Thermodynamics of the Fermi gas in a quantum well

TL;DR

This paper derives the thermodynamic properties of an ideal Fermi gas confined in a quantum well, analyzing how these properties depend on temperature, chemical potential, and the distance between confining planes, revealing oscillatory or monotonic behaviors.

Contribution

It provides a comprehensive general formulation of thermodynamic characteristics for the Fermi gas in a quantum well, including anisotropic pressures and effects of well width as a thermodynamic variable.

Findings

Thermodynamic quantities exhibit jumps at the filling of new discrete levels.

Behavior of thermodynamic quantities varies with the fixed parameter, showing oscillations or monotonic trends.

High-temperature corrections are proportional to the ratio of thermal wavelength to well width.

Abstract

For the ideal Fermi gas that fills a quantum well confined by two parallel planes, there are calculated the thermodynamic characteristics in general form for arbitrary temperatures, namely: the thermodynamic potential, energy, entropy, equations of state, heat capacities and compressibilities. The distance between planes is considered as an additional thermodynamic variable. Owing to the anisotropy, the pressure of the Fermi gas along and transverse to the planes is different, so that the system is characterized by two equations of state and a set of different heat capacities. Limiting cases of low and high temperatures are considered. The temperature dependencies of the entropy and heat capacities at low temperatures remain linear, just as in the volume case, and their dependencies on the chemical potential and density undergo jumps at the beginning of the filling of new discrete levels. It is shown that the behavior of thermodynamic quantities with the distance between plates can be either oscillating or monotonic, depending on what quantity is assumed to be fixed: the volume or surface density. For high temperatures the corrections to thermodynamic quantities are obtained, which are proportional to the ratio of the thermal de Broglie wavelength to the distance between planes.