Chenglong Che(車成龍), Yawei Lv(呂亞威), and Qingjun Tong(童慶軍)

School of Physics and Electronics,Hunan University,Changsha 410082,China

Keywords: moir′e superlattice,valleytronics,transition metal dichalcogenide,quantum anomalous Hall state

1.Introduction

Many two-dimensional (2D) materials have degenerate energy extrema located at inequivalent momentum points in the Brillouin zone,which introduces the valley degree of freedom for quasiparticles.[1,2]This valley index can be used to encode information, forming the basis of valleytronics.[3,4]Because different valleys are separated from each other in the momentum space,the scattering between them is largely suppressed.A prerequisite for the realization of this tempting idea is to manipulate the valley degree of freedom.[5]

The breaking of inversion symmetry has been recognized to be crucial to distinguish and further manipulate the valley degree of freedom.[6]It has been shown that,in Dirac materials without inversion symmetry,there exist valley contrasting Berry curvature and orbital magnetic moment,which result in valley dependent Hall transport and optical selection rules for interband transitions.[7-9]The latter effect means that a circularly polarized light can selectively pump the two valleys,suggesting an optical way to realize valley polarization.Following these theoretical predictions, subsequent experiments have successfully generated valley polarization in monolayer TMDs, which break intrinsically the inversion symmetry in their monolayer limit.[10-12]Besides valley polarization, valley current is another requirement to function the valleytronic devices.Several experimental activities have been taken in TMD materials based on the valley Hall effect.[13-15]However, because of the large intrinsic band gap in TMDs, the Berry curvature is relatively small,which makes the generated valley contrasting physics not robust in practical experiments.We also notice that the quantum valley Hall state and valley polarization quantum anomalous Hall state in-principle can provide topologically protected valley current, however, the realization of which needs carefully engineered spin-orbital coupling(SOC)and applied external magnetic field.

英語里有很多單詞在音和拼寫規(guī)則上是有規(guī)律可循的，可將這些單詞歸為一類來學習。比如在教授四年級新單詞toy時，我讓學生先回憶boy的讀音，通過對boy與toy的拼寫觀察，孩子們能迅速讀出toy。再例如在講解park時，先拿出舊單詞car,學生們即可正確遷移出park的讀音。教授完新單詞后，老師還可以進一步延伸和鞏固音標，創(chuàng)編一些同類的單詞，如：soy、doy、woy、art、arm、dark、gard等等讓學生來拼讀，教授效果相當理想。

Stacking different 2D materials to form van der Waals(vdW)heterostructures offers exciting opportunities for studying low-dimensional physics and functional devices.[16]A general feature of vdW heterostructures is the formation of a long-range quasi-periodic structure, i.e., moir′e pattern, due to the lattice mismatch and twist between the constituent layers.[17-20]For a long-wavelength moir′e pattern, within a length scale much larger than the lattice constant but small compared to the moir′e period, the atomic registry has negligible difference from commensurate bilayers, while varies smoothly over long range.[21]This unique atomic superlattice offers new opportunity to explore novel quantum phases in the 2D limit, such as topological insulator superlattice,[21]moir′e skyrmions,[22]moir′e excitons[23-27]and strongly correlated electronic states.[28-33]These novel quantum phases have opened up a new avenue for studying valleytronics.

In this paper, we show that Dirac fermions with tunable large Berry curvatures can be realized in the moir′e pattern formed by the TMD heterobilayers.In particular, our firstprinciples results show that the moir′e potential has two nearly degenerate extrema, which resembles a magnified gapped graphene lattice and gives rise to valley-indexed moir′e Dirac fermions.Applying on-site potential on the two outmost zigzag edges of the moir′e lattice,we can tune the gapped flatband edge states into gapless helical ones.In addition,we find that in short-period moir′e lattice, the two moir′e valleys become asymmetric due to the triangular wrapping effect,which can give rise to a net spin Hall current in the presence of an in-plane electric field.Interestingly,applying a circularly polarized light can drive the moir′e system into quantum anomalous Hall phase with topological chiral edge states.Our result provides an interesting possibility to explore the moir′e-scale spin and valley physics in vdW heterostructures.

2.Valley contrasting physics in massive Dirac model

對第三層級的研究對象進行建模后，參考埃森曼的探討習慣和伊塔羅·格伯利尼的建筑“構(gòu)成符號”分類[1]41，將對象的建筑符號分為平面符號、連接符號、圍護符號、相互交流符號和屋頂符號等5個部分進行圖解分析。且因為 “能指”與“所指”的任意性，為了更貼近集中含義3），在對應(yīng)語義三角關(guān)系前已經(jīng)對研究對象的設(shè)計語境（其內(nèi)容包括設(shè)計師言論、地方文化、地理氣候等）進行了資料收集。

Before discussing the moir′e Dirac fermions in TMDs heterobilayer, we first review briefly the valley contrasting physics in massive Dirac model that describes the low-energy physics of monolayer TMDs.TMDs have chemical formulaMX2,whereMis a transition metal element(Mo or W)andXis a chalcogenide element (S, Se or Te).Similar to graphite,bulk TMDs are indirect bandgap semiconductors,and formed by stacking different layers together by weak vdW force.Within each layer, the metal atoms are coordinated by three adjacent chalcogenide elements in a triangular prism geometry,together forming a 2D honeycomb lattice from a top view.Unlike bulk TMDs, monolayer TMDs have direct bandgap located at the±Kpoints in the first Brillouin zone, giving the valley degree of freedom.[3,4]We notice that the inversion symmetry is intrinsically broken for monolayer TMDs.Arising from the d orbital nature, a strong SOC of several hundred meV is found in the valence band,resulting in spin-valley locked physics.[9]

ΩwhereHtFis called the Floquet Hamiltonian,which can be diagonalized in the extended space|α,n〉consisting of the electronic basis|α〉and photonic basis|n〉.

whereσis the Pauli matrix for the basis functions consisting of Bloch states at±Kpoints,ais the lattice constant,tis the effective hopping integral,τ=±1 is the valley index,Mis the Dirac mass(energy gap at±Kpoints),andszis the Pauli matrix for spin.In this model,the spin-up(↑)and spin-down(↓)components are completely decoupled andszremains a good quantum number.Figure 1(a) shows the energy spectrum of monolayer MoS2around the two valleys, which has a large SOC-induced splitting of 2λat the valence bands.The Dirac model has nontrivial local band topology, which is characterized by the Berry curvature.In a 2D material,it is defined asΩn(k)= ?z·?k×〈un(k)|i?k|un(k)〉, withnbeing the band index.From Eq.(1), the Berry curvature for the conduction band is

whereM＇=M-τszλ.The distribution of the Berry curvature in momentum space is shown in Fig.1(b).As required by time reversal symmetry, the Berry curvatures fromKvalley and-Kvalley have the same amplitudes but opposite signs,which gives rise to valley contrasting transverse electric current in the presence of an in-plane longitudinal electric field.

Because the Berry curvature in the two moir′e valleys±Kof a long-period moir′e described by Eq.(6)has opposite value,there is no net charge Hall current.However,we find that when reducing the moir′e periodicity, the moir′e valleys in the miniband obtained directly from Eq.(5)become asymmetric,with different alignment and band gap [c.f.Fig.4(a)].This is because the two moir′e valley states atKand-Kare located at opposite directions measured from the originalKpoint,which are inequivalent arising from the triangular wrapping effect.This wrapping effect is proportion tok3,withkbeing the momentum measured fromKpoint in the original Brillouin zone,and is strong for a short-period moir′e pattern with a corresponding large moir′e Brillouin zone.Because of this asymmetry in the two Dirac cones, the Berry curvature from the two moir′e valleys±Kdoes not have opposite values any more,resulting in a charge Hall current at each original valley [c.f.Fig.4(b)].With the increase of moir′e periodL,the triangular wrapping effect fades out gradually.As a consequence, the Berry curvatures at the two moir′e valleys approach opposite values.Considering the strong SOC in the valence band of TMDs,this charge Hall current is spin polarized.In the presence of time-reversal symmetry,there is a net spin Hall current contributed from the two original±Kvalleys,although the total charge Hall current cancels out.Note that in monolayer TMDs,the Berry curvature is very small because of the large band gap.Moir′e Dirac fermions in TMD vdW heterostructure are therefore a potential platform to generate large spin and valley currents.

Fig.1.(a)Schematic diagram of energy bands around the two valleys.The valley and spin indexes are indicated.(b)Distribution of Berry curvature Ω(k) around the two valleys in the conduction band of monolayer MoS2,given in units of °A2.The parameters used are a=3.193 °A,t=1.10 eV,λ =0.075 eV and M=1.66 eV.

First-principles results show that the Bloch states of monolayer TMDs near the band edges are mainly contributed by dz2,dxy,and dx2-y2orbitals of metal atoms.[34]To facilitate the band structure calculation, a tight-binding (TB) model is constructed using these three orbitals, with parameters fitted according to the first-principles results.The TB model that only considers the nearest neighbor hopping between metal atoms can well reproduce the band structures near the±Kpoints,where we mainly focus in this work.

Figure 5 shows the evolution of energy spectrum with the increase of the strength of the applied light for the two moir′e valleys.We find that the circularly polarized light breaks the degeneracy of the two moir′e valleys and leads to qualitatively different gap evolution behavior.In particular,the energy gap ofKvalley increases monotonically,while the one at-Kvalley decreases monotonically first, then closes at the critical valueand finally reopens.The gap evolution behavior at-Kvalley reminds us of a topological phase transition.We then calculate the Berry curvature of the two moir′e valleys under light driving before and after the topological phase transition[c.f.Figs.6(b)and 6(c)],and compare them with those without light driving[c.f.Fig.6(a)].We find that after driving,the Berry curvatures of the two moir′e valleys no longer have time reversal symmetry.In particular,the Berry curvatures of theKvalley before and after the phase transition are both negative, and their magnitudes decrease with the increase of the light driving strength; While for the-Kvalley,the Berry curvature changes from positive value to negative one after the phase transition.After integrating the Berry curvature at each valley,we find that the Chern number is changed from 0.5 to-0.5 at-Kvalley.In contrast, the calculated Chern number is always-0.5 atKvalley independent of the light strength.In total,the Chern number of the whole moir′e system is then changed from 0 to-1 through a light-induced topological phase transition,which means that the strong lightdriving moir′e system is in the quantum anomalous Hall phase.When changing the circulation of the polarized light(η=-1),the energy gap at theKvalley first closes and then reopens,in which the Chern number is changed from-0.5 to 0.5.The total Chern number is then changed from 0 to 1.

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where{m,n}={dz2,dxy,dx2-y2}are the orbital indices andtim,jnis the hopping integral between different orbitals andεmis the on-site energy.The SOC can be incorporated byHSOC=λL·S.The details of parameters for various TMDs are given explicitly in reference.[34]

3.Moir′e Dirac fermions

Form Eq.(2),one can see that the magnitude of the Berry curvature is inversely proportional to the bandgap of the Dirac materials.In the following,we discuss the possibility of using moir′e potential to engineer Dirac fermions with tunable small bandgap.

3.1.Moir′e potential in TMDs heterobilayers

Because of the large number of atoms involved (∝(L/a)2), a direct first-principles calculation of the electronic property of a moir′e pattern is challenging [c.f.Fig.2(a)].However,in a long-period moir′e pattern,each local registry resembles lattice matched commensurate structure,one can first study electronic property at each local region and then map to the whole moir′e pattern using a local approximation.In a semiconductor heterobilayer, the band-edge state is mainly contributed from one layer.The influence from the other layer can be taken into account by defining a moir′e potentialV(r),which depends on the interlayer atomic registryr.Because the interlayer distance is typically much larger than the in-plane variation in a unit cell,one can approximateV(r)by a Fourier expansion that includes only the six moir′e reciprocal lattice vectors in the first shell,[35]

where the summation overbiis restricted to the six moir′e reciprocal lattice vectors as shown in Fig.2(c).Because the potential is real and each TMDs monolayer has threefold-rotation symmetry, we haveν(?R2π/3b) =ν(b) andν(b) =ν?(-b).The three parameters{ν,φ,V0}can be obtained by fitting from first-principles results at the three high-symmetry latticematched configurations[c.f.Fig.2(b)].

Hexagonal lattice with a staggered sublattice potential has been widely explored to study Berry phase supported topological transport.[41-43]The Berry curvature, which leads to an anomalous velocity in the transverse direction in the presence of an in-plane electric field, at the moir′e Dirac point±Kin the conduction band isΩ(τK)=-2τL2t20/Δ2,whereτis the moir′e valley index.Compared with Dirac fermions in monolayer semiconductor,Berry curvature in moir′e Dirac fermions is magnified by a factor ∝(L/a)2.This giant Berry curvature leads to a large moir′e valley Hall current, which can be detected as a giant nonlocal voltage.[44,45]

The first-principles calculations are performed with the open-source QUANTUM ESPRESSO plane-wave density functional theory (DFT) package.[36,37]The Perdew-Burke-Ernzerhof(PBE)exchange-correlation functional with semiempirical DFT-D3 vdW correction method is adopted.[38]The plane-wave cut-off energy is set to 1115 eV.The crystal structures are fully relaxed until the force on each atom and total energy variations are smaller than 2.6×10-3eV/°A and 1.4×10-4eV.To avoid mirror interactions in the thickness direction, a vacuum space of 15 °A is used between adjacent heterostructures.The Brillouin zonek-point samplings are 10×10×1 for both crystal optimizations and electronic structure computations.The obtained band-edge energy of the R-type MoSe2/WS2heterobilayer for different configurations (as referenced by a common vacuum level) is listed in Table 1, from which the parameters{ν,φ,V0}={-6.4 meV,234°,38.3 meV}defined in the moir′e potential of Eq.(4) can be extracted.We find that the moir′e potential has two local maxima for holes located at WS2layer,with an energy difference of about 7 meV [c.f.Fig.2(d)] and forming a honeycomb lattice.Furthermore, we have checked that for the H-type MoSe2/WS2heterobilayer,the moir′e potential forms a triangular lattice for holes located at WS2layer.

Table 1.First-principles calculated band-edge energy for the three highsymmetry configurations of the R-type MoSe2/WS2 heterobilayer[see Fig.2(b)].Energy is given in units of eV.

3.2.Moir′e Dirac fermions in TMDs heterobilayers

Although the moir′e potential breaks the original translation symmetry of each monolayer, the appearance of moir′e periodicity in the heterostructure guarantees the application of Bloch theory and results in the formation of moir′e minibands.We mainly focus on the electronic property of the heterostructure around the valence band, which can be modeled by adding directly a moir′e potentialV(R)on the TB Hamiltonian of monolayer TMDs.We assume that the moir′e pattern is formed between two rigid lattices,where the mapping between the local registryrand locationRis linear.Therefore,therdependence of the interlayer potential in commensurate bilayers can be directly mapped to the location(R)dependence of the interlayer potential in a moir′e.Under this approximation,the moir′e modulated interlayer potential as a function ofRis isomorphic toV(r), i.e.,V(R)≡V(r(R)).[21,35]In terms of this moir′e potential, the TB Hamiltonian of the TMD heterostructure reads

the SOC can also be included in the calculation.We note that because the coupling between the two layers can be modulated by applying external vertical electric field and pressure, the moir′e potential is experimentally tunable.A similar procedure has been introduced for excitons in TMD heterobilayer,[23,24]in which localized excitons have been experimentally observed.[25-27]

For moir′e potential with only one energy minimum,a triangular lattice is formed, which has been proposed to study Hubbard model physics.[35,39]For moir′e potential with two nearly degenerate extrema, a hexagonal lattice is formed,which allows for the creation of moir′e Dirac fermions with additional moir′e valley degree of freedom appearing in the minibands.For an R-type MoSe2/WS2heterobilayer, our firstprinciples results show that the moir′e potential has two local maxima for holes mainly contributed from WS2layer, with an energy difference of about 7 meV [c.f.Fig.2(d)].Figure 3(a) shows the moir′e miniband around the valence band edge,which features two gapped Dirac cones.The two cones locate at in-equivalent±Kcorners of the mini-Brillouin zone,defining a moir′e valley degree of freedom.The miniband structure can be understood from the intra and inter moir′e hopping of the localized state at each confinement center and well fitted by a TB Hamiltonian including only the nearestneighbor hopping on a hexagonal lattice in the moir′e scale

wheret0is the hopping integral andεi=±Δ/2 is the on-site energy for the two sublattices.The fitted hopping parametert0(purple line with triangles) andΔ(black line with disks)as a function of moir′e periodicityLare shown in Fig.3(c).The moir′e hexagonal lattice resembles a magnified graphene,which enables fine tuning on each moir′e site.Applying opposite potentials on the two outmost zigzag edges of a moir′e lattice tunes the gapped flat-band edge states into gapless helical ones[c.f.Fig.3(d)].[40]These in-gap edge states are well separated from the bulk ones and propagate with opposite velocities and moir′e valley indexes,thus opening up a new possibility to manipulate 1D topological state in the moir′e scale.

Fig.3.(a) Minibands with moir′e Dirac cones for an R-type MoSe2/WS2 heterobilayer with L=42a,a being the lattice constant of WS2.Inset: full dispersion near the moir′e Dirac point-K.(b)Wavefunction distribution at moir′e Dirac pointK indicated by the green triangle and dot in (a).(c) Fitted parameters t0 and Δ in Eq.(6) as a function of moir′e periodicity L.(d) The edge states of a moir′e ribbon can be tuned from flat(upper plot)to helical(lower plot)by adding an on-site potential of 1.7t0 and -1.7t0 at the two outmost zigzag edges.The parameters used are t0=2.37 meV,Δ =3.845 meV.

農(nóng)戶滿意就是一個活廣告，在遜克片區(qū)象柞樹崗農(nóng)機合作社這樣體量的有20多個農(nóng)機合作社，今年以來服務(wù)質(zhì)量提升，他們感謝之余，還幫助帶動一些以前在社會加油站的農(nóng)機合作社。截至秋收接近尾聲，大部分分公司兩季銷量均有不同幅度的增長。特別是黑河分公司今年春耕、秋收保供時期的銷量完成最好，9月、10月單月均有不同幅度的增長，特別是10月的前15天，黑河分公司柴油銷量同比增長40%。

Fig.4.(a)Two topmost moir′e minibands for L=27a(left)and L=42a(right).(b)Berry curvature of the topmost band for L=27a.Inset: the asymmetry in the two moir′e valleys leads to a spin Hall current contributed from the two original±K valleys. ζ denotes an in-plane external electric field.The curved red(blue)arrow denotes spin up(down)Hall current.

3.3.Asymmetric moir′e valleys due to triangular wrapping

2.3 多因素分析顯示 以nSLN是否有轉(zhuǎn)移為因變量，將上述差異有統(tǒng)計學意義的指標納入多因素Logistic回歸分析，結(jié)果顯示原發(fā)腫瘤直徑(OR=2.700,P=0.006)、神經(jīng)/脈管等淋巴結(jié)外浸潤(OR=2.759,P=0.008)以及陽性SLN數(shù)目(OR=1.934,P=0.009)均為nSLN轉(zhuǎn)移的獨立影響因素。見表2。

3.4.Light-induced quantum anomalous Hall effect

Light is a powerful tool to engineer physical property in solid state systems.[46-48]In the following,we discuss the effect of light on the moir′e Dirac fermions,focusing on the possible nontrivial global band topology.The light-matter interaction is introduced via the minimal substitution, i.e.,k →k+eA(t)withA(t)being the vector potential of the applied electromagnetic field.For concreteness, we consider a circularly polarized light, which hasA(t)=A0(ηcosΩt,sinΩt)withΩ=2π/Tbeing the frequency of light,A0being its amplitude,andη=±1 denoting its chirality.We note that the linearly polarized light is a superposition of the left and right circularly polarized light,A(t)=A0(0,sinΩt).The light-matter interaction is totally determined by a dimensionless constantA=eA0L/ˉh.We notice that time-reversal symmetry is broken when a circularly polarized light is applied to the system.

Light driving renders the moir′e Dirac Hamiltonian timeperiodic,HL(t) =HL(t+T) with periodicityT, which can be solved using the Floquet theory.This theory states that the solution of the Schr¨odinger equation of a time-periodic Hamiltonian is a product of a phase factor and a time periodic state, i.e.,|ψ(t)〉= e-iεt|u(t)〉, where the time periodic function|u(t)〉=|u(t+2π

）〉is the Floquet state andεis the quasienergy,satisfying

The low-energy physics of monolayer TMDs can be effectively described by a massive Dirac Hamiltonian[9]

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The three-band TB Hamiltonian reads

To further study the light-induced topological phase in the moir′e Dirac system,we construct an effective Haldane model to study the in-gap edge states.The effective TB Hamiltonian reads[50]

where〈i,j〉denotes the nearest-neighbor hopping,and〈〈i,j〉〉denotes the next nearest-neighbor hopping,witht1andt2being their corresponding hopping strengths.We use the convention that the next nearest-neighbor hoppings in a clockwise direction have an extra-φphase while the ones in the counterclockwise direction have+φphase[c.f.Fig.7(a)].The correspondence between the effective Hamiltonian Eq.(8)and TB Hamiltonian Eq.(9) leads to the following relation:t1=t0,,andφ=π/2.

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Without light driving,the bulk states in a zigzag nanoribbon geometry at the two valleys are symmetric and connected by two trivial flat bands [c.f.Fig.3(d)].Upon driving, the two valleys become asymmetric due to the breaking of time reversal symmetry and the bulk gap of the-Kvalley closes at the topological phase transition point [c.f.Fig.7(b)].Interestingly,when further increasing the light driving strength,the gap reopens and a pair of chiral edge states appear in the bulk gap [c.f.Fig.7(c)].These edge states distribute at opposite boundaries and propagate in opposite directions [c.f.Fig.7(d)].

定義圖像對{Ia，Ib}各自有{N，M}個特征點，χ={w1，w2，wi，…，wN}是從圖像 Ia到圖像 Ib 中所有點最近的特征匹配，又定義{a，b}為圖像{Ia，Ib}所分割出來的子區(qū)域，并且在兩個子區(qū)域中各自擁有{n，m}個特征點。vi為第i個特征匹配，fa為在圖像Ia部分區(qū)域a中的一個特征，Tab為區(qū)域{a，b}表示的是同一位置，F(xiàn)ab為{a，b}表示的是不同位置。表示fa的匹配的是一個正確的匹配表示 fa的匹配是錯誤的匹配為最近的一個特征是在b區(qū)域中。

Fig.7.(a) Schematic of a honeycomb lattice with nearest-neighbor hopping t1 and next nearest-neighbor hopping t2 with a phase φ =π/2.(b)Energy spectrum of a zigzag nanoribbon with light driving strength A=A0.(c) The same as (b) with A=.(d) Distribution of wavefunction of the two in-gap states shown in(c).The parameters are the same as those in Fig.5.

4.Conclusions

We have revealed moir′e Dirac fermions with tunable large Berry curvature in a moir′e superlattice of TMD heterobilayer.These novel Dirac fermions are formed by the moir′e potential with two nearly degenerate local maxima.By applying on-site potential on the two outmost zigzag edges of the moir′e lattice,we can tune the gapped flat-band edge states into the gapless helical ones.In short-period moir′e patterns, we find that the two moir′e valleys become asymmetric due to the triangular wrapping effect,which produces a net spin Hall current.More interestingly,a circularly polarized light can drive these topologically trivial moir′e Dirac fermions into a quantum anomalous Hall phase.Our results open up a new possibility to study the Dirac physics in two dimensions, with potential applications in constructing valleytronic devices in moir′e scale.

ψ9、ψ、ψ、μ的最值可能出現(xiàn)在推程和回程的兩端或內(nèi)部。將φ= 0和φ= Φ代入式(9)和式(10)第1式，得ψ9、ψ、ψ、μ在推程和回程的端點處的值；由高等數(shù)學知，從方程ψ= 0、ψ9= 0、ψ9= 0、μ9= 0解出各自φ值，分別代入式(9)和式(10)第1式，可得ψ9、ψ、ψ、μ在推程和回程的內(nèi)部的極值；比較這些端點處的值和內(nèi)部的極值，得ψ9、ψ、ψ、μ在推程和回程里的最值，代入式(18)得Γ、Λ、Σ、T在推程和回程里的最值，它們代入式(19)即得在一個運動循環(huán)上的最值(特性值)。

Acknowledgements

Project supported by the Science Fund for Distinguished Young Scholars of Hunan Province(Grant No.2022J10002),the National Key Research and Development Program of China (Grant No.2021YFA1200503), and the Fundamental Research Funds for the Central Universities from China.