Available at: http://www.ictp.it/~pub−off IC/2005/078

United Nations Educational Scientific and Cultural Organizationand

International Atomic Energy Agency

THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

THERMOELECTRIC FIGURE OF MERIT OF CHIRAL

CARBON NANOTUBE

N.G. MensahDepartment of Mathematics, University of Cape Coast, Cape Coast, Ghana,

G.K. Nkrumah-BuandohDepartment of Physics, University of Ghana, Legon, Accra, Ghana,

S.Y. Mensah1

Department of Physics, Laser and Fibre Optics Centre, University of Cape Coast,

Cape Coast, Ghana

and

The Abdus Salam International Centre for Theoretical Physics, Trieste Italy,

F.K.A. AlloteyInstitute of Mathematical Sciences, Accra, Ghana

and

The Abdus Salam International Centre for Theoretical Physics, Trieste Italy

and

Anthony K. TwumDepartment of Physics, Laser and Fibre Optics Centre, University of Cape Coast,

Cape Coast, Ghana.

MIRAMARE – TRIESTE

September 2005

1Regular Associate of ICTP.

Abstract

We have investigated the thermoelectrical properties of chiral carbon nanotube and numer-

ically evaluated the figure of merit. We observed that the properties are highly anisotropic and

depend on the geometric chiral angle (GCA) θh, temperature and the overlapping integrals (ex-

change energy) for the jumps along the tubular axis ∆z and the base helix ∆s. The thermopower

α exhibited giant values with the peak occuring between 100 K and 150 K. The electron thermal

conductivity showed unusually high value with the peaks shifting towards high temperature.

We attribute the high peak values to electron-phonon interactions. Finally we noted that by

changing the ∆s and ∆z it is possible to get a figure of merit greater than 1.

1

1 Introduction

Tremendous effort was made in thermoelectric materials research in the late 1950s and 1960s after

Ioffe first proposed the investigation of semiconductor materials for utilization in thermoelectric

application [1]. By the 1970s, research on thermoelectric materials had begun a steady decline

and essentially vanished by the 1980s. However, since the early 1990s, there has been a resurgence

of interest in the field of thermoelectric materials research [2-12]. Mensah and Kangah [13]

proposed the use of superlattice as efficient thermoelement. Hicks and Dresselhaus [14] proposed

that the figure of merit of thermoelement might be improved if two dimensional structures were

used. In fact, a steady stream of papers have been published. Some of the theories concentrate

on single quantum well, others discuss multiple quantum wells and others on thin films.

Currently the research on thermoelectrics has been focused on carbon nanotube. This is

mainly because this remarkable material has high electron conductivity, giant thermopower [15-

18] and unusually high thermal conductivity [19-21]. What is even interesting with this material

is its ability to display n-type semiconductor, metallic conductor and p-type conductor by doping

it and hence changing the energy band [16]. This property will enable the production of the

thermoelement with only carbon nanotubes.

Carbon nanotube (CNT) belongs to the family of carbon-based structures formed by wrap-

ping graphite sheets into tube shaped objects. Each CNT is characterized by a chiral index

(n,m) with m and n being two integers which specify it uniquely. In fact, CNTs can also exhibit

metallic or semiconducting properties as stated above also by changing its cross-sectional radius

and geometric chiral angles (GCA) [22-24]. These properties can also be determined using the

chiral index (n,m). When n−m = 3q, (q = 0, 1, 2, . . .), the material exhibits metallic properties.

On the other hand when n−m = 3q +1, (q = 0, 1, 2, . . .) it exhibits semiconducting properties.

In this paper, we calculate the electron conductivity, the Peltier coefficient (and hence the

thermopower), the zero-current density electron thermal conductivity and the figure of merit of

CNT. We use the approach developed in [16, 17, 20, 21]. We noted that the properties mentioned

above strongly depend on the GCA θh, temperature T and real overlapping integrals for jumps

along the tubular axis ∆z and the base helix ∆s. The variation of these parameters can give

rise to giant thermopower, unusual high electron thermal conductivity and the figure of merit

greater than 1, making CNT very good material for the production of thermoelement.

This paper is organised as follows: in section 2 we establish the theory and solution of the

problem, and in section 3 we discuss the results and draw conclusions.

2 Theory and Results

Single walled-carbon nanotube (SWNT) is considered as an infinitely long chain of carbon atoms

wrapped along a base helix. The problem is considered in the semiclassical approximation,

2

starting with the Boltzmann kinetic equation [13],

∂f(r, p, t)

∂t+ v(p)

∂f(r, p, t)

∂r+ eE

∂f(r, p, t)

∂p= −

f(r, p, t) − f0(p)

τ(1)

Here f(r, p, t) is the distribution function, f0(p) is the equilibrium distribution function, v(p)

is the electron velocity, E is a weak constant applied field, r is the electron position, p is the

electron dynamical momentum, τ is the relaxation time and e is the electron charge. The

collision integral is taken in the τ approximation and further assumed constant. Eq. (1) is

solved by perturbation approach treating the second term as the perturbation. In the linear

approximation of ∇T and ∇µ, µ is the chemical potential, we obtain

f(p) = τ−1

∫

∞

0

exp

(

−t

τ

)

f0(p − eEt)dt +

∫

∞

0

exp

(

−t

τ

)

dt

×

(

[ε(p − eEt) − µ]∇T

T+ ∇µ

)

v(p − eEt)∂f0(p − eEt)

∂ε(2)

here ε(p) is the electron energy.

The current density j is defined as

j = e∑

p

v(p)f(p) (3)

and the thermal current density q as

q =∑

p

[ε(p) − µ]v(p)f(p) (4)

Substituting Eq. (2) into Eqs. (3) and (4), and making the transformation

p − eEt → p

we obtain for the current density

j = eτ−1

∫

∞

0

exp

(

−t

τ

)

dt∑

p

v(p − eEt)f0(p) +

+e

∫

∞

0

exp

(

−t

τ

)

dt∑

p

(

[ε(p) − µ]∇T

T+ ∇µ

)

×

×

(

v(p)∂f0(p)

∂ε

)

· v(p − eEt) (5)

and for the thermal current density

q = τ−1

∫

∞

0

exp

(

−t

τ

)

dt∑

p

[ε (p − eEt) − µ]v(p − eEt)fo (p) +

+

∫

∞

0

exp

(

−t

τ

)

dt∑

p

[ε (p − eEt) − µ]

{

[ε (p) − µ]∇T

T+ ∇µ

}

×

×

{

v (p)∂fo (p)

∂ε

}

v(p − eEt) (6)

3

We resolve Eqs. (5) and (6) along the tubular axis (z axis) and the base helix, neglecting the

interference between the axial and the helical paths connecting a pair of atoms, so that transverse

motion quantizatoin is ignored. Then using the following transformation

∑

p

→2

(2π~)2

∫ π

ds

−π

ds

dps

∫ π

dz

−π

dz

dpz

we obtain the electron current density along the tubular axis and the base helix as

Z ′

j =2eτ−1

(2π~)2

∫

∞

0

exp

(

−t

τ

)

dt

∫ π/ds

−π/ds

dps

∫ π/dz

−π/dz

dpzvz(p − eEt)f0 (p) +

+2e

(2π~)2

∫

∞

0

exp

(

−t

τ

)

dt

∫ π/ds

−π/ds

dps

∫ π/dz

−π/dz

dpz

×

{

[ε (p) − µ]∇zT

T+ ∇zµ

}{

vz (p)∂f0 (p)

∂ε

}

vz(p − eEt) (7)

and

S′

j =2eτ−1

(2π~)2

∫

∞

0

exp

(

−t

τ

)

dt

∫ π/ds

−π/ds

dps

∫ π/dz

−π/dz

dpzvs(p − eEt)f0 (p)

+2e

(2π~)2

∫

∞

0

exp

(

−t

τ

)

dt

∫ π/ds

−π/ds

dps

∫ π/dz

−π/dz

dpz

×

{

[ε (p) − µ]∇sT

T+ ∇sµ

}{

vs (p)∂f0 (p)

∂ε

}

vs(p − eEt) (8)

Similarly the thermal current density along the tubular and the base helix base become

Z ′

q =2τ−1

(2π~)2

∫

∞

0

exp

(

−t

τ

)

dt

∫ π

ds

−π

ds

dps

∫ π

dz

−π

dz

dpz [ε (p − eEt) − µ] vz (p − eEt) fo (p) +

+2

(2π~)2

∫

∞

0

exp

(

−t

τ

)

dt

∫ π

ds

−π

ds

dps

∫ π

dz

−π

dz

dpz [ε (p − eEt) − µ] ×

×

{

[ε (p) − µ]∇zT

T+ ∇zµ

}{

vz (p)∂fo (p)

∂ε

}

vz(p − eEt) (9)

and

S′

q =2τ−1

(2π~)2

∫

∞

0

exp

(

−t

τ

)

dt

∫ π

ds

−π

ds

dps

∫ π

dz

−π

dz

dpz [ε (p − eEt) − µ] vs(p − eEt)fo (p) +

+2

(2π~)2

∫

∞

0

exp

(

−t

τ

)

dt

∫ π

ds

−π

ds

dps

∫ π

dz

−π

dz

dpz [ε (p − eEt) − µ] ×

×

{

[ε (p) − µ]∇sT

T+ ∇sµ

}{

vs (p)∂fo (p)

∂ε

}

vs(p − eEt) (10)

The integrations are carried out over the first Brillouin zone. The axial and circumferential

electron current density will be given as follows

jz = Z ′

j + S′

j sin θh; jc = S′

j cos θh (11)

4

and the axial and circumferential thermal current density also as

qz = Z ′

q + S′

q sin θh; qc = S′

q cos θh (12)

where θh is the geometric chiral angle (GCA).

The energy ε(p) of the electrons, calculated using the tight binding approximation is given

as expressed in [25] as follows:

ε(p) = εo − ∆s cospsds

~− ∆z cos

pzdz

~(13)

εo is the energy of an outer-shell electron in an isolated carbon atom, ∆s and ∆z are the

real overlapping integrals for jumps along the respective coordinates, ps and pz are the carrier

momentum along the base helix and the tubular axis respectively, ~ is h/2π and h is Planck’s

constant. ds is the distance between the site n and n + 1 along the base helix and dz is the

distance between the site n and n + N along the tubular axis.

For a non-degenerate electron gas, we use the Boltzmann equilibrium distribution function

f0(p) as expressed in [16], i.e.,

f0(p) = C exp

(

∆s cos psds

~+ ∆z cos pzdz

~+ µ − εo

kT

)

(14)

where C is determined by the condition

C =dsdzno

2 exp(µ−εo

kT

)

I0(∆∗

s)I0(∆∗

z)

and no is charge density, In(x) is the modified Bessel function of order n and k is Boltzmann’s

constant.

The components vs and vz of the electron velocity v are given by

vs(ps) =∂ε(p)

∂ps=

∆sds

~sin

psds

~(15)

and

vz(pz) =∂ε(p)

∂pz=

∆zdz

~sin

pzdz

~(16)

Using Eqs. (7)−(16) and the fact that Es = Ez sin θh, ∇sT = ∇zT sin θh, and E = −∇φ, we

obtain the following expressions

jz =(

σz + σs sin2 θh

)

∇z

(µ

e− φ

)

+

−

{

σzk

e(ξ − ∆∗

zBz − ∆∗

sAs) +

+ σsk

esin2 θh (ξ − ∆∗

sBs − ∆∗

zAz)

}

∇zT (17)

jc = σs sin θh cos θh∇z

(µ

e− φ

)

+

−σsk

esin θh cos θh (ξ − ∆∗

sBs − ∆∗

zAz)∇zT (18)

5

qz =kT

e{σz [ξ − ∆∗

zBz − ∆∗

sAs] +σs sin2 θh (ξ − ∆∗

sBs − ∆∗

zAz)}

∇z

(µ

e− φ

)

+

−k2T

e2

{

σz

[

ξ2 − 2∆∗

zξBz − 2∆∗

sξAs + (∆∗

z)2 Cz +

+2∆∗

z∆∗

sBzAs + (∆∗

s)2

(

1 −As

∆∗

s

)]

+ σs sin2 θh

[

ξ2 − 2∆∗

sξBs +

−2∆∗

zξAz + (∆∗

s)2 Cs + 2∆∗

s∆∗

zBsAz + (∆∗

z)2

(

1 −Az

∆∗

z

)]}

∇zT (19)

qc = σskT

esin θh cos θh {ξ − ∆∗

sBs − ∆∗

zAz}∇z

(µ

e− φ

)

+

−σsk2T

e2sin θh cos θh

{

ξ2 − 2∆∗

sξBs − 2∆∗

zξAz + (∆∗

s)2 Cs +

+2∆∗

s∆∗

zBsAz + (∆∗

z)2

(

1 −Az

∆∗

z

)}

∇zT (20)

Here

ξ =εo − µ

kTAi =

I1(∆∗

i )

I0(∆∗

i ), Bi =

I0(∆∗

i )

I1(∆∗

i )−

2

∆∗

i

Ci = 1 −3

∆∗

i

I0(∆∗

i )

I1(∆∗

i )+

6

(∆∗

i )2,

σi =noe

2∆id2

i τ

~2

I1(∆∗

i )

I0(∆∗

i ), ∆∗

i =∆i

kTwhere i = s, z

The thermal currrent density q given by Eqs. (19) and (20) can be written in terms of

current density j as

qz =k

e

[

σz

σz + σs sin2 θh(ξ − ∆∗

zBz − ∆∗

sAs)

+σs sin2 θh

σz + σs sin2 θh(ξ − ∆∗

sBs − ∆∗

zAz)

]

Tjz

−

[

k2T

e2

(

σz

{

ξ2 − 2∆∗

zξBz − 2∆∗

sξAs + (∆∗

z)2 Cz +

+2∆∗

z∆∗

sBzAs + (∆∗

s)2

(

1 −As

∆∗

s

)}

+ σs sin2 θh

{

ξ2 − 2∆∗

sξBs

−2∆∗

zξAz + (∆∗

s)2 Cs + 2∆∗

s∆∗

zBsAz + (∆∗

z)2

(

1 −Az

∆∗

z

)})

−(

σz + σs sin2 θh

) k2T

e2

(

σz

σz + σs sin2 θh(ξ − ∆∗

zBz − ∆∗

sAs) +

+σs sin2 θh

σz + σs sin2 θh(ξ − ∆∗

sBs − ∆∗

zAz)

)2]

∇zT (21)

and

qc =k

e(ξ − ∆∗

sBs − ∆∗

zAz)Tjc −σsk2T

e2sin θh cos θh

[{

ξ2 − 2∆∗

sξBs − 2∆∗

zξAz +

+(∆∗

s)2 Cs + 2∆∗

s∆∗

zBsAz + (∆∗

z)2

(

1 −Az

∆∗

z

)}

− {ξ − ∆∗

sBs − ∆∗

zAz}2

]

∇zT

(22)

6

Hence from Eqs. (17) and (18) we obtain the axial and circumferential components of the

electrical conductivity σ as follows

σzz = σz + σs sin2 θh (23)

σcz = σs sin θh cos θh (24)

From Eqs. (21) and (22) we also obtain the axial and circumferential components of Peltier

coefficient Π, and the electron thermal conductivity κe when j is zero as follows

Πzz = αzzT

=k

e

[

σz

σz + σs sin2 θh(ξ − ∆∗

zBz − ∆∗

sAs)

+σs sin2 θh

σz + σs sin2 θh(ξ − ∆∗

sBs − ∆∗

zAz)

]

T (25)

Πcz = αczT

=k

e(ξ − ∆∗

sBs − ∆∗

zAz)T (26)

where α is the thermopower or Seebeck coefficient.

κzz =k2T

e2

(

σz

{

ξ2 − 2∆∗

zξBz − 2∆∗

sξAs + (∆∗

z)2 Cz +

+2∆∗

z∆∗

sBzAs + (∆∗

s)2

(

1 −As

∆∗

s

)}

+ σs sin2 θh

{

ξ2 − 2∆∗

sξBs

−2∆∗

zξAz + (∆∗

s)2 Cs + 2∆∗

s∆∗

zBsAz + (∆∗

z)2

(

1 −Az

∆∗

z

)})

−k2T

e2

(

σz + σs sin2 θh

)

(

σz

σz + σs sin2 θh(ξ − ∆∗

zBz − ∆∗

sAs) +

+σs sin2 θh

σz + σs sin2 θh(ξ − ∆∗

sBs − ∆∗

zAz)

)2

(27)

κcz = σsk2T

e2sin θh cos θh

[{

ξ2 − 2∆∗

sξBs − 2∆∗

zξAz +

+(∆∗

s)2 Cs + 2∆∗

s∆∗

zBsAz + (∆∗

z)2

(

1 −Az

∆∗

z

)}

− {ξ − ∆∗

sBs − ∆∗

zAz}2

]

(28)

3 Discussion and Conclusion

In this paper we have calculated the electrical conductivity, Peltier coefficient and electron ther-

mal conductivity. We are interested in results in the axial direction, i.e. along the tubular axis.

We noted that these paraemters are highly anisotropic, depending on the GCA θh, temperature

T and the real overlapping integrals for jumps along the respective coordinates ∆s and ∆z. We

evaluated numerically the thermoelectric figure of merit Z defined by the relation

Z =σα2

κ

7

where κ is the sum of the electron thermal conductivity κzz and lattice thermal conductivity

κlat. In most cases the dimensionless figure of merit ZT is used.

We noted in fig. (1) that electron thermal conductivity behaves in a similar manner as the

lattice conductivity obtained by Berber et al [19]. The data obtained by Berber at al was used

as the lattice thermal conductivity for calculating the figure of merit. It was also noted in fig.

(2) that the electron thermal conductivity κzz when current density j is zero exhibits unusual

high values in the order of 104 W/mK at 100 K and falls off rapidly to 103 W/mK at around

600 K when we vary ∆s and ∆z.

Temperature dependence of the thermopower αzz is presented in fig. (3). We observed that

for ∆z = 0.020 eV and ∆s = 0.010 eV, αzz ∝ 1/T , i.e. it exhibits semiconducting properties [26].

When ∆s is varied from 0.015 eV to 0.020 eV keeping ∆z at 0.020 eV, 0.025 eV and 0.030 eV, the

thermopower αzz rises to a maximum value and then falls off. Such behaviour has been observed

experimentally by Kong et al [18]. They attributed it to quasiparticle tunelling processes at some

blockade sites. Also in [15] Grigorian et al observed this behaviour and attributed it to Kondo

effect. But like Vavro and co-workers [27] we attribute the phenomena to phonon drag effect

which is important in doped SWNTs when electron-phonon scattering is the dominant decay

mechanism for the phonons. We also observed that the peak value of αzz shifts between 100 K

and 150 K as noted in [18].

The thermoelectric figure of merit obtained show very interesting results. It was observed in

fig. (4a) that for ∆s = 0.010 eV and ∆z = 0.020 eV, ZT is greater than 1 at low temperatures.

It then falls rapidly with increasing T and attains a constant value of 0.8 at about 300 K. This

result corroborates with the suggestion made by Small et al. [28] which says that at temperatures

below 30 K, ZT can be greater than 1 for single walled nanotubes (SWNTS). In that same paper

it was suggested that ZT > 1 if α ∼ 200µ V/K. This also agrees with our results (see figs. (3a)

and (3b)). On the other hand, as ∆s changes from 0.018 eV to 0.020 eV at ∆z = 0.020 eV, we

observed that ZT is very small at low temperatures and increases with increasing temperature

to a constant value. As can be seen from figs. (4b) and (4c) the behaviour of the graphs remain

almost the same as we increase the values of ∆z from 0.020 eV to 0.030 eV. However, we noted

that ZT > 1 for values of ∆s = 0.010 eV and ∆z = 0.025 eV and 0.030 eV. In order to get a

further understanding of how ZT depends on T,∆z and ∆s, we made a three dimensional plot

of ZT against ∆s and ∆z at T = 300 K (fig. (5a)), ZT against T and ∆z for ∆s = 0.012 eV

(fig. (5b)) and finally ZT against T and ∆s for ∆z = 0.015 eV (fig. (5c)). We observed that by

varying T,∆s and ∆z it is possible to obtain values of ZT greater than 1.

In conclusion we have studied the thermoelectric effect of CNTs and noted that by optimizing

T,∆z and ∆s, ZT can be made greater than 1. This suggests that CNTs could be used as a

thermoelement.

8

Acknowledgments

This work was performed within the framework of the Associateship Scheme of the Abdus

Salam International Centre for Theoretical Physics, Trieste, Italy. Financial support from the

Swedish International Development Cooperation Agency is acknowledged.

References

[1] A.F. Ioffe, Semiconductor Thermoelements and Thermoelectric Cooling, Infosearch Ltd,

London (1957)

[2] G.D. Mahan, J. Appl. Phys. 70 4551 (1991)

[3] J.O. Sofo and G.D. Mahan, Phys. Rev. B 49 4565 (1994)

[4] G.C. Chirstakudis, S.K. Plachkova, L.E. Shelimova and E.S. Avilov, Phys. Status Solidi A

128 465 (1991)

[5] E.A. Skrabek and D.S. Trimmer, in CRC Handbook of Thermoelectrics, ed. D.M Rowe,

CRC Press, Boca Raton, Fl 267-276 (1995)

[6] G.A. Slack and M.A. Hussain, J. Appl. Phys. 70 2694 (1991)

[7] D.T. Morelli, T. Caillat, J.P. Fleurial, A. Borshchevsky, J. Vandersande, B. Chen and C.

Uher, Phys. Rev. B 51 9622 (1995)

[8] D. Mandrus, A. Migliori, T.W. Darling, M.F. Hundley, E.J. Peterson and J.D. Thompson,

Phys. Rev. B 52 4926 (1995)

[9] J.W. Sharp, E.C. Jones, R.K. Williams, P.M. Martin and B.C. Sales, J. Appl. Phys. 78

1013 (1995)

[10] B.C. Sales, D. Mandrus and R.K. Williams, Science 272 1325 (1996)

[11] J.P. Fleurial, A. Borshchevsky, J. Caillat, D.T. Morelli and G.P Meisner, Proc. 15th Int.

Conf. on Thermoelectrics, Pasadena, CA (March 26-29, 1996)

[12] G.S. Nolas, G.A. Slack, D.T. Morelli, T.M. Tritt and A.C. Ehrlich, J. Appl. Phys. 79 4002

(1996)

[13] S.Y. Mensah and G.K. Kangah, J. Phys. Condens. Matter 4 919 (1992)

[14] L.D. Hicks and M.S. Dresselhaus, Phys. Rev. B 47 12727 (1993)

[15] L. Grogorian, G.U. Sumanasekera, A.L. Loper, S.L. Fang, J.L. Allen and P.C. Eklund,

Phys. Rev. B 60 Number 16, R11309 (1999)

9

[16] S.Y. Mensah, F.K.A. Allotey, N.G. Mensah and G. Nkrumah, J. Phys. Condens. Matter

13 5653 (2001)

[17] S.Y. Mensah, F.K.A. Allotey, N.G. Mensah and G. Nkrumah, Superlattices and Microstruc-

tures 33 173 (2003)

[18] W.J. Kong, L. Lu, H.W. Zhu, B.Q. Wei and D.H. Wu, J. Phys. Condens. Matter 17 1923

(2005)

[19] S. Berber, Y.K. Kwon, D. Tomanek, Phys. Rev. Lett. 84 4613 (2000)

[20] S.Y. Mensah, F.K.A. Allotey, G. Nkrumah, N.G. Mensah, Physica E 152 (2004)

[21] N.G. Mensah, G. Nkrumah, S.Y. Mensah, F.K.A. Allotey, Phys. Lett. A 329 369 (2004)

[22] J.W.G. Wildoer, L.C. Venema, A.G. Rinzler, R.E. Smalley, C. Dekker, Nature (London)

391 59 (1998)

[23] T.W. Odom, J.L. Huang, P. Kim, C.M. Lieber, Nature (London) 391 62 (1998)

[24] G. Ya Slepyan, S.A. Maksimenko, A. Lakhtakia, O. Yevtushenko, A.V. Gusakov, Phys.

Rev. B 60 17136 (1999)

[25] G. Ya Slepyan, S.A. Maksimenko, A. Lakhtakia, O. M. Yevtushenko, A. V. Gusakov, Phys.

Rev. B 57, 9485 (1998)

[26] G.U. Sumanasekera, C.K.W. Adu, S. Fang and P.C. Eklund, Phys. Rev. Lett. 85 1096

(2000)

[27] J. Vavro et al, Phys. Rev. Lett. 90 065503 (2003)

[28] Joshua P. Small, Li Shi and Philip Kim, Solid State Communications 127 181 (2003)

10