Superconducting proximity effect in silicene: Spin-valley-polarized Andreev reflection, nonlocal transport, and supercurrent

We theoretically study the superconducting proximity effect in silicene, which features massive Dirac fermions with a tunable mass (band gap), and compute the conductance across a normal-superconductor (N-S) silicene junction, the nonlocal conductance of an N-S-N junction, and the supercurrent flowing in an S-N-S junction. It is demonstrated that the transport processes consisting of local and nonlocal Andreev reflection may be efficiently controlled via an external electric field owing to the buckled structure of silicene. In particular, we demonstrate that it is possible to obtain a fully spin-valley-polarized crossed Andreev reflection process without any contamination of elastic cotunneling or local Andreev reflection, in stark contrast to ordinary metals. It is also shown that the supercurrent flowing in the S-N-S junction can be fully spin-valley polarized and that it is controllable by an external electric field.

Figure 1

(a) Plot of the insulating gap $E_g$ of normal-state silicene vs an applied electric field $E_z$ perpendicular to the plane. (b) Band structure in an N-S silicene junction. The two conduction bands are split at $\mathbf{k}=0$. The process “1” indicates an incoming quasiparticle from one of the conduction bands, whereas “2” indicates the lost electron in the valence band when Andreev reflection occurs. (c) An effective N-S silicene bilayer where superconductivity is induced via a proximate superconducting lead. (d) An effective N-S-N silicene junction to probe nonlocal transport.

Figure 2

(a)–(e) Normal (dashed lines) and Andreev reflection (solid lines) probabilities for an N-S junction with $\eta =\sigma =+1$, ${\lambda }_{\mathrm{SO}}/{\Delta }_0=5.0$, and $l{E}_z/{\Delta }_0$ ranging from 5.0 to 5.8 from (a) to (e). In (f), the conductance $G/G_N$ (averaged over spin and valleys) is plotted vs bias voltage for the same choices of $l{E}_z/{\Delta }_0$.

Figure 3

Top row: Contour plot of the CAR probability for a bias voltage $\mathrm{eV}/{\Delta }_0=0.9$ vs angle of incidence $\theta$ and junction length $L$. Bottom row: Probabilities for the different scattering events for a fixed junction length $L/\xi =2.1$ for normal incidence. In column (a), we consider scenario (i) as described in the main text (nongapped-superconductor-gapped) whereas in (b) we consider scenario (ii) as described in the main text (gapped-superconductor-gapped). We have considered in all cases a strongly doped superconducting region with ${\mu }_S/{\Delta }_0=20$ and set $m_L/{\Delta }_0=0$ and $m_R/{\Delta }_0=5$ in (a) whereas $m_L/{\Delta }_0=m_R/{\Delta }_0=5$ in (b). The coefficients ($R_e,R_h,T_h$) are the probabilities for normal, Andreev, and crossed Andreev reflection, respectively.

Figure 4

Nonlocal conductance for (a) gapless and (b) gapped silicene on the left side [corresponding to scenarios (i) and (ii) described in the main text].