Inelastic Phonon Scattering in Graphene FETs

Transcription

1 Inelastic Phonon Scattering in Graphene FETs Jyotsna Chauhan and Jing Guo Department o Electrical and Computer Engineering University o Florida, Gainesville, FL, ABSTRACT Inelastic phonon scattering in graphene ield-eect transistors (FETs) is studied by numerically solving the Boltzmann transport equation in three dimensional real and phase spaces (x, k x, k y ). A kink behavior due to ambipolar transport agreeing with experiments is observed. While low ield behavior has previously been mostly attributed to elastic impurity scattering in earlier studies, it is ound in the study that even low ield mobility is aected by inelastic phonon scattering in recent graphene FET experiments reporting high mobilities. As the FET is biased in the saturation regime, the average carrier injection velocity at the source end o the device is ound to remain almost constant with regard to the applied gate voltage over a wide voltage range, which results in signiicantly improved transistor linearity compared to what a simpler model would predict. Physical mechanisms or good linearity are explained, showing the potential o graphene FETs or analogue electronics applications. 1

2 I. Introduction Graphene 1-8 has been one o the most rigorously studied research materials since its inception in There has been a lot o study ocused on low electric ield transport properties o graphene Many issues related to high ield transport properties in graphene ield-eect transistors(fets), however, still remain unclear. Experimental and theoretical studies have shown that even though being a gapless semi metallic material, a graphene FET shows saturating I-V behaviors attributed to inelastic surace polar phonon scattering induced by gate oxide. In this work, inelastic phonon scattering in graphene transistors is studied by numerically solving the Boltzmann transport equation (BTE) in the real and phase spaces. Modeling o inelastic surace polar phonon scattering reveals that low bias mobility is also controlled by inelastic surace polar phonon scattering in addition to elastic scattering. Good linearity is observed in high quality FETs in agreement with recent experiments and it is explained by average carrier injection velocity at the source end remaining nearly constant with the increase in applied gate bias voltage, which makes them desirable or analogue applications. Comparisons to previously reported simpler models clariy validity o those models. To describe semiclassical transport behaviors o graphene FETs at a channel length that quantum mechanical Klein tunneling is not important, a semiclassical Boltzmann transport equation is solved sel consistently with Poisson equation. Though Monte Carlo method and numerical solutions o solving BTE have been implemented in previous studies 17-21, they are limited to two-dimensional k space, which assumes a homogeneous material and thereore it has limitations to describe transport properties and interplay o sel-consistent electrostatics 2

3 and transport in a graphene transistor accurately as discussed in detail later. II. Simulation Approach Top gated graphene FETs as shown in Fig.1 were simulated. The nominal device has a top gate insulator thickness o t ins =10 nm and dielectric constant o ε=3, which results in a gate insulator capacitance o C ins =270nF/cm 2 close to the value in a recent experiment 14. The dielectric constant used here is close to that o hexagonal Boron Nitride (BN), which has been explored as one o the promising gate insulators 22 or graphene FETs. To capture ambipolar transport properties in graphene, we considered the transport in conduction band as well as valence band. A simple linear E-k relation is used, which is valid in the energy range o interest, F 2 x E k k, (1) 2 y where 7 F cm/s is the Fermi velocity in graphene 23. The Boltzmann transport equation (BTE) is solved or a two-dimensional graphene device at the steady state. The Boltzmann Transport equations is given as 24 : where F t k ext k vr t collision is the distribution unction,, (2) t collision is the collision term, F ext is the orce on carrier due to electric ield, v is the group velocity o a particular sub band and is the reduced Planck s constant. Thus BTE involves a collective space containing three dimensions in real space and three dimensions in k or momentum space. However, since graphene is two dimensional and wide enough giving translational symmetry in the y direction, the problem can be easily reduced to three dimensional space o x, k x, k y only. Hence only three 3

4 coordinates and time are needed to speciy the system and BTE or graphene in three dimensional space takes the ollowing orm: F t ext kx vxx t collision F Time evolution o electronic states is described by terms, (3) ext kx vxx. This operator is discretized on inite dierence grid resulting in matrix operator. Backward or orward dierence method is used depending on the direction o lux o carriers. For a charge carrier moving rom let to right with positive velocity vxx operator is approximated by backward dierence scheme and vice versa or charge carriers moving rom right to let with negative velocity. Thus vxx in inite dierence scheme or the carrier at (x i,k xj,k yk ) node is given as: v j x = i,j,k i 1,j,k v j x i v j > 0 = i+1,j,k i,j,k v j x i v j < 0, (4) where v j = k x j k 2 xj +k 2 y k is the velocity o carriers in the transport x direction at (k xj,k yk ) node in the discretization grid and i j, k, is the distribution unction at (x i,k xj,k yk ) node. The F ext kx term includes discretization in k x space because electric ield has only the x component. So dierentiation with respect to k y is eliminated. F ext q is the orce on carrier in presence o electric ield. For >0, electrons experience orce rom right to let and orward dierence scheme is used and vice versa or <0, electrons experience orce rom let to right and dierential operator is approximated using orward dierence scheme. 4

5 Thus or electrons, the equation is solved as : F ext ħ k xj = qℇ ħ k x = i,j 1,k i,j,k qℇ i ħ k xj i ℇ i < 0 = i,j,k i,j +1,k qℇ i ħ k xj i ℇ i > 0, (5) where i j, k, is the distribution unction at (x i,k xj,k yk ) node and ℇ i is the electric ield at x i position along the channel direction. Alternate methods o discretization o dierential operators can also be used to simpliy dierential operators involved. Ater the discretization at all nodes o phase space, the dierential operators in Eq. (3) takes the orm: F ext kx vxx U n B, (6) where U is matrix equivalent or dierential operator in x and k x. The B incorporates the boundary conditions due to x and is physically equivalent to inlux o carriers at right and let contacts. At the let contact i.e. source contact, only positive moving carriers are responsible or inlux into the device, thereore, the boundary condition at x=0 holds true or v x 0 only. Thereby, let contact couples to all the k x, k y nodes at x=0 and carriers are distributed according to source ermi potential. 1 b( x 0, k) or v 0, (7) X 1 (exp(e(k) E (1) ) / k T)) c L Similarly, right contact couples to all k x, k y nodes at x=l or v x 0 and carriers in these states are distributed according to drain ermi level. 1, k) 1 (exp(e(k) E ( Nx) )/ k b( x L or v ch X T)) c R B B 0, (8) where and L are the source and drain ermi levels, Ec( 1) and Ec( N ) are the R x 5

6 conduction band edges at x=0 and x=l respectively and E(k) is energy at every k x, k y node corresponding to x being considered in both conduction and valence band. For kx operator, a periodic boundary condition is used. Physically, periodic boundary conditions ensures conservation o particles and doesn t change B. t collision term representing collision integral is discretized and results in scattering vector C^. t where state k ' collision k ' S( k, k') ( k) 1 k is the distribution unction, Sk, k ' k '. Thus term -S( k, k ') ( k ) 1 ( k ') k ' S( k ', k ) ( k ') 1 ( k ) ( k') S( k', k) ( k') 1 ( k) k ', (9) is the scattering rate rom k state to new represents the out scattering rate and term represents the in scattering rate. As already reported in earlier studies, low ield mobility is controlled by elastic phonon scattering while saturation behavior is largely dominated by inelastic surace polar phonon scattering. So as part o study, elastic phonon scattering is modeled with as the itting parameter, where is the mean ree path (mp) o carrier in presence o elastic scattering. The scattering rate or elastic scattering is calculated as: F E( k) S( k, k'), (10) where E(k ) is the energy o initial state as there is no exchange o energy due to elastic scattering so energy remains the same and λ is used as itting parameter. The surace polar phonon scattering rate is calculated using Fermi s golden rule 17,25,26. For inelastic surace polar phonon emission process, the electron in conduction band can jump to valence band or stay in conduction band while the electron in valence band stays 6

7 only in valence band within modeled energy range as shown in Fig.2(a). In case the electron in valence band jumps outside the considered energy range by emitting a phonon the transition is prohibited. For surace polar phonon absorption process, the electron in conduction band stays within conduction band. In case, transition results in inal energy outside the modeled range, it is prohibited. However, the electron in valence band on absorption can stay in valence band or jump to conduction band as shown in Fig.2(b). Isotropic scattering is assumed or both elastic as well as surace polar phonon scattering. Hence, E-k space is a constant energy sphere and all states lying on constant energy sphere are equally probable. This leads to the assumption that initial state k is coupled to all the states lying on constant energy sphere E(k ') scattering rate rom k to E(k '). k ' with an equal scattering rate. The total is thus, distributed equally among all the states satisying Ater including scattering and all discretization terms the inal BTE can be written as: { n1 } { dt n } U n B ^ C, (11) n1 Where { } and { n } are the distribution unctions at n+1 and n time steps. To reduce computational cost, the device is assumed to be in steady state such that be approximated as zero and equation urther simpliies as: { } { dt n 1 n } can ^ n U B C 0, (12) To start with the simulation, initial guess o distribution unction 0 is calculated assuming ballistic transport conditions which gives 7

8 0 1 U B, (13) Thus, 0 is the equilibrium ballistic distribution unction. The BTE is then solved sel consistently with non linear Poisson equation using Newton Raphson method. III. Results and Discussion The study analyzes the characteristic o the graphene FETs on a micrometer channel length regime. The simulation captures the ambipolar I D -V D behavior observed in graphene FETs which single band Monte Carlo simulations ails to capture. The rest o section addresses the eect o elastic scattering and inelastic surace polar phonon scattering on the transport behavior o graphene FETs. The last section explains the nearly constant transconductance observed experimentally in graphene FETs which makes it good potential or analogue applications. Comparison to previously developed simpler models clariies the validity o these models. We start by simulating I D -V D characteristics o graphene FETs. Figure 1 shows the schematic o the device used in our simulations. Top gated graphene FET with two dimensional graphene as the channel is simulated. A linear E-k or graphene is assumed in two band simulations. The graphene channel is wide enough so that there exists translational symmetry in the y direction, thus helping to reduce the phase space in BTE to (k x, k y, x). The device is simulated or C ins =270 nf/cm 2 with Boron Nitride(BN) as the dielectric which has been predicted to be one o the promising dielectrics or graphene FETs in recent experiments 14,22. All the simulations have been done or channel length L ch= 1µm. The current I D vs. drain voltage V D relation is studied or 1µm device. The I D - V D 8

9 curve in Fig.3(a) shows the characteristic kink due to dirac point entering the drain, thereby leading to ambipolar transport behavior in device. This behavior is in agreement with experimental results 13. Figure 3(b) and Figure 3(c) explains the onset o ambipolar transport near kink drain voltage by plotting electron ( let side plot ) and hole (right side plot) distribution unction near drain end o the channel at V D = 0.25V (beore the kink) and V D =0.7V (ater the kink) respectively. As shown in Fig.3(b) at V D =0.25V beore the onset o kink, the electron distribution unction is quite dominant while hole distribution unction is almost close to zero. The current at this point is thus carried largely by electrons and contribution due to holes to the total current can be neglected. However as shown in Fig.3(c) at V D =0.7V ater the kink, both electron and hole distribution unctions become equally prominent. This explains the ambipolar behavior in the region ollowing the kink. The eect o scattering on mobility is studied next. It has already been observed that elastic impurity scattering and inelastic surace polar phonon scattering due to gate dielectric has been dominant scattering mechanisms in controlling the transport behavior in graphene. The low ield behavior has been attributed to elastic scattering in many earlier studies 10-12,20. Figure 4(a) shows the low ield conductance in presence o elastic scattering. The conductance varies almost linearly with increase in λ, which is used as a itting parameter in our simulations. Figure 4(b) shows the mobility or dierent values o λ as a unction o V G. The mobility increases with the increase in value o λ. The mobility remains almost constant with increase in gate overdrive voltage because low bias conductance or elastic scattering 2 varies linearly with gate overdrive voltage i.e. G E ( V V ). Then we plotted the conductance and mobility values in presence o inelastic surace polar phonon scattering or G T 9

10 dierent Froehlich coupling constants (material parameter) at =40meV with elastic scattering completely turned o. It is observed that low bias conductance as shown in Fig.5(a) and mobility as shown in Fig.5(b), calculated in presence o inelastic surace polar phonon scattering are strongly inluenced by the inelastic surace polar phonon scattering in graphene. Thus even at low bias transport, considering inluence o surace polar phonon scattering is quite important as the values o conductance and mobility in presence o inelastic surace polar phonon scattering are closer to the values calculated or elastic scattering. Hence, in a device when both mechanisms are present we get the mobility behavior controlled by both scattering mechanisms as per Matthiessen's rule. Figure 6(a) shows the simulated I D and V D results. In the simulation, the elastic scattering is turned o by choosing large value o elastic scattering mean ree path. It indicates the device I-V characteristics in the presence o only inelastic phonon scattering. We also perormed simulations by turning on the elastic scattering. I the elastic scattering mp is limited by acoustic phonon scattering, which is considerably longer than the scattering mp by inelastic phonons modeled here, the I-V characteristics shows negligible dierence as shown in Fig. 6(a). The I D -V D characteristics or λ elastic =2µm and 200µm deined at the electron energy o E=0.1eV or elastic scattering remain almost same. The above condition requires a high quality graphene transistor where elastic deect scattering and screened charge impurity scattering are weak. Figure 6(b) plots the average carrier injection velocity (dashed line) at the beginning o channel i.e. near the source end, as a unction o the gate voltage V G, which is ound to remain approximately constant. It is also interesting to compare the detailed numerical 10

11 simulations to previously developed simple models or saturation velocity. The simulated behavior is in contrast to saturation velocity (line with diamonds) behavior which states that 13 v sat v, (14) E and thereby decreases with increase in V G. The dierence is due to two reasons. First, the application o the above equation requires E to be signiicantly larger than, which is not satisied here. Second, the average carrier velocity does not ully reach the saturation velocity value beore the turn-on o ambipolar transport current, especially at low gate overdrive voltages. We also plotted the average carrier velocity (line with crosses) or homogenous graphene material by solving BTE in just two dimensional space (k x, k y ) using same operating conditions as calculated at the source end o the device in previous simulations. It is ound that results or the graphene FET simulations agree with homogenous material simulations within 16 %. In running all the simulations, it is ound that the device does not completely reach saturation regime beore turn on o ambipolar transport. To illustrate this point, we also plotted the saturation velocity (line with squares) calculated by solving BTE in two dimensional space (k x, k y ) or the homogenous material to see how much oset is the device rom saturation regime. The saturation velocity plotted above is the highest velocity achieved by homogenous material 19. It was observed that average carrier injection velocity o carriers at source end (dashed line) is below the saturation velocity (line with squares) o the homogenous material. Hence, it conirms that the device doesn t ully reach velocity saturation beore the turn on o ambipolar transport. In order to investigate the linearity o the transistor, we plotted the transconductance as a 11

12 unction o the gate voltage as shown in Fig.6(c). The line with asterisks, which shows the numerical BTE solution, indicates that the transconductance is nearly constant over a wide applied gate voltage range, indicating excellent linearity o graphene transistors in the saturation drain current regime. On the other hand, a simpler model without numerical solution o the BTE would have predicted much worse linearity, as shown by the line with squares curve in Fig.6(c), which plots g m Cgvsat Cgv as unction o gate voltage. E The simpler model (line with squares) indicates that the transconductance increases as the gate voltage decreases, mostly because the carrier saturation velocity is inversely proportional to Fermi energy level. However, almost constant injection velocity and g m is observed in the device. This is due to the act that device doesn t reach complete saturation beore turn on o ambipolar transport. Figure 6(d) shows the carrier velocity at various gate bias points or the homogenous graphene by solving BTE in two dimensional k space (k x and k y ). On the same plot, the velocity at the source end o the device at V G =1.0V,1.5V,2.0V and 2.5V is highlighted with bold marker points. As can be seen the device is well below the saturation regime and the carrier injection velocity at the source end o the device over the gate bias rom 1.0 to 2.5 V remains almost constant. Since g m = C g v injection, it also remains constant with V G as seen in Fig. 6(c). A numerical solution o the BTE as described in this work, thereore, is necessary or obtaining much accurate prediction o transistor I-V characteristics and clearly explaining excellent linearity observed in recent high-quality experimental devices with BN insulator. High Frequency behavior o graphene FETs is also studied using quasi static treatment 27,28. The intrinsic gate capacitance, C g and the transconductance, g m, are computed 12

13 by calculating derivatives o the charge in the channel and the drain current numerically at slightly dierent gate voltages, C Q ch g, VG V D g I D m, (15) VG V D The intrinsic cut-o requency is computed as, T = 1 g m 2π C g (16) The intrinsic cut o requency T (line with asterisks) is computed by running sel consistent simulations at V D =0.5V and V G =1.1 V. The simulated T increases with increase in phonon energies as maniested in Fig.7. This is due to increase in saturation current with the increase in surace polar phonon energy. However, the increase is not linear. Also, T = 1 <v>, (17) 2π L c where v is the average velocity o carriers, L ch is the length o the channel. I we input v vsat v assuming the device is in saturation, we get E T = 1 v ħω, (18) 2π L c E where 7 F is the Fermi velocity in graphene, is the surace polar phonon energy and E is the Fermi level in graphene depending on the carrier density. The intrinsic cut o requency T calculated using the above equation (with squares) is also plotted in Fig.7 and shows a linear increase with surace polar phonon energy due to inverse dependence on E. The dierence is due to the act that as increases, Eq. (14) breaks down and the carrier velocity does not increase as ast as a proportional relation. For =20meV, the Eq.(14) remains valid and the value calculated is almost equal to simulated value o T.. To summarize, i an applied gate voltage results in a source Fermi Level E with 13

14 reerence to the Dirac point at the source end o the channel, then increasing phonon energy leads to increase o intrinsic cut o requency T i E > which indicates the possibility o improving the high requency perormance by phonon engineering. However, i becomes comparable or larger than E, the T becomes insensitive to phonon energy. IV. Conclusions In Summary, we study the characteristics o graphene FETs in presence o elastic scattering and surace polar phonon by running two band, sel consistent simulations o Boltzmann transport equation. A characteristic kink behavior already seen in experiments is observed in simulations. It is ound that low bias transport regime is aected by both elastic as well as inelastic phonon scatterings. The average velocity at source end remains constant with gate bias as the device doesn t reach ull velocity saturation beore turn on o ambipolar transport which explains or the nearly constant transconductance seen experimentally. A simple relation, v sat v, tends to overestimate the saturation velocity as well as intrinsic cut E o requency o the graphene FETs under certain bias conditions. The intrinsic cut o requency T can be improved by phonon engineering i the condition E > is satisied. However, numerical Boltzmann transport equation being detailed approach captures the device physics o graphene FETs in micrometer regime and explains or the characteristics observed experimentally. 14

15 Acknowledgement The authors would like to thank Dr. Eric Snow o Naval Research Lab or discussions on linearity o graphene FETs, and Pro. Ken Shepard and Inanc Meric o Columbia University or technical discussions. The work is supported by NSF, ONR and ARL. 15

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17 two-dimensional graphene, Phys. Rev. B, vol.75, no.20, p , [10] E.H. Hwang and S. Das Sarma, Acoustic phonon scattering limited carrier mobility in two-dimensional extrinsic graphene, Phys. Rev. B, vol.77, no.11, p , [11] J.H. Chen, C. Jang, S. Xiao, M. Ishigami, and M.S. Fuhrer, Intrinsic and Extrinsic Perormance Limits o Graphene Devices on SiO2, Nat.Nanotech., vol.3, pp , [12] T. Ando, Screening Eect and Impurity Scattering in Monolayer Graphene, J.Phys.Soc.Japan, vol.75, p , [13] I. Meric, M. Y. Han, A. F. Young, B. Oezyilmaz, P. Kim, and K. Shepard, Current saturation in zero-bandgap, top-gated graphene ield-eect transistors, Nat. Nanotech., vol. 3, pp , [14] I. Meric, C. Dean, A.F. Young, J. Hone, P. Kim, and K. Shepard, Graphene ield-eect transistors based on boron nitride gate dielectrics, IEDM Tech. Dig., pp , [15] M. Freitag, M. Steiner, Y. Martin, V. Perebeinos, Z. Chen, J. C. Tsang, and P. Avouris, Energy Dissipation in Graphene Field-Eect Transistors, Nano Lett., vol.9, no.5, pp , [16] V.E Dorgan, M.-H. Bae, E. Pop "Mobility and saturation velocity in graphene on SiO2", App. Phys. Lett., vol.97, no.8, p , [17] V. Perebeinos and P. Avouris, Inelastic scattering and current saturation in graphene, Phys.Rev. B, vol.81, no.18, p , [18] A. M. DaSilva, K. Zou, J. K. Jain, and J. Zhu, Mechanism or current saturation and energy dissipation in graphene transistors, Phys. Rev. Lett., vol.104, no.24, p , [19] X. Li, E. A. Barry, J. M. Zavada, M. Buongiorno Nardelli, and K. W. Kim, Surace polar phonon dominated electron transport in graphene, Appl. Phys. Lett., vol.97, no.23, 17

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19 Figure Captions Fig.1 a) Cross section o the device structure simulated. b) Schematic representation o the E-k diagram o graphene in which two inequivalent valleys are shown. Fig. 2 Schematic sketches o (a) inelastic phonon emission and (b)inelastic phonon absorption or electrons in conduction and valence bands o graphene. Fig.3 a) The current I D vs V D curve or L ch =1µm at V G = 0.6V. A characteristic kink is observed due to the ambipolar transport behavior in graphene FETs, in agreement with experiments 13.Simulated electron (let) and hole (right) distribution unctions in the channel near drain side at (b) V D =0.25V (beore kink) and (c) V D =0.7V (ater kink). Fig.4 a) The simulated low ield channel conductance or elastic scattering as a unction o gate voltage V G at V D =0.1V with λ=40nm(line with squares), λ=80nm(line with asterisks), λ=120nm(line with diamonds), which is deined at the electron energy o E=0.1eV or elastic scattering. (b) Mobility as unction o gate voltage V G in presence o elastic scattering at V D =0.1V. Inelastic surace polar phonon scattering is turned o in all these simulations. Fig. 5 a) The Low ield channel conductance or surace polar phonon scattering as a unction o gate voltage V G at V D =0.1V with Froehlich coupling constant o 1.0eV (line with squares), 1.5eV (line with asterisks), 2.0eV (line with diamonds) at =40meV.(b) Mobility as a unction o gate voltage V G in presence o surace polar phonon scattering at V D =0.1V. Elastic scattering is absent in all these simulations. Fig.6 a) The I D -V D characteristics at V G =0.8,1.5,1.8,2.3 V (bottom to top) with λ elastic =2µm (line with squares) and λ elastic =200µm (line with triangles) deined at the electron 19

20 energy o E=0.1eV or elastic scattering, where the channel length is L ch =1µm, SPP energy =40meV, and gate insulator capacitance C ins =270nF/cm 2. b) The simulated average carrier injection velocity as a unction o the gate voltage V G (dashed line) at V D =0.6 V. For comparison, the ollowing three velocities computed rom simpler models are also plotted. (i)the saturation velocity (line with squares) computed rom a 2D BTE solver in the (k x,k y ) space, (ii) the average velocity computed rom the 2D BTE solver in the (k x,k y ) space at the same electric ield as that at the beginning o the transistor channel at V D =0.6V(line with crosses), and (iii) The saturation velocity(line with diamonds) computed by a simple model in Eq. (14), v sat v. c) The simulated transconductance (line with asterisks) g m at V D =0.6V E vs. the gate voltage V G. For comparison, the line with squares shows g m obtained as the gate capacitance times the saturation velocity computed by Eq. (14) at V D =0.6V. d) Velocity Vs Electric Field at V G =1.0V(line with a triangle), 1.5V( line with a circle), 2.0V(line with a diamond) and 2.5V(line with a square) at V D =0.6V (beore onset o ambipolar regime). Fig. 7 The simulated intrinsic cuto requency T (line with asterisks) vs. phonon energy ( ) curve shows an increase in intrinsic cut o requency with the increase in phonon energy. The plot also shows the intrinsic cuto requency calculated using Eq.(18) under the same operating conditions which shows almost linear dependence on. The intrinsic cut-o requencies are computed at V G =1.1V and V D =0.5V. 20

21 a) b) x y Fig.1 a) Cross section o the device structure simulated. b) Schematic representation o the E-k diagram o graphene in which two inequivalent valleys are shown. 21

22 a) b) Fig.2 Schematic sketches o (a) inelastic phonon emission and (b)inelastic phonon absorption or electrons in conduction and valence bands o graphene 22

23 a) b) c) b) c) Fig. 3 a) The current I D vs V D curve or L ch =1µm at V G = 0.6V. A characteristic kink is observed due to the ambipolar transport behavior in graphene FETs, in agreement with experiments 13.Simulated electron (let) and hole (right) distribution unctions in the channel near drain side at (b) V D =0.25V (beore kink) and (c) V D =0.7V (ater kink). 23

24 a) b) Fig.4 a) The simulated low ield channel conductance or elastic scattering as a unction o gate voltage V G at V D =0.1V with λ=40nm(line with squares), λ=80nm(line with asterisks), λ=120nm ( line with diamonds), which is deined at the electron energy o E=0.1eV or elastic scattering. (b) Mobility as unction o gate voltage V G in presence o elastic scattering at V D =0.1V. Inelastic surace polar phonon scattering is turned o in all these simulations. 24

25 a) b) Fig. 5 a) The low ield channel conductance or surace polar phonon scattering as a unction o gate voltage V G at V D =0.1V with Froehlich coupling constant o 1.0eV (line with squares), 1.5eV (line with asterisks), 2.0eV (line with diamonds) at =40meV. (b) Mobility as unction o gate voltage V G in presence o surace polar phonon scattering at V D =0.1V. Elastic scattering is absent in all these simulations. 25

26 a) b) V G = 1.0, 1.5, 2.0 and 2.5V c) d) Fig.6 a) The I D -V D characteristics at V G =0.8,1.5,1.8,2.3 V (bottom to top) with λ elastic =2µm (line with squares) and λ elastic =200µm (line with triangles) deined at the electron energy o E=0.1eV or elastic scattering, where the channel length is L ch =1µm, SPP energy =40meV, and gate insulator capacitance C ins =270nF/cm 2. b) The simulated average carrier injection velocity as a unction o the gate voltage V G (dashed line) at V D =0.6 V. For comparison, the ollowing three velocities computed rom simpler models are also plotted. (i)the saturation velocity (line with squares) computed rom a 2D BTE solver in the (k x,k y ) space, (ii) the average velocity computed rom the 2D BTE solver in the (k x,k y ) space at the same electric ield as that at the beginning o the transistor channel at V D =0.6V(line with crosses), and (iii) The saturation velocity(line with diamonds) computed by a simple model in Eq. (14), v sat v. c) The simulated transconductance (line with asterisks) g m at E 26

27 V D =0.6V vs. the gate voltage V G. For comparison, the line with squares shows g m obtained as the gate capacitance times the saturation velocity computed by Eq. (14) at V D =0.6V. d)velocity Vs Electric Field at V G =1.0V(line with a triangle), 1.5V( line with a circle), 2.0V (line with a diamond) and 2.5V(line with a square) at V D =0.6V (beore onset o ambipolar regime). 27

28 Fig.7 The simulated intrinsic cuto requency T (line with asterisks) vs. phonon energy ( ) curve shows an increase in intrinsic cut o requency with the increase in phonon energy. The plot also shows the intrinsic cuto requency calculated using Eq.(18) under the same operating conditions which shows almost linear dependence on.the intrinsic cut-o requencies are computed at V G =1.1V and V D =0.5V. 28