Residual resistance of 2D and 3D structures and Joule heat release

We consider a residual resistance and Joule heat release in 2D nanostructures as well as in ordinary 3D conductors. We assume that elastic scattering of conduction electrons by lattice defects is predominant. Within a rather intricate situation in such systems we discuss in detail two cases. (1) The elastic scattering alone (i.e. without regard of inelastic mechanisms of scattering) leads to a transition of the mechanical energy (stored by the electrons under the action of an electric field) into heat in a traditional way. This process can be described by the Boltzmann equation where it is possible to do the configuration averaging over defect positions in the electron-impurity collision term. The corresponding conditions are usually met in metals. (2) The elastic scattering can be considered with the help of the standard electron-impurity collision integral only in combination with some additional averaging procedure (possibly including inelastic scattering or some mechanisms of electron wavefunction phase destruction). This situation is typical for degenerate semiconductors with a high concentration of dopants and conduction electrons. Quite often, heat release can be observed via transfer of heat to the lattice, i.e. via inelastic processes of electron-phonon collisions and can take place at distances much larger than the size of the device. However, a direct heating of the electron system can be registered too by, for instance, local measurements of the current noise or direct measurement of an electron distribution function.