  We consider noninteracting electrons coupled to laser fields, and study
perturbatively the effects of the lattice potential involving disorder on the
harmonic components of the electric current, which are sources of high-order
harmonic generation (HHG). By using the Floquet-Keldysh Green functions, we
show that each harmonic component consists of the coherent and the incoherent
parts, which arise respectively from the coherent and the incoherent
scatterings by the local ion potentials. As the disorder increases, the
coherent part decreases, the incoherent one increases, and the total harmonic
component of the current first decreases rapidly and then approaches a nonzero
value. Our results highlight the importance of the periodicity of crystals,
which builds up the Bloch states extending over the solid. This is markedly
different from the traditional HHG in atomic gases, where the positions of
individual atoms are irrelevant.
