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arXiv:1202.3893v1 [condmat.mtrlsci] 17 Feb 2012
90◦Ferroelectric Domains in PbTiO3: Experimental Observation and Molecular
Dynamics Simulations
Kenta Aoyagi1, Takeshi Nishimatsu2, Takanori Kiguchi2, Toyohiko J. Konno2,
Yoshiyuki Kawazoe2, Hiroshi Funakubo3, Anil Kumar4,5, and Umesh V. Waghmare5
1Department of Materials Science, Tohoku University, Sendai 9808579, Japan
2Institute for Materials Research (IMR), Tohoku University, Sendai 9808577, Japan
3Department of Innovative and Engineered Materials,
Tokyo Institute of Technology, Yokohama 2268503, Japan
4Department of Physics and Astronomy, Rutgers University,
136 Frelinghuysen Road, Piscataway, NJ 085448019,
5Theoretical Sciences Unit, Jawaharlal Nehru Centre for Advanced
Scientific Research (JNCASR), Jakkur, Bangalore, 560 064, India
(Dated: February 20, 2012)
We report observation of 90◦ferroelectric domain structures in transmission electron microscopy
(TEM) of epitaxiallygrown films of PbTiO3. Using molecular dynamics (MD) simulations based on
firstprinciples effective Hamiltonian of bulk PbTiO3, we corroborate the occurance of such domains
showing that it arises as metastable states only in cooling simulations (as the temperature is lowered)
and establish characteristic stability of 90◦domain structures in PbTiO3. In contrast, such domains
do not manifest in similar simulations of BaTiO3. Through a detailed analysis based on energetics
and comparison between PbTiO3 and BaTiO3, we find that 90◦domain structures are energetically
favorable only in the former, and the origin of their stability lies in the polarizationstrain coupling.
Our analysis suggests that they may form in BaTiO3 due to special boundary condition and/or
defectrelated inhomogeneities.
PACS numbers: 64.60.De, 68.37.Lp, 77.80.Dj, 77.80.B, 77.84.s
I.INTRODUCTION
Ferroelectric transitions in perovskite oxides, such as
BaTiO3, are fluctuation driven firstorder phase transi
tions, and hence a state with spatially fluctuating order
parameter can readily form as a result of certain me
chanical and electric boundary conditions1. A common
example of such a state is the one with domains of fer
roelectric polarization with different symmetry equiva
lent orientations of order parameter that are separated
by domain walls. Indeed, many properties of perovskite
ferroelectrics depend on such domain structure, and it is
being increasingly relevant at nanoscale2–5. Naturally,
the properties of a domain wall or an interface between
adjacent ferroelectric domains depend on (a) symmetries
and structural details of ferroelectric phases and (b) mi
croscopic couplings responsible for the ferroelectric phase
transition.
Perovskite oxides such as BaTiO3and PbTiO3are rep
resentative ferroelectric materials, although are quite dif
ferent from each other in terms of their phase transi
tions. While PbTiO3 undergoes a single strongly first
order phase transition from cubic to tetragonal struc
ture as temperature is lowered, BaTiO3 exhibits a se
quence of three relatively weaker firstorder phase transi
tions. A paraelectric phase of BaTiO3with cubic struc
ture transforms into a tetragonal ferroelectric phase at a
Curie temperature, 393 K. Further cooling produces sud
den changes from a tetragonal phase to an orthorhom
bic phase at 278 K and from an orthorhombic phase
to a rhombohedral phase at 203 K6,7. On the other
hand, PbTiO3 exhibits a phase transition from a para
electric cubic phase to a ferroelectric tetragonal phase at
TC= 763 K and remains tetragonal down to 0 K8–10.
While the domain structures in ferroelectrics have been
revealed experimentally7,11–25, detailed in situ experi
mental analysis of the domains and domain walls is quite
challenging and the temperature dependence of dynamics
of domain structure is not well understood. In the case
of PbTiO3, experimental studies of domains are further
more difficult as the sample needs to be heated over its
high transition temperature TC= 763 K and such heating
leads to evaporation of Pb ions changing the composition
of the sample26.
Ferroelectric phase transitions in perovskite oxides
in bulk and thin film have been investigated by com
puter simulations such as phasefield method27, Monte
Carlo simulations28, and molecular dynamics (MD)
simulations29. Recently, Nishimatsu et al. have devel
oped a fast and versatile MD simulator of ferroelectrics
based on firstprinciples effective Hamiltonian30which
can be used in systematic studies of bulk as well as thin
films. They have studied BaTiO3bulk and thinfilm ca
pacitors and obtained results showing good agreement
with experiments. Their MD simulations of BaTiO3un
der periodic boundary condition (PBC) for bulk did not
show any domain structures, as there is no depolariza
tion field in the PBC of bulk. Simulations of thinfilm of
BaTiO3only show 180◦domain structures30, though 90◦
domain structures are widely seen in experiments11,13.
One of the advantages of MD simulations compared
to Monte Carlo simulations is its ability to simulate
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2
timedependent dynamical phenomena, e.g. MD simu
lation can be used to study the evolution of ferroelec
tric domains as a function of time during heatingup and
coolingdown simulations.
In this paper, we report heatingup and coolingdown
moleculardynamics (MD) simulations of bulk PbTiO3
to understand our observation of 90◦domain structures
in epitaxiallygrown sample of PbTiO3. In Sec. II, we
present experimental details for the sample preparation
and we show a transmission electron microscope (TEM)
image of 90◦domain structure in PbTiO3film. We briefly
explain the firstprinciples effective Hamiltonian and de
tails of MD simulations in Sec. III and we present our
results and analysis of heatingup and coolingdown MD
simulations in Sec. IV. We finally summarize our work
and conclusions in Sec. V.
II.EXPERIMENTAL DETAILS AND
OBSERVATIONS
A.Sample preparation and Methods of TEM
An epitaxial PbTiO3thick film, with film thicknesses
of about 1200 nm, was grown on the SrRuO3/SrTiO3
substrate at 873 K by pulsed metal organic chemical va
por deposition (pulsedMOCVD) method. SrRuO3was
deposited on (100) SrTiO3 by rfmagnetron sputtering
method. The detail of film preparation technique is de
scribed elsewhere31,32. The TEM specimens were pre
pared with focused ion beam (FIB) microsampling tech
nique. Damage layers, introduced during FIB microfab
rication, were removed by lowenergy Ar ion milling at
0.3 kV. JEM2000EXII was used for TEM observations.
TEM observations were performed at room temperature.
B. Observed TEM image
In Fig. 1, we show a bright field TEM image of a
PbTiO3thick film, taken with the incident electron beam
parallel to the [100] axis of the PbTiO3. The inset in the
image is the corresponding selectedarea electron diffrac
tion pattern. This brightfield TEM image is taken with
the scattering vector g0¯1¯1excited. Here, a and cdomains
are those with polarization along a and c axes of the
PbTiO3parallel and perpendicular to the substrate, re
spectively. This domain configuration is typical and very
commonly seen for tetragonal PbTiO3 films. The do
main size is about 50200 nm. Such 90◦domain configu
rations, similar to that in Fig. 1, have been also observed
in BaTiO311,13. Boundaries between a and cdomains
are seen as black lines, as the TEM sample is slightly
tilted. From this image, we can not discuss the width of
domain walls between a and cdomain. Highresolution
TEM observation has revealed that the width of 90◦do
main walls is 1.0±0.3 nm33. To this end, highresolution
TEM observation was conducted in order to reveal the
FIG. 1: A brightfield TEM image of a PbTiO3 thick film
taken with an electron incident parallel to the [100] of
PbTiO3. Scripts a and c in this figure denote adomain and
cdomain, respectively.
width of domain walls between a and cdomain. Fig. 2
shows the highresolution TEM image of PbTiO3film, in
dicating that the width of domain walls is corresponding
to 1 or 2 unit cells.
Before our computational study, it should be worth
mentioning that 90◦domains have been often observed
in both BaTiO3and PbTiO3, and are both ferroelectric
and ferroelastic in nature. Typically, 90◦domains are
formed in epitaxial ferroelectric and ferroelastic films in
order to relax the strain resulting from lattice mismatch
with the substrate at and below TC.34,35They nucleate
at misfit dislocations formed above TC. Their growth is
accompanied with the introduction of the additional dis
location perpendicular to the misfit dislocations and the
dissociation of the dislocations into two pairs of partial
dislocations around an antiphase boundary23.
III. MOLECULAR DYNAMICS SIMULATIONS
A.Effective Hamiltonian
Heatingup
(MD) simulations are performed using firstprinciples ef
fective Hamiltonian28–30,36,37,
andcoolingdown moleculardynamics
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3
FIG. 2: A highresolution TEM image of a PbTiO3 thick
film taken with an electron incident parallel to the [100] of
PbTiO3. The 90◦domain boundary is shown by the chain
line.
Heff=M∗
dipole
2
?
R,α
˙ u2
α(R) +M∗
acoustic
2
?
R,α
˙ w2
α(R)
+ Vself({u}) + Vdpl({u}) + Vshort({u})
+ Velas,homo(η1,...,η6) + Velas,inho({w})
+Vcoup,homo({u},η1,...,η6)+Vcoup,inho({u},{w}) ,
(1)
where u = u(R) and w = w(R) are, respectively, the
local dipolar displacement vector and the local acoustic
displacement vector of the unit cell at R in a simulation
supercell. α(= x,y,z) is the Cartesian directions. Braces
{} denote a set of u or w in the supercell. η1,...,η6
are the homogeneous strain components.
M∗
acousticare the effective masses for u and w, therefore,
first two terms in Eq. (1) are kinetic energies of them.
Vself, Vdpl, Vshort, Velas,homo, Velas,inho, Vcoup,homo,
and Vcoup,inhoare a localmode selfenergy, a longrange
dipoledipole interaction, a shortrange interaction, a ho
mogeneous elastic energy, an inhomogeneous elastic en
ergy, a couping between {u} and η1,...,η6, and a couping
between {u} and {w}, respectively. More detailed ex
planation of symbols in the effective Hamiltonian can be
found in Refs. 30 and 37. We take all the parameters of
the firstprinciples effective Hamiltonian for PbTiO3from
the earlier work38. However, the form of the effective
Hamiltonian we use in our simulations30,37is slightly dif
ferent from the one used to get the parameters in Ref. 38.
M∗
dipoleand
TABLE I: The parameters of firstprinciples effective Hamil
tonian for PbTiO3 used in our simulations are given in the
second column and how these parameters are related with
the parameters in the previous work38are shown in the third
column.
parameters value
a0 [˚ A]3.969
B11 [eV] 117.9
B12 [eV]
B44 [eV] 137.0
B1xx [eV/˚ A2] −114.02 2(g0+g0)
B1yy [eV/˚ A2] −13.67
B4yz [eV/˚ A2] −22.67
α [eV/˚ A4] 27.83
γ [eV/˚ A4]
−34.48
k1 [eV/˚ A6]
−42.42
k2 [eV/˚ A6]
k3 [eV/˚ A6]
k4 [eV/˚ A8]156.43
m∗[amu]100.0
Z∗[e]10.02
ǫ∞
κ2[eV/˚ A2]1.170
j1 [eV/˚ A2]
−1.355
j2 [eV/˚ A2]4.986
j3 [eV/˚ A2]0.222
j4 [eV/˚ A2]
−0.018
j5 [eV/˚ A2]0.398
j6 [eV/˚ A2]
−0.083
j7 [eV/˚ A2]
−0.204
relation
a0
C11
C12
C44
51.6
2g0
g2
B + C
−2C
D
0
0
E
Z∗
8.24
ǫ∞
A
2aT
2aL
bl+ bt1
2bt2
bl− bt1
2(cl+2ct)
3
2(cl−2ct)
3
The new parameters can be easily derived from the pre
vious ones. We list values of all the parameters used in
our simulations and how they are related to the previous
work38in Table I.
B.Simulation Details
Heatingup
are performed with our original MD code feram
(http://loto.sourceforge.net/feram/).
the code can be found in Ref. 30.
kept constant in each temperature step in the canon
ical ensemble using the Nos´ ePoincar´ e thermostat.39
This simplectic thermostat is so efficient that we
can set the time step to ∆t = 2 fs.
MD simulations, we thermalize the system for 20,000
time steps, after which we average the properties for
20,000 time steps. We used a supercell of system size
N = Lx×Ly×Lz= 32×32×32 and small temperature
steps in heatingup (+1 K/step) and coolingdown
(−1 K/step) simulations.
from 100 K to 900 K is started from an zpolarized initial
andcoolingdown MDsimulations
Details of
Temperature is
In our present
The heatingup simulation
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4
3.90
3.95
4.00
4.05
4.10
4.15
4.20
4.25
0 200 400 600 800 1000
lattice constants [Å]
T [K]
cubic
a
c
b′
a′
c′
tetragonal
cooling
heating
FIG. 3: (Color online) Simulated temperature dependence of
lattice constants of PbTiO3in heatingup (red solid lines) and
coolingdown (blue dashed lines) moleculardynamics simula
tions.
configuration generated randomly: ?ux? = ?uy? = 0,
?uz? = 0.33˚ A, ?u2
and ?u2
average in supercell ?uα? =
down one from 900 K to 100 K is started from random
paraelectric initial configuration: ?ux? = ?uy? = ?uz? = 0
and ?u2
x?−?ux?2= ?u2
y?−?uy?2= (0.045˚ A)2,
z?−?uz?2= (0.021˚ A)2, where brackets denote R
1
N
?
Ruα(R). The cooling
α? − ?uα?2= (0.15˚ A)2.
IV. RESULTS AND DISCUSSION
From the temperature dependence of averaged lat
tice constants (shown in Fig. 3), a tetragonaltocubic
ferroelectrictoparaelectricphase transition is clearly ob
served in the heatingup simulation at 677 K. However,
a strange behavior in lattice constants is found in the
coolingdown simulation at T=592 K. The average tem
perature of these two transition temperatures (634 K)
is in good agreement with the earlier Monte Carlo sim
ulations and slightly lower than the experimental value
TC= 763 K. Indeed, the observation of an orthorhombic
phase during coolingdown simulations is intriguing.
To understand this interesting behaviour of lattice con
stants in coolingdown simulations, we perform a detailed
analysis of the configurations (snapshots) during our MD
simulations. From a snapshot of dipoles in the supercell
(shown in Fig. 4), we find that the apparently orthorhom
bic nature of the phase is due to a 90◦domain struc
ture. Although the 4 unit cell = 1.6 nm of the domain
size is much smaller than experimentally observed ones
as shown in Sec. IIB, the width of a simulated domain
wall estimated to be ∼ 1 unit cell is in good agreement
with our experiment. Each domain has the a = b < c
tetragonal structure of PbTiO3, but their average value
in whole crystal gives smaller c′than c and larger a′than
a. The lattice constant b′has almost the same values as
a, because the polar directions of two kind of domains
are perpendicular to the b′axis. It should be noted that
this domain structure is found in MD simulations in bulk
under periodic boundary condition (PBC), but we have
not simulated thin films. Under the PBC, there is no
depolarization field inside the bulk. Moreover, this do
main structure can be easily reproduced in coolingdown
simulations from any random paraelectric initial config
urations and any seeds for the pseudo random number
generator40. There was no evidence for such unusual be
havior in simulations of bulk BaTiO330,37.
To understand the reason of stability of the 90◦domain
structure seen here, even in bulk PbTiO3, we compare
“total energy surfaces” between single and 90◦domain
structures. The total energy surface of single domain
structure with [001] polarization is the same as in Refs. 36
and 30. For the total energy surface of 90◦domain struc
ture, we focus on a snapshot of the supercell at 300K
shown in Fig. 4 and represent it with {u90◦,300K(R)}. We
now obtain a sequence of configurations by multiplying
a factor
?u?90◦,300Kfor all R
u
u(R) =
u
?u?90◦,300Ku90◦,300K(R) ,
(2)
and compute total energy as a function of u, where
?u?90◦,300Kis the averaged length of dipoles in the 300 K
snapshot
?u?90◦,300K=
1
N
?
R
u90◦,300K(R) = 0.32˚ A .
(3)
Calculated total energy surfaces of single and 90◦do
main structures for PbTiO3are shown in Fig. 5 with solid
and dashed lines, respectively. For comparison, those for
BaTiO3are also plotted assuming the same 90◦domain
structure by using the set of parameters in Hefflisted in
Ref. 37. While the 90◦domain structure of PbTiO3ex
hibits a minimum at u ?= 0, that of BaTiO3costs energy.
This is why the 90◦domain structures can be found in
simple coolingdown simulations of PbTiO3, but not in
those of BaTiO3.
Minima are indicated with (a)–(c) in Fig. 5. To un
cover the origin of this contrasting behaviour, interac
tion energy terms at the minimums are listed in TA
BLE II. For the BaTiO3, because there is no minimum,
interaction energies of configuration at (d) in Fig. 5 (of
u = 0.10˚ A) are listed.
From TABLE II, It is clear that the energy losses in
Vharmonic, Velas,inho, and Vcoup,homoin forming the 90◦
domain structure in PbTiO3are compensated by the en
ergy gains in Vunharmonic, Velas,homo, and Vcoup,inho. In
contrast, such recoveryis not sufficient in BaTiO3to form
a u ?= 0 minimum or non trivial 90◦domain structures.
In stabilization of the 90◦domain structures in PbTiO3,
our analysis conclusively highlights the role of two mi
croscopic interactions: (a) lower elastic energy cost aris
ing from the smaller strain from compensation along c
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5
a´
b´
c´
b´
FIG. 4: (Color online) Snapshot of three “sides” of the 32 × 32 × 32 supercell at T = 300K in a coolingdown simulation of
PbTiO3. Dipole moments of each site are projected onto the side planes and indicated with arrows. Dipoles of uz > 0.2˚ A are
indicated with red color. Dipoles of uz ≤ 0.2˚ A are indicated with green. Crystalline directions are indicated with a′, b′, and c′
as indicated in Fig. 3.
TABLE II: Comparison of interaction energies Viin the effective Hamiltonian of Eq. (1) for two kinds of domain states of
PbTiO3 and BaTiO3. Vharmonicis the sum of Vdpl, Vshort, and the harmonic terms in Vself. Vunharmonicis the unharmonic
terms in Vself. Unit of energy is eV.
x of
interaction
energy Vx
harmonic
unharmonic
elas,homo
elas,inho
coup,homo
coup,inho
total
PbTiO3
(a) single
domain
−0.22106 −0.14393 +0.07713 −0.03886
0.334350.17249 −0.16186
0.219460.04441 −0.17505
0.000000.05171 +0.05171
−0.43891 −0.08881 +0.35010 −0.05218
0.00000 −0.10342 −0.10342
−0.10616 −0.06755 +0.03861 −0.01741
PbTiO3
(b) 90◦
domain
BaTiO3
(c) single
domain
BaTiO3
(d) 90◦
domain
0.03541 +0.07427
0.00695 −0.04059
0.00131 −0.02478
∆Vi
∆Vi
0.04754
0.02609
0.00000 −0.00262 −0.00262
0.00089 +0.05307
0.00000 −0.00178 −0.00178
0.04016 +0.05757
u [˚ A]0.34 0.290.16 0.10
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−0.12
−0.10
−0.08
−0.06
−0.04
−0.02
0.00
0.02
0.04
0.06
0.0 0.10.2 0.30.40.5
E−E0 [eV]
u [Å]
(a)


(b)
(c)


(d)
BaTiO3 90° domain
BaTiO3 single domain
PbTiO3 90° domain
PbTiO3 single domain
FIG. 5:
for PbTiO3 single domain (solid line), PbTiO3 90◦do
main (dashed line), BaTiO3 single domain (dotted line), and
BaTiO3 90◦domain (chain line).
(Color online) Calculated total energy surfaces
and a axis, and (b) inhomogeneous (local) strain cou
pling with polarization at the domain wall. The latter
does not contribute much to transition behavior in the
bulk, but has rather significant impact on the properties
of domain wall. Noting that a 90◦domain structure is
not energetically favorable in BaTiO3simulated as a per
fect bulk crystal and coupling of soft modes with higher
energy modes is weak, we believe that the experimentally
observed 90◦domain structures in BaTiO3is most likely
due to inhomogeneities in the samples and/or the specific
electric and mechanical boundary conditions.
V.SUMMARY
In this article, TEM observation of PbTiO3thick films
revealed 90◦domain structures, which have been often
observed in BaTiO3. The domain size perpendicular to
domain boundaries was 50–200 nm. The width of domain
wall was corresponding to 1 or 2 unit cell.
We also have performed heatingup and coolingdown
MD simulations of PbTiO3. In coolingdown simulation,
90◦domain structure is found to form spontaneously.
By comparing “total energy surfaces” of single and 90◦
domain structures, we understand that a 90◦domain
structure is metastable in bulk PbTiO3, but not in bulk
BaTiO3. The origin of this contrast is traced to sig
nificantly larger polarizationstran coupling in PbTiO3.
Hence, while 90◦domain structures can form sponta
neously in PbTiO3, they seem to arise in BaTiO3mostly
from special boundary conditions and/or defectrelated
inhomogeneities.
Acknowledgments
This work was supported by Japan Society for the Pro
motion of Science (JSPS) through KAKENHI 23740230
and 21760524. Computational resources were provided
by the Center for Computational Materials Science, Insti
tute for Materials Research (CCMSIMR), Tohoku Uni
versity. We thank the staff at CCMSIMR for their con
stant effort. This study was also supported by the Next
Generation Super Computing Project, Nanoscience Pro
gram, MEXT, Japan. UVW acknowledges an IBM fac
ulty award grant in supporting some of his work.
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