This document discusses ferroelectric materials and molecular dynamics (MD) simulations of their properties using an effective Hamiltonian approach. Key points include: 1) Feram is an MD simulation code that models perovskite ferroelectrics like BaTiO3 using a first-principles based effective Hamiltonian. 2) Simulations of bulk BaTiO3 and PbTiO3 reproduce their phase transitions and domain structures. 3) Simulations of thin-film capacitors show the effect of "dead layers" between ferroelectrics and electrodes on polarization. 4) Direct MD simulations of the electrocaloric effect in BaTiO3 predict a temperature change under applied and removed electric fields.

1. IMR, Tohoku University
Takeshi Nishimatsu

2. dielectrics
piezoelectrics
pyroelectrics
ferroelectrics (switchable)
Because BaTiO3 base and PbZrxTi1-xO3 (PZT) base
ferroelectric materials has good properties for applications
all above, there are many electronics devices using these
materials. Traditionally, Japanese companies have high
market shares in such ferroelectric ceramics industries.
High
Symmetry
Low

3. polarity inversion
#of
classes
crystal family
cubic hexagonal tetragonal
rhombo
hedral
ortho
mono
tri.
nonpolar
crystal
(22)
yes
(11)
11 Oh Th D6h C6h D4h C4h D3d C3i D2h C2h Ci
no
(21)
11
O
T
D6
D3h
C3h
D4
D2d
S4 D3 D2
Td
polar
（pyro）
crystal
(10)
10 C6v C6 C4v C4 C3v C3 C2v
C2
Cs
C1
white background: piezoelectric red letter: pyroelectric

4. } Polarized by an applied
external electric field
} Switch-off electric field
→ polarization vanishes
} Application: multi-layer
ceramics capacitor
(MLCC) BaTiO3-base
} Application: electret
microphone or
capacitor
microphone
Dielectrics

5. } Piezoelectric effect:
polarization develops
in response to applied mechanical stress.
} Inverse piezoelectric effect: strain develops
in response of applied electric field.
} Simple Td GaAs can have piezoelectricity.
} Applications: pressure sensor (Langasite,
La3Ga5SiO14), crystal oscillator (quartz SiO2)
} Ceramics: piezoelectric speaker, ceramic
oscillator, sonar, acceleration and gyro
sensor, supersonic vibrator, etc., etc…..
Piezoelectrics

6. After catalog from muRata P82e.pdf
2011-06-30. Underlines by TN.

7. } Polar crystals
} Polarization can be
slightly changed by IR.
} Application：IR sensor
} Polarization can be
switched by external
electric field →
ferroelectrics
Pyroelectrics IR
IR sensor for
lights (in IMR)
Phase transition of
pyroelectrics.

8. } Polarization can be
switched by external
electric field.
} Spontaneous
polarization (Pr) remains at E=0.
} Polarization can have some easy directions
according to the crystal structure.
} In finite systems, domain structures are
formed to avoid depolarization field.
What are
ferroelectrics?

9. } Ferroelectric RAM is a
random-access memory
using a ferroelectric
capacitor and its
histeresis to achieve
non-volatility.
} Faster and lower voltage
than Flash memory.
} Down sizing →
Higher Density →
Replace DRAM →
Normally-off Computing
(Fast, Non-volatile, Without refresh)

10. What is electrocaloric effect?
Inverse effect of pyroelectric effect
Application: Solid state cooling

11. Experimentally observed temperature dependence of lattice constants
for BaTiO3. There are three first-order phase transitions. the cubic to
tetragonal phase transition is nearly second-order one. After [H. E. Kay
and P. Vousden: Philos. Mag. 40, 1019 (1949)]．
ferroelectric⇔paraelectric

12. After [Gen Shirane, Sadao Hoshino, and Kazuo Suzuki: Phys. Rev. 80, 1105 (1950)]
ferroelectric⇔paraelectric

13. } Various applications of ferroelectric thin-
films: multi-layer ceramics capacitors (MLCC),
nonvolatile FeRAMs, nanoactuators, etc.
} Down-sizing of FeRAMs (nano-capacitors of
ferroelectric thin films) is highly demanded.
After [J. F. Scott: Ferroelectric Memories (Springer, 2000)]

14. } Develope of fast molecular dynamics (MD) code
which can simulate ferroelectric thin-film
capacitors for a realistic system size (> 100 nm)
and a realistic time span (> 100 ns).
} Clarify the effect of dead layers between
ferroelectrics and electrodes.
} Predict properties of ferroelectrics
◦ dielectric constant
◦ domain structures
◦ effects of epitaxial strains
◦ pyroelectric effect
◦ electrocaloric effect
◦ etc, etc…

15. } Parallelized with OpenMP http://OpenMP.org/
} FFTW http://www.fftw.org/
} Object Oriented Programming (OOP) with
Fortran 95/2003
} GNU autotools (autoconf and automake)
◦ Easy to compile feram with ./configure && make
◦ Multi Platform using C preprocessor (CPP)
⇒ Available on PC Linux, Hitachi SR16000,
Fujitsu FX10, etc.
} Source code is version-controlled under
Subversion in http://SourceForge.net/
} It’s free software!

16. Home page of feram

17. Papers using feram

18. } Investigate ferroelectric BaTiO3, PbTiO3, etc. with
first-principles calculations and construct an
effective Hamiltonian with 25 parameters
◦ Using ABINIT http://www.abinit.org/
– Plane wave: Ecut=60 Hartree, on 8x8x8 k-points
– Pseudopotentials http://opium.sf.net/ (Rappe et al.)
– GGA (Wu and Cohen). LDA and GGA (PBE) do not work well.
– Basically, absolute 0 K properties.
} MD: time evolution of the effective Hamiltonian
◦ Original feram http://loto.sf.net/feram/
◦ realistic system size (> 100 nm) and a realistic time span
(> 100 ns)
◦ Temperature, pressure, strain, external electric field
◦ bulk and thin-film

19. After [T. Hashimoto,
T. Nishimatsu et al.:
Jpn. J. Appl. Phys. 43,
6785 (2004)]
electric
polarization
(dipole)
Total-energy surface (can be calculated
with first-principles calculations.)

20. BaTiO3 GGA (Wu and Cohen)

21. } BaTiO3 has shallower
total-energy surface.
} Most stable direction:
[111]
} PbTiO3 has deeper
total-energy surface.
} Most stable direction:
[001]
BaTiO3 vs. PbTiO3
GGA (Wu and Cohen)

22. Other many first-principles calculations
required for determination of parameters
} Lattice constants, elastic coefficients
} Inter-atomic force constant (IFC) matrix Φ(k)
for determination of shor-range interactions.
} etc. etc.
With four parameters I can fit an elephant,
and with five I can make him wiggle his trunk.
-- John von Neumann
It’s quite tough to determine 25
parameters only from first principles.
-- Takeshi Nishimatsu :-)
http://www.lanl.gov/history/atomicbomb/images/NeumannL.GIF

23. } Perovskite ABO3: 15N+6 degrees of freedom
◦ 5 atoms in a unit cell
◦ each atom can move x, y, z 3 directions
◦ N unit cells in a super cell
◦ 6 components of strain
} Coarse-graining: 6N+6 degrees of freedom
◦ One dipole vector Z*u(R) on each unit cell
◦ One acoustic displacement vector w(R) on each u.c.
See [W. Zhong, D. Vanderbilt, and K. M. Rabe, Phys. Rev. B 52, 6301 (1995)].

24. Super cell calculations with
array of dipoles (Periodic
boundary condition)
Parameters for BaTiO3 [Takeshi Nishimatsu et al.: PRB 82, 134106 (2010)]

25. Simplified flow chart for calculating
forces on u(R). FFT and IFFT enable
rapid calculation of long-range
dipole-dipole interactions.
Real space: O(N2)
Reciprocal space with FFT: O(NlogN)
See [U. V. Waghmare et al.: Ferroelectrics 291, 187 (2003)].

26. } Only with long-range
dipole-dipole
interactions, the
minimum is at
M-point (antiferro).
} Adding short-rang
interactions, the
minimum
goes to
Γ-point.
Only with long-range
dipole-dipole interactions
long-rang interactions
+ short-range interactions
M
Γ BaTiO3

27. 3-dimentional simple cubic (sc)
dipole lattice
M-point is the minimum.

28. Electrode can be considered as
electrostatic mirrors
See Nishimatsu et al.: PRB 78, 104104 (2008)

30. RESULTS of MD SIMULATIONS
BaTiO3 (from GGA (Wu & Cohen))
Experiment
} MD simulation

31. €
εαβ = (ε∞)αβ +
V
ε0kBT
Pα Pα − Pα Pα( )

32. -60
-40
-20
0
20
40
-300 -200 -100 0 100 200 300
Pz[µC/cm2
]
E [kV/cm]
bulk 32×32×32, T = 360 K
bulk 32×32×32, T = 460 K
bulk 32×32×32, T = 560 K

33. T = 100 K

34. With dead layers,
the z-polarized
state is no longer
the ground state
for a ferroelectric
capacitor
[BaTiO3, 16x16x
(l=63,d=1)]
Dead layer à

37. PbTiO3

38. Frozen ９０° degree domain structures
of PbTiO3 (snapshot at 300 K)
JPSJ 81, 124702 (2012)

39. Experimentally observed ９０° degree
domain structures of PbTiO3

40. PbTiO3の９０°ドメイン（２）
HAADF-STEM image around edge dislocations with an a-domain.
After [T. Kiguchi et al.: Sci. Technol. Adv. Mater. 12 (2011) 034413].
600℃成膜時ミスフィット大→室温a軸はミスフィット小→転移→aドメイン

41. PbTiO3 90°ドメインのドメイン壁の厚さ
厚さ→１〜２ユニットセル

42. } BaTiO3, supercell of 96×96×96 unit cells
} Constant T canonical ensemble MD
calculation under external electric field Ez
} After that turn off Ez=0, constant energy
micro-canonical ensemble MD calculation
(leap-frog method)
} External electric field: Ez=0〜500 kV/cm
} Coarse graining → under estimate Cv →
over setimate ΔT → correction
Computational conditions of
direct MD simulations of ECE

43. Results of 【direct】 MD calculations
Smaller super cell size,
larger fluctuation of ΔT
BaTiO3
-35
-30
-25
-20
-15
-10
-5
0
300 350 400 450 500 550 600 650 700
∆T[K]
T [K]
BaTiO3 160→60 kV/cm
indirect
MD × 2/5
optimization × 1/5
MD
optimization
x 2/5
correction
x 1/5
correction

44. -50
-40
-30
-20
-10
0
10
20
0 100 200 300 400 500 600 700 800 900
-10
-8
-6
-4
-2
0
2
4∆T[K]
∆Tcorrected[K]
T [K]
BaTiO3 300→0 kV/cm
[001]
[110]
[111]
Anisotropic effect of E

45. Results of direct MD calculations of ECE
BaTiO3
-12
-10
-8
-6
-4
-2
0
300 400 500 600 700 800 900
-60
-50
-40
-30
-20
-10
0
∆Tcorrected[K]
∆T[K]
T [K]
5→0 kV/cm
50→0 kV/cm
100→0 kV/cm
200→0 kV/cm
300→0 kV/cm
400→0 kV/cm
500→0 kV/cm
Effective temperature
rang is narrower for
smaller E field.

46. } We developed "feram", a fast simulator for
perovskite-type ferroelectric bulks and thin films.
} Molecular dynamics (MD) simulation with
first-principles-based effective Hamiltonian.
} Phase transitions of bulk BaTiO3 and PbTiO3.
} Thin-film capacitor with perfect and imperfect
short-circuited electrodes.
} Electrocaloric effect
} etc, etc…
Summary
feram is free software!!! You can freely get it
from http://loto.sourceforge.net/feram/ .