Thermoelectric properties of bismuth telluride quantum wires

Solid State Communications 127 (2003) 649–654 www.elsevier.com/locate/ssc

Thermoelectric properties of bismuth telluride quantum wires M.P. Singha,*, C.M. Bhandarib a

Department of Physics, University of Allahabad, Allahabad 211002, India b Indian 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 ðTh Þ and cold junction ðTc Þ: It also depends upon the properties of materials used. The conversion efficiency of the device is given by

h¼

Th 2 Tc x 2 1 T Th xþ c Th

where rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z x ¼ 1 þ ðTc þ Th Þ 2 and Z ¼ ða2 sÞ=ðkL þ ke Þ: Here a; s; kL ; ke are Seebeck coefficient, electrical conductivity, lattice thermal conductivity and electronic thermal conductivity of the material under consideration. Z is the so-called material parameter * Corresponding author. Tel.: þ91-532-246-0993; fax: þ 91-534246-0993. E-mail addresses: [email protected] (M.P. Singh), [email protected] (C.M. Bhandari).

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 temperatures. [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 anisotropy 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

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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 twoband 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     2 npx lpy sin expðikzÞ ð1Þ Cnlk ¼ pffiffiffiffiffi sin a b abL The energy eigenvalues are given by Enlk ¼ n2 En0 þ l2 El0 þ Ek ;

n; l ¼ 1; 2; 3; …

ð2Þ

where En0 ¼

where Fn ðj1D Þ is Fermi-integral given by ð1 xn dx Fn ðj1D Þ ¼ expðx 2 j1D Þ þ 1 0 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 confinement results in a significant reduction in phonon contribution to thermal conductivity, kph ð1DÞ < 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 of a; 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 ¼

a2e se T kL þ ke

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 contributions from both bands is given by

 ae se þ ah sh a¼ ð9Þ se þ sh Total electrical conductivity written as s ¼ se þ sh :

2 2

p " ; 2mp a2

El0 ¼

2 2

p " ; 2mp b2

2 2

Ek ¼

" k 2mp

ð3Þ

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 Nv2=3 ðm1 m2 m3 Þ1=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 =kB T; is given by " # k F ðje Þ ð5Þ ae ¼ 2 B 2ðEn0 Þ0 2 je1D þ 2 1 1D F0 ðje1D Þ e

se ¼

2 "c11 2 e Nv F0 ðje1D Þ 9p mp E12

2 "c11 4F12 ðje1D Þ ke ¼ NV kB2 T 3F2 ðje1D Þ 2 p 2 9p m E1 F0 ðje1D Þ

ð6Þ ! ð7Þ

Fig. 1. Variation of the electronic thermal conductivity with temperature T at different cross-sectional sizes and concentrations.

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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.

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 2jh1D 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 T



kB e

2

se sh ½d þ dh þ jg 2 se þ sh e

ð11Þ

where de ¼ 2F1 ðje1D Þ=F0 ðje1D Þ; dh ¼ 2F1 ðjh1D Þ=F0 ðjh1D Þ; ZT ¼

½ðs0e

þ

s0h Þð1

þ

ða0h s0h 2 a0e s0e Þ2 þ s0h 6h Þ þ s0e s0h ðde þ dh þ jg Þ2 

s0e 6e

ð12Þ ¼ ðe=kB Þae;h ; ¼ ðkB =eÞ ðT=kL Þse;h ; 6e ¼ where 3F2 ðje1D Þ=F0 ðje1D Þ 2 d2e ; jg ¼ Eg =kB T:

a0e;h

s0e;h

2

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

e 1 (eV)

1.6

0.19

0.51

6

6.3

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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 conductivity 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 crosssectional sizes. For almost all carrier densities the maxima in figureof-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

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

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 1Dsystems 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. 4. Variation of the thermal conductivity with temperature.

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

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.

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