Simulation of Graphene Nanoribbon Field
Effect Transistors
Gianluca Fiori, Giuseppe Iannaccone
Dipartimento di Ingegneria dell’Informazione : Elettronica, Informatica, Telecomunicazioni,
Università di Pisa, Via Caruso 16, 56126 Pisa, Italy.
email : g.fiori@iet.unipi.it; Tel. +39 050 2217638; Fax : + 39 050 2217522
May 28, 2018 DRAFT
http://arxiv.org/abs/0704.1875v2
Abstract
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 Schrödinger 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 prohibitive
effective nanoribbon width of 1-2 nm and atomistic precision, that could in principle be obtained with periodic etch
patterns or stress patterns.
Keyworks: graphene, nanoribbon, NEGF, three-dimensional Poisson, Atomistic tight-binding Hamiltonian.
May 28, 2018 DRAFT
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 pz 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.
May 28, 2018 DRAFT
The considered double-gate GNR-FETs have the structure depicted in the inset of Fig. 1. The gates are metallic,
the oxide thickness tox is equal to 1 nm, the channel is 15 nm long, and W is the channel width. The source and
drain extensions are 10 nm long, and are doped with a molar fraction of fully ionized donors f = 5 × 10−3. The
spacing between parallel GNRs is 4 nm.
In Fig. 1 the transfer characteristics of a (12,0) GNR-FET (W=1.37 nm) for drain-to-source voltage VDS of 0.1
and 0.5 V are shown, and compared to those of a (16,0) CNT-FET with the same geometry (same tox, L and device
spacing), whose energy gap (Egap) is close to that of the GNR-FET and equal to 0.6 eV.
Good control of the channel by the gate potential is shown at VDS= 0.1 V, since the sub-threshold swing (S) for
the GNR-FET and the CNT-FET are 64 and 68 mV/dec, respectively. For VDS= 0.5 V we observe a pronounced
degradation of S, with S=191 mV/dec for the GNR-FET and almost 250 mV/dec for the CNT-FET. This has to be
imputed to Hole-Induced Barrier Lowering (HIBL) [12] : in the sub-threshold regime, when sufficiently high VDS
is applied, confined states in the valence band of the channel align with the occupied states in the drain, leading to
band-to-band injection of holes in the channel.
If only elastic band-to-band tunneling can occur (as assumed in our simulation) the excess of holes in the channel
lowers the channel potential, increasing the off current and S, as shown in Fig. 1: the lower the energy gap and the
higher the VDS , the higher the HIBL effect. HIBL is more pronounced in CNT-FETs than in GNR-FETs, because
the conduction band of CNTs is double degenerate and therefore CNTs have twice the density of states of GNRs
with the same gap.
If, on the other hand, inelastic band-to-band tunneling or Schockley-Read-Hall mechanisms are relevant, holes
can recombine with electrons at the source and, instead of HIBL, we observe a leakage current from source to
drain due to gate-induced drain leakage (GIDL) [13].
In strong inversion, the transconductance gm at VDS=0.1 V is 3600 and 6100 µS/µm for the GNR-FET and the
CNT-FET, respectively, whereas at VDS=0.5 V, we obtain gm=4800 µS/µm for the GNR and a gm=8760 µS/µm for
the CNT. The advantage of CNT-FETs is due to the double degeneracy of the conduction band in carbon nanotubes.
It is known that a variability of the chirality of fabricated CNTs yields metallic nanotubes useless for transistor
applications. For GNRs this problem is mitigated, since all GNRs are semiconducting. In order to investigate
quantitatively the effect of a finite fabrication tolerance on the width of GNRs, we have computed the transfer
characteristics of GNR-FETs with different chiralities : (12,0), (14,0) and (16,0).
As can be seen in Fig. 2a, the three devices behave as transistors, but show very different behavior, even if they
differ by only one carbon atom along the channel width. The problem is that the gap is still largely dependent on
the chirality : the (16,0) GNR (W=1.87 nm) has the larger gap (Egap = 0.71 eV), while the (14,0) (W=1.62 nm)
has the smallest gap (Egap = 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
May 28, 2018 DRAFT
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 on-
current 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 Egap. 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-FETs has been performed by means of the self-consistent solution of
the 3D Poisson and Schrödinger equation with open boundary conditions, within the NEGF formalism. Edge states
have been considered at the lateral ends of the nanoribbon using the model proposed by [7]. GNR-FETs exhibit
performance similar to CNT-FETs, also showing significant band-to-band tunneling when small gap devices are
considered and large VDS is applied. GNR-FETs are more robust than CNT-FETs with respect to variability of the
channel chirality, and edge roughness seems to play an useful averaging effect. Finally we suggest that periodic
strain could in principle represent an alternative to etching for inducing an energy gap in graphene.
ACKNOWLEDGMENT
Authors thank Prof. Massimo Macucci for suggestions and fruitful discussions. Support from the European
Science Foundation EUROCORES Programme Fundamentals of Nanoelectronics, through funding from the CNR
(awarded to IEEIIT-PISA) and the EC Sixth Framework Programme, under project Dewint (Contract N. ERAS-CT-
2003-980409) is gratefully acknowledged.
May 28, 2018 DRAFT
REFERENCES
[1] A. Javey, J. Guo, Q. Wang, M. Lundstrom and H. Dai, “Ballistic carbon nanotube field-effect transistors”, Nature, Vol. 424, No. 9593,
pp.654-657, 2003.
[2] International Technology Roadmap for Semiconductor 2005, Available : http://public.itrs.net.
[3] 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. 306, No. 5696, pp. 666-669, 2004.
[4] M. Y. Han, B. Özyilmaz, Y. Zhang, and P. Kim, “Energy Band Gap Engineering of Graphene Nanoribbons”, available at :
http://arxiv.org/abs/cond-mat/0702511.
[5] M. C. Lemme, T. J. Echtermeyer, M. Baus, and H. Kurz, “A Graphene Field-Effect Device”, Electr. Dev. Lett., Vol. 28, No. 4, pp. 282-284,
2007.
[6] Z.Chen, Y-M. Lin, M. J. Rooks and P. Avouris, “Graphene Nano-Ribbon Electronics”, available at : http://arxiv.org/abs/cond-mat/0701599.
[7] Y.W. Son, M.L. Cohen, and S.G. Louie, “Energy gaps in graphene nanoribbons”, Phys. Rev. Lett., Vol. 97, pp.216803, 2006.
[8] S. Reich, J. Maultzsch, C. Thomsen, P. Ordejon, “Tight-binding description of graphene”, Phys. Rev. B, Vol. 66, p. 035412, 2002.
[9] Y. Ouyang, Y. Yoon, J.K. Fodor, and J. Guo, “Comparison of performance limits for carbon nanoribbon and carbon nanotube transistors”,
Appl. Phys. Lett., Vol. 89, pp.203107-203109, 2006.
[10] G.Liang, N. Neophytou, D.E. Nikonov, M.S. Lundstrom, “Performance projections for ballistic graphene nanoribbon field-effect transistors”,
IEEE Trans. Electr. Dev., Vol. 54, pp.677-682, 2007.
[11] G. Fiori, G. Iannaccone ”Code for the 3D Simulation of Nanoscale Semiconductor Devices, Including Drift-Diffusion and Ballistic Transport
in 1D and 2D Subbands, and 3D Tunneling”, Journal of Computational Electronics, Vol. 4, pp. 63-66, 2005.
[12] G. Fiori, G. Iannaccone and G. Klimeck, “A Three-Dimensional Simulation Study of the Performance of Carbon Nanotube Field-Effect
Transistors With Doped Reservoirs and Realistic Geometry” IEEE Trans. Electr. Dev., Vol. 53, pp.1782-1788, 2006.
[13] J. Chen, T. Y. Chan, I. C. Chen, P. K. Ko and C. Hu, “Subbreakdown drain leakage current in MOSFET”, IEEE El. Dev. Lett., Vol. 8
pp.515-517, 1987.
May 28, 2018 DRAFT
http://public.itrs.net
http://arxiv.org/abs/cond-mat/0702511
http://arxiv.org/abs/cond-mat/0701599
Fig. 1. Transfer characteristics of double-gate CNT and GNR-FETs, with doped source and drain reservoirs, with channel length equal to
15 nm, oxide thickness tox equal to 1 nm and channel width W=1.37 nm.
The lateral space is equal to 2 nm. In the inset, a sketch of the GNR-FET is shown.
Fig. 2. Transfer characteristics in the logarithmic a) and linear b) scale of GNR-FETs with different chiralities (12,0), (14,0) and (16,0) (channel
width W equal to 1.37 nm, 1.62 nm and 1.87 nm, respectively), for VDS=0.1 V. The transfer characteristic for the (16,0) GNR-FET, when
roughness at the lateral edge of the GNR is considered is also shown (dashed line). In the inset, a sketch of the graphene nanoribbon is shown,
where randomly decoupled atoms have been highlighted (thick lines).
Fig. 3. Energy gap of a (24,0) GNR, when tensile and compressive strain is considered in correspondence of the middle of the nanoribbon,
as a function of the strain factor by which the Hamiltonians elements of the strained carbon atoms are multiplied.
May 28, 2018 DRAFT
0 0.2 0.4 0.6 0.8
gate voltage (V)
 = 0.1 V
 = 0.5 V
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10 nm
tox = 1 nm
L = 15 nm
double gate
FIG. 1
Gianluca Fiori, Giuseppe Iannaccone
Electron Device Letters
May 28, 2018 DRAFT
0 0.2 0.4 0.6 0.8
gate voltage (V)
GNR (12,0)
GNR (14,0)
GNR (16,0)
GNR (16,0) with
line edge roughness
0 0.2 0.4 0.6 0.8
gate voltage (V)
 = 0.1 V
a) b)
FIG. 2
Gianluca Fiori, Giuseppe Iannaccone
Electron Device Letters
May 28, 2018 DRAFT
0.5 1.0 1.5 2.0
strain factor
(24,0) GNR
tensile strain
compressive strain
FIG. 3
Gianluca Fiori, Giuseppe Iannaccone
Electron Device Letters
May 28, 2018 DRAFT
	Introduction
	Physical model and Results
	Conclusions
	References

  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.

Introduction
	Physical model and Results
	Conclusions
	References
