Abstract
We rederived the fermion distribution function by considering the effect of assembly size. We did not use Stirling approximation to avoid the deviation generated by this approximation for a small number of constituents and small assembly size. Furthermore, we identified that in small systems, the chemical potential should also depend on the assembly size. We also rederived a general expression for the size-dependent chemical potential from a statistical configuration and showed that it is consistent with the results from previously reported theoretical or simulation methods. Finally, we applied the model to derive a size-dependent thermoelectric power factor of nanostructured materials. One important finding is that the power factor initially increases when reducing the particle size; however, it then reduces to approach zero when further reducing the material size, due to a dramatic change in the material behaviors.
Highlights
Chemical potential plays an essential role in materials physics since it controls several important properties
Takeuchi et al [3] showed that the fine electronic structure near the chemical potential plays an essential role in realizing unusual behavior in thermoelectric power [8,9,10,11]
Using a simple quantum well model, Seki showed that the chemical potential of an electron in a thin slab depends on the slab thickness [29]. From all those reports arise the following questions: How can we prove fundamentally that the chemical potential depends on the assembly size? Can we show from a derivation of the distribution function that the fermion chemical potential depends on the assembly size? The purpose of this work is to show from the initial consideration, i.e. from the derivation of the distribution function, that the fermion chemical potential depends on the assembly size as well as the size dependence of the thermoelectric power factor
Summary
Chemical potential plays an essential role in materials physics since it controls several important properties. Cook and Dickerson explained in a simple way how to understand the chemical potential, especially in small systems [12], starting from a definition of the chemical potential as μ = ∂U/∂N ]s,v , where U is the system internal energy, N is the number of particles in the system, S is the entropy, and V is the system volume. This relation states that the chemical potential is equal to the change in energy when one particle is added to the system while maintaining the entropy. This is not the only relation to define the chemical potential
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