Density of states and supercurrent in diffusive SNS junctions: Roles of nonideal interfaces and spin-flip scattering

We present a theoretical study of the density of states and supercurrent in diffusive superconductor-normal metal-superconductor (SNS) junctions. In particular, we study the influence on these two equilibrium properties of both an arbitrary transparency of the SN interfaces and the presence of spin-flip scattering in the normal wire. We show that the minigap that is present in the spectrum of the diffusive wire is very sensitive to the interface transmission. More importantly, we show that at arbitrary transparency the minigap replaces the Thouless energy as the relevant energy scale for the proximity effect, determining, for instance, the temperature dependence of the critical current. We also study in detail how the critical current is suppressed by the effect of spin-flip scattering, which can be due either to magnetic impurities or, under certain circumstances, to an external magnetic field. Our analysis based on the quasiclassical theory of diffusive superconductors can be very valuable in establishing quantitative comparisons between experiment and theory.

Figure 1

(Color online) Density of states of a SNS junction as a function of energy in the middle of a wire of length $L=2\xi$ without spin-flip scattering $({\Gamma }_{\mathit{sf}}=0)$. The SN interfaces are assumed to be identical and there is no phase difference between the S electrodes. In panel (a), the different curves correspond to different values of the ratio $r=G_{\mathrm{N}}∕G_{\mathrm{B}}$ and a transmission $\tau =1$, while in panel (b), they correspond to different values of the transmission $\tau$ for a ratio $r=1.0$.

Figure 2

(Color online) Minigap ${\Delta }_g$ of a SNS junction as a function of the length of the N wire for different values of $r=G_{\mathrm{N}}∕G_{\mathrm{B}}$ and ${\Gamma }_{\mathit{sf}}=0$. The contact is assumed to be symmetric and the transmission is set to $\tau =1$. Notice that is ${\Delta }_g$ is normalized by the Thouless energy ${ϵ}_T$.

Figure 3

(Color online) Density of states of a SNS junction as a function of energy in the middle of a wire of length $L=2\xi$ $({\Gamma }_{\mathit{sf}}=0)$ for different values of the superconducting phase difference $\varphi$. The interfaces are identical and characterized by a ratio $r=G_{\mathrm{N}}∕G_{\mathrm{B}}=0.0$ in panel (a) and $r=1.0$ in panel (b). In both cases, $\tau =1$.

Figure 4

(Color online) (a) Density of states of a SNS junction as a function of energy in the middle of a wire of length $L=10\xi$ for $r=0$ and $\tau =1.0$. The different curves correspond to different values of the spin-flip scattering rate ${\Gamma }_{\mathit{sf}}$. (b) The same as (a) but for $r=1.0$. Notice that both the energy and the ${\Gamma }_{\mathit{sf}}$ are in units of the Thouless energy.

Figure 5

(Color online) Zero-temperature supercurrent-phase relation of a diffusive SNS junction for $L=8\xi$ $(\Delta =64{ϵ}_T)$ for different values of the ratio $r=G_{\mathrm{N}}∕G_{\mathrm{B}}$ and $\tau =1.0$. In panel (a), we show the results for the product $eRI∕\Delta$, where $R$ is the normal-state resistance, while in panel (b), we have normalized the different curves by the critical current $I_C$.

Figure 6

(Color online) (a) Critical current of a diffusive SNS junction as a function of the temperature for $L=8\xi$ and different values of the ratio $r=G_{\mathrm{N}}∕G_{\mathrm{B}}$ and $\tau =1.0$. (b) The same as (a) but normalized by the critical current at zero temperature.

Figure 7

(Color online) (a) Zero-temperature critical current of a diffusive SNS system with $\tau =1$ as a function of the parameter $r=G_{\mathrm{N}}∕G_{\mathrm{B}}$ for different lengths of the normal wire. (b) Current-phase relation at zero temperature for an asymmetric junction with $L=2\xi$ and $\tau =1$. The asymmetry parameter $\kappa$ is defined as $\kappa =1-r_R∕r_L$ with $r_R⩽r_L$ and $r_L+r_R=r_{LR}=\text{const}$. Here, $r_{LR}=2$ and $r_L=1,1.5,1.7,1.8,1.9,1.95,1.99999$.

Figure 8

(Color online) (a) Zero-temperature critical current of a diffusive SNS as a function of the spin-flip rate for different lengths of the normal wire and ideal interfaces $(r=0)$. (b) The same as (a) but the critical current is now normalized by the zero spin-flip rate value and ${\Gamma }_{\mathit{sf}}$ is normalized by the Thouless energy of the wires. Notice that all the curves collapse into a single one that can be approximately described by the Gaussian function $I_C∕I_C({\Gamma }_{\mathit{sf}}=0)=\exp (-0.145{\Gamma }_{\mathit{sf}}∕{ϵ}_T)$ shown as a black solid line.

Figure 9

(Color online) (a) Zero-temperature critical current for a wire length $L=10\xi$ as a function of the spin-flip rate for different values of the interface resistance and $\tau =0$. (b) The same as (a) but the critical current is normalized by its value at ${\Gamma }_{\mathit{sf}}=0$.

Figure 10

(Color online) (a) Critical current for a wire length $L=10\xi$ as a function of the spin-flip rate for different values of the temperature (in unit of the critical temperature $T_C$). (b) The same as (a) but the critical current is normalized by its value at ${\Gamma }_{\mathit{sf}}=0$. The inset shows the temperature dependence of the constant $a$ used in the Gaussian fits: $I_C∕I_C({\Gamma }_{\mathit{sf}}=0)=\exp (-a{\Gamma }_{\mathit{sf}}∕{ϵ}_T)$.

Figure 11

(Color online) Comparison between the linearized result of Eq. (A9) and the exact numerical result for the temperature dependence of the critical current for a wire with $\Delta =75{ϵ}_T$ in the absence of spin-flip scattering $({\Gamma }_{\mathit{sf}}=0)$. The different curves correspond to different values of the ratio $r=G_{\mathrm{N}}∕G_{\mathrm{B}}$ and in the exact result we have used $\tau =0$.

Figure 12

(Color online) Minigap and critical supercurrent as a function of $r$ for the two lengths $L=2\xi$ and $L=9\xi$. The solid lines show the fitting curves of the numerically calculated data (circles) in units of the Thouless energy for $r⪢1$.