Orbital Hall effect in bilayer transition metal dichalcogenides: From the intra-atomic approximation to the Bloch states orbital magnetic moment approach
Tarik P. Cysne, Sayantika Bhowal, Giovanni Vignale, Tatiana G. Rappoport
Using an effective Dirac model, we study the orbital Hall effect (OHE) in bilayers of transition metal dichalcogenides with 2H stacking (2H-TMD). We use first-order perturbation theory in the interlayer coupling of the bilayer system to obtain analytical expressions for the orbital Hall conductivity in the linear response regime. We use two distinct descriptions of the orbital angular momentum (OAM) operator: The first one is the intra-atomic approximation that considers only the intrasite contribution to the OAM [Cysne et al. Phys. Rev. Lett. 126, 056601 (2021)]. The second one uses the Berry-phase formula of the orbital (valley) magnetic moment to describe the OAM operator [Bhowal and Vignale, Phys. Rev. B 103, 195309 (2021)]. This approach includes both intersite and intrasite contributions to the OAM. Our results suggest that the two approaches agree qualitatively in describing the OHE in bilayers of 2H-TMDs, although they present some quantitative differences. We also show that interlayer coupling plays an essential role in understanding the OHE in the unbiased bilayer of 2H-TMD. This coupling causes the Bloch states to become bonding (antibonding) combinations of states of individual layers, demanding the consideration of the non-Abelian structure of the orbital magnetic moment to the occurrence of OHE. As we discuss throughout the work, the emerging picture of transport of OAM in the unbiased bilayer of 2H-TMDs based on OHE is very different from the usual picture based on the valley Hall effect, shedding new lights on previous experimental results. We also discuss the effect of the inclusion of a gate-voltage bias in the bilayer system. Our work gives support to recent theoretical predictions on OHE in two-dimensional materials.
The geometry of electronic bands in a solid can drastically alter single-particle charge and spin transport. We show here that collective optical excitations arising from Coulomb interactions also exhibit unique signatures of Berry curvature and quantum geometric tensor. A non-zero Berry curvature mixes and lifts the degeneracy of $l \neq 0$ states, leading to a time-reversal-symmetric analog of the orbital Zeeman effect. The quantum geometric tensor, on the other hand, leads to $l$-dependent shifts of exciton states that is analogous to the Lamb shift. Our results provide an explanation of th…
We perform a theoretical study of radiative decay of dark intravalley excitons in transition metal dichalcogenide monolayers. This decay necessarily involves an electronic spin flip. The intrinsic decay mechanism due to interband spin-flip dipole moment perpendicular to the monolayer plane, gives a rate about 100--1000 times smaller than that of bright excitons. However, we find that this mechanism also introduces an energy splitting due to a local field effect, and the whole oscillator strength is contained in the higher-energy component, while the lowest-energy state remains dark and needs a…
Exciton condensation in an electron-hole bilayer system of monolayer transition metal dichalcogenides is analyzed at three different levels of theory to account for screening and quasiparticle renormalization. The large effective masses of the transition metal dichalcogenides place them in a strong coupling regime. In this regime, mean field (MF) theory with either an unscreened or screened interlayer interaction predicts a room temperature condensate. Interlayer and intralayer interactions renormalize the quasiparticle dispersion, and this effect is included in a GW approximation. The renorma…
The weak field magnetoresistance has seen a revived interest due to the distinct role played by the momentum-space Berry curvature of Bloch electrons. While most previous studies in this regard focus on the inter-scattering motion of semiclassical Bloch electrons in electromagnetic fields, the intra-scattering effects of the semiclassical dynamics augmented by the Berry curvature, magnetic moment and shift vector on the magnetoresistance have been largely overlooked. Here we uncover that these intra-scattering effects, which are neglected in the field-independent relaxation time approximation …
In this paper, we study the third-order nonlinear optical response due to transitions between excitonic levels in two-dimensional transition metal dichalcogeniedes. To accomplish this, we use methods not applied to the description of excitons in two-dimensional materials so far and combined with a variational approach to describe the $1s$ excitonic state. The aforementioned transitions allow to probe dark states which are not revealed in absorption experiments. We present general formulas capable of describing any third-order process. The specific case of two-photon absorption in WSe2 is studi…
In an earlier work, it was shown that the dispersion of the van der Waals heterostructures (vdWHs) of graphene monolayer on 2D transition metal dichalcogenide (GrTMD) substrate comprises of the spin-split, gapped bands. Using these it was also shown that the intra-band plasmon dispersion for the finite doping and the long wavelength limit involves the q2/3 (unconventional) behaviour and not the well known q1/2 behaviour. In this paper the optical conductivity and the optical properties of this vdWH are reported. We find that there are no (inter-band) plasmons for electron doping as long as the…
Strain presents a straightforward tool to tune electronic properties of atomically thin nanomaterials that are highly sensitive to lattice deformations. While the influence of strain on the electronic band structure has been intensively studied, there are only few works on its impact on optical properties of monolayer transition metal dichalcogenides (TMDs). Combining microscopic theory based on Wannier and Bloch equations with nearestneighbor tight-binding approximation, we present an analytical view on how uni- and biaxial strain influences the optical fingerprint of TMDs including their exc…
Reflectance and magneto-reflectance experiments together with theoretical modelling based on the $\mathbf{k\cdot p}$ approach have been employed to study the evolution of direct bandgap excitons in MoS$_2$ layers with a thickness ranging from mono- to trilayer. The extra excitonic resonances observed in MoS$_2$ multilayers emerge as a result of the hybridization of Bloch states of each sub-layer due to the interlayer coupling. The properties of such excitons in bi- and trilayers are classified by the symmetry of corresponding crystals. The inter- and intralayer character of the reported excito…
It is assessed in detail both experimentally and theoretically how the interlayer coupling of transition metal dichalcogenides controls the electronic properties of the respective devices. Gated transition metal dichalcogenide structures show electrons and holes to either localize in individual monolayers, or delocalize beyond multiple layers - depending on the balance between spin-orbit inter- action and interlayer hopping. This balance depends on layer thickness, momentum space symmetry points and applied gate fields. A good quantitative agreement of predictions and measurements of the quant…
We study the photogalvanic effect caused by a simultaneous action of circular-polarized interband and linearly-polarized intraband illuminations. It is found that, in such conditions, the steady photocurrent appears. The effect originates from the valley-selective pumping by the circular light, the trigonal asymmetry of the valleys together with the even asymmetry of the linearly-polarized light, that produces a polar in-plane asymmetry of the electron and hole distribution functions, leading to the photocurrent. The approach is based on the solution of the classical kinetic equation for carri…
The semiconducting two-dimensional transition metal dichalcogenides MX$_{2}$ show an abundance of one-dimensional metallic edges and grain boundaries. Standard techniques for calculating edge states typically model nanoribbons, and require the use of supercells. In this paper we formulate a Green$'$s function technique for calculating edge states of (semi-)infinite two-dimensional systems with a single well-defined edge or grain boundary. We express Green$'$s functions in terms of Bloch matrices, constructed from the solutions of a quadratic eigenvalue equation. The technique can be applied to…
In transition metal dichalcogenides the spin-orbit interaction affects differently the conduction and valence band energies as functions of $k$ and the band gap is large. Consequently, when a perpendicular magnetic field $B$ is applied the conduction and valence band Landau levels are also different and this leads to a splitting of the interband optical absorption lines in both the absence and presence of an external electric field $E_{z}$. When $B$ and $E_{z}$ are present the peaks in the imaginary part of the Hall conductivity give two distinct contributions of opposite sign to the interband…
We study both the intrinsic and extrinsic spin Hall effect in spin-valley coupled monolayers of transition metal dichalcogenides. We find that whereas the skew-scattering contribution is suppressed by the large band gap, the side-jump contribution is comparable to the intrinsic one with opposite sign in the presence of scalar and magnetic scattering. Intervalley scattering tends to suppress the side-jump contribution due to the loss of coherence. By tuning the ratio of intra- to intervalley scattering, the spin Hall conductivity shows a sign change in hole-doped samples. Multiband effect in ot…
Contact resistance of semiconducting transition metal dichalcogenides has been shown to decrease in lateral heterojunctions formed with their metallic phases but its origins remain elusive. Here we combine first principles and quantum transport calculations to rationalize the contact resistance of these structures in terms of phase, composition (WTe2, MoTe2, WSe2, and MoSe2), and length of the channel. We find that charge injection in metallic 1T'-WTe2/1T'-MoTe2 junctions is nearly ideal as electrode Bloch states remain delocalized through the channel. Mixtures of 1T' selenides and tellurides …
Excitons dominate the optical properties of monolayer transition metal dichalcogenides (TMDs). Besides optically accessible bright exciton states, TMDs exhibit also a multitude of optically forbidden dark excitons. Here, we show that efficient exciton-phonon scattering couples bright and dark states and gives rise to an asymmetric excitonic line shape. The observed asymmetry can be traced back to phonon-induced sidebands that are accompanied by a polaron redshift. We present a joint theory-experiment study investigating the microscopic origin of these sidebands in different TMD materials takin…
We present a model of the electronic properties of monolayer transition-metal dichalcogenides based on a tight binding approach which includes the effects of strain and curvature of the crystal lattice. Mechanical deformations of the lattice offer a powerful route for tuning the electronic structure of the transition-metal dichalcogenides, as changes to bond lengths lead directly to corrections in the electronic Hamiltonian while curvature of the crystal lattice mixes the orbital structure of the electronic Bloch bands. We first present an effective low energy Hamiltonian describing the electr…
We obtain an analytical expression for the linewidth of the 1s-exciton as a function of temperature in transition metal dichalcogenides. The total linewidth, as function of temperature, is dominated by three contributions: (i) the radiative decay (essentially temperature independent); (ii) the phonon-induced intravalley scattering; (iii) the phonon-induced intervalley scattering. Our approach uses a variational \emph{Ansatz} to solve the Wannier equation allowing for an analytical treatment of the excitonic problem, including rates of the decay dynamics. Our results are in good agreement with …
We use first-principles calculations to investigate the band structure evolution of W$X_2$/Mo$X_2$ ($X$ = S, Se) heterobilayers under a perpendicular electric field. We characterize the extent to which the type-II band alignment in these compounds can be tuned or inverted electrostatically. Our results demonstrate two effects of stacking configuration. First, different stackings produce different net dipole moments, resulting in band offset variations that are larger than 0.1 eV. Second, based on symmetry constraints that depend on stacking, a perpendicular electric field may hybridize W$X_2$ …
Monolayers of transition metal dichalcogenides (TMDCs) exhibit an exceptionally strong Coulomb interaction between charge carriers due to the two-dimensional carrier confinement in connection with weak dielectric screening. High densities of excited charge carriers in the various band-structure valleys cause strong many-body renormalizations that influence both the electronic properties and the optical response of the material. We investigate electronic and optical properties of the typical monolayer TMDCs MoS$_2$, MoSe$_2$, WS$_2$ and WSe$_2$ in the presence of excited carriers by solving sem…
The relative orientation (twist) of successive layers of stacked two-dimensional (2D) materials creates variations in the interlayer atomic registry. The variations often form a super lattice, called a moiré pattern, which can alter electronic properties. In this work we introduce a classification of the single-particle electronic structures that can occur in twisted stacks of 2D layers by characterizing them as "moiré molecules" or "moiré crystals". The molecules generate localized electronic states and moiré flat bands, while the crystals are sometimes unconventional and produce electronic b…
Orbital Hall effect from orbital magnetic moments of Bloch states: the role of a new correction term
Instituto de Física, Universidade Federal Fluminense（弗鲁米嫩塞联邦大学物理研究所）；Centro de Física de Materiales, Universidad del País Vasco（巴斯克大学材料物理中心）；Ikerbasque Foundation（伊克尔巴斯基基金会）；Centro Brasileiro de Pesquisas Físicas（巴西物理研究中心）；Physics Center of Minho and Porto Universities (CF-UM-UP)（米尼奥与波尔图大学物理中心）；International Iberian Nanotechnology Laboratory (INL)（国际伊比利亚纳米技术实验室）
The remarkably strong Coulomb interaction in atomically thin transition metal dichalcogenides (TMDs) results in an extraordinarily rich many-particle physics including the formation of tightly bound excitons. Besides optically accessible bright excitonic states, these materials also exhibit a variety of dark excitons. Since they can even lie below the bright states, they have a strong influence on the exciton dynamics, lifetimes, and photoluminescence. While very recently, the presence of dark excitonic states has been experimentally demonstrated, the origin of these states, their formation, a…
Layered transition metal dichalcogenides (TMDCs) host a variety of strongly bound exciton complexes that control the optical properties in these materials. Apart from spin and valley, layer index provides an additional degree of freedom in a few-layer thick film. Here we show that in a few-layer TMDC film, the wavefunctions of the conduction and valence band edge states contributing to the K (K') valley are spatially confined in the alternate layers - giving rise to direct (quasi-)intra-layer bright exciton and lower-energy inter-layer dark excitons. Depending on the spin and valley configurat…
The valence flat bands in transition metal dichalcogenide (TMD) heterobilayers are shown to exhibit strong intralayer spin-orbit coupling. This is reflected in a simple tight-binding model with spin-dependent complex hoppings based on the continuum model. A perpendicular electric field causes interlayer hybridization, such that the effective model is equivalent to the Kane-Mele model of topological insulators. The proposed model can be used as a starting point to understand interactions and the experimentally observed topological phases.
We theoretically introduce the fundamentally fastest induction of a significant population and valley polarization in a monolayer of a transition metal dichalcogenide (i.e., $\mathrm{MoS_2}$ and $\mathrm{WS_2}$). This may be extended to other two-dimensional materials with the same symmetry. This valley polarization can be written and read-out by a pulse consisting of just a single optical oscillation with a duration of a few femtoseconds and an amplitude of $\sim0.2~\mathrm V/\mathrmÅ$. Under these conditions, we predict a new effect of {\em topological resonance}, which is due to Bloch motio…
A rate equation model for the dark and bright excitons kinetics is proposed which explains the wide variation in the observed degree of circular polarization of the PL emission in different TMDs monolayers. Our work suggests that the dark exciton states play an important, and previously unsuspected role in determining the degree of polarization of the PL emission. A dark exciton ground state provides a robust reservoir for valley polarization, which tries to maintain a Boltzmann distribution of the bright exciton states in the same valley via the intra valley bright dark exciton scattering mec…
An $\it{ab \,\, initio}$ based fully microscopic many-body approach is used to study the carrier relaxation dynamics in monolayer transition-metal dichalcogenides. Bandstructures and wavefunctions as well as phonon energies and coupling matrix elements are calculated using density functional theory. The resulting dipole and Coulomb matrix elements are implemented in the Dirac-Bloch equations to calculate carrier-carrier and carrier-phonon scatterings throughout the whole Brillouin zone. It is shown that carrier scatterings lead to a relaxation into hot quasi-Fermi distributions on a single fem…
Magneto-optical conductivity of monolayer transition metal dichalcogenides in the presence of proximity-induced exchange interaction and external electrical field
Manipulating materials properties far from equilibrium recently garnered significant attention, with experimental emphasis on transient melting, enhancement, or induction of electronic order. A more tantalizing aspect of the matter-light interaction regards the possibility to access dynamical steady states with distinct non-equilibrium phase transitions to affect electronic transport. Here, we show that the interplay of crystal symmetry and optical pumping of monolayer transition-metal dichalcogenides (TMDCs) provides a novel avenue to engineer topologically-protected chiral edge modes. In sta…
It is shown that the strong Coulomb coupling in intrinsic suspended semiconducting transition metal dichalcogenides can exceed the critical value needed for an excitonic ground state. The dipole-allowed optical excitations then correspond to intra-excitonic transitions such that the optically bright excitonic transitions near the Dirac points have a $p$-like symmetry whereas the $s$-like states are dipole forbidden. The large intrinsic coupling strength seems to be a generic property of the semiconducting transition metal dichalcogenides and strong Coulomb-coupling signatures in the form of th…
In this work, we provide an effective model to evaluate the one-electron dipole matrix elements governing optical excitations and the photoemission process of single-layer (SL) and bilayer (BL) transition metal dichalcogenides. By utilizing a $\vec{k} \cdot \vec{p}$ Hamiltonian, we calculate the photoemission intensity as observed in angle-resolved photoemission from the valence bands around the $\bar{K}$-valley of MoS$_2$. In SL MoS$_2$ we find a significant masking of intensity outside the first Brillouin zone, which originates from an in-plane interference effect between photoelectrons emit…
The optical properties of transition metal dichalcogenide monolayers such as the two-dimensional semiconductors MoS$_2$ and WSe$_2$ are dominated by excitons, Coulomb bound electron-hole pairs. The light emission yield depends on whether the electron-hole transitions are optically allowed (bright) or forbidden (dark). By solving the Bethe Salpeter Equation on top of $GW$ wave functions in density functional theory calculations, we determine the sign and amplitude of the splitting between bright and dark exciton states. We evaluate the influence of the spin-orbit coupling on the optical spectra…
Keldysh field theory of spin- and valley-distinguished polariton nonlinearities in transition-metal dichalcogenide monolayers
Abrikosov Center for Theoretical Physics, Moscow Center for Advanced Studies（阿布里科索夫理论物理中心，莫斯科高等研究中心）；National Research Nuclear University MEPhI (Moscow Engineering Physics Institute)（国立研究核大学MEPhI（莫斯科工程物理学院））；Russian Quantum Center, Skolkovo IC（俄罗斯量子中心，斯科尔科沃创新中心）
Geometrical moiré patterns, generic for almost aligned bilayers of two-dimensional (2D) crystals with similar lattice structure but slightly different lattice constants, lead to zone folding and miniband formation for electronic states. Here, we show that moiré superlattice (mSL) effects in $\mathrm{MoSe}_2/\mathrm{WS}_2$ and $\mathrm{MoTe}_2/\mathrm{MoSe}_2$ heterobilayers that feature alignment of the band edges are enhanced by resonant interlayer hybridization, and anticipate similar features in twisted homobilayers of TMDs, including examples of narrow minibands close to the actual band ed…
We study hot-electron cooling by acoustic and optical phonons in monolayer MoS$_2$. The cooling power $P$ ($P_e = P/n$) is investigated as a function of electron temperature $T_e$ (0-500 $\mathrm{K}$) and carrier density $n$ ($10^{10}$-$10^{13}$ $\mathrm{cm}^{-2}$) taking into account all relevant electron-phonon (el-ph) couplings. We find that the cross over from acoustic phonon dominated cooling at low $T_e$ to optical phonon dominated cooling at higher $T_e$ takes place at $T_e \sim 50$-$75$ $\mathrm{K}$. The unscreened deformation potential (DP) coupling to the TA phonon is shown to domina…
We study possible superconducting states in transition metal dichalcogenide (TMD) monolayers, assuming an on-site pairing potential that includes both intra- and inter-orbital terms. We find that if the mirror symmetry with respect to the system's plane is broken (e.g., by a substrate), this type of pairing can give rise to unconventional superconductivity, including time-reversal-invariant nodal and fully gapped topological phases. Using a multi-orbital renormalization group procedure, we show how these phases may result from the interplay between the local Coulomb repulsion, Hund's rule coup…
Angle-resolved photoemission spectroscopy (ARPES) is the most powerful technique to investigate the electronic band structure of crystalline solids. To completely characterize the electronic structure of topological materials, one needs to go beyond band structure mapping and access information about the momentum-resolved Bloch wavefunction, namely orbitals, Berry curvature, and topological invariants. However, because phase information is lost in the process of measuring photoemission intensities, retrieving the complex-valued Bloch wavefunction from photoemission data has yet remained elusiv…
Dzyaloshinskii-Moriya interaction (DMI) is vital to form various chiral spin textures, novel behaviors of magnons and permits their potential applications in energy-efficient spintronic devices. Here, we realize a sizable bulk DMI in a transition metal dichalcogenide (TMD) 2H-TaS2 by intercalating Fe atoms, which form the chiral supercells with broken spatial inversion symmetry and also act as the source of magnetic orderings. Using a newly developed protonic gate technology, gate-controlled protons intercalation could further change the carrier density and intensely tune DMI via the Ruderman-…
Excitons, composite electron-hole quasiparticles, are known to play an important role in optoelectronic phenomena in many semiconducting materials. Recent experiments and theory indicate that the band-gap optics of the newly discovered monolayer transition-metal dichalcogenides (TMDs) is dominated by tightly bound valley excitons. The strong interaction of excitons with long-range electromagnetic fields in these 2D systems can significantly affect their intrinsic properties. Here, we develop a semi-classical framework for intrinsic exciton-polaritons in monolayer TMDs that treats their dispers…
Bilayer transition metal dichalcogenides (TMDs) belong to a class of materials with two unique features, the coupled spin-valley-layer degrees of freedom and the crystal structure that is globally centrosymmetric but locally non-centrosymmetric. In this work, we will show that the combination of these two features can lead to a rich phase diagram for unconventional superconductivity, including intra-layer and inter-layer singlet pairings and inter-layer triplet pairings, in bilayer superconducting TMDs. In particular, we predict that the inhomogeneous Fulde-Ferrell-Larkin-Ovchinnikov state can…
Monolayer transition metal dichalcogenides (TMDs) hold great promise for future information processing applications utilizing a combination of electron spin and valley pseudospin. This unique spin system has led to observation of the valley Zeeman effect in neutral and charged excitonic resonances under applied magnetic fields. However, reported values of the trion valley Zeeman splitting remain highly inconsistent across studies. Here, we utilize high quality hBN encapsulated monolayer WSe$_2$ to enable simultaneous measurement of both intervalley and intravalley trion photoluminescence. We f…
The multivalley band structure of monolayer transition metal dichalcogenides (TMDs) gives rise to intravalley and intervalley excitons. Much knowledge of these excitons has been gained, but fundamental questions remain, such as how to describe them all in a unified picture with their correlations, how are those from different valleys coupled to form the intervalley biexciton? To address the issues, we derive an exciton Hamiltonian from interpair correlations between the constituent carriers-fermions of two excitons. Identifying excitons by irreducible representations of their point symmetry gr…
Magnetic multilayers are promising tuneable systems for hosting magnetic skyrmions at/above room temperature. Revealing their intriguing switching mechanisms and associated inherent electrical responses are prerequisites for developing skyrmionic devices. In this work, we theoretically demonstrate the annihilation of single skyrmions occurring through a multilayer structure, which is mediated by hopping dynamics of topological hedgehog singularities known as Bloch points. The emerging intralayer dynamics of Bloch points are dominated by the Dzyaloshinskii-Moriya interaction, and their propagat…
We propose and analyze a mechanism for inducing spin Hall currents in ordinary (1H phase) monolayer transition metal dichalcogenides (TMDs) due to the nonlinear process of optical rectification. The photo-induced spin current is proportional to the light intensity, and originates from the intrinsic spin-orbit coupling in TMDs. The spin current spectrum is strongly influenced by electron-hole interactions, i.e. excitonic effects, analogous to the optical absorption. Remarkably, excitons change the temperature dependence of the induced spin current, to the point that the current direction can ev…
Twisted bilayer transition metal dichalcogenides are ideal platforms to study flat-band phenomena. In this paper, we investigate flat-band plasmons in the hole-doped twisted bilayer MoS$_2$ (tb-MoS$_2$) by employing a full tight-binding model and the random phase approximation. When considering lattice relaxations in tb-MoS$_2$, the flat band is not separated from remote valence bands, which makes the contribution of interband transitions in transforming the plasmon dispersion and energy significantly different. In particular, low-damped and quasi-flat plasmons emerge if we only consider intra…
