Multiple Andreev reflections in a quantum dot coupled to superconducting leads: Effect of spin-orbit coupling

We study the out of equilibrium current through a multilevel quantum dot contacted to two superconducting leads and in the presence of Rashba and Dresselhaus spin-orbit couplings, in the regime of strong dot-lead coupling. The multiple Andreev reflection (MAR) subgap peaks in the current-voltage characteristics are found to be modified (but not suppressed) by the spin-orbit interaction in a way that it strongly depends on the shape of the dot confining potential. In a perfectly isotropic dot the MAR peaks are enhanced when the strength ${\alpha }_R$ and ${\alpha }_D$ of Rashba and Dresselhaus terms are equal. When the anisotropy of the dot confining potential increases, the dependence of the subgap structure on the spin-orbit angle $\theta =\text{arctan}\left({\alpha }_D/{\alpha }_R\right)$ decreases. Furthermore, when an in-plane magnetic field is applied to a strongly anisotropic dot, the peaks of the nonlinear conductance oscillate as a function of the magnetic-field angle and the location of the maxima and minima allows for a straightforward read-out of the spin-orbit angle $\theta$.

Figure 1

(Color online) Scheme of the setup under investigation: a quantum dot is coupled to two superconducting electrodes biased by a voltage $V$.

Figure 2

(Color online) Sketch of the dot confining potential [Eq. (3)] for different values of the frequencies in the $x$ and $y$ directions. (a) $\hslash {\omega }_{x,y}⪢\left|\Delta \right|$: only one level takes part to transport; (b) $\hslash {\omega }_x=\hslash {\omega }_y\simeq \left|\Delta \right|$: multilevel isotropic dot; (c) $\hslash {\omega }_x\simeq \left|\Delta \right|$ and $\hslash {\omega }_y⪢\left|\Delta \right|$: multilevel strongly anisotropic dot.

Figure 3

(Color online) Pictorial scheme of multiple Andreev reflections in a multilevel quantum dot. The left and right parts represent the energy spectrum of the superconducting leads characterized by a gap $\left|\Delta \right|$, whereas the horizontal lines in the central part of the figure describe the dot levels. The subgap MAR processes related to $\text{eV}/2\left|\Delta \right|=1/n$, with $n=3,5,7$ are depicted.

Figure 4

(Color online) Current (units of $2e\left|\Delta \right|/h$) as a function of the source-drain bias for a three-level quantum dot with isotropic confining potential $\left(\hslash {\omega }_x=\hslash {\omega }_y=20\left|\Delta \right|\right)$ and linewidth ${\Gamma }_{\pm ,i}=\left|\Delta \right|/4\left(i=1,2,3\right)$. The solid curve refers to the case of spin-orbit coupling $\alpha ={10}^{-11}\text{eV}\text{m}\left(\gamma =0.174\right)$ and spin-orbit angle $\theta =\pi /4$. The dashed curve describes the case of vanishing spin-orbit coupling $\alpha =0$. The spin-orbit coupling does not suppress the MAR subgap pattern; it modifies the location and the number of the peaks.

Figure 5

(Color online) Current-voltage characteristics for a three-level quantum dot with isotropic confining potential $\left(\hslash {\omega }_x=\hslash {\omega }_y=4\left|\Delta \right|\right)$ and linewidths ${\Gamma }_{\pm ,i}=\left|\Delta \right|$ $\left(i=1,2,3\right)$ in the presence of a spin-orbit coupling constant $\gamma =0.5$ [see Eq. (31)] and for different values of the Rashba/Dresselhaus angle $\theta$. The solid curve refers to $\theta =\pi /4$, i.e., equal weight of Rashba and Dresselhaus terms, whereas the dashed (dotted) curve refers to $\theta =0\left(\theta =\pi /2\right)$, corresponding to purely Rashba (purely Dresselhaus) coupling. The mixed coupling enhances the MAR peaks, whereas the two other cases turn out to coincide due to symmetry reasons (see text).

Figure 6

(Color online) Current (units of $2e\left|\Delta \right|/h$) as a function of the source-drain bias, for a three-level dot in an anisotropic confining potential $\left({\omega }_y=16{\omega }_x\right)$ with dot-lead linewidths ${\Gamma }_{\pm ,i}=\left|\Delta \right|$ $\left(i=1,2,3\right)$ and spin-orbit coupling constant $\gamma =0.5$ [see Eq. (31)] for different values of the Rashba/Dresselhaus angle $\theta$. The solid curve refers to $\theta =\pi /4$, i.e., equal weight of Rashba and Dresselhaus terms, whereas the dashed (dotted) curve refers to $\theta =0$ $\left(\theta =\pi /2\right)$, corresponding to purely Rashba (purely Dresselhaus) coupling. With respect to the isotropic case, the dependence on $\theta$ is much weaker in the presence of an anisotropic confining potential.

Figure 7

(Color online) Effects of the magnetic field on the MAR subgap structure. The case of an isotropic three-level dot with $E_0=0$, level spacing $\hslash {\omega }_{x,y}=4\left|\Delta \right|$, spin-orbit parameters $\gamma =0.5$ and $\theta =\pi /4$, and dot-lead linewidths ${\Gamma }_{\pm ,i}=\left|\Delta \right|\left(i=1,2,3\right)$. The solid curve describes the case without magnetic field $\left(B=0\right)$, whereas the dashed and dotted curves refer to an applied magnetic field along the $x$ axis with intensity $B=0.5$ and $B=1$, respectively. The presence of a magnetic field suppresses the current at low bias and the MAR peaks.

Figure 8

(Color online) Behavior of (some) dot energy levels as a function of the angle ${\varphi }_H$ of the in-plane magnetic field with dimensionless intensity $B=0.5$ [see Eq. (35)] and for different values of the spin-orbit Rashba/Dresselhaus angle $\theta$ (solid curves refer to $\theta =\pi /4$ and dashed curves to $\theta =\pi /20$). (a) Case of an isotropic three-level dot with $\hslash {\omega }_x=\hslash {\omega }_y=\left|\Delta \right|/2$; the maxima and minima of the dot level oscillations are located at ${\varphi }_H^{\ast }=\pi \left(2n+1\right)/4$ $\left(n=0,1,2,3\right)$ and are independent of the spin-orbit angle $\theta$, which determines only the oscillation amplitudes. (b) Case of a strongly anisotropic three-level dot with $\hslash {\omega }_x=\left|\Delta \right|/2$ and $\hslash {\omega }_y/\left|\Delta \right|\to \infty$. The maxima and minima of the dot level oscillations are located at the spin-orbit angle ${\varphi }_H^{\ast }=\theta \pm n\pi /2\left(n=0,1,2,3\right)$. Vertical dotted lines are guide for the eye.

Figure 9

(Color online) Nonlinear conductance $G=dI/dV$ (in units of $2e^2/h$) as a function of the applied bias $V$ (units of $\left|\Delta \right|/e$) and of the angle ${\varphi }_H$ of the in-plane magnetic field with intensity $B=0.7$ for a strongly anisotropic quantum dot ($\hslash {\omega }_x=\left|\Delta \right|$ and $\hslash {\omega }_y/\left|\Delta \right|\to \infty$) with two levels and dot-lead linewidths ${\Gamma }_{\pm ,1,2}=\left|\Delta \right|/2$. (a) Without spin-orbit coupling; (b) in the presence of a spin-orbit coupling with parameters $\gamma =0.3$ and angle $\theta =0.3\pi$. The spin-orbit coupling yields oscillations of the conductance in ${\varphi }_H$ with minima and maxima given by the spin-orbit angle $\theta$ and $\theta -\pi /2$.