Impurity states in a thin slab of crystal

The energy levels of an impurity atom in a thin slab of crystal are calculated by the method of equivalent Hamiltonian, which includes potential barriers representing the surfaces. For slab thickness smaller than the Bohr radius of the impurity atom, the atom is found to behave like a two dimensional hydrogen atom when projected onto a plane parallel to the surfaces of the slab. Mathematically, the finite thickness of the slab is treated as a perturbation to the case of vanishing thickness. In the direction normal to the surfaces, the electron motion is dominated by the confinement within the potential well formed by the surface barriers rather than by the impurity ion. The energy levels themselves vary significantly with the position of the impurity relative to the surfaces, as demonstrated by a numerical example.