Intrinsic spin Hall effect in monolayers of group-VI dichalcogenides: A first-principlesstudy

Wanxiang Feng,1, 2, 3 Yugui Yao,1, ∗ Wenguang Zhu,3, 2 Jinjian Zhou,4, 1 Wang Yao,5 and Di Xiao2, 6, †

1School of Physics, Beijing Institute of Technology, Beijing 100081, China2Materials Science & Technology Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA

3Department of Physics and Astronomy, University of Tennessee, Knoxville, TN 37996, USA4Institute of Physics, Chinese Academy of Sciences and Beijing National

Laboratory for Condensed Matter Physics, Beijing 100190, China5Department of Physics and Center of Theoretical and Computational Physics,

The University of Hong Kong, Hong Kong, China6Department of Physics, Carnegie Mellon University, Pittsburgh, PA 15213, USA

(Dated: May 24, 2018)

Using first-principles calculations within density functional theory, we investigate the intrinsicspin Hall effect in monolayers of group-VI transition-metal dichalcogenides MX2 (M = Mo, W andX = S, Se). MX2 monolayers are direct band-gap semiconductors with two degenerate valleyslocated at the corners of the hexagonal Brillouin zone. Because of the inversion symmetry breakingand the strong spin-orbit coupling, charge carriers in opposite valleys carry opposite Berry curvatureand spin moment, giving rise to both a valley-Hall and a spin-Hall effect. We also show that theintrinsic spin Hall conductivity in inversion-symmetric bulk dichalcogenides is an order of magnitudesmaller compared to monolayers. Our result demonstrates monolayer dichalcogenides as an idealplatform for the integration of valleytronics and spintronics.

PACS numbers: 72.25.Dc,73.63.-b,75.70.Tj,78.67.-n

I. INTRODUCTION

In crystalline semiconductors, it often happens thatthe conduction band minima and valence band maximaare located at degenerate but inequivalent valleys. Well-known examples include graphene,1 bismuth thin films,2

and AlAs quantum wells.3 Since the valleys are usuallyseparated by a large distance in momentum space, inter-valley scattering is greatly suppressed in the presence ofsmooth scattering potential, rendering the valley indexan intrinsic property for low-energy carriers. Motivatedby this observation, there has been a growing interestin exploiting the valley index in electronic devices, muchin the same way as the spin index is used in spintronicapplications. This is the subject of valleytronics.

Recently, a general scheme based on inversion symme-try breaking has been proposed to generate and manip-ulate the valley polarization.4,5 The central idea is thatunder inversion symmetry breaking, the valley index canbe associated with distinctive physical quantities such asthe Berry curvature and orbital magnetic moment.6 Us-ing graphene as an example, the authors of Ref. 4 and 5showed that inversion symmetry breaking allows a valleyHall effect in which carriers in different valleys flow to op-posite transverse edges when an electric field is applied,leading to a finite valley polarization along the edges.4

Furthermore, it also gives rise to valley-contrasting cir-cular dichroism in the momentum space, which takes theextreme form of optical selection rules at high symmetrypoints.5 Other approaches have also been proposed.1,7

However, they all rely on carefully prepared geometry atthe atomic scale, which is difficult to control in experi-ments.

In general, inversion symmetry breaking also lifts thespin degeneracy of energy bands in the presence of spin-orbit coupling (SOC). As required by time-reversal sym-metry, the spin-splitting in opposite valleys must be op-posite, therefore the valley carriers can be also distin-guished by their spin moments. This is the basis of cou-pled spin and valley physics. However, the SOC is negligi-bly small in graphene,8,9 preventing further investigationalong this direction. In a recent work,10 we have studiedmonolayers of group-VI transition-metal dichalcogenidesfor the following reasons: (i) the inversion symmetry isexplicitly broken in monolayers; (ii) the conduction andvalence bands of these materials harbors a multi-valleyedstructure;11,12 and (iii) the SOC is substantial due to thepresence of heavy metal atoms.13 Therefore these materi-als provide a perfect platform to investigate the interplaybetween spin and valley degrees of freedom. Based on aneffective k · p model, we predicted that the valley Halleffect is accompanied by a spin Hall effect in both n-and p-doped systems, and the valley-dependent opticalselection rule also becomes spin-dependent.10

Monolayers of group-VI dichalcogenides also displayexcellent optical properties for practical applications.Recent experiments have demonstrated that MoS2, a pro-totypical group-VI dichalcogenide, crossovers from an in-directgap semiconductor at multilayers to a direct band-gap one at monolayer.14,15 The direct band-gap is inthe visible frequency range, most favorable for optoelec-tronic applications. Experimental evidence of the valley-dependent optical selection rule in monolayer MoS2 hasbeen recently reported based on polarization-sensitivephotoluminescence measurement.16–18

In this work we present a comprehensive first-principles

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study of the coupled spin and valley physics, focusing onthe Hall effects of valley and spin. We show that, be-cause of the inversion symmetry breaking and the strongSOC, charge carriers in opposite valleys carry oppositeBerry curvature and spin moment, giving rise to both thevalley- and spin-Hall effect. Our first-principles calcula-tions provide a quantitative basis for the k · p model de-rived in Ref. 10, and subtle differences between these twoare discussed. We also show that the intrinsic spin Hallconductivity in inversion-symmetric bulk dichalcogenidesis an order of magnitude smaller compared to monolay-ers. Our result demonstrates monolayer dichalcogenidesas an ideal platform for the integration of valleytronicsand spintronics.

II. METHODOLOGY

The electronic ground-state calculations in this workwere performed using full-potential linearized aug-mented plane-wave method,19 implemented in the pack-age wien2k.20 Exchange-correlation effect was treatedwith the Perdew, Burke, and Ernzerhof parameterizedgeneralized-gradient approximation.21 The crystal struc-ture were adopted from the first-principles optimized re-sults.13 For the slab model, a 20 A thick vacuum layer wasused to avoid the interactions between adjacent mono-layers. The converged ground-states were obtained usingk-mesh 16×16×3 for bulk and 16×16×1 for monolayerin the first Brillouin zone, both with KmaxRMT = 7.0,where RMT represents the smallest muffin-tin radius andKmax is the maximum size of reciprocal-lattice vectors.Wave functions and potentials inside the atomic spherewere expanded in spherical harmonics up to l=10 and4, respectively. Spin-orbit coupling was included by asecond-variational procedure,19 where states up to 9 Ryabove Fermi level were included in the basis expansion.

To calculate the Berry curvature and Hall conductiv-ity, we first computed the Wannier functions by the max-imally localized algorithm,22,23 implemented in the pack-age wannier90.24 The construction of maximally local-ized Wannier functions is a non-self-consistent processon a uniform 8 × 8 × 8 grid of k-point with formerlyconverged self-consistent charge potential. Transition-metal dichalcogenides have the chemical formula MX2

(M = Mo, W and X = S, Se). In the case of monolay-ers, there are 22 bands in the energy range from about -6to 5 eV, mainly formed by M d- and X p-orbitals. Withthis in mind, we chose ten d orbitals on atom M and six porbitals on each atom X as the initial guess of the Wan-nier functions. After less than 200 iterative steps, thetotal Wannier spread was well converged down to 10−7

Bohr2. On the other hand, in bulk systems, the unitcell contains two formula units, and there are 44 bandsalso formed by M d- and X p-orbitals. The constructionprocess in bulk is similar to that in monolayers exceptfor WS2 and WSe2. In these materials, the conductionbands are entangled with higher bands. The disentan-

vdW gap

a

b

kaM

kb

K’

K

Γ

(a) (b)

(c)

M = Mo, W X = S, Se

(d)

dz2 pz

dx2-y2 py

dxy px

dxz dyz

FIG. 1. (Color online) (a) Side view of the unit cell of the 2H-MX2 structure with M=Mo, W and X=S, Se. It containstwo MX2 monolayers separated by a Van der Waals gap. (b)Top view of the MX2 monolayer. The black lines indicatethe unit cell in ab plane. (c) The first Brillouin zone and highsymmetry points of the MX2 monolayer. (d) The Wannierfunctions of the MX2 monolayer, including five d-orbitals onM atom and three p-orbitals on each X atom.

glement approach23 was applied to bulk WS2 and WSe2.Once the Wannier functions were obtained, we followedRef. 25 to calculate the Berry curvature and integrate itover the Brillouin zone to obtain the Hall conductivity.

III. BAND STRUCTURE AND BERRYCURVATURE

In this section we present the electronic band structureand Berry curvature of monolayers of group-VI dichalco-genides, using MoS2 as an example. We show that inver-sion symmetry breaking gives rise to two physical quanti-ties, the spin moment and the Berry curvature, that canbe used to distinguish valley carriers.

Structurally, MoS2 can be regarded as strongly bondedtwo-dimensional S-Mo-S layers that are loosely coupledto one another by Van der Waals interactions. Withineach layer, the Mo and S atoms form hexagonal latticesin separate planes with each Mo atom coordinated by sixnearest-neighboring S atoms in the trigonal prismatic ge-ometry [Fig. 1(a) and (b)]. In its bulk form, MoS2 has theso-called 2H stacking order with space group P63/mmc(D4

6h), which is inversion symmetric [Fig. 1(a)]. Be-cause of the weak interlayer coupling, this layered com-pound can be easily exfoliated into monolayers by me-chanical26,27 and chemical28 means. In monolayer MoS2,the space group is reduced to P 6m2 (D1

3h) with explicitbreaking of inversion symmetry.

Before moving on to the discussion of coupled spin andvalley physics, we briefly study the orbital characters viathe Wannier functions. The partial density of states of

3

Mo-4d S-3p

(a) (b)

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

E

nerg

y (e

V)

1.0

0.0

-1.0

σ+ σ-

024 M MK’ΓK

FIG. 2. (Color online) The calculated electronic structure ofMoS2 with the spin-orbit coupling. (a) The partial density ofstates for Mo-4d and S-3p orbitals, respectively, in the unit ofstates/eV/cell. (b) The band structure with the projection ofspin operator sz (color map). The red and blue colors indi-cate the spin-up and -down states, respectively. The opticaltransitions between the VBM and the CBM are coupled exclu-sively with σ+ (σ−) circular polarizations at the inequivalentvalleys K (K′).10

monolayer MoS2 shown in Fig. 2(a) clearly shows thatin the energy range of -6 to 5 eV the contributions toelectronic states mainly come from the Mo d- and S p-orbitals, whereas other orbitals have vanishing contribu-tions. In Fig. 1(d), we plot the typical Wannier functionsfor five Mo d- and three S p-orbitals together with theunit cell in real space. If the SOC is turned on, the num-ber of orbitals will be doubled and altogether 22 bandsare formed. Under the crystal field of trigonal prismaticcoordination, the d-orbitals split into dz2, dxz, dyz,dx2−y2 , dxy and p-orbitals split into pz, px, py. Atthe two inequivalent valleys, K and K ′, the valence bandmaximum (VBM) is constructed by the Mo dx2−y2 , dxyorbitals with some mixing from the S px, py orbitals,while the conduction band minimum (CBM) is domi-nated by Mo dz2 orbitals. These orbital characters ofband-edges are in line with the analysis of the k ·p modelin Ref. 10.

Figure 2(b) shows the fully relativistic band structureof monolayer MoS2 with the projection of spin operatorsz, i.e., 〈ψnk| sz |ψnk〉, obtained from Wannier interpola-tion.29 We can see that there is a direct band-gap at thetwo inequivalent corners K and K ′ of the Brillouin zone.Furthermore, a large spin splitting (∼ 150 meV) appearsat the VBM with opposite spin moments at the two val-leys, as a result of inversion symmetry breaking.13 Thisindicates that in addition to their valley index, the valleycarriers in the valence bands can be also distinguished bytheir spin index. On the other hand, since the CBM state

-80

-60

-40

-20

0

20

40

60

80

(a)

Ω(k

) (at

omic

uni

t)

K MΓ K'M

-2

0

2

4

6

8

K

Ωs ( k

) (at

omic

uni

t)

MΓ K'M

(c)

(b)

-80-60-40-20-5520406080

Γ

K'

K

M

(d)

-2.0

-0.8

0.5

1.8

3.0

4.3

5.5

6.8

8.0

Γ

K'

K

M

FIG. 3. (Color online) The Berry curvatures of monolayerMoS2 along the high symmetry lines (a) and in the 2D k-plane(b). The spin Berry curvatures of monolayer MoS2 along thehigh symmetry lines (c) and in the 2D k-plane (d). All of theBerry curvatures are in the atomic unit (Bohr2).

is made of the Mo dz2 orbital, SOC is inactive and theCBM remains degenerate (to first order of the SOC).

In the presence of inversion symmetry breaking, thecharge carriers also acquire a valley-contrasting Berrycurvature.4,6 According to the Kubo-formula,30,31 theBerry curvature Ω (k) of the occupied states can be writ-ten as

Ω (k) =∑n

fnΩn (k) , (1)

and

Ωn (k) = −∑n′ 6=n

2Im 〈ψnk|vx |ψn′k〉 〈ψn′k|vy |ψnk〉(En′ − En)

2 ,

(2)where |ψnk〉 is the Bloch function with the eigenvalue En,fn the Fermi-Dirac distribution function, and vx(y) thevelocity operators. Here, we have used the maximallylocalized Wannier functions as the basis to calculate theBerry curvature [Eq. (2)] and the spin Berry curvature[below in Eq. (6)].25 Figure 3(a) shows Ω(k) of mono-layer MoS2 along the high-symmetry lines. We can seethat Ω(k) is significantly peaked at both K and K ′ butwith opposite signs. The k-space contrasting Ω(k) insystems without inversion symmetry is a key quantity tocharacterize the chirality of the Bloch electrons and is thebasis for valley-contrasting phenomena.4,5,10 Away fromthe two valleys, Ω(k) decays rapidly and vanishes at the Γand M points. We also plot the map distribution of Ω(k)in the 2D k-plane, as shown in Fig. 3(b), which clearlyshows the C3 symmetry of the system. This is in con-trast with an energy counter plot, which would displaythe C6 symmetry, i.e., the two valleys are energetically

4

10 Ω

s ( k)

Ene

rgy

(eV

)

2.0

2.2

2.4

0.0

1.0

2.0

P1

M Γ K M

-1.8

-1.6

-1.4

-1.2

0.0

1.0

2.0

P2

M Γ K M

Ene

rgy (

eV)

Ene

rgy (

eV)

Ene

rgy

(eV

)

310

Ωs (k

)3

Doping charge (e)

(a) (d)(b)

(c)

-8 -4 0 4 8 12-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

P2

P1

CBM

VBM

n-doping

p-doping

-0.5 0.0 0.5 1.0-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

P2

P1

CBM

VBM

σsxy (e

2/ )

FIG. 4. (Color online) (a) The intrinsic spin Hall conductivityσsxy (e2/~) as a function of the Fermi energy for monolayer

MoS2. The energy zero point (true Fermi level) is at theVBM. Two large peaks close to the CBM and VBM are de-noted by P1 and P2, respectively. (b)(c) The band structure(up panel) and the spin Berry curvature (down panel) whenthe Fermi level shifts to the positions of the peak P1 and P2,respectively. There is a small peak of Ωs with negative valuealong the M -Γ line in (c). (d) The n- and p-doping chargeas a function of the Fermi energy. The low doping regimesjust above the CBM and below the VBM (indicated by redcircles) are more relevant in experiments.

indistinguishable, but they can be distinguished by theirBerry curvatures.

The Berry curvature drives an anomalous transversevelocity in the presence of an electric field E:6

v⊥ = − e~E ×Ω(k) , (3)

which is responsible for the intrinsic contribution to theanomalous Hall effect.31 However, in our systems, thecharge carriers in the two valleys have opposite trans-verse velocities due to the opposite signs of the Berry

TABLE I. Comparison of the Berry curvatures at VBM andCBM between the present first-principles calculations (thefirst line) and the k · p model in Ref. 10 (the second line).Ωv(c)↑(↓) is the Berry curvature of the valence (conduction)

band with spin ↑ (↓), given in the unit of Bohr2.

MoS2 MoSe2 WS2 WSe2Ωv↑ 38.8 39.7 59.8 64.3

35.3 36.5 55.4 60.0Ωv↓ 31.6 30.0 34.9 34.7

29.5 28.4 34.2 33.3Ωc↑ −35.7 −36.8 −54.7 −59.2

−35.3 −36.5 −55.4 −60.0Ωc↓ −28.8 −27.3 −31.0 −30.8

−29.5 −28.4 −34.2 −33.3

curvatures. Hence, the total anomalous Hall conductiv-ity vanishes because of time reversal symmetry. If a fi-nite valley polarization can be generated, for example, byshining the sample with circularly polarized light, thena charge Hall current will appear.10 On the other hand,since both the valley and spin current remains invariantunder time-reversal, the valley Hall and the spin Halleffect can appear in time-reversal invariant systems, aslong as the inversion symmetry is broken.32

Finally, we compare the value of the Berry curvature atthe VBM and CBM from both the effective k · p model10

and first-principles calculation in Table I. The excellentagreement between them further confirms the validity ofthe k · p model.

IV. THE INTRINSIC SPIN HALL EFFECT

As discussed above, both the valley Hall and spinHall effect exist in MX2 monolayers due to the valley-contrasting Berry curvature. Note that the valley indexis defined only in the vicinity of the valleys, whereas thespin index is defined everywhere in the Brillouin zone.Therefore we will only calculate the intrinsic spin Hallconductivity. For hole-doped samples, when the Fermienergy lies between the spin-split VBM states, the valleyHall conductivity coincides with the spin Hall conductiv-ity.10

At zero-temperature and clean limit, the intrinsic spinHall conductivity (ISHC) tensor is given by

σsxy =

e

~

∫VG

d2k

(2π)2 Ωs (k) . (4)

For the convenience of discussion, in the following, wemultiply a factor 2e/~ to the calculated ISHC to convertits unit to charge conductivity. We can carry out thecalculation of σs

xy again using the Kubo-formulas33,34

Ωs (k) =∑n

fnΩsn (k) , (5)

and

Ωsn (k) = −

∑n′ 6=n

2Im 〈ψnk| jx |ψn′k〉 〈ψn′k|vy |ψnk〉(En′ − En)

2 , (6)

where jx is the spin current operator defined as 12 (szvx +

vxsz). We add a superscript s for the spin Berry curva-ture in order to distinguish them from the ordinary Berrycurvature in Eq. (1) and (2). We can see that Ωs(k) ofmonolayer MoS2 is peaked at both K and K ′ with thesame sign, as shown in Fig. 3(c). This can be under-stood as the following. At the VBM, sz remains a goodquantum number, and the spin Berry curvature is simplygiven by Ωs(K) = szΩ(K). As both sz and Ω(K) flipsign when K → −K, Ωs(K) remains the same. Fig-ure 3(d) shows the map distribution of Ωs(k) in the 2D

5

MoS2 2MoSe WS WSe2 2(a) (b) (c) (d)En

ergy

(eV)

CBM

VBM

MoS(e) 2 - bulk

- ML - ML - ML - ML

σsxy(103Ω-1cm-1)

MoSe(f) 2 - bulk WS(g) 2 - bulk WSe(h) 2 - bulk

σsxy(103Ω-1cm-1) σs

xy(103Ω-1cm-1) σsxy(103Ω-1cm-1)

-0.1 0.0 0.1

0

1

2

-0.1 0.0 0.1

0

1

2

-0.1 0.0 0.1

0

1

2

-0.1 0.0 0.1

0

1

2

-0.5 0.0 0.5-1

0

1

2

-0.5 0.0 0.5-1

0

1

2

-0.5 0.0 0.5-1

0

1

2

-0.5 0.0 0.5-1

0

1

2

CBM

VBM

σsxy (e

2/ ) σsxy (e

2/ ) σsxy (e

2/ ) σsxy (e

2/ )

Ener

gy (e

V)

FIG. 5. (Color online) The intrinsic spin Hall conductivityσsxy in the low doping regimes for the monolayer (a)-(d) and

bulk (e)-(h) of the MoS2, MoSe2, WS2, and WSe2, respec-tively. Dashed lines indicate the band-edges. Note that theband-gaps of bulk are smaller than the monolayer ones. Theunit of σs

xy is e2/~ (∼= 2.43×10−4Ω−1) for 2D monolayer sys-tem, whereas Ω−1cm−1 for 3D bulk system. For quantita-tively comparing the σs

xy in bulk and monolayer, one needs todivide the σs

xy in monolayer by its thickness.

k-plane. We observe that it has a clear C6 symmetryrather than the C3 symmetry in Fig. 3(b).

By integrating Ωs(k) over the occupied states, we ob-tain σs

xy as a function of Fermi level for monolayer MoS2,shown in Fig. 4(a). Here, we set the energy zero point(true Fermi level) at the VBM, and calculate σs

xy byrigidly shifting the Fermi level position. For n-dopedmonolayer MoS2, the calculated σs

xy sharply reaches its

maximum value of 0.89 e2/~ at 2.19 eV. When furtherincreasing the doping concentration, σs

xy displays a com-plex behavior with both dramatic oscillations and signchanges, but it eventually goes to zero above 4.5 eV. Forp-doped monolayer MoS2, σs

xy has two large peaks, re-

spectively, with positive value of 0.29 e2/~ at −1.52 eVand negative value of −0.32 e2/~ at −2.64 eV. In order to

TABLE II. The ISHC σsxy of monolayer MX2 calculated at

hole-doping concentration nh=1.0×1013 cm−2. The slopesof σs

xy when Fermi level lies inside the spin splitting gapsbelow the VBM are also listed. The first and second lines areobtained from first-principles calculation and two-bands k · pmodel10, respectively.

MoS2 MoSe2 WS2 WSe2slope (e2/~/eV) 0.21 0.24 0.17 0.18

0.20 0.23 0.20 0.23σsxy (10−2 e2/~) −0.58 −0.92 −1.13 −1.30

−0.57 −1.00 −1.33 −1.52

analyze the cause of the large peaks, we take P1 and P2

as examples, indicated in Fig. 4(a). When the Fermi levelis located at the positions of P1 and P2, the band struc-tures and spin Berry curvatures are shown in Fig. 4(b)and 4(c), respectively. We can see that Ωs(k) is oftenpeaked at the places where the Fermi level crosses sometiny band-gaps induced by the SOC. Similar behavior ofΩs(k) for the other large peaks of the σs

xy are also found.The appearance of large peaks of the positive (negative)Ωs(k) leads to the positive (negative) peaks of the σs

xy.

Although giant σsxy can be realized at those peak po-

sitions, such a high level of doping is unrealistic in ex-perimental situations. As indicated in Fig. 4(d), theFermi level position of P1 and P2 are 2.19 eV and −1.52eV, respectively, corresponding to electron concentrationne=1.73 e/cell (∼= 1.96×1015 cm−2) and hole concentra-tion nh=3.18 e/cell (∼= 3.60×1015 cm−2). This couldbe difficult in experimental conditions either by chemicaladsorption or by gate voltage. For example, the high-est carrier concentration in two-dimensional graphene isonly up to 1013 cm−2 (Ref. 26 and 35). In contrast, thelow doping regimes just above the CBM and below theVBM are more relevant in experiments, as indicated byred circles in Fig 4(a) and 4(d). In the following, weonly focus on this regime.

Figure 5(a)-(d) show the σsxy in the low doping regimes

for monolayer MoS2, MoSe2, WS2, and WSe2, respec-tively. We can clearly see that σs

xy increases with theSOC strength as the atoms becomes heavier, and the p-doped samples generally has larger σs

xy than the n-dopedsamples. This is due to the large spin splitting at theVBM [see Fig. 2(b)]. Focusing on the valence bands, wefind σs

xy is proportional to the Fermi energy. The ex-tracted slopes are listed in Table II together with theresults from the k · p model.10. When the Fermi level liesinside the spin splitting gaps, the hole concentration is onthe order of 1013 cm−2, which is realistic for experiments.Taken nh=1.0×1013 cm−2 as an example, we list the cal-culated σs

xy from both present first-principles calculation

and the k · p model10 in Table II. The ISHC σsxy listed

in Table II are also comparable to those in p(n)-dopedsemiconductors GaAs, Si, Ge, and AlAs.33,34

Two remarks are in order. First, here we only com-pared the ISHC for p-doped samples for both first-principles calculations and the k ·p method. For n-dopedsamples, the situation is more complicated. As we cansee in Fig. 6, in both WS2 and WSe2 the conduction bandhas a second local minimum between Γ and K, which isvery close to the band-edge at K. Even under light dop-ing (∼ 1.0 × 1013 cm−2), both minima will be occupiedand contribute to the total ISHC, rendering the compar-ison between first-principles and k · p method meaning-less. This shows the limitation of the k · p method: itonly captures the physics around the K point, and first-principles study give us a more complete picture. We alsonote that there is a small spin splitting at the CBM forboth WS2 and WSe2. This is due to the much heavierW atom compared to Mo (SOC scales as Z4, where Z is

6

(a) WS2 (b) WSe2En

ergy

(eV)

-1

0

1

2

M Γ K M

nh=1.0x1013 cm-2

ne=1.0x1013 cm-2 CBM

VBM

-1

0

1

2

nh=1.0x1013 cm-2

ne=1.0x1013 cm-2 CBM

M Γ K M

VBM

FIG. 6. (Color online) The band structure of WS2 and WSe2monolayers. The CBM of WSe2 monolayer still locates at Kpoint. The dashed line indicate the position of the Fermi levelat carrier concentration of 1.0 × 1013 cm−2 for both p- andn-doped samples.

the atomic number). In the k ·p model, this splitting canbe taken into account by considering the second-order ef-fect due to the SOC. Second, one may notice that σs

xy isnonzero in the band-gap. These nonzero values are notdue to numerical errors, but actually reflect the finite hy-bridizations in real materials, similar to what has beenreported in GaAs and Si.33 The MX2 monolayer studiedhere can be viewed as a generalization of the concept ofspin Hall insulator proposed by Murakami et al.,36 suchas PbTe, which is a conventional band insulator but hasnonzero σs

xy without any doping.

As a comparison we also calculate σsxy for bulk MoS2,

MoSe2, WS2, and WSe2, shown in Fig. 5(e)-(h). Theshapes of σs

xy for all bulk dichalcogenides look rather sim-ilar, though the magnitudes and positions of the peaksmay differ. In bulk, σs

xy is zero immediately below theVBM, which is different from the immediate increasingof the σs

xy in monolayers. The reason is that the VBM inbulk always locates at the Γ point, which has negligiblysmall contribution to σs

xy. To compare with monolayers,we divide the σs

xy in monolayer by its thickness and findthat at the same Fermi level the σs

xy in bulk is about anorder of magnitude smaller than that in monolayers.

Finally we mention that here we only considered the

intrinsic contribution to the spin Hall effect, in which thespin Hall current is driven by the Berry curvature of theBloch bands. There are also extrinsic contributions com-ing from scattering of impurities and phonons. When thesample is hole-doped, the effect of phonon scattering onthe SHC should be weak because the phonon scatteringwill mostly contribute to intra-valley scattering, in whichthe spin z-component is nearly conserved due to the largespin-orbit splitting at the valence band top. On the otherhand, impurity scattering can provide the large momen-tum transfer needed for the inter-valley scattering, andtheir effect on valley-dependent transport properties re-mains to be investigated.

V. SUMMARY

In summary, using the first-principles calculations, wehave investigated the intrinsic spin Hall effect in mono-layers MoS2, MoSe2, WS2, as well as WSe2, driven byvalley-contrasting Berry curvature. We find that theISHC is comparable to that in p(n)-doped semiconduc-tors GaAs, Si, Ge, and AlAs.33,34 We show that the ef-fective model may not be adequate to describe the low-energy dynamics in WS2 and WSe2 monolayers. We alsocalculated the ISHC in inversion-symmetric bulk systemsand find that it is about an order of magnitude smallerthan the ISHC in monolayers. The large ISHC, plus otherinteresting physical properties of these materials, such asgiant spin splitting at VBM13 and valley-selective circu-lar dichroism,10,16–18 characterize these materials as anexciting platform for the application of the valleytronicsand spintronics.

ACKNOWLEDGMENTS

W.F. was partly supported by the Laboratory DirectedResearch and Development Program of ORNL. W.Y. wassupported by the Research Grant Council of Hong Kongunder Grants No. HKU706412P, Y.Y. by the MOSTProject of China (Grants No. 2011CBA00100) and NSFof China (Grants No. 10974231 and 11174337), and W.Z.and D.X. by the U.S. Department of Energy, Office of Ba-sic Energy Sciences, Materials Sciences and EngineeringDivision. We also thank Supercomputing Center of Chi-nese Academy of Sciences (SCCAS) and Texas AdvancedComputing Center (TACC) for the computational sup-ports.

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