An obstacle in a quantum film: the density dipole and extra resistance

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Abstract
The additional resistivity of a quantum film due to an obstacle is investigated. The film is treated by a semi-classical formalism whilst the obstacle is characterized by its quantum mechanical scattering cross-section. The film intrinsic scattering mechanism allows for the adaptation of an arbitrary density distribution to a characteristic 'ideal' distribution over the channels. Special attention is paid to the multiple-scattering cycles between the obstacle and its surroundings, i.e., the scattering background of the film. The analysis of these backscattering processes leads to a self-consistent equation for the current density incident on the scatterer. For the general case of an arbitrarily strong obstacle and many conducting channels, this equation system can be solved only numerically. However, the formalism becomes handy if the obstacle scatters only weakly. A condition is found for the obstacle to be considered as weak. On the other hand, if one considers only one conducting channel it is possible to solve the transport problem analytically even for a strong obstacle. In this case, we find an expression for the resistivity which contains the scattering cross-section in a non-linear manner. This non-linearity was already predicted in 1957 by Landauer.