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monolayer graphene has attracted great amount of interest due to its novel electronic properties @xcite . it has perfect two - dimensional ( 2d ) geometry and a linear band structure near dirac points . since its discovery , a substantial body of work has been done on charge transport properties in graphene @xcite . electron mobility as high as @xmath0 120,000 @xmath1/v@xmath2s at @xmath0 240 k has been measured in suspended graphene @xcite . at high field bias , epitaxial graphene field effect transistors ( fets ) on sic substrates have exhibited high current drives , exceeding ~ 3 ma/@xmath3 m @xcite . the current carrying capability of graphene , and the limiting scattering mechanisms have been the focus of recent research . the earliest studies of high - field current drive suggested that monolayer graphene should be able to deliver currents in the range of ~ 4 ma/@xmath3 m by considering the intrinsic optical phonon scattering @xcite . shortly thereafter , current saturation was experimentally achieved in fet structures and the saturation current was observed to be proportional to the square root of the channel carrier density @xcite . surface - optical ( so ) phonon scattering rather than intrinsic optical phonon scattering was suggested as the dominant scattering mechanism for current saturation in graphene on sio@xmath4 substrates . the effect of hot phonons on limiting the current drive in graphene was then considered , but found to be masked by other scattering mechanisms for graphene on sio@xmath4 substrate @xcite . subsequent theoretical and experimental work further confirmed that the limiting mechanism for high - field transport in graphene on sio@xmath4 substrates is so phonon scattering @xcite . the extrinsic scattering from sio@xmath4 could in principle be diminished by using different substrates , or by suspending graphene . the saturation current in _ intrinsic _ graphene is still poorly understood . more recently , optical phonon temperatures as high as 1000 k was reported by raman measurements in graphene sheets under high bias @xcite , and the hot phonon effect was observed in experiments @xcite . these findings show the necessity of considering hot phonon effect on the intrinsic high field transport properties of graphene . moreover , although electron - electron ( e - e ) scattering degrades the low field transport properties of graphene @xcite , its effect on high field transport has not been studied yet . as a result , the high - field current carrying capacity of intrinsic graphene , considering the hot phonon effect and e - e scattering , is not clearly understood .    in this work , high field carrier transport in intrinsic graphene is investigated by monte - carlo ( mc ) simulations , by including hot phonon effects as well as e - e scattering . furthermore , transient high - field transport effects , such as velocity overshoot and transit times are investigated . in the next section , the various scattering mechanisms used in mc simulations are described . in the third section , mc simulation results are presented , followed by discussions of the results . in the last section , we draw conclusions and discuss possible future work . at finite temperature , the _ intrinsic _ scattering mechanisms that affect carrier transport are due to electron - phonon interactions . the phonon modes that need to be accounted for are acoustic and optical . the scattering rates are calculated using fermi s golden rule @xcite . the acoustic phonon scattering rate from state @xmath5 to state @xmath6 is given by @xmath7 where @xmath8 ev is the deformation potential of acoustic phonon scattering , @xmath9 is the boltzmann constant , @xmath10 is the temperature , @xmath11 is the kinetic energy of the carrier and @xmath12 is the wavevector , @xmath13 is the reduced planck constant , @xmath14 cm / s is the fermi velocity in graphene , @xmath15 g/@xmath16 is the 2d mass density of graphene , and @xmath17 km / s is the acoustic phonon ( sound ) velocity in graphene . @xmath18 is the angle of electron velocity between initial state @xmath5 and final state @xmath6 . acoustic phonon scattering in graphene is not isotropic , since the acoustic phonon perturbation is long - range , and the wave functions of carriers in graphene have the ` pseudospin ' symmetry . the angle dependent factor for acoustic phonon scattering is @xmath19 , which prevents backscattering . the acoustic phonon scattering rate for electron in state @xmath5 is calculated by summing over all possible finals in @xmath20 space and it is given by @xmath21 where acoustic phonon scattering is treated as quasi - elastic , and both emission and absorption are considered .    the optical phonon scattering rate from state @xmath5 to state @xmath6 is given by @xmath22,\end{aligned}\ ] ] where @xmath23 is the mode - specific optical deformation potential of graphene , @xmath24 is the optical phonon frequency and @xmath25 is the optical phonon number . similarly , the optical phonon scattering rate for state @xmath5 is the summation of all scatterings to final states , and it is given by @xmath26 where @xmath27 is for phonon absorption and emission respectively . in graphene , the zone edge ( ze ) transverse optical ( to ) mode has the strongest coupling with electrons @xcite and the energy of this mode is around @xmath28 mev . the ` optical ' deformation potential of the mode is @xmath29 ev /   @xcite . the scattering of zone - edge mode is non - isotropic and has the angle dependence factor @xmath30 @xcite . forward scattering ( @xmath31 ) is prohibited and back scattering ( @xmath32 ) is preferred due to this factor ; the situation is opposite of what happens in acoustic phonon scattering . in our simulation , we also include the zone center ( zc ) optical phonon . the zc optical phonon has a higher energy , @xmath33 mev . the deformation potential for the zc phonon is @xmath34 ev / @xcite . the zone center optical phonon scattering is isotropic @xcite . = 2.8 in    the effect of e - e scattering in graphene with linear @xmath35 dispersion is different from traditional semiconductors that have parabolic band structure ( @xmath36 ) @xcite . energy and momentum conservation requirements modify the carrier distribution and center of mass motion in a way that the low - field transport properties ( such as carrier mobility ) is strongly damped , especially at low carrier concentrations . the e - e scattering rate of electron at state @xmath5 with another electron at @xmath37 is given by @xmath38 where @xmath39 and @xmath40 are the finals states . the coulomb interaction matrix @xmath41 is given by @xmath42 where wave vector @xmath43 and @xmath44 is the dielectric function of 2d graphene . @xmath45 and @xmath46 are the scattering angles of the two electrons , which has to be taken into account since coulomb scattering is also long range interaction . similarly , wave vector @xmath47 is for the scattering process of exchange of two electrons . the two delta function shows the conservation of momentum and energy in the scattering . the electron scattering rate is calculated by summation of all @xmath48 state and also all the final states ( @xmath39,@xmath49 ) , and it is given by @xmath50 where the occupation of state @xmath48 is the fermi - dirac distribution and the final states are assumed to be empty . the e - e scattering rate is incorporated into the mc simulation in this work following the procedure of ref.@xcite with one difference : the screening of coulombic forces is treated as temperature - dependent @xcite . the scattering rates at 300 k are plotted in fig.[scatteringrates ] . in order to capture high - field effects , the scattering rate from 0 to 1 ev are calculated for all the scattering mechanisms in intrinsic graphene . the intrinsic optical phonon scattering rate increases linearly with energy at states higher than @xmath51 . acoustic phonon scattering rate is one order of magnitude smaller than the optical phonon scattering at high energies . electrons are accelerated by electric field and populate high - energy states . therefore , optical phonon scattering is the dominant energy relaxation channel for hot electrons rather than acoustic phonon . since the ze optical phonon has lower energy and higher scattering rate , it is the dominant optical mode . the e - e scattering rate at 300 k is higher than optical phonon scattering . this scattering thermalizes the carrier distribution , driving it towards a fermi dirac distribution . the net momentum does not change during e - e collisions , but the average velocity of electrons changes due to the linear band structure of 2d graphene . as a consequence , e - e scattering has an impact on the current . the e - e scattering rate increases with carrier temperature so that it become more important under high field .    from fig.[scatteringrates ] , the phonon scattering rate at high energies exceeds 10 ps@xmath52 . optical phonons emitted by electrons have to relax to other phonon modes by anharmonic interaction before the heat propagates into the substrate or contacts . the characteristic lifetime of optical phonon decay into acoustic phonon modes is @xmath01 to 5ps in carbon - based @xmath53 crystals @xcite . since the optical phonon lifetime is much longer than the phonon generation time ( @xmath0 0.1 ps ) , the optical phonon population in graphene is out of equilibrium . the fast generation rate and slow decay rate of optical phonons result in a hot phonon effect in graphene .    before presenting the numerical results of the ensemble mc calculation , a heuristic picture of the high field transport properties in graphene is developed in the next section . a simple model considering the effect of carrier degeneracy provides analytical estimation of the saturation currents for degenerate electron systems . the electron distribution function in the @xmath20- space in the steady state at high electric fields can be approximated since the optical phonon emission is much faster than competing energy relaxation mechanisms . the steady state electron distribution function @xmath54 under this assumption is shown in fig.[mymodel](a ) . the electrons are distributed between two energy contour circles , the high energy circle @xmath55 and the low energy circle @xmath56 . all electrons move toward @xmath57 direction under the applied electric field . once electrons approach the high energy circle on the right , they are scattered back to low energy circle by emitting optical phonon instantaneously . therefore , the energy difference between the two circles is the optical phonon energy , @xmath58 , as indicated in the figure . the state occupation inside the low energy circle is full so that optical phonon emission by electrons with energy smaller than @xmath59 are prohibited by the pauli exclusion principle . as optical phonon emission continues and scatters the electrons from high energy states back to the low energy states , distribution function @xmath54 is squeezed and elongated along the direction of the electric field . the steady distribution function is called ` streaming ' distribution . this streaming model captures the effect of carrier degeneracy , which has not been considered in high field - transport in traditional non - degenerate semiconductors . = 3.6 in    the carrier concentration is proportional to the area of the distribution function @xmath60 in @xmath20 space , @xmath61 , where @xmath62 and @xmath63 are spin and valley degeneracies in graphene . the area of the distribution function is given by @xmath64 from which the dimensions of the distribution function ( @xmath65,@xmath66 ) can be calculated at any carrier concentration . the current flowing in the direction of the electric field corresponding to the distribution function in fig.[mymodel](a ) is then found by summing the @xmath67directed group velocities of all states inside the shaded region in the @xmath20 space ; the value is found to be @xmath68 where @xmath69 is the electron energy at low energy circle . this streaming model shows that the saturation current is proportional to the optical phonon frequency and radius of low energy circle . in fig.[mymodel](b ) , the ensemble saturation velocity @xmath70 is plotted against the electron density . the saturation velocity varies with carrier concentration and is not a constant . it approaches the fermi velocity in graphene @xmath14 cm / s as @xmath71 , and falls to @xmath72 cm / s at @xmath73/@xmath1 . the dotted line shows the closely related square root relation @xmath74 proposed in ref.@xcite . traditionally , the saturation velocity in non - degenerate semiconductors has been considered a constant , e.g. @xmath75 cm / s in si at room temperature @xcite . the reason for the carrier - density dependent saturation velocity in graphene is the degeneracy of carriers and the pauli exclusion principle . close to inverse square - root dependence of the effective ensemble saturation velocity on the carrier density has been measured in graphene on sio@xmath4 substrate @xcite . the model presented here captures the essence of the high - field transport and is in agreement the experimental results , indicating that the carrier degeneracy plays a major role on high field transport in graphene . however , the model here can stand on a firmer footing by using a numerical monte - carlo approach to find high field transport properties , which is the topic of the next section . in the mc implementation of the high - field transport problem , we have considered electron energies ranging from 0 to 10 @xmath76 in the graphene conduction band . the @xmath77space is divided into around @xmath78 cells . in simulation , ` superelectrons ' move in @xmath20 space according to newton s law @xmath79 @xcite . the number of ` superelectrons ' varies from 1000 to be more than @xmath80 , depending on the carrier density . the current is calculated by summation of the group velocities over all the ` superelectrons ' . the screening of the e - e coulombic interaction is dependent on the carrier temperature . in the mc simulation , the electron temperature @xmath81 is updated every time slot ( 0.5 fs ) . the temperature was calculated from the thermal energy of electron gas @xmath82 , where @xmath83 is wavevector of the ith electron , and the average wavevector of all electrons is @xmath84 . the electron temperature was calculated by assuming a fermi - dirac distribution , whereby the thermal energy of the electron gas at temperature @xmath81 is given by @xmath85 where @xmath86 is the area of graphene sheet and @xmath87 . in simulation , we assume the carrier density in graphene conduction band is fixed by ignoring generation processes . as a result , fermi level changes with the temperature , and both of them can be calculated once we know the thermal energy and carrier concentration .    in order to capture the effect of hot phonons , the phonon @xmath88-space is discretized in the same fashion as the electron @xmath89-space , and the hot - phonon effect is treated as described in @xcite . the optical phonon distribution function is updated based on the phonon wavevectors during the emission and absorption events . optical phonon decay into other phonon modes is governed by the rate equation of phonon occupation function @xmath90 - @xmath91 where @xmath92 is the time slot in mc simulation , @xmath93 is the equilibrium bose - einstein distribution , and @xmath94 is the phonon lifetime . the time slot @xmath92 is around 0.5 fs , corresponding to a scattering rate 2@xmath95 ps@xmath52 , which is higher than the summation of all scattering rates for electrons . this time slot was updated according to the phonon population in simulation , since the maximum scattering rate varies with the phonon population . the degeneracy of carriers is explicitly dealt with by the ` rejection technique ' @xcite . the simulation is continued till a steady state distribution function is obtained . this procedure yields the distribution functions of electrons @xmath96 and phonons @xmath97 at every time step , which is then used to find ensemble velocities , currents , as well as transient effects such as velocity overshoot as a function of the carrier density and the applied electric field . = 3.6 in    the steady - state currents calculated by the mc simulation for various strengths of electric fields are shown in figure [ mcresults](a ) for two carrier densities . the curves of different carrier concentrations show similar behavior . the currents increase with the field and saturate at a field @xmath98 kv / cm . the degradation of the saturation current due to the hot - phonon effect is clearly seen by comparing the open and solid symbol lines , which is indicated by the arrows . hot phonon effect is more significant in degradation of the saturation current at higher carrier concentrations . in order to highlight the effect of e - e scattering at high field , the transport without e - e scattering is also simulated . the dot - dash lines are the currents including hot phonon effect but without e - e scatterings . without e - e scattering , the currents drop at high fields , which is the indication of negative differential resistance ( ndr ) . ndr has been proposed and observed in simulations for carbon naotubes ( cnts ) and graphene @xcite , due to the linear dispersion of band structure . in graphene , the electrons at high energy state move at the same speed with the low energy state , which is the constant fermi velocity in graphene . at high field , more electrons move into high energy states . but the increasing of velocity is not as rapidly as conventional parabolic materials . as more and more electrons occupy high energy states the backscattering is increased , which degrades the average velocity of electron gas and leads to ndr . however , this effect disappears ( or weakened ) if e - e scattering is considered . the e - e scatterings exchange electrons momentum and energy , diving electrons into fermi - dirac distribution . as a result , high energy electrons exchange their momentum and energy with low energy electrons before substantial backscatterings happen . with e - e scatterings , the hot electrons generation and backscattering are weakened and the ensemble velocity of carriers increases with the applied electric field . the current - field curves from the mc simulation therefore show that a ) the hot phonon effect acts as a strong current limiting mechanism at high carrier densities and b ) e - e scattering gets rid of ndr effect in graphene .    in fig.[mcresults](b ) , the saturation current is plotted against the carrier density . the solid black curve is the analytical result from streaming model ( equation [ jsatofstreamingmodel ] ) for comparison . the solid squares are the intrinsic saturation currents without the hot phonon effect . the saturation current is slightly lower than the analytical result due to hot phonon degradation . saturation current density of @xmath03 a / mm has been measured in epitaxial graphene on sic substrate ( @xmath99 @xmath100 ) @xcite , which is consistent with the mc result . the current - carrier concentration curves are not linear , indicating the ensemble saturation velocity is not constant , but decreases with increasing carrier density . the hot phonon effect is better illustrated by the distribution functions of electrons in @xmath89-space , and phonons in @xmath88-space . the electron distribution functions are plotted in fig.[distributionfunctons](a ) along the @xmath57 axis which is along the direction of the electric field . the solid line is the electron distribution at steady state , considering hot phonon and e - e scattering . the peak of the distribution function is close to 1 , showing the high degeneracy of carriers in graphene . for comparison , we turned off the e - e scattering or hot phonon effect in the simulation . the distribution function without e - e scattering has a lower peak and is broader , shown by the dot - dash curve . compared to the solid distribution curve , the distribution without e - e scattering has more electrons at high energy states , but also more backscattered electrons . this highlights the role of e - e scattering in maintaining the shape of carrier distribution close to fermi - dirac like ( the solid line ) and thus preventing the ndr effect . similarly , the effect of hot phonons becomes clear when we compare the distribution functions shown by the solid and dashed curve . the carrier distribution with hot phonons is shifted less along the field direction , so the saturation current is lowered by the hot - phonon effect . = 3.6 in    the effect of hot phonons on the transport can also be seen from the phonon distribution in @xmath88-space . fig.[distributionfunctons](b ) shows the 2d phonon distribution functions @xmath101 . the hot phonon distribution has a peak at positive @xmath102 , which corresponds to the fast optical phonon emission by hot electrons . the phonon occupation number peak exceeds @xmath103 which corresponds to a phonon temperature of @xmath104 k.    besides the steady - state saturation currents , the mc simulation procedure also allows us to study transient phenomena in high - field carrier transport . prior to approaching steady state , carriers moving short distances in traditional semiconductors are known to undergo transient velocity overshoot , attaining higher velocities than what can be attained in the steady state . the hot electrons accelerated by high field keep moving in @xmath20 space without significant phonon emission during timescales shorter than the mean phonon emission time . as a result , a velocity peak is achieved in a short period . in high speed electronic devices , the channel length is scaled to tens of nanometers . the carriers travel from source to drain contacts within ps timescales , and velocity overshoot can occur . this transient velocity overshoot can be utilized to lower the transit time of carriers in graphene devices , thus resulting in an enhanced speed . fig.[vdvstime](a ) shows the ensemble transient velocity response to electric fields . velocity overshoot occurs when the electric field is higher than 5 kv / cm . the higher the field , the higher is the overshoot velocity peak ; it approaches @xmath105 cm / s at a field of 50 kv / cm for the chosen carrier density ( @xmath106/@xmath1 ) . after reaching the peak overshoot velocity , the ensemble drift velocity decreases gradually by dissipating excess energy into the graphene crystal in the form of optical phonon vibrations . the long tail of decreasing velocity is due to the hot phonon effect - optical phonons are slow in decaying into acoustic modes which heat up the lattice . velocity overshoot occurs within a time window of @xmath107 ps . if the channel length ( @xmath108 ) is short enough ( @xmath109 ) , the hot carrier energy is dissipated at the drain contact , and the hot phonon effect can be prevented in a short channel device . thus , the distance @xmath110 that carriers travel in the velocity overshoot period is crucial for the design of high - speed devices . the transit times @xmath111 given by @xmath112 is calculated from the result in fig[vdvstime](a ) and shown for various lengths in [ vdvstime](b ) . for short channel lengths , the transit time is sensitive to the magnitude of the electric field , since velocity overshoot increases with electric field . for long channel lengths , a major portion of the transit time is spent traveling at the steady state drift velocity , and the overshoot velocity makes a minor contribution . in the mc simulation , the transit time is not found to be particularly sensitive to the carrier density , which bodes well for devices requiring high currents and high speeds . in a @xmath113 nm long channel device , the transit time is found to be @xmath114 ps , which is exactly the time window for velocity overshoot . thus , graphene channels @xmath115 nm can fully take advantage of velocity overshoot for enhancing their speed .    the peak overshoot velocity , and overshoot time window ( @xmath116 ) is determined by strength of electron - optical phonon interaction , the optical phonon energy , as well as the low - field mobility . the optical phonon energy and carrier - phonon interaction are intrinsic properties of material , but the peak overshoot velocity peak can be improved by increasing the cleanliness of the substrate or by using gate dielectrics with less impurities . we point out some limitations of this work , and suggest topics that need investigation in the future . in this work we have ignored the band to band processes , e.g. inter - band ( zener ) tunneling and e - e scattering induced carrier generation . the carrier concentration does not change at high electric field under the assumption . however , this is not the case in graphene if the fermi level is close to the dirac point . since the band gap is zero , tunneling and e - e scatterings generate electrons and holes under high fields at very low carrier concentrations . in experimental measurements , the current increases and never saturates when the fermi level is close to the dirac point in intrinsic graphene ( @xmath117 ) @xcite . field - induced carrier generation is also the reason of the low on / off ratio ( @xmath118 ) in graphene fets .    in experiments , intrinsic properties of graphene is difficult to achieve . so phonon scattering limits the high field transport for graphene in close proximity to substrates with low energy phonon modes , like sio@xmath4 . suspended graphene is free from environmental scattering effects , but the carrier density is not easy to modulate . one possible way to measure the intrinsic performance is to use sic substrate graphene which has weak so phonon coupling . we assumed the lattice temperature to be at 300 k in our simulations , although the optical phonon temperature can be much higher than environment temperature 300 k. this assumption could overestimate the saturation current , since the lattice temperature in graphene increases due to the limited thermal conductance of the substrate and contacts . the hot phonon effect is dominant , so to a certain extent ignoring the rise of lattice temperature is justified . the phonon lifetime has been measured for graphene on sic substrate and it was found to be around 2.5 ps @xcite . the hot phonon effect can be diminished by introducing isotopic disorder into graphene @xcite . isotopic disorder introduces phonon modes that are localized in real space , and therefore spread in @xmath119-space , and more phonon modes are thus involved in the cooling of carriers . the saturation currents can be improved by this disorder engineered lifetime of optical phonons , with direct consequences of higher saturation currents .    in conclusion , we investigated the high field transport in intrinsic monolayer graphene . mc simulation results show that both hot - phonon effect and e - e scattering have strong impacts on the saturation current . transient velocity overshoot was studied and the short carrier transit time indicates that graphene is well - suited for high speed electronics . the authors acknowledge kristof tahy and prof . eric pop s group for discussions and experimental data on high field transport in graphene , the nsf career and eccs awards , and the src nri mind center for financial support .
transport of carriers in two dimensional graphene at high electric fields is investigated by combining semi - analytical and monte - carlo methods . 
 a semi - analytical high - field transport model based on the high rate of optical phonon emission provides useful estimates of the saturation currents in graphene . for developing a more accurate picture , 
 the non - equilibrium ( hot ) phonon effect and the role of electron - electron scattering were studied using monte carlo simulations . 
 monte carlo simulations indicate that the hot phonon effect plays a dominant role in current saturation , and electron - electron scattering strongly thermalizes the hot carrier population in graphene . 
 we also find that electron - electron scattering removes negative differential resistance in graphene . 
 transient phenomenon as such as velocity overshoot can be used to speed up graphene - based high speed electronic devices by shrinking the channel length below 80 nm if electrostatic control can be exercised in the absence of a band gap . 
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