Tunneling Planar Hall Effect in Topological Insulators: Spin Valves and Amplifiers

We investigate tunneling across a single ferromagnetic barrier on the surface of a three-dimensional topological insulator. In the presence of a magnetization component along the bias direction, a tunneling planar Hall conductance (TPHC), transverse to the applied bias, develops. Electrostatic control of the barrier enables a giant Hall angle, with the TPHC exceeding the longitudinal tunneling conductance. By changing the in-plane magnetization direction, it is possible to change the sign of both the longitudinal and transverse differential conductance without opening a gap in the topological surface state. The transport in a topological-insulator–ferromagnet junction can, thus, be drastically altered from a simple spin valve to an amplifier.

Figure 1

(a) Schematic setup. (b) Origin of the planar Hall conductance and net Hall voltage, $V_H$, due to asymmetric tunneling. The circle sizes represent the asymmetry in transmission probabilities arising from the interfacial mismatch of spin directions (locked to the velocity). (c) Spin mismatch: Fermi circles in the TI (upper Dirac cone) and the barrier (lower Dirac cone, shifted by a proximity-induced exchange splitting ${\mathrm{\Delta }}_x$). In (b) and (c) violet (black) arrows denote the electron spin orientation (direction of motion).

Figure 2

Dependence of the (a) longitudinal and (b) transverse conductances on $d$ for finite and $\delta$ barriers. (c) Fermi circles in the leads (inner circle) and barrier (outer circle). The arrows denote the wave vectors of states with a positive $x$ component of the velocity, and the vertical dashed line indicates the first-order resonance condition. (d) Transmission $T({ϵ}_F,\theta )$ of a finite barrier as a function of $d$ and $\theta$.

Figure 3

Dependence of the (a) longitudinal and (b) transverse conductances as well as (c) of their ratio on $V_0$ and $\mathrm{\Delta }$ for a finite barrier with $d=50\mathrm{nm}$, ${ϵ}_F=1\mathrm{meV}$, and $v_F=6.0\times 10^5\mathrm{m}/\mathrm{s}$. Green lines: boundaries of regions with negative conductance. (d) Same as in Fig. 2, but for larger $\mathrm{\Delta }$.

Figure 4

Bias dependence of the (a) longitudinal and (b) transverse currents for a finite barrier and different $V_0$. Assuming $D_x=D_y=10\mu \mathrm{m}$, both currents are given in units of $I_0=12\mu \mathrm{A}$. (c) Same as in Fig. 2, but with thickened Fermi circles accounting for a finite energy window around ${ϵ}_F$. (d) $T(E,\theta )$ for $V_0=105\mathrm{meV}$. The inset in (a) shows the appearance of a ND $G_{xx}$ for $V_0=105\mathrm{meV}$ at high bias.