Nanomaterials and Nanotechnology
Numerical Techniques for the Analysis
of Charge Transport and Electrodynamics
in Graphene Nanoribbons
Invited Feature Article
Luca Pierantoni1,2,* and Davide Mencarelli1
1 Università Politecnica delle Marche, Ancona, Italy
2 INFN-Laboratori Nazionali di Frascat, Frascati, Italy
* Corresponding author: email@example.com
Received 18 Oct 2012; Accepted 14 Nov 2012
© 2012 Pierantoni and Mencarelli; licensee InTech. This is an open access article distributed under the terms of the Creative
Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,
distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract In this paper, we report on multiphysics full‐ 1. Introduction
wave techniques in the frequency (energy)‐domain and
the time‐domain, aimed at the investigation of the The theoretical, scientific and technological relevance of
combined electromagnetic‐coherent transport problem in carbon‐based materials (carbon nanotubes, graphene)
carbon based on nano‐structured materials and devices, have been highlighted in a variety of works, both
e.g., graphene nanoribbons. experimental and theoretical [1‐11]. They are fated to
become competitive and compatible with the established
The frequency‐domain approach is introduced in order to silicon technology for applications to electronics. The
describe a Poisson/Schrödinger system in a quasi static analysis of charge transport in carbon nano‐structures can
framework. An example of the self‐consistent solution of be carried out by discrete models, such as tight binding
laterally coupled graphene nanoribbons is shown. (TB), and continuous models, such as effective mass and
k∙p approximations, which stem from the approximation
The time‐domain approach deals with the solution of the of TB around particular points of the dispersion curves.
combined Maxwell/Schrödinger system of equations. The These techniques are suited for the analysis of
propagation of a charge wavepacket is reported, showing CNT/graphene/GNR in a variety of problems such as
the effect of the self‐generated electromagnetic field that bending [17‐18], lattice defects and discontinuities ,
affects the dynamics of the charge wavepacket. and edge terminations [19‐20]. However the latter
methods require high computational resources, and can
Keywords Dirac Equation, Graphene Nanoribbon, hardly include the effect of i) the self‐generated
Quantum Electrodynamics, Transmission Line Matrix electromagnetic field, ii) impinging external EM fields.
Recently, we have introduced full‐wave techniques (fig.
1) both in the frequency (energy)‐domain [21‐26], and the
time‐domain [28‐36] for the investigation of new devices
www.intechopen.com Nanomater. nanotechnol., 2012, Vol. for the Analysis
Luca Pierantoni and Davide Mencarelli: Numerical Techniques 2, Art. 13:2012 1
of Charge Transport and Electrodynamics in Graphene Nanoribbons
based on carbon materials, namely carbon nanotube hopping elements of the Hamiltonian from a unit cell to
(CNT), multiwall (MW) CNT, graphene and graphene the previous one from the left (right), and E is the
nanoribbon (GNR). injection energy.
For both the approaches, the quantum transport is In , we showed that fundamental physical constraints
described by the Schrödinger equation or its Dirac‐like and consistence relations in quantum transport, such as
counterpart, for small energies. The electromagnetic field reciprocity and charge conservation, correspond
provides sources terms for the quantum transport respectively to familiar reciprocity and power
equations that, in turn, provide charges and currents for conservation in a microwave field. We emphasized that
the electromagnetic field. the proposed approach allows handling multiport
graphene systems, where carriers can get into (and out of)
In this contribution, we report some new examples of many different physical ports, each characterized by their
self‐consistent quasi‐static calculations, where charges’ own chirality and possibly by a large number of virtual
transport is affected by the self‐generated potential, in ports, i.e., electronic channels or sub‐bands. Interesting
addition to the electrostatic potential applied by external results involve new concept‐devices, such as GNR nano‐
electrodes, in a typical FET configuration [25,26]. transistors and multipath/multilayer GNR circuits, where
Regarding the time‐domain technique, we show the charges are ballistically scattered among different ports
dynamics of a charge wavepacket from source to drain under external electrostatic control. We developed a in‐
electrodes in a GNR realistic transistor environment. house solver for simulating CNT short‐channel
transistors, with a user friendly interface. The software,
written in Matlab, has been, in particular, focused on the
simulation of GNR short‐channel transistors, as shown in
fig. 1. In modelling the graphene‐metal contact, we
introduce a sort of metal doping of GNR, coherently with
experimental observation; in fact, graphene over metal
seems to preserve its unique electronic structure, and the
metal just shifts the graphene Fermi level with respect to
Figure 1. Frequency‐ and time‐domain techniques. the conical point, by a fraction of eV . Possibly, the
metal contact opens just a small (tens to hundreds eV)
2.1 Frequency‐domain: Poisson‐coherent transport bandgap.
We perform the analysis of self‐consistent charge 2.2 Time‐domain: Maxwell‐coherent transport
transport by using a scattering matrix technique ,
which is physically equivalent to the Green’s function In the time‐domain, a full‐wave approach has been
approach, usually referred to as non‐equilibrium Green’s introduced: the Maxwell equations, discretized by the
function (NEGF) method. In synthesis, each GNR port, transmission line matrix (TLM) method, are self‐
seen as the termination of a semi‐infinite waveguide, is consistently coupled to the Schrödinger/Dirac equations,
described by means of a basis of electronic discretized by a proper finite‐difference time‐domain or a
eigenfunctions, that, in turns, are solution of the GNR TLM scheme [28‐29].
unit‐cell under periodic condition. The analysis is fully
self‐consistent since the solution of the transport The goal is to develop a method that accounts for
equation, and the solution of the Poisson equation for the deterministic electromagnetic eld dynamics, together
electrostatic potential generated by the GNR charge with the quantum coherent transport in the nanoscale
density, are obtained by using an iterative approach. In environment. In [29‐30], we introduced exact boundary
the scattering‐matrix approach, a multimode conditions that rigorously model absorption and injection
transmission matrix model of quantum transport allows of charge at the terminal planes, in a realistic field effect
easy simulation of very large structures, despite the transistor environment.
possibly high number of electronic channels involved.
In order to characterize a GNR, periodic along the z‐ Several examples of the electromagnetics/transport
direction, the Hamiltonian of the unit cell is appropriately dynamics are shown in [28‐29]. It is highlighted that the
rearranged by selecting three consecutive unit cells self‐generated electromagnetic field may affect the
dynamics (group velocity, kinetic energy, etc.) of the
H l l H 0 H r r E (1) quantum transport. This is particularly important in the
analysis of time transients and in describing the
where ψl, ψr, ψ, are the wavefunctions of three behaviour of high energy carrier bands, as well as the
consecutive unit cells and matrix Hl(Hr) denotes the onset of non‐linear phenomena due to external impinging
2 Nanomater. nanotechnol., 2012, Vol. 2, Art. 13:2012 www.intechopen.com
electromagneetic fields. For graphenne/GNR, in the optiics. FDTD is a more gen ue,
neral techniqu suited for r
an EM field, th
presence of a tion reads:
he Dirac equat disc erent kinds of equations, e.g., parabolic,
cretizing diffe f ,
hyp With respect to FDTD, TL is directly
perbolic, etc. W LM y
ie n y,
related to the discretization of, mainly hyperbolic c
i σ p qA c
equations (Maxel Dirac), but it has the addvantages that t
t (2) each portion of t
h d a
the segmented space has an equivalent t
loca electric cir Moreover, TLM can easily
rcuit . M M y
i σ p qA c
t incoorporate extern nal sources as oltage/current
s equivalent vo t
The solution of the Dirac
n c/graphene eq
quation (2) is the
four compon nent spinor commplex wavefuunction ψ(r,t): In TLM, that is c
T considered as the implemen e
ntation of the
Huy ygens principl le, propagatioon and the scaattering of the e
wav amplitudes are express tor
sed by operat equations s
ψ(r, t ) 1 2 3 4 (3)
. The latter property is well illustr rated in the e
Symmmetrical Con ndensed Node (SCN) formulation .
where A and ϕ are vector and scalar p
d r potentials, dirrectly
related to th EM field th hrough the ap ppropriate ga auge,
e.g., the “Lor rentz” gauge, and q is the e electron charge; Vp
is the static p Pauli
file. In eq. (2), are the P
potential prof ,
matrices, p is the canonic cal (linear) mo omentum, k is the
kinematic m momentum, t es
that, include the EM field
p i ˆ ˆ
k p qA r, t (4)
The computa ational scheme e develops as follows: i) the e EM
cretized by t
field is disc the Transmiss sion Line M Matrix
method usin the Symme etrical Conden nsed Node (S SCN)
approach. ii) Quantum ph e
henomena are introduced in a
subregion of the 3D‐dom D‐2D dimensi
main, e.g., a 1D ional
CNT region described b the Schrö ödinger equa ation, Figu 2. Concept o the full‐wave time‐domain technique. The
ure of e e
and/or a 2D graphene/na anoribbon reg gion, described by
tromagnetic fiel provides sou
ld urces for the quantum device
the Dirac equation. iii At each time step, the
i) , in turn, provid
that, mechanical) curr
des (quantum‐m r
rent sources for
Schrödinger/ /Dirac equatio is solved b accounting for
on by g electromagnetic
the e c field.
the quantu um device boundary co onditions, in nitial
conditions (ee.g., injected charge), and additional so ource In [331‐32], we exp plored the corr relation betwe een Dirac and d
terms constit tuted by the E EM field, samp pled in the dom main Max xwell equation ns, in the time e domain; tran nsmission‐linee
of the quan s).
ntum device(s iv) From the wavefunc ction m
equations, valid for both EM and quantum current are e
(charge) solu Schrödinger/D
ution of the S Dirac equation we
n, deriived. This is a step forw ward toward an effective e
derive the qquantum mech hanical (QM) current over the r integration of the Dirac th heory in th numerical
device doma ain. This current is a disstribution of local simu ulation of EM M field problem ms.
sources for the EM field that is inject
t ted into the TTLM
nodes, loca ated only o on the grid points of the In [3
33], we presen nted, for the first time, a TL LM condensed d
Schrödinger/ /Dirac equatio domain. v) At the next time
on ) nod scheme fo solving th Dirac equ
de or he D
uation in 2D
step t+1, the TLM meth hod provides a new upd dated grapphene. This scheme satis sfies the stan ndard charge e
distribution o field values that are, again, sampled over cons servation requ uirement and allows adopting boundary y
domain, and s on, iterativ
the device d so vely. In fig. 2, the cond ditions for graaphene circuit ts.
scheme of the e method is de epicted in the case of graph hene.
The correlation be etween the graphene/Dirac equation and d
The reason for choosing TLM for the discretizatio of on nt
its self‐consisten symmetri ical condens sed node ‐
quations has to be hig ghlighted. Sp pace‐ trannsmission line matrix formu hlighted. This
ulation is high s
discretizing mmethods, like nite‐differe
e ence time‐do‐m main conc o
cept, in turn, is related to the generalized Huygens s
(FDTD) and t transmission l line matrix (TL LM) , are w well‐ prinnciple for the DDirac equation ns.
known techn niques that allo ow the EM full‐wave mode elling
of 3D structtures with ne y
early arbitrary geometry f a for The above techn nique has be een already used for the e
wide range of applicatio M
ons from EM compatibilit to ty inveestigation of realistic and intriguing ap n
www.intechopen.com Luca Pierantoni and Davide Mencarelli: Numerical Techniques for the Analysis 3
of Charge Transport and Electrodynamics in Graphene Nanoribbons
novel areas, bridging nanoscience and engineering can still imply a strong effect when wider GNR, i.e.,
applications. We could define this research area as smaller band gaps, are considered.
“radio‐frequency nanoelectronic engineering”, [39‐40].
In , we analyse the idea of realizing a harmonic radio‐ It is noted that the self‐consistent potential of fig. 4b is
frequency identification (RFID), based on “tag on paper” strongly different from the potential of fig 4a; as largely
with embedded graphene as a frequency multiplier. expected, changing the distance between the GNR does
not simply imply a potential “composition” following a
In [35‐36], we introduce a model for the metal‐carbon superposition of effects ‐ the iterative process develops
contact. The metal‐carbon transition is one of the most very differently in the two cases and the final results are
challenging and not completely understood problems not easily predictable.
that limits production and reproducibility of
nanodevices, arising due to the difficulty of engineering
the contact resistance between metal and nano‐structures.
3.1 Frequency‐domain: Schrödinger‐Poisson
In order to show the potentialities of our approaches, in
the following we show the comparison between the
potential distributions in a region occupied by two
laterally coupled GNR. The coupling takes place by
means of the Coulomb interaction. The schematic view
of the device under study is shown in fig. 3: two
semiconducting GNRs connect the source and drain of a a)
A potential difference of 0.1 V is applied between drain
and source; the source is assumed at 0 V, equipotential
with the lateral gate (G). The nanoribbons are about 2.2
nm wide and the area of the square “window” delimited
by the electrodes is 20x20 nm2.
Figure 4. Self‐consistent potential for different distances of the
two coupled GNR channels: a) d=2.4 nm b) d=0.15 nm.
G source G
3.2 Time‐domain: Dirac‐Maxwell
Figure 3. A two‐channel GNR‐FET; d is the distance between the We analyse the space‐time evolution of a Gaussian charge
two GNR channels. wavepacket ||2, with a broad energy band (up to 1 eV),
propagating on a “metallic” GNR (150x5 nm), as shown
In the following, we report the numerical result in fig. 5. We consider the GNR in a realistic FET
obtained after numerical convergence, expressing the environment, with two metallic source‐drain electrode
self‐consistent potential in the plane of the nanoribbons. contacts. In order to model the injection‐absorption of
We somehow exaggerated the effect of the metal doping charge, we apply absorbing boundary conditions as in
by assuming a 2.9 eV shift of the Dirac point, in order to . In fig. 6 (a), we show the charge wavepacket
place the Fermi level about 0.7eV above the band gap, evolution after t=0, t=20, t=50, t=100 fs, respectively. The
and to have appreciable charge injection from the metal correspondent transversal and longitudinal current
to nanoribbon “bridges”. In practice, a smaller doping components are reported in Fig. 6 (b), for t=20, t=50 fs.
4 Nanomater. nanotechnol., 2012, Vol. 2, Art. 13:2012 www.intechopen.com
5nm 6 fs
Figure 5. Proppagation of a ch harge wavepack nce of
ket in the presen
ial barrier, with E=0.45 eV.
a static potenti
6 nm 6 fs
150 nm 2
t=20 fs t=50 f
ure 7. Spatial di acket at t=0, t=2,
istribution of a charge wavepa ,
f e on b):
t=4 fs. (a): only the Dirac equatio is solved. (b the coupled d
Diraac‐Maxwell system is computed d.
he re, ow
In th same figur (c), we sho the propagation of two o
puls ses launched t through the so ource and dra ain electrodes, ,
propagatio of two pulses for tt=0, t=20, t=50,, t=100 fs.
t=0 fs er ce
We then conside the presenc of a potent f
tial barrier of
0.45 eV with respect to bounding material (e.g., metal l
cont tacts). In fig. 7, we plot the spatial, longitudinal
distributions of th charge wa avepacket in thhree different t
e‐steps, t=2, 4, 8 fs, respectiv
The core point is that in one ca we solve only
ase (fig.7, a), w y
the Dirac equation n and do not c consider
t=100 fs the self‐induced E EM field, whereas in the oth her case (fig.7, ,
we consider th
b), w he coupled Dir rac‐Maxwell s system.
Figure 6. Timee‐evolution of th al and
he wavepacket (a). Transversa , on
We observe that, depending o the initial energy of the e
currents (b). Tw launched p
longitudinal c wo pulses (c) from the
m char rge wavepack ket, the self‐ind
duced electrom magnetic field d
source and draain terminals. affects the propag gation charact teristics.
www.intechopen.com Luca Pierantoni and Davide Mencarelli: Numerical Techniques for the Analysis 5
of Charge Transport and Electrodynamics in Graphene Nanoribbons
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of Charge Transport and Electrodynamics in Graphene Nanoribbons