Nonlocal conductance via overlapping Andreev bound states in ferromagnet-superconductor heterostructures

In a setup where two ferromagnetic electrodes are attached to a superconductor, Andreev bound states are induced at both ferromagnet/superconductor (FM/SC) interfaces. We study how these states propagate through the SC and interact with each other. As a result of this interaction, the energetic positions of the Andreev states are not anymore determined solely by the magnetic properties of a single interface but also depend on the interface distance and the relative magnetization orientation of the FM contacts. We discuss how these bound states show up as distinct peaks in the nonlocal conductance signal and lead to marked asymmetries with respect to the applied voltage. We relate our results to nonlocal crossed Andreev and elastic cotunneling processes.

Figure 1

(Color online) Sketch of the considered FM/SC/FM setup with two tunneling point contacts. A nonlocal scattering event which involves both contacts is indicated.

Figure 2

(Color online) Spin-up (positive axis) and spin-down (negative axis) DOS at different positions $x$ for parallel (left column) and antiparallel magnetization (right column). [(a) and (b)]: $\rho =0.9$, $L=\xi$, and ${\vartheta }_{\text{R}}=\pm {\vartheta }_{\text{L}}=0.4\pi$. [(c) and (d)]: $\rho =0.9$, $L=2\xi$,and ${\vartheta }_{\text{L}}\ne {\vartheta }_{\text{R}}$ with ${\vartheta }_{\text{L}}=\pm 0.4\pi$ and ${\vartheta }_{\text{R}}=0.7\pi$.

Figure 3

(Color online) Spin-up DOS at the right interface vs $L$ (upper panel) and $\vartheta$ (lower panel), both for [(a)–(c)] P and [(b)–(d)] AP magnetization. Dashed lines are analytical approximations (see text). Inset: propagation of the split ABS between the interfaces. Parameters in (a) and (b): $\rho =0.9$ and $\vartheta =0.7\pi$. In (c) and (d): $\rho =0.9$ and $L=2.0\xi$. Inset: $\vartheta =0.7\pi$ and $L=2.0\xi$.

Figure 4

(Color online) CAR and EC contributions for [(a)–(c)] P and [(d)–(f)] AP magnetization of the FM contacts. Transmission coefficients: [(a)–(c)]: $t_{\sigma }^{\text{L}}=t_{\sigma }^{\text{R}}$, [(d)–(f)]: $t_{\sigma }^{\text{L}}=t_{\overline{\sigma }}^{\text{R}}$ with ${\left|t_{\uparrow }^{\text{R}}\right|}^2=0.14$ and ${\left|t_{\downarrow }^{\text{R}}\right|}^2=0.07$. For other parameters see legends of (b) and (e).

Figure 5

(Color online) Sketch of the spin-up (positive axis) and spin-down (negative axis) Andreev states at the FM/SC interfaces for (a) P and (b) AP orientation of the contact magnetizations. CAR and EC processes are shown and expressions for obtaining their relative size are given in the table.