Analytical obtention of eigen-energies for lens-shaped quantum dot with finite barriers

The bound states of a particle in a lens-shaped quantum dot with finite confinement potential are obtained in the envelope function approximation. The quantum dot has circular base with radius $a$ and maximum cap height $b$, and the effective mass of the particle is considered different inside and outside the dot. A 2D Fourier expansion is used in a semi-sphere domain with infinite walls which contains the geometry of the original potential. Electron energies for different values of lens deformation $b/a$, lens radius $a$ and barrier height $V_o$ are calculated. In the very high confinement potential limit, the results for the infinite barrier case are recovered.