The Polylogarithm and the Lambert W Functions in Thermoelectrics

Authors

Muralikrishna Molli, Sri Sathya Sai Institute of Higher Learning, Prasanthinilayam, Andhra Pradesh, IndiaFollow
Kamisetty Venkataramaniah, Sri Sathya Sai Institute of Higher Learning, Prasanthinilayam, Andhra Pradesh, IndiaFollow
Sree Ram Valluri, The University of Western OntarioFollow

Document Type

Article

Publication Date

11-2-2011

Journal

Canadian Journal of Physics

Volume

89

Issue

11

First Page

1171

Last Page

1178

URL with Digital Object Identifier

http://dx.doi.org/10.1139/P11-124

Abstract

In this work, we determine the conditions for the extremum of the figure of merit, theta², in a degenerate semiconductor for thermoelectric (TE) applications. We study the variation of the function theta² with respect to the reduced chemical potential mu* using relations involving polylogarithms of both integral and nonintegral orders. We present the relevant equations for the thermopower, thermal, and electrical conductivities that result in optimizing theta² and obtaining the extremum equations. We discuss the different cases that arise for various values of r, which depends on the type of carrier scattering mechanism present in the semiconductor. We also present the important extremum conditions for theta² obtained by extremizing the TE power factor and the thermal conductivity separately. In this case, simple functional equations, which lead to solutions in terms of the Lambert W function, result. We also present some solutions for the zeros of the polylogarithms. Our analysis allows for the possibility of considering the reduced chemical potential and the index r of the polylogarithm as complex variables.

Notes

This article was published by the NRC Research Press. The authors' final, peer-reviewed version is available for download.