Coupled spin and valley physics in monolayers of MoS2 and other group-VI dichalcogenides
CCoupled spin and valley physics in monolayers of MoS and other group-VI dichalcogenides Di Xiao, ∗ Gui-Bin Liu, Wanxiang Feng,
1, 3, 4
Xiaodong Xu,
5, 6 and Wang Yao † Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee, 37831, USA Department of Physics and Center of Theoretical and Computational Physics, The University of Hong Kong, Hong Kong, China Department of Physics and Astronomy, University of Tennessee, Knoxville, Tennessee 37996, USA Institute of Physics and Beijing National Laboratory for Condensed Matter Physics, Chinese Academy of Sciences, Beijing 100190, China Department of Physics, University of Washington, Seattle, Washington 98195, USA Department of Material Science and Engineering, University of Washington, Seattle, Washington 98195, USA
We show that inversion symmetry breaking together with spin-orbit coupling leads to coupled spin and valleyphysics in monolayers of MoS and other group-VI dichalcogenides, making possible controls of spin andvalley in these 2D materials. The spin-valley coupling at the valence band edges suppresses spin and valleyrelaxation, as flip of each index alone is forbidden by the valley contrasting spin splitting. Valley Hall andspin Hall effects coexist in both electron-doped and hole-doped systems. Optical interband transitions havefrequency-dependent polarization selection rules which allow selective photoexcitation of carriers with variouscombination of valley and spin indices. Photo-induced spin Hall and valley Hall effects can generate long livedspin and valley accumulations on sample boundaries. The physics discussed here provides a route towardsthe integration of valleytronics and spintronics in multi-valley materials with strong spin-orbit coupling andinversion symmetry breaking. PACS numbers: 73.63.-b, 75.70.Tj, 78.67.-n
Since the celebrated discovery of graphene [1–3], there hasbeen a growing interest in atomically thin two-dimensional(2D) crystals for potential applications in next-generationnano-electronic devices [4, 5]. Layered transition-metaldichalcogenides represent another class of materials that canbe shaped into monolayers [4], which display distinct physicalproperties from their bulk counterpart [6–9]. Recent exper-iments have demonstrated that MoS , a prototypical group-VI dichalcogenide, crossovers from an indirect-gap semi-conductor at multilayers to a direct band-gap one at mono-layer [6, 7]. The direct band-gap is in the visible frequencyrange, most favorable for optoelectronic applications. Mono-layer MoS transistor was also realized, demonstrating aroom-temperature mobility over 200 cm /(V · s) [8].In monolayer MoS , the conduction and valence bandedges are located at the corners ( K points) of the 2D hexag-onal Brillouin zone [10–12]. Similar to graphene, the twoinequivalent valleys constitute a binary index for low energycarriers. Because of the large valley separation in momentumspace, the valley index is expected to be robust against scat-tering by smooth deformations and long wavelength phonons.The use of valley index as a potential information carrier wasfirst suggested in the studies of conventional semiconductorssuch as AlAs and Si [13]. With the emergence of graphene,the concept of valleytronics based on manipulating the valleyindex has attracted great interests [14–18].MoS monolayers have two important distinctions fromgraphene. First, inversion symmetry is explicitly broken inmonolayer MoS , which can give rise to the valley Hall effectwhere carriers in different valleys flow to opposite transverseedges when an in-plane electric field is applied [15]. Inversionsymmetry breaking can also lead to valley-dependent opticalselection rules for inter-band transitions at K points [16]. Sec-ond, MoS has a strong spin-orbit coupling (SOC) originatedfrom the d -orbitals of the heavy metal atoms [12], and can be an interesting platform to explore spin physics and spin-tronics applications absent in graphene due to its vanishingSOC [19, 20].In this Letter, we show that inversion symmetry break-ing together with strong SOC lead to coupled spin and val-ley physics in monolayer MoS and other group-VI dichalco-genides, making possible spin and valley control in these 2Dmaterials. We find the conduction and valence band edgesnear K points are well described by massive Dirac fermionswith strong valley-spin coupling in the valence band, whichhas several important consequences. First, the valley Hall ef-fect is accompanied by a spin Hall effect in both electron-doped and hole-doped systems [21–24]. Second, spin and val-ley relaxation are suppressed at the valence band edges as flipof each index alone is forbidden by the valley-contrasting spinsplitting ( ∼ M X ( M = Mo, W, X = S, Se), de-scribed below using MoS as an example. Structurally, MoS can be regarded as strongly bonded 2D S-Mo-S layers that areloosely coupled to one another by Van der Waals interactions.Within each layer, the Mo and S atoms form 2D hexagonallattices, with the Mo atom being coordinated by the six neigh-boring S atoms in a trigonal prismatic geometry (Fig. 1a-b). In a r X i v : . [ c ond - m a t . m e s - h a ll ] M a y (a) (b)(c) FIG. 1. (color online). (a) The unit cell of bulk H -MoS , which hasthe inversion center located in the middle plane. It contains two unitcells of MoS monolayers, which lacks an inversion center. (b) Topview of the MoS monolayer. R i are the vectors connecting nearestMo atoms. (c) Schematic drawing of the band structure at the bandedges located at the K points. its bulk form, MoS has the H stacking order with the spacegroup D h , which is inversion symmetric. When it is thinneddown to a monolayer, the crystal symmetry reduces to D h ,and inversion symmetry is explicitly broken: taking the Moatom as the inversion center, an S atom will be mapped ontoan empty location. As a consequence, the effects we predicthere are expected only in thin films with odd number of lay-ers, since inversion symmetry is preserved in films with evennumber of layers.We start by constructing a minimal band model on thebasis of general symmetry consideration. The band struc-ture of MoS , to a first approximation, consists of partiallyfilled Mo d -bands lying between Mo-S s - p bonding and anti-bonding bands [25]. The trigonal prismatic coordination ofthe Mo atom splits its d -orbitals into three groups: A ( d z ) , E ( d xy , d x − y ) and E (cid:48) ( d xz , d yz ) . In the monolayer limit, thereflection symmetry in the ˆ z direction permits hybridizationonly between A and E orbitals, which opens a band gap atthe K and − K points [25], schematically shown in Fig. 1c.The group of the wave vector at the band edges ( K ) is C h and the symmetry adapted basis functions are | φ c (cid:105) = | d z (cid:105) , | φ τv (cid:105) = 1 √ | d x − y (cid:105) + iτ | d xy (cid:105) ) , (1)where the subscript c ( v ) indicates conduction (valence) band,and τ = ± is the valley index. The valence-band wave func-tions at the two valleys, | φ + v (cid:105) and | φ − v (cid:105) , are related by time-reversal operation. To first order in k , the C h symmetry dic-tates that the two-band k · p Hamiltonian has the form ˆ H = at ( τ k x ˆ σ x + k y ˆ σ y ) + ∆2 ˆ σ z , (2) TABLE I. Fitting result from first-principles band structure calcu-lations. The monolayer is relaxed. The sizes of spin splitting λ at valence band edge were previously reported in the first principlestudies [12]. The unit is ˚A for a , and eV for t , ∆ and λ . Ω ( Ω ) isthe Berry curvature in unit of ˚A , evaluated at − K point for the spinup (down) conduction band. a ∆ t λ Ω Ω MoS where ˆ σ denotes the Pauli matrices for the two basis functions, a is the lattice constant, t the effective hopping integral, and ∆ the energy gap. These parameters are obtained by fittingto first-principles band structure calculations and are listed inTable. I for the four group-VI dichalcogenides [26]. We notethat the same effective Hamiltonian also describes monolayergraphene with staggered sublattice potential [15, 16]. Thisis not surprising, as both systems have the same symmetryproperties. What distinguishes MoS from graphene is thestrong SOC originated from the metal d -orbitals. The con-duction band-edge state is made of d z orbitals and remainsspin-degenerate at K points, whereas the valence band-edgestate splits. Approximating the SOC by the intra-atomic con-tribution L · S , we find the total Hamiltonian given by ˆ H = at ( τ k x ˆ σ x + k y ˆ σ y ) + ∆2 ˆ σ z − λτ ˆ σ z −
12 ˆ s z , (3)where λ is the spin-splitting at the valence band top causedby the SOC and ˆ s z is the Pauli matrix for spin. The spin-up( ↑ ) and spin-down ( ↓ ) components are completely decoupledand s z remains a good quantum number. We emphasize thatthe spin splitting does not depend on the model details; it isa general consequence of inversion symmetry breaking, sim-ilar to the Dresselhaus spin splitting in zinc-blende semicon-ductors [27]. Time-reversal symmetry requires that the spinsplitting at different valleys must be opposite (Fig. 1c) [28].The valley Hall and spin Hall effects are driven by theBerry phase associated with the Bloch electrons. It has beenwell established that in the presence of an in-plane electricfield, an electron will acquire an anomalous velocity propor-tional to the Berry curvature in the transverse direction [29],giving rise to an intrinsic contribution to the Hall conductiv-ity [30], σ int = ( e / (cid:126) ) (cid:82) [ d k ] f ( k )Ω( k ) , where f ( k ) is theFermi-Dirac distribution function, and [ d k ] is a shorthandfor d k / (2 π ) . The Berry curvature is defined by Ω n ( k ) ≡ ˆ z · ∇ k × (cid:104) u n ( k ) | i ∇ k | u n ( k ) (cid:105) , where | u n ( k ) (cid:105) is the periodicpart of the Bloch function and n is the band index. For mas-sive Dirac fermions described by the effective Hamiltonian inEq. (3), the Berry curvature in the conduction band is [15]: Ω c ( k ) = − τ a t ∆ (cid:48) [∆ (cid:48) + 4 a t k ] / . (4)Note that the Berry curvatures have opposite sign in oppositevalleys. In the valence band, we have: Ω v ( k ) = − Ω c ( k ) .In the same valley, the Berry curvature is dependent on spinthrough the spin-dependent band gap: ∆ (cid:48) ≡ ∆ − τ s z λ . Thecurvature is nearly constant in the neighborhood of K pointssince ∆ (cid:29) atk (Table I). The valley Hall conductivity (in unitof e/ (cid:126) ) is then: σ nv = 2 (cid:90) [ d k ]( f n, ↑ ( k )Ω n, ↑ ( k ) + f n, ↓ ( k )Ω n, ↓ ( k )) , and the spin Hall conductivity (in unit of e/ ) is: σ ns = 2 (cid:90) [ d k ]( f n, ↑ ( k )Ω n, ↑ ( k ) − f n, ↓ ( k )Ω n, ↓ ( k )) , where the integration is performed over the neighborhood ofone K point. For moderate hole doping with Fermi energylying between the two split valence band tops (illustrated bythe dot-dashed line in Fig. 2b), the valley and spin Hall con-ductivities of holes are the same, given by σ hs = σ hv = 1 π µ ∆ − λ (5)for µ (cid:28) ∆ − λ , where µ is the Fermi energy measured fromthe valence band maximum. If the system is electron doped,we must consider both conduction bands which are degenerateat K points and have small spin-splitting quadratic in k (seeFig. 2b). We find that σ ev = 1 π ∆∆ − λ µ , σ es = 1 π λ ∆ − λ µ . (6)where µ is the Fermi energy measured from the conductionband minimum. The spin Hall conductivity is about λ/ ∆ ofthe valley Hall conductivity.The robustness of the valley and spin Hall effects is closelyrelated to the relaxation time of the valley and spin index.Flipping of valley index require atomic scale scatters, sincethe two valleys are separated by a wave vector comparablewith the size of Brillouin zone. Spin flips requires the cou-pling with magnetic defects, since s z is a good quantum num-ber at the conduction and valence band edges. In the conduc-tion band, valley scattering could be slow in the bulk at theclean limit, but will be facilitated on the boundaries by valleymixing except with perfect zigzag edge. In the valence band,by the relatively large valley-contrasting spin splitting ( ∼ σ ± circular po-larization is given by P ± ( k ) ≡ P x ( k ) ± i P y ( k ) , where P α ( k ) ≡ m (cid:104) u c ( k ) | (cid:126) ∂ ˆ H∂k α | u v ( k ) (cid:105) is the interband matrix el-ement of the canonical momentum operator and m is the free σ + σ − K - K ω u ω d ↑ ↓ ↓ ↑ (c) (d) E"E" (a) (b)
Hole doped system E" Electron doped system E" FIG. 2. (color online). Coupled spin and valley physics in monolayergroup-VI dichalcogenides. The electrons and holes in valley K aredenoted by white ‘ − ’, ‘ + ’ symbol in dark circles and their counter-parts in valley − K are denoted by inverse color. (a) Valley and spinHall effects in electron and hole doped systems (see text). (b) Val-ley and spin optical transition selection rules. Solid (dashed) curvesdenote bands with spin down (up) quantized along the out-of-planedirection. The splitting in the conduction band is exaggerated. ω u and ω d are respectively the transition frequencies from the two splitvalence band tops to the conduction band bottom. (c) Spin and val-ley Hall effects of electrons and holes excited by linearly polarizedoptical field with frequency ω u . (d) Spin and valley Hall effects ofelectrons and holes excited by two-color optical fields with frequen-cies ω u and ω d and opposite circular polarizations. electron mass [16, 31]. For transitions near K points, we find |P ± ( k ) | = m a t (cid:126) (1 ± τ ∆ (cid:48) √ ∆ (cid:48) + 4 a t k ) . (7)Since ∆ (cid:48) (cid:29) atk , the interband transitions are then coupledexclusively with σ + ( σ − ) circularly polarized optical field atthe K ( − K ) valley. Optical field couples only to the orbitalpart of the wave function and spin is conserved in the opti-cal transitions. By the valley-contrasting spin splitting of thevalence band tops, the valley optical selection rule becomesspin-dependent selection rules, as illustrated in Fig. 2b. ω d and ω u denote here the two band edge excitonic transition fre-quencies from the spin-split valence band tops (see Fig. 2b).Because of the spin-valley coupling and the valley optical se-lection rule from inversion symmetry breaking, spin and lightpolarization are related in the opposite ways at the two fre-quencies, similar to the interband transition involving heavyhole and light hole in III-V semiconductors. One may expecta sign reversal for magneto-optical effects such as Faraday ro-tation and Kerr rotation by spin-polarized electron when thefrequency changes from ω u to ω d .Selective excitation of carriers with various combination ofvalley and spin index becomes possible using optical fields ofdifferent circular polarizations and frequencies. Optical fieldwith σ + circular polarization and frequency ω u ( ω d ) can gen-erate spin up (down) electrons and spin down (up) holes in TABLE II. Photo-induced spin, valley and charge Hall effects. σ e ( h ) s and σ e ( h ) v denote the spin and valley Hall conductivity respectivelycontributed by the photo-excited electrons (holes). σ is the totalcharge Hall conductivity from both carriers. All conductivities arenormalized by the photo-excited carrier density of electron or hole,and only intrinsic contribution is considered.light frequency σ es σ hs σ ev σ hv σ & polarization ( ω u , X or Y ) e Ω e Ω e (cid:126) Ω e (cid:126) Ω ( ω d , X or Y ) − e Ω − e Ω e (cid:126) Ω e (cid:126) Ω ( ω u , σ +) e Ω e Ω e (cid:126) Ω e (cid:126) Ω e (cid:126) Ω ( ω d , σ − ) − e Ω − e Ω e (cid:126) Ω e (cid:126) Ω − e (cid:126) Ω valley K , while the excitation in the − K valley is simplythe time reversal of the above [32]. Such a spin and valleydependent selection rule can be used to generate long livedspin and valley accumulations on sample boundaries in a Hallbar geometry. Consider the photo-excitation of electrons andholes, which are then dissociated by an in-plane electric field,driving a longitudinal charge current (Fig. 2c-d). The photo-excited electrons and holes will also acquire opposite trans-verse velocities because of the Berry curvatures in the conduc-tion and valence band, and moved to the two opposite bound-aries of the sample. This leads to Hall current of valleys, spinsor charges, depending on the polarization and frequency of theoptical field. In Table II, we give the signs and order of mag-nitude estimation of the valley, spin and charge Hall currentsin the clean limit.Excitation with circular polarizations will generate a chargeHall current which can be detected as a voltage. The sign ofthe voltage is exclusively determined by the circular polariza-tion and is independent of the frequency. Excitation with lin-ear polarizations has more interesting consequences. For ex-ample, by excitation with linearly polarized optical field withfrequency ω u , there is a spin Hall current and a valley Hallcurrent in the absence of a charge Hall current. Spin up elec-trons from the K valley and spin up holes from the − K valleyare accumulated on one boundary, while their time reversalsare accumulated on the other boundary (Fig. 2c). Thus, eachboundary can remain charge neutral while carrying a net spinpolarization as well as a net valley polarization. Recombina-tion of these excess electrons and holes are forbidden by theoptical transition selection rules unless assisted by processeswhich flip both the valley and spin index. Holes are expectedto have much longer spin and valley lifetimes on the bound-ary. Thus electrons will get unpolarized first and recombinewith the spin and valley polarized holes, accompanied by theemission of photons with opposite circular polarizations onthe two boundaries. If there is strong valley mixing for elec-trons on the boundary, the decay of the overall spin and valleypolarization is determined by the spin relaxation time of theelectrons.Another interesting excitation scenario is by a non- degenerate optical excitation, consisted of a σ + polarizedcomponent with frequency ω u and a σ − polarized componentwith frequency ω d . This will excite spin up electrons and spindown holes in both valleys. The spin Hall and charge Hall cur-rents from the electrons will largely cancel with those from theholes, while the valley Hall currents from electrons and holesadd constructively (see Table II). The electrons and holes ac-cumulated on the same boundary are of opposite spin and val-ley indice. When electrons get valley unpolarized, they canrecombine with the spin and valley polarized holes, accompa-nied by the photon emission with polarization and frequency ( σ + , ω u ) on one boundary and ( σ − , ω d ) on the other. Thisprocess may provide a direct measurement on the valley life-time of electrons on the boundary.In summary, we have predicted the valley dependent op-tical selection rules for interband transitions near K -pointsin monolayer MoS and other group VI transition metaldichalcogenides. The spin-orbit interaction from the metal d -orbitals further leads to strong coupling of spin and val-ley degrees of freedom, which makes possible selective pho-toexcitation of carriers with various combination of valley andspin indices. We have also predicted the coexistence of valleyHall and spin Hall effects in n -doped and p -doped systems,and proposed photo-induced spin Hall and valley Hall effectsfor generating spin and valley accumulations on edges. Thestrong spin-valley coupling can further protect each index:with the valley dependent spin splitting of O (0 . eV at thevalence band top, flip of spin and valley alone is energeticallyforbidden. These effects suggest the potential of integratedspintronic and valleytronic applications. In hybrid systems ofthese monolayers with other spintronics materials, spin indexmay be used as a universal information carrier across differ-ent materials, while valley index provides a unique ancillaryinformation carrier in the monolayers with logic operationsbetween the two enabled by the spin-valley coupling.We acknowledge useful discussions with D. Mandrus,S. Okamoto, and J.-Q. Yan. We are grateful to W.-G. Zhufor technical support in first-principles band structure calcu-lations. This work was supported by the U.S. Department ofEnergy, Office of Basic Energy Sciences, Materials Sciencesand Engineering Division (D.X.), by Research Grant Councilof Hong Kong (G.B.L. and W.Y.), and by the Laboratory Di-rected Research and Development Program of ORNL (W.F.). Note added. – Recently, experimental evidences on the op-tical selection rules for inter-band transitions at K points arereported in monolayer MoS [33]. We also note an indepen-dent theoretical work discussing the circular dichroism in theentire Brillouin zone [34]. ∗ Corresponding author: [email protected] † Corresponding author: [email protected][1] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang,Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov,Science , 666 (2004). [2] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I.Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov,Nature , 197 (2005).[3] Y. Zhang, Y.-W. Tan, H. L. Stormer, and P. Kim, Nature ,201 (2005).[4] K. S. Novoselov, D. Jiang, F. Schedin, T. J. Booth, V. V. Khotke-vich, S. V. Morozov, and A. K. Geim, Proc. Natl Acad. Sci.USA , 10451 (2005).[5] C. Lee, Q. Li, W. Kalb, X.-Z. Liu, H. Berger, R. W. Carpick,and J. Hone, Science , 76 (2010).[6] A. Splendiani, L. Sun, Y. Zhang, T. Li, J. Kim, C.-Y. Chim,G. Galli, and F. Wang, Nano Lett. , 1271 (2010).[7] K. F. Mak, C. Lee, J. Hone, J. Shan, and T. F. Heinz, Phys. Rev.Lett. , 136805 (2010).[8] B. Radisavljevic, A. Radenovic, J. Brivio, V. Giacometti, andA. Kis, Nat. Nano. , 147 (2011).[9] T. Korn, S. Heydrich, M. Hirmer, J. Schmutzler, and C. Schller,arXiv:1106.2951 (2011).[10] T. Li and G. Galli, J. Phys. Chem. C , 16192 (2007).[11] S. Lebgue and O. Eriksson, Phys. Rev. B , 115409 (2009).[12] Z. Y. Zhu, Y. C. Cheng, and U. Schwingenschl¨ogl, Phys. Rev.B , 153402 (2011).[13] O. Gunawan, Y. P. Shkolnikov, K. Vakili, T. Gokmen, E. P.De Poortere, and M. Shayegan, Phys. Rev. Lett. , 186404(2006).[14] A. Rycerz, J. Tworzydlo, and C. W. J. Beenakker, Nature Phys. , 172 (2007).[15] D. Xiao, W. Yao, and Q. Niu, Phys. Rev. Lett. , 236809(2007).[16] W. Yao, D. Xiao, and Q. Niu, Phys. Rev. B , 235406 (2008).[17] F. Zhang, J. Jung, G. A. Fiete, Q. Niu, and A. H. MacDonald,Phys. Rev. Lett. , 156801 (2011).[18] Z. Zhu, A. Collaudin, B. Fauque, W. Kang, and K. Behnia,Nature Phys. advance online publication , (2011).[19] H. Min, J. E. Hill, N. A. Sinitsyn, B. R. Sahu, L. Kleinman, andA. H. MacDonald, Phys. Rev. B , 165310 (2006). [20] Y. Yao, F. Ye, X.-L. Qi, S.-C. Zhang, and Z. Fang, Phys. Rev.B , 041401 (2007).[21] S. Murakami, N. Nagaosa, and S.-C. Zhang, Science , 1348(2003).[22] J. Sinova, D. Culcer, Q. Niu, N. A. Sinitsyn, T. Jungwirth, andA. H. MacDonald, Phys. Rev. Lett. , 126603 (2004).[23] Y. K. Kato, R. C. Myers, A. C. Gossard, and D. D. Awschalom,Science , 1910 (2004).[24] J. Wunderlich, B. Kaestner, J. Sinova, and T. Jungwirth, Phys.Rev. Lett. , 047204 (2005).[25] L. F. Mattheiss, Phys. Rev. B , 3719 (1973).[26] We have also included second-order terms, but found they arenegligible compared with the first-order terms.[27] G. Dresselhaus, Phys. Rev. , 580 (1955).[28] The valence band spin splitting at K points is the consequenceof spin orbit coupling and inversion symmetry breaking inmonolayer. It is comparable in size with the valence band split-ting at K points seen in the bilayer and in the bulk. But thelatter is from the interlayer coupling [25], and the presence ofboth inversion symmetry and time reversal symmetry forbidsany spin splitting at K points.[29] D. Xiao, M.-C. Chang, and Q. Niu, Rev. Mod. Phys. , 1959(2010).[30] N. Nagaosa, J. Sinova, S. Onoda, A. H. MacDonald, and N. P.Ong, Rev. Mod. Phys. , 1539 (2010).[31] I. Souza and D. Vanderbilt, Phys. Rev. B , 054438 (2008).[32] We use the convention that an unoccupied spin up (down) statein the valence band is referred as a spin down (up) hole. Byexcitation at frequency ω dd