In this paper, we have researched the electronic and optical properties of cylindrical quantum dot structures by selecting four different hyperbolic-type potentials in the axial direction under an axially-applied electric field. We have considered a position-dependent effective mass model in which both the smooth variation of the effective mass in the axial direction adjusted to the way the confining potentials change and its abrupt change in the radial direction have been considered in solving the eigenvalue differential equation. The calculations of the eigenvalue equation have been implemented considering both the Dirichlet conditions (zero flux) and the open boundary conditions (non-zero flux) in the planes perpendicular to the direction of the applied electric field, which guarantees the validity of the results presented in this study for quasi-steady states with extremely high lifetimes. We have used the diagonalization method combined with the finite element method to find the eigenvalues and eigenfunction of the confined electron in the cylindrical quantum dots. The numerical strategies that have been used for the solution of the differential equations allowed us to overcome the multiple problems that the boundary conditions present in the region of intersection of the flat and cylindrical faces that form the boundary of the heterostructure. To calculate the linear and third-order nonlinear optical absorption coefficients and relative changes in the refractive index, a two-level approach in the density matrix expansion is used. Our results show that the electronic and, therefore, optical properties of the structures focused on can be adjusted to obtain a suitable response for specific studies or goals by changing structural parameters such as the widths and depths of the potentials in the axial direction, as well as the electric field intensity.

In this study, we present a detailed theoretical investigation of the effect of an externally applied magnetic field on the energy states $1s$, $1p$, $1d$ and $1f$ in the spherical quantum dot with finite and infinite confinement potentials. For both finite and infinite spherical quantum dot, the first four electron energies, Zeeman transition energies between these electronic states and optical absorption coefficients between the related states with and without magnetic field are investigated. The results show that the confinement potential, magnetic field and dot radius have a strong effect on energy states, Zeeman transition energies and absorption coefficients especially in the large dot radii. In the small dot radii, energy levels are relatively insensitive to the magnetic field because the spatial confinement of the electron prevails over the magnetic confinement. As $\mathrm{\Delta }m$ changes from $-1$ to $+1$, the peak positions of the optical absorptions shift to higher energy values (blue shift). The absorption peaks for the infinite quantum dot are localised in higher photon energies those that of the finite quantum dot. The magnetic field causes that the degeneration of energies to be removed and the peak positions of transitions corresponding to $\mathrm{\Delta }m=+1$ shift towards to blue in contrast to the cases of $\mathrm{\Delta }m=-1$ and $\mathrm{\Delta }m=0$.