  The conduction properties of a grain boundary junction with Fermi velocity
mismatch are analyzed. We provide a generalization of the Dirac Hamiltonian
model taking into account the Fermi velocity gradient at the interface. General
boundary conditions for the scattering problem are derived within the framework
of the matching matrix method. We show that the scattering properties of the
interface, as predicted by the theory, strongly depend on the boundary
conditions used. We demonstrate that when the valley degeneracy is broken a
charge current is established at the grain boundary interface. These findings
provide the working principle of a valley to charge current converter, which is
relevant for the emergent field of valleytronics.
