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Simple expressions for local and total Hall conductivities in finite two dimensional electron systems under magnetic field are obtained from the Kubo formula. The deviations of the Hall conductivity from integer values are always negative and their magnitude is inversely proportional to the effective width of the sample and proportional to the slope of the Landau branch dispersion relation at the Fermi level, Eq. (22). We also calculate the local conductivity in finite samples. The conductivity density is constant in the bulk and sums up to an integer value. Its spatial distribution is terminated in the bulk in a universal manner. Illustrations for simple models of the confinement barrier, as well as relation to recent experimental data for quantum wires are given.

Very-low-energy electron diffraction (VLEED) intensities from a clean Cu (111) surface have been measured in detail in the energy range 15–100 eV by low-energy electron microscope (LEEM). This enabled the elimination of possible disturbances due to stray magnetic fields. Corresponding theoretical I–V curves have been obtained in good agreement with experimental data when an image-type surface barrier and anisotropy of the electron attenuation were taken into account. The reliability factor analysis indicates a slight expansion of the topmost interatomic spacing of Cu (111) relative to its bulk value.

The surface sensitivity of electron diffraction and of electron spectroscopies is determined by the imaginary component of the electron self-energy. In crystals, the energy and direction dependence of the electron attenuation and of the escape depth should be taken into account at very low energies. Strong anisotropy of the electron attenuation has been obtained around 20 eV from peak shapes in VLEED intensity profiles from (111) transition metal surfaces. Extension of the local density approximation in the density functional formalism provides quantitative description of the electron self-energy. The one-step model of angular resolved photoemission incorporating the self-energy predicts a strong energy and angle dependence of the escape depth of low energy photoelectrons emitted from GaAs(110).