We study the influence of pentagons, dislocations and other topological defects breaking the sublattice symmetry on the magnetic properties of a graphene lattice in a Hartree Fock mean field scheme. The ground state of the system with a number of vacancies or similar defects belonging to the same sublattice is known to have total spin equal to the number of uncoordinated atoms in the lattice for any value of the Coulomb repulsion U according to the Lieb theorem. We show that the presence of a single pentagonal ring in a large lattice is enough to alter this behavior and a critical value of U is needed to get the polarized ground state. Glide dislocations made of a pentagon-heptagon pair induce more dramatic changes on the lattice and the critical value of U needed to polarize the ground state depends on the density and on the relative position of the defects. We found a region in parameter space where the polarized and unpolarized ground states coexist.
The derived by us earlier electron scattering matrix for the short-range defects in monolayer graphene is applyed to description of the resistivity electron density dependence. It is argued that large charged defect density is unlikely in the suspended graphene and graphene lying above the trench.
We discuss several mechanisms of magnetism and band opening in graphene produced by its hydrogenation and fluorination that can be examined experimentally.
We present a scanning tunneling microscopy (STM) study of native defects in graphene islands grown by ultra-high vacuum (UHV) decomposition of ethylene on Cu(111). We characterize these defects through a survey of their apparent heights, atomic-resolution imaging, and detailed tunneling spectroscopy. Bright defects that occur only in graphene regions are identified as C site point defects in the graphene lattice and are most likely single C vacancies. Dark defect types are observed in both graphene and Cu regions, and are likely point defects in the Cu surface. We also present data showing the…
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 properties of localized vibrational modes associated with structural defects in a sheet of graphene. For the example of the Stone-Wales defects, one- and two-atom vacancies, many-atom linear vacancies, and adatoms in a honeycomb lattice, we demonstrate that the local defect modes are characterized by stable oscillations with the frequencies lying outside the linear frequency bands of an ideal graphene. In the frequency spectral density of thermal oscillations, such localized defect modes lead to the additional peaks from the right side of the frequency band of the ideal sheet of g…
We present detailed Raman studies of graphene deposited on gallium nitride nanowires with different variations in height. Our results show that different density and height of nanowires being in contact with graphene impact graphene properties like roughness, strain and carrier concentration as well as density and type of induced defects. Detailed analysis of Raman spectra of graphene deposited on different nanowire substrates shows that bigger differences in nanowires height increase graphene strain, while higher number of nanowires in contact with graphene locally reduce the strain. Moreover…
The interaction of Stone-Wales topological defects in graphene has been studied through computer simulation. This simulation has revealed configurations of two defects with energies below the energy of a monolayer with two spaced defects. This indicates the attraction between defects and the possibility of the formation of their clusters. The attraction is due to the interference between defect-induced wavy distortions of the structure of the monolayer. In this case, the amplitude of transverse displacement of atoms near a pair of defects reaches 2-3 angstroms. Such a strong deformation of gra…
Raman Spectroscopy is able to probe disorder in graphene through defect-activated peaks. It is of great interest to link these features to the nature of disorder. Here we present a detailed analysis of the Raman spectra of graphene containing different type of defects. We found that the intensity ratio of the D and D' peak is maximum (~ 13) for sp3-defects, it decreases for vacancy-like defects (~ 7) and reaches a minimum for boundaries in graphite (~3.5).
We investigate theoretically the electronic transport properties in narrow graphene ribbons with an adatom-induced defect. It is found that the lowest conductance step of a metallic graphene nanoribbon may develop a dip even down to zero at certain values of the Fermi energy due to the defect. Accompanying the occurrence of the conductance dip, a loop current develops around the defect. We show how the properties of the conductance dip depend on the parameters of the defect, such as the relative position and severity of the defect as well as the width and edges of the graphene ribbons. In part…
Freestanding graphene displays an outstanding resilience to electron irradiation at low electron energies. Point defects in graphene are, however, subject to beam driven dynamics. This means that high resolution micrographs of point defects, which usually require a high electron irradiation dose might not represent the intrinsic defect population. Here, we capture the inital defects formed by ejecting carbon atoms under electron irradiation, by imaging with very low doses and subsequent reconstruction of the frequently occuring defects via a maximum likelihood algorithm.
With its two degenerate valleys at the Fermi level, the band structure of graphene provides the opportunity to develop unconventional electronic applications. Herein, we show that electron and hole quasiparticles in graphene can be filtered according to which valley they occupy without the need to introduce confinement. The proposed valley filter is based on scattering off a recently observed line defect in graphene. Quantum transport calculations show that the line defect is semitransparent and that quasiparticles arriving at the line defect with a high angle of incidence are transmitted with…
Imperfections change essentially the electronic transport properties of graphene. Motivated by a recent experiment reporting on the possible application of graphene as junctions, we study transport properties in graphene-based junctions with single localized defect. We solve the Dirac-Bogoliubov-de-Gennes equation with a single localized defect superconductor-normal(graphene)-superconductor (SNS) junction. We consider the properties of tunneling conductance and Josephson current through an undoped strip of graphene with heavily doped s-wave superconducting electrodes in the dirty limit. We fin…
We study the effect of extended charge defects in electronic transport properties of graphene. Extended defects are ubiquitous in chemically and epitaxially grown graphene samples due to internal strains associated with the lattice mismatch. We show that at low energies these defects interact quite strongly with the 2D Dirac fermions and have an important effect in the DC-conductivity of these materials.
We study the changes in the electronic structure induced by lattice defects in graphene planes. In many cases, lattice distortions give rise to localized states at the Fermi level. Electron-electron interactions lead to the existence of local moments. The RKKY interaction between these moments is always ferromagnetic, due to the semimetallic properties of graphene
We report on the possibility of valley number fractionalization in graphene with a topological defect that is accounted for in Dirac equation by a pseudomagnetic field. The valley number fractionalization is attributable to an imbalance on the number of one particle states in one of the two Dirac points with respect to the other and it is related to the flux of the pseudomagnetic field. We also discuss the analog effect the topological defect might lead in the induced spin polarization of the charge carriers in graphene.
Stone-Wales (SW) defects, analogous to dislocations in crystals, play an important role in mechanical behavior of $sp^2$-bonded carbon based materials. Here, we show using first-principles calculations that a marked anisotropy in the interaction among the SW defects has interesting consequences when such defects are present near the edges of a graphene nano-ribbon: depending on their orientation with respect to edge, they result in compressive or tensile stress, and the former is responsible to depression or warping of the graphene nano-ribbon. Such warping results in delocalization of electro…
We have used scanning tunneling microscopy (STM) to investigate two types of hydrogen defect structures on monolayer graphene supported by hexagonal boron nitride (h-BN) in a gated field-effect transistor configuration. The first H-defect type is created by bombarding graphene with 1-keV ionized hydrogen and is identified as two hydrogen atoms bonded to a graphene vacancy via comparison of experimental data to first-principles calculations. The second type of H defect is identified as dimerized hydrogen and is created by depositing atomic hydrogen having only thermal energy onto a graphene sur…
We study the electronic transport in a graphene-based superconductor-normal(graphene)-superconductor (SNS) junction by use of the Dirac-Bogoliubov-de Gennes equation. We consider the properties of tunneling conductance through an undoped strip of graphene with heavily doped superconducting electrodes in the dirty limit. We find that spectrum of Andreev bound states are modified in the presence of single localized defect in the bulk. The minimum tunneling conductance remains the same and this result doesn't depend on the actual location of the imperfection.
We study the intervalley scattering in defected graphene by low-temperature transport measurements. The scattering rate is strongly suppressed when defects are charged. This finding highlights "screening" of the short-range part of a potential by the long-range part. Experiments on calcium-adsorbed graphene confirm the role of a long-range Coulomb potential. This effect is applicable to other multivalley systems, provided that the charge state of a defect can be electrically tuned. Our result provides a means to electrically control valley relaxation and has important implications in valley dy…
We investigate the combined influence of structural defects and uniaxial longitudinal strain on the electronic transport properties of armchair graphene nanoribbons using the numerical approach based on the semiempirical tight-binding model, the Landauer formalism, and the recursion method for Green functions. We calculate the conductance of graphene nanoribbons in the quantum coherent regime with different types and concentrations of defects. Further, we apply uniform planar tension to the nonideal graphene ribbons with randomly distributed and oriented single and double vacancies and Stone-W…
Motivated by the state of the art method for fabricating high density periodic nanoscale defects in graphene, the structural, mechanical and electronic properties of defect-patterned graphene nanomeshes including diverse morphologies of adatoms and holes are investigated by means of first-principles calculations within density functional theory. It is found that various patterns of adatom groups yield metallic or semimetallic, even semiconducting behavior and specific patterns can be in a magnetic state. Even though the patterns of single adatoms dramatically alter the electronic structure of …
We theoretically investigate the effects of atomic defect related short-range disorders and electron-electron interactions on Anderson type localization and the magnetic properties of hexagonal armchair graphene quantum dots using an extended mean-field Hubbard model. We observe that randomly distributed defects with concentrations between 1-5\% of the total number of atoms leads to localization alongside magnetic puddle-like structures. We show that localization lenght is not affected by magnetization if there is an even distribution of defects between the two sublattices of the honeycomb lat…
In this paper we propose an analytical method to calculate the band structures of graphene-like nanoribbons of the armchair type with arbitrary line defects or uniaxial strains. The model is based on the tight-binding model and the standing wave assumption for the armchair nanoribbons. It gives accurate band results for large supercell systems. Within this method, we analyze different local states near the line defect in the graphene and boron-nitride nanoribbons. We also derive the analytical expression for the local states in a semi-infinite graphene nanoribbon by this method and the transfe…
The characteristics of energy band spectrum of armchair graphene nanoribbons in presence of line defect are analyzed within a simple non-interacting tight-binding framework. In metallic nanoribbons an energy gap may or may not appear in the band spectrum depending on the location of the defect line, while in semiconducting ribbons the gaps are customized, yielding the potential applicabilities of graphene nanoribbons in nanoscale electronic devices. With a more general model, we also investigate two-terminal electron transport using Green's function formalism.
Electrons in graphene have fourfold spin and valley degeneracies owing to the unique bipartite honeycomb lattice and an extremely weak spin-orbit coupling, which can support a series of broken symmetry states. Atomic-scale defects in graphene are expected to lift these degenerate degrees of freedom at the nanoscale, and hence, lead to rich quantum states, highlighting promising directions for spintronics and valleytronics. In this article, we mainly review the recent scanning tunneling microscopy (STM) advances on the spin and/or valley polarized states induced by an individual atomic-scale de…
It is generally believed that a point defect in graphene gives rise to an impurity state at zero energy and causes a sharp peak in the local density of states near the defect site. We revisit the defect problem in graphene and find the general consensus incorrect. By both analytic and numeric methods, we show that the contribution to the local density of states from the impurity state vanishes in the thermodynamic limit. Instead, the pronounced peak of the zero-bias anomaly is a power-law singularity $1/|E|$ from infinite resonant peaks in the low-energy regime induced by the defect. Our findi…
Ab initio calculations indicate that topological-defect networks in graphene display the full variety of single-particle electronic structures, including Dirac-fermion null-gap semiconductors, as well as metallic and semiconducting systems of very low formation energies with respect to a pristine graphene sheet. Corrugation induced by the topological defects further reduces the energy and tends to reduce the density of states at the Fermi level, to widen the gaps, or even to lead to gap opening in some cases where the parent planar geometry is metallic.
Using calculations on defective graphene from first principles, we herein consider the dependence of the properties of the monovacancy of graphene under isotropic strain, with a particular focus on spin moments. At zero strain, the vacancy shows a spin moment of 1.5 $μ_B$ that increases to $\sim$2 $μ_B$ when the graphene is in tension. The changes are more dramatic under compression, in that the vacancy becomes non-magnetic when graphene is compressed more than 2%. This transition is linked to changes in the atomic structure that occurs around vacancies, and is associated with the formation of…
Using scanning tunneling microscopy (STM) and Fourier Transform STM (FT-STM), we have studied a point defect in an epitaxial graphene sample grown on silicon carbide substrate. This analysis allows us to extract the quasiparticle energy dispersion, and to give a first experimental proof of the validity of Fermi liquid theory in graphene for a wide range of energies from -800 meV to +800 meV. We also find evidence of a strong threefold anisotropy in the standing waves generated by the defect. We discuss possible relations between this anisotropy, the chirality of the electrons, and the asymmetr…
Topological line defects in graphene represent an ideal way to produce highly controlled structures with reduced dimensionality that can be used in electronic devices. In this work we propose using extended line defects in graphene to improve nucleobase selectivity in nanopore-based DNA sequencing devices. We use a combination of QM/MM and non-equilibrium Green's functions methods to investigate the conductance modulation, fully accounting for solvent effects. By sampling over a large number of different orientations generated from molecular dynamics simulations, we theoretically demonstrate t…
Transport properties of single-layer graphene with correlated one-dimensional defects are studied using the time-dependent real-space Kubo-Greenwood formalism. Such defects are present in epitaxial graphene, comprising atomic terraces and steps due to the substrate morphology, and in polycrystalline chemically-vapor-deposited (CVD) graphene due to the grain boundaries, composed of a periodic array of dislocations, or quasi-periodic nanoripples originated from the metal substrate. The extended line defects are described by the long-range Lorentzian-type scattering potential. The dc conductivity…
At high phonon temperature, defect-mediated electron-phonon collisions (supercollisions) in graphene allow for larger energy transfer and faster cooling of hot electrons than the normal, momentum-conserving electron-phonon collisions. Disorder also affects the heat flow between electrons and phonons at very low phonon temperature, where the phonon wavelength exceeds the mean free path. In both cases, the cooling rate is predicted to exhibit a characteristic cubic power law dependence on the electron temperature, markedly different from the T^4 dependence predicted for pristine graphene. The im…
We study the spin relaxation in graphene due to magnetic moments induced by defects. We propose and employ in our studies a microscopic model that describes magnetic impurity scattering processes mediated by charge puddles. This model incorporates the spin texture related to the defect-induced state. We calibrate our model parameters using experimentally-inferred values. The results we obtain for the spin relaxation times are in very good agreement with experimental findings. Our study leads to a comprehensive explanation for the short spin relaxation times reported in the experimental literat…
Evolution of electronic inhomogeneities with back-gate voltage in graphene on SiO$_2$ was studied using room temperature scanning tunneling microscopy and spectroscopy. The reversal of local contrast in some places in the STS maps and sharp changes in cross-correlations between topographic and conductance maps, when graphene Fermi energy approaches its Dirac point, are attributed to change in charge-state of interface defects. The spatial correlations in the conductance maps, described by two different length scales and their growth during approach to Dirac point, show a qualitative agreement …
In this article, we report band structure studies of zigzag graphene nanoribbons (ZGNRs) on introducing defects (sp_3 hybridized carbon atoms) in different concentrations at edges by varying the ratio of sp_3 to sp_2 hybridized carbon atoms. On the basis of theoretical analyses, band gap values of ZGNRs are found to be strongly dependent on relative arrangement of sp3 to sp2 hybridized carbon atoms at the edges for a defect concentration; so the findings would greatly help in understanding band gap of nanoribbons for their electronic applications.
In addition to electron charge and spin, novel materials host another degree of freedom, the valley. For a junction composed of valley filter sandwiched by two normal terminals, we focus on the valley efficiency under disorder with two valley filter models based on monolayer and bilayer graphene. Applying the transfer matrix method, valley resolved transmission coefficients are obtained. We find that: i) under weak disorder, when the line defect length is over about $15\rm nm$, it functions as a perfect channel (quantized conductance) and valley filter (totally polarized); ii) in the diffusive…
We investigate the low-energy electronic transport across grain boundaries in graphene ribbons and infinite flakes. Using the recursive Green's function method, we calculate the electronic transmission across different types of grain boundaries in graphene ribbons. We show results for the charge density distribution and the current flow along the ribbon. We study linear defects at various angles with the ribbon direction, as well as overlaps of two monolayer ribbon domains forming a bilayer region. For a class of extended defect lines with periodicity 3, an analytic approach is developed to st…
We investigate the effect of topological defects on the transport properties of a narrow ballistic ribbon of graphene with zigzag edges. Our results show that the longitudinal conductance vanishes at several discrete Fermi energies where the system develops loop orbital electric currents with certain chirality. The chirality depends on the direction of the applied bias voltage and the sign of the local curvature created by the topological defects. This novel quantum blockade phenomenon provides a new way to generate a magnetic moment by an external electric field, which can prove useful in car…
We propose an extensive report on the simulation of electronic transport in 2D graphene in presence of structural defects. Amongst the large variety of such defects in sp$^2$ carbon-based materials, we focus on the Stone-Wales defect and on two divacancy-type reconstructed defects. First, based on ab initio calculations, a tight-binding model is derived to describe the electronic structure of these defects. Then, semiclassical transport properties including the elastic mean free paths, mobilities and conductivities are computed using an order-N real-space Kubo-Greenwood method. A plateau of mi…
We present a density functional study of graphene adhesion on a realistic SiO$_2$ surface taking into account van der Waals (vdW) interactions. The SiO$_2$ substrate is modeled at the local scale by using two main types of surface defects, typical for amorphous silica: the oxygen dangling bond and three-coordinated silicon. The results show that the nature of adhesion between graphene and its substrate is qualitatively dependent on the surface defect type. In particular, the interaction between graphene and silicon-terminated SiO$_2$ originates exclusively from the vdW interaction, whereas the…
Antidot lattices, defined on a two-dimensional electron gas at a semiconductor heterostructure, are a well-studied class of man-made structures with intriguing physical properties. We point out that a closely related system, graphene sheets with regularly spaced holes ("antidots"), should display similar phenomenology, but within a much more favorable energy scale, a consequence of the Dirac fermion nature of the states around the Fermi level. Further, by leaving out some of the holes one can create defect states, or pairs of coupled defect states, which can function as hosts for electron spin…
With nonequilibrium Green's function approach combined with density functional theory, we perform an ab initio calculation to investigate transport properties of graphene nanoribbon junctions self-consistently. Tight-binding approximation is applied to model the zigzag graphene nanoribbon (ZGNR) electrodes, and its validity is confirmed by comparison with GAUSSIAN03 PBC calculation of the same system. The origin of abnormal jump points usually appearing in the transmission spectrum is explained with the detailed tight-binding ZGNR band structure. Transport property of an edge defect ZGNR junct…
We study the low-energy electronic transport across periodic extended defects in graphene. In the continuum low-energy limit, such defects act as infinitesimally thin stripes separating two regions where Dirac Hamiltonian governs the low-energy phenomena. The behavior of these systems is defined by the boundary condition imposed by the defect on the massless Dirac fermions. We demonstrate how this low-energy boundary condition can be computed from the tight-binding model of the defect line. For simplicity we consider defect lines oriented along the zigzag direction, which requires the consider…
Quantum transport properties of disordered graphene with structural defects (Stone-Wales and divacancies) are investigated using a realistic π-π* tight-binding model elaborated from ab initio calculations. Mean free paths and semiclassical conductivities are then computed as a function of the nature and density of defects (using an order-N real-space Kubo-Greenwood method). By increasing of the defect density, the decay of the semiclassical conductivities is predicted to saturate to a minimum value of 4e^2/πh over a large range (plateau) of carrier density (> 0.5 10^{14}cm^{-2}). Additionally,…
We present numerical studies of conduction in graphene nanoribbons with reconstructed edges based on the standard tight-binding model of the graphene and the extended Huckel model of the reconstructed defects. We performed atomic geometry relaxation of individual defects using density functional theory and then explicitly calculated the tight-binding parameters used to model electron transport in graphene with reconstructed edges. The calculated conductances reveal strong backscattering and electron-hole asymmetry depending on the edge and defect type. This is related to an additional defect-i…
Dynamics of the buckled Stone-Wales defect in graphene is studied by means of computer simulation. Thermally activated switching between two degenerate sine-wave-like configurations of the defect is traced in real time. Transition trajectory is found to be rather complex and pass through a multitude of near-planar, wave-like, and irregular configurations. Surprisingly, the switching time fluctuates strongly and can be up to an order of magnitude longer or shorter than the value given by the Arrhenius formula. This is due to a peculiar shape of the potential relief in the neighborhood of sine-w…
Using density functional theory (DFT), Hartree-Fock, exact diagonalization, and numerical renormalization group methods we study the electronic structure of diluted hydrogen atoms chemisorbed on graphene. A comparison between DFT and Hartree-Fock calculations allows us to identify the main characteristics of the magnetic structure of the defect. We use this information to formulate an Anderson-Hubbard model that captures the main physical ingredients of the system, while still allowing a rigorous treatment of the electronic correlations. We find that the large hydrogen-carbon hybridization put…
We perform {\textit ab initio} calculations for the strain-induced formation of non-hexagonal-ring defects in graphene, graphane (planar CH), and graphenol (planar COH). We find that the simplest of such topological defects, the Stone-Wales defect, acts as a seed for strain-induced dissociation and multiplication of topological defects. Through the application of inhomogeneous deformations to graphene, graphane and graphenol with initially small concentrations of pentagonal and heptagonal rings, we obtain several novel stable structures that possess, at the same time, large concentrations of n…
Correct defect quantification in graphene samples is crucial both for fundamental and applied re-search. Raman spectroscopy represents the most widely used tool to identify defects in graphene. However, despite its extreme importance the relation between the Raman features and the amount of defects in multilayered graphene samples has not been experimentally verified. In this study we intentionally created defects in single layer graphene, turbostratic bilayer graphene and Bernal stacked bilayer graphene by oxygen plasma. By employing isotopic labelling, our study reveals substantial differenc…
Here we report a facile method to generate a high density of point defects in graphene on metal foil and show how the point defects affect the electronic structures of graphene layers. Our scanning tunneling microscopy (STM) measurements, complemented by first principle calculations, reveal that the point defects result in both the intervalley and intravalley scattering of graphene. The Fermi velocity is reduced in the vicinity area of the defect due to the enhanced scattering. Additionally, our analysis further points out that periodic point defects can tailor the electronic properties of gra…
We use our theory of periodized discrete elasticity to characterize defects in graphene as the cores of dislocations or groups of dislocations. Earlier numerical implementations of the theory predicted some of the simpler defect groupings observed in subsequent Transmission Electron Microscope experiments. Here we derive the more complicated defect groupings of three or four defect pairs from our theory, show that they correspond to the cores of two pairs of dislocation dipoles and ascertain their stability.
The role of defects in van der Waals heterostructures made of graphene and hexagonal boron nitride (h-BN) is studied by a combination of ab initio and model calculations. Despite the weak van der Waals interaction between layers, defects residing in h-BN, such as carbon impurities and antisite defects, reveal a hybridization with graphene p$_{\rm z}$ states, leading to midgap state formation. The induced midgap states modify the transport properties of graphene and can be reproduced by means of a simple effective tight-binding model. In contrast to carbon defects, it is found that oxygen defec…
We consider electron transport in a planar fermion model containing various types of line defects modelled by $δ$--function pseudopotentials with different matrix coefficients. The transmission probability for electron transport through the defect line is obtained for various types of pseudopotentials. For the schematic model considered that may describe a graphene structure with different types of linear defects, the valley polarization is obtained.
The effect of contact architecture, graphene defect density and metal-semiconductor work function difference on resistivity of metal-graphene contacts have been investigated. An architecture with metal on the bottom of graphene is found to yield resistivities that are lower, by a factor of 4, and most consistent as compared to metal on top of graphene. Growth defects in graphene film were found to further reduce resistivity by a factor of 2. Using a combination of method and metal used, the contact resistivity of graphene has been decreased by a factor of 10 to 1200 +- 250 Ohm-um using Palladi…
The density functional tight binding approach (DFTB) is well adapted for the study of point and line defects in graphene based systems. After briefly reviewing the use of DFTB in this area, we present a comparative study of defect structures, energies and dynamics between DFTB results obtained using the dftb+ code, and density functional results using the localised Gaussian orbital code, AIMPRO. DFTB accurately reproduces structures and energies for a range of point defect structures such as vacancies and Stone-Wales defects in graphene, as well as various unfunctionalised and hydroxylated gra…
A theoretical study is presented on the scattering of graphene surface plasmons by defects in the graphene sheet they propagate in. These defects can be either natural (as domain boundaries, ripples and cracks, among others) or induced by an external gate. The scattering is shown to be governed by an integral equation, derived from a plane wave expansion of the fields, which in general must be solved numerically but it provides useful analytical results for small defects. Two main cases are considered: smooth variations of the graphene conductivity (characterized by a Gaussian conductivity pro…
