ICSE2010 Proc. 2010, Melaka, Malaysia

Carbon Nanotube Conductance Model in Parabolic

Band Structure

Mohammad Taghi Ahmadi, Zaharah Johari, N.Aziziah Amin, S. Mahdi Mousavi and Razali Ismail Department of Electronic Engineering, Faculty of Electrical Engineering,Universiti Teknologi Malaysia, 81310 Skudai,

Johor Darul Takzim, Malaysia Email: ahmadiph@gmail.com

Abstract-Fermi dirac integral is applied to study the parabolic

band structure of Carbon Nanotube (CNT) which is in the range

of minimum band energy. In this letter electronic transport

property of one dimensional carbon nanotube with parabolic

band structures near the charge neutrality point is investigated.

The temperature dependent conductance model which shows

minimum conductance near the charge neutrality point and

decreases by decreasing the temperature is presented. CNTs with

micrometer length exhibit nondegenerate behavior on

fundamental band structure similar to the conventional long

channel devices.

I. INTRODUCTION

A single wall carbon nanotube can be assumed as a one atom

thickness of graphite (called graphene) rolled up into a cylinder

as shown in fig. 1. This cylinder with diameter less than De-

Broglie wave length indicates one dimensional behavior also

its band structure near the minimum band energy shows

parabolic relation with wave vector [1-3].

Fig.1 A prototype single wall carbon nanotube with length much more than

De-Broglie wave length and diameter less than De-Broglie wave length.

Transport properties as a centre of attraction in the theoretical

and experimental researches creates a lot of efforts on carbon

nanotube carrier statistic study. On the other hand, conductance

is an important subject in carrier transport phenomena have

been studied in nanotechnology aspect [4-5]. Broad-spectrum

study of conductance on single-walled carbon nanotube

(SWNT) and its

simulation indicate discrepancy between the theory and

experiment [6]. Therefore numerical method based on quantum

confinements effect can be used to improve the functionality of

CNTs. Nanotubes with high conductance are appropriate for

large current field emission devices. Besides, they are suitable

as carbon nanotube field effect transistors (CNT-FET) and

sensors as well [7] [8]. In addition, rapid decrease in

conductance at low voltages by decreasing the gate voltage

can be explained by Coulomb blockade effects [9] . In order to

reduce this effect, scandium (Sc) also can be used as a high-

quality Ohmic contact in order to achieve the high-performance

n-type CNT field effect transistors [10] [11].

II. CONDUCTANCE MODELING

Applying the Taylor series expansion on graphene band

structure near the Fermi point, the E (k) relation of the CNT

can be obtained as

22)3

2(

2

3 )( x

CC kd

atkE +±= −

(1)

where A1.42a CC =− is Carbon-Carbon(C-C) bond length,

t=2.7 (eV) is the nearest neighbor C-C tight binding overlap

energy and xk is wave vector component along the length of

the nanotube.

The band gap energy can be assumed as

nm)(in d (ev) 0.8t2a

E CC

G == −

d (2)

Therefore energy relation can be written in simplified form as

2

2

31

2)

dk(

EE(k) xG += (3)

256 978-1-4244-6609-2/10/$26.00 ©2010 IEEE

ICSE2010 Proc. 2010, Melaka, Malaysia

Equation (3) indicates relationship between energy and wave

vector is not parabolic. Applying the square root approximation

leads to parabolic relation between energy and wave vector.

2 2

*2 2

g xE k

Em

≈ +h

(4)

where*m is the effective mass of CNT. Plotting the CNT

energy band of equations (3) and (4) in fig. 3, it is obviously

shown that the band structure is parabolic at certain range of

energy in the E-k relationship.

Fig. 3: The band structure of CNT near the minimum energy is parabolic.

In parabolic part of the band energy, the wave vector can be

extracted as

2

C C

4 8

3a 9

Ek

t d−

= − (5)

Number of actual modes, M(E) at a given energy is dependent

on the sub bands location. For example, if the related energy

includes the bottom of the conduction band then parabolic

approximation of band diagram can be used then the mode

density M(E) increases with energy. In the valence band, any

information related to the sub bands are more difficult to

obtain, because the coupled multiple bands that are increasing

and difficult dispersion relations are needed. By taking the

derivatives wave vector k over the energy E (dk/dE) of

equation (5), the number of the mode M(E) can be written as

12

C C

2

C C

3a 4 8( )

2 3a 9

tE EM E

k L L t d

−

−

∆= = −

∆ ⋅ (6)

where L is the length of the nanotube. Now taking into

consideration of spin degeneracy, the number of conducting

channels can be finalized as

1

2

C C

2

C C

3a 4 8( ) 2

3a 9

tE EM E

k L L t d

−

−

∆= = −

∆ ⋅

(7)

That was not clear till 1980s that there is a maximum

conductance for a channel with one level. That is a

fundamental constant proportional to the Planck’s constant and

electron charge. 2

0

qG

h= (8)

Where q is electron charge and h is Plank constant. In fact

levels of up spin and down spin in the small channels naturally

with same energy as a degenerate level results the maximum

conductance two times larger than this amount which is equal

to 2G0. In the bad contact, measured conductance is always

lower than this value. Based on Landauer formula, the

conductance on large channel can follow the Ohmic scaling

law but in the smaller size, one need to apply two possible

corrections on this law, first we need to work on the interface

resistance which independent of the length. Secondly,

conductance related to the width nonlinearly and depends on

the number of the modes in the conductor which is quantized

parameter in Landauer formula where both of these features are

corporate.

The average probability, T of injected electron at one end will

transmit to the other end and in our ballistic channel this

parameter is equal to one. The expression df

dE is considerable

only near the Fermi energy, indicating that the number of

actual modes at the Fermi energy is two [12].

22( ) ( )

q dfG dEM E T E

h dE

+∞

−∞

= −

∫ (9)

Replacement by the number of sub bands (mode numbers) in

corporate with Fermi – Dirac distribution function conductance

is related to the length of nanoribon as well.

1 122 2

C C C C

2

C C

3a 2a2 4 1

3a 31

F

B

E E

k T

t tqG E d

h L t de

+∞

− −

−

− −∞

= − −

+ ∫

(10)

Temperature effect on nanotube conductance can be seen by

changing the boundary of integral as follow

257

ICSE2010 Proc. 2010, Melaka, Malaysia

( )1 1

2 2 212

C C

0 0

43a

1 1

1 1

B

x x

q x xG tk T dx dx

hL

e eη η

π− −+∞ +∞

−

− +

= + + +

∫ ∫

(11)

where g

B

E Ex

k T

−=

and normalized Fermi energy,

F g

B

E E

k Tη

−= . Presenting Fermi-Dirac integral form of

conductance is useful to understand the role of degenerate and

nondegenerate regimes. Nondegenerate approximation on

Fermi – Dirac integral can be used when Fermi level in band

gap is far from conduction and valence band age more than

3kBT. If the Fermi level lies inside the valance or conduction

band or located 3kBT in the interior of the band edge

degenerate approximation has to be used.

( )2

12

C C 1 1

2 2

43a ( ) ( )B

qG tk T

hLπ η η− − −

= ℑ +ℑ −

(12)

The experimental results are in good agreements with

theoretical calculations presented in this paper as shown in fig.

2 [12]. The presented model here provides possibility towards

emerging carbon nanotube based quantum devices.

Fig.2 Comparison between model (Solid line) and experimental (dotted line)

displays good agreement between theoretical model and experimental data.

As shown in the fig. 2, low conductance with respect to the

gate voltage indicates minimum conductance which is related

to the minimum conductivity at the charge neutrality point

(known as Dirac point) in the neighborhood of the charge-

neutrality point carriers follow linear energy-momentum

dispersion relations. Unlike the three dimensional Graphene,

CNTs show a decrease in minimum conductance more than an

order of magnitude at low temperatures as shown in fig. 3. The

defeat of G near the Dirac point recommends the energy gap in

the CNT.

Fig.3. Conductance of GNRs as a function of gate voltage plotted at different

temperatures Significant shift of the minimum conductance at Dirac point

may be effect on reduction of carrier mobility in CNTs. The

temperature dependence of conductance in CNTs compare to

that of the three dimensional Graphene samples where conductance at Dirac point changes less with variation of

temperature from 0K to 300K shows different behavior .

Also similar to the Anantram et al. work [12] which indicate

that the narrower CNTs demonstrate the greatest dominance of

minimum conductance at Dirac point. The energy gap

decreases with increasing the ribbon width. CNT length effect

can be discussed in terms of degeneracy phenomena as shown

in fig. 4 in which by increasing the length of CNT,

nondegenerate approximation will be dominant. In the

nondegenerate limit, Fermi-Dirac integral can be convert into

the exponential equation this approximation as

( ) [ ])()(21

26 ηηπ −− += eeTtka

hL

qG BCC (13)

258

ICSE2010 Proc. 2010, Melaka, Malaysia

Fig. 4 nondegenerate approximations (dotted line) near the Dirac point is fitted

on real conductance graph

As shown in fig. 5 for the CNT with L=10-20 nm a gap region

appears for 25<Vg< 30V. Near the Dirac point but outside of

the gap region, the conductance balances with the width of the

CNT. Besides, increasing the CNT length effect the gap region

near the charge neutrality point as shown in fig. 5 particularly

in the long channel, nondegenerate approximation can be used

properly. As a device channel, the length and active CNT width

contributing in charge transport, respectively. A narrower CNT

with possible larger band gap makes them as a semiconducting

device component. However, in order to apply the CNT

remarkable electrical appearance in nanoelectronic, ability to

producing a band gap is highly required in CNT, therefore,

strong demand are being made to explore the electrical

properties of one-dimensional carbon nanostructures. The

Fermi level variation and number of conduction modes that

contribute to the transport in CNTs also governed by bias

voltage between the source and drain contacts. Carbon

nanotubes with virtually perfect conductance can be used as a

FET channel. Ineffective surface scattering, disorder, defects

and phonon scattering on conductance, can be explained as

• Larger mean-free-path in micro meter indicates

ineffective acoustic phonon [13]. Also, zone

boundary and optical phonon scattering is ineffective

at room temperature and small biases.

• CNT without disordered boundaries leads to perfect

interface in CNT field effect transistors.

• Finally, reflection probability caused by disorder

defects is very small due to a large velocity of carriers

[14].

III. CONCLUSION

Carbon nanotube conductance in parabolic part of the band

structure is calculated which shows CNTs with enormous

conductance can be assumed as a FET channel for the future

nanoelectronic high speed devices. Based on quantum

confinement effect with parabolic band structure, conductance

in CNTs is a function of Fermi - Dirac integral which is based

on Maxwell approximation in nondegenerate limit especially

for long channel case. The temperature dependence of

conductance in CNTs at Dirac point is presented and CNT

length effect on minimum conductance at neutrality point is

discussed in terms of degeneracy phenomena.

ACKNOWLEDGMENT

The work is supported by postdoctoral fellowship scheme for

the project “Nanoscale device modeling and simulation”,

faculty of Electrical Engineering, managed by the UTM

Research Management Center (RMC).The authors would like

to thank the MOSTI and RMC for cordially sponsoring this

work.

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