Evanescent states and nonequilibrium in driven superconducting nanowires

We study the nonlinear response of current transport in a superconducting diffusive nanowire between normal reservoirs. We demonstrate theoretically and experimentally the existence of two different superconducting states appearing when the wire is driven out of equilibrium by an applied bias, called the global and bimodal superconducting states. The different states are identified by using two-probe measurements of the wire, and measurements of the local density of states with tunneling probes. The analysis is performed within the framework of the quasiclassical kinetic equations for diffusive superconductors.

Figure 1

A superconducting Al nanowire connected to two massive normal reservoirs, consisting of the same Al, covered by a normal metal Cu layer: (a) scanning electron microscopy (SEM) picture, (c) atomic force microscope (AFM) picture, and (d), (e) schematic representation. The thin Al of the pads is driven normal by the inverse proximity effect of the thick normal Cu. Normal tunneling probes are attached for local measurements (b).

Figure 2

(a) The complete wire is in a single superconducting state with order parameter $\Delta \left(x\right)$. However, near the normal reservoirs, the condensate carries only a small fraction $J_s$ of the current as a supercurrent, which results in a resistance and a voltage drop at the ends of the wire, over roughly a coherence length. At the lowest temperatures, a small proximity effect can occur at the connection of the bilayer reservoirs to the wire (schematically illustrated by dotted black lines). (b) Two distinct superconducting domains at the ends of the wire are separated by a normal region in the center of the wire. Due to the small supercurrent, the voltage profile is almost equal to the normal state.

Figure 3

The even mode $f_L$ and odd mode $f_T$ of the nonequilibrium distribution function $f(E,x)$. (a) For the global superconducting state, a two-step distribution is present through the full wire, while the charge mode is only present at the edges. (b) A strong, nonthermal energy mode nonequilibrium $f_L$ suppresses superconductivity at the center of the wire.

Figure 4

(a) The two-probe resistance versus temperature of a 1.4-$\mu$m-long wire (sample 1a). Due to the proximity effect of the wire on the normal reservoirs, the resistance becomes negligible at low temperatures. This weak proximity effect can be suppressed by applying a small bias current (b) or small magnetic field (c) (sample 4, 200 mK). This “corrected” wire resistance is constant down to the lowest temperatures [magenta squares of panel (a)]. A model (dashed line) with rigid normal boundary conditions for the pairing angle $\theta =0$ slightly overestimates the observations. A weaker boundary condition (full line), in which $\theta$ decays gradually to zero over a characteristic length $a$, shows excellent agreement with the experiment.

Figure 5

The differential conductance of the local tunnel probe as a function of applied voltage (magenta triangles). Good agreement between experiment and theory (full line) is obtained when the series resistance of the setup is included (blue squares).

Figure 6

The voltage $V_{12}$ (a) and differential resistance (b) of a 4-$\mu$m-long wire (sample 3b) as a function of bias current $I_{12}$, measured at 200 mK. We define four different regimes with boundaries labeled $I_{c1}-I_{c4}$, each characterized by a nearly constant differential resistance. The critical currents $I_{c2}$ and $I_{c4}$ are defined as the bias currents where the wire switches between the two hysteretic voltage branches. $I_{c1}$ and $I_{c3}$ are the transition points between the two different states of one branch. (c) The apparent resistance of the complete wire $V_{12}/I_{12}$, and of the edge of the wire $V_{13}/I_{12}$ as measured with a voltage probe, multiplied by two for the ease of comparison.

Figure 7

Two-probe voltage (a) and differential resistance (b) as a function of bias current, for a 1.4-$\mu$m wire, at three different bath temperatures. Open symbols: experimental data. Filled symbols: numerical simulations. The critical current as a function of temperature (inset).

Figure 8

The local density of states for (a) the global superconducting state and (b) the bimodal state, for different bias currents $I_{12}$ of the nanowire, measured at 200 mK. For the global superconducting state, the gap is only weakly dependent on the bias current, while for the bimodal state, one observes a DOS gradually changing from a normal into a superconducting state.