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Journal of Physics Condensed Matter

Chemical potential of the low-dimensional multisubband Fermi gas

Elena S. Sokolova, S. S. Sokolov, Nélson Studart

Published Nov 5, 2010• 7 citations
Publisher

Abstract

In this paper, the chemical potential of two-dimensional (2D) and quasi-one-dimensional (Q1D) multisubband charged Fermi gases is evaluated. We start with a rather general formula for the thermodynamic potential of an ideal quantum statistical system with arbitrary occupation-number to calculate, as a particular case, the chemical potential of the multisubband 2D Fermi gas described by a quadratic energy spectrum. The chemical potential is also studied in the case of a low-dimensional Fermi gas in the presence of a quantizing magnetic field. The same approach is used to study the chemical potential of a multisubband Q1D Fermi gas. The influence of temperature is considered. We found interesting analytical results for special limits as well as numerical results that may be important for the description of both the thermodynamic and kinetic properties of Fermi systems in low dimensionality.

DOI
10.1088/0953-8984/22/46/465304

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In this paper, the chemical potential of two-dimensional (2D) and quasi-one-dimensional (Q1D) multisubband charged Fermi gases is evaluated. We start with a rather general formula for the thermodynamic potential of an ideal quantum statistical system with arbitrary occupation-number to calculate, as a particular case, the chemical potential of the multisubband 2D Fermi gas described by a quadratic energy spectrum. The chemical potential is also studied in the case of a low-dimensional Fermi gas in the presence of a quantizing magnetic field. The same approach is used to study the chemical potential of a multisubband Q1D Fermi gas. The influence of temperature is considered. We found interesting analytical results for special limits as well as numerical results that may be important for the description of both the thermodynamic and kinetic properties of Fermi systems in low dimensionality.
Calculations of chemical potentials for ideal monatomic gases with Bose-Einstein and Fermi-Dirac statistics as functions of temperature, across the temperature region that is typical for the collective quantum degeneracy effect, are presented. Numerical calculations are performed without any additional approximations, and explicit dependences of the chemical potentials on temperature are constructed at a fixed density of gas particles. Approximate polynomial dependences of chemical potentials on temperature are obtained that allow for the results to be used in further studies without re-applying the involved numerical methods. The ease of using the obtained representations is demonstrated on examples of deformation of distribution for a population of energy states at low temperatures, and on the impact of quantum statistics (exchange interaction) on the equations of state for ideal gases and some of the thermodynamic properties thereof. The results of this study essentially unify two opposite limiting cases in an intermediate region that are used to describe the equilibrium states of ideal gases, which are well known from university courses on statistical physics, thus adding value from an educational point of view.
The properties of an ideal Fermi gas in an external potential in any dimensional space are studied, based on the semiclassical (Thomas-Fermi) approximation. The general analytical expressions of the total particle number N,N,N, density of states $D(\ensuremath{\varepsilon}),$ Fermi energy EF,{E}_{F},EF​, total energy E,E,E, and heat capacity CCC have been derived, where N,N,N, E,E,E, and CCC are expressed by the Fermi integration. Moreover, the analytical expressions of the total energy E,E,E, chemical potential \ensuremath{\mu}, and heat capacity CCC in the high- and low-temperature approximations are given. From these results, how the characteristics of the Fermi gas depend on an external potential and the dimension of space is discussed.
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.
We present closed-form, analytical expressions for the thermodynamic properties of an ideal, two-dimensional (2D) charged Fermi or Bose gas in the presence of a uniform magnetic field of arbitrary strength. We consider both the homogeneous quantum gas (in which case our expressions are exact) and the inhomogeneous gas within the local-density approximation. Our results for the Fermi gas are relevant to the current-density-functional theory of low-dimensional electronic systems in magnetic fields. For a 2D charged Bose gas (CBG) in a homogeneous magnetic field, we show that the uniform system undergoes a sharp transition at a critical temperature ${T}_{c}^{\ensuremath{\star}},$ below which there is a macroscopic occupation of the lowest Landau level. An examination of the one-body density matrix, however, reveals the absence of long-range order, thereby indicating that the transition cannot be interpreted to a Bose-Einstein condensate. Nevertheless, for $T<{T}_{c}^{\ensuremath{\star}}$ and weak magnetic fields, the system still exhibits magnetic properties which are practically indistinguishable from those of a condensed, superconducting CBG. We therefore conclude that while a condensate is a sufficient condition for the ideal CBG to exhibit a superconducting state, it may not be a necessary condition.