Spin pump in the presence of a superconducting lead

We present a theoretical analysis of a spin pump in the presence of a superconducting lead. The spin pump is facilitated by a rotating magnetic field which provides a spin flip mechanism and hence can generate a spin current without an accompanying charge current. Using a nonequilibrium Green’s function method, we obtain a general solution of the pumped charge current and spin current in both the adiabatic and non-adiabatic regimes. The numerical results for the charge current and spin current are presented as we vary different system parameters such as the gate voltage, the external magnetic field, and the pumping frequency. We find that for a quantum dot with a single resonant level in line with the Fermi energy of the left normal lead, a pure spin current is generated by a rotating magnetic field at any frequency. We have identified two kinds of photon-assisted processes which dominate at low pumping frequencies and high pumping frequencies, respectively.

Figure 1

The charge current $J_L^e$ vs the pumping frequency $\omega$ at different resonant energies $ϵ$: $ϵ=0.2\mathrm{meV}$ (solid line), $ϵ=0.4\mathrm{meV}$ (dashed line), and $ϵ=0.8\mathrm{meV}$ (dotted line). Other parameters: $\mu B_0=0.1\mathrm{meV}$, $\Gamma =0.1\mathrm{meV}$, $\Delta =1.45\mathrm{meV}$ and $\theta =\pi ∕2$.

Figure 2

The spin current of the left lead $J_L^s$ vs the pumping frequency $\omega$ at different $ϵ$: $ϵ=0$ (solid line), $ϵ=0.2\mathrm{meV}$ (dashed line), $ϵ=0.4\mathrm{meV}$ (dotted line), and $ϵ=0.8\mathrm{meV}$ (dash-dotted line). Other parameters are the same as Fig. 1. Inset: the right spin current $J_R^s$ vs the pumping frequency $\omega$ at different $ϵ$. The symbols are the same as the main panel.

Figure 3

The charge current $J_L^e$ vs the pumping frequency $\omega$ at different magnetic fields $\mu B_0$: $\mu B_0=0.05\mathrm{meV}$ (solid line), $\mu B_0=0.1\mathrm{meV}$ (dashed line) and $\mu B_0=0.2\mathrm{meV}$ (dotted line). Other parameters: $ϵ=0.2\mathrm{meV}$, $\Gamma =0.1\mathrm{meV}$, $\Delta =1.45\mathrm{meV}$, and $\theta =0.5\pi$.

Figure 4

The left spin current $J_L^s$ vs $\omega$ at different magnetic fields $\mu B_0$: $\mu B_0=0.05\mathrm{meV}$ (solid line), $\mu B_0=0.1\mathrm{meV}$ (dashed line), and $\mu B_0=0.2\mathrm{meV}$ (dotted line). Other parameters are the same as Fig. 3. Inset: the right spin current $J_R^s$ vs $\omega$ at different magnetic fields. The symbols are the same as the main panel.

Figure 5

The charge current $J_L^e$ vs the resonant energy $ϵ$ at different magnetic fields: $\mu B_0=0.05\mathrm{meV}$ (solid line), $\mu B_0=0.1\mathrm{meV}$ (dashed line) and $\mu B_0=0.2\mathrm{meV}$ (dotted line). Here $\omega =1.8\mathrm{meV}$ and other parameters are the same as Fig. 1. The left inset: $J_L^e$ vs $ϵ$ at different frequencies: $\omega =0.2\mathrm{meV}$ (solid line), $\omega =0.5\mathrm{meV}$ (dashed line), and $\omega =1.4\mathrm{meV}$ (dotted line). The right inset: $J_L^e$ vs $ϵ$ at much larger frequencies: $\omega =1.5\mathrm{meV}$ (solid line), $\omega =2.0\mathrm{meV}$ (dashed line), $\omega =2.5\mathrm{meV}$ (dotted line), and $\omega =3.0$ (dash-dotted line). The other parameters are the same as Fig. 1.

Figure 6

$J_L^s$ vs $ϵ$ at different $\omega$: $\omega =0.01\mathrm{meV}$ (solid line), $\omega =0.1\mathrm{meV}$ (dashed line), $\omega =0.5\mathrm{meV}$ (dotted line), and $\omega =1.0\mathrm{meV}$ (dash-dotted line). Other parameters are the same as Fig. 1. The right inset: $J_R^s$ vs $ϵ$ at much larger frequencies: $\omega =1.5\mathrm{meV}$ (solid line), $2.0\mathrm{meV}$ (dashed line), and $2.5\mathrm{meV}$ (dotted line). The left inset: $J_L^s$ vs $ϵ$ at different $\omega$. The symbols are the same as the right inset.