Thermoelectric properties of bismuth telluride quantum wires

M.P. Singha,*, C.M. Bhandarib

aDepartment of Physics, University of Allahabad, Allahabad 211002, IndiabIndian Institute of Information Technology, Allahabad.211002, India

Received 9 April 2003; received in revised form 22 May 2003; accepted 25 June 2003 by C.N.R. Rao

Abstract

Electrical and thermal properties of rectangular quantum wires of polycrystalline bismuth telluride have been investigated in

the framework of the relaxation time approximation. Electrical conductivity, electronic thermal conductivity and thermopower

have been obtained in the temperature range 200–600 K for two cross-sectional sizes (10 and 20 nm), and for different carrier

densities at and around optimal doping levels. Finally the thermoelectric figure of merit has been estimated in the entire

temperature range.

q 2003 Elsevier Ltd. All rights reserved.

PACS: 68.66.La

Keywords: A. Nanostructures; A. Quantum wires; D. Electronic transport

1. Introduction

Thermoelectric materials are used in power generation

for specialized applications in thermoelectric generators and

also in refrigerators. Thermoelectric efficiency is a function

of temperatures of the hot junction ðThÞ and cold junction

ðTcÞ: It also depends upon the properties of materials used.

The conversion efficiency of the device is given by

h ¼Th 2 Tc

Th

x2 1

xþTc

Th

where

x ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 þ

Z

2ðTc þ ThÞ

rand Z ¼ ða2sÞ=ðkL þ keÞ: Here a; s; kL; ke are Seebeck

coefficient, electrical conductivity, lattice thermal conduc-

tivity and electronic thermal conductivity of the material

under consideration. Z is the so-called material parameter

and there is need to obtain largest possible values. Bismuth

telluride has long been known as a good thermoelectric

material for application at relative lower range of tempera-

tures. [1–5]

During the last two decades low dimensional systems

have been the subject matter of many investigations. It was

natural to expect an upsurge of interest in quantum wells and

wires [6–9]. Electronic transport in Q1D structures has been

studied using relaxation time approach. Amongst the

electron scattering mechanisms, which, are likely to limit

its mean free path are acoustic phonons, optical phonons,

impurities and boundaries [10–13]. In the present paper, we

present the results of investigation of the temperature

dependence of electronic and thermoelectric properties; the

properties relevant for thermoelectric application are

electrical conductivity, electronic and lattice thermal

conductivity, and Seebeck coefficient. This work deals

with polycrystalline bismuth telluride and therefore aniso-

tropy effects have been excluded. An extension of this kind

of work to single crystal is expected to show significant

anisotropic effects. However, it has pointed out [14] that

figure-of-merits may still be fairly isotropic provided lattice

contribution is negligible compared with electronic thermal

conductivity. As the thermoelectric materials are relatively

0038-1098/03/$ - see front matter q 2003 Elsevier Ltd. All rights reserved.

doi:10.1016/S0038-1098(03)00520-9

Solid State Communications 127 (2003) 649–654

www.elsevier.com/locate/ssc

* Corresponding author. Tel.: þ91-532-246-0993; fax: þ91-534-

246-0993.

E-mail addresses: singhmps74@rediffmail.com (M.P. Singh),

cmbhandari@yahoo.com (C.M. Bhandari).

heavily doped ke < kL; and hence the assumption is valid to

an extent.

Lattice thermal conductivity estimates for the present

work are based on the recent researches related to phonon

confinement effects. Taking into account the possible

change in lattice thermal transport as compared to the

bulk values we have obtained realistic estimates of

thermoelectric figure-of-merit in the framework of two-

band conduction model. This takes care of possible minority

carrier effects, which are likely to be significant in all small

band gap semiconductors particularly at higher

temperatures.

2. Theory

The system under consideration is in the form of wire of

length L along the z-axis having rectangular cross-section

with transverse dimensions a and b: Assuming a single

spherical electron energy band, electronic wave function is

given by

Cnlk ¼2ffiffiffiffiffiabL

p sinnpx

a

� �sin

lpy

b

� �expðikzÞ ð1Þ

The energy eigenvalues are given by

Enlk ¼ n2E0n þ l2E0

l þ Ek; n; l ¼ 1; 2; 3;… ð2Þ

where

E0n ¼

p2"2

2mpa2; E0

l ¼p2"2

2mpb2; Ek ¼

"2k2

2mpð3Þ

We consider the size-quantum-limit (SQL) [15,16] and

assume all electrons to be in the ground state, n ¼ l ¼ 1:

The energy band structure is assumed to be independent of

wire dimensions and multivallied structure of energy bands

has been taken into account. The total density of states

effective mass is given by

mpd ¼ N2=3

v ðm1m2m3Þ1=3 ð4Þ

m1; m2; m3 are components of the effective mass tensor

along principal axes.

Considering a square cross-section ða ¼ bÞ Seebeck

coefficient, electrical conductivity and electronic thermal

conductivity in terms of reduced Fermi energy je1D ¼

EF=kBT ; is given by

ae ¼ 2kB

e2ðE0

nÞ0 2 je

1D þ 2F1ðj

e1DÞ

F0ðje1DÞ

" #ð5Þ

se ¼2

9p

"c11

mpE21

e2NvF0ðje1DÞ ð6Þ

ke ¼2

9p

"c11

mpE21

NVk2BT 3F2ðj

e1DÞ2

4F21ðj

e1DÞ

F0ðje1DÞ

!ð7Þ

where Fnðj1DÞ is Fermi-integral given by

Fnðj1DÞ ¼ð1

0dx

xn

expðx 2 j1DÞ þ 1

At relatively lower temperatures single-band conduction

model is applicable and total thermal conductivity is given by

k ¼ kL þ ke

Thermal conductivity studies on silicon quantum wires of

20 nm cross-sectional diameters reveal that phonon confine-

ment results in a significant reduction in phonon contribution

to thermal conductivity, kphð1DÞ < ð1=10Þ £ kðBulkÞ [12,13].

Assuming a similar situation for bismuth telluride we estimate

a highly approximate value of phonon thermal conductivity

<0.16 W m21 K21. From the calculated values ofa;s; and ke

from Eqs. (5)–(7) and using the approximate estimate for kL;

dimensionless figure-of-merit is in the single band conduction

model is given by

ðZTÞ ¼a2

ese

kL þ ke

T ð8Þ

At relatively higher temperature effect of minority carrier

become significant and two band conduction model has to be

employed [17–19]. The Seebeck coefficient with contri-

butions from both bands is given by

a ¼aese þ ahsh

se þ sh

�ð9Þ

Total electrical conductivity written as s ¼ se þ sh:

Fig. 1. Variation of the electronic thermal conductivity with

temperature T at different cross-sectional sizes and concentrations.

M.P. Singh, C.M. Bhandari / Solid State Communications 127 (2003) 649–654650

Expressions for ah; sh and kh (contributions to Seebeck

coefficient, electrical conductivity and thermal conductivity

from the hole band) are given by Eqs. (5)–(7) with jh1D

replacing je1D; where jg ¼ 2jh

1D 2 je1D:

Thermal transport of the material is also enhanced with

both bands contributing and thermal conductivity is written

as a sum of several contributions

k ¼ kL þ ke þ kh þ kb ð10Þ

The bipolar contribution to thermal conductivity is written

as

kb ¼ TkB

e

� �2 sesh

se þ sh

½de þ dh þ jg�2 ð11Þ

where de ¼ 2F1ðje1DÞ=F0ðj

e1DÞ; dh ¼ 2F1ðj

h1DÞ=F0ðj

h1DÞ;

ZT ¼ða0

hs0h 2 a0

es0eÞ

2

½ðs0e þ s0

hÞð1 þ s0e6e þ s0

h6hÞ þ s0es

0hðde þ dh þ jgÞ

2�

ð12Þ

where a0e;h ¼ ðe=kBÞae;h; s0

e;h ¼ ðkB=eÞ2ðT=kLÞse;h; 6e ¼

3F2ðje1DÞ=F0ðj

e1DÞ2 d2

e ; jg ¼ Eg=kBT :

Table 1

Physical parameters of polycrystalline bismuth telluride used in the

calculations (values refer to 300 K)

kL (W m21 K21) c11 (1011 N m22) mpd=m0 Nv e1 (eV)

1.6 0.19 0.51 6 6.3

Fig. 2. Variation of electrical conductivity with temperature. (a) and (b) refers to the single band model and (c) refers to the two band conduction

model.

M.P. Singh, C.M. Bhandari / Solid State Communications 127 (2003) 649–654 651

3. Results and discussion

The purpose of the paper is to evaluate thermoelectric

efficiency of bismuth telluride wire as thermoelements in

thermoelectric applications. The electrical and thermal

properties relevant for this are also calculated and displayed.

Various physical parameters used in the calculation are

displayed in Table 1.

Fig. 1 shows the temperature dependence of electronic

contribution to thermal conductivity for two cross-section

sizes 20 and 10 nm and for two different carrier densities.

In the temperature range studied the electronic thermal

conductivity increases with temperature and with carrier

density. Reduction in cross-sectional size by a factor 1/2

(from 20 to 10 nm) results in a reduction in ke by 50–

60% at 300 K. Fig. 2(a)–(c) shows the temperature

variation of electrical conductivity. Electrical conduc-

tivity decreases with increase in temperature for a given

carrier density. Larger carrier densities result in larger

value of s and cross-sectional size reduction causes its

reduction. Fig. 3 shows the Seebeck coefficient against

temperature for various carrier densities. Total thermal

conductivity needs to be evaluated for an assessment of

the figure-of-merit. Fig. 4 shows various contributions to

it along with the total thermal conductivity. Fig. 5(a) and

(b) shows the dimensionless figure-of-merit against

temperature for various carrier densities and cross-

sectional sizes.

For almost all carrier densities the maxima in figure-

of-merit occur near 300 K. A reduction in cross-sectional

size from 20 to 10 nm results in a significant increase in

ZT with a relatively minor shift in the maxima to lower

temperature side. Towards higher temperature side of the

maxima ZT falls off faster for narrower wires. From the

present study, it can be concluded that the best

thermoelectric performance of bismuth telluride is

expected near room temperature. For larger diameter

wires fall of ZT with T beyond the maximum is slower

and at 500 K it may fall by 20% of the peak value. On

the other hand the rapid fall in ZT for 10 nm diameter

reduces ZT by 50%. Shorter diameters although result in

a better peak performances their range of usefulness is

narrower. A compromise is to be sought between the

two; to have a large ZT and a wider peak so that

usefulness of a thermoelement is maintained over the

temperature range of operation.

It is difficult to seek an experimental confirmation of

theoretical results on the thermoelectric properties of

quantum wires. Theoretical results are sensitive to the

model being considered; the model considered here is

idealized due to approximations regarding the relaxation

time as also size quantum limit (SQL). On the other

hand, measured values are sensitive to the properties of

materials and process of preparation. For 2D-systems

measurement of thermoelectric performance is in an

advance stage and improved performance has been

demonstrated. However, this could not be said for 1D-

systems and to the best of authors’ know-how no

detailed experimental work is available in bismuth

telluride nanowires for a direct comparison. However,

some justification can be sought by comparing these

results with available information. Sun et al. [6] Lin

et al. [20] have reported a theoretical model for

transport properties of cylindrical Bi wires. For a

Fig. 3. Variation of Seebeck coefficient with temperature for the two

band model. Fig. 4. Variation of the thermal conductivity with temperature.

M.P. Singh, C.M. Bhandari / Solid State Communications 127 (2003) 649–654652

10 nm wire diameter maximum of ZT reported is around

2.0 at nopt h 1 £ 1024 m23: Our calculations indicate an

optimum n almost same as that of Bi nanowires and our

calculated maximum ZT is around 1.75. Without

seeking a direct correspondence there is an agreement

in the order of (ZT)max and nopt in the two models.

Available experimental results on PbTe at 300 K give a

ZT for quantum dot structures a value around 0.8,

whereas corresponding bulk is 0.4 showing a 100%

improvement over the bulk value [21]. Our values

ðZTÞmax ¼ 1:75 at 300 K and ðZTÞmax ¼ 1:2 give

approximately 46% increase in ZT over the bulk values

[22]. At the first glance our results are not inconsistent

with the available data and other calculations.

Acknowledgements

M.P. Singh thankfully acknowledges financial assistance

from the Council of Scientific and Industrial Research, New

Delhi, India. Authors are thankful to Prof. G.K. Pandey and

Dr M.D. Tiwari for their interest and support.

Fig. 5. (a) Dimensionless figure-of-merit ZT plotted against temperature at different concentrations at cross-sectional size, a ¼ 100 �A and for

ne ¼ 1 £ 1023 m23; ne ¼ 5 £ 1023 m23 ne ¼ 1 £ 1024 m23 ne ¼ 5 £ 1024 m23; (b) ZT versus temperature at ne ¼ 5 £ 1023 m23; a ¼ 10 and

20 nm.

M.P. Singh, C.M. Bhandari / Solid State Communications 127 (2003) 649–654 653

References

[1] H.J. Goldsmid, Elctronic Refrigeration, Pion, London, 1986.

[2] C.M. Bhandari, D.M. Rowe, Thermal Conduction in Semi-

conductor, Wiley, New Delhi, 1988, Chapters 3 and 4.

[3] L.D. Hicks, M.S. Dresselhaus, Phys. Rev. B 47 (24) (1993)

16631.

[4] L.D. Hicks, M.S. Dresselhaus, Phys. Rev. B 47 (19) (1993)

12727.

[5] T. Koga, X. Sun, S.B. Cronin, M.S. Dressellhaus, Appl. Phys.

Lett. 75 (16) (1999) 2438.

[6] X. Sun, Z. Zhang, M.S. Dresselhaus, Appl. Phys. Lett. 74 (26)

(1999) 4005.

[7] V.L. Gurevich, A. Thellung, Phys. Rev. B 65 (2002) 153313.

[8] X.R. Wang, Y. Wang, Z.Z. Sun, Phys. Rev. B 65 (2002)

193402.

[9] M.S. Dresselhaus, G. Dresselhaus, X. Sun, Z. Zange, S.B.

Cronin, T. Koga, Conf. Proc. Phys. Solid State 41 (5) (1999)

679.

[10] A. Balandin, K.L. Wang, J. Appl. Phys. 84 (11) (1999) 6149.

[11] S.G. Walkauskas, D.A. Broido, K. Kempa, T.L. Reinecke,

J. Appl. Phys. 85 (5) (1999) 2579.

[12] J. Zou, A. Balandin, MRS Spring Meeting, SanFrancisco,

2001.

[13] A. Balandin, Phys. Low-dim. Struct. (2000) 1/2.

[14] W.E. Bies, R.J. Radtke, H. Ehrenreich, Phys. Rev. B 65 (2002)

085208.

[15] S.S. Kubakaddi, B.G. Mulimani, J. Appl. Phys. 58 (9) (1985)

3643.

[16] G.B. Ibragimov, J. Phys. C: Condens. Matter 14 (2002)

8145.

[17] (a) C.M. Bhandari, in: D.M. Rowe (Ed.), CRC Hand Book of

Thermoelectric, vol. 27, CRC, Boca Raton, 1995, Chapter 4.

(b) R.W. Ure Jr., Energy Convers., 12, 1972, pp. 45.

[18] C.M. Bhandari, V.K. Agrawal, Indian J. Pure Appl. Phys. 28

(1990) 448.

[19] C.M. Bhandari, D.M. Rowe, in: D.M. Rowe (Ed.), Proceed-

ings of The First European Conference on Thermoelectrics,

1987, pp. 178, Chapter 16.

[20] Y.M. Lin, X. Sun, M.S. Dresselhaus, Phys. Rev. B 62 (2001)

4610.

[21] T.C. Harman, P.J. Taylor, D.L. Spears, M.P. Walsh,

J. Electron. Mater. 29 (2000) L1.

[22] G.D. Mahan, in: H. Ehrenreich, F. Saepen (Eds.), Good

Thermoelectrics in Solid State Physics, vol. 51, Academic

Press, San Diego, 1998.

M.P. Singh, C.M. Bhandari / Solid State Communications 127 (2003) 649–654654