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

Olendski O - Ann Phys (2015)

Bottom Line: 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.

View Article: PubMed Central - PubMed

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.


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The same as in Fig.5 but for the heat capacity . Note different c and  scales for different panels.
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fig06: The same as in Fig.5 but for the heat capacity . Note different c and scales for different panels.

Mentions: A comparison of these remarkable results with Eqs. (16), (21) and (23) confirms the general property, which states that for the large temperatures there is no difference between canonical and grand canonical distributions [2]. However, for the small T these two statistics produce very different features. Fig.6 shows the FD heat capacity of the pure Dirichlet QW in terms of the temperature and electric field for the different N corresponding to their counterparts from Fig.5. It is seen that, for the larger number of the particles, the asymptotics from Eq. (43b) is achieved at the higher T. At the zero field, a prominent characteristic of the heat capacity dependence for the one particle (top left panel of Fig.6) is a salient maximum observed at , i.e., at the right edge of the plateau from Eq. (38). Accordingly, we attribute this extremum to the different behavior of the chemical potential for and ; namely, as it was mentioned during discussion of Fig.5, for one particle the Fermi energy decreases after the flat part from Eq. (38) while for any other number N it grows with T. Thus, their contributions to the heat capacity from Eq. (32) are opposite to each other what results in the resonance that is observed for the one particle only. Even though the shape of this maximum is quite similar to its NN counterpart for the canonical ensemble, see Sec. 2, its physical explanation is completely different. First, we point out that the very similar extrema are calculated also for the ND (with and ) and pure Neumann ( and ) QWs too. The fact that the three are almost the same and the ratios of the three temperatures are practically equal to those of Δ0(0) from Eq. (25), undoubtedly proves that the origin of this effect is the BC independent one and that the interplay between the two lowest states plays a dominant role in it. To understand these resonances, let us recall that, for the very small temperatures, the properties of the FD well are determined only by the highest occupied level and its interaction with the nearest (empty at ) above lying state, what is reflected in the extremely rapid approach by the chemical potential to the energy from Eq. (38) that is located exactly in the middle between them. For , a contribution from the lower lying members in this regime is negligibly small and can be safely neglected, while for the one-electron QW this addition is absent by definition. Further growth of the temperature increases thermal energy but it is still too “weak” to compel the corpuscles, which at lied below the Fermi energy, to contribute to the heat capacity. Only at the right edge of the plateau, the thermodynamic quantum becomes strong enough and forces other particles to donate to and . Therefore, for the heat capacity is a quite smoothly varying function of the temperature. However, for there are no such additional donors that aid to support the continuous growth of the heat capacity, which can not be sustained by the one particle only. As a result, the specific heat reaches maximum and drops.


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

Olendski O - Ann Phys (2015)

The same as in Fig.5 but for the heat capacity . Note different c and  scales for different panels.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4402035&req=5

fig06: The same as in Fig.5 but for the heat capacity . Note different c and scales for different panels.
Mentions: A comparison of these remarkable results with Eqs. (16), (21) and (23) confirms the general property, which states that for the large temperatures there is no difference between canonical and grand canonical distributions [2]. However, for the small T these two statistics produce very different features. Fig.6 shows the FD heat capacity of the pure Dirichlet QW in terms of the temperature and electric field for the different N corresponding to their counterparts from Fig.5. It is seen that, for the larger number of the particles, the asymptotics from Eq. (43b) is achieved at the higher T. At the zero field, a prominent characteristic of the heat capacity dependence for the one particle (top left panel of Fig.6) is a salient maximum observed at , i.e., at the right edge of the plateau from Eq. (38). Accordingly, we attribute this extremum to the different behavior of the chemical potential for and ; namely, as it was mentioned during discussion of Fig.5, for one particle the Fermi energy decreases after the flat part from Eq. (38) while for any other number N it grows with T. Thus, their contributions to the heat capacity from Eq. (32) are opposite to each other what results in the resonance that is observed for the one particle only. Even though the shape of this maximum is quite similar to its NN counterpart for the canonical ensemble, see Sec. 2, its physical explanation is completely different. First, we point out that the very similar extrema are calculated also for the ND (with and ) and pure Neumann ( and ) QWs too. The fact that the three are almost the same and the ratios of the three temperatures are practically equal to those of Δ0(0) from Eq. (25), undoubtedly proves that the origin of this effect is the BC independent one and that the interplay between the two lowest states plays a dominant role in it. To understand these resonances, let us recall that, for the very small temperatures, the properties of the FD well are determined only by the highest occupied level and its interaction with the nearest (empty at ) above lying state, what is reflected in the extremely rapid approach by the chemical potential to the energy from Eq. (38) that is located exactly in the middle between them. For , a contribution from the lower lying members in this regime is negligibly small and can be safely neglected, while for the one-electron QW this addition is absent by definition. Further growth of the temperature increases thermal energy but it is still too “weak” to compel the corpuscles, which at lied below the Fermi energy, to contribute to the heat capacity. Only at the right edge of the plateau, the thermodynamic quantum becomes strong enough and forces other particles to donate to and . Therefore, for the heat capacity is a quite smoothly varying function of the temperature. However, for there are no such additional donors that aid to support the continuous growth of the heat capacity, which can not be sustained by the one particle only. As a result, the specific heat reaches maximum and drops.

Bottom Line: 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.

View Article: PubMed Central - PubMed

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