Nonequilibrium electron transport in a hybrid superconductor–normal metal entangler in a dissipative environment

We consider a three-terminal Cooper-pair splitting device with a superconducting electrode tunnel coupled to two normal metal electrodes. We employ the Nambu-Gor'kov and Schwinger-Keldysh formalisms to describe the nonequilibrium transport properties of the device for arbitrary transmissions of the barriers and for a general electromagnetic environment. We derive the analytic expressions for the current and the nonlocal differential conductance, and analyze the limits of clean and dirty superconductivity.

Figure 1

Schematic geometry of the device under consideration: a bulk BCS superconductor is coupled to two normal metal leads. Voltages between the leads (with impedances $Z_a$ and $Z_b)$ and the superconductor are $V_a$ and $V_b$. We consider the clean and dirty limits of the superconductor. In the dirty limit, we average over disorder the products of retarded and advanced Green's functions corresponding to crossed-Andreev reflection (pair splitting) and elastic cotunneling processes (between the normal metal electrodes); these Green's functions are represented by the red and blue lines inside the superconductor.

Figure 2

An example of a diagram obtained in the expansion of ${\check{\mathbf{G}}}_{c\tilde{a}11}$. The upper (lower) horizontal line corresponds to the “$-$” (“$+$”) branch of the Schwinger-Keldysh contour. The empty (filled) triangles represent the creation (annihilation) operators in the normal metal electrode $a$; analogously squares represent the operators for the electrode $b$. Light (dark) blue dots correspond to the creation (annihilation) operators for the electrons in the supeconducting electrode. Wick contractions are represented by solid lines connecting the operators of the same normal metal electrode, and by dashed lines connecting superconducting electrode operators. Wiggly lines (red) represent phase-phase correlation functions corresponding to the electromagnetic environment. While we show only a couple of such lines, every pair of shapes (triangles or squares) corresponding to the same normal metal electrode is joined by such lines.

Figure 3

Diagrammatic representation of the coordinate-energy space contribution of the last term in (14) for $\mu =b$. The diagram on the left depicts elastic cotunneling $\left(i=1\right)$, while the one on the right depicts crossed Andreev reflection $\left(i=2\right)$. The solid lines stand for the bare lesser and greater Green's functions for electrons in the normal metal electrodes, wiggly lines represent phase correlators, and dotted (dash-dotted) lines represent full retarded (advanced) Green's functions in the superconducting electrode. These are normal and anomalous Green's functions in the case of elastic cotunneling and crossed Andreev reflection, respectively. In the case of the diffusive superconducting electrode (Sec. 2b), we consider impurity scattering (indicated on the diagrams by dashed lines with crosses). The probability of quantum diffusion is obtained by iterating to infinity a series of these scattering processes.

Figure 4

The numerical results from formula (14) for the elastic cotunneling current, $I_{EC}^{\left(a\right)}$ (top curve) and the crossed Andreev current, $I_{CAR}^{\left(a\right)}$ (middle curve) as functions of the voltage across the junction $a$; the analytic result for these currents given by the formula (16), valid in the low voltage limit, is shown as a dashed curve. Parameter values are $e{V}_b=0,k_F\delta r=10,z=0.6,{\mathrm{\Gamma }}^{\left(a\right)}={\mathrm{\Gamma }}^{\left(b\right)}=0.1,\mathrm{\Delta }/E_c=1.25,\mathrm{\Delta }/E_F=2\times {10}^{-5}$.

Figure 5

Nonlocal differential conductance ${\mathcal{G}}_{ab}$, in units ${\mathcal{G}}_0$, as a function of voltage $e{V}_b$, for $\delta r\ll {\xi }_d$. In three dimensions ${\mathcal{G}}_0=e^2/2h{\left(k_F\delta r\right)}^2$ in the ballistic case and ${\mathcal{G}}_0=3e^2/2h{k}_F^2\delta r{l}_e$ in the diffusive case. By decreasing the barrier transmission the curves get closer to the horizontal axis, $T_{\mathrm{eff}}^{\left(a\right)}=T_{\mathrm{eff}}^{\left(b\right)}=1,0.9,0.75,0.55$ (main plot), and 0.35,0.25,0.15 (inset).

Figure 6

Nonlocal differential conductance ${\mathcal{G}}_{ab}$, in units ${\mathcal{G}}_0=2e^2\overline{\mathcal{P}}(\delta r,2\mathrm{\Delta }/\hslash )/h^2{\nu }_c$, as a function of the effective transmission coefficient $T_{\mathrm{eff}}^{\left(a\right)}=T_{\mathrm{eff}}^{\left(b\right)}=T_{\mathrm{eff}}$, for $z=0.001$ (top plot), $z=0.06$ (middle plot), and $z=0.6$ (bottom plot); $e{V}_a=0.2\mathrm{\Delta },e{V}_b=0.1\mathrm{\Delta }$. In the top plot, the blue line is the result from Ref. [24]. Dotted lines represent contributions from the crossed-Andreev reflection, dash-dotted lines those of the elastic cotunneling, and solid (red) lines the total nonlocal differential conductance. The inset in the middle plot shows a zoom-in in the region where ${\mathcal{G}}_{ab}$ changes sign, and the inset in the bottom plot shows the crossed-Andreev reflection and elastic cotunneling contributions; in both insets the dashed lines show the corresponding analytic results valid in the limit of low interface transmissions.