Spin-filtered and spatially distinguishable crossed Andreev reflection in a silicene-superconductor junction

We theoretically investigate the quantum transport in a junction between a superconductor and a silicene nanoribbon, under the effect of a magnetic exchange field. We find that for a narrow nanoribbon of silicene, remarkable crossed Andreev reflection (with a fraction $>50\%$) can be induced in the energy window of the elastic cotunneling, by destroying some symmetries of the system. Since the energy responses of electrons to the exchange field are opposite for opposite spins, these transport channels can be well spin polarized. Moreover, due to the helicity conservation of the topological edge states, the Andreev reflection, the crossed Andreev reflection, and the elastic cotunneling are spatially separated in three different locations of the device, making them experimentally distinguishable. This crossed Andreev reflection is a nonlocal quantum interference between opposite edges through evanescent modes. If two superconducting leads with different phases are connected to two edges of the silicene nanoribbon, the crossed Andreev reflection can present Josephson type oscillations, with a maximal fraction $\sim 100\%$.

Figure 1

Schematic of the proposed experimental setups. After the silicene nanoribbon is deposited on a FM substrate, (a) the central region is entirely covered by a single SC lead or (b) two edges of the central region are covered by two SC leads with phase ${\varphi }_i$. In (a), we also illustrate the channel configuration at energy $E_0$ marked in Figs. 2 and 2, where up and down arrows denote spin direction, and plus and minus signs represent hole and electron edge states.

Figure 2

(a) and (b) are BdG spectrums of the silicene ribbon in the “pair method” for the spin up (down) component respectively. (c) and (d) are the spin resolved transmission coefficients calculated from the “pair method”. (e) to (h) are those calculated from the “lead method,” with zero $U_S$ [(e) and (f)] and nozero $U_S$ [(g) and (h)], respectively. The upper (lower) row corresponds the spin up (down) component. The parameters are $L=40, W=22, M=2$ meV, and $U_S=1.5$ meV is only turned on in (g) and (h).

Figure 3

(a) The transmission coefficients as functions of the onsite energy $U_S$ of the central region. Other parameters are $L=20, W=26, M=1.7$ meV, and $E=0.66$ meV. Notice that the $U_S$ axis is logarithmic. (b) The real space charge density distribution in the device with $U_S=1$ meV, corresponding to the dashed line in (a), where the optimal CAR occurs. Some part of the left and right leads (marked as “left” and “right”) are included in (b). Positive and negative numbers correspond to electron and hole states, respectively.

Figure 4

(a) The transmission coefficients as a function of length with fixed width $W=26$. (b) The transmission coefficients as a function of width with fixed length $L=20$. Other parameters are same with Fig. 3.

Figure 5

Disorder averaged (over 50 disorder configurations) transmission coefficients in the presence of disorder with $D=11$ meV. Other parameters are same with Figs. 2 and 2 in the main text.

Figure 6

Transmission coefficients for the case of two edge SC leads in Fig. 1. (a) The spin-resolved transmission coefficients as functions of energy with SC phase difference $\mathrm{\Delta }\varphi =\pi$ for the component of spin up (a) and spin down (b). Other parameters are $L=40, W=22, M=2$ meV, and $U_S=0$ meV. (c) The transmission coefficients as functions of the phase difference with energy $E=0.66$ meV. Other parameters are same with (a) and (b).

Figure 7

(a) The CAR $T_{\mathrm{CAR}}$ for different ${\lambda }_{\mathrm{SO}}$. (b) The CAR $T_{CAR}$ for different $\mathrm{\Delta }$. (c) The distributions of wavefunctions along the transverse direction near the lower edge for different ${\lambda }_{\mathrm{SO}}$ at $E_1=-0.87$ meV, corresponding to the orange vertical line in (a). Here ${\lambda }_{\mathrm{SO}}^0=3.9$ meV and ${\mathrm{\Delta }}_0=1$ meV, same with those taken to be constans in the main text. Other parameters are same with Fig. 2 in the main text, except that the width $W=24$.