Vol.2, No.9, 979-983 (2010) Natural Science
http://dx.doi.org/10.4236/ns.2010.29119
Impact of phonon-assisted tunneling on electronic
conductivity in graphene nanoribbons and oxides ones
Povilas Pipinys, Antanas Kiveris*
Department of Physics, Vilnius Pedagogical University, Vilnius, Lithuania; [e-mail], *Corresponding Author:
Received 25 May 2010; revised 13 July 2010; accepted 20 July 2010.
ABSTRACT
Phonon-assisted tunneling (PhAT) model is ap-
plied for explication of temperature-dependent
conductivity and I-V characteristics measured
by various investigators for graphene nanorib-
bons and oxides ones. Proposed model de-
scribes well not only current dependence on
temperature but also the temperature-dependent
I-V data using the same set of parameters
characterizing material under investigation. The
values of active phonons energy and field
strength for tunneling are estimated from the fit
of current dependence on temperature and I-V/T
data with the phonon-assisted tunneling theory.
Keywords: Electronic Transport in Graphene;
Phonon-Assisted Tunneling; Electron-Phonon
Interaction
1. INTRODUCTION
Graphene systems, consisting of one or a few monolayer
of carbon atoms connected via covalent bonds in the
hexagonal lattice, have attracted great interest of re-
searchers because of their peculiar physical and electri-
cal properties and their potential for applications in
nanoelectronics [1-3]. Graphene, in its basic two-dimen-
sional (2D) form, does not have an energy gap separat-
ing the valence and conduction bands of graphene,
which is an essential ingredient for making electronic
devices. However, it is possible to induce the gap pat-
tering single-layer graphene into nanometer size ribbons
creating in this way one dimensional (1D) system similar
to 1D carbon nanotubes [4,5].
This energy gap depends on the width and crystallo-
graphic orientation of the graphene nanoribbon (GNR),
and as was shown in [5] the induced gap’s width varied
from 300 meV to 4 meV with changing nanoribbons
width from 15 nm to 90 nm. Electronic conduction in
GNR is a subject of intense study, both theoretical and
experimental, during the past few years [5-19].
The charge carriers transport through GNR’s shows
some of intriguing peculiarities [19] and due to the band
gap exhibits thermal activation behavior [5,13]. Thus,
from this viewpoint, the conduction in GNR is similar to
the conduction in conventional semiconductors. Ther-
mally activated electrical conductivity is also observed
in graphene oxide sheets [20-24], possessing also the
energy band gap. The thermally activated conductivity in
GNR and oxidized graphene by different authors is ex-
plained in a different way. For instance, Han et al. [13]
thermal behavior of the conductivity in GNR at higher
temperatures explained by thermal emission of the carri-
ers from localized states in the band gap and by one di-
mensional variable range hopping (1D-VRH) at lower
temperatures. Gómez-Navarro et al. [21] suggested that
charge transport in individual chemically reduced gra-
phene oxide sheet occurs via two-dimensional VRH be-
tween intact graphene islands. Kaiser et al. [22] the
temperature-dependent intrinsic electrical conduction in
individual monolayers of chemically reduced graphene
oxide interpreted in the framework of the 2D-VRH in
parallel with electric-field-driven tunneling at low tem-
peratures. Graphene oxide thin film field effect transis-
tors (GO-FET) fabricated by Jin et al. [23] on Si sub-
strates showed p-type semiconducting behavior. The
temperature dependence of the conductance of these
films the authors [23] have explained by VRH with 2 + 3
dimensionality, however, temperature-dependent I-V
characteristics were not explained. Thus, a variety of the
interpretation of the temperature-dependent conductivity
implies that the conduction mechanism is not fully un-
derstood.
We want to note that such behavior of the thermal-
activated conductivity and temperature-dependent I-V
characteristics is usual for polymers and carbon nano-
tubes and in [25,26] has been properly explained by the
phonon-assisted tunneling (PhAT) model based on the
quantum-mechanics [27]. Therefore, in this article we
explore the PhAT model, which account phonon activa-
tion of the electric field stimulated tunneling emission of

P. Pipinys et al. / Natural Science 2 (2010) 979-983
980
electrons from the local states to the conduction band, to
describe the temperature-dependent conductance and I-V
characteristics observed in GNR structures and graphene
oxides.
2. THE PHONON-ASSISTED MODEL
AND COMPARISON OF
EXPERIMENTAL DATA
We suggest that the source of the carriers is the elec-
tronic levels in the band gap of NRB at the metal-nanori-
bbon interface, the electrons from which enter into the
conduction band due to the tunneling stimulated by
phonons under action of the electric field. Assuming that
due to the tunneling released electrons are transferred
through the layer, the current will be equal to:
,I=eNWS (1)
where W is the phonon-assisted tunneling rate, e is elec-
tronic charge unit, N is the surface density of localized
electrons, and S is the area of the barrier electrode. On
this basis we can compare the experimental data on cur-
rent/conduction dependence upon applied voltage and
temperature with computed tunneling rate dependence
on field strength E and temperature W(E,T). For this
purpose we will employ expression presented in [25,26]
which for the tunneling rate from the level of εT depth
gives:


 
1/2
1/21/4
22
1/2
*
1/2
*2
1/2
3/2 2
1/2
11
8
2
4
exp 1
3
1
12
T
T
eE
W
m
m
eE
 



















 



 (2)
where

1/2
*2
1/2
2m
8T eE


 is a parameter, which provides
the temperature dependence for tunneling process. Here
 
2
2821an
   is the width of the absorption
band of the states broadened by the phonons,

1
exp /1
B
nkT 

 

 is the temperature distribution
of phonons,

 is the energy phonon taken part in the
tunneling process, m* is the electron effective mass in
the GNR lattice, and a is the electron-phonon coupling
constant,


2
28
o
a   .
At first in Figure 1 we present the fit of the experi-
mental results on the temperature dependence of the
conductivity in the temperature range from 4 K to 300 K
measured by Han et al. [13] for GNR with the PhAT
-10
-8
-6
-4
-2
0
2
14
16
18
20
22
24
26
1.5 MV/m
ln(dI/dV) (S)
1/T
(
K-1
)
Han et al., 2010,
Fig. 1b, GNR
1.35 MV/m
()
0.0 0.05 0.10 0.15 0.20 0.25
      
T (K)
lnW (s-1)
200 50 20 10 5
(a)
-6
-4
-2
0
2
18
20
22
24
26
ln(dI/dV) (S)
1/T
(
K-1
)
back gated
Han
et al. , 2010, Fig. 4
Graphene nanoribons
E = 1.6 MV/m
0.0 0.02 0.04 0.06 0.08 0.10

lnW (s-1)
200 100 50 25 12.5 10
T (K)
(b)
Figure 1. (a) The temperature dependence of the conductance
minimum of GNR in the temperature range from 4 to 300 K,
extracted from [13, Figure 1(b)] (symbols) fitted to theoretical
lnW(E,T) vs. 1/T dependences (solid lines); (b) The same for
back gated GNRs from Figure 4 in [13]. Data for computation:
εT = 24 meV, 1

 = 5 meV, a1 = 0.1, 2

 = 1 meV, a2 = 6,
m* = 0.8me.
model. The authors [13] suggested thermally activated
behavior at higher temperatures and 1D-VRH at lower
temperatures. The computation of W(E,T) for fitting with
experimental data was performed using for εT the value
of 24 meV assessed in [13], the effective mass of elec-
tron m* was taken to be equal to 0.8 me.
In carbon nanotubes there are exist a large variety of
phonons [28]. We believe that this is and in GNRs sam-
ples. However, the energy of phonons taken part in the
tunneling is unknown. Since the temperature dependence
of the conductivity persists into low temperatures (4 K)
and levels depth is small (24 meV), the energy of pho-
nons should be not be large. The phonons of higher en-
ergy, which probably dominate at higher temperatures,
can be frozen in the low temperature range and therefore

P. Pipinys et al. / Natural Science 2 (2010) 979-983
981
981
the phonons of low energy must be effective. For the
calculation of W(E,T) in this case we used phonons of 1
meV and 5 meV energy, and the total W(E,T) was ex-
pressed as a sum of W1(E,T) and W2(E,T) with 1

 = 1
meV and 2

= 5 meV, respectively. The electron-
phonon coupling constants a1 and a2 were chosen so that
the best fit of the experimental data with the calculated
dependences could be achieved. The fit of the ln(dI/dV)
dependence on 1/T extracted from [13, Figure 1(b) and
Figure 4] with the theoretical dependence ln[W1(E,T) +
W2(E,T)] on 1/T is shown by the solid line in Figures 1
(since the W(E,T) is computed for one value of E it was
not divided by E). As is seen in Figure 1(a) and Figure
1(b), the theoretical curve describes well the temperature
behavior of the conductivity in the entire range of the
measured temperatures.
In Figure 2 the results on temperature dependence of
the current measured by Kaiser et al. [22] for the mono-
layers of chemically reduced graphene oxide in the plot
of lnI vs. 1/T1/3 extracted from [22, Figures 3(a), (b), (c)]
for three values of drain-source voltage Vds and for two
values of gate voltages Vg = 0 and Vg = – 20 V are ex-
posed. For the lowest value of bias voltage (Vds = 0.1 V),
the measured data followed the 2D-VRH law (from 216
K down to 34 K). For the largest bias voltage (Vds = 2 V),
for all values of gate voltage there was a flattening below
~100 K, with temperature-independent behavior below
25 K down to the temperature of 2 K. Such behavior of
the experimental data the authors of [22] have described
by the expression 1/3
10
() exp()GT GBTG, where
the first term represents the usual 2D-VRH conduction
expression and the second term represents purely tun-
neling conduction, i.e., independent of temperature. As
can be seen in Figure 2, the theoretical curves computed
using Eq.2 and phonon of 5 meV energy only describe
both the temperature-dependent and independent of
temperature part of measured data equally well. This is
because the phonons of 5 meV are “frozen” at low
temperatures and independent of temperature pure tun-
neling determines free carrier generation process. At
higher temperatures the tunneling is by phonons stimu-
lated process, consequently, temperature-dependent
process.
Jin at al. [23] the temperature dependence of the con-
ductance in graphene oxide thin films has explained by
2D- and 3D-VRH model. These experimental data in
Figure 3(a) and Figure 3(b) are fitted to tunneling rate
dependence on temperature also in natural logarithm of
the lnW(E,T) vs. T-1/3 and lnW(E,T) vs. T-1/4 plots.
A good fit with experimental data is achieved over all
temperatures with parameters of a = 1.5, m* = 0.8me and

 = 12 meV, for εT using the value of 0.1 eV assessed
in [23].
-26
-24
-22
-20
-18
-16
-14
-12
14
16
18
20
22
24
26
28
E1 = 1.2 MV/m
E2 = 3.6 MV/m
lnI (A)
T-1/3 (K-1/3)
2 V
0.5 V
0.1 V
Kaiser et al. , 2009
graphene oxide
E3 = 7 MV/m
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
     
lnW (s-1)
200 100 50 20 10 5 2
T (K)
(a)
-26
-24
-22
-20
-18
-16
-14
-12
14
16
18
20
22
24
26
28
lnI (A)
T -1/3 (K -1/3)
2 V
0.5 V
0.1 V
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
 
200 100 50 20 10 5 2
T (K)
E1= 1.2 MV/m
E2= 3.6 MV/m
E3= 7 MV/m
lnW (s-1)
(b)
Figure 2. Natural logarithm of the source-drain current I vs.
1/T1/3 extracted from Figures 3(a), (b), (c ) in [22], for different
value of source-drain voltages Vds: (a) at the gate voltage Vg =
0 (symbols), fitted to the theoretical PhAT lnW(E,T) depend-
ences (solid lines); (b) The same at the gate voltage Vg = – 20
V. Parameters for PhAT computation: ћ = 5 meV, a = 1.5, m*
= 0.8me. and (a) εT = 40 meV, (b) εT = 33 meV. Note that in this
case the W(E,T) was computed using one value of the phonon
energy.
Hence, all three models, i.e. 2D-VRH, 3D-VRH and
PhAT in the temperature range from 280 K to 80 K ex-
plain the observed temperature dependence of the con-
ductance well, but at lower temperatures, as can be seen
in Figure 2 the VRH model mismatches. A deviation
from the 2D-VRH model one can see at lower tempera-
tures also in Figure 3. Thus, only from temperature de-
pendence of the conductance cannot be resolved the
conduction mechanism.
For decision on the dominant transport mechanisms
more reliable are the I-V characteristics and their varia-
tion with temperature. In particular such results from [23]

P. Pipinys et al. / Natural Science 2 (2010) 979-983
982
-24
-22
-20
-18
16
18
20
22
lnG
T -1/3 (K -1/3)
Experimental data
2D-VRH model
 PhAT model
Jin et al. , 2009, GO, Fig.3a
0.14 0.16 0.16 0.20 0.22 0.24

lnW (s-1)
(a)
300 250 200 150 100 75
T (K)
(a)
-24
-22
-20
-18
16
18
20
22
lnG
T-1/4
(
K-1/4
)
Experimental data
 3D-VRH
PhAT
Jin et al. , 2009, GO, Fig.3b
T = 0.1 eV
m*= 0.8me
hw = 12 meV
a = 1.5
0.24 0.26 0.28 0.30 0.32 0.34

T (K)
lnW (s-1)
300 250 200 150 100 75
ћ
(b)
Figure 3. Natural logarithm of the measured conductance G
versus (a) 1/T-1/3 and (b) T-1/4 extracted from [23, Figure 3]
(symbols), fitted to (a) 2D VRH, (b) 3D VRH model (solid line)
and comparison with theoretical PhAT (a) lnW(E,T-1/3); (b)
lnW(E,T-1/4) dependences (dashed lines). Parameters for com-
putation: εT = 0.1 eV, ћ = 12 meV, a = 1.5, m* = 0.8me. E = 7
MV/m.
we represent in the Figure 4. As can be seen in Figure 4,
the theoretical dependences W(E,T) computed using for
three different temperatures and for the same parameters
as in Figure 3 match very well with experimental data.
We want to note that the difficulty arises in the frame-
work of the VRH model explaining conductivity de-
pendence on electric-field strength [29] therefore, for
this dependence other models are used [29,30]. Thus, the
results of the discussions can be consistently interpreted
in the framework of the PhAT model.
3. CONCLUSIONS
In conclusion, the thermally activated conductivity in
-2
-1
0
1
2
3
19
20
21
22
23
24
lnISD (nA)
ln(VSD)1/2 (V)
180 K
240 K
300 K
Jin et al., 2009, Fig 2c
lnE (MV/m)
graphene oxide thin film
-1.5 -1.0 -0.5 0.0
lnW (s -1)
1.0 1.5 2.0 2.5
Figure 4. Natural logarithm of the measured ISD versus
ln(VSD)1/2 extracted from [23, Figure 3(c)] (symbols), fit-
ted to theoretical PhAT lnW(E,T) dependences (lines) at
three different temperatures. Parameters for computation:
εT = 0.1 eV, ћ = 12 meV, a = 1.5, m* = 0.8me.
graphene nanoribbons and oxidized graphene can be
explained by the temperature-dependent charge carrier
generation thanks to phonon-assisted tunneling initiated
by electrical field. In GNRs for conductance the phonons
of low energy (1 meV) are also effective. Contrariwise,
in graphene oxides the influence of phonons of low en-
ergy is not noticed. The phonon-assisted tunneling model
describes also the temperature-dependent I-V character-
istics measured in oxidized graphene using the same set
of parameters characterizing the material. From the fit of
experimental data with the PhAT theory the field strength
at which the tunneling occurs and participating in this
process phonon’s energy can be evaluated.
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