=begin
= Accurate total-energy surfaces for zone-center distortions of BaTiO<sub>3</sub>, PbTiO<sub>3</sub> and SrTiO<sub>3</sub>
Takeshi Nishimatsu


http://loto.sourceforge.net/feram/

CLICK anywhere to go to the next page.

# I entitled my talk "Accurate total-energy surfaces for zone-center distortions of BaTiO3, PbTiO3 and SrTiO3"
# This study is done by my original MD code, feram.
# You can download the code from here.


== How to use Slidy
This presentation is written with Slidy http://www.w3.org/Talks/Tools/Slidy/ .
Use Firefox web browser.
 * Advance to next slide with mouse click or [Space]
 * Move forward/backward between slides with [Left], [Right], [Pg Up] and [Pg Dn] keys
 * [Home] key for first slide, [End] key for last slide
 * The [C] key for an automatically generated table of contents (or click on "contents" on the toolbar)
 * Function [F11] toggles between the full screen mode and the normal mode
 * [F] key toggles the display of the tool bar below.
 * [A] key toggles the presentation mode and the printing mode
 * [S] and [B] keys can control the size of fonts


== Classification of insulators by their dielectric properties
Fig. 1 figures/classification.jpg
  All insulators can be classified by their dielectric properties.
  All insulators are dielectrics.
  Some of them are piezoelectrics.
  Some of them are pyroelectrics.
  Some of them are ferroelectrics,
  though difference between pyroelectrics and ferroelectrics is controversial.
  Especially, perovskite-type ferroelectrics based on BaTiO3 and Pb(Zr$_{(1-x)}$Ti$_x$)O$_3$ (PZT)
  have good properties of all of them for applications.
/Fig.

== What is ferroelectric?
Fig. 1 figures/ferroelectrics.jpg
  Spontaneous polarization of ferroelectrics
  can have some directions according to their crystal structures.
  Therefore, the spontaneous polarization can be
  switchable by an external electric field,
  the $P$-$E$ curve has hysteresis,
  and finite-size ferroelectrics exhibit domain structures.
  Ferroelectrics also have phase transitions and
  large piezoelectricity and inverse-piezoelectricity.
/Fig.


== Procedure of Molecular dynamics simulations of ferroelectrics
 * Target ferroelectrics = BaTiO3.
 * Investigate it with first-principles calculation.
   * Construct an effective Hamiltonian with 25 parameters.
   * Using ABINIT http://www.abinit.org/ with an original patch.
   * Limited small cell size, 1-8 unit cells.
   * Only properties under zero Kelvin can be calculated.
 * Molecular dynamics (MD) simulation with the first-principles-based effective Hamiltonian.
   * Original feram code http://loto.sf.net/feram/ .
   * Realistic system size (up to 100 nm).
   * Realistic time span ($>$ 1 ns).
   * Various conditions: temperature, pressure, bulk or thin-film, externally applied electric field


== Molecular dynamics simulations of ferroelectrics with feram code
Fig. 1 ../logo.jpg
/Fig.
 * Scientific features
   * Molecular dynamics (MD) simulation with first-principles-based effective Hamiltonian
   * $AB$O<sub>3</sub>-type perovskite ferroelectrics and relaxors
   * Uses supercell; ($AB$O<sub>3</sub>)$_N$, $N=32\times 32\times 512$ $\Rightarrow$ 2,621,440 atoms
   * Coarse-graining; reduction of the number of degree of freedom
   * Long-range dipole-dipole interaction is treated in reciprocal-space; k-locality of the force matrix
   * Applications:
     * Phase transition of bulk ferroelectrics
     * Capacitor, <b>ferroelectric thin film</b> is sandwiched between short-circuited electrodes

Eq. 1
    H^{\rm eff}
    = \frac{M^*_{\rm dipole}}{2} \sum_{\bm{R},\alpha}\dot{u}_\alpha^2(\bm{R})
    + \frac{M^*_{\rm acoustic}}{2}\sum_{\bm{R},\alpha}\dot{w}_\alpha^2(\bm{R})
    + V^{\rm self}(\{\bm{u}\})+V^{\rm dpl}(\{\bm{u}\})+V^{\rm short}(\{\bm{u}\})
    + V^{\rm elas,\,homo}(\eta_1,...,\eta_6)+V^{\rm elas,\,inho}(\{\bm{w}\})
    + V^{\rm coup,\,homo}(\{\bm{u}\}, \eta_1,...,\eta_6)+V^{\rm coup,\,inho}(\{\bm{u}\}, \{\bm{w}\})
    - Z^*\sum_{\bm{R}} \mathscr{E} \cdot \bm{u}(\bm{R})
/Eq.

Fig. 1 figures/arrows.jpg
  Snapshot of the supercell. Strengths and directions of $\{\bm{u}(\bm{R})\}$
  are indicated with arrows. red and blue colors represent $+z$-polarized and $-z$-polarized sites.
/Fig.

 * Technical features
   * Fast
     * FFTW library version 3  http://www.FFTW.org/
     * Parallelized with OpenMP http://www.OpenMP.org/
   * Multi-platform
     * Linux PC
     * HITACHI SR11000 Super Technical Server
     * SONY PLAYSTATION3 (ongoing)
   * Object oriented programming (OOP) in Fortran 95
   * GNU autotools  http://www.gnu.org/software/autoconf/
   * Free software (GPLv3) http://loto.sf.net/feram/

# Other features of feram program are:
# Its Molecular dynamics (MD) simulation uses first-principles-based effective Hamiltonian.
# It can simulate ABO3-type perovskite ferroelectrics and relaxors
# It can use big size of supercell up to 32x32x512. It corresponds to 2,621,440 atoms.
# It can do such a huge calculation because it employs some kind of Coarse-graining that
# reduces the number of degree of freedom of the system.
# and because Long-range dipole-dipole interaction is treated
# in reciprocal-space using k-locality of the force matrix
#
# Using this program, you can simulate
# Phase transition of bulk ferroelectrics and properties of
# capacitors in which ferroelectric thin film is sandwiched between short-circuited electrodes.
# Today, I will mainly show you our study of ferroelectric thin films.
#
# This program is fast, because it uses a fast FFT library and
# it is parallelized with OpenMP.
# I usually use normal Linux personal computers.
# But you can also use feram on some supercomputers.
# You can freely download feram from this site.


== Background and Purpose of this study
=== Background of this study
 * LDA underestimate lattice constants slightly (1-2%)
 * Double-well total-energy surfaces for zone-center distortions of
   $AB$O$_3$ perovskite-type ferroelectrics are shallower
   than the depths expected from their observed transition
   temperatures ($T_{\rm C}=403 \rm{K}=0.0347 {\rm eV}$ for BaTiO$_3$).
 * Wu and Cohen introduced a new accurate GGA
   and obtained good agreement between calculated and experimentally observed lattice
   constants.

=== Purpose of this study
 * Calculate accurate total-energy surfaces for zone-center distortions of BaTiO$_3$, PbTiO$_3$ and SrTiO$_3$
   with GGA (Wu and Cohen).
 * Construct new set of parameters for effective Hamiltonian of BaTiO$_3$
 * Perform new MD simulation of  BaTiO$_3$




== Soft-mode displacements of perovskite type ferroelectrics <em>AB</em>O<sub>3</sub>
# [King-Smith and Vanderbilt: Phys. Rev. B <b>49</b> 5828 (1994)] Parametrization of coupling between soft-mode displacement and strains.

Fig. 1 figures/perovskite-contours.jpg
  Coupling between atomic displacements and strains. After [T. Hashimoto, T. Nishimatsu et al.: Jpn. J. Appl. Phys. <b>43</b>, 6785 (2004)].
/Fig.

Eq. 1
    u_{\alpha}^2 = \{v_{\alpha}^A\}^2 + \{v_{\alpha}^B\}^2 + \{v_{\alpha}^{\rm O_{I}}\}^2 + \{v_{\alpha}^{\rm O_{II}}\}^2 + \{v_{\alpha}^{\rm O_{III}}\}^2
/Eq.



== Calculated total-energy surfaces soft-mode displacements of perovskite type ferroelectrics BaTiO<sub>3</sub>
Fig. 1 figures/BaTiO3WuCohenGGA/BaTiO3WuCohenGGA.jpg
/Fig.




== Total-energy surfaces of the zone-center distortion for BaTiO<sub>3</sub>
Fig. 1 figures/comp-bto.jpg
  Calculated total-energy surfaces of the zone-center distortion for BaTiO$_3$ with various $E_{\rm xc}$.
/Fig.

== Total-energy surfaces of the zone-center distortion for PbTiO<sub>3</sub>
Fig. 1 figures/comp-pto.jpg
  Calculated total-energy surfaces of the zone-center distortion for PbTiO$_3$ with various $E_{\rm xc}$.
/Fig.

== Total-energy surfaces of the zone-center distortion for SrTiO<sub>3</sub>
Fig. 1 figures/comp-sto.jpg
  Calculated total-energy surfaces of the zone-center distortion for SrTiO$_3$ with various $E_{\rm xc}$.
/Fig.


== Comparison of calculated equilibrium cubic lattice constants <em>a</em><sub>0</sub>
Fig. 1 figures/a0.jpg
/Fig.



== Heating-up and Cooling-down MD simulations
Fig. 1 figures/compare-p.jpg
  Heating-up and Cooling-down MD simulations under
  (a) constant negative pressure and (b) $T$-dependent negative pressure to simulate thermal expansion.
/Fig.

== Experimentally observed lattice parameters for BaTiO<sub>3</sub>
Fig. 1 figures/BaTiO3-abc-T-dependence.jpg
  Experimentally observed lattice parameters as functions of temperature,
  after [H. E. Kay and P. Vousden: Philos. Mag. <b>40</b>, 1019  (1949)].
  The relatively weak first-order nature of the cubic-to-tetragonal phase
  transition can be compared with the first-order tetragonal-to-orthorhombic
  and orthorhombic-to-rhombohedral phase transitions.
/Fig.

== Summary
 * LDA calculations under certain negative pressures and GGA (Wu and Cohen)
   calculations are capable of yielding accurate total-energy surfaces of zone-center
   distortions.
 * New set of parameters of effective Hamiltonian for BaTiO$_3$.
 * Effect of thermal expansion cannot be ignored
   to reproduce properties in the whole range of temperature ($30-450$ K).

=end
#
# for ulmul.rb
#
require 'rubygems'
require 'ulmul'
u=Ulmul.new(1..0,'ulmul2xhtml')
u.subs_rules << [  /(heating-up)\s*(\S*\/?\.gif)(\s|$)/,'<a href="../animations/\2">\1</a>\3']
u.subs_rules << [/(cooling-down)\s*(\S*\/?\.gif)(\s|$)/,'<a href="../animations/\2">\1</a>\3']
u.parse(ARGF)
puts u.html(["feram-presentation.css"],["ulmul-slidy.js"],"Takeshi Nishimatsu","en")

# Local variables:
#   mode: RD
#   compile-command: "ruby accurate.txt accurate.txt > accurate.xhtml"
# End:
