Spin-charge filtering through a spin-orbit coupled quantum dot controlled via an Aharonov-Bohm interferometer

We show that a strongly correlated quantum dot embedded in an Aharonov-Bohm interferometer can be used to filter both charge and spin at zero voltage bias. The magnitude with which the Aharonov-Bohm arm is coupled to the system controls the many-body effects on the quantum dot. When the quantum dot is in the Kondo regime, the flow of charge through the system can be tuned by the phase of the Aharonov-Bohm arm, ${\phi }_{\mathit{AB}}$. Furthermore, when a spin-orbit interaction is present on a Kondo quantum dot, we can control the flow of spin by the spin-orbit phase, ${\phi }_{\mathit{SO}}$. The existence of the Kondo peak at the Fermi energy makes it possible to control the flow of both charge and spin in the zero voltage bias limit.

Figure 1

Experimental setup of the quantum dot embedded AB interferometer. Two infinite electron reservoirs (Left, Right) are connected by two arms. The bottom arm contains the embedded QD with real tunneling coefficients $t_L$ and $t_R$. The upper arm is a direct connection between the left and right leads with a complex tunneling coefficient $t_0=\mid t_0\mid e^{i{\phi }_{\mathit{AB}}}$, where ${\phi }_{\mathit{AB}}$ is a phase factor controlled by the magnetic field, $B$. The sign of ${\phi }_{\mathit{AB}}$ is positive for electrons traveling from the left to the right, as the arrow indicates. ${\mathrm{I}}_L$, ${\mathrm{I}}_R$, and ${\mathrm{I}}_{\mathit{\text{Ring}}}$ are the Left, Right, and Ring currents, respectively.

Figure 2

QD spectral function for different magnitudes of $\mid t_0\mid$ at ${\phi }_{\mathit{AB}}=0$. As $\mid t_0\mid$ is increased, the electron becomes more localized on the QD and the distribution of energies shifts away from the Fermi energy.

Figure 3

Noninteracting transmission function for different values of ${\phi }_{\mathit{AB}}$. When ${\phi }_{\mathit{AB}}=(-\frac{\pi }{2},\frac{\pi }{2})$, we see a resonance at the QD energy level. The ${\phi }_{\mathit{AB}}=(-\frac{\pi }{2},\frac{\pi }{2})$ curves coincide; away from these curves, the interference effects become more pronounced and the transmission function becomes asymmetrical. When ${\phi }_{\mathit{AB}}=(0,\pi )$, the interference effects become most pronounced and a very strong antiresonance emerges, which is shifted to the left or right of the QD energy level. In the limit $\mid ϵ\mid \to \infty$, the transmission converges to a finite value, due to the AB arm.

Figure 4

Self-energy expanded to second order in $U$.

Figure 5

Interacting QD spectral function for different magnitudes of $\mid t_0\mid$ when ${\phi }_{\mathit{AB}}=\frac{\pi }{2}$, $U=0.6$, and $\beta =160$. We see that as $\mid t_0\mid$ is increased, the electron becomes more localized on the QD and, as a result, the correlations due to the Coulomb interaction become more pronounced.

Figure 6

QD spectral function and transmission amplitude for $U=0.6$ and $\beta =160$. The ${\phi }_{\mathit{AB}}=\frac{\pi }{2}$ and $\frac{-\pi }{2}$ curves are identical, and at these values, the spectral function is symmetric and the transmission amplitude displays resonant behavior. On the other hand, when ${\phi }_{\mathit{AB}}=0$ and $\pi$, the spectral function is asymmetric and the transmission amplitude has an antiresonance.

Figure 7

Zero bias conductance as a function of ${\phi }_{\mathit{AB}}$ for $t_L=t_R=0.15$, $U=0.5$, and $\beta =160$. The conductance approaches unity when ${\phi }_{\mathit{AB}}=\frac{2n+1}{2}\pi$. Conversely, the conductance approaches a minimum when ${\phi }_{\mathit{AB}}=n\pi$, with integer $n$.

Figure 8

Spin dependent spectral and transmission functions for ${\phi }_{\mathit{SO}}=\frac{\pi }{8}$ and $\frac{\pi }{4}$, $U=0.6$, and $\beta =160$. For ${\phi }_{\mathit{SO}}=\frac{\pi }{4}$, $A_{\uparrow }$ $\left(A_{\downarrow }\right)$ is symmetric (asymmetric), while $T_{\uparrow }$ $\left(T_{\downarrow }\right)$ displays resonance (antiresonance) behavior. In addition, $T_{\uparrow }\left(0\right)\sim 1$ and $T_{\downarrow }\left(0\right)\sim 0$. For ${\phi }_{\mathit{SO}}=\frac{\pi }{8}$, both $A_{\uparrow }$ and $A_{\downarrow }$ are asymmetrical and $A_{\downarrow }$ lies on the curve of $A_{\downarrow }({\phi }_{\mathit{SO}}=\frac{\pi }{4})$. $T_{\uparrow }$ and $T_{\downarrow }$ both display antiresonant phenomena.

Figure 9

SO oscillations for ${\phi }_{\mathit{AB}}=\frac{\pi }{8}$ and $\frac{\pi }{4}$, $U=0.6$, and $\beta =160$. The minima and maxima of the SO oscillations approach 0 and 1, respectively.

Figure 10

Spin-polarized conductance for different AB phases when $U=0.6$ and $\beta =160$. Here, the spin polarization is defined as $\eta =[G_{\uparrow }\left(0\right)-G_{\downarrow }\left(0\right)]∕[G_{\uparrow }\left(0\right)+G_{\downarrow }\left(0\right)]$. We see that the spin polarization may be fully controlled by tuning ${\phi }_{\mathit{AB}}$ and ${\phi }_{\mathit{SO}}$. The ${\phi }_{\mathit{AB}}=0$ and $\frac{\pi }{2}$ curves are identical.