While thermoelectric transport theory is well established and widely applied, there remains some degree of confusion on the proper thermodynamic definition of the Seebeck coefficient (or thermoelectric power) which is a measure of the strength of the mutual interaction between electric charge transport and heat transport. Indeed, as one considers a thermoelectric system, it is not always clear whether the Seebeck coefficient is to be related to the gradient of the systems chemical potential or to the gradient of its electrochemical potential. This pedagogical article aims to shed light on this confusion and clarify the thermodynamic definition of the thermoelectric coupling. First, we recall how the Seebeck coefficient is experimentally determined. We then turn to the analysis of the relationship between the thermoelectric power and the relevant potentials in the thermoelectric system: As the definitions of the chemical and electrochemical potentials are clarified, we show that, with a proper consideration of each potential, one may derive the Seebeck coefficient of a non-degenerate semiconductor without the need to introduce a contact potential as seen sometimes in the literature. Furthermore, we demonstrate that the phenomenological expression of the electrical current resulting from thermoelectric effects may be directly obtained from the drift-diffusion equation.
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