  The extension of the adiabatic regularization method to spin-$1/2$ fields
requires a self-consistent adiabatic expansion of the field modes. We provide
here the details of such expansion, which differs from the WKB ansatz that
works well for scalars, to firmly establish the generalization of the adiabatic
renormalization scheme to spin-$1/2$ fields. We focus on the computation of
particle production in de Sitter spacetime and obtain an analytic expression of
the renormalized stress-energy tensor for Dirac fermions.
