Parameters in feram

This document describes how to determine parameters for the effective Hamiltonian in the feram code http://loto.sourceforge.net/feram/ . We determine the parameters for BaTiO3 from first-principles calculations. Theories in background are written in [Takeshi Nishimatsu, Masaya Iwamoto, Yoshiyuki Kawazoe, and Umesh V. Waghmare: "First-principles accurate total-energy surfaces for polar structural distortions of BaTiO3, PbTiO3, and SrTiO3: consequences to structural transition temperatures", Phys. Rev. B, vol.82, p.134106 (2010) http://dx.doi.org/10.1103/PhysRevB.82.134106 ]. Equations referred in this document are those ones in the PRB article.

You may find latest version of this document in http://loto.sourceforge.net/feram/parameters/ . You can also find this document and input files in the feram source package, feram-X.YY.ZZ.tar.gz.


Table of Contents:

List of required free or open source software

Pseudopotentials and GGA functional

Pseudopotentials BaTiO3-WuCohenGGA/ba.wcgga.fhi, BaTiO3-WuCohenGGA/ti.wcgga.fhi, and BaTiO3-WuCohenGGA/o.wcgga.fhi are generated by Opium http://opium.sourceforge.net/ . BaTiO3-WuCohenGGA/ba.param, BaTiO3-WuCohenGGA/ti.param, and BaTiO3-WuCohenGGA/o.param are input parameter files for Opium.

Wu and Cohen's GGA [Z. G. Wu and R. E. Cohen, Phys. Rev. B 73, 235116 (2006) http://dx.doi.org/10.1103/PhysRevB.73.235116 ] is employed.

Equilibrium lattice constant a0

With BaTiO3-WuCohenGGA/perovskite-a0.in and BaTiO3-WuCohenGGA/perovskite-a0.files, we can determine the equilibrium lattice constant a0. For example, execute abinit with

% mpirun -np 4 ./abinit < perovskite-a0.files > perovskite-a0.log

or

% sh perovskite-a0.nqs

where BaTiO3-WuCohenGGA/perovskite-a0.nqs is a system-dependent queueing script. With this calculation, we determine that

a0 = 3.98596 [Angstrom] = 7.5323692530 [Bohr].

Elastic constants B11, B12 and B44

With BaTiO3-WuCohenGGA/perovskite-B1112.in, BaTiO3-WuCohenGGA/perovskite-B1112.files, BaTiO3-WuCohenGGA/perovskite-B1112.nqs and BaTiO3-WuCohenGGA/perovskite-B1112.gp files, we can determine elastic constants B11 and B12. Note that the previous result a0=7.5323692530 [Bohr] is used in BaTiO3-WuCohenGGA/perovskite-B1112.in.

% emacs perovskite-B1112.in   # Write a0 for the acell parameter.
% sh perovskite-B1112.nqs     # perovskite-B1112.dat1 and perovskite-B1112.dat1 will be made.
% gnuplot perovskite-B1112.gp
      :
DATASET 1 -- 9
a0 = 7.5323909483511 [Bohr] = 3.98596961387263 [Angstrom]
Emin = -3574.58001772059 [eV]
B = 177.307024161185 [GPa]
DATASET 10 -- 19
a0 = 7.53357884227272 [Bohr] = 3.98659822026182 [Angstrom]
Emin = -3574.5800592618 [eV]
B11 = 126.731671475652 [eV]
C11 = 320.626887681826 [GPa]
Compute B12
B12 = 41.7582963902597 [eV]
C12 = 105.647092400865 [GPa]
% gv perovskite-B1112.eps
BaTiO3-WuCohenGGA/perovskite-B1112.jpg
Figure 1: (a) Calculated volume V dependence of Etot-E_0. (b) Strain e_xx dependence of Etot-E_0.

We can also check a value of B11-B12 with files of BaTiO3-WuCohenGGA/perovskite-B11-12.in, BaTiO3-WuCohenGGA/perovskite-B11-12.files, BaTiO3-WuCohenGGA/perovskite-B11-12.nqs, BaTiO3-WuCohenGGA/perovskite-B11-12.delta and BaTiO3-WuCohenGGA/perovskite-B11-12.gp as follows:

% emacs perovskite-B11-12.in   # Write a0 for the acell parameter.
% sh perovskite-B11-12.nqs     # perovskite-B11-12.dat will be made.
% gnuplot perovskite-B11-12.gp
      :
B11-B12 = 81.7753050280732 [eV]
% gv perovskite-B11-12.eps
BaTiO3-WuCohenGGA/perovskite-B11-12.jpg
Figure 2: Quadratic fitting of calculated strain dependence of Etot-E_0. From the centrosymmetric cubic structure, constraint-tetragonal strain is applied.

B44 can be calculated with BaTiO3-WuCohenGGA/perovskite-B44.in, BaTiO3-WuCohenGGA/perovskite-B44.files, BaTiO3-WuCohenGGA/perovskite-B44.nqs, BaTiO3-WuCohenGGA/perovskite-B44.delta and BaTiO3-WuCohenGGA/perovskite-B44.gp files as follows:

% emacs perovskite-B44in   # Write a0 for the acell parameter.
% sh perovskite-B44.nqs    # perovskite-B44.dat will be made.
% gnuplot perovskite-B44.gp
      :
B44 = 49.2408864348646 [eV]
% gv perovskite-B44.eps
BaTiO3-WuCohenGGA/perovskite-B44.jpg
Figure 3: Quadratic fitting of calculated strain dependence of Etot-E_0. From the centrosymmetric cubic structure, rhombohedral strain is applied.

Polynomial coefficients P_* and coupling constants B1xx, B1yy and B4yz

We determine the potential surface of ABO3 with the method described in [T. Hashimoto, T. Nishimatsu, H. Mizuseki, Y. Kawazoe, A. Sasaki and Y. Ikeda: Jpn. J. Appl. Phys. 43, 6785-6792 (2004) http://dx.doi.org/10.1143/JJAP.43.6785 ].

Patch for ABINIT

We apply our original patch to ABINIT, rename from abinit to abinit-xyz, then use it. This brdmin-6.2.3-2011-07-01.patch is applicable to abinit-6.2.3.

% tar zxf abinit-6.2.3.tar.gz
% cd abinit-6.2.3
% mkdir x86_64-Linux-mpif90-gfortran-4.3.3-O3-perovskite-xyz
% cd x86_64-Linux-mpif90-gfortran-4.3.3-O3-perovskite-xyz/
% ../configure FC=mpif90 --enable-mpi --with-mpi-level=2 --disable-netcdf --disable-libxc --disable-etsf-io
% cd src/95_drive/
% cp ../../../src/21drive/brdmin.F90 .
% cp ../../../src/21drive/brdmin_init.F90 .
% cp ../../../src/21drive/interfaces_95_drive.F90 .
% patch -p0 < SOMEWERE/brdmin-6.2.3-2011-07-01.patch
% cd ../..
% make
% cd src/main
% mv abinit abinit-xyz

Input file generator

Generate input files with each ruby script in the directory BaTiO3-WuCohenGGA.

% emacs perovskite-optcell*.rb      # Write a0 for the acell parameter.
% ruby perovskite-optcell2-001.rb   # => perovskite-optcell2-001.in
% ruby perovskite-optcell2-110.rb   # => perovskite-optcell2-110.in
% ruby perovskite-optcell2-111.rb   # => perovskite-optcell2-111.in

Computation

% sh perovskite-optcell2-001.nqs   # results in perovskite-optcell2-001.dat
% sh perovskite-optcell2-110.nqs   # results in perovskite-optcell2-110.dat
% sh perovskite-optcell2-111.nqs   # results in perovskite-optcell2-111.dat

B1xx, B1yy, B4yz, P_k1, P_k2, P_k3, P_k4, P_alpha and P_gamma

It is quite difficult to express the total-energy surfaces even with up to 8th order polynomial in wide range of u. Therefore, we fit Eqs. (14a)--(14c) only to the calculated data points within narrow range of u. E0, a0, B11, B12, and B44 must be written in BaTiO3-WuCohenGGA/perovskite-optcell2.gp.

% emacs perovskite-optcell2-001-narrow.dat   # narrow range of u of perovskite-optcell2-001.dat
% emacs perovskite-optcell2-110-narrow.dat   # narrow range of u of perovskite-optcell2-110.dat
% emacs perovskite-optcell2-111-narrow.dat   # narrow range of u of perovskite-optcell2-111.dat
% ./perovskite-optcell2.gp
     :
B1xx = -185.347187551195 [eV/Angstrom^2]
B1yy = -3.28092949275452 [eV/Angstrom^2]
B4yz = -14.5501738943852 [eV/Angstrom^2]
P_k1 = -267.980139917128 [eV/Angstrom^6]
P_k2 = 197.500718362569 [eV/Angstrom^6]
P_k3 = 830.19997929324 [eV/Angstrom^6]
P_k4 = 641.968099408291 [eV/Angstrom^8]
P_alpha = 78.9866142426711 [eV/Angstrom^4]
P_gamma = -115.484148812671 [eV/Angstrom^4]
#kappa = -1.51821042113559 [eV/Angstrom^2]   <=== We will also use this value to determine P_kappa2.
% gv perovskite-optcell2.eps
BaTiO3-WuCohenGGA/perovskite-optcell2.jpg
Figure 4: GNUPLOT drawing to determine polynomial coefficients P_* and coupling constants B1xx, B1yy and B4yz. Filled points are selected data in narrow ranges of u.

Direction to the minimum

The direction to the minimum can be calculated with input files of BaTiO3-WuCohenGGA/perovskite-tetragonal.in and BaTiO3-WuCohenGGA/perovskite-tetragonal.files. The program feram-X.YY.ZZ/src/displacement.F help us to calculate the normalized vector.

% ./abinit < perovskite-tetragonal.files > perovskite-tetragonal.log
% feram-X.YY.ZZ/src/displacement < perovskite-tetragonal.out
xred     0.000  0.000  0.000  0.000  0.000     0.000  0.000  0.000  0.000  0.000     0.291  0.718 -0.235 -0.235 -0.538    removed translations:  0.000000000  0.000000000 -0.000000000
% feram-X.YY.ZZ/src/displacement < perovskite-optcell2-001.out > perovskite-optcell2-001.dsp   # check the xi_z(u)

Eigenvalues and eigenvectors of IFC matrix at the Gamma point

Eigenvalues and eigenvectors of IFC matrix at the Gamma point can be calculated with frozen phonon calculations at the Gamma. The program feram-X.YY.ZZ/src/frozen_phonon_Gamma.F help us to do it. These eigenvalues and eigenvectors must be similar to those of calculated from following response-function calculations.

% sh perovskite-frozen-phonon-Gamma.nqs   # perovskite-frozen-phonon-Gamma.dat will be made.
% feram-X.YY.ZZ/src/frozen_phonon_Gamma   # this program reads perovskite-frozen-phonon-Gamma.dat
a0 =   3.9859580 [Angstrom]
eigenvalues [eV/Angstrom^2] and eigenvectors of force_constant_matrix
1  -3.982367   0.1648   0.7726  -0.2003  -0.2003  -0.5438 <=== Gamma_15 soft mode
2   0.058552   0.4511   0.4482   0.4458   0.4458   0.4451
3   4.682338  -0.0000   0.0000   0.7071  -0.7071   0.0000
4   8.125787   0.8555  -0.3118  -0.2915  -0.2915   0.0309
5  13.605841  -0.1936   0.3240  -0.4197  -0.4197   0.7108

Response-function calculations

We perform some response-function calculations with ABINIT (RF calculations, See http://www.abinit.org/documentation/helpfiles/for-v6.8/tutorial/lesson_rf1.html and [Xavier Gonze and Chngyol Lee: Phys. Rev B vol.55, pp.10355-10368 (1997) http://dx.doi.org/10.1103/PhysRevB.55.10355 ]) to determine optical dielectric constant epsilon_inf, effective charge Z_star, effective mass mass_amu, self interaction P_kappa2, and short range interactions j1, ..., j7. We do NOT move ions explicitly.

Input files and execution

We calculate interatomic force constant (IFC) matrices at the Gamma, X, M and R points and at the center of the Sigma axis. Input fies are BaTiO3-WuCohenGGA/force-constant-matrix/perovskite-Gamma.in, BaTiO3-WuCohenGGA/force-constant-matrix/perovskite-M.in, BaTiO3-WuCohenGGA/force-constant-matrix/perovskite-R.in, BaTiO3-WuCohenGGA/force-constant-matrix/perovskite-Sigma.in and BaTiO3-WuCohenGGA/force-constant-matrix/perovskite-X.in. Write determined a0 for the acell parameters in each .in file. Adding "rfasr 1" in the input files may be a good idea. First, we need results of the Gamma point, perovskite-Gamma_o_DS1_WFK, then we can calculate other points.

% ./abinit < perovskite-Gamma.files > perovskite-Gamma.log
% ./abinit < perovskite-M.files     > perovskite-M.log
% ./abinit < perovskite-R.files     > perovskite-R.log
% ./abinit < perovskite-Sigma.files > perovskite-Sigma.log
% ./abinit < perovskite-X.files     > perovskite-X.log

Eigenvalues and eigenvectors of IFC matrices

Using feram-X.YY.ZZ/src/diagonalize15x15.F, eigenvalues and eigenvectors of IFC matrices of each k-point are calculated. Dashed lines in Fig. 5 (B) is a plot of the eigenvalues of IFC matrices along symmetric axes in the first Brillouin zone. How to plot eigenvalues of IFC matrices is described in http://forum.abinit.org/viewtopic.php?f=12&t=1273 .

% ./diagonalize15x15 < perovskite-Gamma_o_DS3_DDB
acell =    3.98596   3.98596   3.98596 [Angstrom]
eigenvalues [eV/Angstrom^2] and eigenvectors of the matrix of interatomic force constants (IFCs)
 1  -3.812330     0.001 -0.000  0.166     0.005 -0.000  0.770    -0.004  0.000 -0.202    -0.001  0.000 -0.202    -0.001  0.000 -0.546 <=== Gamma_15 soft mode
 2  -3.812330     0.166 -0.000 -0.001     0.770 -0.000 -0.005    -0.546  0.000  0.001    -0.202  0.000  0.001    -0.202  0.000  0.004
 3  -3.812320     0.000  0.166  0.000     0.000  0.770  0.000    -0.000 -0.202 -0.000    -0.000 -0.546 -0.000    -0.000 -0.202 -0.000
 4   0.179744    -0.000  0.459 -0.000    -0.000  0.448 -0.000    -0.000  0.443 -0.000    -0.000  0.442 -0.000    -0.000  0.443 -0.000
 5   0.179753     0.005  0.000  0.459     0.005  0.000  0.448     0.005  0.000  0.443     0.005  0.000  0.443     0.005  0.000  0.442
 6   0.179753    -0.459 -0.000  0.005    -0.448 -0.000  0.005    -0.442 -0.000  0.005    -0.443 -0.000  0.005    -0.443 -0.000  0.005
 7   4.818055     0.000  0.000  0.000     0.000  0.000  0.000    -0.000 -0.000  0.707     0.000 -0.000 -0.707    -0.000  0.000  0.000
 8   4.818055     0.000 -0.000 -0.000     0.000 -0.000  0.000     0.000  0.000  0.000    -0.707  0.000 -0.000     0.707 -0.000  0.000
 9   4.818055    -0.000 -0.000 -0.000    -0.000 -0.000  0.000     0.000  0.707  0.000     0.000  0.000 -0.000    -0.000 -0.707  0.000
10   8.165232     0.003 -0.000 -0.852    -0.001  0.000  0.319     0.000  0.000  0.293    -0.001 -0.000  0.293    -0.001  0.000 -0.027
11   8.165232    -0.852  0.000 -0.003     0.319 -0.000  0.001    -0.027 -0.000  0.001     0.293  0.000  0.001     0.293 -0.000 -0.000
12   8.165234     0.000  0.852 -0.000    -0.000 -0.319  0.000     0.000 -0.293  0.000    -0.000  0.027  0.000    -0.000 -0.293 -0.000
13  13.740829     0.000  0.190 -0.000    -0.000 -0.325  0.000    -0.000  0.420 -0.000     0.000 -0.711 -0.000     0.000  0.420  0.000
14  13.740832    -0.190  0.000 -0.004     0.325 -0.000  0.007     0.711  0.000 -0.010    -0.420 -0.000 -0.010    -0.420  0.000  0.016
15  13.740832     0.004 -0.000 -0.190    -0.007  0.000  0.325    -0.016 -0.000 -0.420     0.010  0.000 -0.420     0.010 -0.000  0.711

% ./diagonalize15x15 < perovskite-X_o_DS1_DDB 
acell =    3.98596   3.98596   3.98596 [Angstrom]
eigenvalues [eV/Angstrom^2] and eigenvectors of the matrix of interatomic force constants (IFCs)
 1  -2.844890    -0.000 -0.000  0.000    -0.000 -0.808 -0.000    -0.000 -0.000  0.000    -0.000  0.540  0.000    -0.000  0.237  0.000
 2  -2.844881    -0.000 -0.000 -0.000    -0.000  0.000 -0.808    -0.000 -0.000  0.000     0.000 -0.000  0.237    -0.000 -0.000  0.540
 3   3.293016     0.000 -0.000  0.000     0.000 -0.000 -0.501     0.000 -0.000 -0.000    -0.000 -0.000 -0.759     0.000 -0.000 -0.415
 4   3.293026    -0.000  0.000 -0.000    -0.000  0.501 -0.000     0.000  0.000  0.000     0.000  0.415 -0.000    -0.000  0.759 -0.000
 5   5.647355    -0.000 -0.064  0.832    -0.000  0.000  0.000    -0.000 -0.042  0.550    -0.000 -0.000  0.000    -0.000  0.000  0.000
 6   5.647355    -0.000 -0.832 -0.064     0.000  0.000  0.000    -0.000 -0.550 -0.042     0.000 -0.000  0.000     0.000  0.000 -0.000
 7   6.306471    -0.000  0.000  0.000    -0.000 -0.000  0.000    -0.000 -0.000  0.000    -0.707 -0.000  0.000     0.707  0.000  0.000
 8   6.422750     0.000 -0.000 -0.000    -0.377  0.000 -0.000     0.000  0.000  0.000    -0.655  0.000 -0.000    -0.655 -0.000 -0.000
 9   7.106444     0.000  0.007 -0.552     0.000  0.000 -0.000     0.000 -0.010  0.834     0.000  0.000 -0.000     0.000 -0.000 -0.000
10   7.106444     0.000  0.552  0.007    -0.000  0.000  0.000     0.000 -0.834 -0.010    -0.000 -0.000  0.000    -0.000  0.000 -0.000
11  10.605361    -0.922  0.000  0.000    -0.000 -0.000 -0.000    -0.388 -0.000 -0.000    -0.000 -0.000  0.000    -0.000  0.000 -0.000
12  11.865325    -0.000 -0.000  0.000     0.000  0.311  0.000    -0.000 -0.000 -0.000    -0.000  0.732 -0.000     0.000 -0.606  0.000
13  11.865326    -0.000  0.000 -0.000    -0.000 -0.000  0.311    -0.000 -0.000  0.000     0.000 -0.000 -0.606    -0.000  0.000  0.732
14  27.604766    -0.388 -0.000  0.000    -0.000 -0.000 -0.000     0.922  0.000  0.000     0.000  0.000 -0.000     0.000 -0.000  0.000
15  34.256065    -0.000 -0.000  0.000    -0.926 -0.000  0.000    -0.000 -0.000 -0.000     0.266  0.000  0.000     0.266 -0.000 -0.000

% ./diagonalize15x15 < perovskite-M_o_DS1_DDB
acell =    3.98596   3.98596   3.98596 [Angstrom]
eigenvalues [eV/Angstrom^2] and eigenvectors of the matrix of interatomic force constants (IFCs)
 1  -2.285745    -0.000  0.000  0.000     0.000 -0.000  0.885     0.000  0.000 -0.000     0.000  0.000 -0.000    -0.000 -0.000 -0.466
 2   2.701916    -0.000  0.000  0.000     0.000 -0.000 -0.000    -0.000 -0.707 -0.000     0.707 -0.000  0.000    -0.000 -0.000  0.000
 3   4.181197     0.000  0.563 -0.000     0.398  0.000 -0.000     0.000  0.000 -0.000    -0.000 -0.000  0.000     0.724  0.000  0.000
 4   4.181213    -0.563  0.000  0.000     0.000 -0.398 -0.000    -0.000  0.000 -0.000    -0.000  0.000  0.000     0.000 -0.724  0.000
 5   5.930853    -0.000 -0.000 -1.000    -0.000 -0.000  0.000    -0.000  0.000  0.000     0.000 -0.000  0.000     0.000 -0.000 -0.000
 6   7.209251    -0.726  0.000 -0.000     0.000 -0.181  0.000    -0.000 -0.000 -0.000    -0.000  0.000  0.000    -0.000  0.664  0.000
 7   7.209255     0.000  0.726 -0.000     0.181  0.000  0.000     0.000  0.000 -0.000    -0.000 -0.000  0.000    -0.664 -0.000  0.000
 8   7.259426     0.000  0.000  0.000     0.000  0.000 -0.000     0.000 -0.000  0.705     0.000 -0.000  0.709    -0.000 -0.000 -0.000
 9   7.259426    -0.000  0.000  0.000     0.000 -0.000 -0.000     0.000 -0.000  0.709     0.000 -0.000 -0.705    -0.000  0.000 -0.000
10   8.728420     0.000 -0.000 -0.000    -0.000  0.000  0.466     0.000  0.000  0.000     0.000 -0.000  0.000     0.000  0.000  0.885
11   9.261847    -0.000 -0.000 -0.000    -0.000 -0.000 -0.000     0.707 -0.000 -0.000    -0.000 -0.707  0.000     0.000  0.000 -0.000
12  12.688868    -0.000  0.000  0.000     0.000  0.000 -0.000     0.000  0.707  0.000     0.707 -0.000  0.000    -0.000 -0.000 -0.000
13  29.698188    -0.000  0.000 -0.000    -0.000  0.000 -0.000     0.707 -0.000  0.000     0.000  0.707  0.000     0.000 -0.000  0.000
14  32.665709    -0.395 -0.000  0.000     0.000  0.899  0.000    -0.000 -0.000 -0.000    -0.000 -0.000 -0.000    -0.000 -0.187  0.000
15  32.665721     0.000 -0.395  0.000     0.899 -0.000  0.000     0.000 -0.000 -0.000    -0.000  0.000 -0.000    -0.187  0.000  0.000

% ./diagonalize15x15 < perovskite-R_o_DS1_DDB
acell =    3.98596   3.98596   3.98596 [Angstrom]
eigenvalues [eV/Angstrom^2] and eigenvectors of the matrix of interatomic force constants (IFCs)
 1   2.094246     0.000 -0.000 -0.000     0.000 -0.000  0.000    -0.000 -0.500 -0.460     0.500  0.000  0.196     0.460 -0.196  0.000
 2   2.094246    -0.000 -0.000  0.000    -0.000  0.000 -0.000     0.000 -0.000  0.278     0.000 -0.000  0.650    -0.278 -0.650 -0.000
 3   2.094246    -0.000 -0.000  0.000     0.000 -0.000 -0.000     0.000 -0.500  0.460     0.500 -0.000 -0.197    -0.460  0.197  0.000
 4   7.635658    -0.000 -0.000 -0.000    -0.000  0.000  0.000    -0.740  0.000  0.000     0.000  0.071 -0.000     0.000 -0.000  0.669
 5   7.635658    -0.000 -0.000 -0.000     0.000  0.000  0.000    -0.345  0.000  0.000     0.000  0.813 -0.000     0.000 -0.000 -0.468
 6   8.895905    -0.000  0.000  0.978    -0.000  0.000  0.000     0.000  0.146  0.000     0.146  0.000 -0.000     0.000 -0.000  0.000
 7   8.895914     0.700 -0.683  0.000     0.000  0.000  0.000    -0.000  0.000 -0.102     0.000 -0.000  0.105    -0.102  0.105 -0.000
 8   8.895914    -0.683 -0.700 -0.000     0.000 -0.000 -0.000     0.000 -0.000 -0.105    -0.000  0.000 -0.102    -0.105 -0.102  0.000
 9  10.557109     0.147 -0.146  0.000     0.000  0.000  0.000    -0.000 -0.000  0.488    -0.000 -0.000 -0.490     0.488 -0.490 -0.000
10  10.557109     0.146  0.147  0.000    -0.000  0.000  0.000    -0.000 -0.000 -0.490    -0.000 -0.000 -0.488    -0.490 -0.488 -0.000
11  10.557118     0.000 -0.000 -0.207     0.000  0.000  0.000     0.000  0.692  0.000     0.692 -0.000 -0.000     0.000 -0.000 -0.000
12  27.741413     0.000 -0.000 -0.000    -0.707 -0.500 -0.500     0.000  0.000  0.000     0.000  0.000 -0.000     0.000 -0.000  0.000
13  27.741413     0.000 -0.000  0.000    -0.001 -0.707  0.707     0.000  0.000  0.000     0.000  0.000  0.000     0.000 -0.000  0.000
14  27.741413    -0.000 -0.000 -0.000    -0.707  0.500  0.500     0.000  0.000  0.000     0.000 -0.000  0.000     0.000  0.000 -0.000
15  32.573120    -0.000  0.000  0.000    -0.000 -0.000 -0.000    -0.577 -0.000 -0.000    -0.000 -0.577  0.000    -0.000  0.000 -0.577

% ./diagonalize15x15 < perovskite-Sigma_o_DS1_DDB
acell =    3.98596   3.98596   3.98596 [Angstrom]
 1  -2.839858    0.00 0.00 -0.00 0.00  0.11-0.00    0.00 0.00  0.00-0.00  0.00 0.83   -0.00-0.00 -0.00-0.00 -0.09-0.09   -0.00-0.00 -0.00-0.00 -0.09-0.09    0.00-0.00  0.00-0.00 -0.00-0.51
 2   1.994966    0.34 0.00 -0.34 0.00 -0.00-0.00    0.00 0.28 -0.00-0.28  0.00-0.00    0.15 0.15 -0.28-0.28 -0.00-0.00    0.28 0.28 -0.15-0.15 -0.00-0.00   -0.00 0.32 -0.00-0.32  0.00-0.00
 3   2.828540   -0.00 0.00 -0.00 0.00 -0.44 0.00    0.00 0.00 -0.00 0.00 -0.00-0.38   -0.00-0.00  0.00 0.00 -0.33-0.33   -0.00-0.00  0.00 0.00 -0.33-0.33   -0.00-0.00 -0.00 0.00 -0.00-0.46
 4   3.849726    0.35 0.00 -0.35 0.00  0.00 0.00    0.00 0.29 -0.00-0.29 -0.00 0.00   -0.20-0.20  0.32 0.32  0.00 0.00   -0.32-0.32  0.20 0.20  0.00 0.00   -0.00 0.12 -0.00-0.12 -0.00 0.00
 5   4.930208   -0.21 0.00  0.21 0.00  0.00-0.00    0.00-0.13  0.00 0.13  0.00-0.00    0.20 0.20  0.21 0.21 -0.00-0.00   -0.21-0.21 -0.20-0.20 -0.00-0.00   -0.00 0.52  0.00-0.52  0.00-0.00
 6   5.203297   -0.08 0.00 -0.08 0.00  0.00 0.00   -0.00-0.25 -0.00-0.25 -0.00-0.00   -0.11-0.11 -0.05-0.05 -0.00-0.00   -0.05-0.05 -0.11-0.11 -0.00-0.00   -0.00-0.63 -0.00-0.63  0.00-0.00
 7   6.142422    0.00 0.00 -0.00 0.00  0.00-0.00    0.00 0.00  0.00 0.00 -0.00-0.00   -0.00-0.00 -0.00-0.00 -0.59-0.40   -0.00-0.00  0.00 0.00  0.59 0.40   -0.00-0.00  0.00 0.00 -0.00-0.00
 8   7.000273   -0.00 0.00 -0.00 0.00 -0.89-0.00    0.00 0.00 -0.00 0.00 -0.00 0.26   -0.00-0.00  0.00 0.00  0.19 0.19    0.00 0.00 -0.00-0.00  0.19 0.19   -0.00-0.00 -0.00-0.00 -0.00 0.09
 9   7.184468    0.67-0.00  0.67-0.00 -0.00-0.00   -0.00-0.21 -0.00-0.21 -0.00 0.00    0.05 0.05  0.04 0.04  0.00 0.00    0.04 0.04  0.05 0.05  0.00 0.00   -0.00-0.03 -0.00-0.03  0.00 0.00
10   8.237461   -0.01 0.00 -0.01 0.00  0.00 0.00    0.00 0.20  0.00 0.20  0.00-0.00    0.07 0.07  0.46 0.46 -0.00-0.00    0.46 0.46  0.07 0.07 -0.00-0.00   -0.00-0.17 -0.00-0.17 -0.00 0.00
11  10.064120   -0.33 0.00  0.33-0.00 -0.00 0.00   -0.00 0.14 -0.00-0.14 -0.00-0.00   -0.35-0.35 -0.15-0.15  0.00 0.00    0.15 0.15  0.35 0.35  0.00 0.00    0.00 0.29 -0.00-0.29  0.00-0.00
12  12.050401    0.00 0.00  0.00 0.00 -0.10-0.00    0.00-0.00  0.00-0.00 -0.00 0.32   -0.00-0.00 -0.00-0.00 -0.31-0.31   -0.00-0.00  0.00 0.00 -0.31-0.31   -0.00 0.00  0.00-0.00 -0.00 0.72
13  18.583901    0.05 0.00  0.05-0.00  0.00 0.00    0.00 0.32  0.00 0.32  0.00-0.00    0.37 0.37 -0.17-0.17  0.00 0.00   -0.17-0.17  0.37 0.37  0.00 0.00   -0.00-0.24 -0.00-0.24  0.00-0.00
14  19.658994    0.33 0.00 -0.33 0.00  0.00-0.00   -0.00-0.54  0.00 0.54  0.00-0.00   -0.17-0.17 -0.07-0.07  0.00 0.00    0.07 0.07  0.17 0.17  0.00 0.00    0.00 0.17 -0.00-0.17  0.00-0.00
15  40.181463   -0.21-0.00 -0.21 0.00 -0.00-0.00    0.00-0.50  0.00-0.50  0.00 0.00    0.30 0.30  0.08 0.08 -0.00-0.00    0.08 0.08  0.30 0.30 -0.00-0.00    0.00 0.10 -0.00 0.10 -0.00 0.00

Optical dielectric constant epsilon_inf

Optical dielectric constant tensor is in BaTiO3-WuCohenGGA/force-constant-matrix/perovskite-Gamma.out.

% less perovskite-Gamma.out
      :
  Dielectric tensor, in cartesian coordinates,
     j1       j2             matrix element
  dir pert dir pert     real part    imaginary part
  
   1    7   1    7         6.8691464565         0.0000000000
   1    7   2    7         0.0000000000         0.0000000000
   1    7   3    7         0.0000000000         0.0000000000
  
   2    7   1    7         0.0000000000         0.0000000000
   2    7   2    7         6.8691464565         0.0000000000
   2    7   3    7         0.0000000000         0.0000000000
  
   3    7   1    7         0.0000000000         0.0000000000
   3    7   2    7         0.0000000000         0.0000000000
   3    7   3    7         6.8691464565         0.0000000000
      :

Effective charge Z_star and effective mass mass_amu

We can find the calculated Born effective charges for each atom in the file of BaTiO3-WuCohenGGA/force-constant-matrix/perovskite-Gamma.out. There are two ways of calculations of effective charges, (from electric field response) and (from phonon response). They must be identical within some error.

% less perovskite-Gamma.out
      :
  Effective charges, in cartesian coordinates,
  (from electric field response) 
   if specified in the inputs, asr has been imposed
     j1       j2             matrix element
  dir pert dir pert     real part    imaginary part
  
   1    1   1    7         2.7419582957         0.0000000000
   2    1   1    7         0.0000000000         0.0000000000
   3    1   1    7         0.0000000000         0.0000000000
   1    2   1    7         7.4934093087         0.0000000000
   2    2   1    7         0.0000000000         0.0000000000
   3    2   1    7         0.0000000000         0.0000000000
   1    3   1    7        -5.9318277473         0.0000000000
   2    3   1    7         0.0000000000         0.0000000000
   3    3   1    7         0.0000000000         0.0000000000
   1    4   1    7        -2.1492074350         0.0000000000
   2    4   1    7         0.0000000000         0.0000000000
   3    4   1    7         0.0000000000         0.0000000000
   1    5   1    7        -2.1492081943         0.0000000000
   2    5   1    7         0.0000000000         0.0000000000
   3    5   1    7         0.0000000000         0.0000000000
      :

The direction of the Gamma_15 soft mode and The direction to the minimum are not same in principle. Using one of them, we determine the effective charge Z_star and the effective mass mass_amu.

Z_star = 10.33 =
 2.7419582957 *  0.166 +
 7.4934093087 *  0.770 +
-2.1492074350 * -0.202 +
-2.1492074350 * -0.202 +
-5.9318277473 * -0.546

mass_amu = 38.24 = 137.327*0.166**2 + 47.867*0.770**2 + 15.9994*(2*0.202**2 +0.546**2)

Self interaction P_kappa2 and nearest neighbor interactions j1, ..., j7

We select eigenvalues and make BaTiO3-WuCohenGGA/force-constant-matrix/eigenvalues2j.in. Using feram-X.YY.ZZ/src/eigenvalues2j.F, we can calculate P_kappa2 and j1, ..., j7.

% cat eigenvalues2j.in
DDB_a = -3.812330
DDB_b = 34.256065
DDB_c = -2.844881
DDB_d = -2.285745
DDB_e = 32.665721
DDB_f = 27.741413
DDB_g =  0.0
a0          = 3.98596961387263
Z_star      = 10.33
epsilon_inf = 6.8691464565
% feram-X.YY.ZZ/src/eigenvalues2j < eigenvalues2j.in
../../src/eigenvalues2j.F: 43: BEGIN: Version 0.14.07
  ../../src/eigenvalues2j.F: 61: FILENAME: stdin
  ../../src/param_module.F:109: BEGIN: read_Param().
    DDB_a = -3.812330
    DDB_b = 34.256065
    DDB_c = -2.844881
    DDB_d = -2.285745
    DDB_e = 32.665721
    DDB_f = 27.741413
    DDB_g =  0.0
    a0          = 3.98596961387263
    Z_star      = 10.33
    epsilon_inf = 6.8691464565
  ../../src/param_module.F:284: END:   read_Param().
  ../../src/param_module.F:290: BEGIN: make_Param().
  ../../src/param_module.F:313: END:   make_Param().
       j_1 =   -2.0840250430 [eV/Angstrom^2] =   -0.0214464062 [Hartree/Bohr^2]
       j_2 =   -1.1290411983 [eV/Angstrom^2] =   -0.0116188029 [Hartree/Bohr^2]
       j_3 =    0.6894579816 [eV/Angstrom^2] =    0.0070951143 [Hartree/Bohr^2]
       j_4 =   -0.6113408159 [eV/Angstrom^2] =   -0.0062912216 [Hartree/Bohr^2]
       j_5 =    0.0000000000 [eV/Angstrom^2] =    0.0000000000 [Hartree/Bohr^2]
       j_6 =    0.2768966803 [eV/Angstrom^2] =    0.0028495045 [Hartree/Bohr^2]
       j_7 =    0.0000000000 [eV/Angstrom^2] =    0.0000000000 [Hartree/Bohr^2]
  P_kappa2 =    8.1460516421 [eV/Angstrom^2] =    0.0838298622 [Hartree/Bohr^2]
  j =  -2.08403 -1.12904  0.68946 -0.61134  0.00000  0.27690  0.00000    [eV/Angstrom^2]
  a0          =   3.98597    [Angstrom]
  Z_star      =  10.33000
  epsilon_inf =   6.86915
../../src/eigenvalues2j.F:126: END
P_kappa2(old) + [kappa - kappa(Gamma_TO)]
= 8.1460516421 + [-1.51821042113588 - (-3.812330/2)]
= 8.1460516421 + 0.38795457886412
= 8.53400622096412
= P_kappa2(new)
../doc/figures/BaTiO3-000-dispersion-with-ifc/BaTiO3-000-dispersion-with-ifc.jpg
Figure 5: (A) Half of eigenvalues of the 3x3 long-range dipole-dipole interaction matrix Phi(k) are plotted along symmetric axes in the the first Brillouin zone of the simple-cubic lattice. Special points and k/(2pi)=(1/4, 1/4, 0) (the center of the Sigma axis) are indicated with vertical dotted lines. Labels (a)--(g) corresponds to Eqs. (15a)--(15g), respectively. Tics in the unit of Z^2/epsilon/a^3 is placed in left side. Tics in the unit of eV, in the case of the parameter set of [Takeshi Nishimatsu, Masaya Iwamoto, Yoshiyuki Kawazoe, and Umesh V. Waghmare: Phys. Rev. B 82, 134106 (2010)] is placed in right side. (B) Half of eigenvalues of the calculated 15x15 inter-atomic force constant (IFC) matrix (dashed black lines). Half of eigenvalues of the total (long-range + short-range) interaction matrix Phi^quad(k) (solid red line). Difference between them is the elevation of 0.38795.

Author of this document

Takeshi Nishimatsu (t-nissie{at}imr.tohoku.ac.jp)


Copyright © 2011 Takeshi Nishimatsu