Crossed Andreev Reflection and Charge Imbalance in Diffusive Normal-Superconducting-Normal Structures

We formulate a microscopic theory of nonlocal electron transport in three-terminal diffusive normal-superconducting-normal (NSN) structures with arbitrary interface transmissions. At low energies $\epsilon$, we predict strong enhancement of nonlocal spectral conductance $g_{12}\propto 1/\epsilon$ due to quantum interference of electrons in disordered N terminals. In contrast, nonlocal resistance $R_{12}$ remains smooth at small $\epsilon$ and, furthermore, is found to depend neither on parameters of normal-superconducting interfaces nor on those of N terminals. At higher temperatures, $R_{12}$ exhibits a peak caused by the trade-off between charge imbalance and Andreev reflection. Our results are in a good agreement with recent experimental observations and can be used for quantitative analysis of future experiments.

Figure 1

NSN structure under consideration.

Figure 2

Nonlocal spectral conductance $g_{12}(\epsilon )$ (normalized by $G_0=r_{x_1}{r}_{L-x_2}/r_L{R}_1{R}_2$) for diffusive NSN structures. We set $L=10\sqrt{2}{\xi }_S(0)$, $x_1=0.45L$, and $x_2=0.55L$. (a) The case of two identical barriers with resistances $R_1=R_2=\pi /e^2{N}_{\mathrm{ch}}\tau$ ($N_{\mathrm{ch}}$ is the number of channels and $\tau$ is the barrier transmission) and for $r_{{\mathrm{N}}_1}=r_{{\mathrm{N}}_2}=0$. (b) The case of two tunnel barriers with $T_{1,n}$, $T_{2,n}\ll 1$ and for $L_1=L_2=L_{\mathrm{N}}$, $r_{{\mathrm{N}}_1}=r_{{\mathrm{N}}_2}$, $R_1=R_2$, $r_{{\mathrm{N}}_1}{\xi }_{\mathrm{N}}/L_1{R}_1=0.0025$, and $\sqrt{2}{r}_L{\xi }_S/L{R}_1=0.005$.

Figure 3

(a) Nonlocal resistance $R_{12}(T)$ (18), normalized by $R_0=r_{x_1}{r}_{L-x_2}/r_L$, at different barrier transmissions $\tau$ and at ${\tau }_{Q^{*}}$, ${\tau }_{\phi }\to \infty$. We chose $R_1=R_2=\pi /e^2{N}_{\mathrm{ch}}\tau =h/20e^2$ and set $L=20\sqrt{2}{\xi }_S(0)$, $x_1=9\sqrt{2}{\xi }_S(0)$, $x_2=11\sqrt{2}{\xi }_S(0)$, $r_L=200$ $\Omega$, and $r_{{\mathrm{N}}_1}=r_{{\mathrm{N}}_2}=0$. (b) $R_{12}(T)/R_0$ at different $|x_2-x_1|$. Other parameters are the same as in Fig. 2b.