High-frequency capability of Schottky-barrier carbon nanotube FETs

The high-frequency capability of carbon nanotube field-effect transistors is investigated by simulating the small-signal performance of a device with negative-barrier Schottky contacts for the source and drain, and with a small, ungated region of nanotube between the end contacts and the edge of the wrap-around gate electrode. The overall structure is shown to exhibit resonant behaviour, which leads to a significant bias dependence of the small-signal capacitances and transconductance. This could lead to high-frequency figures of merit (fT and fmax) in the terahertz regime. Introduction Carbon nanotube field-effect transistors (CNFETs) are predicted to have superior DC characteristics to those of foreseeable silicon MOSFETs [1,2]. Experimentally, impressive values for drain current and transconductance have already been reported in prototype devices [3]. However, the AC performance of CNFETs has not yet received much attention [4], and the likelihood of high-frequency operation needs to be established. Here, we take a step in this direction by predicting two useful figures of merit for high-frequency transistors, namely: the unity current-gain and unity power-gain frequencies, fT and fmax, respectively. We employ the standard small-signal method [5], abetted by a self-consistent SchrödingerPoisson solver [6], in order to obtain preliminary estimates of these valuable figures of merit. In particular, we consider the cylindrically gated device depicted in Fig. 1. The CNFET geometry consists of a nanotube with radius rt, coated by an insulator of thickness tins, wrapped by a cylindrical gate of thickness tg, and capped at the ends by planar source and drain contacts of radius tins+tg+rt. These end contacts are separated longitudinally from the gate by a distance Lgap. Figure 1. Structure of the modeled CNFET. Model Since the CNFET is much like a traditional field-effect transistor, we may employ the standard equivalent circuit model for this device [5], where the small-signal parameters themselves, such as the transconductance and the various transcapacitances, are computed by taking numerical derivatives based on the results of self-consistent DC charge-voltage calculations [6,7]. Through our use of a Schrödinger-Poisson solver for these DC results, we are able to include the effects of geometry, quantum capacitance, and spatial non-uniformity of the charge in our calculations. While we have previously presented results for the intrinsic fT with only three circuit elements [7], here we consider the additional effects of series resistances associated with each of the terminals, and we include the remaining capacitance components. This allows us to compute the extrinsic fT, and also to consider fmax. For clarity, we include the standard equivalent circuit in Fig. 2, where we note that the subscripts s, d and g refer to the source, drain and gate, respectively. In the usual way, we consider the fT and fmax expressions, which may be arrived at by extrapolating the characteristic decay in gain, from its value at an appropriate low frequency, to the 0dB point. Figure 2. Small signal equivalent circuit. The transcapacitance Cm = Cdg Cgd. Results and Discussion The model device in this study employs a (16,0) nanotube of radius 0.63nm, length 30nm, bandgap 0.62eV, unity relative permittivity, and effective mass 0.06m0, where m0 is the free electron rest mass. Moreover, tins is 2.5nm, tg is 3nm, the insulator permittivity is 25 as is appropriate for zirconia [8], and Lgap is 5nm for this initial investigation. The work function of the nanotube is taken to be 4.5eV, while that of the end-contacts is 3.9eV, yielding negativeSchottky barrier, n-type transistor operation. All results are for a drain-source voltage of 0.5V. Fig. 3 illustrates the intrinsic parameters for our device. The oscillatory peaks in the capacitances and transconductance are related to the formation of quasi-bound states in the short channel [7,9]. With the modulation of the applied voltage, the states, indicated by the bright patches in Fig. 4, are shifted in energy by band bending in the channel. As they cross the source or drain Fermi level, they become populated, and we obtain peaks in the charge accumulation and, consequently, in the capacitances. Since the charge accumulation affects the amount of band bending in the channel through our self-consistent DC calculations, we also see peaks in the transconductance gm. The gm behaviour is complicated due to its dependence both on the charge and on the Fermi distributions at the injecting contacts. In Fig. 5, we present our main results, the predictions of fT and fmax for our model CNFET. Fig. 5(a) shows fT for Rs = Rd set to 1kΩ, 10kΩ, and 100kΩ, and we recall that Rg has no influence on this figure of merit. Note that the values of the parasitic resistors Rs and Rd are chosen to be comparable to the contact resistance that results from mode constriction when carriers pass from a many-moded material to a material with only a few modes [10]. In the carbon nanotube case, we consider the lowest two degenerate modes in energy for an equivalent contact resistance of around 6-7kΩ. Note that the contact resistance is automatically included in our self-consistent DC calculations, so an explicit resistor is not needed to represent it in the equivalent circuit.. Thus, the resistors shown in Fig. 2 are solely parasitic. It is evident that the maximum value of fT occurs at the first peak in gm. Turning now to fmax, shown in Fig. 5(b), we focus on the effect of Rg, and show results for Rg set to 100Ω, 1kΩ, and 10kΩ, with Rs = Rd = 10kΩ. Again, a pronounced peak coincides with the first peak in gm. Figure 3. Capacitances and transconductance for the model device. Figure 4. Charge density for the model device subject to a gate-source voltage of 0.38V. Brighter patches indicate higher charge density, while a portion of the conduction band edge is shown by the white line. The energy values are referenced to the source Fermi level. Figure 5. Extrapolated figures of merit: (a) fT with Rs = Rd set to 1kΩ (solid), 10kΩ (dashed), and 100kΩ (dotted), and (b) fmax with Rs = Rd = 10kΩ for Rg set to 100Ω (solid), 1kΩ (dashed), and 10kΩ (dotted). ConclusionsFrom this simulation of the high-frequency performance of CNFETs, it can be concluded that, inshort-channel devices, with negative-barrier Schottky contacts for the source and drain, and withvery short ungated regions, and where coherent transport is possible, resonance effects can leadto a strong bias dependence of the high-frequency figures of merit, fT and fmax. At the resonancepeaks, these frequencies may reach the THz level. References[1] L.C. Castro, D.L. John and D.L. Pulfrey: Smart Mater. Struct. Accepted Aug. 13, 2004.[Online.] Available: http://nano.ece.ubc.ca/pub/publications.htm. [2] J. Guo, M. Lundstrom and S. Datta: Appl. Phys. Lett. Vol. 80 (2002), p. 3192. [3] A. Javey, J. Guo, Q. Wang, M. Lundstrom and H. Dai: Nature Vol. 424 (2003), p. 654. [4] D.V. Singh, K.A. Jenkins, J. Appenzeller, D. Neumayer, A. Gill and H.-S.P. Wong: IEEETrans. Nanotechnol. Vol. 3 (2004), p. 383. [5] Y.P. Tsividis: Operation and Modeling of the MOS Transistor, Chapter 9, (McGraw-Hill,Toronto 1987). [6] D.L. John, L.C. Castro, P.J.S. Pereira and D.L. Pulfrey: Proc. NSTI Nanotech Vol. 3 (2004),p. 65. [7] L.C. Castro, D.L. John, D.L. Pulfrey, M. Pourfath, A. Gehring and H. Kosina: IEEE Trans.Nanotechnol. Submitted Nov. 16, 2004. [8] A. Javey, R. Tu, D. Farmer, J. Guo, R. Gordon and H. Dai: Nano Lett. Vol. 5 (2005),p. 345. [9] D.L. John, L.C. Castro and D.L. 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