Using the finite element method, we investigate the lowest and first few excited state energies in a two- dimensional GaAs quantum ring （QR） with a hydrogenic donor impurity and effective mass approximation under a uniform magnetic field perpendicular to the ring plane. We study in detail the dependence of the energy spectrum with different angular momentum on the inner radius, the outer radius and width of the QR, the magnetic field and impurity position. The results reveal that the electron energies increase with the inner radius while decrease with the outer radius and width of the QR; for a fixed ring, the magnetic field induces the increase of the electron energies. Moreover, the existence of impurity reduces energy levels, and the energy levels depend highly on the impurity position, which decreases as the impurity is far away from the center of the QR. Also, the dependence of the angular momentum on the energy spectrum is analyzed in detail.
The ground and few excited states of the electrons confined in a square GaAs quantum wire （QW） subjected to an external transverse electric field are investigated using the finite difference method within the effective-mass approximation. When the transverse electric field is applied along a side of the square quantum wire, the calculation of the eigenstates of the quantum wire has an exactly solvable problem whose solutions involve the linear combinations of two independent Airy functions. Compared with the exact analytical results using Airy functions, the results obtained by the use of the finite difference method in terms of the eigenstates of the particle in the QW are in excellent agreement. Subsequently, it is considered that the eigenstates of the particle depend on the orientations of the electric field with respect to the center axis of the QW. It is interesting that the peak value of the energy is found for the field directed along the diagonal in the QW, which can lead to a large energy shift. Meanwhile, dependence of the optical absorption phenomenon in the square QW on the optical field and applied electric field is investigated. It is shown that the optical absorption spectrum depends highly upon the polarization of the optical field and the applied electric field intensities and orientations.
