.e., $J_x>0$.