Light-modulated 0-$\pi$ transition in a silicene-based Josephson junction

We investigate the Andreev bound states (ABSs) and Josephson current in a silicene-based superconductor-normal-superconductor junction modulated by a perpendicular electric field and an off-resonant circularly polarized light. Based on the Dirac–Bogoliubov–de Gennes equation, we analytically derive the ABS levels and show they have different phase-difference dependences, which will remarkably influence the velocity of Cooper pairs and then the Josephson current. In the pristine or gated silicene, the ABS levels always show negative slope, which means that the Josephson current is irreversible because of the time-reversal symmetry. When an off-resonant circularly polarized light is applied, whether or not there is a perpendicular electric field, the ABS levels will have positive slope, leading to the emergence of reversed Josephson current due to the nonzero center-of-mass wave vector of Cooper pairs. In this light-modulated silicene-based Josephson junction, valley polarization provides an alternative mechanism for 0-$\pi$ transition, very different from that for the conventional ferromagnetic Josephson junctions where the spin polarization is essential.

Figure 1

Schematic diagrams for (a) the top view of a silicene-based Josephson junction and (b) the side view of silicene. An applied external electric field and an off-resonant right-circularly polarized light in the normal region are also shown.

Figure 2

(a)–(f) Andreev-bound-state level versus the phase difference with different $l{E}_z,F_{\omega }$, and $L$. The Fermi energy $E_{\mathrm{F}}=120$, and the unit for the junction length is 1 nm.

Figure 3

(a)–(d) Josephson current versus the phase difference with different perpendicular electric fields, off-resonant lights, and junction lengths but fixed Fermi level $E_{\mathrm{F}}=120$.

Figure 4

(a)–(d) the critical Josephson current versus the junction length with different perpendicular electric fields and off-resonant lights in the case of $E_{\mathrm{F}}=120$.

Figure 5

(a) and (b) Josephson current versus the phase difference with different perpendicular electric fields at zero and nonzero temperatures, respectively. Also a comparison is made for Josephson current between the nonzero and zero spin-orbit couplings with $l{E}_z=50$. Free-energy-phase relation at different temperatures for (c) $l{E}_z=20$ and (d) $l{E}_z=30$. The other parameters are $E_{\mathrm{F}}=120,F_{\omega }=50,G_0=\frac{W{E}_{\mathrm{F}}}{\pi \hslash v_{\mathrm{F}}}{\mathrm{\Delta }}_0$, and $L=20$ nm.

Figure 6

The contour plot for critical current $J_{\mathrm{c}}$ versus (a) $E_{\mathrm{F}}$ and $F_{\omega }$ with $L=30$ nm and $l{E}_z=0$, (b) $l{E}_z$ and $F_{\omega }$ with $L=30$ nm and $E_{\mathrm{F}}=120$, and (c) and (d) $l{E}_z$ and $F_{\omega }$ with $L=100$ nm, $E_{\mathrm{F}}=20$, and ${\lambda }_{\mathrm{so}}=3.9$ and 0, respectively.