Adiabatic pumping through an interacting quantum dot with spin-orbit coupling

We study adiabatic pumping through a two-level quantum dot with spin-orbit coupling. Using a diagrammatic real-time approach, we calculate both the pumped charge and spin for a periodic variation of the dot's energy levels in the limit of weak tunnel coupling. Thereby, we compare the two limits of vanishing and infinitely large charging energy on the quantum dot. We discuss the dependence of the pumped charge and pumped spin on gate voltages, the symmetry in the tunnel-matrix elements, and spin-orbit coupling strength. We identify the possibility to generate pure spin currents in the absence of charge currents.

Figure 1

Energy scheme of the two-level quantum dot. The two orbital, spin-degenerate levels can be varied in time. They are tunnel coupled to the left (L) and the right (R) lead, with tunnel-matrix elements $V_{\lambda \alpha }$. The leads have the same chemical potential $\mu$.

Figure 2

Scheme of different relevant isospin quantization axes. The vector $\mathit{n}$ represents the quantization where the orbital levels $\left|1\sigma \right.〉$ and $\left|2\sigma \right.〉$ are the eigenstates of the ${\widehat{I}}_z$ operator of the isospin. The two axes ${\tilde{\mathit{n}}}_{\sigma }$ are the quantization axes where the eigenstates of ${\widehat{I}}_z$ are the eigenstates of $H^{\mathrm{dot}}$ for single occupation. In a ferromagnetic analogy, the leads are fully isospin polarized along the axes ${\mathit{m}}_{\lambda }$.

Figure 3

Pumped charge (spin) with finite SO coupling, ${\alpha }_{\mathrm{so}}=\Gamma /10$, in the $U=0$ and the $U=\infty$ limit depending on the time-averaged orbital level positions. There are two sets of coupling parameters: First, ${\varphi }_{\mathrm{L}}=\pi /2$ and ${\varphi }_{\mathrm{R}}=\pi /4$ for panels (a)–(d), and second, the antisymmetric combination, ${\varphi }_{\mathrm{L}}=-\pi /4$ and ${\varphi }_{\mathrm{R}}=\pi /4$, for panels (e) and (f). For all panels we chose ${\Gamma }_{\text{L}}={\Gamma }_{\text{R}}$. The latter, antisymmetric combination leads to vanishing pumped charge.

Figure 4

Pumped charge (spin) with finite SO coupling, ${\alpha }_{\mathrm{so}}=\Gamma /10$, in the $U=0$ and the $U=\infty$ limit depending on the averaged level spacing of both orbital levels. In the $U=0$ limit we choose $\overline{\epsilon }=0$, which is the position of the maximum value. In the limit of strong Coulomb interaction, we use $\overline{\epsilon }=k_{B}T$ as an approximation to the position of the maximum value. The two sets of coupling parameters (1) and (2) are the ones given in the text [see Eq. (31)]. The vertical dotted lines indicate pure spin pumping. In the weak-coupling limit, this is only possible for pumping with Coulomb interaction.

Figure 5

Pumped charge and pumped spin in the $U=0$ and the $U=\infty$ limit depending on the orbital-coupling configuration for fixed ${\Gamma }_{\mathrm{L}}={\Gamma }_{\mathrm{R}}$. The illustrated function shows the maximum value of the pumped charge (spin) in the $(\overline{\epsilon },\overline{\Delta \epsilon })$ parameter space for ${\alpha }_{\mathrm{so}}=\Gamma /10$. The dotted lines represent coupling configurations where the pumped charge (spin) is zero. Along the dashed lines, for $U=0$, the pumped charge is always zero while the spin is still finite. For strong Coulomb interaction, $U=\infty$, only the line ${\varphi }_{\mathrm{L}}=-{\varphi }_{\mathrm{R}}$ leads to vanishing pumped charge.

Figure 6

Pumped charge and pumped spin in the $U=0$ and $U=\infty$ limits depending on the orbital-coupling configuration for fixed ${\Gamma }_{\mathrm{L}}=2{\Gamma }_{\mathrm{R}}$. The illustrated functions show the maximum value of the pumped charge (spin) in the $(\overline{\epsilon },\overline{\Delta \epsilon })$ parameter space for ${\alpha }_{\mathrm{so}}=\Gamma /10$.

Figure 7

Pumped charge and pumped spin in the $U=0$ and $U=\infty$ limits depending on the strength of the SO coupling. Additionally, the green dotted line shows the pumped charge (spin) for $U=\infty$ in the case that the exchange field is turned off by hand. The illustrated functions are the maximum value of the pumped charge (spin) for given SO strength, ${\alpha }_{\mathrm{so}}$, in the $(\overline{\epsilon },\overline{\Delta \epsilon })$ parameter space. The coupling parameters are ${\Gamma }_{\mathrm{L}}={\Gamma }_{\mathrm{R}}$, ${\varphi }_{\mathrm{L}}=\pi /2$, and ${\varphi }_{\mathrm{R}}=\pi /4$.