Spintronics in MoS${}_2$ monolayer quantum wires

We study analytically and numerically spin effects in ${\mathrm{MoS}}_2$ monolayer armchair quantum wires and quantum dots. The interplay between intrinsic and Rashba spin orbit interactions induced by an electric field leads to helical modes, giving rise to spin filtering in time-reversal-invariant systems. The Rashba spin orbit interaction can also be generated by spatially varying magnetic fields. In this case, the system can be in a helical regime with nearly perfect spin polarization. If such a quantum wire is brought into proximity to an $s$-wave superconductor, the system can be tuned into a topological phase, resulting in midgap Majorana fermions localized at the wire ends.

Figure 1

(a) The ${\mathrm{MoS}}_2$ monolayer lattice consists of $\mathrm{Mo}$ atoms (large green dots) each connected to six $\mathrm{S}$ atoms (small blue dots). The armchair quantum wire in the monolayer can be formed by metallic gates (yellow area) that fix the propagation direction (defined as the $y$ axis) to be perpendicular to one of the lattice translation vectors, say ${\mathbf{a}}_{\mathbf{1}}$ (red arrow). (b) Brillouin zone where the valleys $K$ and $-K$ lie on the $k_x$ axis which is perpendicular to the direction of propagation given by the $k_y$ axis. Note that the boundaries of a ${\mathrm{MoS}}_2$ flake are not important for this setup.

Figure 2

(a) The energy spectrum of $H_0$ [$\epsilon \left({\tilde{k}}_y\right)\equiv E\left(\sqrt{3}{k}_{y}a\right)-\Delta$] for a quantum wire of width $W=50a$ as obtained by numerical diagonalization. Parameters are chosen as $t=1.27\mathrm{eV}$, $\Delta =0.83\mathrm{eV}$, and $a=0.32\mathrm{nm}$. All levels are degenerate only in spin. This spectrum is in good agreement with our analytical predictions; see Eq. (3). (b) The Rashba SOI term $H_{Rx}$, ${\alpha }_R=10\mathrm{meV}$, lifts the spin degeneracy and results in the spin-dependent shift of the wave vector ${\tilde{k}}_y$. (c) The remaining SOI terms—intrinsic SOI $H_{so}$, $\alpha =38\mathrm{meV}$, and Rashba SOI $H_{Ry}$, ${\alpha }_R=10\mathrm{meV}$—lead to the anticrossings in the spectrum (red dashed circles).

Figure 3

The spin polarization $\langle s_x\rangle$ along $x$ direction as function of momentum ${\tilde{k}}_y$ for the $n$th level defined in Fig. 2c. The spin projections onto the $y$ and $z$ directions vanish. Away from the anticrossings the spin is almost perfectly aligned along $x$. If the chemical potential $\mu$, defining the Fermi momentum ${\tilde{k}}_F^{\left(n\right)}$ for the $n$th subband, is tuned close to the anticrossing [see Fig. 2c], the total polarization $\langle s_x\rangle$ of a left (right) propagating electron is nonzero.

Figure 4

(a) The two lowest energy levels of $H_0+H_{Rn}+H_{so}+H_Z$; cf. Fig. 2. The Zeeman term lifts the Kramers degeneracy at ${\tilde{k}}_y=0$ and opens a gap $2{\Delta }_Z$. If $\mu$ is tuned inside the gap, the system is in a helical regime with a left (right) propagating mode with spin down (up). (b) The spin polarization $\langle s_z\rangle$ as function of the momentum ${\tilde{k}}_y$ for the lowest level. The parameters are chosen as $W=50a$, ${\alpha }_{Rn}=10\mathrm{meV}$, and ${\Delta }_Z=0.05\mathrm{meV}$.