Acoustic phonon scattering limited carrier mobility in 2D extrinsic graphene

Abstract

We theoretically calculate the phonon scattering limited electron mobility in extrinsic (i.e. gated or doped with a tunable and finite carrier density) 2D graphene layers as a function of temperature $(T)$ and carrier density $(n)$. We find a temperature dependent phonon-limited resistivity ${\rho }_{ph}(T)$ to be linear in temperature for $T\agt 50 K$ with the room temperature intrinsic mobility reaching values above $10^5$ cm${}^2/Vs$. We comment on the low-temperature Bloch-Grüneisen behavior where ${\rho }_{ph}(T)\sim T^4$ for unscreened electron-phonon coupling.

arXiv comment6 pages, 5 figures

References in corpus (16)

• Electric Field Effect in Atomically Thin Carbon Films
• Experimental Observation of Quantum Hall Effect and Berry's Phase in Graphene
• Giant Intrinsic Carrier Mobilities in Graphene and Its Bilayer
• Dielectric function, screening, and plasmons in 2D graphene
• Carrier transport in 2D graphene layers
• A self-consistent theory for graphene transport
• Measurement of Scattering Rate and Minimum Conductivity in Graphene
• Quantum Hall Ferromagnetism in Graphene
• Electronic transport in graphene: A semi-classical approach including midgap states
• Friedel oscillations, impurity scattering and temperature dependence of resistivity in graphene
• Temperature Dependent Resistivity of Single Wall Carbon Nanotubes
• Transport in chemically doped graphene in the presence of adsorbed molecules
• Voltage and temperature dependencies of conductivity in gated graphene
• Scattering mechanisms and Boltzmann transport in graphene
• Transport and percolation in a low-density high-mobility two-dimensional hole system
• Statistics of random voltage fluctuations and the low-density residual conductivity of graphene

Cited by in corpus (293)

• High-mobility transport anisotropy and linear dichroism in few-layer black phosphorus
• Electronic transport in two dimensional graphene
• Intrinsic and Extrinsic Performance Limits of Graphene Devices on SiO2
• Micrometer-scale ballistic transport in encapsulated graphene at room temperature
• Carbon nanomaterials for electronics, optoelectronics, photovoltaics, and sensing
• Temperature dependent transport in suspended graphene
• Controlling electron-phonon interactions in graphene at ultra high carrier densities
• Thermoelectric and Magnetothermoelectric Transport Measurements of Graphene
• Graphene Schottky diodes: an experimental review of the rectifying graphene/semiconductor heterojunction
• Cleaning Interfaces in Layered Materials Heterostructures
• Substrate limited electron dynamics in graphene
• Carrier scattering, mobilities and electrostatic potential in mono-, bi- and tri-layer graphenes
• Superballistic flow of viscous electron fluid through graphene constrictions
• Graphene-Based Integrated Photonics For Next-Generation Datacom And Telecom
• Hydrodynamics of electrons in graphene
• Phonon scattering dominated electron transport in twisted bilayer graphene
• Limits on electron quality in suspended graphene due to flexural phonons
• Novel effects of strains in graphene and other two dimensional materials
• First-principles calculations of charge carrier mobility and conductivity in bulk semiconductors and two-dimensional materials
• Utilization of a Buffered Dielectric to Achieve High Field-Effect Carrier Mobility in Graphene Transistors
• Imaging viscous flow of the Dirac fluid in graphene
• Ballistic transport exceeding 28 μm in CVD grown graphene
• Effect of high-K dielectrics on charge transport in graphene
• Inelastic Scattering and Current Saturation in Graphene
• Acoustic phonon limited mobility in two-dimensional semiconductors: Deformation potential and piezoelectric scattering in monolayer MoS2 from first principles
• First-principles method for electron-phonon coupling and electron mobility: Applications to 2D materials
• Screening induced temperature dependent transport in 2D graphene
• The role of contact resistance in graphene field-effect devices
• Non-local transport and the hydrodynamic shear viscosity in graphene
• Carrier Recombination and Generation Rates for Intravalley and Intervalley Phonon Scattering in Graphene
• Gate Tunable Quantum Oscillations in Air-Stable and High Mobility Few-Layer Phosphorene Heterostructures
• Electron-phonon heat transfer in monolayer and bilayer graphene
• Coulomb drag
• The electronic thermal conductivity of graphene
• First Principles Analysis of Electron-Phonon Interaction in Graphene
• High-Mobility Few-Layer Graphene Field Effect Transistors Fabricated on Epitaxial Ferroelectric Gate Oxides (Supplementary Information)
• High-Mobility Few-Layer Graphene Field Effect Transistors Fabricated on Epitaxial Ferroelectric Gate Oxides
• Phonon-induced giant linear-in-$T$ resistivity in magic angle twisted bilayer graphene: Ordinary strangeness and exotic superconductivity
• Thermoelectric properties of atomic-thin silicene and germanene nano-structures
• Energy Relaxation of Hot Dirac Fermions in Graphene
• Fluidity Onset in Graphene
• Observation of Low-frequency Interlayer Breathing Modes in Few-layer Black Phosphorus
• Slow imbalance relaxation and thermoelectric transport in graphene
• Enhanced Thermoelectric Power in Graphene: Violation of the Mott Relation By Inelastic Scattering
• Theory of thermopower in 2D graphene
• Electrically tunable transverse magnetic focusing in graphene
• Out-of-plane heat transfer in van der Waals stacks: electron-hyperbolic phonon coupling
• Transport in inhomogeneous quantum critical fluids and in the Dirac fluid in graphene
• Electron-Phonon Interactions and the Intrinsic Electrical Resistivity of Graphene
• Graphene-Ferroelectric Hybrid Structure for Flexible Transparent Electrodes
• The Deposition of High-Quality HfO2 on Graphene and the Effect of Remote Oxide Phonon Scattering
• Hydrodynamics in graphene: Linear-response transport
• Semiconducting-to-metallic photoconductivity crossover and temperature-dependent Drude weight in graphene
• Structural and optical properties of single- and few-layer magnetic semiconductor CrPS4
• Temperature-dependent resistivity of suspended graphene
• Unraveling the acoustic electron-phonon interaction in graphene
• The ultrafast dynamics and conductivity of photoexcited graphene at different Fermi energies
• Hydrodynamic model for electron-hole plasma in graphene
• Phonon-limited resistivity of graphene by first-principle calculations: electron-phonon interactions, strain-induced gauge field and Boltzmann equation
• Cooling of photoexcited carriers in graphene by internal and substrate phonons
• Mobility of 2D materials from first principles in an accurate and automated framework
• Anisotropic carrier mobility of distorted Dirac cones: theory and application
• Diffusive Charge Transport in Graphene on SiO2
• Ballistic Hot Electron Transport in Graphene
• Tunable Carrier Multiplication and Cooling in Graphene
• Flexural mode of graphene on a substrate
• Plasmon losses due to electron-phonon scattering: the case of graphene encapsulated in hexagonal Boron Nitride
• Disorder-induced temperature-dependent transport in graphene: Puddles, impurities, activation, and diffusion
• Theory of charged impurity scattering in two dimensional graphene
• Surface Polar Phonon Dominated Electron Transport in Graphene
• Hot-electron cooling by acoustic and optical phonons in monolayers of MoS${}_2$ and other transition-metal dichalcogenides
• Temperature dependent resistivity in bilayer graphene due to flexural phonons
• Carrier screening, transport, and relaxation in 3D Dirac semimetals
• Acoustic-phonon-mediated superconductivity in rhombohedral trilayer graphene
• Mechanism for current saturation and energy dissipation in graphene transistors
• Hydrodynamic theory of transport in doped graphene
• Intrinsic Electron-Phonon Resistivity in Bi2Se3 in the Topological Regime
• Electron-electron interactions in the conductivity of graphene
• Electron spin relaxation in graphene with random Rashba field: Comparison of D'yakonov-Perel' and Elliott-Yafet--like mechanisms
• Massless Dirac fermions in a laser field as a counterpart of graphene superlattices
• Non-monotonic temperature dependent transport in graphene grown by Chemical Vapor Deposition
• Theory of 2D transport in graphene for correlated disorder
• Electron-phonon scattering in topological insulator thin films
• Excess resistivity in graphene superlattices caused by umklapp electron-electron scattering
• Electron-phonon scattering in topological insulators
• Finite temperature inelastic mean free path and quasiparticle lifetime in graphene
• Ultra-sensitive and Wide Bandwidth Thermal Measurements of Graphene at Low Temperatures
• Acoustic-phonon-mediated superconductivity in Bernal bilayer graphene
• Electron-phonon mediated heat flow in disordered graphene
• Stokes paradox in electronic Fermi liquids
• First-Principles Electron Transport with Phonon Coupling: Large-Scale at Low Cost
• Magnetic noise spectroscopy as a probe of local electronic correlations in two-dimensional systems
• Intrinsic Charge Transport in Stanene: Roles of Bucklings and Electron-Phonon Couplings
• High-quality electrical transport using scalable CVD graphene
• Linear-in-T resistivity in dilute metals: A Fermi liquid perspective
• Energy Flows in Graphene: Hot Carrier Dynamics and Cooling
• Interplay between interferences and electron-electron interactions in epitaxial graphene
• Surface polar optical phonon interaction induced many-body effects and hot-electron relaxation in graphene
• Giant Plasmon Instability in Dual-Grating-Gate Graphene Field-Effect Transistor
• Universal density scaling of disorder-limited low-temperature conductivity in high-mobility two-dimensional systems
• Boundary Scattering in Ballistic Graphene
• Transport measurements in twisted bilayer graphene: Electron-phonon coupling and Landau level crossing
• Density dependent electrical conductivity in suspended graphene: Approaching the Dirac point in transport
• Graphene Transistor Based on Tunable Dirac-Fermion-Optics
• Chirality-dependent phonon-limited resistivity in multiple layers of graphene
• First-principles study of the electrical and lattice thermal transport in monolayer and bilayer graphene
• Universal linear and nonlinear electrodynamics of the Dirac fluid
• Many-body effects and possible superconductivity in the 2D metallic surface states of 3D topological insulators
• Super-Planckian electron cooling in a van der Waals stack
• Coherent Electron-Phonon Coupling in Tailored Quantum Systems
• Disorder by order in graphene
• Phonon bottleneck in graphene-based Josephson junctions at millikelvin temperatures
• Transport coefficients of graphene: Interplay of impurity scattering, Coulomb interaction, and optical phonons
• Electron-phonon scattering from Green's function transport combined with Molecular Dynamics: Applications to mobility predictions
• Microscopic description of intraband absorption in graphene: the occurrence of transient negative differential transmission
• Migdal's theorem and electron-phonon vertex corrections in Dirac materials
• Scattering by flexural phonons in suspended graphene under back gate induced strain
• Dynamics of photoexcited carriers in graphene
• Integrating Functional Oxides with Graphene
• Temperature dependence of the conductivity of graphene on boron nitride
• Non-linear resistivity and heat dissipation in monolayer graphene
• Acoustic-phonon-mediated superconductivity in moiréless graphene multilayers
• Minibands in twisted bilayer graphene probed by magnetic focusing
• Commensurability Oscillations in One-Dimensional Graphene Superlattices
• Gauge phonon dominated resistivity in twisted bilayer graphene near magic angle
• Piezoelectric surface acoustical phonon limited mobility of electrons in graphene on a GaAs substrate
• Disorder From the Bulk Ionic Liquid in Electric Double Layer Transistors
• Flexural phonon scattering induced by electrostatic gating in graphene
• Theoretical analysis of high-field transport in graphene on a substrate
• Electron spin relaxation in graphene from a microscopic approach: Role of electron-electron interaction
• Transport scattering time probed through rf admittance of a graphene capacitor
• Electron spin diffusion and transport in graphene
• Sound waves and resonances in electron-hole plasma
• Measuring Phonon Dispersion and Electron-Phason Coupling in Twisted Bilayer Graphene with a Cryogenic Quantum Twisting Microscope
• Collision dominated, ballistic, and viscous regimes of terahertz plasmonic detection by graphene
• Temperature Dependence of Electric Transport in Few-layer Graphene under Large Charge Doping Induced by Electrochemical Gating
• Two-dimensional electronic transport on the surface of 3D topological insulators
• Strong magnetophonon oscillations in extra-large graphene
• Unusual magnetotransport in twisted bilayer graphene from strain-induced open Fermi surfaces
• Localization of charge carriers in monolayer graphene gradually disordered by ion irradiation
• Hot carrier and hot phonon coupling during ultrafast relaxation of photoexcited electrons in graphene
• Theory of the spontaneous buckling of doped graphene
• Carrier transport theory for twisted bilayer graphene in the metallic regime
• Prediction of group IV-V hexagonal binary monolayers
• Growth and Magnetotransport in Thin Film α-Sn on CdTe
• A DFT study on the mechanical properties of hydrogenated and fluorinated germanene sheets
• Electronic transport in two dimensional Si:P $\delta$-doped layers
• Phonon scattering induced carrier resistivity in twisted double bilayer graphene
• Gate-Voltage Tunability of Plasmons in Single and Multi-layer Graphene Structures: Analytical Description and Concepts for Terahertz Devices
• Weak Localization in Electric-Double-Layer Gated Few-layer Graphene
• Enhanced electron-phonon coupling in doubly aligned hexagonal boron nitride bilayer graphene heterostructure
• Direct probing of phonon mode specific electron-phonon scatterings in two-dimensional semiconductor transition metal dichalcogenides: Symmetry and Berry phase
• Effect of charged impurity correlation on transport in monolayer and bilayer graphene
• Coulomb interaction effects in graphene bilayers: electron-hole pairing and plasmaron formation
• Direct Visualization of Native Defects in Graphite and Their Effect on the Electronic Properties of Bernal-Stacked Bilayer Graphene
• Vertex renormalization in dc conductivity of doped chiral graphene
• Thermoelectric transport properties of borophane
• Quantum optomechanical straight-twin engine
• Extraordinary high room-temperature carrier mobility in graphene-WSe${}_2$ heterostructures
• Raise and collapse of strain-induced pseudo-Landau levels in graphene
• Phonon-limited carrier mobility and resistivity from carbon nanotubes to graphene
• Negative photoconductivity and hot-carrier bolometric detection of terahertz radiation in graphene-phosphorene hybrid structures
• S-shaped current-voltage characteristics of n+-i-n-n+ graphene field-effect transistors due the Coulomb drag of quasi-equilibrium electrons by ballistic electrons
• Wiedemann-Franz law in graphene
• Scaling of graphene field-effect transistors supported on hexagonal boron nitride: radio-frequency stability as a limiting factor
• Hopping-resolved electron-phonon coupling in bilayer graphene
• Disorder induced Dirac-point physics in epitaxial graphene from temperature-dependent magneto-transport measurements
• Coherent charge carrier dynamics in the presence of thermal lattice vibrations
• Lippmann-Schwinger theory for two-dimensional plasmon scattering
• Correlation-Driven Electron-Hole Asymmetry in Graphene Field Effect Devices
• Moiré phonons and impact of electronic symmetry breaking in twisted trilayer graphene
• Microscopic theory of ultrafast dynamics of carriers photoexcited by THz and near-infrared linearly-polarized laser pulses in graphene
• Electron cooling with graphene-insulator-superconductor tunnel junctions and applications to fast bolometry
• Temperature-linear Resistivity in Twisted Double Bilayer Graphene
• Gate tunable quantum transport in double layer graphene
• Magnetophonon spectroscopy of Dirac Fermion scattering by transverse and longitudinal acoustic phonons in graphene
• Strange metallicity of moiré twisted bilayer graphene
• Interplay between in-plane and flexural phonons in electronic transport of two-dimensional semiconductors
• Observation of superconductivity in bilayer graphene/hexagonal boron nitride superlattices
• Phonon limited thermoelectric transport in Pb
• Magnetoresistance of disordered graphene: from low to high temperatures
• Phonon-mediated room-temperature quantum Hall transport in graphene
• Long-Range Propagation and Interference of $d$-wave Superconducting Pairs in Graphene
• Current distribution in a slit connecting two graphene half-planes
• Electrically-driven amplification of terahertz acoustic waves in graphene
• Nonequilibrium mesoscopic conductance fluctuations as the origin of 1/f noise in epitaxial graphene
• Interplay between phonon and impurity scattering in 2D hole transport
• Impurity scattering induced carrier transport in twisted bilayer graphene
• Room-temperature quantum transport signatures in graphene/LaAlO3/SrTiO3 heterostructures
• Tuning the scattering mechanism in three-dimensional Dirac semimetal Cd${}_3$As${}_2$
• Inelastic vibrational signals in electron transport across graphene nanoconstrictions
• Ab initio electron dynamics in high electric fields: accurate predictions of velocity-field curves
• Superballistic conduction in hydrodynamic antidot graphene superlattices
• Theory of phonon side jump contribution in Anomalous Hall transport
• Electron spin relaxation in bilayer graphene
• Effect of uniaxial strain on the optical Drude scattering in graphene
• Planckian properties of 2D semiconductor systems
• Nonlinear magnetotransport in a dc-current-biased graphene
• Dephasing time in graphene due to interaction with flexural phonons
• Electron phonon coupling in the topological heavy fermion model of twisted bilayer graphene
• Phonon dispersion and electron-phonon interaction in peanut-shaped fullerene polymers
• Infrared spectroscopic study of carrier scattering in gated CVD graphene
• Inelastic Phonon Scattering in Graphene FETs
• Nonequilibrium Excitations and Transport of Dirac Electrons in Electric-Field-Driven Graphene
• Resonant plasmonic detection of terahertz radiation in field-effect transistors with the graphene channel and the black-As${}_x$P${}_{1-x}$ gate layer
• Anomalous charge transport of superconducting Cu${}_x$PdTe${}_2$ under high pressure
• Effect of electron thermal conductivity on resonant plasmonic detection in the metal/black-AsP/graphene FET terahertz hot-electron bolometers
• Extended linear-in-$T$ resistivity due to electron-phason scattering in moiré superlattices
• Enhanced exciton-exciton collisions in an ultra-flat monolayer MoSe2 prepared through deterministic flattening
• Inelastic carrier lifetime in a coupled graphene electron-phonon system: Role of plasmon-phonon coupling
• Parallel transport and layer-resolved thermodynamic measurements in twisted bilayer graphene
• Quantum Lifetime Spectroscopy and Magnetotunneling in Double Bilayer Graphene Heterostructures
• A Fitting Model for Asymmetric I-V Characteristics of Graphene Field-Effect Transistors for Extraction of Intrinsic Mobilities
• Role of remote interfacial phonons in the resistivity of graphene
• Electron spin relaxation in rippled graphene with low mobilities
• Infrared study of carrier scattering mechanism in ion-gated graphene
• Specular electron focusing between gate-defined quantum point contacts in bilayer graphene
• Optical Transparency in an effective model for Graphene
• Epitaxial graphene morphologies probed by weak (anti)-localization
• Two-Dimensional Graphene: Theoretical Study of Multi-photon Non-linear Absorption Coefficient of a Strong Electromagnetic Wave by Using Quantum Kinetic Equation
• A ballistic electron source with magnetically-controlled valley polarization in bilayer graphene
• Transport evidence of superlattice Dirac cones in graphene monolayer on twisted boron nitride substrate
• Full consideration of acoustic phonon scatterings in two-dimensional Dirac materials
• Bogolon-mediated electron scattering in graphene in hybrid Bose-Fermi systems
• Theoretical study of scattering in graphene ribbons in the presence of structural and atomistic edge roughness
• Role of acoustic phonons in frequency dependent thermal conductivity of graphene
• Inelastic scattering and cooling of photoexcited electrons through coupling with acoustic, optic and surface polar optic phonons in graphene
• Phonon-limited resistivity of multilayer graphene systems
• ${\sigma }_h$ symmetry and electron-phonon interaction in two-dimensional crystalline systems
• Acoustic phonon contribution to the resistivity of twisted bilayer graphene
• Role of acoustic phonons in exotic conductivity of two-dimensional Dirac electrons
• Bloch-Grüneisen temperature and universal scaling of normalized resistivity in doped graphene revisited
• Relationship between conductance fluctuation and weak localization in graphene
• Optical Conductivity in an effective model for Graphene: Finite temperature corrections
• Wiedemann-Franz law in graphene in the presence of a weak magnetic field
• Power loss of hot Dirac fermions in silicene and its near equivalence with graphene
• Nonlinear vs. bolometric radiation response and phonon thermal conductance in graphene-superconductor junctions
• Theory of the effective Seebeck coefficient for photoexcited 2D materials: the case of graphene
• Enhanced photoenergy harvesting and extreme Thomson effect in hydrodynamic electronic systems
• Emergence of Fourier's law of heat transport in quantum electron systems
• Quantum-phase two-dimensional materials
• An analytic virtual-source-based current-voltage model for ultra-thin black phosphorus field-effect transistors
• Phonon-Driven Electron Scattering and Magnetothermoelectric Effect in Two-Dimensional Tin Selenide
• Phonon-limited valley life times in single-particle bilayer graphene quantum dots
• Quasiclassical nonlinear plasmon resonance in graphene
• Temperature dependence of universal conductance fluctuation due to development of weak localization in graphene
• Photoconductivity calculations of bilayer graphene from first principles and deformation-potential approach
• Anomalous thermodiffusion of electrons in graphene
• Hall viscosity and hydrodynamic inverse Nernst effect in graphene
• Remote Surface Optical Phonon Scattering in Ferroelectric Ba${}_{0.6}$Sr${}_{0.4}$TiO${}_3$ Gated Graphene
• Pseudospin entanglement and Bell test in graphene
• Ballistic magnetotransport in graphene
• Attractive polaron formed in doped nonchiral/chiral parabolic system within ladder approximation
• Universal Scaling in Intrinsic Resistivity of Two-Dimensional Metal Borophene
• Hysteresis-Free High Mobility Graphene Encapsulated in Tungsten Disulfide
• Thermal-radiation-induced nonequilibrium carriers in an intrinsic graphene
• Koshino-Taylor effect in graphene
• Sub-Kelvin Lateral Thermal Transport in Diffusive Graphene
• Radiation-induced magnetoresistance oscillations with massive Dirac fermions
• Second and third harmonics generation by coherent sub-THz radiation at induced Lifshitz transitions in gapped bilayer graphene
• Non-equilibrium electron relaxation in Graphene
• The role of vector potential coupling in hot electron cooling power in bilayer graphene at low temperature
• Ambipolar Electric Field Effect in Metallic Bi2Se3
• Voltage Tunable Plasmon Propagation in Dual Gated Bilayer Graphene
• Intrinsic and extrinsic effects on intraband optical conductivity of hot carriers in photoexcited graphene
• Tunable spectral narrowing enabling the functionality of graphene qubit circuits at room temperature
• The kinetic theory of ultra-subsonic fermion systems and applications to flat band magic angle twisted bilayer graphene
• Fractal non-Fermi liquids from moiré-Hofstadter phonons
• Modeling charge relaxation in graphene quantum dots induced by electron-phonon interaction
• Signatures of electronic ordering in transport in graphene flat bands
• Acoustic phonon-limited resistivity in spin-orbit coupled 2DEG: Deformation potential and piezoelectric scattering
• Charge carrier mobility degradation in graphene sheet under induced strain
• Unusual spin-triplet superconductivity in monolayer graphene
• Phonon-limited transport coefficients in extrinsic graphene
• Performance limits due to thermal transport in graphene single-photon bolometers
• Enhanced Terahertz Photoresponse via Acoustic Plasmon Cavity Resonances in Scalable Graphene
• Graphene-based Bolometers
• Robust $T$-Linear Resistivity due to SU(4) Valley + Spin Fluctuation Mechanism in Magic Angle Twisted Bilayer Graphene
• On the Wiedemann-Franz law violation in Graphene and quark-gluon plasma systems
• Electron viscosity and device-dependent variability in four-probe electrical transport in ultra-clean graphene field-effect transistors
• Hexagonal boron nitride/bilayer graphene moiré superlattices in the Dirac-material family: energy-band engineering and carrier doping by dual gating
• Electric field tunable bands in doubly aligned bilayer graphene hBN moire superlattice
• Negative intercept of the apparent zero-temperature extrapolated linear-in-$T$ metallic resistivity
• Graphene resistivity in diffusive limit due to scalar and vector potential electron-phonon scattering
• Resonant optical gating of suspended carbon nanotube transistor
• Induced by coherent THz-radiation high harmonics generation in bilayer graphene at high Fermi energies
• The effect of bilayer domains on electronic transport properties of epitaxial graphene on SiC
• Optical conductivity of a Dirac-Fermi liquid
• Mott and Wiedemann Franz Law for Monolayer Graphene for Different Scattering Mechanisms
• Non-Adhesive Transfer Process of Carbon Nanotube Forests onto Flexible Kapton Substrates
• Path-Integral Approach to Quantum Acoustics
• Hall field-induced magneto-oscillations near charge neutrality point in graphene
• Theory and computation of Hall scattering factor in graphene