A semi-analytical model of Bilayer-Graphene Field Effect Transistor

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1 A semi-analytical model of Bilayer-Graphene Field Effect Transistor arxiv:8.4739v [cond-mat.mes-hall] 5 Sep 9 Martina Cheli, Gianluca Fiori and Giuseppe Iannaccone Dipartimento di Ingegneria dell Informazione: Elettronica, Informatica, Telecomunicazioni, via Caruso 6, 56 Pisa, Italy {martina.cheli, g.fiori, g.iannaccone}@iet.unipi.it Abstract Bilayer graphene has the very interesting property of an energy gap tunable with the vertical electric field. We propose an analytical model for a bilayer-graphene field-effect transistor, suitable for exploring the design parameter space in order to design a device structure with promising performance in terms of transistor operation. Our model, based on the effective mass approximation and ballistic transport assumptions, takes into account bilayer-graphene tunable gap and self-polarization, and includes all band-to-band tunneling current components, which are shown to represent the major limitation to transistor operation, because the achievable energy gap is not sufficient to obtain a large I on/i off ratio. Keywords - Graphene Bilayer, FETs, analytical model, band-to-band tunneling. I. INTRODUCTION The progress of CMOS technology, with the pace foreseen by the International Technology Semiconductor Roadmap ITRS [], cannot be based only on the capability to scale down device dimensions, but requires the introduction of new device architectures [] and new materials for the channel, the gate stack and the contacts. This trend has already emerged for the recent technology nodes, and will hold probably requiring more aggressive innovations for devices at the end of the Roadmap. In the last decade carbon allotropes have attracted the attention of the scientific community, first with carbon nanotubes [3] and, since its isolation in 4, with graphene [4], which has shown unique electronic [5] and physical properties [6], such as unconventional integer quantum Hall effect [7], [8], high carrier mobility [4] at room temperature, and potential for a wide range of applications [9], [], [], like nanoribbon FETs []. Despite graphene is a zero gap material, an energy gap can be engineered by rolling it in carbon nanotubes [3] or by the definition of lateral confinement like in graphene nanoribbons [4]. However, theoretical [5] and experimental [6] works have shown that significant gap in nanoribbons is obtained for widths close to - nm, which are prohibitive for fabrication technology on the scale of integrated circuits, at least in the medium term. Recently, theoretical models [7], [8], [9] and experiments [] have shown that bilayer graphene has the interesting property of an energy gap tunable with an applied vertical electric field. Anyway, the largest attainable gap is of few hundreds of mev, which make its use questionable for nanoelectronics applications: limits and potentials of bilayer graphene still have to be shown. From this point of view, device simulations can greatly help in assessing device performance. Bilayer-graphene FETs BG-FETs have been compared against monolayer FETs, by means of the effective mass approximation [] and Monte Carlo simulations [] in the ballistic limit, showing really poor potential as compared to ITRS requirements []. These approaches, however, did not take into account some of the main specific and important properties of bilayer graphene, such as the possibility of tuning the band gap and the dispersion relation with the vertical electric field, and dielectric polarization in the direction perpendicular to the D sheet. Such problems have been overcome in Ref. [3], using a real space Tight-Binding approach. However, for the limited set of device structures considered, the small band gap does not allow a proper on and off switching of the transistor. One limitation of detailed physical simulations is that, despite their accuracy, they are typically too demanding from a computational point of view for a complete investigation of device potential. Analytical approaches could help in this case. One example has been proposed in Ref. [4], but it has serious drawbacks, because it completely neglects band-to-band tunneling and the dependence of the effective mass on the vertical electric field, providing a unrealistic optimistic picture of the achievable performance. In this work, we have developed a semi-analytical model for a bilayer-graphene FET with two gates to study August 9, 8

2 the possibility of realizing an FET by tuning the gap with a vertical electric field. The model has been validated through comparison with results obtained by means of a full 3D atomistic Poisson-Schrödinger solver, showing good agreement in the applied bias range [3], [5]. Interband tunneling proves to be the main limiting factor in device operation, as demonstrated by the device analysis performed in the parameter space. II. MODEL In this section we provide a detailed description of the developed model, which is based both on a top of the barrier model [6] and on the calculation of all the interband tunneling components. In particular we adopt the ballistic transport and the effective mass approximation, whose main electrical quantities, such as the effective mass and the energy gap, have been extracted from the energy bands obtained from a p z -orbital Tight Binding TB Hamiltonian. Since we want to address long channel devices, short channel effects have been completely neglected, as well as inelastic scattering mechanisms, which are expected to be negligible in this kind of material [6]. With respect to more accurate atomistic models, the followed approach may underestimate the actual concentration of carriers in the channel, especially for large drain-to-source V DS and gate voltages V GS, when parabolic band misses to match the exact dispersion relation. We however believe that the developed model represents a good trade-off between accuracy and speed. A. Effective mass approximation In order to proceed with the definition of an analytical model based on the effective mass approximation, we first need an expression for the energy bands of bilayer graphene. The top view of the bilayer-graphene lattice structure with carbon-carbon distance a =.44 Å is shown in Fig. a: A-B atoms lay on the top layer, while A-B on the bottom layer. The energy dispersion relation can be computed by means of a p z -Tight Binding TB Hamiltonian [7] considering two layers of graphene coupled in correspondence of the overlaying atoms A and A. The energy dispersion relation reads [7] : Ek = U +U ± fk + U 4 + t ± 4U +t fk +t 4, where U and U are the potential energies on the first and second layer, respectively, U = U U, t =-.35 ev is the inter-layer hopping parameter [7], k = k xˆkx +k yˆky and [7]: fk = te [cos ikxa/ k y a ] 3 +e i3kxa/, which is the well known off-diagonal element of the graphene p z -Hamiltonian, where t is the in-plane hopping parameter t=-.7 ev. In Fig. b the band diagram for U =.5 ev is shown. As can be seen, bilayer graphene has four bands, symmetric with respect to the coordinate axis. For large U, the mexican-hat behavior in correspondence of the band minima can be observed, as detailed in Fig. c. Let us now consider the third band Fig. b, which corresponds to the conduction band same considerations follow for the valence band, i.e. second band and apply a parabolic band approximation in correspondence of the minimum k min, which reads [7]: k min = The dispersion relation can now be expressed as [7] U +t U +t U v F. 3 where Ek = E gap + m = t U +t 3/ UU +t v F m k k min + U +U, 4 ; E gap = Ut, 5 U +t August 9, 8

3 3 a U=.5 ev b 4 c E ev E ev K - K.5 k y nm K -.5 k y nm - Fig.. a Real space lattice structure of bilayer graphene. The bilayer consists of two coupled hexagonal lattices with inequivalent sites A, B and A, B in the first and in the second sheet, respectively, arranged according to Bernal A-B stacking. b Tight-Binding band structure of bilayer graphene for U=U -U =.5 ev. c Detail of the band structure in correspondence of band minimum k min : K is the Dirac point. v F = 3at is the Fermi velocity and is the reduced Planck s constant. As can be observed in 5, the effective mass m has a singularity for U =, which is clearly unphysical. In order to avoid such an issue, energy bands in the range U [,.4] have been fitted with the parabolic expression in 4, within an energy range of k B T from the band minimum where k B is the Boltzmann constant and T is the room temperature, and using m as a fitting parameter. In Figs. a, b, we show, for two different inter-layer potential energies U= ev and U=. ev, the TB energy bands as well as the parabolic bands exploiting the analytical expression in 5 and the fitted values for m, respectively. As can be seen, the fitted effective mass manages to better match the TB band in the specified energy range. In Fig. c, we show the fitted effective mass for different U. In particular, for U <.4 ev, m can be expressed as: while for larger values eq. 5 recovers. B. Electrostatics m =.9U +.43, 6 Once obtained the expression for m, the electron concentration n can be expressed as: n = ν + E c DE[f E E FS +f E E FD ]de, 7 August 9, 8

4 4 Fig.. Comparison between energy dispersions obtained by means of the analytical effective mass dashed-dotted line, fitted effective mass solid line and TB Hamiltonian circle for an inter-layer potential equal to a U= ev and b U=. ev. c Analytical and fitted relative effective mass as a function of the inter-layer potential U. m e is the free electron mass. d α cond U and α val U as a function of the inter-layer potential U. where f is the Fermi-Dirac occupation factor, E FS and E FD are the Fermi energies of the source and drain, respectively, and ν= is band degeneracy. DE is the total density of states per unit area for the complete calculation see the Appendix, which reads: DE = π where E c is the conduction band edge. If we define: f n E f = m π k BT ln m m + k min, 8 E E c k min m k B T π [ +exp ] Ec E f + k B T F / Ec E f k B T where F / is the Fermi-Dirac integral of order /, the electron concentration reads:, 9 n = [f n E FS +f n E FD ]. Analogous considerations can be made for the hole concentration p, which reads: where p = [f p E FS +f p E FD ], [ f p E f = m π k BT ln +exp + k min m k B T π ] Ef E v k B T F / Ef E v k B T, August 9, 8

5 5 and E v is the valence band edge. Once n and p are computed, attention has to be posed on how charge distributes on the two layers i.e. on dielectric polarization. To this purpose, we have numerically extracted from TB simulations α val U and α cond U, that represent the fraction of the total states in the valence band and of electrons in the conduction band, respectively, on layer [3]. We computed α cond U for a particular bias U = U = U/ and E F = ev and made the assumption that its dependence on the bias can be neglected. As far as α val is concerned, we assumed in our considered bias range, that all electron states in the valence band are fully occupied and therefore fe =. Fig. d, shows α cond U and α val U as a function of the inter-layer potential U. The charge density ρ j per unit area on layer j j=, is expressed as the sum of the polarization charge, electrons and holes and finally reads: ρ U = q{[ α val U]N tot n[ α cond U]+pα cond U}; 3 ρ U = q{[α val U ]N tot nα cond U+p[ α cond U]}, where q is the electron charge and N tot is the concentration of ions per unit area. The considered device structure is a double-gate FET embedded in SiO. The bilayer graphene inter-layer distance d is equal to.35 nm, while two different oxide thicknesses t and t have been considered Fig. 3a. An air interface between bilayer graphene and oxide has also been taken into account t sp =.5 nm [8]. For such a system, ] we can define an equivalent capacitance circuit as in Fig. 3b, where C = ǫ d =[, C t, [ ] ǫ + tsp ǫ C = t ǫ + tsp ǫ and ǫ = ǫ = 3.9ǫ, while ǫ = 8.85 F/m. V Tg and V Bg are the top gate and back gate voltage respectively, V U q and V U q. In Fig. 3c, the flat band diagram along the transverse direction y axis is shown. Metal work functions for the back gate and top gate are equal to 4. ev [Φ Bg =Φ Tg =4. ev], while the graphene work function Φ gra is equal to 4.5 ev [9]. E FTg, E FBg are the Fermi level of the top and of the back gate, respectively. The conduction band edge inserted in eq. 9, can be expressed as: E c = Φ Bg +E FBg Φ gra + U +U + E gap. 4 Applying the Gauss theorem, we obtain the following expression: { C V Tg V +V V C = ρ 5 C V V +V Bg V C = ρ. Eqs. 3 and 5 are then solved self-consistently till convergence on V and V is achieved. C. Current Drain-to-source J TOT current is computed at the end of the self-consistent scheme. As depicted in Fig. 3d, J TOT consists of three different components: the first is due to the thermionic current J th over the barrier [6], whereas the second J TS and the third J TD to band-to-band tunneling. In the same picture, we sketch the conduction band edge E CS E CD and the valence band edge E V S E V D at the source drain. Assuming reflectionless contacts, the thermionic current is due to electrons injected from the source with positive velocity v x > and to electrons injected from the drain with v x < : = q + [ E dk y J th π k < x k > x E k x fe E FD dk x E k x fe E FS dk x + ], 6 where E = E c + m k k min, v x = k x is the group velocity and k x > k< x is the wavevector range for which v x > v x <. For the complete derivation see the Appendix. August 9, 8

6 6 y V Tg Top gate V Tg Source t SiO t sp t sp n + n + t SiO x Drain V DS V V C C C V Bg Back gate a b SOURCE V Bg CHANNEL DRAIN J th Φ Tg t ε t sp Φ gra Φ Bg tsp t ε E E V FS E CS E VS x E C E gap J TS E FTg E c E E FBg i d E v c d W S J TD E FD E CD E VD Fig. 3. a Sketch of the considered bilayer-graphene FET: d is the inter-layer distance, t and t are the top and back oxide thicknesses. b Equivalent capacitance circuit of the simulated device. c Flat band diagram of the BG-FET along the y direction. d Conduction and valence band edge profiles in the longitudinal direction; we assume that deep in the source and drain regions, the electric field induced by the gate vanishes and the gap gradually reduces to zero. Let us now discuss the band-to-band tunneling current due to the barrier at sourcedrain contact, which reads: J Ti = q E π T i k y [fe E FS k x k y k > x fe E FD ] k x k y i = S,D, 7 wheres refers to the source andd to the drain, whilet i k y is the transmission coefficient at the different reservoirs. The key issue in computing 7 is the definition of an expression for T i k y, which accounts for band-to-band tunneling process. We have assumed a non charge-neutrality region of fixed width x at the contact/channel interface and an electric field E i =E c E Fi /q x with i = S,D. For what concern the J TS term, electrons emitted with electrochemical potential E FS see two triangular barriers, one at the source junction and one in correspondence of the drain Fig. 3d, whose heights are equal to E gap and width W i = E gap /qe i. Assuming the same x for both source and drain junctions, the drain barrier is transparent with respect to the source barrier, since, for large V DS, the electric field at the source is smaller than the electric field at the drain barrier: T S k y is therefore essentially given by the source junction barrier. Same considerations follow for the other band-to-band tunneling current component J TD, flowing only through the drain-channel contact. In this case E D = E c E FD = Ec EFS+qVDS xq. August 9, 8

7 7 Assuming the WKB approximation, the transmission coefficient can be expressed as: T i k y = e R [W b ] Im{kx} dx, i = S,D, 8 where Im{k x } is the imaginary part of k x and is obtained from: m k k min = qe i x E gap, i = S,D. 9 Finally, J Ti is computed performing the integral 7 numerically. III. EXPLORATION OF THE DESIGN SPACE In order to validate our model, we have first compared analytical results with those obtained by means of numerical NEGF Tight Binding simulations [5], considering a test structure with t = t =.5 nm, t sp =.5 nm, Φ gra = Φ G = Φ Bg =4. ev, V DS =. V and V Bg = V. In Fig. 4a-b the electron concentrations ρ, ρ and Charge Density /m 3e+6 e+6 e+6 ρ num. ρ num. ρ an. ρ an..5 VoltageV V num. V num. V an. V an. E gap ev.5..5 a b c V Tg V V Tg V V Tg V Fig. 4. Comparison between analytical and numerical simulation of a ρ and ρ and b V and V as a function of V Tg. V Bg = V and V DS = V. c Energy gap as a function of top gate voltage, with V Bg = V. the electrostatic potentials V, V on layer and are shown, as a function of V Tg, for V DS = V and V Bg = V. As can be seen, results are in good agreement. Some discrepancies however occur for larger V DS V DS >. V, where the parabolic band approximation misses to reproduce band behavior for large k y. In Fig. 4c the energy gap is plotted as a function of V Tg. As can be seen, even for large V Tg, the biggest attainable E gap is close to.5 ev. Let us now consider the different contributions of the three current components J th, J TS and J TD to the total current J TOT Fig. 5d. For each of these components, we can define a sort of threshold voltage, above which their contribution is not negligible. In particular, J th starts to be relevant as soon as E c E FS. We then define V th as the V Tg for which E c = E FS. Similarly, interband current J TS is not zero when E v E CS so we define V TS the top-gate voltage for which E v =E CS. Finally, J TD is not zero in the energy range E CD < E < E V S : we define V > TD and V < TD the top-gate voltages for which E V S=E CD ; thanks to these definitions, we can qualitatively evaluate current contribution by observing the band structure. Our goal is indeed to obtain the largest value for the I on /I off ratio, and this is only possible if the band-to-band component of the current is suppressed. We have considered three different solutions to accomplish this task: by varying the back gate oxide t, by varying the E FS E CS or E FD E CD difference, or by simply varying the back gate voltage. If otherwise specified, x =.7 nm, as obtained from TB simulations of an abrupt junction with the same doping of the considered BG-FET. In Fig. 5a-b-c the above-defined thresholds are shown for the three considered cases. As shown in Fig. 5a, back gate oxide thickness has no effect in our case, since the top layer screens the electric field induced by the top gate, as can also be seen from Fig. 4b, where V remains almost constant. Fig. 5b shows thresholds as a function of E FS E CS, and therefore as a function of dopant concentration. We observe that for E FS E CS = ev, V > TD = V < TD, so that J TD is practically eliminated, while J TS increases since V TS August 9, 8

8 8 Fig. 5. a Thresholds as a function of the oxide thickness t for V DS =.5 V, V BG = V, E FS E CS =.9 ev. b Thresholds as a function of E FS E CS for t =.5 nm, V DS =.5 V, V BG = V. c Thresholds as a function of V Bg for t =.5 nm, V DS =.5 V and E FS E CS = ev. d Total current J TOT and its three components J TS, J TD, J th for V DS =. V, V Bg = V and E FS E CS =.5 ev. Thresholds are shown along the coordinate axis. e Total current for t =.5 nm, V DS =.5 V, E FS E CS = ev and V BG = V and different x. becomes larger. In Fig. 5c, we show V th and V TS, for E FS E CS = ev, as a function of V Bg. Unfortunately the two curves have the same behavior, so that V TS cannot be reduced to values smaller than V th, or in other words we cannot suppress current due to interband tunneling at source contact. We have then computed the transfer characteristics forv Bg = V,t = t =.5 nm,e FS E CS = ev. In Fig. 5e J TOT is shown. As can be seen, poor I on /I off ratio can be obtained since band-to-band tunneling at source contact is too high as also observed in graphene FET [3]. Reducing E, i.e. Tk y, could lead to a reduction of J TS and consequently to an improvement of the I on /I off ratio. As can be seen in Fig. 5e, an improved I on /I off is obtained increasing x to 5- nm, but it is still lower than the ITRS requirements 4 for digital circuits. In Fig. 6, we also sketch the simulated band edges for three different cases: a when tunneling is negligible J TOT J Ti for V Tg = V, b when tunneling weakly affect the total current J TOT J TS +J TD for V Tg =.4 V and c when tunneling represents the predominant component J TOT J TS +J TD for V Tg = V. IV. CONCLUSION We have developed an analytical model for bilayer-graphene field effect transistors, suitable for the exploration of the design parameter space. The model is based on some simplifying assumptions, such as the effective mass approximation, but includes all the relevant physics of bilayer graphene. First and foremost, it includes the tunable gap of bilayer graphene with the vertical electric field, which is exploited in order to induce the largest gap, when the device is in the off state. It also fully includes polarization of bilayer graphene in response to a vertical electric field. As far as transport is concerned, it includes the thermionic current components and all interband tunneling August 9, 8

9 9 Fig. 6. Simulated band edges when: a Tunnel current weakly affects J TOT ; b J TS +J TD is one order of magnitude smaller than the total current; c Inter band tunneling current is the dominant component. components, which are the main limiting factor in achieving a large I on /I off ratio. Significant aspects of the model have been validated through comparisons with numerical TB NEGF simulations. Due to the small computational requirements, we have been able to explore the parameter design space of bilayergraphene FETs in order to maximize the I on /I off ratio. Despite applied vertical field manages to induce an energy gap of the order of one hundred mev, band-to-band tunneling greatly affects device performance, limiting its use for device applications. A larger gap must be induced to make bilayer graphene a useful channel material for digital applications, probably by combining different options, such as using bilayer graphene in addition to limited lateral confinement, stress, or doping. A. Density of states APPENDIX I The total density of states can be computed as follow, performing the integral over the first Brillouin zone BZ: DE = π δ E c + m k k min E πkdk. BZ If we apply the following property of the delta function δ[fx] = n where x n are the zeroes of the function fx, eq. reads: DE = m π E E c m r+k min dr where r= k -k m min. = π δx x n f, x n BZ δ r E E c m m + k min, E E c August 9, 8

10 B. Thermionic Current In order to derive the expression of the thermionic current, we have first to compute the group velocity v x, which reads: Ek x,k y = Egap + k x k x m k k min Replacing 3 in 6 we obtain: J th = q π m + k < x + = k x m dk y [ k > x k x k min k k min k k x k min k. 3 fe E FS dk x fe E FD dk x ]. 4 In order to remove the singularity in eq. 4 for k x = k y =, we can use cylindrical coordinates, i.e. k x =kcosθ, k y =ksinθ and k =k. In this representation, the condition v x > translates in: { kcosθ > k k min > { kcosθ < k k min < The integral 4 becomes: C. Transmission coefficient J th = q π m kmax kk k min [f S E,k f D E,k]dk. 5 The tunneling transmission probabilitytk y has been computed through the WKB approximation. The Im{k x } in eq. 8 is computed from the energy dispersion relation as follows: from 9 we can write kx +k y k m min = qex E gap. 6 Defining m Egap qex βx = > 7 and inserting eq. 7 in eq. 6, we obtain: k x +ky k min = iβx, 8 which reads: If we expressed k x as k x =a+ib a,b R, k x reads: k x = k min βx k y +iβxkmin. 9 k x = a b +iab. 3 By comparing eq. 9 and eq. 3, Im{k x } simply reads: C + C Im{k x } = +4βx kmin, 3 with C = k min +β x+k y. Acknowledgements The work was supported in part by the EC Seventh Framework Program under project GRAND Contract 575 and by the Network of Excellence NANOSIL Contract 67. Authors gratefully acknowledge Network for Computational Nanotechnology NCN for providing computational resources at nanohub.org, through which part of the results here shown has been obtained. August 9, 8

11 REFERENCES [] International Technology Roadmap for Semiconductor 7. [Online]. Available: [] Z. F. Wang, H. Zheng, Q. W. Shi, and J. Chen, Emerging nanocircuit paradigm: Graphene-based electronics for nanoscale computing, in Nanoscale Architectures, 7. NANOARCH 7. IEEE International Symposium on, Oct. 7, pp. 93. [3] S. Iijima, Helical microtubules of graphitic carbon, Nature, vol. 354, no. 6348, pp , Nov. 99. [4] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, Electric field effect in atomically thin carbon films, Science, vol. 36, no. 5696, pp , Oct. 4. [5] A. H. C. Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, The electronic properties of graphene, Rev. Mod. Phys., vol. 8, pp. 9 63, Jan. 9. [6] A. K. Geim and K. S. Novoselov, The rise of graphene, Nat Mater, vol. 6, no. 3, pp. 83 9, Mar. 7. [7] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, Twodimensional gas of massless Dirac fermions in graphene, Nature, vol. 438, no. 765, pp. 97, Nov. 5. [8] V. P. Gusynin and S. G. Sharapov, Unconventional Integer Quantum Hall effect in graphene, Phys. Rev. Lett., vol. 95, no. 4, pp , Sep. 5. [9] J. C. Meyer, A. K. Geim, M. I. Katsnelson, K. S. Novoselov, T. J. Booth, and S. Roth, The structure of suspended graphene sheets, Nature, vol. 446, no. 73, pp. 6 63, Dec. 6. [] J. S. Bunch, A. M. Van der Zande, S. S. Verbridge, I. W. Frank, D. M. Tanenbaum, J. M. Parpia, H. G. Craighead,, and P. L. McEuen, Electromechanical Resonators from Graphene Sheets, Science, vol. 35, no. 58, pp , Jan. 7. [] D. A. Dikin, S. Stankovich, E. J. Zimney, R. D. Piner, G. H. B. Dommett, G. Evmenenko, S. T. Nguyen, and R. S. Ruoff, Preparation and characterization of graphene oxide paper, Nature, vol. 448, pp , Jul. 7. [] G. Fiori and G. Iannaccone, Simulation of Graphene Nanoribbon Field-Effect Transistors, IEEE Electron Device Letters, vol. 8, no. 8, pp , Aug. 7. [3] C. Zhou, J. Kong, and H. Dai, Electrical measurements of individual semiconducting single-walled nanotubes of various diameters, Appl. Phys. Lett., vol. 76, no., pp , Mar.. [4] Z. Chen, Y.-M. Lin, M. J. Rooks, and P. Avouris, Graphene nano-ribbon electronics, Phys. E, vol. 4, no., pp. 8 3, Jun. 7. [5] Y.-W. Son, M. L. Cohen, and S. G. Louie, Energy gaps in graphene nanoribbons, Phys. Rev. Lett., vol. 97, no., pp , Nov. 6. [6] M. Y. Han, B. Özyilmaz, Y. Zhang, and P. Kim, Energy Band-Gap Engineering of Graphene Nanoribbons, Phys. Rev. Lett., vol. 98, no., pp , May 7. [7] J. Nilsson, C. A. H. Neto, F. Guinea, and N. M. R. Peres, Electronic properties of bilayer and multilayer graphene, Phys. Rev. B, vol. 78, no. 4, pp , Jul. 8. [8] E. V. Castro, K. S. Novoselov, S. V. Morozov, N. M. R. Peres, J. M. B. L. dos Santos, J. Nilsson, F. Guinea, A. K. Geim, and A. H. C. Neto, Biased bilayer graphene: semiconductor with a gap tunable by the electric field effect, Phys. Rev. Lett., vol. 99, no., p. 68, Nov. 7. [9] E. McCann, Asymmetry gap in the electronic band structure of bilayer graphene, Phys. Rev. B, vol. 74, no. 6, pp , Nov. 6. [] T. Ohta, A. Bostwick, T. Seyller, K. Horn, and E. Rotenberg, Controlling the Electronic Structure of Bilayer Graphene, Science, vol. 33, no. 5789, pp , Aug. 6. [] Y. Ouyang, P. Campbell, and J. Guo, Analysis of ballistic monolayer and bilayer graphene field-effect transistors, Appl. Phys. Lett., vol. 9, no. 9, pp , Feb. 8. [] N. Harada, M. Ohfuti, and Y. Awano, Performance estimation of Graphene Field-Effect-Transistors using semiclassical Monte Carlo Simulation, Appl. Phys. Exp., vol., no., pp , Feb. 8. [3] G. Fiori and G. Iannaccone, On the possibility of tunable-gap bilayer graphene FET, IEEE Electron Device Letters, vol. 3, no. 3, pp. 6 64, Mar. 9. [4] V. Ryzhii, M. Ryzhii, A. Satou, T. Otsuji, and N. Kirova, Device model for graphene bilayer field-effect transistor, J. App. Phys., vol. 5, no., p. 45, Dec. 9. [5] Nanotcad vides. [Online]. Available: Codeand{D}ocumentationcanbefoundattheurl: [6] K. Natori, Ballistic metal-oxide-semiconductor field effect transistor, J. App. Phys., vol. 76, no. 8, pp , Jul [7] P. R. Wallace, The band theory of graphite, Phys. Rev., vol. 7, no. 9, pp , May 947. [8] X. Li, X. Wang, L. Zhang, S. Lee, and H. Dai, Chemically Derived, Ultrasmooth Graphene Nanoribbon Semiconductors, Science, vol. 39, no. 9, p. 5878, Feb. 8. [9] J. W. G. Wildoer, L. C. Venema, A. G. Rinzler, R. E. Smalley, and C. Dekker, Electronic structure of atomically resolved carbon nanotubes, Nature, vol. 39, pp. 59 6, Jan [3] V. Ryzhii, M. Ryzhii, and T. Otsuji, Thermionic and tunneling transport mechanism in graphene field-effect transistors, Phys. Stat. Sol. a, vol. 5, no. 7, pp , Mar. 8. August 9, 8