12th Joint European Thermodynamics Conference Brescia, July 1-5, 2013 INTRODUCTION Since the low dimensional behaviours of the thermoelectric materials are different than the bulk ones, higher thermoelectric efficiencies can be achieved by using low dimensional structures like quantum wells, wires and dots [1-3]. The conversion efficiency of thermoelectric devices depends on the properties of the materials and determined by the well known figure of merit relation,   2 S Z  , where S is the Seebeck coefficient,  is the electrical conductivity and  is the thermal conductivity. Since the all variables in the figure of merit relation are dependent, they adversely affect each other so that the figure of merit does not vary significantly for the bulk thermoelectric materials. However, in the literature, it is proposed that quantum confinement plays an important role to enhance the figure of merit by using low dimensional nanostructures [4-8]. In contrast to the macroscopic approaches, transport properties of materials become size and shape dependent when the domain size is comparable to the characteristic length scale of the problem. Therefore, size and shape of the system are considered as additional control parameters on transport properties of nanoscale devices [9, 10]. In sufficiently small structures, the wave character of particles significantly changes the transport properties by modifying probability density distribution, the smallest values of the momentum components and momentum spectrum of particles [9, 11]. Some contributions arise when the thermal de Broglie wavelength of particles are not negligible in comparison with the size of the domain. These contributions are called as quantum size effects (QSE) in general [9, 11]. In the calculations here, free electrons in the conduction band of conductors and semi-conductors are considered as an ideal Fermi gas and Seebeck coefficient is analytically derived by considering QSE. Through the confinement, sizes of the domain are smaller than the mean free path of the particles but equal to or greater than the thermal de Broglie wave length. Therefore conventional definitions of particle flux and transport coefficients are used. Dependencies of Seebeck coefficient on domain size and quantum degeneracy are investigated. Particle flux equation is derived in terms of external potential, chemical potential and temperature gradients. Relaxation time approximation is used to obtain non-equilibrium distribution function. Furthermore, relaxation time is assumed to be equal to the inverse of the collision frequencies of the particles. The summations inside the particle flux expression are replaced by the Poisson summation formula to consider QSE. Since the system size in at least one direction is comparable to the thermal de Broglie wavelength, the contribution of zero correction term of the Poisson summation formula is included. On the other hand, discreetness correction term is neglected since all the sizes are equal to or larger than the thermal de Broglie wavelength of particles. In order to obtain an analytically solvable problem, rectangular domain geometry is considered. DERIVATION OF PARTICLE FLUX AND SEEBECK COEFFICIENT Transport domain is a rectangular structure with dimensions of z y x L L L , , and the domain is strongly confined through the direction x. The particle flux in direction x is written as, f v J ijk x w x   , N (1) QUANTUM CONFINEMENT EFFECTS ON SEEBECK COEFFICIENT Z.Fatih Ozturk, Altug Sisman * and Sevan Karabetoglu Istanbul Technical University, Energy Institute, 34469, Istanbul/Turkey * Corresponding Author: sismanal@itu.edu.tr ABSTRACT Seebeck coefficient is analytically derived for Fermi gas by considering quantum size effects to investigate the dependencies of Seebeck coefficient on domain size and quantum degeneracy. Under the relaxation time approximation, particle flux equation is expressed in terms of external potential, chemical potential and temperature gradient. The summations in the expression of particle flux are replaced by the Poisson summation formula to consider quantum size effects. Since the system size considered here is comparable to the thermal de Broglie wavelength, the contribution of zero correction term of the Poisson summation formula is included. In order to obtain an analytically solvable problem, rectangular domain geometry is considered. The results show that quantum size effects cause a significant improvement on Seebeck coefficient when the system size in one direction approach to thermal de Broglie wave length. This improvement is due to transition from bulk to quantum well behaviour. Variation of quantum size effects with quantum degeneracy, which is represented by dimensionless chemical potential, is also examined. It is seen that the chemical potential value for the maximum dimensionless Seebeck coefficient decreases with decreasing system size while the Seebeck coefficient increases. 567