Impact of phonon-assisted tunneling on electronic conductivity in graphene nanoribbons and oxides ones

doi:10.4236/ns.2010.29119

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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; [email protected], *Corresponding Author: [email protected]

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

1/2

3/2 2

1/2

exp 1

 





































 







 (2)

where



1/2





 is a parameter, which provides

the temperature dependence for tunneling process. Here

 

2821an



  is the width of the absorption

band of the states broadened by the phonons,



exp /1

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,





 .

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

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

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

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

() 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

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

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

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

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

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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