Topological charge, spin, and heat transistor

Spin pumping consists in the injection of spin currents into a nonmagnetic material due to the precession of an adjacent ferromagnet. In addition to the pumping of spin the precession always leads to pumping of heat, but in the presence of spin-orbital entanglement it also leads to a charge current. We investigate the pumping of charge, spin, and heat in a device where a superconductor and a quantum spin Hall insulator are in proximity contact with a ferromagnetic insulator. We show that the device supports two robust operation regimes arising from topological effects. In one regime, the pumped charge, spin, and heat are quantized and related to each other due to a topological winding number of the reflection coefficient in the scattering matrix formalism, translating to a Chern number in the case of Hamiltonian formalism. In the second regime, a Majorana zero mode switches off the pumping of currents owing to the topologically protected perfect Andreev reflection. We show that the interplay of these two topological effects can be utilized so that the device operates as a robust charge, spin, and heat transistor.

Figure 1

Sketch of the proposed device consisting of a QSHI in proximity contact with FI of length $L$ (induced energy gap ${\mathrm{\Delta }}_F=\left|\mathbf{m}\right|sin\theta$) and SC (induced energy gap ${\mathrm{\Delta }}_0$). The monodomain magnetization $\mathit{m}\left(t\right)$ precesses at an angle $\theta$ around the axis perpendicular to the QSHI driving charge, spin, and heat currents to the drain (left), which can be controlled with potential at the FI region $V_g$, the precession angle $\theta$, temperature and drain voltage $V_d$. The system harbors a zero-energy MF at the FI-SC interface that affects the pumped currents via the scattering coefficient $r_{ee}^{\downarrow \uparrow }\left(E\right)$.

Figure 2

The dimensionless charge $\mathcal{Q}$ in the adiabatic limit [Eq. (9)] as a function of $T$ for various values of (a) gate voltages $e{V}_g$ (in units ${\mathrm{\Delta }}_0$) and (b) angles $\theta$. The insets are zooms into the suppression ($k_{B}T\ll \mathrm{\Gamma }, \mathcal{Q}\approx 0$) and quantization ($k_{B}T\gg \mathrm{\Gamma }, \mathcal{Q}\approx 1$) regimes. The asymptotic expressions of $\mathcal{Q}$ [Eq. (11)] are shown in the suppression (circles) and quantization (squares) regimes. The other parameters are $L=400$ nm, ${\mathrm{\Delta }}_0=1\mathrm{meV}, m_0/{\mathrm{\Delta }}_0=2$, and $v_F=2.7\times {10}^5$ m/s.

Figure 3

The dimensionless charge $\mathcal{Q}$ as a function of $\omega$ for $V_d=T=0$ and various values of (a) $e{V}_g$ (in units of ${\mathrm{\Delta }}_0$) and (b) $\theta$. The insets are zooms into the suppression ($\hslash \omega \ll \mathrm{\Gamma }, \mathcal{Q}\approx 0$) and quantization ($\hslash \omega \gg \mathrm{\Gamma }, \mathcal{Q}\approx 1$) regimes. The asymptotic expressions of $\mathcal{Q}$ [Eq. (12)] are shown in the suppression (circles) and quantization (squares) regimes. The other parameters are the same as in Fig. 2.

Figure 4

The dimensionless charge $\mathcal{Q}$ as a function of drain voltage $V_d$ for various values of (a) gate voltages $V_g$ and (b) angles $\theta$. Two different limits for $\mathcal{Q}$ are evident in the plots. The insets are zooms into the suppression (low $V_d$) and quantization (high $V_d$) regimes, respectively. The asymptotic limits of the pumped charge are also shown in the suppression (circles) and quantization (squares) regimes. The charges in the bottom insets have been shifted evenly for the sake of visualization.