Valley Hall Conductivity Research Articles

Theory of Valley Hall Conductivity in Graphene with Gap

The valley Hall conductivity, having opposite signs between the K and K′ valleys, is calculated in disordered monolayer graphene with gap. In ideal graphene without disorder, it is quantized into ±e2/2h within the gap and its absolute value decreases in proportion to the inverse of the Fermi energy in the band continuum. In the presence of scatterers, the Hall conductivity in the band continuum is strongly enhanced. This enhancement depends on explicit form of scattering potential even in the clean limit where the concentration and strength of scatterers are vanishingly small. Numerical calculations performed within the self-consistent Born approximation for scatterers with Gaussian potential and for charged impurities show that the valley Hall conductivity remains appreciable in the presence of large disorder and exhibits double-peak structure near zero energy.

Controllable intrinsic DC spin/valley Hall conductivity in ferromagnetic silicene: Exploring a fully spin/valley polarized transport

We study intrinsic DC spin and valley Hall conductivity in doped ferromagnetic silicene in the presence of an electric filed applied perpendicular to silicene sheet. By calculating its energy spectrum and wavefunction and by making use of Kubo formalism, we obtain a general relation for the transverse Hall conductivity which can be used to obtain spin- and valley-Hall conductivity. Our results, in the zero limit of the exchange field, reduces to the previous results. Furthermore we discuss electrically tunable spin and valley polarized transport in ferromagnetic silicene and obtain the necessary conditions for observing a fully spin or valley polarized transport.

Valley Hall effect in disordered monolayerMoS2from first principles

Electrons in certain two-dimensional crystals possess a pseudospin degree of freedom associated with the existence of two inequivalent valleys in the Brillouin zone. If, as in monolayer ${\mathrm{MoS}}_{2}$, inversion symmetry is broken and time-reversal symmetry is present, equal and opposite amounts of $k$-space Berry curvature accumulate in each of the two valleys. This is conveniently quantified by the integral of the Berry curvature over a single valley---the valley Hall conductivity. We generalize this definition to include contributions from disorder described with the supercell approach, by mapping (``unfolding'') the Berry curvature from the folded Brillouin zone of the disordered supercell onto the normal Brillouin zone of the pristine crystal, and then averaging over several realizations of disorder. We use this scheme to study from first principles the effect of sulfur vacancies on the valley Hall conductivity of monolayer ${\mathrm{MoS}}_{2}$. In dirty samples the intrinsic valley Hall conductivity receives gating-dependent corrections that are only weakly dependent on the impurity concentration, consistent with side-jump scattering and the unfolded Berry curvature can be interpreted as a $k$-space resolved side jump. At low impurity concentrations skew scattering dominates, leading to a divergent valley Hall conductivity in the clean limit. The implications for the recently observed photoinduced anomalous Hall effect are discussed.

Topological Valley Currents in Gapped Dirac Materials.

Gapped 2D Dirac materials, in which inversion symmetry is broken by a gap-opening perturbation, feature a unique valley transport regime. Topological valley currents in such materials are dominated by bulk currents produced by electronic states just beneath the gap rather than by edge modes. The system ground state hosts dissipationless persistent valley currents existing even when topologically protected edge modes are absent. Valley currents induced by an external bias are characterized by a quantized half-integer valley Hall conductivity. The undergap currents dominate magnetization and the charge Hall effect in a light-induced valley-polarized state.

Low-temperature linear transport of two-dimensional massive Dirac fermions in silicene: Residual conductivity and spin/valley Hall effects

Considering finite-temperature screened electron-impurity scattering, we present a kinetic equation approach to investigate transport properties of two-dimensional massive fermions in silicene. We find that the longitudinal conductivity is always nonvanishing when chemical potential lies within the energy gap. This residual conductivity arises from interband correlation and strongly depends on strength of electron-impurity scattering. We also clarify that the electron-impurity interaction makes substantial contributions to the spin- and valley-Hall conductivities, which, however, are almost independent of impurity density. The dependencies of longitudinal conductivity as well as of spin- and valley-Hall conductivities on chemical potential, on temperature, and on gap energy are analyzed.

Valley Hall Conductivity in Gapped Graphene Enhanced by Scattering Even in the “Clean Limit”

The valley Hall conductivity in gapped graphene has been found to be significantly enhanced by scattering in the band continuum, even for vanishingly small concentration and strength of scatterers.

Dc and ac transport in silicene

We investigate dc and ac transport in silicene in the presence of a perpendicular electric field Ez that tunes its band gap, finite temperatures, and level broadening. The interplay of silicene's strong spin-orbit interaction and the field Ez gives rise to topological phase transitions. We show that at a critical value of Ez the dc spin-Hall conductivity undergoes a transition from a topological insulator phase to a band insulator one. We also show that the spin- and valley-Hall conductivities exhibit a strong temperature dependence. In addition, the longitudinal conductivity is examined as a function of the carrier density ne, for screened Coulomb impurities of density ni, and found to scale linearly with ne/ni. It also exhibits an upward jump at a critical value of ne that is associated with the opening of a new spin subband. Furthermore, the contributions of the spin-up and spin-down carriers to the power absorption spectrum depend sensitively on the topological phase and valley index. Analytical results are presented for both dc and ac conductivities in the framework of linear response theory.

Conductivity of Dirac fermions with phonon-induced topological crossover

We study the Hall conductivity in single layer gapped Dirac fermion materials including coupling to a phonon field, which not only modifies the quasi-particle dynamics through the usual self-energy term but also renormalizes directly the gap. Consequently the Berry curvature is modified. As the temperature is increased the sign of the renormalized gap can change and the material can cross over from a band insulator to a topological insulator at higher temperature (T). The effective Chern numbers defined for valley and spin Hall conductivity show a rich phase diagram with increasing temperature. While the spin and valley DC Hall conductivity is no longer quantized at elevated temperature a change in sign with increasing T is a clear indication of a topological crossover. The chirality of the circularly polarized light which is dominantly absorbed by a particular valley can change with temperature as a result of a topological crossover.

AC/DC spin and valley Hall effects in silicene and germanene

The intrinsic spin and valley Hall conductivities of silicene, germanene, and other similar two-dimensional crystals are explored theoretically. Particular attention is given to the effects of the intrinsic spin-orbit coupling, electron doping, and the type of insulating phase of the system (i.e., a topological insulator or a band insulator) which can be tuned by a perpendicular electric field. At finite frequency, the transverse edge to which carriers of a particular spin and valley label flow can be controlled such that an accumulation of a particular combination of spin and valley index can be obtained. The direction of flow is found to be dependent on the type of insulating phase. The magnitude of the Hall conductivity response is enhanced from the DC values at certain incident photon frequencies associated with the onset of interband transitions. Analytic results are presented for both the DC and finite-frequency results.

Gate-induced Dirac cones in multilayer graphenes

We study the electronic structures of ABA (Bernal) stacked multilayer graphenes in uniform perpendicular electric field, and show that the interplay of the trigonal warping and the potential asymmetry gives rise to a number of emergent Dirac cones nearly touching at zero energy. The band velocity and the energy region (typically a few tens of meV) of these gate-induced Dirac cones are tunable with the external electric field. In ABA trilayer graphene, in particular, applying an electric field induces a non-trivial valley Hall state, where the energy gap at the Dirac point is filled by chiral edge modes which propagate in opposite directions between two valleys. In four-layer graphene, in contrast, the valley Hall conductivity is zero and there are no edge modes filling in the gap. A nontrivial valley Hall state generally occurs in asymmetric odd layer graphenes and this is closely related to a hidden chiral symmetry which exists only in odd layer graphenes.

Unbalanced edge modes and topological phase transition in gated trilayer graphene

Gapless edge modes hosted by chirally-stacked trilayer graphene display unique features when a bulk gap is opened by applying an interlayer potential difference. We show that trilayer graphene with half-integer valley Hall conductivity leads to unbalanced edge modes at opposite zigzag boundaries, resulting in a natural valley current polarizer. This unusual characteristic is preserved in the presence of Rashba spin-orbit coupling that turns a gated trilayer graphene into a ${Z}_2$ topological insulator with an odd number of helical edge mode pairs.

Transport Properties of Graphene Nanoroads in Boron Nitride Sheets

We demonstrate that the one-dimensional (1D) transport channels that appear in the gap when graphene nanoroads are embedded in boron nitride (BN) sheets are more robust when they are inserted at AB/BA grain boundaries. Our conclusions are based on ab initio electronic structure calculations for a variety of different crystal orientations and bonding arrangements at the BN/C interfaces. This property is related to the valley Hall conductivity present in the BN band structure and to the topologically protected kink states that appear in continuum Dirac models with position-dependent masses.

Spontaneous Quantum Hall States and Novel Luttinger Liquids in Chiral Graphene

Chirally stacked neutral N-layer graphene systems with N ≥ 2 exhibit spontaneous inversion symmetry breaking and interaction induced charge gaps due to interlayer exchange interactions. A variety of distinct broken symmetry states are distinguished by their charge, spin, and valley Hall conductivities, by their orbital magnetizations, and by their edge state properties. In the spinless case, quantum anomalous and valley Hall states are favored by a weak magnetic field and by an electric field between the graphene layers, respectively. At interfaces between different phases one dimensional gapless modes exhibit novel Luttinger liquid behaviors.

Berry-curvature-mediated valley-Hall and charge-Hall effects in graphene via strain engineering

We point out a practical way to induce a finite charge-Hall conductivity in graphene via an off-diagonal strain. Our conclusions are based on a general analysis and supported by numerical examples. The interplay between a substrate-induced gap and the strain is discussed. We show that the substrate-induced gap itself does not break the symmetry that maps one valley onto the other, and hence does not yield a net charge-Hall effect. The net charge-Hall conductivity is found to be sizable and should be observable in the presence of both the gap and the strain. The sign of the Hall conductivity can be tuned by varying the strain parameters. The valley-Hall conductivity is studied as well. The valley and the orbital magnetic moment accumulations on the sample edges may be detected in a Hall conductivity experiment.

Valley-Hall kink and edge states in multilayer graphene

We report on a theoretical study of one-dimensional (1D) states localized at few-layer graphene system ribbon edges, and at interfaces between few-layer graphene systems with different valley Hall conductivities. These 1D states are topologically protected when valley mixing is neglected. We address the influence on their properties of stacking arrangement, interface structure, and external electric field perpendicular to the layers. We find that 1D states are generally absent at multilayer ribbon armchair direction edges, but present irrespective of crystallographic orientation at any internal valley-Hall interface of an ABC stacked multilayer.

Spontaneous Quantum Hall States in Chirally Stacked Few-Layer Graphene Systems

Chirally stacked N-layer graphene systems with N≥2 exhibit a variety of distinct broken symmetry states in which charge density contributions from different spins and valleys are spontaneously transferred between layers. We explain how these states are distinguished by their charge, spin, and valley Hall conductivities, by their orbital magnetizations, and by their edge state properties. We argue that valley Hall states have [N/2] edge channels per spin valley.