Bilayer Heisenberg ferromagnetic insulators hold degenerate terahertz Dirac magnon modes associated with two opposite valleys of the hexagonal Brillouin zone. We show that this energy degeneracy can be removed by breaking of the inversion symmetry (I), leading to a topological magnon valley current. We show furthermore that this current leads to valley Seebeck effect for magnons and is thereby detectable. We perform calculations in the specific example of bilayer CrBr3, where I can be broken by electrostatic doping.
We propose a scheme to trap and filter electrons, valley dependently, on a scale beyond the diffraction limit, in a gapped Dirac system using a circularly polarized light beam and a microscale metallic resonator. The main mechanism allowing the selection is the valley dependent break of the time reversal symmetry. Indeed in one valley, the light partially closes the already existing gap while it opens it in the other one. This difference in the band structure close to the Dirac points, induces a change in the dynamics of the electrons, leading to a valley router behaviour of the system.
Gapped 2D Dirac materials, in which inversion symmetry is broken by a gap-opening perturbation, feature a unique valley transport regime. The system ground state hosts dissipationless persistent valley currents existing even when topologically protected edge modes are absent or when they are localized due to edge roughness. Topological valley currents in such materials are dominated by bulk currents produced by electronic states just beneath the gap rather than by edge modes. Dissipationless currents induced by an external bias are characterized by a quantized half-integer valley Hall conducti…
Recent experiments have revealed that novel nonequilibrium states consistent with the hydrodynamic description of electrons are realized in ultrapure graphene, which hosts the valley degrees of freedom. Here, we formulate a theory of electron hydrodynamics including dissipation processes of the valley angular momentum by employing the concept of micropolar fluids. As a result, our theory proposes a novel strategy to generate a valley polarization by the microrotation. We uncover that the rotational viscosity induces longitudinal valley currents which are second order in electric fields.
Motivated by recent experimental work, we consider spin transport between a normal metal and a gapped quantum paramagnet. We model the latter as the magnonic Mott-insulating phase of an easy-plane ferromagnetic insulator. We evaluate the spin current mediated by the interface exchange coupling between the ferromagnet and the adjacent normal metal. For the strongly interacting magnons that we consider, this spin current gives rise to a spin Hall magnetoresistance that strongly depends on the magnitude of the magnetic field, rather than its direction. This Letter may motivate electrical detectio…
We investigate the Goos-Hänchen shift for ballistic electrons (i) reflected from a step-like inhomogeneity of the potential energy and (or) effective mass, and (ii) transmitted through a ferromagnetic barrier region in monolayer silicene or gapped graphene. For the electrons reflected from a single interface we found that the Goos-Hänchen shift is valley-polarized for gapped graphene structure, and valley- and spin-polarized for silicene due to the spin-valley coupling. Incontrast, for example, to gapless graphene the lateral beam shift in gapped structures occurs not only in the case of total…
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 va…
For gapped graphene, we predict that an intense ultrashort (single-oscillation) circularly-polarized optical pulse can induce a large population of the conduction band and a large valley polarization. With an increase in the bandgap, the magnitude of the valley polarization gradually increases from zero (for the native gapless graphene) to a value on the order of unity. The energy bandwidth of the electrons excited into the conduction band can be very large ($\gtrsim 10$ eV for a reasonable pulse amplitude of $\sim 0.5$ $\mathrm{V/Å}$). These phenomena are due to the effect of topological reso…
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…
Owing to its charge-free property, magnon is highly promising to achieve dissipationless transport without Joule heating and thus potentially applicable to energy-efficient devices. Moreover, a kagome lattice, as stacking layers of many magnon ferromagnets, also exhibits valley structure in quasiparticle spectra, which are likely to add a new dimension to magnon excitation. Here, we investigate valley magnon and associated valley modulation in a kagome lattice, with staggered exchange interaction and Dzyaloshinskii-Moriya interaction. The staggered exchange interaction breaks spatial inversion…
We present a low-energy effective-mass theory to describe chiral orbital current and anomalous magnetic moment in graphenes with band gap and related materials. We show that a Bloch electron generally contains an anomalous current density due to inter-band matrix elements, which describes a chiral current circulation associated with magnetic moment. In gapped graphenes, the chiral current is opposite between different valleys, and corresponding magnetic moment accounts for valley splitting of Landau levels. In gapped bilayer graphene, in particular, the valley-dependent magnetic moment is resp…
We study a gapped graphene monolayer in a combination of uniform magnetic field and strain-induced uniform pseudomagnetic field. The presence of two fields completely removes the valley degeneracy. The resulting density of states shows a complicated behaviour that can be tuned by adjusting the strength of the fields. We analyze how these features can be observed in the sublattice, valley and full density of states. The analytical expression for the valley DOS is derived.
Valley Hall Viscosity in Gapped Graphene with and without a Magnetic Field
Kavli Institute for Theoretical Sciences, University of Chinese Academy of Sciences（中国科学院大学 Kavli理论科学研究所）；Center for Spintronics Research Network, Keio University（庆应义塾大学自旋电子学研究网络中心）；Department of Physics, Nagoya University（名古屋大学物理学系）；Department of Physics, Kyoto University（京都大学物理学系）；Institute of Solid State Physics, University of Tokyo（东京大学固体物理研究所）；CAS Center for Excellence in Topological Quantum Computation, University of Chinese Academy of Sciences（中国科学院大学 中科院拓扑量子计算卓越创新中心）；RIKEN Center for Emergent Matter Science (CEMS)（理化学研究所 新兴物质科学中心）；Advanced Science Research Center, Japan Atomic Energy Agency（日本原子力研究机构 先进科学研究中心）
In this work, we provide a detailed derivation of Foldy-Wouthuysen (FW) transformation for two-dimensional (2D) gapped Dirac fermions under external fields and apply the formalism to study valley excitons in 2D semiconducting materials. Similar to relativistic quantum few-body problem, the gapped Dirac equation can be transformed into a Schrödinger equation with "relativistic" correction terms. In this 2D materials system, the correction terms can be interpreted as the Berry-curvature effect. The Hamiltonian for a valley exciton in external fields can be written based on the FW transformed Dir…
Valleytronics is a pioneering technological field relying on the valley degree of freedom to achieve novel electronic functionalities. Topological valley-polarized electrons confined to domain walls in bilayer graphene were extensively studied in view of their potentials in valleytronics. Here, we study the magnonic version of domain wall excitations in 2D honeycomb ferromagnetic bilayers (FBL) with collinear order. In particular, we explore the implications of Dzyaloshinskii-Moriya interaction (DMI) and electrostatic doping (ED) on the existence and characteristics of 1D magnons confined to l…
The rich and unconventional physics in layered 2D magnets can open new avenues for topological magnonics and magnon valleytronics. In particular, two-dimensional (2D) bilayer quantum magnets are gaining increasing attention due to their intriguing stacking-dependent magnetism, controllable ground states, and topological excitations induced by magnetic spin-orbit couplings (SOCs). Despite the substantial research on these materials, their topological features remain widely unexplored to date. The present study comprehensively investigates the magnon topology and magnon valley-polarization in ho…
Magnonic excitations in the two-dimensional (2D) van der Waals (vdW) ferromagnet CrI3 are studied. We find that bulk magnons exhibit a non-trivial topological band structure without the need for Dzyaloshinskii-Moriya (DM) interaction. This is shown in vdW heterostructures, consisting of single-layer CrI3 on top of different 2D materials as MoTe2, HfS2 and WSe2. We find numerically that the proposed substrates modify substantially the out-of-plane magnetic anisotropy on each sublattice of the CrI3 subsystem. The induced staggered anisotropy, combined with a proper band inversion, leads to the o…
Berry curvature is analogous to magnetic field but in momentum space and is commonly present in materials with non-trivial quantum geometry. It endows Bloch electrons with transverse anomalous velocities to produce Hall-like currents even in the absence of a magnetic field. We report the direct observation of in situ tunable valley-selective Hall effect (VSHE), where inversion symmetry, and thus the geometric phase of electrons, is controllable by an out-of-plane electric field. We use high-quality bilayer graphene with an intrinsic and tunable bandgap, illuminated by circularly polarized mid-…
We calculate the Hall viscosity and the nonlocal (i.e., dependent on wave vector $\bm q$) Hall conductivity of "gapped graphene" (a non-topological insulator with two valleys) in the presence of a strong perpendicular magnetic field. Using the linear-response theory at zero temperature within the Dirac approximation for the Landau levels, we present analytical expressions for both valley and total Hall viscosity and conductivity up to $\bm q^2$ at all frequencies. Although the final formulas for total Hall viscosity and conductivity are similar to the ones previously obtained for gapless graph…
The rise of graphene opens a new door to qubit implementation, as discussed in the recent proposal of valley pair qubits in double quantum dots of gapped graphene (Wu et al., arXiv: 1104.0443 [cond-mat.mes-hall]). The work here presents the comprehensive theory underlying the proposal. It discusses the interaction of electrons with external magnetic and electric fields in such structures. Specifically, it examines a strong, unique mechanism, i.e., the analogue of the 1st-order relativistic effect in gapped graphene. This mechanism is state mixing free and allows, together with the electrically…
We demonstrate theoretically that the interaction of electrons in gapped Dirac materials (gapped graphene and transition-metal dichalchogenide monolayers) with a strong off-resonant electromagnetic field (dressing field) substantially renormalizes the band gaps and the spin-orbit splitting. Moreover, the renormalized electronic parameters drastically depend on the field polarization. Namely, a linearly polarized dressing field always decreases the band gap (and, particularly, can turn the gap into zero), whereas a circularly polarized field breaks the equivalence of valleys in different points…
We present a microscopic theory for collective excitations of quantum anomalous Hall ferromagnets (QAHF) in twisted bilayer graphene. We calculate the spin magnon and valley magnon spectra by solving Bethe-Salpeter equations, and verify the stability of QAHF. We extract the spin stiffness from the gapless spin wave dispersion, and estimate the energy cost of a skyrmion-antiskyrmion pair, which is found to be comparable in energy with the Hartree-Fock gap. The valley wave mode is gapped, implying that the valley polarized state is more favorable compared to the valley coherent state. Using a no…
We develop a microscopic theory of an unconventional photogalvanic effect in two-dimensional materials with the Dirac energy spectrum of the carriers of charge under strong driving. As a test bed, we consider a layer of a transition metal dichalcogenide, exposed to two different electromagnetic fields. The first pumping field is circularly-polarized, and its frequency exceeds the material bandgap. It creates an extremely nonequilibrium distribution of electrons and holes in one valley (K) and opens dynamical gaps, whereas the other valley (K') remains empty due to the valley-dependent interban…
We show that the spectrum of subbands in an electrostatically defined quantum wire in gapped bilayer graphene directly manifests the minivalley structure and reflects Berry curvature via the associated magnetic moment of the states in the low-energy bands of this two-dimensional material. We demonstrate how these appear in degeneracies of the low-energy minibands and their valley splitting, which develops linearly in a weak magnetic field. Consequently, magneto-conductance of a ballistic point contact connecting two non-gapped areas of a bilayer through a gapped (top and bottom gated) barrier …
We consider the anomalous thermoelectric transport in gapped single and bilayer graphene where the gap may be due to broken inversion symmetry. In the presence of the gap, non-trivial Berry phase effects can be shown to mediate a transverse thermoelectric voltage in response to an applied temperature gradient even in the absence of a perpendicular magnetic field. This spontaneous, anomalous Nernst effect is nonzero for non-uniform chemical potential in the two inequivalent valleys in the graphene band structure. Conversely, the Nernst response can be used to create a valley-index polarization …
The valley-contrasting orbital magnetic moment of Bloch electrons allows the lifting of valley degeneracy by an out-of-plane magnetic field. We demonstrate that this leads to negative magnetoresistance, utilizing a gapped two-dimensional multi-valley model as an example. An intuitive physical picture in terms of the increased carrier density from a magnetic gating effect is proposed for this negative magnetoresistance. In particular, giant negative magnetoresistance is achieved after one of the two valleys is depleted by the magnetic field. This new mechanism of negative magnetoresistance is a…
Topological domain walls in dual-gated gapped bilayer graphene host edge states that are gate- tunable and valley polarized. Here we predict that plasmonic collective modes can propagate along these topological domain walls even at zero bulk density, and possess a markedly different character from that of bulk plasmons. Strikingly, domain wall plasmons are extremely long-lived, with plasmon lifetimes that can be orders of magnitude larger than the transport scattering time in the bulk. While most pronounced at low temperatures, long domain wall plasmon lifetimes persist even at room temperatur…
We study the momentum alignment of photoexcited carriers and the optical control of valley population in gapless and gapped two-dimensional Dirac materials. The trigonal warping effect leads to the spatial separation of charge carriers belonging to different valleys upon linearly polarized high-frequency photoexcitation. Valley separation in gapped materials can be detected by measuring the degree of circular polarization of band-edge photoluminescence at different sides of the sample or light spot (optical valley Hall effect). We demonstrate that the celebrated Rashba effect, caused by substr…
Electronic states at domain walls in bilayer graphene are studied by analyzing their four and two band continuum models, by performing numerical calculations on the lattice, and by using quantum geometric arguments. The continuum theories explain the distinct electronic properties of boundary modes localized near domain walls formed by interlayer electric field reversal, by interlayer stacking reversal, and by simultaneous reversal of both quantities. Boundary mode properties are related to topological transitions and gap closures which occur in the bulk Hamiltonian parameter space. The import…
We propose a device for the generation of valley polarized electronic current in bilayer graphene. By analyzing the response of this material to intense terahertz frequency light in the presence of a transverse electric field, we demonstrate that dynamical states are induced in the gapped energy region, and if the system parameters are properly tuned, these states exist only in one valley. The valley polarized states can then be used to filter an arbitrary electron current, so generating a valley polarized current.
We study the topological properties of magnon excitations in a wide class of three dimensional (3D) honeycomb lattices with ferromagnetic ground states. It is found that they host nodal ring magnon excitations. These rings locate on the same plane in the momentum space. The rings can be gapped by Dzyaloshinskii-Moriya (DM) interactions to form two Weyl points with opposite charges. We explicitly discuss these physics in the simplest 3D honeycomb lattice, the hyperhoneycomb lattice and show drumhead and arc surface states in the nodal ring and Weyl phases, respectively, due to the bulk-boundary…
We study the correspondence between the non-trivial topological properties associated with the individual valleys of gapped bilayer graphene (BLG), as a prototypical multi-valley system, and the gapless modes at its edges and other interfaces. We find that the exact connection between the valley-specific Hall conductivity and the number of gapless edge modes does not hold in general, but is dependent on the boundary conditions, even in the absence of intervalley coupling. This non-universality is attributed to the absence of a well-defined topological invariant within a given valley of BLG; ye…
Berry curvature hot spots in two-dimensional materials with broken inversion symmetry are responsible for the existence of transverse valley currents, which give rise to giant nonlocal dc voltages. Recent experiments in high-quality gapped graphene have highlighted a saturation of the nonlocal resistance as a function of the longitudinal charge resistivity $ρ_{{\rm c}, xx}$, when the system is driven deep into the insulating phase. The origin of this saturation is, to date, unclear. In this work we show that this behavior is fully compatible with bulk topological transport in the regime of lar…
An anomaly in the magnon dispersion of the amorphous ferromagnet Co$_{4}$P, often referred to as a `roton-like' excitation, attracted much attention half a century ago. With the current interest in heat and spin currents in amorphous magnets, we apply modern simulation methods, combining reverse Monte Carlo to build the atomic structure and the stochastic Landau-Lifshitz equation for spin dynamics, to re-investigate the magnetic excitation spectrum. We find two magnon valleys, one at the origin and another at a finite wavenumber close to the observations, but without a magnon gap. We conclude …
In the model of gapped graphene, we have shown how the recently predicted topological resonances are solely related to the presence of an energy band gap at the $K$ and $K^\prime$ points of the Brillouin zone. In the field of a strong single-oscillation chiral (circularly-polarized) optical pulse, the topological resonance causes the valley-selective population of the conduction band. This population distribution represents a chiral texture in the reciprocal space that is structured with respect to the pulse separatrix as has earlier been predicted for transition metal dichalcogenides. As the …
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 …
We study transport across p-n junctions of gapped two-dimensional semi-Dirac materials: nodal semimetals whose energy bands disperse quadratically and linearly along distinct crystal axes. The resulting electronic properties --- relevant to materials such as TiO$_2$/VO$_2$ multilayers and $α$-(BEDT-TTF)$_2$I$_3$ salts --- continuously interpolate between those of mono- and bi-layer graphene as a function of propagation angle. We demonstrate that tunneling across the junction depends on the orientation of the tunnel barrier relative to the crystalline axes, leading to strongly non-monotonic cur…
We report on Landau level spectroscopy studies of two HgTe quantum wells (QWs) near or at the critical well thickness, where the band gap vanishes. In magnetic fields up to $B$=16T, oriented perpendicular to the QW plane, we observe a $\sqrt{B}$ dependence for the energy of the dominant cyclotron resonance (CR) transition characteristic of two-dimensional Dirac fermions. The dominant CR line exhibits either a single or double absorption lineshape for the gapless or gapped QW. Using an effective Dirac model, we deduce the band velocity of single valley Dirac fermions in gapless HgTe quantum wel…
The recent observation [R. V. Gorbachev et al., Science {\bf 346}, 448 (2014)] of nonlocal resistance $R_\mathrm{NL}$ near the Dirac point (DP) of multiterminal graphene on aligned hexagonal boron nitride (G/hBN) has been interpreted as the consequence of topological valley Hall currents carried by the Fermi sea states just beneath the bulk gap $E_g$ induced by the inversion symmetry breaking. However, the valley Hall conductivity $σ^v_{xy}$, quantized inside $E_g$, is not directly measurable. Conversely, the Landauer-Büttiker formula, as numerically exact approach to observable nonlocal trans…
Valley pseudospin, the quantum degree of freedom characterizing the degenerate valleys in energy bands, is a distinct feature of two-dimensional Dirac materials. Similar to spin, the valley pseudospin is spanned by a time reversal pair of states, though the two valley pseudospin states transform to each other under spatial inversion. The breaking of inversion symmetry induces various valley-contrasted physical properties; for instance, valley-dependent topological transport is of both scientific and technological interests. Bilayer graphene (BLG) is a unique system whose intrinsic inversion sy…
Magnons dominate the magnetic response of the recently discovered insulating ferromagnetic two dimensional crystals such as CrI$_3$. Because of the arrangement of the Cr spins in a honeycomb lattice, magnons in CrI$_3$ bear a strong resemblance with electronic quasiparticles in graphene. Neutron scattering experiments carried out in bulk CrI$_3$ show the existence of a gap at the Dirac points, that has been conjectured to have a topological nature. Here we propose a theory for magnons in ferromagnetic CrI$_3$ monolayers based on an itinerant fermion picture, with a Hamiltonian derived from fir…
We consider the combined effect of a gap and a Zeeman interaction on the helical Dirac fermions which exist on the surface of a topological insulator. Magneto-optical properties, the magnetization, Hall effect and the density of states are considered with emphasis on the particle-hole asymmetry which arises when a subdominant Schrödinger piece is included along with the dominant Dirac part of the Hamiltonian. When appropriate, we compare our results with those of a single valley gapped graphene system for which Zeeman splitting behaves differently. We provide a derivation of the phase offset i…
In bilayer graphene, electrostatic confinement can be realized by a suitable design of top and back gate electrodes. We measure electronic transport through a bilayer graphene quantum dot, which is laterally confined by gapped regions and connected to the leads via p-n junctions. Single electron and hole occupancy is realized and charge carriers $n = 1, 2,\dots 50$ can be filled successively into the quantum system with charging energies exceeding $10 \ \mathrm{meV}$. For the lowest quantum states, we can clearly observe valley and Zeeman splittings with a spin g-factor of $g_{s}\approx 2$. In…
We propose a novel method for the generation of the transverse valley currents, which is based on the pure crossed Andreev reflection (pCAR) in the superconducting hybrid junctions composed of the gapped $α-\mathcal{T}_3$ lattices with ferromagnet-induced exchange interaction. The angle-resolved pCAR probability is asymmetric for a given valley, resulting in the transverse valley currents with zero net charge. This pCAR assisted charge-valley conversion is highly efficient with the valley Hall angle reaching an order of unity, suggesting potential applications for valleytronic devices.
Topologically protected gapless edge states are phases of quantum matter which behave as massless Dirac fermions, immunizing against disorders and continuous perturbations. Recently, a new class of topological insulators (TIs) with topological corner states have been theoretically predicted in electric systems, and experimentally realized in two-dimensional (2D) mechanical and electromagnetic systems, electrical circuits, optical and sonic crystals, and elastic phononic plates. Here, we demonstrate a pseudospin-valley-coupled phononic TI, which simultaneously exhibits gapped edge states and to…
We study the electromagnetic response and surface electromagnetic modes in a generic gapped Dirac material under pumping with circularly polarized light. The valley imbalance due to pumping leads to a net Berry curvature, giving rise to a finite transverse conductivity. We discuss the appearance of nonreciprocal chiral edge modes, their hybridization and waveguiding in a nanoribbon geometry, and giant polarization rotation in nanoribbon arrays.
We investigate the emergence of topological valley Hall and kink states in a two-dimensional topolectrical (TE) model as a result of broken chiral and reflection symmetries. The TE system consists of two segments hosting distinct topological states with opposite signs of the valley Hall index, and separated by a heterojunction. In the practical circuit, the valley Hall index can be flipped between the two segments by modulating the onsite potential on the sublattice nodes of the respective segments. The presence of resistive coupling, which introduces non-Hermiticity in the system, subsequentl…
Understanding how the electron spin is coupled to orbital degrees of freedom, such as a valley degree of freedom in solid-state systems is central to applications in spin-based electronics and quantum computation. Recent developments in the preparation of electrostatically-confined quantum dots in gapped bilayer graphene (BLG) enables to study the low-energy single-electron spectra in BLG quantum dots, which is crucial for potential spin and spin-valley qubit operations. Here, we present the observation of the spin-valley coupling in a bilayer graphene quantum dot in the single-electron regime…
The recent realizations of topological valley phase in photonic crystal, an analog of gapped valleytronic materials in electronic system, are limited to the valley Chern number of one. In this letter, we present a new type of valley phase that can have large valley Chern number of two or three. The valley phase transitions between the different valley Chern numbers (from one to three) are realized by changing the configuration of the unit cell. We demonstrate that these new topological phases can guide the wave propagation robustly along the domain wall of sharp bent. Our results are promising…
