This investigation is a theoretical study of the influence of defects of finite volume on the electrical conductivity in the quantum size effect regime. Correction terms to existing equations are derived, and a physical explanation of the results is given. Many macroscopic properties of films exhibit an oscillatory dependence on thickness when the thickness is comparable to the de Broglie wavelength of an electron at the Fermi surface. This behavior is called the quantum size effect. In very thin films, scattering from surfaces, phonons, and crystal defects plays an increasingly important role. In this investigation the influence of scattering centers …
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This investigation is a theoretical study of the influence of defects of finite volume on the electrical conductivity in the quantum size effect regime. Correction terms to existing equations are derived, and a physical explanation of the results is given. Many macroscopic properties of films exhibit an oscillatory dependence on thickness when the thickness is comparable to the de Broglie wavelength of an electron at the Fermi surface. This behavior is called the quantum size effect. In very thin films, scattering from surfaces, phonons, and crystal defects plays an increasingly important role. In this investigation the influence of scattering centers (defects) in semimetal films on the electrical conductivity is explored by extending existing work to include scattering centers of finite range. The purpose of this study is to determine the overall change in the conductivity and the alteration of the amplitude of the oscillations. The Boltzmann transport equation is the starting point for the calculation. An equation for the vector mean free path is derived, and a solution is obtained by the iterative process. The relaxation approximation need not be made since the vector mean free path is determined. The sample is a thin slab that is infinite in two dimensions. The assumption is made that the electron wave function is zero at the walls of the sample. It is further assumed that there is a known number of randomly located defects within the slab. The noninteracting electrons are considered free except in the vicinity of the scattering centers. The defects are characterized by a potential that is constant within a small cube and zero outside of it. This approach allows the potential matrix elements to be evaluated by expanding in a power series. The electrical conductivity is calculated for three defect sizes, and a comparison is made to 3-function (infinitely small) scattering centers. An overall decrease in the conductivity is found in each case, and the absolute magnitude of the oscillations is decreased. The percentage of oscillation, however, is increased. The general conductivity decrease is attributed to the increase in the scattering range. The change in the amplitude of the oscillations is explained by analyzing the transition probabilities to available energy states at critical film thicknesses. The oscillations are found to be a result of transitions from states with large energies in the plane of the film to states with small energies in the plane of the film. The number of electrons occupying the various states is determined at critical film thicknesses, and a comparison with the conductivity equation shows excellent agreement.
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