Spin currents in semiconductors: Redefinition and counterexample

We show that effective Hamiltonians in solids lead to specific problems when dealing with spin-orbit interaction. We prove that the usual construction, where the velocity operator is simply deduced from the velocity calculated using the Hamilton relation, has a restricted validity and that there is an absolute necessity to modify the standard current-tensor definition. We derive proper symmetrized expressions in the case of a Hamiltonian which has linear and cubic dependence versus momentum components.