Probing the exchange field of a quantum-dot spin valve by a superconducting lead

Electrons in a quantum-dot spin valve, consisting of a single-level quantum dot coupled to two ferromagnetic leads with magnetizations pointing in arbitrary directions, experience an exchange field that is induced on the dot by the interplay of Coulomb interaction and quantum fluctuations. We show that a third, superconducting lead with large superconducting gap attached to the dot probes this exchange field very sensitively. In particular, we find striking signatures of the exchange field in the symmetric component of the supercurrent with respect to the bias voltage applied between the ferromagnets already for small values of the ferromagnets’ spin polarization.

Figure 1

(Color online) Schematic model of a quantum-dot spin valve with an additional superconducting electrode attached. A single-level quantum dot with excitation energies $\epsilon$ and $\epsilon +U$ is coupled to noncollinearly magnetized ferromagnets and a superconductor via tunnel barriers.

Figure 2

(Color online) Amplitude of the exchange field ${\mathbf{B}}_r\cdot {\mathbf{n}}_r$ in units of $p_r{\Gamma }_r$ for $k_{\text{B}}T=0.01U$, ${\Gamma }_{\text{S}}=0.4U$ as a function of the chemical potential ${\mu }_r$ and detuning $\delta$. The peaks and dips map out the Andreev bound states whose energies are indicated by dotted lines.

Figure 3

(Color online) Transport processes in the proximized quantum-dot spin valve. In the first process, the $|-⟩$ state is projected onto the $|0⟩$ state, pushing a Cooper pair into the superconductor as indicated by the double arrow. Then an electron tunnels in from the left lead, leaving the dot in the singly occupied state. Similarly, in the second process, an electron leaves the singly occupied dot to the right lead. Then, a Cooper pair enters from the superconductor to bring the dot in the state $|-⟩$. As both processes probe the empty contribution to $|-⟩$, their rates are proportional to $1+\delta /\left(2{\epsilon }_{\text{A}}\right)$. Similarly, the other two processes probe the doubly occupied component of $|-⟩$ such that their rates are proportional to $1-\delta /\left(2{\epsilon }_{\text{A}}\right)$.

Figure 4

(Color online) Current into the superconductor for a symmetric coupling, ${\Gamma }_{\text{L}}={\Gamma }_{\text{R}}$, ${\mu }_{\text{L}}=-{\mu }_{\text{R}}\equiv \mu$ and perpendicularly magnetized ferromagnets, $\varphi =\pi /2$. In the upper panel, we have $p=0.3$, while in the lower panel $p=0.95$. The solid (black) curves take into account the exchange field, it is neglected in the dashed (red) curves. Other parameters are $\delta =0.2U$, ${\Gamma }_{\text{S}}=0.4U$, and $k_{\text{B}}T=0.01U$.

Figure 5

(Color online) When the third lead is a normal metal instead of a superconductor, a finite current flows for any magnetic configuration, thereby completely obscuring the exchange field effects for small polarizations. Parameters are $\delta =0.4U$, ${\Gamma }_r={\Gamma }_{\text{N}}$, $p=0.3$, and $k_{\text{B}}T=0.01U$.

Figure 6

(Color online) Influence of spin relaxation on the supercurrent in the noncollinear configuration. As the spin accumulation is reduced, the supercurrent is decreased. However, the exchange field effects still prevail. Polarization is $p=0.95$, other parameters as in Fig. 3.

Figure 7

(Color online) Symmetrized and antisymmetrized current into the superconductor as a function of bias voltage for different magnetic configurations and small ($p=0.3$, upper panel) and large ($p=0.95$, lower panel) polarizations. The asymmetry is $a=0.05$, other parameters as in Fig. 4.