Thermodynamic properties of the one-dimensional Robin quantum well

TL;DR

This paper investigates the thermodynamic behavior of a Robin quantum well, revealing how energy spectrum modifications due to Robin boundary conditions influence heat capacity, phase transitions, and ensemble-specific properties.

Contribution

It provides a comprehensive theoretical analysis of thermodynamic properties of Robin quantum wells, highlighting the effects of split-off energy levels and Robin distance variations on different statistical ensembles.

Findings

Heat capacity exhibits nonmonotonic behavior with temperature, with maxima diverging as Robin length approaches zero.

Fermi-Dirac ensemble shows nonmonotonic specific heat with a plateau and extremum depending on Robin length.

Bose-Einstein ensemble displays a cusp indicating a phase transition to condensate state, influenced by Robin boundary conditions.

Abstract

Thermodynamic properties of Robin quantum well with extrapolation length $\Lambda$ are analyzed theoretically both for canonical and two grand canonical ensembles with special attention being paid to situation when energies of one or two lowest-lying states are split-off from rest of spectrum by large gap that is controlled by varying $\Lambda$. For single split-off level, which exists for the geometry with equal magnitudes but opposite signs of Robin distances on confining interfaces, heat capacity $c_V$ of canonical averaging is a nonmonotonic function of temperature $T$ with its salient maximum growing to infinity as $\ln^2\Lambda$ for decreasing to zero extrapolation length and its position being proportional to $1/(\Lambda^2\ln\Lambda)$. Specific heat per particle $c_N$ of Fermi-Dirac ensemble depends nonmonotonically on temperature too with its pronounced extremum being foregone on $T$ axis by plateau whose value at dying $\Lambda$ is $(N-1)/(2N)k_B$, with $N$ being a number of fermions. Maximum of $c_N$, similar to canonical averaging, unrestrictedly increases as $\Lambda$ goes to zero and is the largest for one particle. Most essential property of Bose-Einstein ensemble is a formation, for growing number of bosons, of sharp asymmetric shape on the $c_N-T$ characteristics that is more protrusive at the smaller Robin distances. This cusp-like structure is a manifestation of the phase transition to the condensate state. For two split-off orbitals, one additional maximum emerges whose position is shifted to colder temperatures with increase of energy gap between these two states and their higher-lying counterparts and whose magnitude approaches $\Lambda$-independent value. All these physical phenomena are qualitatively and quantitatively explained by variation of energy spectrum by Robin distance.