An analytical solution for quantum size effects on Seebeck coefficient

There are numerous experimental and numerical studies about quantum size effects on Seebeck coefficient. In contrast, in this study, we obtain analytical expressions for Seebeck coefficient under quantum size effects. Seebeck coefficient of a Fermi gas confined in a rectangular domain is considered. Analytical expressions, which represent the size dependency of Seebeck coefficient explicitly, are derived in terms of confinement parameters. A fundamental form of Seebeck coefficient based on infinite summations is used under relaxation time approximation. To obtain analytical results, summations are calculated using the first two terms of Poisson summation formula. It is shown that they are in good agreement with the exact results based on direct calculation of summations as long as confinement parameters are less than unity. The analytical results are also in good agreement with experimental and numerical ones in literature. Maximum relative errors of analytical expressions are less than 3% and 4% for 2D and 1D cases, respectively. Dimensional transitions of Seebeck coefficient are also examined. Furthermore, a detailed physical explanation for the oscillations in Seebeck coefficient is proposed by considering the relative standard deviation of total variance of particle number in Fermi shell.