Spin-orbit-coupled quantum wires and Majorana fermions on zigzag edges of monolayer transition-metal dichalcogenides

Majorana fermions, quantum particles with non-Abelian exchange statistics, are not only of fundamental importance, but also building blocks for fault-tolerant quantum computation. Although certain experimental breakthroughs for observing Majorana fermions have been made recently, their conclusive detection is still challenging due to the lack of proper material properties of the underlined experimental systems. Here we propose a platform for Majorana fermions based on edge states of certain nontopological two-dimensional semiconductors with strong spin-orbit coupling, such as monolayer group-VI transition-metal dichalcogenides (TMDs). Using first-principles calculations and tight-binding modeling, we show that zigzag edges of monolayer TMD can host a well isolated single edge band with strong spin-orbit-coupling energy. Combining with proximity induced $s$-wave superconductivity and in-plane magnetic fields, the zigzag edge supports robust topological Majorana bound states at the edge ends, although the two-dimensional bulk itself is nontopological.

Figure 1

Isolated single edge band in monolayer MX${}_2$. (a) Side and top view of a monolayer MX${}_2$ zigzag ribbon. $R_i$ are the vectors connecting the nearest M atoms. The ribbon is infinite in the $R_1$ and $R_4$ directions. The lower and upper edges of the ribbon are referred to as the X edge and M edge, respectively. The ribbon's width is measured by the number of zigzag chains $N_c$ in width. (b) DFT band structure of a WSe${}_2$ zigzag ribbon with $N_c=8$, not including SOC. The grayscale bar represents the total orbital ($d_{z^2}+d_{xy}+d_{x^2-y^2}$) weight of the band. The color dots represent the orbital ($d_{z^2}+d_{xy}+d_{x^2-y^2}$) weight (with larger dot for larger weight) from the M atoms on the X or M edge. Three pairs (marked 1 to 3) of well localized in-gap edge bands can be identified, one (red dot) is on the X edge and the other two (blue dot and light gray) are on the M edge. The two inequivalent valleys $K$ and $K^{\prime }$ are located at $ka/2\pi =1/3$ and $2/3$, respectively. $a=3.31\AA$ is the bulk's lattice constant. Fermi surface is at $E=0$. Plots for MoS${}_2$, MoSe${}_2$, and WS${}_2$ have similar patterns. (c) Same as (b) with tight-binding model and $N_c=20$.

Figure 2

X-edge band minimum. (a) Tight-binding band structure for zigzag nanoribbon with SOC turned on. The color marks spin direction with $\text{red}=\text{spin}\text{up}$, $\text{blue}=\text{spin}\text{down}$. $N_c=20$, $\lambda =460\text{meV}$. (b) Zoom in of the X-edge band minimum of (a) at $ka=\pi$. The black line is without SOC. The dashed line marks the chemical potential used in the BdG calculation in Fig. 3. (c) Same as (b) for Fig. 1 with SOC turned on, where the dots are DFT results and the lines represents the best fit from Eq. (3).

Figure 3

Majorana zero-energy mode. (a) A monolayer MX${}_2$ zigzag ribbon deposited on top of an $s$-wave superconductor. A top gate can be applied to tune the chemical potential. An external magnetic field is applied parallel to the zigzag edge in order to make the system a topological superconductor. (b) The emergence of the zero-energy mode. $L/a=400$. (c) Evolution of low energy spectrum with ribbon's length $L$. $V_z=2.0\text{meV}$. In (b) and (c) only six modes closest to zero energy are plotted. (d) Real-space distribution of the zero-energy mode over the ribbon for $L/a=300$. The 3D view angle is set to be the same as that in (a). Other parameters are $N_c=10$, $\Delta =1.0\text{meV}$, and $\mu =0.4364\text{eV}$.

Figure 4

Effect of edge and bulk disorders. In each panel 50 random disorder configurations are collected and the six lowest energy modes for each disorder configuration are plotted. Red and black curves represent the clean and disordered case, respectively. (a)–(c) Disorders are put on both the ribbon's bulk and edge. (a) $W=5\text{meV}$; (b) $W=10\text{meV}$; (c) $W=20\text{meV}$; (d) disorders are put only on the ribbon's bulk but not on the edge, $W=200\text{meV}$. $\Delta =1\text{mev}$, $V_z=2\text{meV}$, $L/a=400$.