=begin
= Simulations of hysteresis loops for thin-film ferroelectric capacitors with feram code
Takeshi Nishimatsu

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# Today, I will introduce you my new simulation program "feram", that is for
# "Ferroelectrics and Relaxor Analyzing Machine. In second half of my talk,
# I will introduce a planning study of relaxors using this "feram".


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== Purpose of feram: ferroelectric thin-film capacitors
 * Down-sizing of non-volatile FeRAM
 * Fatigue of electrodes of capacitors
   [M. Dawber et al.: J. Phys. Condes. Matter vol.15 p.L393 (2003)]
 * Hysteresis, polarization switching, dynamics of domain wall
   and their nano-size effects are not well known (Figure below).
 * <em>in situ</em> observations are difficult
$\to\to\to$ Computational simulation with "feram" program.


figures/Fong.jpg


== Background: effective Hamiltonian
King-Smith and Vanderbilt: Phys. Rev. B Vol.49 p.5828 (1994): Parametrization of coupling between soft-mode displacement and strains.

figures/perovskite-contours.jpg Coupling between atomic displacements and strains. After Hashimoto et al.: Jpn. J. Appl. Phys. Vol.43 p.6785 (2004).

# feram can simulate Perovskite type ferroelectrics which have ABO3 formula.
# In perovskite type ferroelectrics ABO$_3$, atomic displacements
# or polarization and strains are strongly coupled.
# This is the cubic high temperature crystal structure of ABO3.
# Unit cell has 5 atoms. So atomic displacements spans a 3x5=15-dimensional space.
# This is a 2-dimensional subspace of the 15-dimensional space of atomic displacements.
# In the 2-dimensional subspace, potential surface is ploted.
# If there is strain in crystal, the potential minimum goes far from its origin and
# get deeper.
#
# In 1994, my current boss David Vanderbilt parametrized this coupling.


== Coarse-graining: reduction of the number of degree of freedom
 * Real perovskite-like system has $15N+6$ degree of freedom
   * $N$ unit cells in supercell
   * $AB$O$_3$ perovskite-type ferroelectrics have 5 atoms per unit cell
   * Each atom can move along 3 directions
   * 6 components of strain
 * Simplified model has $6N+6$ degree of freedom
   * Define 1 dipole $Z^*\bm{u}(\bm{R})$ per unit cell
   * 1 acoustic displacement (Inhomogeneous strain) $\bm{w}(\bm{R})$ per unit cell

figures/CoarseGraining.jpg

# Real perovskite-like system has 15N+6 degree of freedom,
# where N is the number of unit cells in supercell,
# because each unit cell has 5 atoms, each atom can move along 3 directions,
# and there are 6 components of homogeneous strain.
#
# For Fast and Simple simulation, we reduce the number of degree of freedom to 6N+6,
# by replacing 5 atoms to 2 vectors, 1 dipole Z^*u(R) and 1 acoustic displacement
# per unit cell.


== MD simulation with first-principles-based effective Hamiltonian
 * Parametrization 1: [R. D. King-Smith and David Vanderbilt: Phys. Rev. B, vol.49, pp.5828-5844 (1994)].
 * Parametrization 2: [Zhong, Vanderbilt, and Rabe: Phys. Rev. B Vol.52 p.6301 (1995)].
 * [New] External electric field $\bm{\mathscr{E}}$.
 * [New] Kinetic energy terms are added to Monte Carlo Hamiltonian with effective masses.

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}}\bm{\mathscr{E}}\cdot\bm{u}(\bm{R})
/Eq.





== Features of feram
 * Ferroelectric thin films with perfect and imperfect electrodes (explained in the next slide).
 * feram is fast!!!
   * Efficient treatment of long-range dipole-dipole interaction (explained in the next next slide).
   * Nosé-Poincaré thermostat $\Rightarrow$ $\Delta t$ = 2 fs.
   * FFTW library version 3 http://www.FFTW.org/
   * Parallelized with OpenMP http://www.OpenMP.org/
   $\Rightarrow\Rightarrow\Rightarrow$ Large scale and realistic time-dependent MD simulations.
   * Spercell: ($AB$O$_3$)$_N$, $N=32\times 32\times 512$ $\approx$ 2,621,440 atoms
   * Up to 0.1-1$\mu$s
 * Multi-platform
   * Linux PC (Intel and AMD multi-core CPUs)
   * HITACHI SR11000 Super Technical Server
   * SONY PLAYSTATION3 (ongoing)
 * Object oriented programming (OOP) in Fortran 2003
 * GNU autotools are employed $\Rightarrow$ easy to compile
 * Free software under the GNU General Public License version 3 (the "GPLv3"); DOWNLOAD: http://loto.sourceforge.net/feram/


# Before going to physics, Let me introduce some technical features of ferams.
# It's fast. For speeding up, I use some computational technique as well as
# physics of mathematics. I employed FFTW library, .....



== Electrostatic boundary conditions for capacitors in simulations
figures/capacitor.jpg (a) A thin film in vacuum. (b) A thin film with short-circuited perfect electrodes. Doubly periodic boundary conditions for simulations of films sandwiched between (c) perfect and (d) imperfect short-circuited electrodes. In (a) depolarization field $E_{\rm d}= -4\pi P_z$, in (b) and (c) $E_{\rm d}=0$, and in (d) $E_{\rm d}=-4\pi P_z \frac{d}{l+d}$.


== Forces on dipoles are calculated in the reciprocal space
figures/flow.jpg Simplified flow chart for calculating forces on dipoles. Long-range dipole-dipole interaction is treated in reciprocal-space using $\bm{k}$-locality of the force matrix [U.V. Waghmare, E. J. Cockayne, and B. P. Burton: Ferroelectrics vol.291, p.187 (2003)]. Fast Fourier transformation (FFT) and inverse FFT (IFFT) enable the rapid calculation.

# Time consuming part in simulation is the calculation of forces of dipoles because
# dipole-dipole interactions are long-range. ..... You can reduce computational time
# from O(N^2) to O(NlogN).



== Time-dependent external electric field
figures/step.jpg Schematic illustrations of triangle-wave electric field for measuring hysteresis loops of ferroelectrics experimentally (inset) and that for present simulations.

Frequency may be
Eq. 4
  f=\frac{\Delta\mathscr{E}}{4 \Delta t n_{\rm steps} \mathscr{E}_0} .
/Eq.
When
$\Delta t = 2$ fs,
$n_{\rm steps}=50000$,
$\mathscr{E}_0 = 0.040$ V/Å and
$\Delta\mathscr{E} = 0.001$ V/Å,
$f = 62.5$ MHz.




== Parameters for BaTiO<sub>3</sub> and an input file
 * King-Smith and Vanderbilt: Phys. Rev. B Vol.49 p.5828 (1994)
 * Zhong, Vanderbilt and Rabe: Phys. Rev. B Vol.52 p.6301 (1995)

Tricky shell script:
 #!/bin/sh
 # externalE.sh
 # Time-stamp: <2008-02-05 19:07:16 takeshi>
 # Author: Takeshi NISHIMATSU
 ##
 rm -f externalE.avg
 
 temperature=100
 
 externalE_start=-0.04
 externalE_goal=0.04
 externalE_step=0.001
 
 n_thermalize=400
 n_average=100
 n_coord_freq=`expr $n_thermalize + $n_average`
 
 dt=0.002
 
 frequency=`perl -e "print $externalE_step/(4.0e-6*$dt*$n_coord_freq*$externalE_goal)"`
 printf "# dt = %.6f [pico second]\n"                       $dt             >> externalE.avg
 printf "# n_steps = %d\n"                                  $n_coord_freq   >> externalE.avg
 printf "# E_0 = %.5f [V/Angstrom]\n"                       $externalE_goal >> externalE.avg
 printf "# dE  = %.5f [V/Angstrom]\n"                       $externalE_step >> externalE.avg
 printf "# frequency = dE/(4 dt n_steps E_0)= %.6f [MHz]\n" $frequency      >> externalE.avg
 
 i=0
 externalE=$externalE_start
 while [ `perl -e "print $externalE <= $externalE_goal || 0"` = "1" ] ; do
     i=`expr $i + 1`
     filename=externalE"$temperature"K`printf '%.3d%+.5f' $i $externalE`
     cat > $filename <<-EOF
         #--- Method, Temperature, and mass ---------------
         method = 'md'
         GPa = -5.0
         kelvin = $temperature
         mass_amu = 39.0
         Q_Nose = 0.05
         
         #--- System geometry -----------------------------
         bulk_or_film = 'epit'
         L = 16 16 32
         gap = 1
         a0 =  3.94         latice constant a0 [Angstrom]
         epi_strain = -0.01
         #--- Time step -----------------------------------
         dt = $dt [pico second]
         n_thermalize = $n_thermalize
         n_average    = $n_average
         n_coord_freq = $n_coord_freq
         
         #--- External electric field ---------------------
         external_E_field = 0.00 0.00 $externalE
         
         #--- On-site (Polynomial of order 4) -------------
         P_kappa2 =    5.502  [eV/Angstrom^2] # P_4(u) = kappa2*u^2 + alpha*u^4
         P_alpha  =  110.4    [eV/Angstrom^4] #  + gamma*(u_y*u_z+u_z*u_x+u_x*u_y),
         P_gamma  = -163.1    [eV/Angstrom^4] # where u^2 = u_x^2 + u_y^2 + u_z^2
         
         #--- Inter-site ----------------------------------
         j = -2.648 3.894 0.898 -0.789 0.562 0.358 0.179    j(i) [eV/Angstrom^2]
         
         #--- Elastic Constants ---------------------------
         B11 = 126.
         B12 =  44.9
         B44 =  50.3  [eV]
         
         #--- Elastic Coupling ----------------------------
         B1xx = -211.   [eV/Angstrom^2]
         B1yy =  -19.3  [eV/Angstrom^2]
         B4yz =   -7.75 [eV/Angstrom^2]
         
         #--- Dipole --------------------------------------
         init_dipo_avg = 0.00  0.00 -0.10   [Angstrom]  # Average   of initial dipole displacements
         init_dipo_dev = 0.02  0.02  0.02   [Angstrom]  # Deviation of initial dipole displacements
         Z_star        = 9.956
         epsilon_inf   = 5.24
 EOF
     echo 1 > FILES
     echo $filename >> FILES
     ../feram $filename
     # OMP_NUM_THREADS=6 ./feram $filename > /dev/null
     ln -sf $filename.`printf '%.7d' $n_coord_freq`.coord restart.coord
     cat $filename.avg >> externalE.avg
     externalE=`perl -e "print $externalE + $externalE_step"`
 done


== Time evolution and fluctuation
figures/plot.epit-16x16-15-01-5.0GPa-0.01-cooling.jpg Energy versus time-step plot of a cooling simulation. You can see Nose-Poincare Hamiltonian remains zero.

# This figure shows the time steps of simulations.
# First, the system exchanges energy between the
# one particle heat bath, then it reaches the equilibrium.
# You can see Nose-Poincare Hamiltonian remains zero.



== Hysteresis loops of bulk BaTiO<sub>3</sub> at various temperatures
figures/bulk-two-box.jpg Paraelectric/ferroelectric transition ($T_{\rm C}\approx 320$ K (underestimated)) can be clearly seen.


== epitaxially constraint and "free" ferroelectric films
figures/epit-vs-free.jpg


== Hysteresis loops of epitaxially constraint and "free" capacitors
figures/epit-vs-free-hysteresis-box.jpg



== Summary
 * "feram" is a fast simulator for perovskite-type ferroelectric bulk and thin film
 * Molecular dynamics (MD) simulation with first-principles-based effective Hamiltonian
 * "feram" can simulate temperature, thickness, frequency and electrode dependences of ferroelectric properties
 * Phase transition of bulk ferroelectrics
 * Thin-film capacitor: perfect and imperfect short-circuited electrodes
 * Striped domain structure is predicted
 * Hysteresis loops for bulk ferroelectrics and ferroelectric capacitors
 * We confirmed that the imperfect screening of electrodes
   decreases the coercive field as the film thicknesses decreases,
   as described in [M. Dawber et al.: J. Phys. Condes. Matter vol.15 p.L393 (2003)].
 * We also found that compressive strain arising from epitaxial constraints
   prevents the polarization switching, while the inclusion of
   inhomogeneous strain (i.e., acoustic displacements) eases the switching.


== Future work
 * first-principles-based (non-empirical) and semi-empirical potentials for other $AB$O$_3$ perovskite-type ferroelectrics
 * Key-Dunn scaling $E_{\rm c} \propto l^{\frac{2}{3}}$, $l$: film thickness
 * Frequency dependence of susceptibility of relaxors


== Dielectric constant can be calculated from the fluctuation of dipoles
Eq. 2
\epsilon_{\alpha\beta}^{\rm r}=(\epsilon_{\infty}^{\rm r})_{\alpha\beta}+\frac{1}{\epsilon_0 V k_B T}
                       (<P_\alpha P_\beta>-<P_\alpha><P_\beta>)
/Eq.
figures/32x32xL032-D0-susceptibility.jpg Calculated dielectric constants (in-plane and out-of-plane) of epitaxially grown BaTiO$_3$ with perfect electrodes.



== ---------------------------- Relaxor ----------------------------
Relaxors Pb$B'_x$$B''_{(1-x)}$O$_3$: Pb$^{2+}$
 * Pb(Sc$_{1/2}$Nb$_{1/2}$)O$_3$ (PSN): Sc$^{3+}$, Nb$^{5+}$
 * Pb(Sc$_{1/2}$Ta$_{1/2}$)O$_3$ (PST): Sc$^{3+}$, Ta$^{5+}$
 * Pb(Mg$_{1/3}$Nb$_{2/3}$)O$_3$ (PMN:) Mg$^{2+}$, Nb$^{5+}$
 * $(1-x)$Pb(Mg$_{1/3}$Nb$_{2/3}$)O$_3$-xPbTiO$_3$ (PMN-$x$PT): Mg$^{2+}$, Nb$^{5+}$, Ti$^{4+}$
 * Pb(Mg$_{1/3}$Ta$_{2/3}$)O$_3$ (PMT): Mg$^{2+}$, Ta$^{5+}$
 * Pb(Zn$_{1/3}$Nb$_{2/3}$)O$_3$ (PZN): Zn$^{2+}$, Nb$^{5+}$
The averaged valence number of B-site ions $B'_x$ and $B''_{(1-x)}$ is $4+$.


# There are some Pb-based relaxors PbB'B''O$_3$.





== Frequency dependence of dielectric constant of relaxors
figures/TsurumiFig3.jpg After [Takaaki Tsurumi, Kouji Soejima, Toshio Kamiya and Masaki Daimon: Jpn. J. Appl. Phys. Vol.33pp. 1959-1964 (1994)].

# Most characteristic and still mysterious property of relaxors is 
#Eq.1
#\tan\delta=\tan()
#/Eq.






== Crystal structures of relaxors
figures/RandallBhallaFig3.jpg (a) Ordered B-site structure of $AB'_xB''_{(1-x)}$O$_3$. In the case of Pb(Sc$_{1/2}$Nb$_{1/2}$)O$_3$, Sc and Nb has the NaCl structure. In the case of Pb(Mg$_{1/3}$Nb$_{2/3}$)O$_3$, the Na-site is randomly occupied by Mg$_{1/3}$Nb$_{1/6}$ and the Cl-site is occupied by  Nb$_{1/2}$.  (b) Disordered structure. B-sites is occupied by $B'$ and $B''_{(1-x)}$ randomly. After [C. A. Randall and A. S. Bhalla: Jpn. J. Appl. Phys. Vol. 29 pp.327-333 (1990)].







== Local field on Pb-site
Displacement of Pb is the main source of dipole moment.
figures/TwoDoubleWells.jpg Symmetric potential around Pb at the chemically ordered region (COR). Asymmetric one at the chemically disordered region (CDR).








== Chemically ordered and disordered regions of relaxors
figures/RandallBhallaFig8.jpg The quenched sample has chemically ordered regions (&sim;100&Aring;) in a chemically disordered matrix. After [C. A. Randall and A. S. Bhalla: Jpn. J. Appl. Phys. Vol. 29 pp.327-333 (1990)].






== Modeling of relaxors
figures/DoubleWells.jpg Schematic illustration of modeling of relaxor. Bubble-like Chemically ordered regions (COR, red symmetric double wells) is created in a chemically disordered matrix (blue asymmetric double wells)





== Snapshot of a simulation of PSN relaxor
figures/SilviaFig1.jpg A (110) plane through the PSN simulation box representing the projected local field (arbitrary units) at each Pb site in the plane. Chemically ordered regions (approximately circular) have small approximately homogeneous fields, and chemically disordered regions have larger more varied and disordered local fields. After [Tinte, Burton, Cockayne and Waghmare: Phys. Rev. Lett. Vol.97 p.137601 (2006)].






== Susceptibility tensor
Now you can calculate $\bm{S}(\bm{R},t)$ in some periods (1-100 ns) with feram.

Susceptibility tensor $\chi(\bm{R}-\bm{R}',t-t')$
Eq. (Real space)
\bm{S}(\bm{R},t) = \sum_{\bm{R}'}\int dt' \chi(\bm{R}-\bm{R}',t-t')\,\mathscr{E}(\bm{R}',t')
/Eq.
where $\bm{S}(\bm{R},t)=Z^* \bm{u}(\bm{R},t)$ and $\mathscr{E}(\bm{R}',t')$ is external electric field.
Its Fourier transformation
Eq. (q space)
\bm{S}(\bm{q},\omega) = \chi(\bm{q},\omega)\,\mathscr{E}(\bm{q},\omega) .
/Eq.
Using the fluctuation-dissipation theorem,
Eq. (The fluctuation-dissipation theorem)
{\rm Im} \chi_{\alpha\beta}(\bm{q},\omega) = \frac{1}{2k_BT}\int_{-\infty}^{\infty}dt
                            <S_\alpha(-\bm{q},0) \,\, S_\beta(\bm{q},t)> e^{-i\omega t}
/Eq.
Then, with Kramers-Kronig relation,
Eq. (Kramers-Kronig relation)
{\rm Re} \chi_{\alpha\beta}(\bm{q},\omega) = ...
/Eq.


# If you apply external electric field at site R' at time t', dopole at R,t sespinses
# through this equation with real-space-and-time usceptibility tensor chi(R-R',t-t').
# You can simplify it by Fourier transformation and get q-omega expression.
# And then, if you apply the fluctuation-dissipation theorem to it, you may get
# the imaginary part of chi. Then with Kramers-Kronig relation, you can get the real part.
#
# G. L. Squires: Introduction to the Theory of Thermal Neutron Scattering (Dover Publications, 1996).




== Estimation of computational time
32x32x32 unit cells, $\Delta t = 2$ fs, [one dual core 1.8GHz x86_64] or [SR11000 1 node = 16 cores]

1THz, $T=1\,$ps,            500 steps,  54 sec  or 8.4 sec.

1GHz, $T=1\,$ns,        500,000 steps, 900 min. or 140 min.

1MHz, $T=1\mu$s,    500,000,000 steps, 620 days or  97 days

So, I am planning to calculate and compare $\chi_{\alpha\beta}(q,\omega)$ for $\omega/(2\pi)$ = 10M ... 10GHz.

=end
#
# for ulmul.rb
#
require 'rubygems'
require 'ulmul'
u=Ulmul.new()
u.subs_rules << [/(\S*\/?\.jpg)\s*(.*)$/, '<div class="figure">
  <img src="\1" alt="\1" />
  <div class="caption">
    \2
  </div>
</div>']

u.parse(ARGF)
puts u.html(["feram-presentation.css"],["slidy.js"],"Takeshi Nishimatsu")

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