Valley Hall Conductivity Research Articles

Intrinsic anomalous, spin and valley Hall effects in ’ex-so-tic’ van-der-Waals structures

We consider the anomalous, spin, valley, and valley spin Hall effects in a pristine graphene-based van-der-Waals (vdW) heterostructure consisting of a bilayer graphene (BLG) sandwiched between a semiconducting van-der-Waals material with strong spin-orbit coupling (e.g., WS2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\hbox {WS}_2$$\\end{document}) and a ferromagnetic insulating vdW material (e.g. Cr2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\hbox {Cr}_2$$\\end{document}Ge2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\hbox {Ge}_2$$\\end{document}Te6\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\hbox {Te}_6$$\\end{document}). Due to the exchange proximity effect from one side and spin-orbit proximity effect from the other side of graphene, such a structure is referred to as graphene based ’ex-so-tic’ structure. First, we derive an effective Hamiltonian describing the low-energy states of the structure. Then, using the Green’s function formalism, we obtain analytical results for the Hall conductivities as a function of the Fermi energy and gate voltage. For specific values of these parameters, we find a quantized valley Hall conductivity.

Optical control of berry curvature in gated WSe2 bilayers

Abstract Unlike single layers of 2H transition metal dichalcogenides (TMDCs), bilayers of 2H TMDCs maintain inversion and time reversal (TR) symmetries, resulting in a vanishing Berry curvature ( Ω ( k ) ∼ 0 ) that inhibits various potential transport phenomena. A nonzero Berry curvature ( Ω ( k ) ≠ 0 ) is imperative for the occurrence of several unconventional transport phenomena, including the anomalous Hall effect and the anomalous Nernst effect. To overcome this limitation, we break these symmetries in bilayer TMDCs using electrostatic gating and circularly polarized light as external means. For non-gated WSe2 bilayers, circularly polarized light breaks TR symmetry, creating a finite Berry curvature signal in both conduction and valence bands, controllable by light intensity and its polarity. In gated WSe2 bilayers, where inversion symmetry is also broken, we observe a sign reversal in Berry curvature within the conduction bands, the extent of which depends on the relative strengths of the electric gating and light intensity. Overall, under finite bias and light intensity, the 2H bilayers of WSe2 exhibits finite spin Hall, valley Hall, and anomalous Hall conductivities, which depend on the strengths of the applied perturbations.

Renormalization of the valley Hall conductivity due to interparticle interaction

Renormalization of the valley Hall conductivity due to interparticle interaction

Gate-Tunable Asymmetric Quantum Dots in Graphene-Based Heterostructures: Pure Valley Polarization and Confinement

We explore the possibility of attaining valley-dependent tunnelling and confinement using proximity-induced spin-orbit couplings (SOCs) in graphene-based heterostructures. We consider gate-tunable asymmetric quantum dots (AQDs) on graphene heterostructures and exhibiting a C3v and/or C6v symmetry. By employing a tight-binding model, we explicitly reveal a pure valley confinement and valley signal in AQDs by streaming the valley local density, leading to valley-charge separation in real space. The confinement of the valley quasi-bound states is sensitive to the locally induced SOCs and to the spatial distribution of the induced AQDs; it is also robust against on-site disorder. The adopted process of attaining a pure valley-Hall conductivity and confinement with zero charge currents is expected to provide more options towards valley-dependent electron optics.

Strain-tuned spin polarization and optical conductivity in MoS2/EuS heterostructures

We investigate strain-tuned spin transport and optical conductivity in a monolayer (ML) $n$-type ${\mathrm{MoS}}_{2}/\mathrm{EuS}$ heterostructure under linearly polarized terahertz radiation using a low-energy effective Hamiltonian, which takes displacement of the Dirac point induced by strain into account. Through modifying the strain modulus, we find an adjustable relationship between the band edge energy and the strain. Thus the threshold and range of allowed angles are strongly dependent on the strength of strain $\ensuremath{\epsilon}$, whereas the angle $\ensuremath{\theta}$ along which the strain is applied only acts on the former. Due to spin-flipped scattering generated by Rashba spin-orbit coupling, the in-plane spin polarizations (i.e., ${P}_{X}$ and ${P}_{Y}$) can be achieved, and their intensity can be enhanced by increasing $\ensuremath{\epsilon}$. Interestingly, we find that the out-of-plane spin polarization (i.e., ${P}_{Z}$) is related to the synergy of $\ensuremath{\epsilon}$ and the Rashba parameter ${\ensuremath{\lambda}}_{R}$. Especially, ${P}_{Z}$ can approach $100%$ when $\ensuremath{\epsilon}=9%$ and ${\ensuremath{\lambda}}_{R}=2$ meV. Furthermore, the optical transition between the spin-splitting subbands of the conduction band under the strain is studied. It is shown that by varying $\ensuremath{\epsilon}$ we can control the magnitude of both absorption peaks and absorption valleys in optical conductivity. Moreover, the presence of ${\ensuremath{\lambda}}_{R}$ can be used to further enhance the absorption peaks and valleys because ${\ensuremath{\lambda}}_{R}$ can tune the energy difference between the spin-splitting subbands within a specific valley and thus affect the optical transition channels. Our findings demonstrate that the strain can influence the tunneling behavior and valley Hall conductivity in a ML $n$-type ${\mathrm{MoS}}_{2}/\mathrm{EuS}$ heterostructure, and these features are applicable to other two-dimensional ML transition metal dichalcogenides, which is promising for terahertz and valleytronic devices.

Probing Rashba spin-orbit coupling by subcycle lightwave control of valley polarization

We perform nonperturbative calculations of light-field driven valley-polarization process in monolayer ${\mathrm{MoS}}_{2}$ which has additional Rashba spin-orbit couplings (SOCs). The ultrafast electron dynamics is simulated within the independent particle picture by solving density-matrix equations in the basis of linear combination of atomic orbitals, where tight-binding (TB) models including both intrinsic atomic and Rashba SOCs are used to calculate relevant matrix elements. We demonstrate that the Rashba-type SOCs can be manifested by suboptical-cycle control of valley selectivity excitations, in particular necessary via using few-cycle linearly polarized pulse with controlled carrier-envelope phase (CEP). This procedure will lead to a CEP-dependent valley Hall conductivity (VHC), which exhibits an important phase shift among different Rashba coupling strengths. The additional analysis shows that this phase shift is mainly determined by the ${d}_{z}^{2}$-orbital TB Rashba parameter from Mo atom and originates from contribution of conduction bands to VHC, where the Berry curvature modified by Rashba SOC plays a crucial role. Moreover, we also provide a qualitative interpretation on the Rashba-dependent VHC in terms of suboptical-cycle Landau-Zener-St\"uckelberg interference. Our results suggest a feasible approach for probing Rashba SOCs in hexagonal two-dimensional materials, and might pave the way of achieving more controls in the future valleytronics application.

Valley and spin quantum Hall conductance of silicene coupled to a ferroelectric layer

We study the quantum valley and Hall conductances in silicene coupled to a ferroelectric (FE) layer. The spin orbit interaction in silicene couples the valley, pseudospin, and real spin degrees of freedom resulting in a topological Berry curvature in the system. The finite Berry curvature in turn induces a transverse Hall conductance. In particular, if the Fermi level Ef is within the bulk energy gap, the Hall conductance is quantized to integer multiples of π. We study the quantum spin and valley Hall conductivities (QSH and QVH) as functions of the applied out-of-plane electric field for different values of Ef and temperature. Both conductivities vary linearly as 1/|Ef| when Ef is within the conduction or valence bands but reach a quantized plateau value when Ef is within the bulk gap. Further, by coupling silicene to a FE layer, the QSH and QVH signals can be modulated by means of the coupling strength. This can potentially provide a robust topological memory read-out with distinct binary outputs over a wide temperature range.

All-optical valley switch and clock of electronic dephasing.

2D materials with broken inversion symmetry posses an extra degree of freedom, the valley pseudospin, that labels in which of the two energy-degenerate crystal momenta, K or K', the conducting carriers are located. It has been shown that shining circularly-polarized light allows to achieve close to 100% of valley polarization, opening the way to valley-based transistors. Yet, switching of the valley polarization is still a key challenge for the practical implementation of such devices due to the short valley lifetimes. Recent progress in ultrashort laser technology now allows to produce trains of attosecond pulses with controlled phase and polarization between the pulses. Taking advantage of such technology, we introduce a coherent control protocol to turn on, off and switch the valley polarization at faster timescales than electron-hole decoherence and valley depolarization, that is, an ultrafast optical valley switch. We theoretically demonstrate the protocol for hBN and MoS2 monolayers calculated from first principles. Additionally, using two time-delayed linearly-polarized pulses with perpendicular polarization, we show that we can extract the electronic dephasing time T2 from the valley Hall conductivity.

Strain engineering of electronic properties and anomalous valley hall conductivity of transition metal dichalcogenide nanoribbons

Strain engineering is a powerful technique for tuning electronic properties and valley degree of freedom in honeycomb structure of two-dimensional crystals. Carriers in + k and − k (opposite Berry curvature) in transition metal dichalcogenide (TMD) with broken inversion symmetry act as effective magnetic fields, where this polarized valleys are suitable for encoding information. In this work, we study the strained TMD nanoribbons by Slater-Koster tight-binding model, which acquires electronic bands in whole Brillouin zone. From this, we derive a generic profile of strain effect on the electronic band structure of TMD nanoribbons, which shows indirect band gap, and also exhibits a phase transition from semiconductor to metallic by applying uniaxial X-tensile and Y-arc type of strain. Midgap states in strained TMD nanoribbons are determined by calculation of localized density of electron states. Moreover, our findings of anomalous valley Hall conductivity reveal that the creation of pseudogauge fields using strained TMD nanoribbons affect the Dirac electrons, which generate the new quantized Landau level. Furthermore, we demonstrate in strained TMD nanoribbons that strain field can effectively tune both the magnitude and sign of valley Hall conductivity. Our work elucidates the valley Hall transport in strained TMDs due to pseudo-electric and pseudo-magnetic filed will be applicable as information carries for future electronics and valleytronics.

Topological insulator phase transition in 8-Pmmn borophene

We have investigated topological phase transition in 8-Pmmn borophene. This transition from band insulating to topological insulating phase can be tuned by an applied electric field which can vary the band gap induced by spin-orbit coupling. Signatures of this topological phase transition are exhibited in the Hall conductivities. Charge, spin and valley Hall conductivities in the two phases are determined using linear response theory.

Orbital Hall effect as an alternative to valley Hall effect in gapped graphene

Gapped graphene has been proposed to be a good platform to observe the valley Hall effect, a transport phenomenon involving the flow of electrons that are characterized by different valley indices. In the present work, we show that this phenomenon is better described as an instance of the orbital Hall effect, where the ambiguous "valley" indices are replaced by a physical quantity, the orbital magnetic moment, which can be defined uniformly over the entire Brillouin zone. This description removes the arbitrariness in the choice of arbitrary cut-off for the valley-restricted integrals in the valley Hall conductivity, as the conductivity in the orbital Hall effect is now defined as the Brillouin zone integral of a new quantity, called the orbital Berry curvature. This reformulation in terms of OHE provides the direct explanation to the accumulated opposite orbital moments at the edges of the sample, observed in previous Kerr rotation measurements.

Trigonal warping, satellite Dirac points, and multiple field tuned topological transitions in twisted double bilayer graphene

We show that the valley Chern number of the low energy band in twisted double bilayer graphene can be tuned through two successive topological transitions, where the direct bandgap closes, by changing the electric field perpendicular to the plane of the graphene layers. The two transitions with Chern number changes of -3 and +1 can be explained by the formation of three satellite Dirac points around the central Dirac cone in the moir\'e Brillouin zone due to the presence of trigonal warping. The satellite cones have opposite chirality to the central Dirac cone. Considering the overlap of the bands in energy, which lead to metallic states, we construct the experimentally observable phase diagram of the system in terms of the indirect bandgap and the anomalous valley Hall conductivity. We show that while most of the intermediate phase becomes metallic, there is a narrow parameter regime where the transition through three insulating phases with different quantized valley Hall conductivity can be seen. We systematically study the effects of variations in the model parameters on the phase diagram of the system to reveal the importance of particle-hole asymmetry and trigonal warping in constructing the phase diagram. We also study the effect of changes in interlayer tunneling on this phase diagram.

Optical properties of massive anisotropic tilted Dirac systems

We explore the effect of valley-contrasting gaps in the optical response of two-dimensional anisotropic tilted Dirac systems. We study the spectrum of intraband and interband transitions through the joint density of states (JDOS), the optical conductivity tensor, and the Drude spectral weight. The energy bands present an indirect gap in each valley. Thus a new possibility opens for the position of the Fermi level (an ``indirect zone''), and for the momentum space for allowed transitions. The JDOS near each gap displays a set of three van Hove singularities which are in contrast to the case of gapped graphene (an absorption edge only) or 8-$Pmmn$ borophene (two interband critical points due to the tilt). For the Fermi level lying in the gap, the JDOS shows the usual linear dependence on frequency, while when above an indirect zone it looks similar to the borophene case. These spectral characteristics determine the prominent structure of the optical conductivity. The longitudinal conductivity illustrates the strong anisotropy of the optical response. Similarly, the Drude weight is anisotropic and shows regions of nonlinear dependence on the Fermi level. The breaking of valley symmetry leads to a finite Hall response and associated optical properties. The anomalous and valley Hall conductivities present graphene-like behavior with characteristic modifications due to the indirect zones. Almost perfect circular dichroism can be achieved by tuning the exciting frequency with an appropriate Fermi level position. We also calculate the spectra of optical opacity and polarization rotation, which can reach magnitudes of tenths of radians in some cases. The spectral features of the calculated response properties are signatures of the simultaneous presence of tilt and mass and suggest optical ways to determine the formation of different gaps in such class of Dirac systems.

Valley Hall effect and nonlocal resistance in locally gapped graphene

We report on the emergence of bulk, valley-polarized currents in graphene-based devices, driven by spatially varying regions of broken sublattice symmetry, and revealed by non-local resistance ($R_\mathrm{NL}$) fingerprints. By using a combination of quantum transport formalisms, giving access to bulk properties as well as multi-terminal device responses, the presence of a non-uniform local bandgap is shown to give rise to valley-dependent scattering and a finite Fermi surface contribution to the valley Hall conductivity, related to characteristics of $R_\mathrm{NL}$. These features are robust against disorder and provide a plausible interpretation of controversial experiments in graphene/hBN superlattices. Our findings suggest both an alternative mechanism for the generation of valley Hall effect in graphene, and a route towards valley-dependent electron optics, by materials and device engineering.

Imaging the valley and orbital Hall effect in monolayer MoS2.

The topological properties of a material's electronic structure are encoded in its Berry curvature, a quantity which is intimately related to the transverse electrical conductivity. In transition metal dichalcogenides with broken inversion symmetry, the nonzero Berry curvature results in a valley Hall effect. In this paper we identify a previously unrecognized consequence of Berry curvature in these materials: an electric field-induced change in the electrons' charge density orientation. We use first principles calculations to show that measurements of the electric field-induced change in the charge density or local density of states in MoS2 can be used to measure its energy-dependent valley and orbital Hall conductivity.

Magnon magic angles and tunable Hall conductivity in 2D twisted ferromagnetic bilayers

Twistronics is currently one of the most active research fields in condensed matter physics, following the discovery of correlated insulating and superconducting phases in twisted bilayer graphene (tBLG). Here, we present a magnonic analogue of tBLG. We study magnons in twisted ferromagnetic bilayers (tFBL) with collinear magnetic order, including exchange and weak Dzyaloshinskii-Moriya interactions (DMI). For negligible DMI, tFBL presents discrete magnon magic angles and flat moiré minibands analogous to tBLG. The DMI, however, changes the picture and renders the system much more exotic. The DMI in tFBL induces a rich topological magnon band structure for any twist angle. The twist angle turns to a control knob for the magnon valley Hall and Nernst conductivities. Gapped flat bands appear in a continuum of magic angles in tFBL with DMI. In the lower limit of the continuum, the band structure reconstructs to form several topological flat bands. The luxury of twist-angle control over band gaps, topological properties, number of flat bands, and valley Hall and Nernst conductivities renders tFBL a novel device from fundamental and applied perspectives.

Graphene under uniaxial inhomogeneous strain and an external electric field: Landau levels, electronic, magnetic and optical properties

We investigate graphene under an inhomogeneous uniaxial strain and an in-plane electric field. We examine in detail the effect of strain and the electric field on relativistic Landau levels, Hall conductivity, de Haas-van Alphen oscillation and optical conductivity. Using Lorentz transformation in combination with supersymmetric quantum mechanics, we examine three different structures of Landau levels induced by three different profiles of inhomogeneous uniaxial strain and external electric fields. It is shown that strain-induced pseudomagnetic field forms Landau levels while electric field opposes formation of these levels. Besides the collapse of strain induced Landau levels, the influences of electric field on the quantization of strain dependent valley Hall conductivity, de Haas-van Alphen quantum oscillation of magnetization as well as optical conductivity have been investigated.

Influence of interface induced valley-Zeeman and spin-orbit couplings on transport in heterostructures of graphene on WSe2

We investigate the electronic dispersion and transport properties of graphene/${\mathrm{WSe}}_{2}$ heterostructures in the presence of a proximity induced spin-orbit coupling (SOC) using a low-energy Hamiltonian, with different types of symmetry breaking terms, obtained from a four-band, first and second nearest-neighbour tight-binding (TB) one. The competition between different perturbation terms leads to inverted SOC bands. Further, we study the effect of symmetry breaking terms on ac and dc transport by evaluating the corresponding conductivities within linear response theory. The scattering-independent part of the valley-Hall conductivity, as a function of the Fermi energy ${E}_{F}$, is mostly negative in the ranges $\ensuremath{-}{\ensuremath{\lambda}}_{R}\ensuremath{\leqslant}{E}_{F}$ and ${E}_{F}\ensuremath{\geqslant}{\ensuremath{\lambda}}_{R}$ when the strength ${\ensuremath{\lambda}}_{R}$ of the Rashba SOC increases except for a very narrow region around ${E}_{F}=0$ in which it peaks sharply upward. The scattering-dependent diffusive conductivity increases linearly with electron density, is directly proportional to ${\ensuremath{\lambda}}_{R}$ in the low- and high-density regimes, but weakens for ${\ensuremath{\lambda}}_{R}=0$. We investigate the optical response in the presence of a SOC-tunable band gap for variable ${E}_{F}$. An interesting feature of this SOC tuning is that it can be used to switch on and off the Drude-type intraband response. Furthermore, the ac conductivity exhibits interband responses due to the Rashba SOC. We also show that the valley-Hall conductivity changes sign when ${E}_{F}$ is comparable to ${\ensuremath{\lambda}}_{R}$ and vanishes at higher values of ${E}_{F}$. It also exhibits a strong dependence on temperature and a considerable structure as a function of the frequency.

Control of spin and valley Hall effects in monolayer transition metal dichalcogenides by magnetic proximity effect

Monolayer transition metal dichalcogenides have strong spin–orbit coupling and broken space inversion symmetry, which enable them to be the key building blocks in realizing spin and valley-related effects. Here, we report the spin and valley Hall conductivities of monolayer transition metal dichalcogenides in the presence of the magnetic proximity effect, which is introduced by a ferromagnetic substrate. It is found that the profile and magnitude of the spin and valley Hall conductivities in monolayer transition metal dichalcogenides are different with and without magnetic exchange interactions. This difference can be attributed to the asymmetrical band structure of monolayer transition metal dichalcogenides and chemical potential-dependent interband transitions. The former comes from the fact that the magnetic proximity effect can effectively break the time reversal symmetry and thus lead to the asymmetry of the band structures between K+ and K− valleys, which causes the final changes in the spin and valley Hall conductivities. Our findings demonstrate that the magnetic proximity effect can affect the spin as well as valley Hall behaviors in monolayer transition metal dichalcogenides, and this strategy is applicable for other two-dimensional layered materials, which is promising for spintronic and valleytronic devices.

Tunable valley Hall effect in gate-defined graphene superlattices

We theoretically investigate gate-defined graphene superlattices with broken inversion symmetry as a platform for realizing tunable valley dependent transport. Our analysis is motivated by recent experiments [C. Forsythe et al., Nat. Nanotechnol. 13, 566 (2018)] wherein gate-tunable superlattice potentials have been induced on graphene by nanostructuring a dielectric in the graphene/patterneddielectric/gate structure. We demonstrate how the electronic tight-binding structure of the superlattice system resembles a gapped Dirac model with associated valley dependent transport using an unfolding procedure. In this manner we obtain the valley Hall conductivities from the Berry curvature distribution in the superlattice Brillouin zone, and demonstrate the tunability of this conductivity by the superlattice potential. Finally, we calculate the valley Hall angle relating the transverse valley current and longitudinal charge current and demonstrate the robustness of the valley currents against irregularities in the patterned dielectric.