Javier Junquera

How to compute the

Born effective charge tensor

Definition of the Born effective charges, also known as dynamical charges

Units: electron charges

For periodic solids, the Born effective charge of atom is a tensor defined as the coefficient of proportionality at the linear order and under the condition of zero macroscopic electric field, between the macroscopic polarization per unit cell

created in direction and a cooperative displacements of atoms in direction

Read pages 106 and following of Philippe Ghosez’s PhD thesis http://www.phythema.ulg.ac.be/webroot/misc/books/PhD-Ph.Ghosez.pdf

In siesta computed from finite differences of the bulk spontaneous polarization

SrTiO3 in the centrosymmetric bulk cubic structure

To compute the phonons, the atomic masses are introduced

in this block We are going to displace all the atoms in the unit cell 0.01 Bohrs

along the x, y, and z direction For each atomic configuration, we compute the macroscopic

polarization with this Polarization Grid in reciprocal space

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SystemName Bulk SrTiO3# Centrosymmetric paraelectric configuration# LDA-CA# 1200 Ry# 6 x 6 x 6; 0.5 0.5 0.5 MP mesh

SystemLabel SrTiO3NumberOfAtoms 5NumberOfSpecies 3%block ChemicalSpeciesLabel1 38 Sr2 22 Ti3 8 O

%endblock ChemicalSpeciesLabelLatticeConstant 3.874 Ang%block LatticeVectors

1.000 0.000 0.0000.000 1.000 0.0000.000 0.000 1.000

%endblock LatticeVectorsAtomicCoordinatesFormat Fractional%block AtomicCoordinatesAndAtomicSpecies

0.00000000 0.00000000 0.00000000 1 87.62 Sr0.50000000 0.50000000 0.50000000 2 47.867 Ti0.50000000 0.50000000 0.00000000 3 15.9994 O0.50000000 0.00000000 0.50000000 3 15.9994 O0.00000000 0.50000000 0.50000000 3 15.9994 O

%endblock AtomicCoordinatesAndAtomicSpeciesXC.functional LDAXC.authors CAMeshCutoff 1200 RyMD.TypeOfRun FCMD.FCDispl 0.01 bohrBornCharge .true.WriteForces .true.WriteCoorStep .true.%block PolarizationGrids

20 4 4 yes4 20 4 yes4 4 20 yes

%endblock PolarizationGridsEigenvectors .true.LatticeDielectricConstant .true.%block BandLines1 0.0 0.0 0.0 \Gamma # Compute eigenvectors only at \Gamma%endblock BandLines

Born effective charges dumped into a file: SystemLabel.BC

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BC matrixx y z

x 2.5275365 -0.0000002 0.0000013 |y 0.0010035 2.5302200 -0.0000002 | Born effective charges for Baz -0.0000137 0.0003847 2.5285002 |--------------------------------------------------x 7.5581447 -0.0000001 0.0001447 |y -0.0004625 7.5559190 0.0000031 | Born effective charges for Tiz -0.0000003 0.0002686 7.5581385 |--------------------------------------------------x -2.0648235 0.0000017 -0.0007149 |y 0.0003726 -2.0647665 -0.0000099 | Born effective charges for O1z 0.0000077 0.0002320 -5.9525109 |--------------------------------------------------x -2.0647275 0.0000029 0.0004251 |y 0.0002713 -5.9519215 -0.0000003 | Born effective charges for O2z -0.0000006 0.0004387 -2.0647094 |--------------------------------------------------x -5.9521785 0.0000059 0.0003053 |y 0.0004152 -2.0646781 0.0000069 | Born effective charges for O3z 0.0000021 0.0003887 -2.0646981 |--------------------------------------------------

Acoustic sum rule

It is important that the acoustic sum rule is preserved by the Born effective charges

(if we displace the whole solid rigidly, no polarization should be generated)

In our simulation, taking , and rounding to the third significant digit

Substracting this same amount to all the atoms

To fulfill the acoustic sum rule, we divide the sum by the number of atoms

2

2.527 + 7.558 -2.065 -2.065 -5.952 = 0.003

0.003 / Number of atoms = 0.003 / 5 = 0.0006

2.526 + 7.557 - 2.066 - 2.066 - 5.951 = 0

2

2.527 + 7.558 -2.065 -2.065 -5.952 = 0.003

0.003 / Number of atoms = 0.003 / 5 = 0.0006

2.526 + 7.557 - 2.066 - 2.066 - 5.951 = 0

2

2.527 + 7.558 -2.065 -2.065 -5.952 = 0.003

0.003 / Number of atoms = 0.003 / 5 = 0.0006

2.526 + 7.557 - 2.066 - 2.066 - 5.951 = 0

Comparison with previous results

First-principles studies of ferroelectric oxides K. M. Rabe and Ph. Ghosez, included in

Physics of Ferroelectrics. A Modern Perspective. Topics in Applied Physics

K. Rabe, Ch. Ahn, and J. –M. Triscone (Editors) Springer-Verlag, Heidelberg (2007)

Adapted from Ph. Ghosez et al. ,

Phys. Rev. B 58, 6224 (1998)

Phonon frequencies and eigenvectors at the Γ-point

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Computing Eigenvalues and Eigenvectorseigenvalue # 1 omega= -2.7961828519320900E-002eigenvalue # 2 omega= -2.0814810919826041E-002eigenvalue # 3 omega= 1.9265020396784287E-002eigenvalue # 4 omega= 101.04549449236812eigenvalue # 5 omega= 101.16550726308604eigenvalue # 6 omega= 101.24387730397110eigenvalue # 7 omega= 193.79205340562504eigenvalue # 8 omega= 193.81052897990472eigenvalue # 9 omega= 193.90404052238060eigenvalue # 10 omega= 233.35853422015677eigenvalue # 11 omega= 233.37431843154627eigenvalue # 12 omega= 233.41735049898216eigenvalue # 13 omega= 576.15638028855415eigenvalue # 14 omega= 576.20570609763286eigenvalue # 15 omega= 576.37908201058326

num_wann = 6!here we exclude all bands except the Mn 3p bandsexclude_bands : 1,2

dis_win_min = -60.0d0dis_win_max = -50.0d0dis_num_iter = 500dis_mix_ratio = 0.15d0

begin projectionsbohrMn:pend projections

Three frequencies are zero, They correspond to translational

modes

- Go to the directory $cd <your_siesta_path>/Util/Vibra/Src -  To compile the vibra suite with the same arch.make as in siesta, type $make -  Go back to the directory where you are running the exercise and type $ <your_siesta_path>/Util/Vibra/Src/vibrator < SrTiO3.fdf -  You get the file SrTiO3.bands with the eigenvalues and SrTiO3.vectors with the eigenvectors