The effect of strong long-range disorder on the quantization of the Hall conductivity $σ_{xy}$ in graphene is studied numerically. It is shown that increasing Landau-level mixing progressively destroys all plateaus in $σ_{xy}$ except the plateaus at $σ_{xy}=\mp e^2/2h$ (per valley and per spin). The critical state at the charge-neutral Dirac point is robust to strong disorder and belongs to the universality class of the conventional plateau transitions in the integer quantum Hall effect. We propose that the breaking of time-reversal symmetry by ripples in graphene can realize this quantum critical point in a vanishing magnetic field.
We derive the effective field theory from the microscopic Hamiltonian of interacting two-dimensional (pseudo) Dirac electrons by performing a statistic gauge transformation. The quantized Hall conductance are expected to be $σ_{xy}=\frac{e^2}{h}(2k-1)$ with $k$ is arbitrary integer. There are also topological excitations which have fractional charge and obey fractional statistics.
The fractal spectrum of magnetic minibands (Hofstadter butterfly), induced by the moiré super- lattice of graphene on an hexagonal crystal substrate, is known to exhibit gapped Dirac cones. We show that the gap can be closed by slightly misaligning the substrate, producing a hierarchy of conical singularities (Dirac points) in the band structure at rational values Phi = (p/q)(h/e) of the magnetic flux per supercell. Each Dirac point signals a switch of the topological quantum number in the connected component of the quantum Hall phase diagram. Model calculations reveal the scale invariant cond…
It is proposed that the new generation of spintronics should be ideally massless and dissipationless for the realization of ultra-fast and ultra-low-power spintronic devices. We demonstrate that the spin-gapless materials with linear energy dispersion are unique materials that can realize these massless and dissipationless states. Furthermore, we propose four new types of spin Hall effects which consist of spin accumulation of equal numbers of electrons and holes having the same or opposite spin polarization at the sample edge in Hall effect measurements, but with vanishing Hall voltage. These…
We report on magneto-transport measurements of a two-dimensional electron gas confined in a Cd$_{0.997}$Mn$_{0.003}$Te quantum well structure under conditions of vanishing Zeeman energy. The electron Zeeman energy has been tuned via the $s-d$ exchange interaction in order to probe different quantum Hall states associated with metallic and insulating phases. We have observed that reducing Zeeman energy to zero does not necessary imply the disappearing of quantum Hall states, i.e. a closing of the spin gap. The spin gap value under vanishing Zeeman energy conditions is shown to be dependent on t…
For the two-dimensional ideal electron gas with the Rashba spin-orbit interaction and in the presence of non-magnetic short-ranged potential impurities the spin-Hall conductivity $σ_{sH}$ is found by direct microscopic calculation. Within the semiclassical approximation $\hbar/τ\ll ε_F$ and $Δ\ll ε_F$ the value of $σ_{sH}$ is zero for arbitrary ratio of the spin-orbit splitting $Δ$ and the inverse elastic scattering time $1/τ$.
We present the phase diagram of a ferromagnetic quantum Hall effect liquid in a narrow quantum well with vanishing single-particle Zeeman splitting, $ε_{\rm Z}$ and pronounced spin-orbit coupling. Upon decreasing $ε_{\rm Z}$, the spin-polarization field of a liquid takes, first, the easy-axis configuration, followed by the formation of a helical state, which affects the transport and NMR properties of a liquid and the form of topological defects in it. The analysis is extended over high odd integer filling factors.
At total Landau level filling factor $ν_{tot}=1$ a double layer two-dimensional electron system with small interlayer separation supports a collective state possessing spontaneous interlayer phase coherence. This state exhibits the quantized Hall effect when equal electrical currents flow in parallel through the two layers. In contrast, if the currents in the two layers are equal, but oppositely directed, both the longitudinal and Hall resistances of each layer vanish in the low temperature limit. This finding supports the prediction that the ground state at $ν_{tot}=1$ is an excitonic superfl…
Spin currents that flow perpendicular to the electric field direction are generic in metals and doped semiconductors with spin-orbit coupling. It has recently been argued that the spin Hall conductivity can be dominated by an intrinsic contribution which follows from Bloch state distortion in the presence of an electric field. Here we report on an numerical demonstration of the robustness of this effect in the presence of disorder scattering for the case of a two-dimensional electron-gas with Rashba spin-orbit interactions (R2DES).
Starting from the photon self-energy tensor in a magnetized medium, the 3D complete antisymmetric form of the conductivity tensor is found in the static limit of a fermion system $C$ non-invariant under fermion-antifermion exchange. The massless relativistic 2D fermion limit in QED is derived by using the compactification along the dimension parallel to the magnetic field. In the static limit and at zero temperature the main features of quantum Hall effect (QHE) are obtained: the half-integer QHE and the minimum value proportional to $e^2/h$ for the Hall conductivity . For typical values of gr…
We study the vanishing of the topological properties of a quantum spin Hall insulator induced by a deformation of the band structure that interpolates between the Dirac and the semi-Dirac limits of a tight-binding model on a honeycomb lattice. The above scenario is mimicked in a simple model, where there exists a differential hopping along one of the three neighbours (say, $t_1$) compared to the other two (say, $t$). For $t_1 = t$, the properties of the quantum spin Hall phase is described by the familiar Kane Mele model, while $t<t_1<2t$ denotes a situation in which the spin resolved bands ar…
Observations of $ν=1$ Quantum Hall Effect and Inter-Band Effects of Magnetic fields on Hall Conductivity in Organic Massless Dirac Fermion System $α$-(BETS)$_2$I$_3$ under Pressure
本文在压力下的 α-(BETS)₂I₃ 有机无质量 Dirac 费米子体系中观测到 ν=1 量子 Hall 态，并通过化学势跨越 Dirac 点揭示了磁场对 Hall 电导率的带间效应。
Employing the quantum Liouville equation with phenomenological dissipation, we investigate the transport properties of massless and massive Dirac fermion systems that mimics graphene and topological insulators, respectively. The massless Dirac fermion system does not show an intrinsic Hall effect, but it shows a Hall current under the presence of circularly-polarized laser fields as a nature of a optically-driven nonequilibrium state. Based on the microscopic analysis, we find that the light-induced Hall effect mainly originates from the imbalance of photocarrier distribution in momentum space…
We revisit the $θ$ dependence in $CP^{N-1}$ model with large $N$. We study the consequences of a recently discovered new ingredient of the instanton vacuum, i.e. the massless chiral edge excitations. Contrary to the previous believes, our results demonstrate that the large $N$ expansion displays all the fundamental features of the quantum Hall effect. This includes the {\em robust} quantization of the Hall conductance, as well as the appearance of massless bulk excitations at $θ= π$. We conclude that the quantum Hall effect is a generic feature of the instanton vacuum, i.e. for any number of f…
Theory predicts for the two-dimensional electrons gas with only Rashba spin-orbit interaction a vanishing spin Hall conductivity and at the same time a finite inverse spin Hall effect. We show how these seemingly contradictory results are compatible with the Onsager relations: the latter do hold for spin and particle (charge) currents in the two-dimensional electron gas, although (i) their form depends on the experimental setup and (ii) a vanishing bulk spin Hall conductivity does not necessarily imply a vanishing spin Hall effect. We also discuss the situation in which extrinsic spin orbit fr…
We propose a nonequilibrium Green's function approach to study the spin-Hall effect in a two-dimensional electron system with both the Rashba and Dresselhaus spin-orbit couplings. By taking into account the long-range electron-impurity scattering, the derived kinetic equations are solved numerically. It is found the vanishing of the total zero-temperature dissipationless spin-Hall effect, contributing from the intrinsic and disorder-mediated processes. This result has been examined in the wide ranges of spin-orbit coupling constants and electron density.
We have discovered two-dimensional zero-gap material with a layered structure in the organic conductor $α$-(BEDT-TTF)$_2$I$_3$ under high hydrostatic pressure. In contrast to graphene, the electron-hole symmetry is not good except at the vicinity of the Dirac points. Thus, temperature dependence of the chemical potential, $μ$, plays an important role in the transport in this system. The experimental formula of $μ$ is revealed. We succeeded in detecting the inter-band effects of a magnetic field on the Hall conductivity when $μ$ passes the Dirac point.
The quantum spin Hall insulator (QSHI) state has been demonstrated in two semiconductor systems - HgTe/CdTe quantum wells (QWs) and InAs/GaSb QW bilayers. Unlike the HgTe/CdTe QWs, the inverted band gap in InAs/GaSb QW bilayers does not open at the $Γ$ point of the Brillouin zone, preventing the realization of massless Dirac fermions. Here, we propose a new class of semiconductor systems based on InAs/Ga(In)Sb multilayers, hosting a QSHI state, a graphene-like phase and a bilayer graphene analogue, depending on their layer thicknesses and geometry. The QSHI gap in the novel structures can reac…
We study the current and charge distribution of a two dimensional electron gas under strong perpendicular magnetic fields within the linear response regime. We show within a self-consistent screening theory that incompressible strips with integer values of local Landau-level filling factor exist for finite intervals of the magnetic field strength $B$. Within an essentially local conductivity model, we find that the current density in these $B$ intervals is confined to the incompressible strips of vanishing local longitudinal resistivity. This leads to vanishing longitudinal and exactly quantiz…
We investigate the unusual properties of quasirelativistic massless fermions under a magnetic or electric field by means of nonminimal couplings. Within this approach, the spin-orbit coupling (SOC) effects are properly generated in the energy spectrum of the quasiparticles. By including a magnetic field, $B$, we show that the spin splitting of Landau Levels (LL) obeys a $\sqrt{B}$ linear dependence with SOC, typical of relativistic particles. Moreover, our calculated spectrum of LLs resembles the behavior of the three-dimensional (3D) massless Kane fermions. Using a nonminimal coupling with an…
We present a quantum Boltzmann equation analysis of the spin-Hall effect in a diffusive Rashba two-dimensional electron system. Within the framework of the self-consistent Born approximation, we consider the roles of disorder-induced quasiclassical relaxation, collisional broadening of the quasiparticles, and the intracollisional field effect in regard to spin-Hall dynamics. We present an analytical proof that the spin-Hall current vanishes, independently of the coupling strength, of the quasiparticle broadening, of temperature and of the specific form of the isotropic scattering potential. A …
In contrast to the case of ordinary quantum Hall effect, the resistance of ballistic helical edge channels in typical quantum spin-Hall experiments is non-vanishing, additive and poorly quantized. Here we present a simple argument connecting this qualitative difference with a spin relaxation in the current/voltage leads in an experimentally relevant multi-terminal bar geometry. Both the finite lead resistance and the spin relaxation contribute to a non-vanishing four-terminal edge resistance, explaining poor quantization quality. We show that corrections to the four-terminal and two-terminal r…
We discuss models for massless Dirac fermions being subject to a perpendicular magnetic field in spherical geometry. These models are analogues of Haldane`s spherical construction for massful charge carriers. The single particle states constructed here are easily implemented in existing numerical code for many-body problems in conventional quantum Hall systems. Moreover, the many-body states of fully filled sublevels in the subspace of lowest Landau level index are skyrmions with respect to the layer spin.
We study the Hall response of quasi-two-dimensional lattice systems of charged fermions under a weak transverse magnetic field, in the ballistic coherent limit. We identify a setup in which this response vanishes over a wide range of parameters: the paradigmatic "Landauer-Büttiker" setup commonly studied for coherent quantum transport, consisting of a strip contacted to biased ideal reservoirs of charges. We show that the effect does not rely on particle-hole symmetry, and is robust to a variety of perturbations including variations of the transverse magnetic field, chemical potential, and tem…
Charged particles exhibit the Hall effect in the presence of magnetic fields. Analogously, ferromagnetic skyrmions with non-zero topological charges and finite fictitious magnetic fields exhibit the skyrmion Hall effect, which is detrimental for applications. The skyrmion Hall effect has been theoretically predicted to vanish for antiferromagnetic skyrmions because the fictitious magnetic field, proportional to net spin density, is zero. We experimentally confirm this prediction by observing current-driven transverse elongation of pinned ferrimagnetic bubbles. Remarkably, the skyrmion Hall eff…
The exsitance of three-dimensional Hall effect (3DQHE) due to spontaneous Fermi surface instabilities in strong magnetic field was proposed decades ago, and has stimulated recent progress in experiments. The reports in recent experiments show that the Hall plateaus and vanishing transverse magneto-resistivities (TMRs) (which are two main signatures of 3DQHE) are not easy to be observed in natural materials. And two main different explanations of the slow varying slope like Hall plateaus and non-vanishing TMRs (which can be called as quasi-quantized Hall effect (QQHE)) have been proposed. By st…
Transport through edge-channels is responsible for conduction in quantum Hall (QH) phases. Topology dictates quantization of both charge and thermal transport coefficients. These turn out to approach robust quantized values when incoherent equilibration processes become dominant. Here, we report on measurements of both electrical and thermal conductances of integer and fractional quantum Hall (FQH) phases, realized in hBN encapsulated graphite gated bilayer graphene devices. Remarkably, for the complex edge at filling factors $ν=5/3$ and $ν=8/3$, which correspond to the paradigmatic hole-conju…
We investigate the classical ground state configurations of a collection of skyrmions in the limit of vanishing Lánde $g$ factor. We show that in this limit, the skyrmions which are large and overlapping, behave qualitatively differently from small skyrmions. We investigate the system in various regimes of spin stiffness. We find that in all regimes, at $g=0$, the ground state configuration is a rectangular skyrmion lattice with Néel orientational ordering. The charge distribution has a higher symmetry of a face-centered rectangular lattice and the $z$-component of the spin polarization is zer…
The half-quantized Hall conductance is characteristic of quantum systems with parity anomaly. Here we investigate topological and transport properties of a class of parity anomalous semimetals, in which massive Dirac fermions coexist with massless Dirac fermions in momentum space or real space, and uncovered a distinct bulk-edge correspondence that the half-quantized Hall effect is realized via the bulk massless Dirac fermions while the nontrivial Berry curvature is provided by the massive Dirac fermions. The spatial distribution of the edge current decays away from the boundary in a power law…
Three dimensional (3D) topological insulators (TIs) are an important class of materials with applications in electronics, spintronics and quantum computing. With the recent development of truly bulk insulating 3D TIs, it has become possible to realize surface dominated phenomena in electrical transport measurements e.g. the quantum Hall (QH) effect of massless Dirac fermions in topological surface states (TSS). However, to realize more advanced devices and phenomena, there is a need for a platform to tune the TSS or modify them e.g. gap them by proximity with magnetic insulators, in a clean ma…
We consider a honeycomb network built of quantum wires, with each node of the network having a Y-junction of three wires with a ring through which flux can be inserted. The junctions are the basic circuit elements for the network, and they are characterized by 3 x 3 conductance tensors. The low energy stable fixed point tensor conductances result from quantum effects, and are determined by the strength of the interactions in each wire and the magnetic flux through the ring. We consider the limit where there is decoherence in the wires between any two nodes, and study the array as a network of …
We compute the entanglement entropy, in real space, of the ground state of the integer Quantum Hall states for three different domains embedded in the torus, the disk and the sphere. We establish the validity of the area law with a vanishing value of the topological entanglement entropy. The entropy per unit length of the perimeter depends on the filling fraction, but it is independent of the geometry.
We investigate the spin-Hall effect of both electrons and holes in semiconductors using the Kubo formula in the correct zero-frequency limit taking into account the finite momentum relaxation time of carriers in real semiconductors. This approach allows to analyze the range of validity of recent theoretical findings. In particular, the spin-Hall conductivity vanishes for vanishing spin-orbit coupling if the correct zero-frequency limit is performed.
Graphene is a two-dimensional carbon material with a honeycomb lattice and Dirac-type low-energy spectrum. In a strong magnetic field, where Coulomb interactions dominate against disorder broadening, quantum Hall ferromagnetic states realize at integer fillings. Extending the quantum Hall ferromagnetism to the fractional filling case of massless Dirac fermions, we study the elementally charge excitations which couple with the valley degrees of freedom (so-called valley skyrmions). With the use of the density matrix renomalization group (DMRG) method, the excitation gaps are calculated and extr…
We calculate the spin-Hall conductivity for a two-dimensional electron gas within the self-consistent Born approximation, varying the strength and type of disorder. In the weak disorder limit we find both analytically and numerically a vanishing spin-Hall conductivity even when we allow a momentum dependent scattering. Separating the reactive from the disspative current response, we find the universal value $σ^R_{sH} = e/8 π$ for the reactive response, which cancels however with the dissipative part $σ^D_{sH} = -e/8 π$.
We examine the radiation induced modification of the Hall effect in high mobility GaAs/AlGaAs devices that exhibit vanishing resistance under microwave excitation. The modification in the Hall effect upon irradiation is characterized by (a) a small reduction in the slope of the Hall resistance curve with respect to the dark value, (b) a periodic reduction in the magnitude of the Hall resistance, $R_{xy}$, that correlates with an increase in the diagonal resistance, $R_{xx}$, and (c) a Hall resistance correction that disappears as the diagonal resistance vanishes.
Massless Dirac fermions in graphene at charge neutrality form a strongly interacting system in which both charged and neutral (energy) modes play an important role. These modes are essentially decoupled in the absence of a magnetic field, but become strongly coupled when a field is applied. We show that these ideas explain the recently observed giant magnetodrag, arising in classically weak fields when electron density is tuned near charge neutrality. We predict strong Hall drag in this regime, which is in stark departure from the weak coupling regime, where theory predicts the absence of Hall…
The energy spectrum of massless Dirac fermions in graphene under two dimensional periodic magnetic modulation having square lattice symmetry is calculated. We show that the translation symmetry of the problem is similar to that of the Hofstadter or TKNN problem and in the weak field limit the tight binding energy eigenvalue equation is indeed given by Harper Hofstadter hamiltonian. We show that due to its magnetic translational symmetry the Hall conductivity can be identified as a topological invariant and hence quantized. We thus extend the idea of Quantum Hall Effect to magnetically modulate…
We demonstrate a device concept to fabricate resistance standards made of quantum Hall series arrays by using p-type and n-type graphene. The ambipolar nature of graphene allows fabricating series quantum Hall resistors without complex multi-layer metal interconnect technology, which is required when using conventional GaAs two-dimensional electron systems. As a prerequisite for a precise resistance standard we confirm the vanishing of longitudinal resistance across a p-n junction for metrological relevant current levels in the range of a few \muA.
We study the realization in a model of graphene of the phenomenon whereby the tendency of gauge-field mediated interactions to break chiral symmetry spontaneously is greatly enhanced in an external magnetic field. We prove that, in the weak coupling limit, and where the electron-electron interaction satisfies certain mild conditions, the ground state of charge neutral graphene in an external magnetic field is a quantum Hall ferromagnet which spontaneously breaks the emergent U(4) symmetry to U(2)XU(2). We argue that, due to a residual CP symmetry, the quantum Hall ferromagnet order parameter i…
Recent reports of spin-orbit coupling enhancement in chemically modified graphene have opened doors to studies of the spin Hall effect with massless chiral fermions. Here, we theoretically investigate the interaction and impurity density dependence of the extrinsic spin Hall effect in spin-orbit coupled graphene. We present a nonperturbative quantum diagrammatic calculation of the spin Hall response function in the strong-coupling regime that incorporates skew scattering and anomalous impurity density-independent contributions on equal footing. The spin Hall conductivity dependence on Fermi en…
We investigate electronic transport in high mobility (\textgreater 100,000 cm$^2$/V$\cdot$s) trilayer graphene devices on hexagonal boron nitride, which enables the observation of Shubnikov-de Haas oscillations and an unconventional quantum Hall effect. The massless and massive characters of the TLG subbands lead to a set of Landau level crossings, whose magnetic field and filling factor coordinates enable the direct determination of the Slonczewski-Weiss-McClure (SWMcC) parameters used to describe the peculiar electronic structure of trilayer graphene. Moreover, at high magnetic fields, the d…
An asymmetric triangular potential well provides the simplest model for the confinement of mobile electrons at the interface between two insulating oxides, such as LaAlO_3 and SrTiO_3 (LAO/STO). These electrons have been recently shown to exhibit a large spin-orbit coupling of the Rashba type, i.e., linear in the in-plane momentum. In this paper we study the intrinsic spin Hall effect due to Rashba coupling in an asymmetric triangular potential well. Besides splitting each subband into two branches of opposite helicity, the spin-orbit interaction causes the wave function in the direction perpe…
We report on the unusual nature of nu=0 state in the integer quantum Hall effect (QHE) in graphene and show that electron transport in this regime is dominated by counter-propagating edge states. Such states, intrinsic to massless Dirac quasiparticles, manifest themselves in a large longitudinal resistivity rho_xx > h/e^2, in striking contrast to rho_xx behavior in the standard QHE. The nu=0 state in graphene is also predicted to exhibit pronounced fluctuations in rho_xy and rho_xx and a smeared zero Hall plateau in sigma_xy, in agreement with experiment. The existence of gapless edge states p…
We report the consequences of a new interaction symmetry that protects the renormalization of the electron gas in low dimensions in general, and in the quantum Hall regime in particular. We introduce a generalized Thouless' criterion for localization to include the effects of the Coulomb interaction and establish the quantization of the Hall conductance as well as the theory of massless edge bosons.
Quite generally, an insulator is theoretically defined by a vanishing conductivity tensor at the absolute zero of temperature. In classical insulators, such as band insulators, vanishing conductivities lead to diverging resistivities. In other insulators, in particular when a high magnetic field (B) is added, it is possible that while the magneto-resistance diverges, the Hall resistance remains finite, which is known as a Hall insulator. In this letter we demonstrate experimentally the existence of another, more exotic, insulator. This insulator, which terminates the quantum Hall effect series…
An unbounded massless Dirac model with two nondegenerate Dirac cones is the simplest model for Weyl semimetals, which show the anomalous electromagnetic response of chiral magnetic effect (CME) and anomalous Hall effect (AHE). However, if this model is naively used to analyze the electromagnetic response within a linear response theory, it gives the result apparently inconsistent with the persuasive prediction based on a lattice model. We show that this serious difficulty is related to the breaking of current conservation in the Dirac model due to quantum anomaly and can be removed if current …
We study the quantum Hall effect(QHE) on the Kagomé lattice with anisotropy in one of the hopping integrals. We find a new type of QHE characterized by the quantization rules for Hall conductivity $σ_{xy}=2ne^{2}/h$ and Landau Levels $E(n)=\pm v_{F}\sqrt{(n+1/2)\hbar Be}$ ($n$ is an integer), which is different from any known type. This phase evolves from the QHE phase with $σ_{xy}=4(n+1/2)e^{2}/h$ and $E(n)=\pm v_{F}\sqrt{2n\hbar Be}$ in the isotropic case, which is realized in a system with massless Dirac fermions (such as in graphene). The phase transition does not occur simultaneously in a…
We predict the occurrence of a novel type of atomic-scale spin lattice in an Fe monolayer on the Ir(001) surface. Based on density functional theory calculations we parametrize a spin Hamiltonian and solve it numerically using Monte-Carlo simulations. We find the stabilization of a three-dimensional spin structure arranged on a 3x3 lattice. Despite an almost vanishing total magnetization we predict the emergence of a large anomalous Hall effect, to which there is a significant topological contribution purely due to the real space spin texture at the surface.
We show that, when graphene is subjected to an appropriate one-dimensional external periodic potential, additional branches of massless fermions are generated with nearly the same electron-hole crossing energy as that at the original Dirac point of graphene. Because of these new zero-energy branches, the Landau levels at charge neutral filling becomes 4(2N+1)-fold degenerate (with N=0,1,2,..., tunable by the potential strength and periodicity) with the corresponding Hall conductivity $σ_{xy}$ showing a step of size 4(2N+1)$e^2/h$. These theoretical findings are robust against variations in the…
We examine the long wave length limit of the self-consistent Hartree-Fock approximation irreducible static density-density response function by evaluating the charge induced by an external charge. Our results are consistent with the compressibility sum rule and inconsistent with earlier work that did not account for consistency between the exchange-local-field and the disorder potential. We conclude that the thermodynamic density of states is finite, in spite of the vanishing tunneling density of states at the critical energy of the integer quantum Hall transition.
We demonstrate evidences of electronic transport via topological Dirac surface states in a thin film of strained HgTe. At high perpendicular magnetic fields, we show that the electron transport reaches the quantum Hall regime with vanishing resistance. Furthermore, quantum Hall transport spectroscopy reveals energy splittings of relativistic Landau levels specific to coupled Dirac surface states. This study provides new insights in the quantum Hall effect of topological insulator (TI) slabs, in the cross-over regime between two- and three-dimensional TIs, and in the relevance of thin TI films …
Photovoltaic Hall effect -- the Hall effect induced by intense, circularly-polarized light in the absence of static magnetic fields -- has been proposed in Phys. Rev. B 79, 081406R (2009) for graphene where a massless Dirac dispersion is realized. The photovoltaic Berry curvature (a nonequilibrium extension of the standard Berry curvature) is the key quantity to understand this effect, which appears in the Kubo formula extended to Hall transport in the presence of strong AC field backgrounds. Here we elaborate the properties of the photovoltaic curvature such as the frequency and field strengt…
