PRL 113, 077201 (2014)

week ending 15 AUGUST 2014

PHYSICAL REVIEW LETTERS

Quantum Spin Hall Effect in Two-Dimensional Crystals of Transition-Metal Dichalcogenides M. A. Cazalilla,1,2 H. Ochoa,3 and F. Guinea3

1

Department of Physics, National Tsing Hua University, and National Center for Theoretical Sciences (NCTS), Hsinchu City, Taiwan 2 Donostia International Physics Center (DIPC), Manuel de Lardizabal 4, 20018 San Sebastian, Spain 3 Instituto de Ciencia de Materiales de Madrid (ICMM), CSIC, Cantoblanco, E-28049 Madrid, Spain (Received 25 December 2013; published 11 August 2014) We propose to engineer time-reversal-invariant topological insulators in two-dimensional crystals of transition-metal dichalcogenides (TMDCs). We note that, at low doping, semiconducting TMDCs under shear strain will develop spin-polarized Landau levels residing in different valleys. We argue that gaps between Landau levels in the range of 10–100 K are within experimental reach. In addition, we point out that a superlattice arising from a moiré pattern can lead to topologically nontrivial subbands. As a result, the edge transport becomes quantized, which can be probed in multiterminal devices made using strained 2D crystals and/or heterostructures. The strong d character of valence and conduction bands may also allow for the investigation of the effects of electron correlations on the topological phases. DOI: 10.1103/PhysRevLett.113.077201

PACS numbers: 85.75.−d, 73.43.Qt

There is currently much interest in two-dimensional crystals of transition-metal dichalcogenides (TMDC) such as MoS2 or WSe2 [1]. Compared to graphene, many of these materials are semiconductors with sizeable gaps (≈1 eV), which makes them good candidates for applications in conventional electronics such as the manufacture of transistors [2]. By contrast, the quantum spin Hall (QSH) effect [3] has been postulated as the paradigm for future nondissipative nanoelectronics [4]. Indeed, provided that magnetic impurities (or other time-reversal symmetry-breaking perturbations) can be eliminated, electron transport through the edge of a QSH insulator becomes ballistic. This is because counterpropagating channels of the same spin are spatially separated. Nevertheless, the QSH effect has been observed in very few materials [5,6] so far, making the search for new systems exhibiting this effect a major scientific endeavor. In the context of 2D crystals, Weeks and coworkers have recently proposed an approach to enhance the spin-orbit coupling in graphene by heavy adatom deposition [7], which could lead to a realization of the Kane-Mele model [8] and the QSH effect. The availability of 2D materials exhibiting this effect may allow for the construction of flexible electronic devices with low power consumption. In this Letter, we present a proposal for engineering the QSH effect in strained 2D crystals and heterostructures made using TMDCs. Strain is known to induce pseudomagnetic fields in 2D crystals [9–11]. These fields have been predicted [9] and experimentally found [10] to produce Landau levels (LLs) in the electronic spectrum of 2D crystals such as graphene. As explained below, it is also possible to exploit these pseudomagnetic fields to engineer time-reversal invariant topological phases in other 0031-9007=14=113(7)=077201(6)

2D crystals such as TMDCs. This approach has several attractive features. Using MoS2 as an example, we find that gaps between LLs scale as ℏωc =kB ≃ 2.7B0 ½T K, where B0 ½T is the strength of the pseudomagnetic field in Tesla. In the case of graphene fields B0 ½T ∼ 10–102 T have been experimentally demonstrated [9,10], which, if achieved for TMDCs, could lead to LL gaps of up to ≈100 K. A more accurate estimation should take into account the maximum tensile strength of these materials. We argue that LL gaps of 20 K can be realistically achieved in strained MoS2 samples. By comparison, the gaps achievable by straining GaAs are in the range of tens of millikelvin [12]. The much larger magnitude of the LLs gaps in the present proposal stems the strain coupling to the electron current, rather than the electron spin current as in Ref. [12]. These two coupling constants are, in general, vastly different in order of magnitude. However, in multivalley systems like graphene, strain only leads to (spin-) unpolarized LLs. What sets semiconducting TMDCs apart is the spin-orbit coupling that produces a large spin-splitting of the valence (and to a smaller extent, the conduction) band. For small doping, this leads to spin-polarized LLs in different valleys, which opens the possibility of realizing time-reversal invariant (TRI) topological phases. Furthermore, the valence and conduction band of semiconducting TMDC have strong d character [13], which, along with the poor screening of the Coulomb interaction in 2D, means that electron interactions can have interesting effects on properties of the TRI topological phases realized in TMDCs. Bulk TMDC are composed of X-M-X layers stacked on top of each other and coupled by weak van der Waals forces. Therefore, like graphite, these materials can be exfoliated down to a single layer. The transition-metal atoms (M) are arranged in a triangular lattice and each one

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© 2014 American Physical Society

PRL 113, 077201 (2014)

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PHYSICAL REVIEW LETTERS X

M

X

K'

Irrep

K M

A01 A02 0

FIG. 1 (color online). Top view of the 2D lattice of TMDC. The unit cell and Brillouin zone are shown in the right panel. M corresponds to the transition-metal atom, and X to the chalcogen atoms.

is bonded to six chalcogen atoms (X) (see Fig. 1 for a top view of the lattice). The point group of the 2D crystal is D3h . The Fermi level lies at the two inequivalent corners of the Brillouin zone, K and K 0 points, as in the case of graphene, but the spectrum is gapped. The character of the conduction and valence bands is dominated by d3z2 −r2 and dx2 −y2  idxy orbitals from M atoms, respectively, where the sign þ (−) holds for K (K 0 ) point. Furthermore, both bands have a non-negligible hybridization with the bonding combination of px  ipy (in the conduction band) and px ∓ipy (in the valence band) orbitals from X atoms. In the continuum limit, the system is well described by the following two-band Hamiltonian [13–15]: H0 ¼ vðτz σ x px þ σ y py Þ þ ðΔ=2Þσ z þ ðλSO =2Þτz sz ð1 − σ z Þ;

TABLE I. Symmetry classification of the electronic operators and strain tensor components according to the irreducible representations (Irrep) of D3h and time reversal operation.

ð1Þ

where p ¼ ðpx ; py Þ is the electron crystal momentum referred to the K (K 0 ) point, for which we introduce τz ¼ þ1 (τz ¼ −1); σ α are Pauli matrices acting on the space span by conduction and valence band states, Ψ ¼ ðψ c ; ψ v Þ; Δ is the band gap and v ¼ ta=ℏ, where a is the lattice parameter and t is a phenomenological hybridization parameter; finally, sz is the spin projection along the z axis perpendicular to the 2D crystal. The Hamiltonian (1) can be constructed phenomenologically by considering scalar invariants of D3h formed from operators σ α , p, and spin sα . The symmetry properties of these operators are listed in Table I. Besides the gap, another important difference with graphene is the spin-splitting of the bands due to the strong spin-orbit interaction provided by M atoms. Below, we focus on p-doped TMDCs, which allows us to neglect the smaller spin-splitting of the conduction band [15]. For the valence band, the latter is in the range of 100–400 meV, depending on the material [16–18]. Taking MoS2 as a reference, we take Δ ¼ 1.66 eV and t ¼ 1.1 eV [13]. In order to turn a TMDC into a TRI topological insulator, we need to consider the effect of strain on the band structure. In the theory of elasticity [19], strain is described by a rank-2 tensor, uαβ ¼ 12 ½∂ α uβ þ ∂ β uα þ ð∂h=∂xα Þð∂h=∂xβ Þ, where

E E00

TR Even P 1; σ z , α uαα  ðσ x ; τz σ y Þ, ðuxx − uyy ; −2uxy Þ 

TR Odd  τz ; sz ðτz σ x ; σ y Þ, p ¼ ðpx ; py Þ ðsx ; sy Þ

u ¼ ðux ; uy Þ (h) is the in-plane (out-of-plane) displacement of the unit cell in the long-wave length limit. The three components of uαβ can be split according to irreducible representations of D3h as shown in Table I. Thus, the coupling with strain to the leading order, assuming juαβ j ≪ 1, reads Hstrain ¼ β0 t

X X uαα þ β1 t uαα σ z α

α

þ β2 t½ðuxx − uyy Þσ x − 2uxy τz σ y :

ð2Þ

Microscopically, the origin of these couplings is the change in the hybridization between d orbitals from M atoms and p orbitals from X atoms due to the distortion of the lattice [20]. For completeness, we provide a derivation of Eqs. (1) and (2) starting from a simplified tight-binding model of TMDC [21], which serves as a qualitative derivation to understand the microscopic origin of the electron-strain coupling. The trace of the strain tensor generates scalar potentials of different strength in the valence and conduction bands. In addition, strain can be introduced in Eq. (1) as a minimal coupling p → p − eA to a vector potential eA ¼ ðℏβ2 =aÞτz ðuyy − uxx ; 2uxy Þ. The presence of τz indicates that the pseudomagnetic field has an opposite sign on different valleys, which is necessary as strain does not violate TRI. Henceforth, we assume that the system is doped with holes and therefore the Fermi level crosses the valence band. Integrating out the conduction band to leading order in Δ−1 yields H v ¼ −½Πþ ðτz ÞΠ− ðτz Þ=2m þ UðrÞ þ λSO sz τz ;

ð3Þ

where m ¼ Δ=2v2 ≃ 0.5 [32], UðrÞ ¼ gðuxx þ uyy Þ with g ¼ tðβ0 − β1 Þ, and Π ðτz Þ ¼ ðτz px ipy Þ−eðτz Ax iAy Þ, which obey the commutation relation ½Πþ ðτz Þ; Π− ðτz Þ ¼ 2eℏτz BðrÞ, where BðrÞ ¼ ∂ x Ay − ∂ y Ax . Corrections of OðΔ−2 Þ have been also obtained and can lead to mixing of the Landau levels, but they can be neglected because, for typical parameters, ℏωc ≲ 10−2 Δ. In the absence of strain (i.e., A ¼ U ¼ 0), Eq. (3) describes the Bloch states at the top of the valence band near K; K 0. Owing to the spin-orbit coupling (∝ λSO ), for

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PRL 113, 077201 (2014)

small hole doping (i.e., jϵF j ≪ λSO ), the spin and valley spin of the holes are locked to each other, i.e., only holes with either ðK; ↑Þ or ðK 0 ; ↓Þ can exist (see Fig. 2). Furthermore, the states at K and K 0 are Kramers pairs. This feature is crucial for the realization of a time-reversal invariant topological phase, as we argue below. Applying a pure shear strain, i.e., uxx þ uyy ¼ 0 (i.e., U ¼ 0), and uxx ¼ −uyy ¼ −Cy, uxy ¼ −Cx, we have Bz ðrÞ ¼ ∂ x Ay − ∂ y Ax ¼ −τz B0 with B0 ¼ 4Cℏβ2 =ea > 0 [33]. The Hamiltonian in (3) canpbe diagonalized after ffiffiffiffiffiffiffiffiffiffiffiffiffi introducing apKffiffiffiffiffiffiffiffiffiffi ¼ Π− ðτz ¼ þ1Þ= 2eℏB0 and aK0 ¼ Π− ðτz ¼ −1Þ= 2eB0 , H ¼ −ℏωc ða†K aK þ a†K0 aK0 Þ þ λSO sz τz ;

ð4Þ

where ωc ¼ eB0 =m . As stated above, even for the largest achievable pseudomagnetic fields (B0 ∼ 102 T) ℏωc ≪ λSO , and therefore, provided jϵF j ≪ λSO , LLs at different valleys are occupied by holes with opposite spin. The resulting model has been shown by Bernevig and Zhang [12] to display the QSH effect, meaning that the Hall conductivity xy σ xy H is zero but the spin-Hall conductivity σ sH is quantized in units of 2e=4π. The strain configuration described above can be created by the methods described in Ref. [9]. For a TMDC 2D crystal flake under trigonal strain [9] (see Fig. 2), one concern is the sample size. The latter is limited by the maximum tensile strength of the TMDCs, T max . For MoS2, the thermodynamic relation σ αβ ¼ 2μuαβ (σ αβ being the stress tensor and μ the shear modulus) allows us to estimate the maximum sample size L ≈ 2T max =μC. The values of μ ≃ 50.4 N=m and T max ≃ 16.5 N=m can be obtained from density-functional calculations [34]. This yields a relation between the maximum pseudomagnetic field (in T) and the sample size L in μm: B0 ≈ 8=L. Using ℏωc =kB ¼ 2.7B0 and taking L ≈ 1, we estimate ℏωc =kB ≃ 20 K for MoS2. Note, however, that deviations from a pure shear strain configuration may affect these estimates [21]. For small strained 2D crystal flakes, it is necessary to take into account the effect of an inhomogeneous pseudomagnetic field resulting from a nonuniform strain distribution. In this regard, we note that the lowest LL eigenfunctions are null eigenvectors of Π− ðτz Þ, Π− ðτz ÞψðrÞ ¼ 0:

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PHYSICAL REVIEW LETTERS

ð5Þ

Therefore, following [35], we write AðrÞ ¼ τz ½^z× ∇χðrÞ þ ∇ϕðrÞ, which allows us to solve (5) for K [K 0 ] and yields ψðrÞ ¼ fðz Þeð2πΦ0 Þ½χðrÞþiϕðrÞ fψðrÞ ¼ fðzÞeð2π=Φ0 Þ½χðrÞ−iϕðrÞ g, where fðz Þ [fðzÞ] is a polynomial  of z ¼ ðx þ iyÞ [z R ¼ ðx − iyÞ] of maximum degree N ¼ ½Φ=Φ0 , Φ ¼ drB0 ðrÞ > 0 being the total flux and Φ0 ¼ h=e the flux quantum [36]. Hence, the wave function

(a) K

(b)

K

K

K LLs

B0

B0

E eV 0.90

(c) K Κ

Μ

K

(d)

0.95

Κ

Μ Κ

1.00 1.05 1.10

Κ

1.15

K

Κ

Μ

Κ'

K

k

FIG. 2 (color online). (a) Low-energy spectrum of a semiconducting transition-metal dicalchogenide around the K and K 0 points of the first Brillouin zone described by the continuum-limit Hamiltonian in Eq. (3). The inset shows the orientation of the stress tensor field discussed through the text with respect to the lattice. (b) Schematic representation of the LLs induced in the valence band when strain is applied. (c) A superlattice can be also used to create a topologically nontrivial subband structure. A triangular superlattice produces replicas of K and K 0 by mixing the valence and conduction band states at the mini-Dirac points as indicated by the arrows. (d) Highest valence subbands of a 20 × 20 MX 2 superlattice with Δ ¼ 1 eV, v ¼ 6.6 eV A, V s ¼ −0.1 eV, and V g ¼ 0.05 eV, where V s , V g are the scalar and gauge potentials discussed through the text. Here κ, κ 0 , and μ label the two inequivalent corners and the intermediate point between them of the superlattice Brillouin zone as indicated in (c).

describing N ↑ ¼ N ↓ ¼ N (noninteracting) electrons in the lowest LL reads [35] Φ0 ðfriα gÞ ¼ e−F

Y ðzi↑ − zj↑ Þðzi↓ − zj↓ Þ;

ð6Þ

i 0 and assume β2 ≈ 3 according to CastellanosGomez, Ref. [29e]. [34] R. C. Cooper, C. Lee, C. A. Marianetti, X. Wei, J. Hone, and J. W. Kysar, Phys. Rev. B 87, 035423 (2013). The value of μ is obtained from the calculated Young’s modulus E and Poisson’s ratio ν as μ ¼ E=½2ð1 þ νÞ. [35] Y. Aharonov and A. Casher, Phys. Rev. A 19, 2461 (1979); M. Bander, Phys. Rev. B 41, 9028 (1990). [36] Φ is assumed R to be positive as in the uniform case. Note that χðrÞ ¼ − ðdr0 =2πÞ lnðjr − r0 jÞB0 ðr0 Þ ∼ −ðΦ=2πÞ lnðjrjÞ as jrj → ∞, and therefore ψðrÞ ∼ fðx þ iτz yÞjrj−ðΦ=Φ0 Þ . [37] M. Taillefumier, V. K. Dugaev, B. Canals, C. Lacroix, and P. Bruno, Phys. Rev. B 78, 155330 (2008). [38] T. Low, F. Guinea, and M. I. Katsnelson, Phys. Rev. B 83, 195436 (2011). [39] I. Snyman, Phys. Rev. B 80, 054303 (2009).

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[40] M. Yankowitz, J. Xue, D. Cormode, J. D. Sanchez-Yamagishi, K. Watanabe, T. Taniguchi, P. Jarillo-Herrero, P. Jacquod, and B. J. LeRoy, Nat. Phys. 8, 382 (2012). [41] L. A. Ponomarenko et al., Nature (London) 497, 594 (2013). [42] C. R. Dean et al., Nature (London) 497, 598 (2013). [43] B. Hunt et al., Science 340, 1427 (2013). [44] O. P. Sushkov and A. H. Castro Neto, Phys. Rev. Lett. 110, 186601 (2013). [45] C. Xu and J. E. Moore, Phys. Rev. B 73, 045322 (2006); C. Wu, B. A. Bernevig, and S.-C. Zhang, Phys. Rev. Lett. 96, 106401 (2006). [46] A. Shitade, H. Katsura, J. Kuneš, X.-L. Qi, S.-C. Zhang, and N. Nagaosa, Phys. Rev. Lett. 102, 256403 (2009) [47] R. Roldán, E. Cappelluti, and F. Guinea, Phys. Rev. B 88, 054515 (2013).

077201-6

Quantum spin Hall effect in two-dimensional crystals of transition-metal dichalcogenides.

We propose to engineer time-reversal-invariant topological insulators in two-dimensional crystals of transition-metal dichalcogenides (TMDCs). We note...
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