Majorana Fermions on Zigzag Edge of Monolayer Transition Metal Dichalcogenides

Majorana Fermions on Zigzag Edge of Monolayer Transition Metal Dichalcogenides

Rui-Lin Chu    Gui-Bin Liu    Wang Yao    Xiaodong Xu    Di Xiao    Chuanwei Zhang Department of Physics, the University of Texas at Dallas, Richardson, TX 75080 USA
School of Physics, Beijing Institute of Technology, Beijing 100081, China
Department of Physics and Center of Theoretical and Computational Physics, The University of Hong Kong, Hong Kong, China
Department of Physics, University of Washington, Seattle, Washington, USA
Department of Material Science and Engineering, University of Washington, Seattle, Washington, USA
Department of Physics, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA
Abstract

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 dection is still challenging due to the lack of proper material properties of the underlined experimental systems. Here we propose a new platform for Majorana fermions based on edge states of certain non-topological two-dimensional semiconductors with strong spin-orbit coupling, such as monolayer group-VI transition metal dichalcogenides (TMD). Using first-principles calculations and tight-binding modeling, we show that zigzag edges of monolayer TMD can host 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 non-topological. Our findings points to a controllable and integrable platform for searching and manipulating Majorana fermions.

thanks: chuanwei.zhang@utdallas.edu

Majorana fermionsWilczek () are quantum particles that are their own anti-particles, and satisfy non-Abelian exchange statistics. The latter is the key for their potential use in fault-tolerant topological quantum computationTQC (), which makes their experimental realization an extremely important task from the long-term technological perspective. In the past two decades, some exotic condensed matter/cold atom systemsTQC (); Fuliang-majo (); CW (); Sau-prl (); Lutchyn-prl (); Oreg-prl (); Alicea-prb (); Mao () have been proposed to support Majorana fermions. The experimental breakthrough occurs only recentlyKouwenhoven (); Deng-nano (); Das-nat (); Rokhinson (), using heterostructures consisting of conventional s-wave superconductors and semiconducting nanowires subjected to an external magnetic field, where certain signatures of Majorana fermions were observed. However, there are a few material complications inherent to semiconductor nanowires that may prevent the experimental signature from being conclusiveLiujie-prl (); Altland-prl (); Kells-prb (); JHLee-prl (); Pikulin-njp (); Sarma-rapid (): (i) The large diameters of the nanowires yield multiple occupied transversal sub bands, resulting in complications for the superconductor proximity effect and the chemical potential levelWimmer-10prl (); Potter-10prl (); Lutchyn-11prl (); (ii) The spin-orbit coupling in these wires is rather weak, which renders the Majorana physics extremely vulnerable to disorder, making it challenging to exclude alternative interpretations of the experimental signature based on disorder effectStanescu-11prb (); Potter-11prb (); Liujie-prl (); (iii) The random growth process of nanowires also makes it hard to build a nanowire networkAlicea-nat () to detect the statistics of Majorana fermions.

Amid the above difficulty, it is critically important to look for other 1D conducting states to realize Majorana fermions. A natural and more controllable way is to consider 1D edge states of a 2D material. In this context, helical edge states of 2D quantum spin Hall insulators (QSHIs) have been proposed to support Majorana fermionsFuliang-qshi-majo (); du1 (); xlq (). However, so far the QSHIs have only been realized in semiconductor heterostructures and are subjected to stringent growth conditions. Furthermore, the bulk itself of a QSHI is generally not a good insulator because of the relatively small band gap (about 10 meV). It is therefore natural to ask whether the edge states of non-topological 2D materials with a large bulk band gap can support Majorana fermions. While in pursuit of such platforms a few key material properties are of particular interest: (i) The compounds must have heavy elements that can generate strong spin-orbit coupling (SOC) necessary for robust 1D topological superconductorsPotter-12prb (); sau2 (); and (ii) 2D atomically thin materials with honeycomb-like lattice structures (i.e., similar as graphene), which are more likely to support single band edge states.

In this article we demonstrate this idea by showing that 1D zigzag edges of a new class of 2D semiconductors, monolayer group-IV transition metal dichalcogenides (TMDs), provide a promising new platform for studying 1D topological superconductors with a single transversal band, strong SOC energy, and controlled Majorana network generation. Using both first-principles calculations and tight-binding modeling, we show that the chalcogen-terminated zigzag edges of these 2D semiconductors support edge bands with strong Rashba-type SOC and are well separated from the large bulk bands (1.5 to 1.8 eV). By utilizing a minimal realistic tight-binding model, we numerically confirm the existence of zero-energy Majorana states at the two ends of the edge in the presence of proximity induced s-wave superconductivity, and their robustness against disorders. Our findings point out a new pathway for searching for Majorana fermions using edge states of widely existing 2D non-topological semiconductors.

Results


Figure 1: Isolated single edge band in monolayer MX. (a) Side and top view of a monolayer MX zigzag ribbon. are the vectors connecting the nearest M atoms. The ribbon is infinite in and directions. The lower and upper edges of the ribbon are referred to as X-edge and M-edge, respectively. The ribbon’s width is measured by the number of zigzag chains in width. (b) DFT band structure of a WSe zigzag ribbon with , not including SOC. The grey scale bar represents the total orbital () weight of the band. The color dots represent the orbital () weight (with larger dot for larger weight) from the M atoms on X-edge or M-edge. Three pairs of well localized in-gap edge states can be identified, one (red dot) is on the X-edge and the other two (blue dot and light gray) on the M-edge. The two inequivalent valleys and are located at and , respectively. is the bulk’s lattice constant. Fermi surface is at . See supplementary note for similar plots for MoS, MoSe and WS. (c) Same as (b) with tight-binding model and .

Isolated single edge band. We consider four different TMDs: MoS, MoSe, WS and WSe, but will present our results in the following mainly using WSe as a representative because of its much larger SOC energy. Monolayer TMDs are atomically thin 2D direct-bandgap semiconductors with exotic coupled spin and valley physics xiaodi-mo () and excellent optical properties, as demonstrated in recent experimentsMak-10prl (); AKis (); cao-nat-com (); Mak2 (); xiaodongcui (); xu1 (); xu2 (). Structurally monolayer MX is a tri-layer X-M-X sandwich. Within each layer, M and X atoms form 2D hexagonal lattices. When viewed from top it shows a honeycomb structure. The 2D bulk of monolayer MX has a direct band gap of 1.5 1.8 eV located at the corners of its 2D hexagonal Brillouin zone called valleysxiaodi-mo (). The bulk’s edges can be classified as zigzag and armchair types like in graphene. Due to the lacking of inversion symmetry in the monolayer, the zigzag edges can be further classified as X-terminated and M-terminated, which correspond to the lower and upper edges of the ribbon shown in Fig. 1a. We refer to them as X-edge and M-edge respectively. It is already known from STM measurements that the zigzag edges of triangular shaped monolayer MoS nanoflakes support multiple pairs of 1D metallic edge statesHelveg (). The edges of these nanoflakes are later identified as Mo-edge with passivated S atomsBollinger (). For a zigzag MX ribbon shown in Fig. 1a, the edge states exist on both the M-edge and X-edge.

In addition to the density functional theory (DFT) calculations, insight into the underlying physics can be obtained from a minimal tight-binding model that is constructed by considering the lattice symmetry and the corresponding crystal field splitting. It is known that the valence band maximum and conduction band minimum of monolayer MX consist mainly of M atom’s d orbitals. Thus to describe the low energy band structure of the monolayer’s bulk it is sufficient to consider the d orbitals from the M atomsxiaodi-mo (). The trigonal prismatic coordination of the M atom splits its d orbitals into three groups: , , ) and , . The monolayer’s mirror symmetry in the direction permits hybridization only between the and groups. This allows us to consider three orbitals of , , and for a minimal tight-binding model. We refer the readers to RefLiuguibin () for detailed descriptions of this tight-binding model including the symmetry analysis and material specific parameters fitted from first-principle calculations. The tight-binding model is able to capture the essential physics of the monolayer, including the direct band gaps at the and valleys, the degeneracy of the band edges and the valley contrast spin splitting of the valence band due to SOC, etcxiaodi-mo (); Liuguibin ().

In simple languages this tight-binding model only considers M atoms’ on-site energies and electron hopping along the six vectors connecting the nearest M atoms (marked as in Fig. 1a). Without considering SOC and the spin degree of freedom, the tight-binding Hamiltonian can be written as a matrix

in the -space. Here the basis is taken as and represents the matrix element between and , and is obtained from the Fourier transformation of the real space tunneling matrix between neighboring sites.

Figure 2: X-Edge band minimum. (a) Tigh-binding band structure for zigzag nanoribbon with SOC turned on. The color marks spin direction with red=spin up, blue=spin down. , . (b) Zoom in of the X-edge band minimum of (a) at . The black line is without SOC. The dash line marks the chemical potential used in the BdG calculation in Fig.3. (c) Same as (b) for Fig.1b with SOC turned on, where the dots are DFT results and the lines represents the best fit from Eq.(2). See supplementary note for similar plots for MoS, MoSe and WS.

Rashba-type SOC in edge band. We proceed to include SOC and demonstrate how the Rashba-type SOC in the edge band is generated. The type SOC in MX originates from the d orbitals of the heavy M atoms (Mo or W)zhuzhiyong (); xiaodi-mo (). In monolayer;s Bulk, the spi-oribt term can be described as

(1)

where and represent the spin and orbital angular momentum operator respectively. It turns out in the basis of the and are both matrices, which enables us to write the total Hamiltonian in a spin-decoupled form

Here the upper and lower sub-blocks represent the spin up and down respectively. It is noted that is a time-reversal-invariant Bloch Hamiltonian meaning , where is the time-reversal operator. The total Hamiltonian also respects the time-reversal symmetry(TRS) in this sense. However, when view each spin block individually, the TRS is explicitly broken by the SOC term .

In Figs. 1b and 1c and Fig. 2 we present both the DFT and tight-binding band structure of the WSe zigzag ribbon, where the edge states localized on different edges are marked correspondingly. The electrons from these edge bands dominantly resides on the M atoms of the zigzag edges. Comparing with the DFT band structure, we see the tight-binding model can successfully captures the parabolic edge bands on both the X-edge and M-edge. It’s worth to mention that the effective SOC we find in the parabolic M-edge band is generally larger than in the X-edge band. Nevertheless the X-edge is preferable for two main reasons. First, X-edge hosts a single edge band while the M-edge hosts multiple edge bands; Second, the X-edge is structurally very stable while the M-edge can be dramatically affected by edge passivationsSeifert (); PanHui (); Wang-jacs (); Chen-jacs (). Here after, we will focus on the X-edge band.

In Fig.2b we compare the X-edge band before and after turning on the SOC. Apparently the can be viewed as a TRS breaking perturbation term, whose effect is slightly shifting the spin up branch to the left and spin down branch to the right. The whole band structure nevertheless remains symmetric about because of the TRS. Accordingly the low energy effective 1D Hamiltonian for the X-edge band can be written as

(2)

where is the component of the Pauli matrix, , and is a constant. Up to a unitary transformation, this Hamiltonian is equivalent to that for the semiconductor nanowires with Rashba-type SOCLutchyn-prl (); Alicea-nat (); Wimmer-10prl (); Potter-10prl (); Lutchyn-11prl (); Kouwenhoven (); Deng-nano (); Das-nat (). Here is the effective Rashba velocity. In Table 1 the effective mass and fitted from our first-principle calculations for MoS, MoSe, WS and WSe are listed. These parameters are orders of magnitude larger than their semiconductor nanowire counterparts, especially for WS and WSe.

MoS MoSe WS WSe InAsAlicea-prb () InSbKouwenhoven ()
0.28 0.24 0.33 0.31 0.04 0.015
0.12 0.11 0.33 0.46 0.06 0.2
0.26 0.2 2.3 4.3 0.01 0.05
Table 1: Effective mass and Rashba velocity for TMD’s S(Se)-edge band and semiconductor nanowires. SOC energy is defined as .

Figure 3: Majorana zero-energy mode. (a) A monolayer MX 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. ; (c) Evolution of low energy spectrum with ribbon’s length . . In (b) and (c) only 6 modes closest to zero-energy are plotted. (d) Real-space distribution of the zero-energy mode over the ribbon for . The 3-D view angle is set to be the same as that in (a). Other parameters are: , , .

Existence of Majorana end states. To create a 1D topological superconductor, we introduce superconducting pairing through proximity effects by depositing the MX monolayer on top of a conventional s-wave superconductor (Nb, NbSe, etc.), as illustrated in Fig. 3a. The MX monolayers have X-M-X layer thickness 3 Å, which is well within the superconducting coherence length of a typical s-wave superconductor. The selected monolayer can either be (1) a zigzag nanoribbon or (2) a large monolayer sample with an identified X-edge. A top gate can then be applied locally to tune the chemical potential. We note that in the first case there are coexisting edge states on the M-edge as shown in the ribbon’s band structures (Figs. 1b and 1c). These M-edge states are well localized on the M-edge’s M atoms. To realize a topological superconducting state, the chemical potential need be tuned to be around the X-edge band bottom near (see Fig. 2b). The corresponding M-edge states at this chemical potential occur at momenta far away from . In that region, even number of Mo-edge bands are cut at the Fermi surface, which do not affect the topological properties of the systemLutchyn-prl (). Therefore the M-edge states do not interfere with the topological superconducting state on the X-edge, which is also confirmed in our following numerical simulations. Such coexistence of edge bands is completely eliminated for the second case, where the X-edge can be well isolated. For this case the gate is only required to cover the selected segment of the edge since the bulk maintains a large band gap.

To demonstrate the functionality of the proposed setup, we carry out a numerical simulation in 2D with the tight-binding model. We adopt the ribbon structure for this purpose as illustrated in Fig. 3a. Because of their excellent structural stability, TMD zigzag nanoribbons can be synthesized with uniform width and smooth edges without defects Wang-jacs (); Wang-JMC (); ZhengLiu (). To drive the system into a topological superconducting state, an in-plane magnetic field is applied to create a Zeeman splitting gap at the band crossing point.

The Zeeman term induced by the magnetic field reads

(3)

where is the creation operator for electron on site with orbital index and spin index and . We have assumed the magnetic field is in the direction, but nevertheless any in-plane magnetic field would work equivalently. The proximity effect induced superconducting paring term writes

(4)

where for simplicity we assume a uniform intra-orbit pairing strength. Denote the lattice version of the ribbon’s Hamiltonian in Eq. LABEL:tot-H as , we then solve the corresponding BdG equation for the full Hamiltonian (see Methods)

(5)

to get the low energy spectrum. The emergence of the zero-energy mode with increasing Zeeman splitting is shown in Fig. 3b. In Fig. 3c we show the evolution of the 6 lowest energy modes with the ribbon’s length. As a signature of Majorana fermionsTudor-13prb (), the zero-energy modes show an oscillating energy splitting with an exponentially decaying envelope. When the ribbon is sufficiently long the zero-energy mode as well as the excitation gap become well defined. We notice that the fast alternating modes in the excitation gap of a short ribbon are contributions from the coexisting M-edge bands, which confirms our prediction that they do not affect the topological superconductor on the X-edge. It is well established that in a 1D topological superconductor the Majorana fermions appear as end states in real spaceSau-prl (); Lutchyn-prl (). We confirm this by plotting the particle component of the zero-energy mode wave function in Fig.3d, where the ribbon’s size is in length and in width.


Figure 4: Effect of edge and bulk disorders. In each panel 50 random disorder configurations are collected and the 6 lowest energy modes for each disorder configuration are plotted. Red and black curve represent clean and disordered case respectively. (a-c) Disorders are put on both the ribbon’s bulk and edge. (a) ; (b) ; (c) ; (d) Disorders are put only on the ribbon’s bulk but not on the edge, . , , .

Effect of disorder. It is important that Majorana fermions can sustain certain amount of disorders since in realistic experimental conditions disorders are unavoidable. To explore the disorder effect in this system, we add random on-site potential

(6)

to the tight-binding model, where are normally distributed in the range [-,]. We have simulated two kinds of disorders: (1) disorder covers both bulk and edge; (2) disorder only covers the bulk but not the edge. As shown in Fig.4a-c, the zero-energy modes as well as the excitation gap are robust against edge disorders up to ( can be much larger for a smaller ). For even stronger edge disorders the excitation gap starts to diminish and the zero-energy modes gain splitting. An important advantage of this proposed system is that the topological superconductor resides only on the edge, as a result it gains strong immunity from bulk disorders. This is demonstrated in Fig.4d where the zero-energy modes and excitation gap remain totally intact despite of the strong disorder in the bulk.

Material considerations. Although both the zigzag and armchair edges of monolayer MX support edge states, the zigzag edges are generally more stable than the armchair edges. Consequently during nanoribbon synthesis the zigzag nanoribbons dominatesWang-jacs (); Wang-JMC (); ZhengLiu (). In particular, the X-edge shows maximal stability among all the edge configurations of monolayer MXWang-jacs (); PanHui (); Chen-jacs (). In the experimentally grown WS nanoribbon, the S-edge was found to be perfect without defects ZhengLiu (). Furthermore, edge states have been observed in MoS nanoflakes using STMHelveg (); Bollinger ().

In the DFT band structures of MX zigzag ribbon (Fig.1b and supplementary note), three pairs of in-gap edge states exist connecting the two inequivalent valleys. All of them consists dominantly of the d orbitals from the M atoms of the ribbon’s outermost zigzag chain. Notably two of these edge states have parabolic dispersions at their band minimum/maximum near , which suits our purpose to find an effective Hamiltonian like Eq.2. The two parabolic bands are also fully captured in the tight-binding model. The edge state phenomenon is similar in all MX zigzag ribbons we have calculated. After including SOC, the Rashba-type SOC in the edge band is evident when zooming in the band bottom at (Fig.2c and supplementary note)). We have also calculated ribbons with different width. We find these well localized edge states start to exist in very narrow ribbons ().

The Fermi surface in DFT calculations of suspended MX is typically in the band gap and close to the valence band top. However, both n-typeAKis (); Xu-acsnano (); Zande-natm () and p-typeptype1 (); ptype2 () conductivities have been reported in transport measurements, suggesting a wide-range tunability for the chemical potentialsDolui (). The material synthesis and device fabrication of monolayer MX are rapidly developing because of their potential applications in the next generation of electronics. The nanoribbon samples for a possible experimental realization for our proposed setup have become readily availableWang-jacs (); Wang-JMC (); ZhengLiu (). The monolayer’s 2D nature and similarity with graphene also makes many well developed 2D device engineering and fabrication techniques directly applicable.

Discussion

The advantage of the proposed platform can be summarized in such a few aspects: (1) The single edge band is well isolated in the middle of a relatively large bulk band gap, which would lead to a minimal background signal in the zero-bias peak measurements for detecting Majorana fermions. The well known multiband problem in the semiconductor nanowires is successfully avoidedPotter-10prl (). More importantly, the platform is a true 1D system localizing on a single atomic chain resulting in strong immunity from bulk’s disorders as we have demonstrated; (2) The system is atomically thick, which would lead to robust and uniform superconducting proximity effect when placed on top of the s-wave superconductor. It would also result in efficient gate tunability, which has already been demonstrated in transport measurements with the monolayerAKis (); YZhang (); (3) The large effective mass and large effective Rashba-SOC in this platform is unparalleled to the conventional semiconductor nanowires. As a result, the proximity induced superconducting pairing and the associated Majorana fermions are robust against disordersStanescu-11prb (); Potter-11prb (); Potter-12prb (); sau2 (); (4) The 1D edge states can be controllably generated from 2D materials, making it possible to construct a Majorana networkAlicea-nat () for studying the non-Abelian statistics of Majorana fermions and implementing topological quantum computation.

Methods

Real-space BdG equation. Here we elaborate how the BdG calculation for Fig. 3 is implemented on the zigzag ribbon’s lattice. Denote the spin-independent hopping matrix from the nearest M atoms at site to as and the on-site potential matrices as diagLiuguibin (). Then the real-space BdG equation can be written as

where , when and , . and each being a vector are the components of the th quasiparticle wave function at site , is the corresponding energy eigenvalue. The low energy spectrum and wave function is then obtained by using the sparse matrix eigensolver in MATLAB.

DFT calculation. Our first-principle DFT calculations are performed using the all-electron full-potential linearized augmented-planewave (FP-LAPW) methodELK (). The SOC is included in terms of the second-variational method with scalar-relativistic orbitals as a basis. Construction of our tight-binding model is described in details in RefLiuguibin ().

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Acknowledgements

R.C., and C.Z. are supported by ARO (W911NF-12-1-0334), AFOSR (FA9550-13-1-0045), and NSF-PHY (1104546). G.L acknowledges support by the Basic Research Funds of Beijing Institute of Technology with Grant No. 20121842003 and the National Basic Research Program of China 973 Program with Grant No. 2013CB934500. W.Y. acknowledges support by HKSAR Research Grant Council (HKU705513P), and Croucher Foundation under the Croucher Innovation Award. X.X is supported by US DoE, BES, Division of Materials Sciences and Engineering (DE-SC0008145). D.X. acknowledges support by the U.S. Department of Energy, Office of Basic Energy Sciences, Materials Sciences and Engineering Division, and AFOSR (FA9550-12-1-0479).

Author contributions

All authors made critical contributions to the work.

Competing financial interests

The authors declare no competing financial interests.

Supplementary Materials

.1 Isolated S(Se)-edge band in MoSe, Ws and WSe from DFT calculations


Figure 5: Left: DFT MoS zigzag ribbon band structure, without SOC; Right: Zoom in of the Se-edge band bottom, with SOC.

Figure 6: Left: DFT MoSe zigzag ribbon band structure, without SOC; Right: Zoom in of the Se-edge band bottom, with SOC.

Figure 7: Left: DFT WS zigzag ribbon band structure, without SOC; Right: Zoom in of the S-edge band bottom, with SOC.
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