Valley- and spin-switch effects in molybdenum disulfide superconducting spin valve

We propose a hole-doped molybdenum disulfide (${\mathrm{MoS}}_2$) superconducting spin valve (F/S/F) hybrid structure in which the Andreev reflection process is suppressed for all incoming waves with a determined range of the chemical potential in ferromagnetic (F) region and the cross-conductance in the right F region depends crucially on the configuration of magnetizations in the two F regions. Using the scattering formalism, we find that the transport is mediated purely by elastic electron cotunneling (CT) process in a parallel configuration and changes to the pure crossed Andreev reflection (CAR) process in the low-energy regime, without fixing of a unique parameter, by reversing the direction of magnetization in the right F region. This suggests both valley- and spin-switch effects between the perfect elastic CT and perfect CAR processes and makes the nonlocal charge current to be fully valley- and spin-polarized inside the right F region where the type of the polarizations can be changed by reversing the magnetization direction in the right F region. We further demonstrate that the presence of the strong spin-orbit interaction $\lambda$ and an additional topological term ($\beta$) in the Hamiltonian of ${\mathrm{MoS}}_2$ result in an enhancement of the charge conductance of the CT and CAR processes and make them to be present for long lengths of the superconducting region. Besides, we find that the thermal conductance of the structure with a small length of the highly doped superconducting region exhibits linear dependence on the temperature at low temperatures, whereas it enhances exponentially at higher temperatures. In particular, we demonstrate that the thermal conductance versus the strength of the exchange field ($h$) in F region displays a maximum value at $h<\lambda$, which moves towards larger exchange fields by increasing the temperature.

Figure 1

(a) Schematic illustration of the proposed superconducting spin valve in a $p$-doped monolayer molybdenum disulfide with antiparallel configuration: the left and right regions are proximity induced ferromagnetic (F) regions, respectively, with the exchange fields ${\mathit{h}}_L=h\widehat{z}$ and ${\mathit{h}}_R=-{\mathit{h}}_L$, and the intermediate region is in the superconducting (S) state (caused by proximity to a top S electrode). (b),(c) The dispersion relation in momentum space of $p$-doped F regions with the magnitude of the exchange field $h<\lambda$ for parallel (P) and antiparallel (AP) alignments of magnetizations at the $K$ and $K^{\prime }$ valleys. The conduction and the valence bands are separated by a large band gap $\Delta =1.9\mathrm{eV}$. The energies of the valence band edges for different spin subbands of two valleys are ${\mu }_1=-\Delta /2+\lambda +h$, ${\mu }_2=-\Delta /2+\lambda -h$, ${\mu }_3=-\Delta /2-\lambda +h$, and ${\mu }_4=-\Delta /2-\lambda -h$, which are measured from the center of the gap $\Delta$ (zero-energy point).

Figure 2

Probabilities of the AR ($|r_h{|}^2$), CT ($|t_e{|}^2$), and CAR ($|t_h{|}^2$) processes as a function of the chemical potential of the F region, ${\mu }_F$, for normal incidence (${\theta }_e=0$) to the ${\mathrm{MoS}}_2$-based F/S/F structure with (a) parallel (${\mathit{h}}_R={\mathit{h}}_L$) and (b) antiparallel (${\mathit{h}}_R=-{\mathit{h}}_L$) alignments of magnetizations, when $h_L=0.5\lambda$, $L/{\xi }_S=0.5$, ${\mu }_S=-1\mathrm{eV}$, and $eV/{\Delta }_S=0$.

Figure 3

Cross conductance of the superconducting spin valve structure versus the exchange field of the right F region ${\mathit{h}}_R/\lambda$ (in units of the spin-orbit coupling constant $\lambda$) for three different lengths of the S region, when $\alpha =0$, $\beta =2.21$, $eV/{\Delta }_S=0$, ${\mu }_F=-\Delta /2+\lambda +eV-0.001$, ${\mu }_S=1.1{\mu }_F=-1\mathrm{eV}$, and ${\Delta }_S=0.01\mathrm{eV}$.

Figure 4

Behavior of the charge conductance of the pure CT (left panel) and pure CAR (right panel) processes, respectively, in the parallel and antiparallel alignments of magnetizations versus the length of the S region $L/{\xi }_S$ (in units of the superconducting coherence length ${\xi }_S=\hslash v_{\mathrm{F}}/{\Delta }_S$) for gapped graphene ($\lambda =0,\beta =0$) and ${\mathrm{MoS}}_2$ ($\lambda =0.08$) with $\alpha =0$ and $\beta =0$ and $2.21$, when $\Delta =1.9\mathrm{eV}$ (a),(b) and for ${\mathrm{MoS}}_2$ with $\Delta =1.9$, and $1.7\mathrm{eV}$, $\alpha =0$, and $\beta =0$ and $2.21$ (c),(d), when ${\Delta }_S=0.01\mathrm{eV}$, ${\mu }_F=-\Delta /2+\lambda -h+eV+0.001$, ${\mu }_S=1.1{\mu }_F=-1\mathrm{eV}$, $h=0.5\lambda$, and $eV/{\Delta }_S=0$.

Figure 5

Plots of the charge conductance of the CT (left panel) and CAR (right panel) processes, respectively, in the parallel and antiparallel configurations versus $L/{\xi }_S$ for two values of the bias voltage $eV/{\Delta }_S=0$ and 1, in ${\mathrm{MoS}}_2$-based structure with $\alpha =0$ and $\beta =0$ (a),(b) and $2.21$ (c),(d), when $\Delta =1.9\mathrm{eV}$, ${\mu }_F=-\Delta /2+\lambda -h+eV+0.001$, ${\mu }_S=1.1{\mu }_F=-1\mathrm{eV}$, ${\Delta }_S=0.01\mathrm{eV}$, and $h=0.5\lambda$.

Figure 6

Charge conductance of the CT and CAR processes as a function of $L/{\xi }_S$ for different values of the chemical potential ${\mu }_S$ in ${\mathrm{MoS}}_2$-based structure with $\alpha =0$, $\beta =2.21$ (a),(b) and $\beta =0$ (c),(d), when $\Delta =1.9\mathrm{eV}$, ${\mu }_F=-\Delta /2+\lambda -h+eV+0.001$, ${\Delta }_S=0.05\mathrm{eV}$, $h=0.5\lambda$, and $eV/{\Delta }_S=0$.

Figure 7

Dependence of the charge conductance of the CT and CAR processes on the magnitude of the exchange field in the right F region $|{\mathit{h}}_R|/\lambda =h/\lambda$ (in units of the spin-orbit coupling constant $\lambda$) for different lengths of the S region, when $\beta =0$ (a),(b) and $2.21$ (c),(d), $\alpha =0$, $eV/{\Delta }_S=0$, ${\mu }_F=-\Delta /2+\lambda +eV-0.001$, ${\mu }_S=1.1{\mu }_F=-1\mathrm{eV}$, and ${\Delta }_S=0.01\mathrm{eV}$.

Figure 8

Thermal conductance of the ${\mathrm{MoS}}_2$-based F/S/F structure with parallel alignment of magnetizations as a function of $T/T_C$ for different lengths and chemical potentials of the S region, when ${\mu }_F=-\Delta /2+\lambda -h+0.001$, and $h=0.5\lambda$ (a),(b) and for different values of the exchange field in F region, when $L/{\xi }_S=0.1$, ${\mu }_F=-\Delta /2+\lambda$, and ${\mu }_S=5{\mu }_F$ (c). (d) The thermal conductance as a function of $h/\lambda$ for different values of $T/T_C$, when $L/{\xi }_S=0.1$, ${\mu }_F=-\Delta /2+\lambda$, and ${\mu }_S=5{\mu }_F$.