Thermodynamical Study of the Thermoelectric Effect for Magnesium Silicide † Z. H. Zhu ‡ and Chaoyuan Zhu* ,§ Institute of Atomic and Molecular Physics, Sichuan UniVersity, Chengdu 610065, China, and Department of Applied Chemistry, Institute of Molecular Science and Center for Interdisciplinary Molecular Science, National Chiao-Tung UniVersity, Hsinchu 300, Taiwan ReceiVed: May 19, 2007; In Final Form: August 3, 2007 The thermoelectric effect of magnesium silicide is studied by using a thermodynamical method in the presence of an electric field. The thermoelectric potential is evaluated from the partial derivative of free energy with respect to charge in which the free energy is calculated at the B3LYP/6-31G(d,p) level of density functional theory. This free energy is also utilized to determine the average dipole moment from which the polarizability, R; molar polarization, Ψ; and dielectric constant can be computed. The present calculation for the dielectric constant (∼24-20) is in very good agreement with the experimental value (20). This accurate dielectric constant can be used to derive the relation of the thermoelectric potential with respect to temperature, from which the thermoelectric power or the Seebeck coefficients are calculated. The present result shows good agreement with experiment measurement for the Seebeck coefficients. In comparison, that calculation from the energy band structure theory is far off from the experimental values. I. Introduction Material with thermoelectric energy conversion has attracted a great deal of attention for its applications to temperature measurement; cooling of laser modules; computer chip coolers; and particularly, for the recovery of exhaust heat sources to generate electric power and to reduce global warming due to the greenhouse effect. For thermoelectric energy conversion to be highly efficient, the thermoelectric conversion efficiency, defined by a dimensionless figure of merit, ZT, which is relevant to 10% conversion efficiency, must be greater than unity. The figure of merit is defined by where S is called Seebeck coefficient, σ is the electrical conductivity, k is the thermal conductivity, and T is temperature. The ideal thermoelectric material would have a large S and small k, like an insulator, but would also have a high σ, like a metal. Magnesium silicide (Mg 2 Si), an alkali monosilicide that has the space group Fm3m with crystal parameter 6.351, is an inexpensive, promising, n-type semiconductor material at tem- peratures ranging from 500 to 773 K 1-6 for its narrow-band gap of 0.78 eV 16 , and is notable for the abundance of its constituent elements in the earth’s crust and its lack of toxicity. Its density is 1.88 g/cm 3 . 7 The Seebeck coefficient is conven- tionally based on the calculation of the carrier motion of the energy band, which includes two parts: diffusion thermoelectric power and phonon drag power. 8 For example, for an n-type semiconductor, the diffusion thermoelectric power is defined by where k B is the Boltzmann constant, h is the Planck constant, q is the electron charge, m n / is the effective electron mass, and n is the effective electron concentration. Accurate calculations based on eq 1 cannot be easily performed. The present work suggests another alternative way to study the thermoelectric effect based on thermodynamic theory for the systems that are in equilibrium with an electric field. The first step is to calculate the Helmholtz free energy and dipole moment at different electric fields; then it is possible to obtain the polarizability, distortion polarizability, and dielectric constant for Mg 2 Si. Thermodynamics can be considered as an exact theory in which all microscopic quantities are systematically averaged out so that it is possible to perform a more accurate calculation. However, some of experimental data are needed: for instance, the density of solid Mg 2 Si. In Section II, the thermodynamic method in the presence of an electric field is briefly introduced. Section III is concerned with the thermodynamical study of the free energy, polariz- ability, and dielectric constant for solid Mg 2 Si by combining the molecular structure calculation of density functional theory. The thermoelectric effect and calculation of the Seebeck coefficient is presented using three different methods in Section IV. Concluding remarks are given in Section V. II. Thermodynamic Method For the self-contained purpose of the present study, we briefly review the thermodynamic method in the presence of an electric field by using Samoluoviqi’s reference book 9 and its notations. Following the Gibbs phase rule, the equilibrium state for a one- component homogeneous system is defined by two intensive † Part of the “Sheng Hsien Lin Festschrift”. * To whom correspondence should be addressed. E-mail: cyzhu@ mail.nctu.edu.tw. ‡ Sichuan University. § National Chiao-Tung University. ZT ) S 2 σT k R n )- k B q [ 3 2 + ln 2(2πm n / k B T) 3/2 nh 3 ] (1) 9362 J. Phys. Chem. A 2007, 111, 9362-9366 10.1021/jp073880d CCC: $37.00 © 2007 American Chemical Society Published on Web 08/22/2007