Band-by-band decompositions of the Born effective charges

The Born effective charge, Z*, that describes the polarization created by collective atomic displacements, can be computed from first principles following different techniques. We establish the connections existing between these different formulations, and analyse the related band-by-band decompositions. We show that unlike for the full Z*, the different band-by-band values are not equal, and emphasize that one of them has a natural physical meaning in terms of Wannier functions.