Pure valley- and spin-entangled states in a ${\mathrm{MoS}}_2$-based bipolar transistor

In this study, we show that the local Andreev reflection not only can be tuned largely by the type of the normal metal electrode, it also is related to the electrostatic potential in the superconductor region in a ${\mathrm{MoS}}_2$-based $n\left(p\right)$-type metal/superconductor junction. In a ${\mathrm{MoS}}_2$-based $n$-type metal/$n\left(p\right)$-type superconductor/$p$-type metal $\left(\mathrm{n}\text{S}\mathrm{p}\right)$ transistor, nonlocal pure valley- and spin-entangled current can be tuned by the length and local gate voltage of a superconductor region. In particular, switching the quasiparticle type in both structures results in a series of intriguing features. Such an effect is not attainable in a graphene-based junction where the electron-hole symmetry enables the symmetry results to be observed. Besides, we have shown that the crossed Andreev reflection exhibits a maximum around $\xi /2$ instead of the exponential decay behavior in conventional superconductors and a maximum around $\xi$ in the graphene material. The proposed straightforward experimental design and pure valley- and spin-entangled state can pave the way for a wider use in the entanglement based on material group-VI dichalcogenides.

Figure 1

Schematic diagram of a $n\left(p\right)\text{S}$ junction. The yellow and the blue blocks denote a local gate voltage electrode and an $s$-wave superconductor electrode which are deposited on the surface of the single-layer ${\mathrm{MoS}}_2$, respectively. By the proximity effect, a superconducting gap ${\Delta }_s$ is induced in the single-layer ${\mathrm{MoS}}_2$. Schematic representation of energy bands for $n\text{S}$ and $p\text{S}$ junctions corresponds to left and right figures in the lower panel, respectively. Black (blue) curves denote spin-up (-down) bands. Black dashed line represents the Fermi energy $E_F$ measured from the middle of the gap ${\Delta }_s$.

Figure 2

Dependence of the Andreev conductance on the magnitude of the chemical potential for $n\text{S}$ and $p\text{S}$ is plotted in (a) and (b), respectively. Here, ${\Delta }_N=1.66\mathrm{eV},t=1.1\mathrm{eV},\lambda =75\mathrm{meV},a=0.3193$ nm, $V=0$, and ${\Delta }_s=1\mathrm{meV}$. The other parameters are shown in the figure. Solid lines correspond to spin-up electron and dashed lines to spin-down electron.

Figure 3

Andreev conductance of a $n\left(p\right)\text{S}$ junction as a function of incident energy $\mathrm{eV}$ for $n\text{S}$ and $p\text{S}$ is plotted in (a) and (b), respectively. The parameters are similar to Fig. 2 and the differences are shown in the figure.

Figure 4

Schematic diagram of a $\mathrm{n}\text{S}\mathrm{p}$ transistor in the top panel. In the lower panel, left and right figures correspond to the schematic representation of energy bands for $\mathrm{n}\text{S}\mathrm{p}$ with $n$-type and $p$-type S electrodes, respectively. By applying gate voltages, the bands in each electrode can be tuned with respect to the Fermi energy. In a $\mathrm{n}\text{S}\mathrm{p}$ junction, it is possible to achieve a perfect nonlocal entangled states splitting.

Figure 5

Crossed Andreev conductivity as a function of $\mathrm{eV}$ and potential strength $U$ for $l/\xi =1$. (a) and (b) correspond to $n$-type and $p$-type superconductors, respectively.

Figure 6

(a) and (b) show the crossed Andreev conductivity as a function of $\mathrm{eV}$ and $l$ for cases of $n$-type $(U=0.4$ eV) and $p$-type $(U=-2$ eV) superconductors $\mathrm{n}\text{S}\mathrm{p}$ transistor, respectively.

Figure 7

Crossed Andreev conductivity as a function of the superconducting length with $U=0.4$ eV and $U\approx -2$ eV corresponding to $n$-type and $p$-type superconductors, respectively. The other parameters are the same as Fig. 6.