Berry-phase theory of proper piezoelectric response

Recent theoretical advances have established that the electric polarization in an insulating crystal can be viewed as a multivalued quantity that is determined by certain Berry phases associated with the occupied Bloch bands. The application of this approach to the computation of piezoelectric coefficients is not entirely straightforward, since a naive determination of the (``improper'') piezoelectric coefficients from finite differences of the polarization at nearby strain states leads to a dependence upon the choice of ``branch'' of the polarization. The purpose of the present paper is to clarify that if one calculates instead the ``proper'' piezoelectric response, the branch dependence is eliminated. From this analysis, a simplified recipe for the direct finite-difference computation of the proper piezoelectric coefficients emerges naturally.