Self-consistent microscopic calculations for nonlocal transport through nanoscale superconductors

We implement self-consistent microscopic calculations in order to describe out-of-equilibrium nonlocal transport in normal-metal-superconductor-normal-metal hybrid structures in the presence of a magnetic field and for arbitrary interface transparencies. A four-terminal setup simulating usual experimental situations is described by means of a tight-binding model. We present results for the self-consistent order-parameter and current profiles within the sample. These profiles illustrate a crossover from a quasiequilibrium to a strong nonequilibrium situation when increasing the interface transparencies and the applied voltages. We analyze in detail the behavior of the nonlocal conductance in these two different regimes. While in quasiequilibrium conditions this can be expressed as the difference between elastic cotunneling and crossed Andreev transmission coefficients, in a general situation additional contributions due to the voltage dependence of the self-consistent order parameter have to be taken into account. The present results provide a first step toward a self-consistent theory of nonlocal transport including nonequilibrium effects and describe qualitatively a recent experiment [P. Cadden-Zimansky and V. Chandrasekhar, Phys. Rev. Lett. 97, 237003 (2006)].

Figure 1

(Color online) Schematic representation of the electrical circuit used in a nonlocal conductance experiment. The lowest-order processes are shown in the figure: (a) local Andreev reflection, (b) crossed Andreev reflection, and (c) EC.

Figure 2

(Color online) Schematic representation of the tight-binding model used in our simulations and of the electrical circuit (green lines). The self-consistent gap is evaluated on a square lattice with $M=R_y/a_0$ transverse channels and a length $R_x=N{a}_0$, with $a_0$ the lattice spacing. Normal electrodes $N_b$ $\left(N_a\right)$ are connected to the square lattice at left (right). Superconducting reservoirs (with zero superconducting phase) are connected on top and bottom. The normal and superconducting electrodes are modeled by a collection of one-dimensional channels.

Figure 3

(Color online) Self-consistent gap profile at zero voltage for a system size $R_x/a_0=R_y/a_0=20$ and two different values of the normal transparency.

Figure 4

(Color online) Order-parameter and current profiles for a sample size $R_x/a_0=12$ and $R_y/a_0=20$ under an applied voltage $V_b$. The upper panels show the profiles of the self-consistent gap. The gap is normalized to its valued ${\Delta }_0$ in the superconducting reservoirs. The voltage is indicated by an arrow on the vertical axis. The middle panels show the phase profile and the lower ones show the corresponding current maps. The size of the arrows in the three lower panels is proportional to the value of the local current, with the same scaling factor for the three panels. The values of the parameters $T_N$ and $V_b$ are indicated on top of each panel.

Figure 5

(Color online) Same as Fig. 4 for sample dimensions $R_x/a_0=30$ and $R_y/a_0=20$. The scaling factor between current and size of arrow is not the same as in Figs. 4g, 4h, 4i, and it is different for panels e and f.

Figure 6

(Color online) (a) Surface map of the self-consistent phase under magnetic field, for sample dimensions $R_x/a_0=12$ and $R_y/a_0=20$. The corresponding current map is shown in (b). The flux is $\varphi /{\varphi }_0=1$. The scaling factor between current and size of arrow is the same as in Figs. 4g, 4h, 4i.

Figure 7

(Color online) Current map for increasing values of the magnetic flux $\varphi /{\varphi }_0$ and for $T_N=0.64$, $R_x/a_0=12$, and $R_y/a_0=20$. The applied voltage $V_b$ is set to $0.9{\Delta }_0$. An arrow with the same length corresponds to the same current for all panels. The scaling factor between current and size of arrow is the same as in Figs. 4g, 4h, 4i.

Figure 8

(Color online) Same as in Fig. 7 for $T_N=0.64$, $e{V}_b=0.9{\Delta }_0$, $R_x/a_0=30$, and $R_y/a_0=20$.

Figure 9

(Color online) Variation in the quantities $F_{\text{EC}}\left(T_N\right)$ and $F_{\text{CAR}}\left(T_N\right)$ defined in text, as a function of the normal transparency $T_N$. A predominance of EC over CAR is obtained for high values of interface transparencies. The chosen system dimensions are $R_x=28a_0$ and $R_y=20a_0$.

Figure 10

(Color online) Dependence of the EC transmission coefficient on the length $R_x=N{a}_0$ of the superconductor. Panel (a) corresponds to low transparency and two different values of the transverse dimension $R_y=15$ and $25a_0$. The sample dimension $R_x$ on panel (a) is normalized to the value ${\xi }_{\text{eff}}\left(0\right)$ of the coherence length for $R_y=15a_0$. Panel (b) corresponds to high transparency and two different values of the magnetic flux. The sample dimension on panel (b) is normalized to the value ${\xi }_{\text{eff}}\left(0\right)$ of the coherence length for $\varphi /{\varphi }_0=0$. Out-of-equilibrium effects are negligible for both panels (see insets of Fig. 12).

Figure 11

(Color online) Dependence of the transmission coefficients $T_{\text{EC}}\left(\varphi \right)/T_{\text{EC}}\left(\varphi =0\right)$ and $T_{\text{CAR}}\left(\varphi \right)/T_{\text{EC}}\left(\varphi =0\right)$ on the magnetic flux $\varphi /{\varphi }_0$. We have chosen $T_N=0.64$, $R_x/a_0=20$, and $R_y/a_0=14$. The broken and dotted lines indicate the quadratic fit of Eq. (19).

Figure 12

(Color online) Voltage dependence of the nonlocal conductance (upper panels) and of nonlocal resistance (lower panels) normalized to their value at zero bias in zero applied magnetic field. The values of $T_N$ and $\varphi$ are indicated on each panel. The insets of the upper panels illustrate the comparison between the full calculation ${\mathcal{G}}_{a,b}=\partial I_a/\partial V_b$ (solid line) and the expression approximation ${\mathcal{G}}_{a,b}=e^2/h\left(T_{\text{CAR}}-T_{\text{EC}}\right)$ with the transmission coefficients defined by Eqs. (15, 16) (dashed line). The nonlocal conductance ${\mathcal{G}}_{a,b}$ in the insets is given in units of $e^2/h$.

Figure 13

(Color online) Voltage dependence of the local and nonlocal conductances (upper panels) and of the nonlocal resistance (lower panels) for an asymmetric junction with $T_N=1$ at the b interface and $T_N=0.1$ at the a side, for $R_y=40a_0$, and two values of $R_x:13a_0$ [panels (a) and (c)] and $15a_0$ [panels (b) and (d)].

Figure 14

(Color online) Dependence of the normalized EC and CAR transmission coefficients $T\left(R_x\right)/T\left(0\right)$ on $2R_x/{\xi }_0$, where $R_x$ is the dimension of the superconductor along $x$ axis and ${\xi }_0$ is the bulk coherence length. The applied voltage is small compared to the gap $\left(e{V}_b/{\Delta }_0=0.02\right)$. The coherence length is larger than for the corresponding parameters that in the presence of superconducting leads. Panel (a) corresponds to $R_y=5a_0$ and panel (b) to $R_y=25a_0$. The symbols correspond to EC and the bold red lines to CAR.