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Simulation of Graphene Nanoribbon Field Effect Transistors

We present an atomistic three-dimensional simulation of graphene nanoribbon field effect transistors (GNR-FETs), based on the self-consistent solution of the 3D Poisson and Schroedinger equation with open boundary conditions within the non-equilibrium Green's Function formalism and a tight-binding hamiltonian. With respect to carbon nanotube FETs, GNR-FETs exhibit comparable performance, reduced sensitivity on the variability of channel chirality, and similar leakage problems due to band-to-band tunneling. Acceptable transistor performance requires effective nanoribbon width of 1-2 nm, that could be obtained with periodic etching patterns or stress patterns.


I. INTRODUCTION
In the last decade, Carbon NanoTubes (CNT) have attracted extraordinary interest for their extremely interesting physical and electrical properties [1], and their potential as an alternative to silicon as channel material for transistors beyond CMOS technology [2]. Recent experiments by Novoselov et al. [3] demonstrated the possibility of fabricating stable single atomic layer graphene sheets, with remarkable electrical properties, that have brought new excitation to the field of carbon electronics. Two-dimensional graphene is a zero gap material, which makes it not suitable for transistor applications.
Energy gap can however be induced by means of lateral confinement [4], realized for example by lithography definition of narrow graphene stripes, the so-called graphene nanoribbons.
Experiments on graphene-based devices [5] and Graphene NanoRibbon FETs [6] (GNR-FETs) have appeared only very recently, and demonstrate limited capability to modulate the conductance of a graphene channel at room temperature. The main problem is the need to fabricate extremely narrow nanowires (of the order of 1 nm) with atomic precision to obtain an energy gap adequate for room temperature operation.
Since at the moment the fabrication technology is at its very first steps, computer simulations can be very useful to provide physical insights of GNR-FETs and to estimate the attainable performance. Recent theoretical works have shown that graphene nanoribbons have an energy gap which has an oscillating behavior as a function of width, with average roughly proportional to the inverse width, and that edge states play a very important role in inhibiting the existence of fully metallic nanoribbons [7]. Such behavior cannot be reproduced if one does not consider edge effects [8].
Also from the simulation point of view, research on GNR-FETs is at an embryonic stage: the only works available in the literature [9], [10] are based on a semiclassical analytical top-of-the-barrier model. For short-channel transistors only a three-dimensional simulation is suitable for an accurate evaluation of the electrostatics and of intraband and interband tunneling.
To this purpose, we have developed a code for the simulation of GNR-FETs, based on the Non-Equilibrium Green's Function formalism (NEGF), with a tight-binding Hamiltonian built from p z orbital basis set in the real space, which has been included in our in-house three-dimensional device simulator NANOTCAD ViDES [11]. We will show that GNR-FETs have performance comparable with CNT-FETs, and that can be greatly affected by the channel width and edge roughness.

II. PHYSICAL MODEL AND RESULTS
Our approach is based on the self-consistent solution of the three-dimensional Poisson and Schrödinger equations with open boundary conditions [12], which is able to take into account fully ballistic transport, in order to outline the higher limits of device performance, as well as elastic scattering due to line edge roughness. The Hamiltonian is taken from [7], in which edge states at the nanoribbon lateral ends have been considered. In this work we refer to (N ,0) armchair graphene nanoribbons, which consist of an unrolled (n,0) zig-zag nanotube with N = 2n. has the smallest gap (E gap = 0.13 eV). As a consequence, the (16,0) device show the best gate control over the channel potential, while the (14,0) the worst : the energy gap is so small that elastic band-to-band tunneling occurs at the source and current is dominated by GIDL.
Such problem is reduced if rough edges are considered. We have considered the impact of line edge roughness in a (16,0) GNR-FET device, by randomly decoupling carbon atoms on the lateral boundaries of the GNR. The transfer characteristic for one example is shown in Fig. 2 (dashed line). Since the channel consists of several hundreds of rings, the rough GNR behaves as a GNR with an intermediate effective gap. More statistical simulations would be needed to assess the dispersion of the electrical characteristics, but the typical GNRs is probably long enough to provide sufficient averaging to suppress inter device dispersion. Rough edge scattering strongly affects the oncurrent and the transconductance suppressing it by about 30% with respect to fully ballistic transistors. Additional suppression in realistic GNR-FETs can be due to defects, ionized impurities and phonon scattering.
From the above simulations, it is clear that lateral confinement way beyond state-of-the-art etching capabilities would be needed to obtain adequate E gap . We also found that electrostatic periodic potential modulation with a peak-to-peak value of few Volts is not sufficient to induce the required gap of few hundreds mV.
In order to evaluate whether a periodic strain pattern can allow to engineer the GNR gap, we have computed the energy gap in a (24,0) GNR (W =5.86 nm), multiplying the overlap integral of the element of the Hamiltonian in correspondence of the couple of atoms in the middle of the GNR by a "strain factor" σ : σ is larger than one for compressive strain, and smaller than one for tensile strain. As can be seen in Fig. 3, compressive strain seems to be able to increase the energy gap of the nanowire by a significant amount. Of course, we can only suggest to experimentalists to evaluate the option.

III. CONCLUSIONS
In this work, a simulation study of GNR-  The lateral space is equal to 2 nm. In the inset, a sketch of the GNR-FET is shown.