Superconducting proximity effect and zero-bias anomaly in transport through quantum dots weakly attached to ferromagnetic leads

The Andreev transport through a quantum dot coupled to two external ferromagnetic leads and one superconducting lead is studied theoretically by means of the real-time diagrammatic technique in sequential and cotunneling regimes. We show that the tunnel magnetoresistance (TMR) of the Andreev current displays a nontrivial dependence on the bias voltage and the level detuning and can be described by analytical formulas in the zero-temperature limit. The cotunneling processes lead to a strong modification of the TMR, which is most visible in the Coulomb blockade regime. We find a zero-bias anomaly of the Andreev differential conductance in the parallel configuration, which is associated with a nonequilibrium spin accumulation in the dot triggered by Andreev processes.

Figure 1

Schematic of a quantum dot strongly coupled to an s-wave superconductor and weakly coupled to two ferromagnetic leads. The coupling to the superconductor is described by ${\Gamma }_S$, while the spin-dependent couplings to ferromagnetic leads are denoted ${\Gamma }_L^{\sigma }$ and ${\Gamma }_R^{\sigma }$, respectively. Magnetizations of the ferromagnets can form either a parallel or an antiparallel magnetic configuration, as indicated. The superconducting gap $\Delta$ is assumed to be the largest energy scale in the problem, $\left|\Delta \right|\to \infty$, and the electrochemical potential of the superconducting lead is set to 0, ${\mu }_S=0$. $\varepsilon$ denotes the dot level energy and $U$ is the Coulomb repulsion on the dot.

Figure 2

Absolute value of the total Andreev current for (a) the parallel ($I_S^{\mathrm{P}}$) and (b) the antiparallel ($I_S^{\mathrm{AP}}$) magnetic configuration as a function of the detuning $\delta =2\varepsilon +U$ and applied bias voltage $V$. Dashed lines indicate the position of the respective Andreev bound states [see Eq. (7)] as denoted in (a). Parameters are ${\Gamma }_S=0.4$, $\Gamma =0.01$, and $T=0.02$, with $U\equiv 1$ the energy unit, and $p=0.5$. The current is plotted in units of $I_0=e\Gamma /\hslash$. The same color scale is used in (a) and (b) to enable direct comparison.

Figure 3

Total differential conductance $G_S=d{I}_S/dV$ due to the Andreev current for (a) the parallel ($G_S^{\mathrm{P}}$) and (b) the antiparallel ($G_S^{\mathrm{AP}}$) magnetic configuration as a function of the detuning $\delta =2\varepsilon +U$ and applied bias voltage $V$. Dashed lines indicate the position of the respective Andreev bound states. Parameters are as in Fig. 2.

Figure 4

The tunnel magnetoresistance (TMR) of the Andreev current as a function of the detuning $\delta$ and the bias voltage $V$ calculated by using only (a) sequential tunneling processes [${\mathrm{TMR}}^{\left(1\right)}$] and (b) the first- and second-order tunneling processes. Parameters are the same as in Fig. 2. Dashed lines show the position of Andreev bound states. The same color scale is used in (a) and (b) to facilitate comparison.

Figure 5

(a) The Andreev current in the parallel (solid line) and antiparallel (dashed line) magnetic configurations, (b) the corresponding differential conductance, and (c) the tunnel magnetoresistance (TMR). For comparison in (c) we also show the TMR calculated using only sequential tunneling processes (dashed line). Parameters are the same as in Fig. 2.

Figure 6

The same as Fig. 5, calculated for detuning $\delta =0$.

Figure 7

Bias dependence of the Andreev differential conductance for the parallel and antiparallel magnetic configurations calculated for the Coulomb blockade regime with $\delta =0$. Inset: Relevant occupation probabilities for spin-up and spin-down levels. Parameters are the same as in Fig. 2.

Figure 8

Bias dependence of the Andreev differential conductance in the parallel configuration for different (a) spin polarization $p$ of the ferromagnets, (b) detuning $\delta$, and (c) temperature $T$. Parameters being changed are specified in each panel, while other parameters are the same as in Fig. 2 with $\delta =0$. Arrows indicate the direction of change of the parameters that are tuned in each panel.