Energy levels of a particle confined in an ellipsoidal potential well

Physica E: Low-dimensional Systems and Nanostructures

Nadareishvili

et al. 2016

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A charged particle confined in a strongly prolate ellipsoidal shaped finite potential well is studied. In the case when a distance R between foci is large and accordingly 1 R  is small, the asymptotic solutions of quasiradial and quasiangular equations in prolate spheroidal coordinates are found. We demonstrate that quasiangular wave functions inside and outside of the potential well coincide on the entire surface of strongly prolate ellipsoid if separation parameters are chosen appropriately. This allows us to obtain the transcendental equation for the energy levels by equating the quasiradial wave function and its derivative on the surface of ellipsoid.The obtained equation is solved numerically and algebraically. The calculated energies are in good qualitative and quantitative agreement with the results obtained earlier for the infinitely high ellipsoidal potential well via a numerical solution of the quasiradial and quasiangular equations. An importance of the actual shape of ellipsoidal potential well for calculation of the energy spectrum for the trapped particle is shown. A dependence of the energy spectrum on the effective mass when it is a different constant inside and outside of the ellipsoid is addressed.

Section: Results Of Calculation and Discussioncontrasting

confidence: 92%

Section: Resultssupporting

confidence: 87%

Section: Results Of Calculation and Discussionmentioning

confidence: 99%

Section: Results Of Calculation and Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Energy spectra of a particle confined in a finite ellipsoidal shaped potential well

Physica E: Low-dimensional Systems and Nanostructures

Nadareishvili

et al. 2016

Self Cite

Discontinuity of dipole-moment matrix elements in ellipsoidally shaped nanoparticles and profiles of spectral lines

Opto-Electronics Review

Devdariani

2018

The behaviour of energy levels and optical spectra of a charged particle (electron or hole) confined within a potential well of ellipsoidal shape is investigated as a function of the shape-anisotropy parameter. If two energy levels of the same symmetry intersect in a perturbation-theory approximation, they move apart on direct diagonalization of the appropriate Hamiltonian. The intersection of the energy levels leads to a discontinuity of the corresponding dipole-moment matrix element. The discontinuity of matrix elements is not reflected in the behaviour of transition probabilities which are continuous functions of the shapeanisotropy parameter. The profiles of a spectral line emitted or absorbed by an ensemble of ellipsoidally shaped nanoparticles with a Gaussian distribution of size are calculated and discussed.

Interband optical transitions in ellipsoidal shaped nanoparticles

Physica B: Condensed Matter

Devdariani

2017

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The optical properties of crystalline semiconductor nanoparticles with ellipsoidal shape are investigated and discussed as a function of the shape-anisotropy parameter. The optical transitionmatrix elements are calculated in the dipole approximation using perturbation theory and with a direct diagonalization of the appropriate Hamiltonian. The matrix elements involving the ground and first excited states are monotonic functions of the shape-anisotropy parameter, whereas matrix elements involving the highly excited states have zeros and extrema that are reflected in the behaviour of the corresponding transition probabilities. Moreover, some matrix elements involving the excited states have discontinuity. We demonstrate that, nanoparticles with ellipsoidal shape can be grown with the infrared as well as ultraviolet features.