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Comment: 4 pages, two-column style with 2 postscript figures embedded. Uses REVTEX and epsf macros. Also available at
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 emerge...
Comment: 8 pages, with 3 postscript figures embedded. Uses REVTEX and epsf macros. Also available at
Comment: Replaced with extended version
Comment: Changes to the manuscript (clarifications and correction of typos) made in response to referee's comments
Comment: The issue with the sign of polarization is resolved now
Comment: 15 pages, 9 figures and 7 tables. Supplementary video: or
Comment: 4 pages, 3 figures (small corrections)
Comment: 4 pages, 2 figures, submitted to EPJ B (MEIPIC6 proceedings)
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