Comparative analysis of electric field influence on the quantum wells with different boundary conditions. II. Thermodynamic properties

TL;DR

This paper investigates how electric fields affect the thermodynamic properties of 1D quantum wells with various boundary conditions, revealing unique spectral features, phase transition shifts, and dipole moment behaviors.

Contribution

It provides analytical expressions and detailed analysis of thermodynamic responses of quantum wells under electric fields with different boundary conditions, including effects on heat capacity and Bose-Einstein condensation.

Findings

Electric field modifies heat capacity and energy spectrum features.

Maximum of heat capacity depends on boundary conditions and electric field.

Critical temperature for Bose-Einstein condensation increases with applied voltage.

Abstract

Thermodynamic properties of the one-dimensional (1D) quantum well (QW) with miscellaneous permutations of the Dirichlet (D) and Neumann (N) boundary conditions (BCs) at its edges in the perpendicular to the surfaces electric field $\mathscr{E}$ are calculated. For the canonical ensemble, analytical expressions involving theta functions are found for the mean energy and heat capacity $c_V$ for the box with no applied voltage. Pronounced maximum accompanied by the adjacent minimum of the specific heat dependence on the temperature $T$ for the pure Neumann QW and their absence for other BCs are predicted and explained by the structure of the corresponding energy spectrum. Applied field leads to the increase of the heat capacity and formation of the new or modification of the existing extrema what is qualitatively described by the influence of the associated electric potential. A remarkable feature of the Fermi grand canonical ensemble is, at any BC combination in zero fields, a salient maximum of $c_V$ observed on the $T$ axis for one particle and its absence for any other number $N$ of corpuscles. Qualitative and quantitative explanation of this phenomenon employs the analysis of the chemical potential and its temperature dependence for different $N$. It is proved that critical temperature $T_{cr}$ of the Bose-Einstein (BE) condensation increases with the applied voltage for any number of particles and for any BC permutation except the ND case at small intensities $\mathscr{E}$ what is explained again by the modification by the field of the interrelated energies. It is shown that even for the temperatures smaller than $T_{cr}$ the total dipole moment $\langle P\rangle$ may become negative for the quite moderate $\mathscr{E}$. For either Fermi or BE system, the influence of the electric field on the heat capacity is shown to be suppressed with $N$ growing.