Full counting statistics of crossed Andreev reflection

We calculate the full transport counting statistics in a three-terminal tunnel device with one superconducting source and two normal-metal or ferromagnet drains. We obtain the transport probability distribution from direct Andreev reflection, crossed Andreev reflection, and electron transfer which reveals how these processes’ statistics are determined by the device conductances. The cross-correlation noise is a result of competing contributions from crossed Andreev reflection and electron transfer, as well as antibunching due to the Pauli exclusion principle. For spin-active tunnel barriers that spin polarize the electron flow, crossed Andreev reflection and electron transfer statistics exhibit different dependencies on the magnetization configuration and can be controlled by relative magnetization directions and voltage bias.

Figure 1

Circuit theory representation of the considered beam splitter where supercurrent flows from a superconducting reservoir $\left(\mathrm{S}\right)$ into normal-metal drains $\left({\mathrm{N}}_n\right)$. Tunnel barriers between cavity (c) and the drains can be spin active and are characterized by the conductances $g_n$ and spin polarizations $g_{\text{MR}n}/g_n$.

Figure 2

(Color online) Transport processes in the three-terminal device when $e{V}_1=0$. (a) Crossed Andreev reflection: a Cooper pair from $\mathrm{S}$ is converted into an electron-hole pair in $\mathrm{c}$ by Andreev reflection, and the electron with energy $+E$ is transferred into ${\mathrm{N}}_1$ and the hole with energy $-E$ is transferred into ${\mathrm{N}}_2$. Tunnel barriers between the reservoirs may be spin active and are described by magnetization vectors $\mathbf{m}$ that in this paper are considered collinear. (b) Electron transfer: a particle from ${\mathrm{N}}_1$ tunnels through the cavity $\mathrm{c}$ into ${\mathrm{N}}_2$. The density of states in the cavity c is suppressed due to the proximity effect from the superconducting terminal.

Figure 3

(Color online) Probability distribution for transport of $N_1$ electrons into terminal $N_1$, $P\left(N_1;t_0\right)$ [Eq. (19)]. We show distributions for two different values of the parameter $g_S/g=5.00$ (red [gray] solid impulses) and 0.05 (blue [dark gray] dotted impulses). We have chosen the parameter $\alpha =g_1{g}_2{V}_2{t}_0/eg=20$, which gives the mean value ${\overline{N}}_1$ for small $g_S/g$; see Eq. (20).