Comparative analysis of electric field influence on the quantum wells with different boundary conditions: II. Thermodynamic properties.
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 [Formula: see text] are calculated. For the canonical ensemble, analytical expressions involving theta functions are found for the mean energy and heat capacity [Formula: see text] for the box with no applied voltage. Pronounced maximum accompanied by the adjacent minimum of the specific heat dependence on the temperatureTfor 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 [Formula: see text] observed on theTaxis for one particle and its absence for any other numberNof corpuscles. Qualitative and quantitative explanation of this phenomenon employs the analysis of the chemical potential and its temperature dependence for differentN. It is proved that critical temperature [Formula: see text] 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 [Formula: see text] what is explained again by the modification by the field of the interrelated energies. It is shown that even for the temperatures smaller than [Formula: see text] the total dipole moment [Formula: see text] may become negative for the quite moderate [Formula: see text]. For either Fermi or BE system, the influence of the electric field on the heat capacity is shown to be suppressed withNgrowing. Different asymptotic cases of, e.g., the small and large temperatures and low and high voltages are derived analytically and explained physically. Parallels are drawn to the similar properties of the 1D harmonic oscillator, and similarities and differences between them are discussed.
King Abdullah Institute for Nanotechnology, King Saud University P.O. Box 2455, Riyadh, 11451, Saudi Arabia.
Oleg
O
Olendski
Bose-Einstein condensation; Electric field; Fermi-Dirac statistics; boundary conditions; heat capacity; quantum well