Thermionic emission is a classical process （no tunneling） by which a carrier transfers from an allowed state in one material to an allowed state in another material with ideally no change in total energy. The general model, valid for semiconductor-semiconductor or metal-semiconductor interfaces, for the net current density JOB coming over the barrier at an interface, may be written for electrons, for example, as.In this expression, EFn（0+） = Ointhe metal （i.e., the electron quasi-Fermi level is at the Fermi level） because the high population of carriers in the metal is not disturbed by the flow of current.
In these expressions, A* is the effective Richardson constant30; it is a function of the materials involved in the interface. For example, in crystalline and polycrystalline materials, electrons crossing the interface in conservative processes must preserve their total energy as well as the component of their k-vector that lies in the plane of the junction. The number of electrons in a materials system that are able to do this will depend on the E = E（k） functions of the materials involved, and this
is reflected in the value of A*. There can be materials systems where electrons must interact with phonons to adjust their k-vectors so they can fit into the E = E（k） of their new host. In so doing, their energy is changed. For amorphous materials, this k conservation is relaxed, since k is not a “good quantum number.” The effective A* must incorporate these variations from materials system to materials system.