MOSFET-Like Carbon Nanotube Field Effect Transistor Model Mohammad Taghi Ahmadi, Yau Wei Heong, Ismail Saad, Razali Ismail Faculty of Electrical Engineering Universiti Teknologi Malaysia 81310 Skudai, Johor, Malaysia, [email protected] ABSTRACT curiosity towards ballistic nature of the carriers is elevated. Initially, it was in the work of Arora[1] that the possibility An analytical model that captures the essence of of ballistic nature of the transport in a very high electric physical processes in a CNTFET’s is presented. The model field for a nondegenerate semiconductor was indicated. In covers seamlessly the whole range of transport from drift- this article, the work of Arora is extended to embrace diffusion to ballistic. It has been clarified that the intrinsic degenerate domain in the carbon nanotube where electrons speed of CNT’s is governed by the transit time of electrons. have analog type classical spectrum only in one direction Although the transit time is more dependent on the while the other two directions are quantum confined or saturation velocity than on the weak-field mobility, the digital in nature. When only the lowest digitized quantum feature of high-electron mobility is beneficial in the sense state is occupied (quantum limit), a carbon nanotube shows that the drift velocity is maintained always closer to the distinct one-dimensional character. saturation velocity, at least on the drain end of the transistor where electric field is necessarily high and controls the 2 DISTRIBUTION FUNCTION saturation current. The results obtained are applied to the modeling of the current-voltage characteristic of a carbon In one dimensional nanotube with diameter around nanotube field effect transistor. The channel-length nanometer (see Figure. 1), only one of the three Cartesian modulation is shown to arise from drain velocity becoming directions is much larger than the De - Broglie wavelength closer to the ultimate saturation velocity as the drain (Taken is x direction). Since the current is independent of voltage is increased. band structure a MOSFET-like CNTFET in the quantum limit can be described by the same theory for 1 INTRODUCTION semiconductor nanowires MOSFETs. For the Carbon nanotube near the k = 0 band structure is parabolic and the There is an intensive search and research for high-speed energy spectrum is analog-type only in x-direction. devices is an ongoing process. Two important factors that determine the speed of a signal propagating through a = kE 22 G x (1) conducting channel. One is the transit-time or gate delay E +≈ * that depends on the length of the channel and the other is 2 2m the wire delay that is due to finite RC time constants. The two factors are intertwined as in each one the ultimate with saturation of velocity plays a predominant role. The higher mobility brings an electron closer to saturation as high 2 ta ⎛ i −−− )1(36 i ⎞ electric field is encountered, saturation velocity remaining −cc ⎜ ⎟ (2) EG ×≈ ⎜ ⎟ i = 1,2,3,..... the same [Mohammad]. The reduction in conducting d ⎝ 4 ⎠ channel length of the device results in reduced transit-time- delay and hence enhanced operational frequency. There is no clear consensus on the interdependence of saturation Here EG is the band gap for ith sub bands and a −cc is velocity on low-field mobility that is scattering-limited. In the Carbon –Carbon band in the quantum limit of a any solid state device, it is very clear that the band structure nanotube and d is the diameter of the Carbon nanotube, and parameters, doping profiles (degenerate or nondegenerate), t=2.7 (eV) is the nearest neighbor C-C tight binding overlap and ambient temperatures play a variety of roles in energy [5, 6].In the y, z-direction where the length determining performance behavior. The outcome that the L ,zy << λD , the De - Broglie wavelength λD with a higher mobility leads to higher saturation is not supported by experimental observations [1, 2]. This paper focuses on typical value of 10 nm. kx is the momentum wave vector in * the process controlling the ultimate saturation. It has been the x-direction. = .0 05mm o [7]. confirmed in a number of works that the low-field mobility is a function of quantum confinement [3, 4]. In the following, the fundamental processes that limit drift velocity are delineated. As devices are being scaled down in all dimensions, the NSTI-Nanotech 2009, www.nsti.org, ISBN 978-1-4398-1784-1 Vol. 3, 2009 574 1 G (Ef k ) = G (6) Fk +− qEE ε .A e BTk +1 This distribution has a simple interpretation [8, 9] as given in the tilted band diagram In an extremely large electric field, virtually all the electrons are traveling in the positive x-direction (opposite to the electric field). This is Figure 1. A prototype CNT’s (9, 2) with R << λ and D what is meant by conversion of otherwise completely . L >> λD random motion into a streamlined one with ultimate velocity per electron equal to the intrinsic velocity v . The quantum wave in the x-direction is a traveling wave i and that in the y (z) - direction is the standing wave. Hence the ultimate velocity is ballistic independent of scattering interactions. The ballistic motion in a mean-free path is interrupted by the onset of a quantum emission of j(k .x) ⎛πy ⎞ ψ (x,y,z)= 2 e x sin sin⎛πz ⎞ (3) energy =ω . This quantum may be an optical phonon or a k Ω ⎜ L ⎟ ⎜ L ⎟ 0 ⎝ y ⎠ ⎝ z ⎠ photon or any digital energy difference between the quantized energy levels with or without external stimulation In carbon nanotube near the minimum band energy present. The mean-free path with the emission of a similar to one dimensional nanowire we obtain following quantum of energy is related to A (zero-field mean free equation for density of state (DOS). 0 path) by an expression. 1 * 1 − Δnx 1 EG 2m 2 2 EQ AQ DOOS = (E −= () 2 ) (4) − − ΔELx 2π 2 = qεA 0 A 0 = AA 0 1[ − e A 0 1[] −= e ] (7) The distribution function of the energy E is given by the Fermi-Dirac distribution function qεA Q = Q = (NE 0 + )1 =ω0 (8) 1 1 f (Ek ) = N = Ek −EF (5) 0 =ω0 (9) k T e B +1 e BTk −1 where EF is the one-dimensional Fermi energy at which the Here N0 + )1( gives the probability of a quantum probability of occupation is half and T is the ambient emission. N is the Bose-Einstein distribution function temperature. In non- degenerately doped semiconductors o determining the probability of quantum emission. The the ‘1’ in the denominator of Eq. (5) is neglected as compared to the large exponential factor. In this degraded mean free path A is now smaller than the low- approximation, the distribution is Maxwellian, which is field mean free path A o . ≈ AA 0 in the Ohmic low-field used in determining the transport parameters extensively. regime as expected. In the high electric field This simplification is good for a nondegenerately-doped semiconductor. However, most nanoelectronic devices ≈ AA Q = Q qE ε . The inelastic scattering length during these days are degenerately doped. Hence any design based which a quantum is emitted is given by .Obviously on the Maxwellian distribution is not strictly correct and A = ∞ in zero electric field and will not modify the often leads to errors in our interpretation of the Q experimental data. In the other extreme, for strongly traditional scattering described by mean free path A 0 as degenerate carriers, the probability of occupation is 1 when A Q >> A 0 . The low-field mobility and associated drift Ek < EF and is zero if Ek > EF.Arora modified the equilibrium distribution function of Eq. (4) by replacing EF motion is therefore scattering-limited. The effect of all (the chemical potential) with the electrochemical potential possible scattering interactions is now buried in the mean G G G E + qε .A ε free path A . This by itself may be enough to explain the F where is the applied electric field, q the 0 G degradation of mobility μ in a high electric field; [10] electronic charge and A the mean free path. Arora’s distribution function is given by 575 NSTI-Nanotech 2009, www.nsti.org, ISBN 978-1-4398-1784-1 Vol. 3, 2009 qτ qA qA Q 2k T N μ = c = ≈ B c * * * (10) vi = * × ℑ0 ()ηF (16) m m vi m vi πm n Here τ c is the mean free time (collision time) during With which the electron motion is ballistic. v i is the mean 1 * 2 intrinsic velocity for 1D carbon nanotube that is discussed ⎛ 2m kB T ⎞ N = ⎜ ⎟ (17) in the next section. c ⎜ π= 2 ⎟ ⎝ ⎠ 3 INTRINSIC VELOCITY where Nc is the effective density of states for the The intrinsic velocity is given by conduction band with m* now being the density-of-states effective mass. n1 is the carrier concentration per unit 1 ℑ0 ()η F vi = vth (11) length. Figure 2 indicates the ultimate velocity as a function π ℑ 1 ()ηF of temperature for (9,2) CNT with effective mass − * 2 m =0.099m0 . Also shown is the graph for nondegenerate approximation. The velocity for low carrier concentration With follows T 2/1 behavior independent of carrier concentration. However for high concentration (degenerate carriers) the E E − G velocity depends strongly on concentration and becomes 2k T F independent of the temperature. The ultimate saturation v = B , η = 2 (12) th m* F k T velocity is thus the thermal velocity appropriate for 1D B carrier motion: 1 ∞ x j 1 2k T ℑ ()η = dx (13) v = v= v = B (18) j ∫ ()x−η iND thd th * Γ( j + 1) 0 e + 1 π πmd Here, ℑ j ()η is the Fermi-Dirac (FD) integral of order j and Γ(j + 1) is a Gamma function.