The Chemical Potential

TL;DR

This paper clarifies the correct definition of chemical potential in finite systems, emphasizing the grand canonical ensemble's role, and discusses finite size effects with applications to semiconductors and quantum dots.

Contribution

It establishes the grand canonical chemical potential as the unique correct definition for all finite systems and extends the zero-temperature limit derivation to semiconductors.

Findings

Grand canonical chemical potential is the only correct finite system definition.

Finite size corrections can be significant even in macroscopic samples.

Applications to semiconductors and quantum dots illustrate the concepts.

Abstract

The definition of the fundamental quantity, the chemical potential, is badly confused in the literature: there are at least three distinct definitions in various books and papers. While they all give the same result in the thermodynamic limit, major differences between them can occur for finite systems, in anomalous cases even for finite systems as large as a cm$^3$. We resolve the situation by arguing that the chemical potential defined as the symbol $\mu$ conventionally appearing in the grand canonical density operator is the uniquely correct definition valid for all finite systems, the grand canonical ensemble being the only one of the various ensembles usually discussed (microcanonical, canonical, Gibbs, grand canonical) that is appropriate for statistical thermodynamics, whenever the chemical potential is physically relevant. The zero-temperature limit of this $\mu$ was derived by Perdew et al. for finite systems involving electrons, generally allowing for electron-electron interactions; we extend this derivation and, for semiconductors, we also consider the zero-T limit taken after the thermodynamic limit. The enormous finite size corrections (in macroscopic samples, e.g. 1 cm$^3$) for one rather common definition of the c.p., found recently by Shegelski within the standard effective mass model of an ideal intrinsic semiconductor, are discussed. Also, two very-small-system examples are given, including a quantum dot.