Notes written by DRHamann, related to the computation of the piezoelectric tensor. Later, these notes will be placed inside a help file. Copyright (C) 2004-2018 ABINIT group (DRH) This file is distributed under the terms of the GNU General Public License, see ~abinit/COPYING or http://www.gnu.org/copyleft/gpl.txt . For the initials of contributors, see ~abinit/doc/developers/contributors.txt . ======================================== 11/4/03 Modifications by D. R. Hamann to add calculations of the rigid-ion piezoelectric tensor to the existing response function calculations for the strain perturbation. This tensor is computed by setting rfeld = 3 and rfstrs = 1, 2, or 3. The d/dk wave functions should be available from a preceding calculation. Note that even if only a limited number of piezoelectric tensor terms are wanted (as determined by rfstrs and idir in this calculation) it is necessary to set idir = 1 1 1 in the d/dk calculation for most structures. The only obvious exception to this requirement is cases in which the primitive lattice vectors are all aligned with the cartesian axes. The code will omit terms in the output piezoelectric tensor for which the available d/dk set is incomplete. The full (relaxed-ion) piezoelectric tensor is obtained by adding the contribution from internal strains weighted by Born effective charges. This requires combining force-constant matrix, internal strain coupling parameter, and Born effective charge information from the DDB files, and should be automated in anaddb. The effective charges are calculated when rfeld = 2, and thus are obtained simultaneously with the rigid-ion piezoelectric tensor. Note that the piezoelectric terms in the 2nd-order matrix output (as well as DDB) correspond to derivatives of the Berry phase for the reciprocal lattice direction conjugate to the leading idir index with respect to the strain components, and thus do not have the expected dimensions (bohr). Also note that the response function piezoelectric tensor is the "proper" piezoelectric tensor, while numerical differentiation of the ground-state polarization yields the "improper" tensor (see D. Vanderbilt, J. Phys. Chem. Solids 61, 147 (2000)). Note further that comparisons of the response-function piezoelectric tensor results with finite-difference ground-state polarization calculations are not straightforward. The README notes for Test_v4#65 and 66 discuss the relevant issues in more detail. Rigid-ion piezoelectric tensor test 11/04/03 AlP with randomly distorted zincblende structure: rprim -0.007 0.548 0.473 0.468 0.038 0.512 0.481 0.514 -0.032 xred 0.0000 0.0000 0.0000 0.2715 0.2465 0.2540 Finite-difference d/dk used for response-function calculation. "Improper" to "Proper" corrections applied to numerical derivatives. "idir" below refers to cartesian coordinates. "istr" = (1,...,6) is a strain index. i i i i response i i numerical delta d p d p function d s derivative i e i e piezoelectric i t of r r r r tensor r r polarization t t 1 4 1 5 0.0003662414 1 1 3.66241383E-04 -1.7000e-11 1 4 2 5 -0.0015088664 1 2 -1.50886638E-03 2.0000e-11 1 4 3 5 0.0019905373 1 3 1.99053730E-03 0.0000e+00 1 4 1 6 -0.0121920868 1 4 -1.21920868E-02 0.0000e+00 1 4 2 6 -0.0005417539 1 5 -5.41753867E-04 3.3000e-11 1 4 3 6 -0.0001328516 1 6 -1.32851555E-04 4.5000e-11 2 4 1 5 0.0012894250 2 1 1.28942506E-03 6.0000e-11 2 4 2 5 -0.0015381276 2 2 -1.53812756E-03 4.0000e-11 2 4 3 5 0.0003834910 2 3 3.83491031E-04 3.1000e-11 2 4 1 6 -0.0005521035 2 4 -5.52103529E-04 -2.9000e-11 2 4 2 6 -0.0115938482 2 5 -1.15938481E-02 1.0000e-10 2 4 3 6 -0.0016492703 2 6 -1.64927028E-03 2.0000e-11 3 4 1 5 0.0003691675 3 1 3.69167513E-04 1.3000e-11 3 4 2 5 0.0002598317 3 2 2.59831666E-04 -3.4000e-11 3 4 3 5 -0.0007158592 3 3 -7.15859169E-04 3.1000e-11 3 4 1 6 -0.0009563458 3 4 -9.56345781E-04 1.9000e-11 3 4 2 6 0.0016306286 3 5 1.63062861E-03 1.0000e-11 3 4 3 6 -0.0116492025 3 6 -1.16492026E-02 -1.0000e-10 Finite-difference d/dk versus analytic d/dk piezoelectric tensor comparison: ngkpt = 4 4 4 (above) 7.82E-4 rms difference ngkpt = 8 8 8 1.01E-4 rms difference