Comparative analysis of electric field influence on the quantum wells with different boundary conditions: II. Thermodynamic properties.
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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].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.
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Affiliation: King Abdullah Institute for Nanotechnology, King Saud University P.O. Box 2455, Riyadh, 11451, Saudi Arabia.
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 [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 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 [Formula: see text] 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 [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 with N growing. 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. No MeSH data available. Related in: MedlinePlus |
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Mentions: Applied electric field modifies the energy spectrum what, in turn, affects the thermodynamic properties of the wells. It was shown that the voltage increases the difference between the ground and first excited levels for any permutation of the BCs (the only exception is the ND case at the small fields, see equations (50) in [1]); accordingly, the larger temperature is needed to push out the electron from its lowest state. This is reflected in Figs.2 and 3 where the energy and heat capacity , respectively, are shown. It is seen that the range where the mean energy does not change appreciably from the ground-state value gets wider for the stronger intensities . The same is true for the heat capacity where the plateau with its almost zero value grows with the field. The increasing voltage wipes out the NN minimum of the heat capacity simultaneously moving the maximum to the higher temperatures and increasing its magnitude. For each of the mixed BCs, it also creates a maximum that was absent at . Mentioned above DD extremum of the heat capacity gets narrower and its peak increases with the field growing. Recalling again the language of the classical statistical mechanics [2], one qualitatively explains the larger heat capacities at the nonzero fields by the contribution of the electric potential; namely, the thermally averaged value of the potential energy is: |
View Article: PubMed Central - PubMed
Affiliation: King Abdullah Institute for Nanotechnology, King Saud University P.O. Box 2455, Riyadh, 11451, Saudi Arabia.
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 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 [Formula: see text] 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 [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 with N growing. 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.
No MeSH data available.