Electronic States in Crystals under Large Over-All Perturbations

Solutions of the three-dimensional Schr\"odinger equation are discussed for a potential which is the sum of a potential with the periodicity of the crystal lattice plus a perturbing potential. A general theory of large over-all perturbations, such that the energy lies close to one permitted band in one region of the crystal and close to a second permitted band in another, is developed. The theory is then applied to a one-dimensional crystal in a uniform electric field, using the narrow band approximation; the probability for an electron to cross a forbidden energy band is calculated. These results are considered in connection with the interpretation of the current-voltage characteristic of an N-P junction of germanium at high electric fields.