The complex nature of filling factor $ν$=0 of monolayer graphene is studied in magnetotransport experiments. As function of perpendicular magnetic field a metal-insulator transition is observed, which is attributed to disorder-induced Landau level broadening in the canted antiferromagnetic phase. In the metallic regime a separation of the zeroth Landau level appears and signs of the quantum spin Hall effect are seen near $ν$=0. In addition to local transport, nonlocal transport experiments show results being consistent with helical edge transport.
Monolayer WTe2 is predicted to be a quantum spin Hall insulator (QSHI) and electron transport along its edges has been experimentally observed. However, the 'smoking gun' of QSHI, spin momentum locking of the edge electrons, has not been experimentally demonstrated. We propose a model to establish the relationship between the anisotropic magnetoresistance (AMR) and spin orientation of the helical electrons in WTe2. Based on the predictions of the model, angular dependent magnetoresistance measurements were carried out. The experimental results fully supported the model and the spin orientation…
Recent research on disorder effects in topological phases in quasicrystalline systems has received much attention. In this work, by numerically computing the (spin) Bott index and the thermal conductance, we reveal the effects of disorder on a class D chiral topological superconductor and a class DIII time-reversal-invariant topological superconductor in a two-dimensional Ammann-Beenker tiling quasicrystalline lattice. We demonstrate that both the topologically protected chiral and helical Majorana edge modes are robust against weak disorder in the quasicrystalline lattice. More fascinating is…
We study analytically and numerically spin effects in MoS_2 monolayer armchair quantum wires and quantum dots. The interplay between intrinsic and Rashba spin orbit interactions induced by an electric field leads to helical modes, giving rise to spin filtering in time-reversal invariant systems. The Rashba spin orbit interaction can also be generated by spatially varying magnetic fields. In this case, the system can be in a helical regime with nearly perfect spin polarization. If such a quantum wire is brought into proximity to an s-wave superconductor, the system can be tuned into a topologic…
It is well known that quantum spin Hall (QSH) edge modes being helical are immune to backscattering due to non-magnetic disorder within the sample. Thus, quantum spin Hall edge modes are non-localized and show a vanishing Hall resistance along with quantized 2-terminal, longitudinal and non-local resistances even in presence of sample disorder. However, this is not the case for contact disorder. This paper shows that when all contacts are disordered in an N-terminal quantum spin Hall sample, then transport via these helical QSH edge modes can have a significant localization correction. All the…
The integration of diverse electronic phenomena, such as magnetism and nontrivial topology, into a single system is normally studied either by seeking materials that contain both ingredients, or by layered growth of contrasting materials. The ability to simply stack very different two dimensional (2D) van der Waals materials in intimate contact permits a different approach. Here we use this approach to couple the helical edges states in a 2D topological insulator, monolayer WTe2, to a 2D layered antiferromagnet, CrI3. We find that the edge conductance is sensitive to the magnetization state of…
Two-dimensional materials such as graphene or hexagonal boron nitride are indispensable in industry. The recently discovered 2D ferromagnetic materials also promise to be vital for applications. In this work, we develop a phenomenological description of non-centrosymmetric 2D ferromagnets with trigonal prismatic crystal structure. We chose to study this special symmetry group since these materials do break inversion symmetry and therefore, in principle, allow for chiral spin structures such as magnetic helices and skyrmions. However, unlike all non-centrosymmetric magnets known so far, we show…
Topological insulators (TI) are materials having an energy band gap in the bulk and conducting helical electronic states on the surface. The helical states are protected by time reversal symmetry thus are expected to be robust against static disorder scattering. In this work we report atomistic first principles analysis of disorder scattering in two-probe transport junctions made of three dimensional TI material Bi2Se3. The robustness of the device against disorder scattering is determined quantitatively. Examining many different scattering configurations, a general trend emerges on how strong…
Transport by helical edge states of a quantum spin Hall insulator is experimentally characterized by a weakly temperature-dependent mean free path of a few microns and by reproducible conductance oscillations, challenging proposed theoretical explanations. We consider a model where edge electrons experience spatially random Rashba spin-orbit coupling and couple to a magnetic impurity with spin S >= 1/2. In a finite bias steady state, we find for S > 1/2 an impurity induced resistance with a temperature dependence in agreement with experiments. Since backscattering is elastic, interference betw…
A tunnel junction between helical edge states, realized via a constriction in a Quantum Spin Hall system, can be exploited to steer both charge and spin current into various terminals. We investigate the effects of disorder on the transmission coefficient $T_p$ of the junction, by modelling disorder with a randomly varying (complex) tunneling amplitude $Γ_p=|Γ_p| \exp[i ϕ_p]$. We show that, while for a clean junction $T_p$ is only determined by the absolute value $|Γ_p|$ and is independent of the phase $ϕ_p$, the situation can be quite different in the presence of disorder: phase fluctuations …
Robustness of helical hinge states of three-dimensional weak second-order topological i sulators (WSOTIs) against disorders is studied. The pure WSOTI is obtained from a weak $\mathbb{Z}_2$ first-order topological insulator through a surface band inversion. Both bulk states and surface states in the WSOTI are gapped, and in-gap valley-momentum locked helical hinge states are topologically protected by the surface valley-Chern number. In the presence of weak disorders, helical hinge states are robust against disorders while the quantized conductance of the states is fragile due to the inter-val…
In an ideal topological insulator, the helical spin structure of surface electrons suppresses backscattering and thus can enhance surface conductivity. We investigate the effect of perpendicular magnetic field on the spin structure of electrons at the Fermi energy and calculate a magnetic-field dependent topological enhancement factor for different disorder potentials, ranging from short-range disorder to screened Coulomb potential. Within the Boltzmann approximation, the topological enhancement factor reaches its maximum value of 4 for a short-range disorder at zero magnetic field and approac…
The optical spectra of transition metal dichalcogenide (TMDC) monolayers are dominated by excitons and trions. Here we establish the dependences of these optical transitions on disorder from hyperspectral imaging of h-BN encapsulated monolayer MoSe2. While both exciton and trion energies vary spatially, these two quantities are almost perfectly correlated, with spatial variation in the trion binding energy of only ~0.18 meV. In contrast, variation in the energy splitting between the two lowest energy exciton states is one order of magnitude larger at ~1.7 meV. Statistical analysis and theoreti…
In this work, we study the disorder effect on topological metals that support a pair of helical edge modes deeply embedded inside the gapless bulk states. Strikingly, we predict that a quantum spin Hall (QSH) phase can be obtained from such topological metals without opening a global band gap. To be specific, disorder can lead to a pair of robust helical edge states which is protected by an emergent $Z_2$ topological invariant, giving rise to a quantized conductance plateau in transport measurements. These results are instructive for solving puzzles in {various transport experiments on QSH mat…
Hydrodynamic behavior in electronic systems is commonly accepted to be associated with extremely clean samples such that electron-electron collisions dominate and total momentum is conserved. Contrary to this, we show that in monolayer graphene the presence of disorder is essential to enable an unconventional hydrodynamic regime which exists near the charge neutrality point and is characterized by a large enhancement of the Wiedemann-Franz ratio. Although the enhancement becomes more pronounced with decreasing disorder, the very possibility of observing the effect depends crucially on the pres…
Atomic disorder is a common limiting factor for the low-temperature mobility in monolayer transition-metal dichalcogenides (TMDs; MX2). Here, we study the effect of often occurring atomic vacancies on carrier scattering and transport in p- and n-type monolayer MoS2. Due to charge trapping in vacancy-induced in-gap states, both neutral and charged vacancies resembling, respectively, short-range and combined short-range and long-range Coulomb scatterers, must be considered. Using the T-matrix formalism, we demonstrate a strong renormalization of the Born description of short-range scattering, ma…
Ridged, orthorhombic two-dimensional atomic crystals with a bulk {\em Pnma} structure such as black phosphorus and monochalcogenide monolayers are an exciting and novel material platform for a host of applications. Key to their crystallinity, monolayers of these materials have a four-fold degenerate structural ground state, and a single energy scale $E_C$ (representing the elastic energy required to switch the longer lattice vector along the $x-$ or $y-$direction) determines how disordered these monolayers are at finite temperature. Disorder arises when nearest neighboring atoms become gently …
Gradual localization of charge carriers was studied in a series of micro-size samples of monolayer graphene fabricated on the common large scale film and irradiated by different doses of C$^+$ ions with energy 35 keV. Measurements of the temperature dependence of conductivity and magnetoresistance in fields up to 4 T showed that at low disorder, the samples are in the regime of weak localization and antilocalization. Further increase of disorder leads to strong localization regime, when conductivity is described by the variable-range-hopping (VRH) mechanism. A crossover from the Mott regime to…
We calculate the conductance of a two-dimensional bilayer with inverted electron-hole bands, to study the sensitivity of the quantum spin Hall insulator (with helical edge conduction) to the combination of electrostatic disorder and a perpendicular magnetic field. The characteristic breakdown field for helical edge conduction splits into two fields with increasing disorder, a field $B_{c}$ for the transition into a quantum Hall insulator (supporting chiral edge conduction) and a smaller field $B'_{c}$ for the transition to bulk conduction in a quasi-metallic regime. The spatial separation of t…
We study the evolutions of the nonmagnetic disorder-induced edge states with the disorder strength in the HgTe/CdTe quantum wells. From the supercell band structures and wave-functions, it is clearly shown that the conducting helical edge states, which are responsible for the reported quantized conductance plateau, appear above a critical disorder strength after a gap-closing phase transition. These edge states are then found to decline with the increase of disorder strength in a stepwise pattern due to the finite-width effect, where the opposite edges couple with each other through the locali…
The hydrogenated $\rm{Sb}$ monolayer epitaxially grown on a $\rm{LaFeO_3}$ substrate is a novel type of two-dimensional material hosting quantum spin-quantum anomalous Hall (QS-QAH) states. For a device formed by $\rm{Sb}$ monolayer ribbon, the QAH edge states, belong to a single valley, are located at opposite edges of the ribbon. The QSH edge states, on the other hand, belong to the other valley and are distributed in a very narrow region at the same edge. In this paper, we find such material can be used to fabricate perfect valley filter. Adopting scattering matrix method and Green's functi…
Cascade of topological phase transitions and revival of topological zero modes in imperfect double helical liquids
Institute of Physics, Academia Sinica（中央研究院物理研究所）；Department of Electronics and Electrical Engineering, National Yang Ming Chiao Tung University（国立阳明交通大学电子工程学系）；Physics Division, National Center for Theoretical Sciences（国家理论科学中心物理组）
A brief review of experiments directed to study a gradual localization of charge carriers and metal-insulator transition in samples of disordered monolayer graphene is presented. Disorder was induced by irradiation with different doses of heavy and light ions. Degree of disorder was controlled by measurements of the Raman scattering spectra. The temperature dependences of conductivity and magnetoresistance (MR) showed that at low disorder, conductivity is governed by the weak localization and antilocalization regime. Further increase of disorder leads to strong localization of charge carriers,…
We study supercurrent and proximity vortices in a Josephson junction made of disordered surface states of a three-dimensional topological insulator with a proximity induced in-plane helical magnetization. In a regime where the rotation period of helical magnetization is larger than the junction width, we find supercurrent 0-π crossovers as a function of junction thickness, magnetization strength, and parameters inherent to the helical modulation and surface states. The supercurrent reversals are associated with proximity induced vortices, nucleated along the junction width, where the number of…
We propose a physical model based on disordered (a hole punched inside a material) monolayer transition metal dichalcogenides (TMDs) to demonstrate a large-gap quantum valley Hall insulator. We find an emergence of bound states lying inside the bulk gap of the TMDs. They are strongly affected by spin-valley coupling, rest- and kinetic- mass terms and the hole size. In addition, in the whole range of the hole size, at least two in-gap bound states with opposite angular momentum, circulating around the edge of the hole, exist. Their topological insulator (TI) feature is analyzed by the Chern num…
We investigate the contribution of charge puddles to the non-vanishing conductivity minimum in disordered graphene flakes at the charge neutrality point. For that purpose, we study systems with a geometry that suppresses the transmission due to evanescent modes allowing to single out the effect of charge fluctuations in the transport properties. We use the recursive Green's functions technique to obtain local and total transmissions through systems that mimic vanishing density of states at the charge neutrality point in the presence of a local disordered local potential to model the charge pud…
Transport properties of disordered quantum confined helical Dirac systems are investigated in the large energy limit. As long as the 2D transport length is larger than the perimeter of the nanowire, the conductance and the Fano factor are sensitive to disorder only when the Fermi energy is close to an opening of a transverse mode. In the limit of a large number of transverse modes, transport properties are insensitive to the geometry of the nanowire or the nature and strength of the disorder but, instead, are dominated by the properties of the interface between the ohmic contact and the nanowi…
We study Aharonov-Bohm effect in a two-terminal helical ring with long-range hopping in presence of Rashba spin-orbit interaction. We explore how the spin polarization behavior changes depending on the applied magnetic flux and the incoming electron energy. The most interesting feature that we articulate in this system is that zero-energy crossings appear in the energy spectra at $Φ=0$ and also at integer multiples of half-flux quantum values ($n Φ_0/2$, $n$ being an integer) of the applied magnetic flux. We investigate the transport properties of the ring using Green's function formalism and …
Quantum spin Hall insulators are characterized by topologically protected counterpropagating edge states. Here we study the dynamical response of these helical edge states under a time-dependent flux biasing, in the presence of a heat bath. It is shown that the relaxation time of the edge carriers can be determined from a measurement of the dissipative response of topological insulator disks. The effects of various perturbations, including Zeeman coupling and disorder, are also discussed.
We study charge transport in a monolayer molybdenum disulfide nanoflake over a wide range of carrier density, temperature, and electric bias. We find that the transport is best described by a percolating picture in which the disorder breaks translational invariance, breaking the system up into a series of puddles, rather than previous pictures in which the disorder is treated as homogeneous and uniform. Our work provides insight to a unified picture of charge transport in monolayer molybdenum disulfide nanoflakes and contributes to the development of next-generation molybdenum disulfide based …
We show that when graphene monolayers are disordered, the conductance exhibits a metallic-to-insulating transition, which opens the door to new electronic devices. The transition can be observed by driving the density or Fermi energy through the mobility edge. At the Dirac point the system is localized, whereas at higher densities there is a region of metallic behavior before the system becomes insulating again at higher densities. The region of metallic behavior depends on the disorder strength and eventually vanishes at high disorder. This result is quite unexpected since in square lattices,…
Elastic backscattering of electrons moving along the helical edge is prohibited by time-reversal symmetry (TRS). We demonstrate, however, that an ensemble of magnetic impurities may cause TRS-preserving quasi-elastic backscattering, resulting in interference effects in the conductance. The characteristic energy transferred in a backscattering event is suppressed due to the RKKY interaction of localized spins (the suppression is exponential in the total number of magnetic impurities). We predict the statistics of conductance fluctuations to differ from those in the conventional case of a one-di…
Bulk chromium triiodide (CrI$_3$) has long been known as a layered van der Waals ferromagnet. However, its monolayer form was only recently isolated and confirmed to be a truly two-dimensional (2D) ferromagnet, providing a new platform for investigating light-matter interactions and magneto-optical phenomena in the atomically thin limit. Here, we report spontaneous circularly polarized photoluminescence in monolayer CrI$_3$ under linearly polarized excitation, with helicity determined by the monolayer magnetization direction. In contrast, the bilayer CrI$_3$ photoluminescence exhibits vanishin…
We present magnetotransport measurements at classical magnetic fields for three graphene monolayers with various levels of disorder. A square root magnetoresistance (SRMR) behavior is observed in one sample which has the characteristic sub-linear conductivity signaling on the presence of short-range disorder in this sample. No square root MR was observed in other samples where short-range scattering is inessential as it is evident from the gate voltage dependences of their conductivities. Comparing our experimental data for the sample with theoretical calculations we found a good qualitative a…
The quantum spin Hall (QSH) state was recently demonstrated in monolayers of the transition metal dichalcogenide 1T'-WTe$_2$ and is characterized by a band gap in the two-dimensional (2D) interior and helical one-dimensional (1D) edge states. Inducing superconductivity in the helical edge states would result in a 1D topological superconductor, a highly sought-after state of matter. In the present study, we use a novel dry-transfer flip technique to place atomically-thin layers of WTe$_2$ on a van der Waals superconductor, NbSe$_2$. Using scanning tunneling microscopy and spectroscopy (STM/STS)…
Theoretically, the helical edge states of two-dimensional topological insulators are protected from coherent backscattering due to nonmagnetic disorder provided electron interactions are not too strong. Experimentally, the edges typically do not demonstrate the systematic and robust quantization, at the same time little is known about the sub-Kelvin temperature behavior. Here, we report the surprising localization of the edge states in an 8 nm HgTe quantum well in zero magnetic field at millikelvin temperatures. Additionally, the magnetoresistance data at 0.5 K for the edges few micrometers lo…
Recently, the chiral-induced spin selectivity in molecular systems has attracted extensive interest among the scientific communities. Here, we investigate the effect of the gate voltage on spin-selective electron transport through the $α$-helical peptide/protein molecule contacted by two nonmagnetic electrodes. Based on an effective model Hamiltonian and the Landauer-Büttiker formula, we calculate the conductance and the spin polarization under an external electric field which is perpendicular to the helix axis of the $α$-helical peptide/protein molecule. Our results indicate that both the mag…
We discuss transport through interferometer formed by helical edge states tunnel-coupled to metallic leads. We focus on the experimentally relevant case of relatively high temperature as compared to the level spacing and discuss a response of the setup to the external magnetic flux $ϕ$ (measured in units of flux quantum) piercing the area encompassed by edge states. We demonstrate that tunneling conductance of the interferometer is structureless in ballistic case but shows a sharp antiresonances, as a function of magnetic flux $ϕ$ - with the period 1/2 - in the presence of magnetic impurity. W…
We theoretically obtain the phase diagram of localized magnetic impurity spins arranged in a one-dimensional chain on top of a one- or two-dimensional electron gas with Rashba spin-orbit coupling. The interactions between the spins are mediated by the Ruderman-Kittel-Kasuya-Yosida (RKKY) mechanism through the electron gas. Recent work predicts that such a system may intrinsically support topological superconductivity when a helical spin-density wave is formed in the spins, and superconductivity is induced in the electron gas. We analyze, using both analytical and numerical techniques, the cond…
We report low-temperature transport measurements in strained InAs/Ga0.68In0.32Sb quantum wells, which supports time-reversal symmetry-protected helical edge states. The temperature and bias voltage dependence of the helical edge conductance for devices of various sizes are consistent with the theoretical expectation of a weakly interacting helical edge state. Moreover, we found that the magnetoresistance of the helical edge states is related to the edge interaction effect and the disorder strength.
We show that burying of the Dirac point in semiconductor-based quantum-spin-Hall systems can generate unexpected robustness of edge states to magnetic fields. A detailed ${\bf k\cdot p}$ band-structure analysis reveals that InAs/GaSb and HgTe/CdTe quantum wells exhibit such buried Dirac points. By simulating transport in a disordered system described within an effective model, we further demonstrate that buried Dirac points yield nearly quantized edge conduction out to large magnetic fields, consistent with recent experiments.
A two-dimensional topological insulator (2DTI) is guaranteed to have a helical 1D edge mode in which spin is locked to momentum, producing the quantum spin Hall effect and prohibiting elastic backscattering at zero magnetic field. No monolayer material has yet been shown to be a 2DTI, but recently the Weyl semimetal WTe2 was predicted to become a 2DTI in monolayer form if a bulk gap opens. Here, we report that at temperatures below about 100 K monolayer WTe2 does become insulating in its interior, while the edges still conduct. The edge conduction is strongly suppressed by in-plane magnetic fi…
Using time-resolved optical Kerr rotation, we measure the low temperature valley dynamics of resident electrons and holes in exfoliated WSe$_2$ monolayers as a systematic function of carrier density. In an effort to reconcile the many disparate timescales of carrier valley dynamics in monolayer semiconductors reported to date, we directly compare the doping-dependent valley relaxation in two electrostatically-gated WSe$_2$ monolayers having different dielectric environments. In a fully-encapsulated structure (hBN/WSe$_2$/hBN, where hBN is hexagonal boron nitride), valley relaxation is found to…
Scientific curiosity to uncover original optical properties and functionalities of atomically thin semiconductors, stemming from unusual Coulomb interactions in the two-dimensional geometry and multi-valley band structure, drives the research on monolayers of transition metal dichalcogenides (TMDs). While recent works ascertained the exotic energetic schemes of exciton complexes in TMDs, we here employ four-wave mixing microscopy to indicate that their subpicosecond dynamics is determined by the surrounding disorder. Focusing on a monolayer WS$_2$, we observe that exciton coherence is lost pri…
The Raman scattering spectra (RS) of two series of monolayer graphene samples irradiated with various doses of C$^{+}$ and Xe$^{+}$ ions were measured after annealing in high vacuum, and in forming gas (95\%Ar+5\%H$_{2}$). It was found that these methods of annealing have dramatically different influence on the RS lines. Annealing in vacuum below 500$^{\circ}$C leads to significant decrease of both D-line, associated with defects, and 2D-line, associated with the intact lattice structure, which can be explained by annealing-induced enhanced doping. Further annealing in vacuum up to 1000$^{\cir…
The recent discovery of exciton quantum emitters in transition metal dichalcogenides (TMDCs) has triggered renewed interest of localized excitons in low-dimensional systems. Open questions remain about the microscopic origin previously attributed to dopants and/or defects as well as strain potentials. Here we show that the quantum emitters can be deliberately induced by nanobubble formation in WSe2 and BN/WSe2 heterostructures. Correlations of atomic-force microscope and hyperspectral photoluminescence images reveal that the origin of quantum emitters and trion disorder is extrinsic and relate…
Altermagnets Enable Gate-Switchable Helical and Chiral Topological Transport with Spin-Valley-Momentum-Locked Dual Protection
Eastern Institute of Technology（东方理工大学）；University of Science and Technology of China（中国科学技术大学）；Suzhou University of Science and Technology（苏州科技大学）；University at Buffalo, State University of New York（纽约州立大学布法罗分校）
The layered semimetal WTe_2 has recently been found to be a two-dimensional topological insulator (2D TI) when thinned down to a single monolayer, with conducting helical edge channels. We report here that intrinsic superconductivity can be induced in this monolayer 2D TI by mild electrostatic doping, at temperatures below 1 K. The 2D TI-superconductor transition can be easily driven by applying a just a small gate voltage. This discovery offers new possibilities for gate-controlled devices combining superconductivity and topology, and could provide a basis for quantum information schemes base…
We discuss transport through an interferometer formed by helical edge states of the quantum spin Hall insulator. Focusing on effects induced by a strong magnetic impurity placed in one of the arms of interferometer, we consider the experimentally relevant case of relatively high temperature as compared to the level spacing. We obtain the conductance and the spin polarization in the closed form for arbitrary tunneling amplitude of the contacts and arbitrary strength of the magnetic impurity. We demonstrate the existence of quantum effects which do not show up in previously studied case of weak …
The electronic properties of a graphene monolayer in a magnetic and a strain-induced pseudo-magnetic field are studied in the presence of spin-orbit interactions (SOI) that are artificially enhanced, e.g., by suitable adatom deposition. For the homogeneous case, we provide analytical results for the Landau level eigenstates for arbitrary intrinsic and Rashba SOI, including also the Zeeman field. The edge states in a semi-infinite geometry are studied in the absence of the Rashba term. For a critical value of the magnetic field, we find a quantum phase transition separating two phases with spin…
We study the kinetic magnetoelectric effect (current-induced magnetization including both the orbital and spin contributions) in three-dimensional conductors, specializing to the case of p-doped trigonal tellurium. We include both intrinsic and extrinsic contributions to the effect, which stem from the band structure of the crystal, and from disorder scattering, respectively. Specifically, we determine the dependence of the kinetic magnetoelectric response on the hole doping in tellurium, and show that the intrinsic and extrinsic effects dominate for low and high levels of doping, respectively…
We develop a theory of finite-temperature momentum-resolved tunneling spectroscopy (MRTS) for disordered, interacting two-dimensional topological-insulator edges. The MRTS complements conventional electrical transport measurement in characterizing the properties of the helical Luttinger liquid edges. Using standard bosonization technique, we study low-energy spectral function and the MRTS tunneling current, providing a detailed description controlled by disorder, interaction, and temperature, taking into account Rashba spin orbit coupling, interedge interaction and distinct edge velocities. Ou…
A monolayer of the high-$T_c$ superconductor FeTe$_{1-x}$Se$_x$ has been predicted to realize a topologically non-trivial state with helical edge modes at its boundary, providing a novel intrinsic system to search for topological superconductivity and Majorana zero modes. Evidence in favor of a topological phase transition and helical edge modes has been identified in recent experiments \cite{Peng2019}. We propose to create Majorana zero modes by applying an in-plane magnetic field to the FeTe$_{1-x}$Se$_x$ monolayer and by tuning the local chemical potential via electric gating. Owing to the …
