We study the anomalous Hall Effect (AHE) of single-crystalline Co$_3$Sn$_{2-x}$In$_x$S$_2$ over a large range of indium concentration x from 0 to 1. Their magnetization reduces progressively with increasing x while their ground state evolves from a ferromagnetic Weyl semimetal into a nonmagnetic insulator. Remarkably, after systematically scaling the AHE, we find that their intrinsic anomalous Hall conductivity (AHC) features an unexpected maximum at around x = 0.15. The change of the intrinsic AHC corresponds with the doping evolution of Berry curvature and the maximum arises from the magnetic topological nodal-ring gap. Our experimental results show a larger AHC in a fundamental nodal-ring gap than that of Weyl nodes.
Adler-Bell-Jackiw chiral anomaly is a representative feature arising from the topological nature in topological semimetal. We report the first experimental observation of giant planar Hall effect in type-II Weyl semimetal WTe2. Our comprehensive analyes of the experimental data demonstrate that the detected planar Hall effect is originated from the chiral anomaly of Weyl fermions. Unlike the somewhat elusive negative magnetoresistance, the planar Hall effect is robust and easy to be detected in type-II Weyl semimetal. This work reveals that the planar Hall effect is an effective transport prob…
We report a comprehensive study on the intrinsic spin Hall conductivity (SHC) of semimetals MoTe2 and WTe2 by ab initio calculation. Large SHC and desirable spin Hall angles have been discovered, due to the strong spin orbit coupling effect and low charge conductivity in semimetals. Diverse anisotropic SHC values, attributed to the unusual reduced-symmetry crystalline structure, have been revealed. We report an effective method on SHC optimization by electron doping, and exhibit the mechanism of SHC variation respect to the energy shifting by the spin Berry curvature. Our work provides insight…
The magnetotransport properties in antiferromagnetic half-Heusler single crystals of TbPtBi, a magnetic-field-induced topological semimetal with simple band structure, are investigated. We found that a nonmonotonic magnetic field dependence of the anomalous Hall resistivity in a high magnetic field (B>7T), which come from the change of band structure induced by the Zeeman-like splitting when applying the external magnetic field. The experiment results show that credible anomalous Hall resistivity and conductivity reach up to 0.6798mΩcm and 125Ω-1cm-1, respectively. A large AHA up to 33% is obt…
In ferromagnetic solids, even in absence of magnetic field, a transverse voltage can be generated by a longitudinal temperature gradient. This thermoelectric counterpart of the Anomalous Hall effect (AHE) is dubbed the Anomalous Nernst effect (ANE). Expected to scale with spontaneous magnetization, both these effects arise because of the Berry curvature at the Fermi energy. Here, we report the observation of a giant ANE in a newly-discovered magnetic Weyl semimetal Co$_3$Sn$_2$S$_2$ crystal. Hall resistivity and Nernst signal both show sharp jumps at a threshold field and exhibit a clear hyste…
Topological materials ranging from topological insulators to Weyl and Dirac semimetals form one of the most exciting current fields in condensed-matter research. Many half-Heusler compounds, RPtBi (R= rare earth) have been theoretically predicted to be topological semimetals. Among various topological attributes envisaged in RPtBi, topological surface states, chiral anomaly and planar Hall effect have been observed experimentally. Here, we report on an unusual intrinsic anomalous Hall effect (AHE) in the antiferromagnetic Heusler Weyl semimetal compounds GdPtBi and NdPtBi that is observed over…
Anomalous Nernst effect generates a transverse voltage perpendicular to the temperature gradient. It has several advantages compared with the longitudinal thermoelectricity for energy conversion, such as decoupling of electronic and thermal transports, higher flexibility, and simpler lateral structure. However, a design principle beyond specific materials systems for obtaining a large anomalous Nernst conductivity (ANC) is still absent. In this work, we theoretically demonstrate that a pair of Dirac nodes under a Zeeman field manifests a double-peak anomalous Hall conductivity curve with respe…
In this paper, we report on the investigation of (1) the transport properties of multi-carriers in semi-metal Bi and (2) the spin conversion physics in this semimetal system in a ferrimagnetic insulator, yttrium-iron-garnet. Hall measurements reveal that electrons and holes co-exist in the Bi, with electrons being the dominant carrier. The results of a spin conversion experiment corroborate the results of the Hall measurement; in addition, the inverse spin Hall effect governs the spin conversion in the semimetal/insulator system. This study provides further insights into spin conversion physic…
Topological materials are expected to show distinct transport signatures due to their unique band-inversion character and band-crossing points. However, the intentional modulation of such topological responses by experimentally feasible means is less explored. Here, an unusual elevation of anomalous Hall effect (AHE) is obtained in electron(Ni)-doped magnetic Weyl semimetal Co3-xNixSn2S2, showing peak values of anomalous Hall-conductivity, Hall-angle and Hall-factor at a relatively low doping level of x = 0.11. The separation of intrinsic and extrinsic contributions to total AHE using TYJ scal…
Weyl semimetals (WSM) have been extensively studied due to their exotic properties such as topological surface states and anomalous transport phenomena. Their band structure topology is usually predetermined by material parameters and can hardly be manipulated once the material is formed. Their unique transport properties appear usually at very low temperature, which sets challenges for practical device applications. In this work, we demonstrate a way to modify the band topology via a weak magnetic field in a ferromagnetic topological semimetal, Co2MnAl, at room temperature. We observe a tunab…
The nonlinear Hall effect (NLHE), which can produce a transverse voltage without any magnetic field, is a potential alternative for rectification or frequency doubling. However, the low temperature detection of NLHE limits its applications. Here, we report the room-temperature NLHE in a type-II Weyl semimetal TaIrTe4, which hosts a robust NLHE due to substantial broken inversion symmetry and large band overlapping at the Fermi level. We also observe a temperature-induced sign inversion of NLHE in TaIrTe4. Our theoretical calculations suggest that the observed sign inversion is a result of temp…
Intrinsic anomalous Nernst effect (ANE), like its Hall counterpart, is generated by Berry curvature of electrons in solids. Little is known about its response to disorder. In contrast, the link between the amplitude of the ordinary Nernst coefficient and the mean-free-path is extensively documented. Here, by studying Co$_3$Sn$_2$S$_2$, a topological half-metallic semimetal hosting sizable and recognizable ordinary and anomalous Nernst responses, we demonstrate an anti-correlation between the amplitude of ANE and carrier mobility. We argue that the observation, paradoxically, establishes the in…
Magnetic Weyl semimetals with broken time-reversal symmetry are expected to generate strong intrinsic anomalous Hall effects, due to their large Berry curvature. Here, we report a magnetic Weyl semimetal candidate Co3Sn2S2 with a quasi-two-dimensional crystal structure consisting of stacked Kagome lattices. This lattice provides an excellent platform for hosting exotic quantum topological states. We observe a negative magnetoresistance that is consistent with the chiral anomaly expected from the presence of Weyl fermions close to the Fermi level. The anomalous Hall conductivity is robust again…
Topological materials have recently attracted considerable attention among materials scientists as their properties are predicted to be protected against perturbations such as lattice distortion and chemical substitution. However, any experimental proof of such robustness is still lacking. In this study, we experimentally demonstrate that the topological properties of the ferromagnetic kagome compound Co3Sn2S2 are preserved upon Ni substitution. We systematically vary the Ni content in Co3Sn2S2 single crystals and study their magnetic and anomalous transport properties. For the intermediate Ni…
The electronic anomalous Hall effect (AHE), where charge carriers acquire a velocity component orthogonal to an applied electric field, is one of the most fundamental and widely studied phenomena in physics. There are several different AHE mechanisms known, and material examples are highly sought after, however in the highly conductive (skew scattering) regime the focus has centered around ferromagnetic metals. Here we report the observation of a giant extrinsic AHE in KV$_3$Sb$_5$, an exfoliable, Dirac semimetal with a Kagome layer of Vanadium atoms. Although there has been no reports of magn…
We predict a large anomalous Nernst effect in the inverse Heusler compensated ferrimagnets Ti$_2$Mn$X$ ($X$=Al,Ga,In) with vanishing net magnetic moments. Though the net magnetic moment is zero, the Weyl points in these systems lead to a large anomalous Nernst conductivity (ANC) due to the lack of a magnetic sublattice that inverses the sign of the Berry curvature. In comparison to the noncollinear antiferromagnets Mn$_3$Sn and Mn$_3$Ge, the high ANC stems almost entirely from the Weyl points in this class of compounds, and thus, it is topologically protected. This work shows for the first tim…
Spin-1 chiral semimetal is a new state of quantum matter hosting unconventional chiral fermions that extend beyond the common Dirac and Weyl fermions. B20-type CoSi is a prototypal material that accommodates such an exotic quasiparticle. To date, the spin transport properties in the spin-1 chiral semimetals, have not been explored yet. In this work, we fabricated B20-CoSi thin films on sapphire c-plane substrates by magnetron sputtering and studied the spin Hall effect (SHE) by combining experiments and first-principles calculations. The SHE of CoSi using CoSi/CoFeB/MgO heterostructures was in…
Synthesis of crystallographically well-defined thin films of topological materials is important for unraveling their mesoscale quantum properties and for device applications. Mn$_3$Ge, an antiferromagnetic Weyl semimetal with a chiral magnetic structure on a Kagome lattice, is expected to have enhanced Berry curvature around Weyl nodes near the Fermi energy, leading to large anomalous Hall / Nernst effects and a large spin-Hall effect. Using magnetron sputtering, we have grown epitaxial thin films of hexagonal D0$_{19}$ Mn$_3$Ge that are flat and continuous. Large anomalous Nernst and inverse …
Topological magnetic semimetals promise large Berry curvature through the distribution of the topological Weyl nodes or nodal lines and further novel physics with exotic transport phenomena. We present a systematic study of the structural and magnetotransport properties of Co$_2$MnGa films from thin (20 nm) to bulk like behavior (80 nm), in order to understand the underlying mechanisms and the role on the topology. The magnetron sputtered Co$_2$MnGa films are $L$$2_{\mathrm {1}}$-ordered showing very good heteroepitaxy and a strain-induced tetragonal distortion. The anomalous Hall conductivity…
Anomalous Hall effect (AHE) can be induced by intrinsic mechanism due to the band Berry phase and extrinsic one arising from the impurity scattering. The recently discovered magnetic Weyl semimetal Co3Sn2S2 exhibits a large intrinsic anomalous Hall conductivity (AHC) and a giant anomalous Hall angle (AHA). The predicted energy dependence of the AHC in this material exhibits a plateau at 1000 Ω-1 cm-1 and an energy width of 100 meV just below EF, thereby implying that the large intrinsic AHC will not significantly change against small-scale energy disturbances such as slight p-doping. Here, we …
A giant planar Hall effect (PHE) and anisotropic magnetoresistance (AMR) is observed in TaP, a nonmagnetic Weyl semimetal with ultrahigh mobility. The perpendicular resistivity (i.e., the planar magnetic field applied normal to the current) far exceeds the zero-field resistivity, which thus rules out the possible origin of negative longitudinal magnetoresistance. The giant PHE/AMR is finally attributed to the large anisotropic orbital magnetoresistance that stems from the ultrahigh mobility. Furthermore, the mobility-enhanced current jetting effects are found to strongly deform the line shape …
Magnetic Weyl semimetals with spontaneously broken time-reversal symmetry exhibit a large intrinsic anomalous Hall effect originating from the Berry curvature. To employ this large Hall current for room temperature topo-spintronics applications, it is necessary to fabricate these materials as thin or ultrathin films. Here, we experimentally demonstrate that Weyl semimetal Co2MnGa thin films (20-50 nm) show a very large anomalous Hall angle ~11.4% at low temperature and ~9.7% at room temperature, which can be ascribed to the nontrivial topology of the band structure with large intrinsic Berry c…
We predict a nonlinear Hall effect in certain Weyl semimetals with broken inversion symmetry. When the energy dispersions about pairs of Weyl nodes are skewed -- the Weyl cones are "tilted" -- the concerted actions of the anomalous velocity and the chiral anomaly give rise to the nonlinear Hall effect. This Hall conductivity is linear in both electric and magnetic fields, and depends critically on the tilting of the Weyl cones. We also show that this effect does not rely on a finite Berry curvature dipole, in contrast to the intrinsic quantum nonlinear Hall effect that was recently observed in…
Observation of Weyl and Dirac Fermions in condensed matter systems is one of the most important discoveries. Among the very few available tools to characterize Weyl semimetals through electrical transport, negative magnetoresistance is most commonly used. Considering shortcomings of this method, new tools to characterize chiral anomaly in Weyl semimetals are desirable. We employ planar Hall effect as an effective technique in half Heusler Weyl semimetal GdPtBi to study chiral anomaly. This compound exhibits a large value of 1.5 mohm cm planar Hall resistivity at 2 K and in 9 T. Our analysis re…
Exploration of the novel relationship between magnetic order and topological semimetals has received enormous interest in a wide range of both fundamental and applied research. Here we predict that soft ferromagnetic (FM) material EuB6 can achieve multiple topological semimetal phases by simply tuning the direction of the magnetic moment. Explicitly, EuB6 is a topological nodal-line semimetal when the moment is aligned along the [001] direction, and it evolves into a Weyl semimetal with three pairs of Weyl nodes by rotating the moment to the [111] direction. Interestingly, we identify a novel …
Recently, giant planar Hall effect originating from chiral anomaly has been predicted in nonmagnetic Dirac/Weyl semimetals. ZrTe5 is considered to be an intriguing Dirac semimetal at the boundary of weak topological insulators and strong topological insulators, though this claim still remains controversial. Here, we report the observation in ZrTe5 of the giant planar Hall resistivity that shows two different magnetic-field dependences as predicted by theory and a maximum at the Lifshitz transition temperature. We found that the giant planar Hall resistivity fades out with decreasing the thickn…
Since their discovery, topological insulators have been expected to be ideal spintronic materials owing to the spin currents carried by surface states with spin--momentum locking. However, the bulk doping problem remains an obstacle that hinders such application. In this work, we predict that a newly discovered family of topological materials, the Weyl semimetals, exhibits large intrinsic spin Hall effects that can be utilized to generate and detect spin currents. Our $ab~initio$ calculations reveal a large spin Hall conductivity that is comparable to that of $4d$ and $5d$ transition metals. T…
We study the second-order nonlinear optical response of a Weyl semimetal (WSM), i.e. a three-dimensional metal with linear band touchings acting as point-like sources of Berry curvature in momentum space, termed "Weyl-Berry monopoles". We first show that the anomalous second-order photocurrent of WSMs can be elegantly parametrized in terms of Weyl-Berry dipole and quadrupole moments. We then calculate the corresponding charge and node conductivities of WSMs with either broken time-reversal invariance or inversion symmetry. In particular, we predict a universal dissipationless second-order anom…
PbTaSe$_2$ is a unique topological material, in which the number of nodal lines is expected to change at the structural transition between the lower temperature/pressure "L" phase and the higher temperature/pressure "H" phase. We report the anomalous Nernst effect attributed to the Berry curvature of nodal lines and its change with the structural transition. In the L phase, the Nernst coefficient ($S_{yx}$) shows the step-like magnetic field dependence reminiscent of the anomalous Nernst effect of nonmagnetic Dirac/Weyl semimetals. By applying hydrostatic pressure, we discovered that the ampli…
Giant Domain-Wall Hall Magnetoresistance in Magnetic Topological Semimetal
Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences（中国科学院物理研究所）；School of Physical Sciences, University of Chinese Academy of Sciences（中国科学院大学物理科学学院）；Guangdong Provincial Key Laboratory of Optical Information Materials and Technology, Institute for Advanced Materials, South China Academy of Advanced Optoelectronics, South China Normal University（华南师范大学）
We report the synthesis and magneto-transport measurements on the single crystal of Dirac semimetal PdTe$_2$. The de Haas-van Alphen oscillations with multiple frequencies have been clearly observed, from which the small effective masses and nontrivial Berry phase are extracted, implying the possible existence of the Dirac fermions in PdTe$_2$. The planar Hall effect and anisotropic longitudinal resistivity originating from the chiral anomaly and nontrivial Berry phase are observed, providing strong evidence for the nontrivial properties in PdTe$_2$. With the increase of temperature up to 150 …
We theoretically investigated the magnetoresistance (MR) and planar Hall effect (PHE) in Weyl semimetals based on the semiclassical Boltzmann theory, focusing on the fine structure of the band dispersion. We identified that the negative longitudinal MR and sign change in the PHE occur because of the negative off-diagonal effective-mass with no topological effects or chiral anomaly physics. Our results highlight the crucial role of the off-diagonal effective-mass, which can cause anomalous galvanomagnetic effects. We propose that the PHE creates a dip in their temperature dependence, which enab…
The semiclassical transport theory is especially powerful for investigating galvanomagnetic effects. Generally, the semiclassical theory is applicable only in weak fields because it does not consider Landau quantization. Herein, we extend the conventional semiclassical theory by considering Landau quantization through the field dependence of carrier density in semimetals. The extended semiclassical theory is applicable even in strong fields, where Landau quantization is noticeable. Using this new approach, we explain the qualitative change in the angular dependence of transverse magnetoresista…
The magnetotransport properties of epitaxial films of Cd3As2, a paradigm three-dimensional Dirac semimetal, are investigated. We show that an energy gap opens in the bulk electronic states of sufficiently thin films and, at low temperatures, carriers residing in surface states dominate the electrical transport. The carriers in these states are sufficiently mobile to give rise to a quantized Hall effect. The sharp quantization demonstrates surface transport that is virtually free of parasitic bulk conduction and paves the way for novel quantum transport studies in this class of topological mate…
We investigate the pressure evolution of the anomalous Hall effect in magnetic topological semimetal Co3Sn2S2 in diamond anvil cells with pressures up to 44.9-50.9 GPa. No evident trace of structural phase transition is detected through synchrotron x-ray diffraction over the measured pressure range of 0.2-50.9 GPa. We find that the anomalous Hall resistivity and the ferromagnetism are monotonically suppressed as increasing pressure and almost vanish around 22 GPa. The anomalous Hall conductivity varies non-monotonically against pressure at low temperatures, involving competition between origin…
Owing to the coupling between open Fermi arcs on opposite surfaces, topological Dirac semimetals exhibit a new type of cyclotron orbit in the surface states known as Weyl orbit. Here, by lowering the carrier density in Cd3As2 nanoplates, we observe a crossover from multiple- to single-frequency Shubnikov-de Haas (SdH) oscillations when subjected to out-of-plane magnetic field, indicating the dominant role of surface transport. With the increase of magnetic field, the SdH oscillations further develop into quantum Hall state with non-vanishing longitudinal resistance. By tracking the oscillation…
In this paper, the chiral Hall effect of strained Weyl semimetals without any external magnetic field is proposed. Electron-phonon coupling emerges in the low-energy fermionic sector through a pseudogauge potential. We show that, by using chiral kinetic theory, the chiral Hall effect appears as a response to a real time-varying electric field in the presence of structural distortion and it causes spatial chirality and charges separation in a Weyl system. We also show that the coupling of the electrons to acoustic phonons as a gapless excitation leads to emerging an optical absorption peak at $…
Thermoelectrics are promising by directly generating electricity from waste heat. However, (sub-)room-temperature thermoelectrics have been a long-standing challenge due to vanishing electronic entropy at low temperatures. Topological materials offer a new avenue for energy harvesting applications. Recent theories predicted that topological semimetals at the quantum limit can lead to a large, non-saturating thermopower and a quantized thermoelectric Hall conductivity approaching a universal value. Here, we experimentally demonstrate the non-saturating thermopower and quantized thermoelectric H…
We study the quantum Hall effect of 2D electron gas in black phosphorus in the presence of perpendicular electric and magnetic fields. In the absence of a bias voltage, the external magnetic field leads to a quantization of the energy spectrum into equidistant Landau levels, with different cyclotron frequencies for the electron and hole bands. The applied voltage reduces the band gap, and eventually a semiconductor to semimetal transition takes place. This nontrivial phase is characterized by the emergence of a pair of Dirac points in the spectrum. As a consequence, the Landau levels are not e…
The quantum Hall effect is usually observed in 2D systems. We show that the Fermi arcs can give rise to a distinctive 3D quantum Hall effect in topological semimetals. Because of the topological constraint, the Fermi arc at a single surface has an open Fermi surface, which cannot host the quantum Hall effect. Via a "wormhole" tunneling assisted by the Weyl nodes, the Fermi arcs at opposite surfaces can form a complete Fermi loop and support the quantum Hall effect. The edge states of the Fermi arcs show a unique 3D distribution, giving an example of (d-2)-dimensional boundary states. This is d…
Dirac and Weyl semimetals are new discovered topological nontrivial materials with the linear band dispersions around the Dirac/Weyl points. When applying non-orthogonal electric current and magnetic field, an exotic phenomenon called chiral anomaly arises and negative longitudinal resistance can be detected. Recently, a new phenomenon named planer Hall effect (PHE) is considered to be another indication of chiral anomaly which has been observed in many topological semimetals. However, it still remains a question that is the PHE only attributed to chiral anomaly? Here we demonstrate the PHE in…
We investigate electrical transport in a three-dimensional massless Dirac fermion model that describes a Dirac semimetal state realized in topological materials. We derive a set of interdependent diffusion equations with eight local degrees of freedom, including the electric charge density and the spin density, that respond to an external electric field. By solving the diffusion equations for a system with a boundary, we demonstrate that a spin Hall effect with spin accumulation occurs even though the conventional spin current operator is zero. The Noether current associated with chiral symmet…
Valley-based electronics, known as valleytronics, is one of the keys to break through to a new stage of electronics. The valley degree of freedom is ubiquitous in the honeycomb lattice system. The honeycomb lattice structure of silicon called silicene is an fascinating playground of valleytronics. We investigate topological phases of silicene by introducing different exchange fields on the $A$ and $B$ sites. There emerges a rich variety of topologically protected states each of which has a characteristic spin-valley structure. The single Dirac-cone semimetal is such a state that one gap is clo…
Discovery of topological Weyl semimetals has revealed the opportunities to realize several extraordinary physical phenomena in condensed matter physics. Specifically, these semimetals with strong spin-orbit coupling, broken inversion symmetry and novel spin texture are predicted to exhibit a large spin Hall effect that can efficiently convert the charge current to a spin current. Here we report the direct experimental observation of a large spin Hall and inverse spin Hall effects in Weyl semimetal WTe2 at room temperature obeying Onsager reciprocity relation. We demonstrate the detection of th…
The recent discovery of topological Dirac semimetals (DSM) has provoked intense curiosity not only on Weyl physics in solids, but also on topological phase transitions originating from DSM. One example is controlling the dimensionality to realize two-dimensional quantum phases such as quantum Hall and quantum spin Hall states. For investigating these phases, the Fermi level is a key controlling parameter. From this perspective, we report here the carrier-density control of quantum Hall states realized in thin films of DSM Cd3As2. Chemical doping of Zn combined with electrostatic gating has ena…
An exotic anomalous plasmon mode is found in strained Weyl semimetals utilizing the topological Landau Fermi liquid and chiral kinetic theories, in which quasiparticle interactions are modeled by long-range Coulomb and residual short-range interactions. The gapped collective mode is derived from the dynamical charge pumping between the bulk and the surface and behaves like $k_{\rm F}^{-1}$. The charge oscillations are accurately determined by the coupling between the induced electric field and the background pseudofields. This novel mode unidirectionally disperses along the pseudomagnetic fiel…
Weyl semimetals are promising quantum materials that offer unique topological properties. Lately, it has been shown that laser-driven electron dynamics have characteristic signatures in two-dimensional and three-dimensional Dirac semimetals. The transition from Dirac to Weyl semimetal requires the breaking of either inversion or time-reversal symmetry. The present work shows that the laser-driven electron dynamics in a Weyl semimetal with broken time-reversal symmetry has intriguing features in its high-harmonic spectrum. It is found that the parity and magnitude of the non-zero Berry curvatur…
The recent discovery of magnetic Weyl semimetal Co3Sn2S2 opens up new avenues for research into the interactions between topological orders, magnetism, and electronic correlations. Motivated by the observations of large anomalous Hall effect because of large Berry curvature, we investigate another Berry curvature-induced phenomenon, the anomalous thermal Hall effect in Co3Sn2S2. We study it with and without strain, using a Wannier tight-binding Hamiltonian derived from first principles density functional theory calculations. We first identify this material as a tilted type-I Weyl semimetal bas…
A well known semiconductor Cd3As2 has reentered the spotlight due to its unique electronic structure and quantum transport phenomena as a topological Dirac semimetal. For elucidating and controlling its topological quantum state, high-quality Cd3As2 thin films have been highly desired. Here we report the development of an elaborate growth technique of high-crystallinity and high-mobility Cd3As2 films with controlled thicknesses and the observation of quantum Hall effect dependent on the film thickness. With decreasing the film thickness to 10 nm, the quantum Hall states exhibit variations such…
