Proximity effect on spin-dependent conductance and thermopower of correlated quantum dots

We study the electric and thermoelectric transport properties of correlated quantum dots coupled to two ferromagnetic leads and one superconducting electrode. Transport through such hybrid devices depends on the interplay of ferromagnetic-contact-induced exchange field, superconducting proximity effect, and correlations leading to the Kondo effect. We consider the limit of large superconducting gap. The system can be then modeled by an effective Hamiltonian with a particle-nonconserving term describing the creation and annihilation of Cooper pairs. By means of the full density-matrix numerical renormalization group method, we analyze the behavior of electrical and thermal conductances, as well as the Seebeck coefficient as a function of temperature, dot level position, and strength of the coupling to the superconductor. We show that the exchange field may be considerably affected by the superconducting proximity effect and is generally a function of Andreev bound-state energies. Increasing the coupling to the superconductor may raise the Kondo temperature and partially restore the exchange-field-split Kondo resonance. The competition between ferromagnetic and superconducting proximity effects is reflected in the corresponding temperature and dot level dependence of both the linear conductance and the (spin) thermopower.

Figure 1

Schematic of the considered system. Quantum dot, with dot level energy $\varepsilon$ and Coulomb correlation $U$, is connected to two ferromagnetic leads and one superconducting lead. ${\Gamma }_L^{\sigma }$ (${\Gamma }_R^{\sigma }$) describes the spin-dependent coupling between the dot and the left (right) ferromagnetic lead and ${\Gamma }_S$ is the coupling to the superconductor. The magnetizations of ferromagnets can form either parallel or antiparallel magnetic configuration, as indicated. There is a temperature gradient $\delta T$ applied between the two ferromagnetic leads.

Figure 2

The energy dependence of the normalized spectral function in the AP magnetic configuration ${\mathcal{A}}^{\mathrm{AP}}$ calculated for $\delta =0$ and different couplings to superconducting lead ${\Gamma }_S$. The parameters are $U=1$, $\Gamma =U/12$, $T=0$, and $p=0.4$.

Figure 3

The normalized spectral function ${\mathcal{A}}^{\mathrm{P}}$ in the P magnetic configuration as a function of energy $\omega$ and level detuning $\delta =\varepsilon +U/2$ for different couplings to superconducting lead: (a) ${\Gamma }_S=0$, (b) ${\Gamma }_S/U=0.2$, (c) ${\Gamma }_S/U=0.3$, (d) ${\Gamma }_S/U=0.4$. The dashed lines show the exchange field $\Delta {\varepsilon }_{\mathrm{exch}}$ obtained from the analytical formula (20). The parameters are the same as in Fig. 2.

Figure 4

The exchange field in the P magnetic configuration for a few values of ${\Gamma }_S$ obtained from Eq. (20) at $T=0$. The other parameters are as in Fig. 2.

Figure 5

The level detuning dependence of the linear conductance in the AP $\left(G^{\mathrm{AP}}\right)$ (a)–(c) and P $\left(G^{\mathrm{P}}\right)$ (d)–(f) magnetic configurations, as well as the resulting TMR (g)–(i) for different couplings to the superconductor and temperatures. The left, middle, and right columns correspond to $T/U={10}^{-4}$, $T/U={10}^{-2}$, and $T/U={10}^{-1}$, respectively. The other parameters are as in Fig. 2.

Figure 6

The temperature dependence of $G^{\mathrm{AP}}$ (a)–(c), $G^{\mathrm{P}}$ (d)–(f), and the TMR (g)–(i) for different couplings ${\Gamma }_S$. The left, middle, and right columns correspond to $\delta =0$, $\delta =U/6$, and $\delta =U/2$, respectively. The other parameters are the same as in Fig. 2.

Figure 7

The thermal conductance $\kappa$ as a function of detuning $\delta$ for the AP (left column) and P (right column) configurations and for different couplings to the superconductor ${\Gamma }_S$. Each row corresponds to different temperature, as indicated. The parameters are the same as in Fig. 2.

Figure 8

The thermopower as a function of $\delta$ for the AP (left column) and P (right column) configurations and for different ${\Gamma }_S$ and $T$, as indicated. The other parameters are as in Fig. 2.

Figure 9

The spin thermopower in the P configuration as a function of $\delta$ for different ${\Gamma }_S$ and for (a) $T/U={10}^{-1}$, (b) $T/U={10}^{-2}$, (c) $T/U={10}^{-3}$, and (d) $T/U={10}^{-4}$. The other parameters are the same as in Fig. 2.

Figure 10

The thermopower in the P configuration as a function of $\delta$ for different ${\Gamma }_S$ and for different temperatures in the case of finite spin accumulation. The parameters are the same as in Fig. 2.

Figure 11

The spin thermopower (a),(b) and thermopower (c),(d) in the P configuration as a function of temperature for different ${\Gamma }_S$ in the case of finite spin accumulation. The left column corresponds to $\delta =U/6$, while the right column to $\delta =U/2$. The parameters are as in Fig. 2.