Coulomb blockade of a three-terminal quantum dot

We study an interacting single-level quantum dot weakly coupled to three electrodes. When two electrodes are biased by voltages with opposite polarities, while keeping the third lead (the stem) grounded, the current through the stem is a measure of electron-hole asymmetry of the dot. In this setup, we calculate the stem current for both metallic and ferromagnetic (collinearly polarized) leads and discuss how the three-terminal device gives additional information compared to the usual two-terminal setup. We calculate both the sequential and cotunneling contributions for the currents. For the latter part, we include a regularization procedure for the cotunneling current, which enables us to also describe the behavior at the charge degeneracy points.

Figure 1

The three-terminal device. The occupation number is controlled via the gate by applying the voltage $V_g$. The stem chemical potential, ${\mu }_s=0$, serves as the reference level. The left and right leads are biased in push-pull manner with ${\mu }_l=eV$ and ${\mu }_r=-eV$.

Figure 2

(Color online) The stem current (solid, red) being the sum of the sequential part (dotted, blue) and the cotunneling (dashed, green). The sequential contribution gives sufficient qualitative description. For this plot, leads are unpolarized with ${\Gamma }_l={\Gamma }_r={\Gamma }_s=\Gamma ∕3$, $U=20\Gamma$, $B=5\Gamma$, $k_{B}T=2\Gamma$, and $V∕\Gamma =1$.

Figure 3

The comparison of a cotunneling event for the dot occupied by a spin-up electron with the spin-flip process (on the right) and without (on the left). In case of inelastic processes, the final state of the dot and the leads is the same for both paths resulting in the interference.

Figure 4

The stem current in function of the bias voltage $V$ and the gate voltage ${ϵ}_0$. The leads are unpolarized and equally coupled to the dot, with coupling’s strength $\Gamma ∕3$. The picture without the magnetic field (on the left) differs from this with field $B=5\Gamma$ (on the right). The interaction energy $U=20\Gamma$ and temperature $k_{B}T=\Gamma$. The gray scale in the far right describes the stem current value in units $e\Gamma ∕\hslash$.

Figure 5

(Color online) The stem current for different values of magnetic field $B$ is plotted. Interestingly, there is a sign change in the vicinity of $V=0$ for $B={ϵ}_0$. See text for explanation. The gate voltage is ${ϵ}_0=6\Gamma$. Other parameters are as in the previous figure.

Figure 6

The stem current for different polarizations of leads and magnetic field $B=5\Gamma$. The central branch is always unpolarized. In the upper row, both left and right leads are spin-up polarized, $P_l=P_r=1$ (the left plot), or spin-down polarized, $P_l=P_r=-1$ (the right plot). Due to nonzero magnetic field, the spin symmetry is broken and the stem current is no longer an odd function of the gate voltage with respect to ${ϵ}_0=-U∕2$. In the lower row, polarization of the left and right leads is antiparallel ($P_l=1$, $P_r=-1$ on the left and $P_l=-1$, $P_r=1$ on the right). In this case, the current symmetry in $V$ is violated because the dot couples to the left and right leads asymmetrically. The other parameters are as in the previous figure.

Figure 7

(Color online) The strict result for the stem current (thick, black) compared to the perturbative result: sequential (dotted, blue), cotunneling (dashed, green), and their sum (thin, red). Parameters are as in Fig. 2 (except for $U=0$).