Subgap transport in silicene-based superconducting hybrid structures

We investigate the influences of exchange field and perpendicular electric field on the subgap transport in silicene-based ferromagnetic/superconducting (FS) and ferromagnetic/superconducting/ferromagnetic (FSF) junctions. Owing to the unique buckling structure of silicene, the Andreev reflection and subgap conductance can be effectively modulated by a perpendicular electric field. It is revealed that the subgap conductance in the FS junction can be distinctly enhanced by an exchange field. Remarkably, resorting to the tunable band gap of silicene, an exclusive crossed Andreev reflection (CAR) process in the FSF junction can be realized within a wide range of related parameters. Moreover, in the FSF junction the exclusive CAR and exclusive elastic cotunneling processes can be switched by reversing the magnetization direction in one of the ferromagnetic regions.

Figure 1

Schematic of a silicene-based FS junction.

Figure 2

AR patterns in a silicene-based FS junction, where $\epsilon =0$ and $\eta =\sigma =1$. The regions ${\mathrm{R}}_{1\left(2\right)}$ and ${\mathrm{S}}_{1\left(2\right)}$ indicate the regimes of retro AR and specular AR, respectively. The black (light gray) lines denote the critical situations with $\left|{\theta }_{11}^h\right|=0$ $(\pi /2)$, and $N_{1\left(2\right)}$ labels the region where the AR is forbidden. In addition, the white areas represent the regimes where there is no propagating incident mode.

Figure 3

Conductance spectra of a silicene-based FS junction, where ${\mu }_F=0,{\lambda }_{SO}=5{\mathrm{\Delta }}_0$, and $h_0={\lambda }_{SO}$. For comparison, the conductance spectra for $h_0=0$ (open-circle curves) are illustrated here and have been discussed in the context of a silicene-based NS junction in Ref. [23].

Figure 4

(a) and (b) Band structures in the F region of a silicene-based FS junction, where the shaded areas denote the subgap energy regimes and $(\mathrm{e}/\mathrm{h},\mathrm{C}/\mathrm{V},\uparrow /\downarrow )$ indicates the electronlike/holelike conduction/valence band with up/down spin. (c) and (d) Contour plots of conductance as functions of $h_0$ and $m_{\eta \sigma }$ $\left(l{E}_z\right)$. The other parameters are chosen as ${\mu }_F=0$ and ${\lambda }_{SO}=5{\mathrm{\Delta }}_0$.

Figure 5

Schematic of a silicene-based FSF junction.

Figure 6

Band structures in the left (L) and right (R) ferromagnetic regions of a silicene-based FSF junction with AP configuration. In the L region only the $K$-valley band structures are exhibited as an example to illustrate the scattering process. Here $\left(\mathrm{e}/\mathrm{h},\mathrm{C}/\mathrm{V},\uparrow /\downarrow \right)$ represents the electronlike/holelike conduction/valence band with up/down spin, and the horizontal solid and dashed lines indicate the Fermi level. The related parameters are fixed as $l{E}_z^L=l{E}_z^R=0,{\mu }_F^L={\mu }_F^R={\lambda }_{SO}$, and $h_0={\lambda }_{SO}$.

Figure 7

The exclusive CAR conductance as a function of (a) incident energy and (b) length of the S region, where $l{E}_z^L=l{E}_z^R=0$ and $h_0={\lambda }_{\mathrm{SO}}$. (c) and (d) Contour plots of CAR conductance as functions of $h_0$ and $l{E}_z$ $\left(l{E}_z^L=l{E}_z^R\equiv l{E}_z\right)$, where $d=0.9\xi$. The other parameters are chosen as ${\mu }_S=10{\lambda }_{\mathrm{SO}}$ and ${\mu }_F^L={\mu }_F^R={\lambda }_{\mathrm{SO}}$.

Figure 8

Band structures in the left (L) and right (R) ferromagnetic regions of a silicene-based FSF junction with P configuration. Here $\left(\mathrm{e}/\mathrm{h},\mathrm{C}/\mathrm{V},\uparrow /\downarrow \right)$ represents the electronlike/holelike conduction/valence band with up/down spin, and the horizontal solid and dashed lines indicate the Fermi level. The corresponding parameters are the same as those in Fig. 6.

Figure 9

EC conductance as a function of (a) incident energy and (b) length of the S region, where $l{E}_z^L=l{E}_z^R=0$ and $h_0={\lambda }_{\mathrm{SO}}$. (c) and (d) Contour plots of EC conductance as functions of $h_0$ and $l{E}_z$ $\left(l{E}_z^L=l{E}_z^R\equiv l{E}_z\right)$, where $d=0.5\xi$. The other parameters are chosen as ${\mu }_S=10{\lambda }_{\mathrm{SO}}$ and ${\mu }_F^L={\mu }_F^R={\lambda }_{\mathrm{SO}}$.