Simulation of phonon-assisted band-to-band tunneling in carbon nanotube field-effect transistors

Electronic transport in a carbon nanotube (CNT) metal-oxide-semiconductor field effect transistor (MOSFET) is simulated using the non-equilibrium Green's functions method with the account of electron-phonon scattering. For MOSFETs, ambipolar conduction is explained via phonon-assisted band-to-band (Landau-Zener) tunneling. In comparison to the ballistic case, we show that the phonon scattering shifts the onset of ambipolar conduction to more positive gate voltage (thereby increasing the off current). It is found that the subthreshold swing in ambipolar conduction can be made as steep as 40mV/decade despite the effect of phonon scattering.

2 Electronic transport in conventional metal-oxide-semiconductor field-effect transistors (MOSFETs) is limited at room temperature primarily by surface roughness and phonon scattering. However carbon nanotubes, due to their high mobility, are believed to be operating in the ballistic regime under low source-drain bias. Carbon nanotube field effect transistors (CNTFET) 1 2 3 have been recently demonstrated. Non-equilibrium Green's function simulations of carbon nanotubes 4 5 have been performed in the ballistic limit. It was demonstrated that the simulation results are in good agreement with experiments 6 at least over a wide range of parameters. Monte Carlo simulations 7 showed that in some regimes scattering produces only a small effect on current.
In this letter we introduce into the formalism a model of scattering due to electronphonon interaction and explore the range of validity of the previous quasi-ballistic results. We show that the most dramatic difference occurs for the off-current, i.e. at the cross-over from the direct (over the barrier) conduction to ambipolar (tunneling) conduction b . The effects of band-to-band (Landau-Zener) tunneling 8 in CNTFETs have been experimentally investigated by Appenzeller et al. 9 Here we examine ambipolar conduction in the presence of phonon scattering. We explore the physics of the very steep subthreshold swing in ambipolar conduction as recently reported by Appenzeller et al. [9]. We describe regimes where it is preserved or destroyed by phonon scattering.
We apply the non-equilibrium Green's functions method, as described by Datta 10 to simulation of the carbon nanotubes transistors and extend the method of [4,5]. It involves solution of the equation for the Green's function, G , in the mode-space approach 11 b Ambipolar conduction is a well-understood effect in CNT Schottky-barrier FETs. We show that it also occurs in CNT MOSFETs.
where E is the energy of quantum states, the imaginary part of self energy, , contains the contribution of respectively in-and -out scattering and contacts c . The real part of the self energy is obtained via the Hilbert transform 12 and corresponds to a shift of the energies of quantum states; we disregard it in this letter. The Hamiltonian, H , of the device includes the tight-binding coupling between carbon atoms with the bonding energy, 3eV t = , at the nearest neighbor distance, 0.142nm cc a = , as well as the electrostatic potential energy. The latter is expressed via the potential,V , from a solution of the Poisson's equation with electric charges given by the electron and hole densities, n and p , respectively; see [4,5] for the geometry of the device and the details of the computation. The carrier densities are obtained via integrals over energy of the electron and hole correlation functions, they express the density of filled and unfilled states, respectively, over the energy and the coordinate. All the G -functions are matrices in the basis of the discrete points, z , along the nanotubes. The current at any point along the tube is 13 c We disregard the effects of Coulomb blockade which are only significant for shorter channel length, thicker gate oxide, and lower gate dielectric constant, than in practically important devices considered in this paper.
We will be plotting the energy spectrum of the current, which is the expression under the integral.
In addition to commonly included contributions to self-energy from the source and drain electrodes, here we also introduce the contributions due to electron-phonon scattering where the diagonal elements of the electron and hole correlation function are implied, the energy of a phonon mode is ω , number of phonons in a mode, N ω , is given by the Bose-Einstein distribution. The electron-phonon coupling, ph R , is proportional to the square of the deformation potential. In this letter for simplicity we take the fixed energy of phonons, 160meV, and set an independent of energy coupling constant which is related to the mean free path as We model a CNTFET with a (n,m)=(13,0) carbon nanotube surrounded by a gate in a cylindrically symmetric geometry. The bandgap energy for this nanotube is,  14 15 .
This insight into the processes governing conduction is confirmed by the current distribution (Fig. 3). It has a series of sharp peaks each corresponding to a discrete state in the channel. As the coordinate varies from the source to the drain, the current distribution shifts to lower energies due to emission of phonons. Uninterrupted lines of current distribution correspond to a fraction of electrons tunneling without scattering. with phonon scattering it never exceeds 140mV/decade. The reason is that in all cases the turn-on of ambipolar conduction is less sharp than one expects from the ballistic calculations: smaller contribution to conduction exist at higher gate voltages when multiphonon scattering becomes resonant with discrete states.
These results have significant consequences for the device design. One has to include phonon scattering to get the correct value of the gate voltage at which the current is minimal and to predict its exact value. In the presence of phonon scattering, one can obtain a sharp turn-on of current in ambipolar conduction, but at lower doping density and consequently smaller "on"-current.
In summary, we demonstrated that the ambipolar conduction in carbon nanotubes is ruled by phonon-assisted band-to-band tunneling. Phonon scattering shifts of the onset of ambipolar conduction to more positive gate voltage and thereby sets a lower limit for the off current. The steep subthreshold swing expected in tunneling conduction occurs for lower doping of the source and drain and is destroyed by phonon scattering for higher doping levels.
We acknowledge the support of this work by the National Science Foundation's Network for Computational Nanotechnology, NASA, and Intel Corporation.