Proximity effect and multiple Andreev reflections in diffusive superconductor–normal-metal–superconductor junctions

We present a theory of the current-voltage characteristics in diffusive superconductor–normal-metal–superconductor junctions. By solving the time-dependent Usadel equations we are able to describe the phase-coherent transport for arbitrary length of the normal wire. We show how the interplay between proximity effect and multiple Andreev reflections gives rise to a rich subgap structure in the conductance and how it is revealed in the nonequilibrium distribution function.

Figure 1

(a) Normalized density of states in the middle of the wire $(z=0.5)$ as a function of energy for different wire lengths. The inset shows the minigap ${\Delta }_g$ as a function of the length. (b) Normalized density of states as a function of the energy in different positions $\left(z\right)$ along a wire of length $L=4\xi$.

Figure 2

Zero-temperature dc current as a function of the voltage for different wire lengths $L$. Upper inset: zero-temperature critical current as a function of $L$. Lower inset: excess current as a function of $L$.

Figure 3

Zero-temperature differential conductance as a function of the voltage for different wire lengths. The vertical lines indicate the position of $eV=2\Delta ∕n$ with $n=1,\dots ,6$.

Figure 4

Finite-bias spectral function (see text for definition) for two wires of different lengths at $eV=2.5\Delta$. The different curves correspond to different positions along the wire. The vertical dotted lines indicate the positions of the chemical potentials of both electrodes.

Figure 5

(a) dc component of the distribution function in the middle of the wire $(z=1∕2)$ for three different wire lengths $L$ and $eV=2.5\Delta$. The positions of the chemical potentials are indicated by vertical lines. The dotted lines are the results of the incoherent model (see text). (b) dc component of the renormalization of the diffusion constant for the cases shown in (a). (c) The same as (a) for $eV=0.5\Delta$. (d) Spatial variation along the wire of the distribution functions in (a) for fixed energies. The curves in (a)–(d) have been shifted by multiples of 0.5 for convenience.