Hydrogenic impurity states in quantum dots and quantum wires

The energies of hydrogenic impurity states with an impurity atom located at the center of a quantum dot and on the axis of a quantum-well wire are studied. These two systems are all assumed to have an infinite confining potential. In the case of the quantum dot, the impurity eigenfunctions are expressed in terms of Whittaker functions and Coulomb scattering functions. The calculated ground-state energy of the impurity approaches the correct limit of three-dimensional hydrogen atom as the radius of the quantum dot becomes very large. In the case of the quantum-well wire, analytical solutions can be obtained if we divide the space into a two-dimensional subspace (perpendicular to the axis of the quantum-well wire) and a one-dimensional subspace (parallel to the axis of the quantum-well wire). The calculated ground-state energy of the quantum-well wire approaches the ground-state energy of the shallow-impurity atom located on the surface as the radius of the wire becomes infinite. Variations of the state energies with the radius of the quantum dot and the quantum-well wire are obtained.