Thermionic emission in heterosystems with different effective electronic masses

It is shown that the usual way of calculating thermionic emission takes into account incorrectly the conservation law of the quasimomentum of the electrons on the boundary separating two regions with different effective masses of electrons. Therefore, at a thermodynamic equilibrium the opposite flows of the electrons crossing this boundary are different (because the Richardson constants are different). This paradox disappears if one takes into account that not all electrons with larger effective mass can cross the boundary even if the part of their kinetic energy perpendicular to the interface is larger than the barrier height. In fact this condition is contained in the quantum-mechanical coefficient of electron transmission. It is shown that in the spherical and parabolic effective-mass approximation the Richardson constant is defined by the smallest effective mass. This fact is important in the calculations of the thermionic currents in semiconductor contact phenomena especially in heterostructures. It makes clear what effective mass must be used in the definition of the pre-exponent factor of the thermionic current.