Band-by-band decompositions of the Born effective charges

The Born effective charge, Z*, that describes the polarization created by c ollective atomic displacements, can be computed from first principles follo wing different techniques. We establish the connections existing between th ese different formulations, and analyse the related band-by-band decomposit ions. We show that unlike for the full Z*, the different band-by-band value s are not equal, and emphasize that one of them has a natural physical mean ing in terms of Wannier functions.