## Abstract

Quantum spin Hall (QSH) effect materials feature edge states that are topologically protected from backscattering. However, the small band gap in materials that have been identified as QSH insulators limits applications. We use first-principles calculations to predict a class of large-gap QSH insulators in two-dimensional transition metal dichalcogenides with 1T′ structure, namely, 1T′-MX_{2} with M = (tungsten or molybdenum) and X = (tellurium, selenium, or sulfur). A structural distortion causes an intrinsic band inversion between chalcogenide-*p* and metal-*d* bands. Additionally, spin-orbit coupling opens a gap that is tunable by vertical electric field and strain. We propose a topological field effect transistor made of van der Waals heterostructures of 1T′-MX_{2} and two-dimensional dielectric layers that can be rapidly switched off by electric field through a topological phase transition instead of carrier depletion.

## Predicting an exotic state of matter

Much like graphene, twodimensional flakes of transition metal dichalcogenides have appealing electronic properties. Qian *et al.* now find that certain structures of these materials may also exhibit the so-called spin Hall effect. The spin Hall effect represents an exotic state of matter in which a 2D material conducts electricity along its edge in a way that drastically reduces dissipation. To show this, the researchers used first-principle calculations and found that the materials also feature a large band gap, which reduces undesirable conduction through the bulk. Their proposed device could be switched on and off quickly using an electric field.

*Science*, this issue p. 1344

The discovery of graphene (*1*) has fuelled vigorous investigation of two-dimensional (2D) materials (*2*), revealing a wide range of extraordinary properties (*3*–*5*) and functionalities (*6*, *7*). Owing to their atomic thickness, 2D materials can be horizontally patterned through chemical and mechanical techniques (*8*). Moreover, the weak van der Waals (vdW) interaction between adjacent layers enables vertical stacking of different 2D materials, forming vdW heterostructures (*9*), which offer unprecedented opportunities for exploring quantum electronics at the nanoscale.

Quantum spin Hall (QSH) insulators (*10*–*16*) have an insulating bulk but conducting edge states that are topologically protected from backscattering by time-reversal symmetry. Quantized conductance through QSH edge states have been experimentally demonstrated in HgTe/CdTe (*13*, *14*) and InAs/GaSb (*17*, *18*) quantum wells. This could in principle provide an alternative route to quantum electronic devices with low dissipation. However, the realization of such QSH-based devices for practical applications is impeded by three critical factors: (i) band gaps of existing QSH insulators are too small, which limits the operating regime to low temperatures. This has motivated efforts to search for large-gap QSH insulators (*19*–*26*); (ii) the small number of conducting channels (*e*^{2}/*h* per edge, where *e* is the elementary charge and *h* is Planck’s constant) results in a small signal-to-noise ratio; and (iii) efficient methods of fast on/off switching are lacking.

Here, we use first-principles calculations to show that 2D materials can provide a practical platform for developing topological electronic devices that may potentially overcome the above hurdles. Specifically, we predict a class of large-gap (~0.1 eV) QSH insulators in 2D transition metal dichalcogenides (TMDCs) MX_{2} with M = (W, Mo) and X = (Te, Se, S). We demonstrate the feasibility of a vdW-heterostructured topological field-effect transistor (vdW-TFET) made of 2D atomic layer materials. The proposed device exhibits a parametrically enhanced conductance through QSH edge channels in the “on” state and can be rapidly switched off via a topological phase transition by applying a vertical electric field. Our findings have potential applications in low-power quantum electronics and spintronics and may enable topological quantum computing based on Majorana fermions (*27*–*30*).

Monolayer TMDCs (*31*)—MX_{2} with M = (W, Mo) and X = (Te, Se, S)—possess a variety of polytypic structures such as 1H, 1T, and 1T′ (*32*, *33*) shown in Fig. 1. The most-studied 1H structure is a sandwich of three planes of 2D hexagonally packed atoms, X-M-X, forming Bernal (ABA) stacking. In contrast, the three atomic planes in the 1T structure form rhombohedral (ABC) stacking. It has been known that the 1T structure in MX_{2} is typically unstable in free-standing condition and undergoes a spontaneous lattice distortion in the *x* direction to form a period-doubling 2 × 1 distorted structure—i.e., the 1T′ structure, consisting of 1D zigzag chains along the *y* direction (Fig. 1C) (*33*); this structure is the subject of our work.

The electronic structures of various 1T′-MX_{2} materials were obtained by many-body perturbation theory calculations (*34*). Figure 2A shows a typical band structure of 1T′-MX_{2} using 1T′-MoS_{2} as an example, and the results of the other five compounds are shown in fig. S1. Unlike its 1H or 1T counterparts, 1T′-MoS_{2} is a semiconductor with a fundamental gap (*E*_{g}) of about 0.08 eV, located at (red dots in Fig. 2B). The conduction and valence bands display a camelback shape near Γ in the 2D Brillouin zone (BZ) (see Fig. 2B), suggestive of band inversion with a large inverted gap (2δ) at Γ of about 0.6 eV. The fundamental gap (*E*_{g}) and inverted gap (2δ) of all six 1T′-MX_{2} are shown in Fig. 2C and listed in table S1. The inverted gap (2δ) was later used in the fitting of a low-energy model (*34*). Because the 1T′ structure has inversion symmetry, we calculated the *Z*_{2} index (0 or 1, indicating trivial or nontrivial topology, respectively) based on the parity criterion (*35*) of valence bands (fig. S2) and found *Z*_{2} = 1, indicating that 2D 1T′-MX_{2} materials are in the QSH insulator phase.

To understand the origin of the inverted band structure near Γ, we analyzed the orbital character of the bands and found that the valence band mainly consists of metal *d-*orbitals and the conduction band mainly consists of chalcogenide *p* orbitals, as expected. The unexpected band inversion arises from the period doubling of the metal chain in the 1T′ structure, which lowers the metal *d* orbital below chalcogenide *p* orbital (Fig. 2A) around Γ. The large band inversion at Γ (~0.6 eV) is an intrinsic characteristic of the 1T′ structure and already takes place without including spin-orbit interaction in the first-principles calculations. In the absence of spin-orbit coupling, this band inversion leads to the appearance of two Dirac cones centered at finite momenta on Y-Γ-Y in 2D BZ. Spin-orbit coupling then opens up a fundamental gap of 0.08 eV at the Dirac points (Fig. 2A, inset, and fig. S3), leading to a QSH insulator through a similar mechanism as the Kane-Mele model for graphene (*10*); we show that this is the case by using a low-energy Hamiltonian (*34*) (fig. S4). In addition to MoS_{2}, we found that all other five 1T′-MX_{2} have *Z*_{2} nontrivial band topology.

The QSH insulator phase in 2D 1T′-MX_{2} leads to helical edge states that are protected from localization and elastic backscattering by time-reversal symmetry. Figure 2D displays the edge states of 1T′-MoS_{2} obtained from surface Green’s function calculations (*34*), showing a pair of counter-propagating edge modes with a high velocity of ~ that cross at Γ and carry opposite spin polarizations. From the local density of states in Fig. 2E, we find the decay length of these helical edge states to be as short as 5 nm. The short localization length can greatly reduce scattering with bulk states (*36*), hence increasing the transport lifetime. Similar results were found for five other TMDCs (figs. S5 and S6).

The inverted bands between chalcogenide’s *p* and metal’s *d* orbitals located on well-separated planes offer a facile mechanism to control topological electronic properties by electric field, which is highly desirable for vdW devices. Based on first-principles calculations, we found that a vertical electric field indeed induces a topological phase transition in 1T′-MX_{2}, as shown in Fig. 3 for 1T′-MoS_{2}. Here, the electric field breaks inversion symmetry and introduces a strong Rashba splitting of the doubly degenerate bands near the fundamental gap *E*_{g} at the Λ points. As the field increases, *E*_{g} first decreases to zero at a critical field strength of 0.142 V/Å and then reopens (see Fig. 4A). This gap-closing transition induces a topology change to a trivial phase (*11*, *37*), leading to the destruction of helical edge states.

The above findings point to the possibility of an all-electrical control of the on/off charge/spin conductance of helical edge states, which would have substantial implications for QSH-based devices (*38*–*43*). Here, we propose a TFET based on vdW heterostructures of 2D TMDCs and 2D wide-gap insulators. The proposed device is sketched in Fig. 4B, where the top and bottom gates supply the vertical electric field to control the on/off function. Two-dimensional wide-gap insulators such as hexagonal boron nitride (hBN) (*44*) electrically insulate adjacent QSH layers, hence protecting parallel helical edge channels from being gapped by interlayer hybridization. This can parametrically increase the number of edge transport channels. Under ideal conditions, this device will support dissipationless charge/spin transport in the “on” state (*Z*_{2} = 1), with a quantized conductance of 2*Ne*^{2}/*h*, where *N* is the number of QSH layers. Applying a moderate electric field will transform 1T′-MX_{2} into an ordinary insulator (*Z*_{2} = 0) and turn the edge conduction off (Fig. 4A).

The proposed vdW-TFET may possess important technical advantages, thanks to an operation mechanism that is fundamentally different from traditional metal-oxide-semiconductor FETs (MOS-FET). MOS-FET works by injecting/depleting free carrier beneath the gate, with a resistor–capacitor-type response time influenced by carrier concentration and mobility in the working channel, whereas our vdW-TFET works by switching the presence/absence of topologically protected conduction channels. The electric field–induced topological phase transition can happen very rapidly, with an electronic response time scale (*45*, *46*). In addition, the vdW heterostructure and the 2D nature of vdW-TFET make it convenient to both vertically and horizontally pattern devices at a large scale. The short decay length of helical edge states allows the minimum horizontal feature size of vdW-TFET to be ~20 nm, enabling high device density on a chip. Horizontally patterned TFET devices of such nanoscale size have large edge-to-area ratios, which will greatly reduce the contribution of thermally populated bulk carriers to the total electric current and hence enhance the on/off ratio. The predicted QSH insulators and their vdW heterostructures may provide a platform for realizing low-dissipation quantum electronics and spintronics (*38*–*43*).

From a materials perspective, the 1T′ structure in monolayer MoS_{2} has recently been observed in high-resolution transmission electron microscopy experiments (*33*). We have checked the structural stability of 1T′ by carrying out first-principles calculations (fig. S7). Although MoS_{2}, MoSe_{2}, MoTe_{2}, WS_{2}, and WSe_{2} are more stable in the 1H structure, we find that there exists a large energy barrier of more than 1 eV/MX_{2} between 1T′ and 1H for all MX_{2}, suggesting that the 1T′ structure can be stabilized under appropriate chemical (*47*), thermal, or mechanical conditions. Moreover, we found that the 1T structure is unstable and will spontaneously relax to 1T′, like in a Peierls distortion. We also verified the stability of the 1T′ structure by computing its phonon dispersion (fig. S8). The absence of imaginary frequencies throughout the 2D BZ provides direct evidence of the structural stability in all 1T′-MX_{2}.

Encouragingly, the desired 1T′ structure with intrinsic band inversion is the natural ground-state structure of WTe_{2} (*31*), also confirmed by our stability calculation. Although monolayer WTe_{2} was found to be a semimetal from our calculation, the small negative fundamental gap can be easily lifted by a tensile strain as small as ~1% (fig. S9). We also found that hBN used as a dielectric in the aforementioned vdW-TFET device has little effect on the electronic structure of 1T′-WTe_{2} near the Fermi level (fig. S10). These attractive features make WTe_{2} potentially the most promising 2D TMDC for realizing the QSH effect and related vdW devices.

## Supplementary Materials

www.sciencemag.org/content/346/6215/1344/suppl/DC1

Supplementary Methods

Figs. S1 to S10

Tables S1 and S2

## References and Notes

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**Acknowledgments:**We acknowledge support from NSF under award DMR-1120901 (X.Q. and J.L.), U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under award DE-SC0010526 (L.F.), and the Science and Technology Center on Integrated Quantum Materials, NSF grant no. DMR-1231319 (J.W.L). Computational time on the Extreme Science and Engineering Discovery Environment (XSEDE) under grant nos. TG-DMR130038 and TG-DMR140003 is gratefully acknowledged. X.Q. and J.W.L. contributed equally to this work. MIT has applied for a provisional patent based on the content of this paper. Both atomic and electronic structures of 1T′-MX_{2}can be found in the supplementary materials.