Voltage-induced Shapiro steps in a superconducting multiterminal structure

When a superconducting tunnel junction at a finite voltage is irradiated with microwaves, the interplay between the alternating Josephson current and the ac field gives rise to steps in the dc current known as Shapiro steps. In this work, we predict that in a mesoscopic structure connected to several superconducting terminals, one can induce Shapiro-like steps in the absence of any external radiation simply by tuning the voltages of the leads. To illustrate this effect, we make quantitative predictions for a three-terminal structure which comprises a diffusive superconductor–normal-metal–superconductor junction and a tunneling probe, a setup which can be realized experimentally.

Figure 1

(Color online) Schematic representation of the setup: a metallic wire (N) is connected at its ends to superconducting reservoirs ${\mathrm{S}}_l$ and ${\mathrm{S}}_r$, biased at potentials $-U∕2$ and $U∕2$, respectively. An additional superconducting probe $\left({\mathrm{S}}_p\right)$ at a potential $V$ is attached to the wire through a tunnel junction.

Figure 2

(Color online) Zero-temperature dc current in the probe electrode ${\mathrm{S}}_p$ at zero bias $(U=0)$ for different lengths. The probe is located at $x=L∕2$. The inset shows the critical current as a function of the wire length.

Figure 3

(Color online) (a) Zero-temperature dc current in the probe ${\mathrm{S}}_p$ located at $x=L∕2$ for $L=2\xi$ and three different values of the voltage $U$: from bottom to top $eU∕\Delta =0.3,0.6,0.9$. (b) dc part of the distribution function of the normal wire at $x=L∕2$ as a function of the energy for the three voltages above. (c) Corresponding nonequilibrium spectral density in the normal wire $(x=L∕2)$ as a function of the energy. In the three panels, the two upper curves have been shifted upward for clarity.

Figure 4

(Color online) Height of the Shapiro steps at zero temperature as a function of the position of the probe $\left(x\right)$ for $eU=0.5\Delta$ and different wire lengths.

Figure 5

(Color online) Height of the first two Shapiro steps for positive voltages in $x=L∕2$ at zero temperature as a function of $U$ for different lengths $L$.