We report on infrared spectroscopy of bilayer graphene integrated in gated structures. We observed a significant asymmetry in the optical conductivity upon electrostatic doping of electrons and holes. We show that this finding arises from a marked asymmetry between the valence and conduction bands, which is mainly due to the inequivalence of the two sublattices within the graphene layer. From the conductivity data, the energy difference of the two sublattices is determined.
We present a self-consistent calculation of the interlayer asymmetry in bilayer graphene caused by an applied electric field in magnetic fields. We show how this asymmetry influences the Landau level spectrum in bilayer graphene and the observable inter-Landau level transitions when they are studied as a function of high magnetic field at fixed filling factor as measured experimentally by E.A. Henriksen et al., Phys. Rev. Lett. 100 (2008), 087403. We also analyze the magneto-optical spectra of bilayer flakes in the photon-energy range corresponding to transitions between degenerate and split b…
Inter-Landau-level transitions in the bilayer graphene at high perpendicular magnetic field at the filling-factor v<<1 have been studied. The next-nearest-neighbor transitions, energy difference between dimer and non-dimer sites and layer asymmetry are included. The influence of Coulomb interaction is taken into account. The magnetoplasmon excitations in bilayer graphene at small momenta are considered in the frame of the Hartree-Fock approximation. It is shown that asymmetry in cyclotron resonance of clean bilayer graphene depends on magnetic field. At lower magnetic fields the energy splitti…
In this paper, we study the quantum properties of a bilayer graphene with (asymmetry) line defects. The localized states are found around the line defects. Thus, the line defects on one certain layer of the bilayer graphene can lead to an electric transport channel. By adding a bias potential along the direction of the line defects, we calculate the electric conductivity of bilayer graphene with line defects using Landauer-Büttiker theory, and show that the channel affects the electric conductivity remarkably by comparing the results with those in a perfect bilayer graphene. This one-dimension…
We study the superlattice minibands produced by the interplay between moiré pattern induced by hexagonal BN substrate on graphene layer and the interlayer coupling in bilayer graphene with Bernal stacking (BLG). We compare moiré miniband features in BLG, where they are affected by the interlayer asymmetry of BLG-hBN heterostructure and trigonal warping characteristic for electrons in Bernal-stacked bilayers with those found in monolayer graphene.
Cyclotron resonance in bilayer graphene is studied with weak electron-hole asymmetry, suggested by experiment, taken into account and with the focus on many-body corrections that evade Kohn's theorem. It is shown by direct calculation that the theory remains renormalizable to O(e^2) in the presence of electron-hole asymmetry parameters, and a general program to carry out renormalization for graphene under a magnetic field is presented. Inclusion of electron-hole asymmetry in part improves the theoretical fit to the existing data and the data appear to indicate the running of the renormalized v…
A tight binding model is used to calculate the band structure of bilayer graphene in the presence of a potential difference between the layers that opens a gap $Δ$ between the conduction and valence bands. In particular, a self consistent Hartree approximation is used to describe imperfect screening of an external gate, employed primarily to control the density $n$ of electrons on the bilayer, resulting in a potential difference between the layers and a density dependent gap $Δ(n)$. We discuss the influence of a finite asymmetry gap $Δ(0)$ at zero excess density, caused by the screening of an …
We show that Friedel charge oscillation near an interface opens a gap at the Fermi energy for electrons with wave vectors perpendicular to the interface. If the Friedel gaps on two sides of the interface are different, a nonequlibrium effect - shifting of these gaps under bias - leads to asymmetric transport upon reversing the bias polarity. The predicted transport asymmetry is revealed by scanning tunneling potentiometry at monolayer-bilayer interfaces in epitaxial graphene on SiC (0001). This intriguing interfacial transport behavior opens a new avenue towards novel quantum functions such as…
We report an experimental study of temperature-dependent spectral lineshapes of phonon side-band emission stemming from dark excitons in monolayer and bilayer WSe$_{2}$. Using photoluminescence spectroscopy in the range from 4 to 100K, we observe a pronounced asymmetry in the phonon-assisted luminescence from momentum-indirect exciton reservoirs. We demonstrate that the corresponding spectral profiles are distinct from those of bright excitons with direct radiative decay pathways. The lineshape asymmetry reflects thermal distribution of exciton states with finite center-of-mass momenta, charac…
We investigate the band structure and the optical absorption spectrum of twisted bilayer graphenes with changing interlayer bias and Fermi energy simultaneously. We show that the interlayer bias lifts the degeneracy of the superlattice Dirac point, while the amount of the Dirac point shift is significantly suppressed in small rotation angles, and even becomes opposite to the applied bias. We calculate the optical absorption spectrum in various asymmetric potentials and Fermi energies, and associate the characteristic spectral features with the band structure. The spectroscopic features are hig…
We derive an effective Hamiltonian at low energies for bilayer graphene when Fermi velocity manufactured on each layer is different of the velocity measured in pristine graphene. Based on the effective Hamiltonian, we investigate the influence of Fermi velocity asymmetry on the band structure of trigonally warped bilayer graphene in the presence of interlayer applied bias. In this case, the Fermi line at low energies is still preserved its threefold rotational symmetry appearing as the three pockets. Furthermore, the interlayer asymmetry in Fermi velocities leads to an indirect band gap which …
The electronic and optical response of Bernal stacked bilayer graphene with geometry modulation and gate voltage are studied. The broken symmetry in sublattices, one dimensional periodicity perpendicular to the domain wall and out-of-plane axis introduces substantial changes of wavefunctions, such as gapless topological protected states, standing waves with bonding and anti-bonding characteristics, rich structures in density of states and optical spectra. The wavefunctions present well-behaved standing waves in pure system and complicated node structures in geometry-modulated system. The optic…
Under a magnetic field perpendicular to an monolayer graphene, the existence of a two-dimensional periodic scatter array can not only mix Landau levels of the same valley for displaying split electron-hole Hofstadter-type energy spectra, but also couple two sets of Landau subbands from different valleys in a bilayer graphene. Such a valley mixing effect with a strong scattering strength has been found observable and studied thoroughly in this paper by using a Bloch-wave expansion approach and a projected $2\times 2$ effective Hamiltonian including interlayer effective mass, interlayer coupling…
In this work we performed density functional theory calculations for a twisted bilayer graphene (BLG). Several conmensurable rotation angles were analyzed and for each one a constant height mode STM image was obtained. These STM images, calculated under the Tersoff-Hamman theory, reproduce the main features experimentally observed, paticularly superstructures and giant corrugations. In this way we confirm that STM characterization of twisted BLG can produce superstructures whose tunneling current intensity maxima occur over regions with $AA$ stacking. Additionally we give new evidence in favou…
We report strong electron-phonon coupling in magic-angle twisted bilayer graphene (MA-TBG) obtained from atomistic description of the system including more than 10000 atoms in the moire supercell. Electronic structure, phonon spectrum, and electron-phonon coupling strength lambda are obtained before and after atomic-position relaxation both in and out of plane. Obtained lambda is very large for MA-TBG, with lambda > 1 near the half-filling energies of the flat bands, while it is small (lambda ~ 0.1) for monolayer and unrotated bilayer graphene. Significant electron-hole asymmetry occurs in the…
We investigate the magnetic minibands of a heterostructure consisting of bilayer graphene (BLG) and hexagonal boron nitride (hBN) by numerically diagonalizing a two-band Hamiltonian that describes electrons in BLG in the presence of a moire potential. Due to inversion-symmetry breaking characteristic for the moire potential, the valley symmetry of the spectrum is broken, but despite this, the zero-energy Landau level in BLG survives, albeit with reduced degeneracy. In addition, we derive effective models for the low-energy features in the magnetic minibands and demonstrate the appearance of se…
When a bilayer of graphene is placed in a suitably configured field effect device, an asymmetry gap can be generated and the carrier concentration made different in each layer. This provides a tunable semiconducting gap, and the valence and the conductance band no longer meet at the two Dirac points of the graphene Brillouin zone. We calculate the optical conductivity of such a semiconductor with particular emphasis on the optical spectral weight redistribution brought about by changes in gap and chemical potential due to charging. We derive an algebraic formula for arbitrary value of the chem…
We investigated the valley depolarization and carrier relaxation process of monolayer and bilayer WS2 at 10 K by using time-resolved Kerr rotation (TRKR) and differential reflectance measurement simultaneously. Two decay processes extracted from TRKR signals were observed on both monolayer and bilayer WS2. In monolayer WS2, the initial ultrafast decay component (less than 1 ps) was interpreted as the stimulated emission or Pauli-blocking of electrons and holes in one valley by comparing with carrier decay process. The relatively slow component was around 4 ps under low excitation energy (less …
Spontaneous orbital magnetism observed in twisted bilayer graphene (tBG) on nearly aligned hexagonal boron nitride (BN) substrate builds on top of the electronic structure resulting from combined G/G and G/BN double moire interfaces. Here we show that tBG/BN commensurate double moire patterns can be classified into two types, each favoring the narrowing of either the conduction or valence bands on average, and obtain the evolution of the bands as a function of the interlayer sliding vectors and electric fields. Finite valley Chern numbers $\pm 1$ are found in a wide range of parameter space wh…
We report precision measurements of the effective mass m* in high-quality bilayer graphene using the temperature dependence of the Shubnikov-de Haas oscillations. In the density range of 0.7 x 10^12/cm^2 < n < 4.1 x 10^12 /cm^2, both the hole mass m*_h and the electron mass m*_e increase with increasing density, demonstrating the hyperbolic nature of the bands. The hole mass m*_h is approximately 20-30% larger than the electron mass m*_e. Tight-binding calculations provide a good description of the electron-hole asymmetry and yield an accurate measure of the inter-layer hopping parameter v_4 =…
The spin polarization of Pt in Pt/NiFe2O4 and Pt/Fe bilayers is studied by interface-sensitive x-ray resonant magnetic reflectivity to investigate static magnetic proximity effects. The asymmetry ratio of the reflectivity was measured at the Pt L3 absorption edge using circular polarized x-rays for opposite directions of the magnetization at room temperature. The results of the 2% asymmetry ratio for Pt/Fe bilayers are independent of the Pt thickness between 1.8 and 20 nm. By comparison with ab initio calculations, the maximum magnetic moment per spin polarized Pt atom at the interface is dete…
We report precision measurements of the effective mass m* in high-quality bilayer graphene using the temperature dependence of the Shubnikov-de Haas oscillations. In the density range of 0.7 x 10^12/cm^2 < n < 4.1 x 10^12 /cm^2, both the hole mass m*_h and the electron mass m*_e increase with increasing density, demonstrating the hyperbolic nature of the bands. The hole mass m*_h is approximately 20-30% larger than the electron mass m*_e. Tight-binding calculations provide a good description of the electron-hole asymmetry and yield an accurate measure of the inter-layer hopping parameter v_4 =…
Twisted bilayer graphene (TBG) provides an example of a system in which the interplay of interlayer interactions and superlattice structure impacts electron transport in a variety of non-trivial ways and gives rise to a plethora of interesting effects. Understanding the mechanisms of electron scattering in TBG has, however, proven challenging, raising many questions about the origins of resistivity in this system. Here we show that TBG exhibits high-temperature magnetooscillations originating from the scattering of charge carriers between TBG minivalleys. The amplitude of these oscillations re…
We review the electronic properties of bilayer graphene, beginning with a description of the tight-binding model of bilayer graphene and the derivation of the effective Hamiltonian describing massive chiral quasiparticles in two parabolic bands at low energy. We take into account five tight-binding parameters of the Slonczewski-Weiss-McClure model of bulk graphite plus intra- and interlayer asymmetry between atomic sites which induce band gaps in the low-energy spectrum. The Hartree model of screening and band-gap opening due to interlayer asymmetry in the presence of external gates is present…
Bilayer graphene can exhibit deformations such that the two graphene sheets are locally detached from each other resulting in a structure consisting of domains with different inter-layer coupling. Here we investigate how the presence of these domains affect the transport properties of bilayer graphene. We derive analytical expressions for the transmission probability, and the corresponding conductance, across walls separating different inter-layer coupling domain. We find that the transmission can exhibit a valley-dependent layer asymmetry and that the domain walls have a considerable effect o…
We calculate the electronic band structure of ABA-stacked trilayer graphene in the presence of external gates, using a self-consistent Hartree approximation to take account of screening. In the absence of a gate potential, there are separate pairs of linear and parabolic bands at low energy. A gate field perpendicular to the layers breaks mirror reflection symmetry with respect to the central layer and hybridizes the linear and parabolic low-energy bands, leaving a chiral Hamiltonian essentially different from that of monolayer or bilayer graphene. Using the self-consistent Born approximation,…
AB-stacked bilayer graphene with a tunable electronic bandgap in excess of the optical phonon energy presents an interesting active medium, and we consider such theoretical possibility in this work. We argue the possibility of a highly resonant optical gain in the vicinity of the asymmetry gap. Associated with this resonant gain are strongly amplified plasmons, plasmons with negative group velocity and superluminal effects, as well as directional leaky modes.
We present a theoretical study of the local optical conductivity, plasmon spectra, and thermoelectric properties of twisted bilayer graphene (TBG) at different filling factors and twist angles $θ$. Our calculations are based on the electronic band structures obtained from a continuum model that has two tunable parameters, $u_0$ and $u_1$, which parametrize the intra-sublattice inter-layer and inter-sublattice inter-layer tunneling rate, respectively. In this Article we focus on two key aspects: i) we study the dependence of our results on the value of $u_0$, exploring the whole range $0\leq u_…
Using the method developed in the companion paper, we construct the effective continuum theories for two different microscopic tight binding models of the twisted bilayer graphene at the twist angle of $1.05^\circ$, one Slater-Koster based and the other ab-initio Wannier based. The energy spectra obtained from the continuum theory -- either for rigid twist or including lattice relaxation -- are found to be in nearly perfect agreement with the spectra from the tight binding models when the gradient expansion is carried out to second order, demonstrating the validity of the method. We also analy…
We report on our studies of interacting electrons in bilayer graphene in a magnetic field. We demonstrate that the long range Coulomb interactions between electrons in this material are highly important and account for the band asymmetry in recent optical magneto-absorption experiments. We show that in the unbiased bilayer (where both layers are at the same electrostatic potential), the interactions can cause mixing of Landau levels in moderate magnetic fields. For the biased bilayer (when the two layers are at different potentials), we demonstrate that the interactions are responsible for a c…
We study the energy gap opening in the electronic spectrum of graphene bilayers caused by asym- metric doping. Both substitutional impurities (boron acceptors and nitrogen donors) and adsorbed potassium donors are considered. The gap evolution with dopant concentration is compared to the situation in which the asymmetry between the layers is induced by an external electric field. The effects of adsorbed potassium are similar to that of an electric field, but substitutional impurities behave quite differently, showing smaller band gaps and a large sensitivity to disorder and sublattice occupati…
We investigate the ultralow-frequency Raman response of atomically thin ReS2, a special type of two-dimensional (2D) semiconductors with unique distorted 1T structure. Bilayer and few-layer ReS2 exhibit rich Raman spectra at frequencies below 50 cm-1, where a panoply of interlayer shear and breathing modes are observed. The emergence of these interlayer phonon modes indicate that the ReS2 layers are coupled and stacked orderly, in contrast to the general belief that the ReS2 layers are decoupled from one another. While the interlayer breathing modes can be described by a linear chain model as …
We utilize transverse ac susceptibility measurements to characterize magnetic anisotropy in archetypal exchange-bias bilayers of ferromagnet Permalloy (Py) and antiferromagnet CoO. Unidirectional anisotropy is observed for thin Py, but becomes negligible at larger Py thicknesses, even though the directional asymmetry of the magnetic hysteresis loop remains significant. Additional magnetoelectronic measurements, magneto-optical imaging, as well as micromagnetic simulations show that these surprising behaviors are likely associated with asymmetry of spin flop distribution created in CoO during P…
The electronic structure of the zig-zag bilayer strip is analyzed. The electronic spectra of the bilayer strip is computed. The dependence of the edge state band flatness on the bilayer width is found. The density of states at the Fermi level is analytically computed. It is shown that it has the singularity which depends on the width of the bilayer strip. There is also asymmetry in the density of states below and above the Fermi energy.
We demonstrate that, at the onset of conduction, an electrostatically defined quantum wire in bilayer graphene (BLG) with an interlayer asymmetry gap may act as a 1D semimetal, due to the multiple minivalley dispersion of its lowest subband. Formation of a non-monotonic subband coincides with a near-degeneracy between the bottom edges of the lowest two subbands in the wire spectrum, suggesting an $8e^2/h$ step at the conduction threshold, and the semimetallic behaviour of the lowest subband in the wire would be manifest as resonance transmission peaks on an $8e^2/h$ conductance plateau.
We model optical absorption of monolayer and bilayer graphene on hexagonal boron nitride for the case of closely-aligned crystal lattices. We show that perturbations with different spatial symmetry can lead to similar absorption spectra. We suggest that a study of the absorption spectra as a function of the doping for almost completely full first miniband is necessary to extract meaningful information about the moire characteristics from optical absorption measurements and to distinguish between various theoretical proposals for the physically realistic interaction. Also, for bilayer graphene,…
We present measurements of the electronic compressibility, $K$, of bilayer graphene in both zero and finite magnetic fields up to 14 T, and as a function of both the carrier density and electric field perpendicular to the graphene sheet. The low energy hyperbolic band structure of bilayer graphene is clearly revealed in the data, as well as a sizable asymmetry between the conduction and valence bands. A sharp increase in $K^{-1}$ near zero density is observed for increasing electric field strength, signaling the controlled opening of a gap between these bands. At high magnetic fields, broad La…
Layer-Asymmetry-Induced Topological Superconductivity in High-T$_c$ Bilayer Nickelates
Hong Kong University of Science and Technology（香港科技大学）；RIKEN Interdisciplinary Theoretical and Mathematical Sciences (iTHEMS)（理化学研究所交叉理论与数学科学中心）；Institute of Physics, Chinese Academy of Sciences（中国科学院物理研究所）；CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences（中国科学院理论物理研究所理论物理重点实验室）；School of Physics, Beijing Institute of Technology（北京理工大学物理学院）
The quantum Hall system can be used to study many-body physics owing to its multiple internal electronic degrees of freedom and tunability. While quantum phase transitions have been studied intensively, research on the temperature-induced phase transitions of this system is limited. We measured the pure bulk conductivity of a quantum Hall antiferromagnetic state in bilayer graphene over a wide range of temperatures and revealed the two-step phase transition associated with the breaking of the long-range order and short-range antiferromagnetic order. Our findings are fundamental to understandin…
We investigate the electronic confinement in bilayer graphene by topological loops of different shapes. These loops are created by lateral gates acting via gap inversion on the two graphene sheets. For large-area loops the spectrum is well described by a quantization rule depending only on the loop perimeter. For small sizes, the spectrum depends on the loop shape. We find that zero-energy states exhibit a characteristic pattern that strongly depends on the spatial symmetry. We show this by considering loops of higher to lower symmetry (circle, square, rectangle and irregular polygon). Interes…
We consider the orbital effect of an in-plane magnetic field on electrons in bilayer graphene, deriving linear-in-field contributions to the low-energy Hamiltonian arising from the presence of either skew interlayer coupling or interlayer potential asymmetry, the latter being tunable by an external metallic gate. To illustrate the relevance of such terms, we consider the ratchet effect in which a dc current results from the application of an alternating electric field in the presence of an in-plane magnetic field and inversion-symmetry breaking. By comparison with recent experimental observati…
We investigate the effect of eddy currents on ferromagnetic resonance (FMR) in ferromagnet-normal metal (FM/NM) bilayer structures. Eddy-current effects are usually neglected for NM layer thicknesses below the microwave (MW) skin depth (approx. 800 nm for Au at 10 GHz). However, we show that in much thinner NM layers (10-100 nm of Au or Cu) they induce a phase shift in the FMR excitation when the MW driving field has a component perpendicular to the sample plane. This results in a strong asymmetry of the measured absorption lines. In contrast to typical eddy-current effects, the asymmetry is l…
Electron transport in bilayer graphene placed under a perpendicular electric field is revealed experimentally. Steep increase of the resistance is observed under high electric field; however, the resistance does not diverge even at low temperatures. The observed temperature dependence of the conductance consists of two contributions: the thermally activated (TA) conduction and the variable range hopping (VRH) conduction. We find that for the measured electric field range (0 - 1.3 V/nm) the mobility gap extracted from the TA behavior agrees well with the theoretical prediction for the band gap …
Despite the now vast body of two-dimensional materials under study, bilayer graphene remains unique in two ways: it hosts a simultaneously tunable band gap and electron density; and stems from simple fabrication methods. These two advantages underscore why bilayer graphene is critical as a material for optoelectronic applications. In the work that follows, we calculate the one- and two-photon absorption coefficients for degenerate interband absorption in a graphene bilayer hosting an asymmetry gap and adjustable chemical potential--all at finite temperature. Our analysis is comprehensive, char…
In a magnetic field bilayer graphene supports, at the lowest Landau level, eight characteristic zero-energy levels with an extra twofold degeneracy in Landau orbitals $n=\{0,1\}$. They, under general one-body and many-body interactions, evolve into pseudo-zero-mode (PZM) levels. A close look is made into the detailed structure and characteristics of such PZM levels and cyclotron resonance they host, with full account taken of spin splitting, interlayer bias, weak electron-hole asymmetry and Coulomb interactions. It is pointed out that the PZM levels generally undergo orbital level mixing (in o…
The temperature dependence of the relaxation dynamics in the bilayer thin film heterostructure composed of multiferroic BiFeO$_{3}$ (BFO) and superconducting YBa$_{2}$Cu$_{3}$O$_{7}$ (YBCO) grown on a (001) SrTiO$_{3}$ substrate is studied by a time-resolved pump-probe technique, and compared with that of pure YBCO thin film grown under the same growth conditions. The superconductivity of YBCO is found to be retained in the heterostructure. We observe a speeding up of the YBCO recombination dynamics in the superconducting state of the heterostructure, and attribute it to the presence of weak f…
We theoretically study the electronic structure of small-angle twisted bilayer graphene with a large potential asymmetry between the top and bottom layers. We show that the emergent helical states known to appear on the triangular AB-BA domain boundary do not actually form a percolating network, but instead they provide independent, perfect one-dimensional channels propagating in three different directions. Using the continuum-model Hamiltonian, we demonstrate that an applied bias causes two well-defined energy windows which contain sparsely distributed one-dimensional channels. The origin of …
When surface states (SSs) form in topological insulators (TIs), they inherit the properties of bulk bands, including the electron-hole (e-h) asymmetry but with much more profound impacts. Here, via combining magneto-infrared spectroscopy with theoretical analysis, we show that e-h asymmetry significantly modifies the SS electronic structures when interplaying with the quantum confinement effect. Compared to the case without e-h asymmetry, the SSs now bear not only a band asymmetry as that in the bulk but also a shift of the Dirac point relative to the bulk bands and a reduction of the hybridiz…
We demonstrate that AA-stacked bilayer graphene (AA-BLG) encapsulated by dielectric materials can possess an energy gap due to the induced mass term. Using the four-band continuum model, we evaluate transmission and reflection probabilities along with the respective conductance. Considering interlayer mass-term difference opens a gap in the energy spectrum and also couples the two Dirac cones. This cone coupling induces an inter-cone transport that is asymmetric with respect to the normal incidence in the presence of asymmetric mass-term. The energy spectrum of the gapped AA-BLG exhibits elect…
Magnetic state of exchanged biased CoO(20nm)/Co($d_F$) bilayer ($d_F$=5-20nm) was studied by means of polarized neutron reflectometry. By spacing of CoO/Co bilayer and Al$_2$O$_3$ substrate with Nb(20nm) layer we created waveguide structure which allowed us to significantly enhance intensity of spin-flip (SF) scattering in the position of optical resonances. For the trained sample with thinnest Co(5nm) we detected strong SF scattering at the resonance position (up to 30\% of incoming intensity) speaking about high non-collinearity of the system. As $d_F$ increases, the intensity of SF scatteri…
We study the effects of disorder on bilayer graphene using four different microscopic models and directly compare their results. We compute the self-energy, density of states, and optical conductivity in the presence of short-ranged scatterers and screened Coulomb impurities, using both the Born approximation and self-consistent Born approximation for the self-energy. We also include a finite interlayer potential asymmetry which generates a gap between the valence and conduction bands. We find that the qualitative behavior of the two scattering potentials are similar, but that the choice of ap…
We investigate conductance fluctuations as a function of carrier density $n$ and magnetic field in diffusive mesoscopic samples made from monolayer and bilayer graphene. We show that the fluctuations' correlation energy and field, which are functions of the diffusion coefficient, have fundamentally different variations with $n$, illustrating the contrast between massive and massless carriers. The field dependent fluctuations are nearly independent of $n$, but the $n$-dependent fluctuations are not universal and are largest at the charge neutrality point. We also measure the second order conduc…
