An exact solution to the problem of current transport through the grain boundary in a metal

The problem of electrical resistance of a planar boundary between two crystalline grains of a polycrystalline metal is solved analytically. The solution is derived using the Case method. The electric field and the electron distribution function are represented in the form of an eigenfunction expansion of the corresponding characteristic equation. It is demonstrated that the electric field at the interface increases drastically when the Debye frequency substantially exceeds the frequency of electron collisions in the bulk of the metal. It is found that the resistance of the grain boundary does not depend on the frequency of electron collisions in the bulk, i.e., on their mean free path.