Atomistic Simulation of Surface Effects
in BaTiO3
S.Tinte and M.G.Stachiotti
Instituto de Fı́sica Rosario, Universidad Nacional de Rosario,
27 de Febrero 210 Bis, 2000 Rosario, Argentina
Abstract.
Interatomic potentials for BaTiO3 are determined, in the framework of a shell model,
by mapping first-principles potential energy surfaces for various ferroelectric distortions. Several bulk properties, such as lattice parameters, phase transition sequence,
thermal expansivity, etc., are correctly reproduced by molecular dynamics simulations.
To investigate whether the model will also prove successful for describing surface properties, such as structural relaxations and surface energies, we perform static calculations
on periodic slabs to compare with recent ab-initio studies of BaTiO3 surfaces, which
are taken as benchmark results. The agreement is quite good in spite that the model
was developed for the bulk material. The good description of the static surface properties constitutes an important first step to a further finite-temperature simulation study
of the thickness dependence of ferroelectric transitions in BaTiO3 thin films.
INTRODUCTION
In recent years, a large amount of research worldwide was focused on the growth
and characterization of ferroelectric thin films. On one hand, their ferroelectric, dielectric, and piezoelectric properties were found to be promising for microelectronic
and micromechanical applications [1]. On the other hand, the physical properties
of ferroelectric thin films were found to be substantially different from those of bulk
materials. The ferroelectric properties are known to degrade in thin films [2], and
it is very important to understand the origin of this effect.
First-principles calculations have contributed greatly to the understanding of the
origins of the structural phase transitions in ABO3 perovskites. Although a large
number of calculations were performed to elucidate bulk properties, studies on surfaces have not been very extensive. For the case of BaTiO3 , Cohen [3] presented
Linear Augmented Plane Wave (LAPW) calculations performed for slabs with (001)
and (111) surfaces, using both symmetrical and asymmetrical terminations. Although some relaxations were allowed, the atomic positions were not fully relaxed.
Later on, Padilla and Vanderbilt [4] carried out first-principles total-energy calculations of (001) surfaces of the tetragonal and cubic phases. Both BaO-terminated
and TiO2 -terminated surfaces were considered, and the atomic configurations were
fully relaxed. As in Cohen’s work, the surface relaxation energies were found to
be substantial, i.e., many times larger than the bulk ferroelectric well depth. For
the tetragonal phase, they considered only the case of the tetragonal c axis (i.e.,
polarization) parallel to the surface. They found that ferroelectricity is strongly
enhanced at the TiO2 -terminated surface and suppressed at the BaO-terminated
surface.
Although first-principles methods are extremely precise, they are restricted to investigate zero-temperature properties of perovskites. For the study of the thermal
behaviour, other methods are necessary. A successful approach has been developed on the grounds of effective Hamiltonians. This approach has been applied
with considerable success to several ferroelectric materials, including BaTiO3 [5].
Atomistic simulation methods, on the other hand, are well known to play an important role in solid-state and material sciences. The traditional approach consist
in adjusting unknown model parameters to macroscopic crystal properties. The
availability of accurate first-principles methods makes possible the derivation of reliable interatomic potentials. However, obtaining a model which is able to describe
the structural instabilities of ABO3 perovskites constitutes a challenging problem.
The first goal of our work was to obtain an atomistic model for bulk BaTiO3 by
mapping first-principles potential energy surfaces. This model allowed us to investigate finite-temperature properties by means of molecular dynamics simulations.
The goal now is to use the same model to simulate the surface and size effects on
the phase-transition properties of BaTiO3 thin films. To this end, a necessary
step is to check if the model for the bulk material proves also successful for describing surface properties, such as structural relaxations and surface energies. So, in
this paper we present static calculations on periodic slabs of BaTiO3 to compare
atomic relaxations and surface energies with the above mentioned ab-initio studies
of BaTiO3 surfaces. This constitutes a very important step in order to validate the
model to perform a further finite-temperature simulation, which will elucidate the
ferroelectric properties of BaTiO3 thin crystalline films.
MODEL
The model was developed from first-principles calculations by mapping the potential energy surface for various configurations of some carefully selected atomic
displacements. The potential parameters were obtained by performing a fit of
interatomic potentials to this energy surface. The first-principles total energy calculations were done within the Local Density Approximation to Density Functional
Theory, using the full-potential LAPW method. We used the WIEN97 implementation of the method [6].
For the atomistic simulation we chose the nonlinear oxygen polarizability model
previously applied to BaTiO3 [7], since this model provided an accurate description
of its lattice dynamics. Here each ion is modeled as a massive core linked to a mass-
less shell. An anisotropic core-shell interaction is considered at the O−2 ions, with
a fourth-order core-shell interaction along the O-Ti bond. Besides the coulombic
interactions, the
model contains pairwise short-range potentials which are defined
r
as V (r) = ae(− ρ ) − rc6 . The potential parameters were initially determined from the
corresponding transverse and longitudinal harmonic force constants, taken from set
II in Ref. [7]. Then, they were readjusted in order to fit the model energy behavior
to the ab-initio results. While doing this, we also took care that the equilibrium
lattice constant of the model in the cubic phase reproduces the extrapolation to
0K of the experimental cubic lattice constant.
By modifying the potential parameters initially determined, the model yields
clear ferroelectric instabilities with similar energetics compared with the LAPW
calculations. Energy lowerings of ≈ 1.2, 1.65 and 1.9 mRy/cell are obtained for
the (001), (011) and (111) ferroelectric mode displacements, respectively; which is
consistent with the experimentally observed phase transition sequence. The effects
of the lattice strain are also correctly simulated. See Reference [8] for more details
of the model building.
RESULTS
Bulk properties
Although the results of the simulations for the bulk material have been recently
published [8], we would like to highlight the main achievements of our modelling
approach.
The temperature-dependent properties of the material were investigated by
constant-pressure molecular dynamics (MD) simulations, which were carried out
using the DL-POLY package [9]. The runs were performed employing a Hoover
constant-(σ̄,T) algorithm with external stress set to zero; all cell lengths and cell
angles were allowed to fluctuate. Periodic boundary conditions over 7x7x7 primitive
cells were considered.
The MD-results for the different phases are compared with experimental data in
Table 1. An excellent overall agreement is obtained for the structural parameters,
showing that the model reproduces the delicate structural changes involved in the
transitions. The spontaneous polarizations, on the other hand, are understimated.
This could be related with the fact that the equilibrium values for the transition
metal-oxygen relative displacements obtained by our fitting procedure are smaller
than the LAPW results.
Although the phase transition sequence is correctly reproduced, the theoretically
determined transition temperatures tend to be too small compared with experiments. The orthorhombic and tetragonal phases are stabilized over a temperature
range of only ≈ 30 K and ≈ 70 K, respectively. Similar features are obtained
with the effective Hamiltonian approach [5]. Since, however, the thermal expansivity is correctly reproduced in our case, the underestimation of the ferroelectric
TABLE 1. Transition temperatures, structural
parameters, cube edge component of spontaneous polarization and expansion coefficient for
the different phases of BaTiO3 .
Parameter
Rhombohedral
a(Å)
α (deg)
P(µC/cm2 )
α (10−6 /Co )
TO−R (K)
Orthorhombic
a(Å)
b=c(Å)
P(µC/cm2 )
α (10−6 /Co )
TT −O (K)
Tetragonal
a(Å)
c(Å)
P(µC/cm2 )
α (10−6 /Co )
TC−T (K)
Cubic
a(Å)
α (10−6 /Co )
MD simulation
Experiment
4.012
89.81
12.5
7.2
90
4.003 [10]
89.84 [10]
19 [12]
5.2 [13]
183
3.995
4.022
14
4.3
120
3.987 [10]
4.018 [10]
25 [12]
4.6 [13]
278
4.002
4.043
17
7.7
190
3.999 [10]
4.036 [10]
27 [12]
6.5 [13]
393
4.016
8.5
4.012 [11]
9.8 [13]
instabilities in perovskites seems to be a failure of the LDA methods.
It is worth mentioning that our model for bulk BaTiO3 also reproduces several
zero-temperature properties which are relevant for this material: bulk modulus,
Born effective charges, the presence of two-dimensional instabilities in the phonon
dispersion curves, Γ-phonon frequencies and eigenvectors, etc [8]. It also provides a
good description of the nature of the dynamics in each phase, where order-disorder
is found to be the dominant dynamical mechanism for the transitions. This mechanism leads to the presence of a rhombohedral local enviroment of Ti in all phases.
This local structure appears because of the slow dynamics associated with a relaxational motion of local polarizations, which correlate within chain-like precursor
domains in the paraelectric phase [14].
Surface properties
The achievements reached by our model in the description of the bulk properties
do not necessarily indicate that the same model (i.e. with the same parameters) is
able to reproduce BaTiO3 surface effects. As our final goal is to perform finitetemperature simulations on thin films, it is very important to check if the model
describes properly the static surface properties of BaTiO3 . Although a comparison
with experimental data is, of course, the best way to validate a model, experimental studies of perovskites surfaces are complicated by the difficulties of preparing
clean and defect-free surfaces. Therefore, most experimental investigations have
not been very conclusive. In the case of SrTiO3 , for example, the reported experimental studies [15,16] are in poor agreement with each other. Furthermore, we are
not aware of experimental studies of BaTiO3 surfaces. So, we take recent firstprinciples total-energy calculations on periodic slabs [3,4] as benchmark results to
compare with. To this end, we determine the equilibrium atomic positions for the
same kind of slabs used in the ab-initio calculations.
Firstly, we have performed a simple test by allowing to relax, along the direction perpendicular to the surface (z axis), the first-layer atoms on both sides of an
asymmetrically terminated (001) slab. By symmetry, there are no forces along the
x and y directions for the cubic surfaces, so we only compare the atomic displacements along the z direction, which is perpendicular to the surface. As in Cohen’s
calculations, the slab contains 15 atoms (three BaO layers and three TiO2 layers)
and the vacuum region is three lattice constants thick. The results are listed and
compared with the LAPW calculations in Table 2. It is seen that the model reproduces satisfactorily the relaxation of the suface-layer atoms, which move inwards
the slab (this is indicated by a negative sign in the table). While a very good
quantitative agreement is obtained for the atomic relaxations of the TiO2 surface,
the diplacements for the BaO surface are understimated with respect to the LAPW
results. Consequently, the relaxation energy obtained with our model is lower than
the ab-initio one. In spite of that, the energy gain on displacing the surface ions
is, in both approaches, much greater than the bulk ferroelectric well depth, which
indicates that ionic motions on surfaces could indeed dominate the bulk energetics
for thin slabs.
TABLE 2. Atomic relaxations (relative to ideal atomic positions)
of first-layer atoms on both sides of an asymmetrically terminated
(001) slab. The values are given in percent of the lattice constant
a=4.0058Å. For comparison, LAPW results [3] are shown in parenthesis.
surface
BaO
BaO
TiO2
TiO2
Relaxation energy
Atom
Ba
OIII
Ti
OI , OII
21.3 mRy/cell (31.5 mRy/cell)
δ z (C)
-1.8 (-4.3)
-1.98 (-3.26)
-4.72 (-4.79)
-3.52 (-2.69)
Padilla and Vanderbilt [4], as was already mentioned, studied symmetrically
BaO-terminated and TiO2 -terminated slabs of tetragonal and cubic BaTiO3 whose
coordinates have been fully relaxed by minimizing the total energy. Therefore, a
comparison with their results could be considered as a very exigent test for our
model. Following their steps, we first determine the equilibrium atomic positions
for the two types of slabs in the cubic phase, starting from the ideal structure. We
TABLE 3. Atomic relaxations of the TiO2 - (left
pannel) and BaO-terminated (right pannel) surfaces in the cubic (C) phase, given as percent of
theoretical unit cell parameter a, with repect to
ideal positions. For comparison, ab-initio results
[4] are shown in parenthesis.
Atom
Ti(1)
OI (1)
OII (1)
Ba(2)
OIII (2)
Ti(3)
OI (3)
OII (3)
δz
-4.14
-2.74
-2.74
2.36
-0.50
-0.81
-0.72
-0.72
(C)
(-3.89)
(-1.63)
(-1.63)
(1.31)
(-0.62)
(-0.75)
(-0.35)
(-0.35)
Atom
Ba(1)
OIII (1)
Ti(2)
OI (2)
OII (2)
Ba(3)
OIII (3)
-
δz
-0.72
-1.09
1.70
2.75
2.75
-0.69
-0.28
(C)
(-2.79)
(-1.40)
(0.92)
(0.48)
(0.48)
(-0.53)
(-0.26)
-
set a=3.99 Å which is the equilibrium lattice parameter yielded by the model calculation [8]. The lattice parameter used in the ab-initio calculation is a=3.948(Å).
The results obtained for both slabs are listed in Table 3, together with the ab-initio
results which are shown in parenthesis. It is worth mentioning that the relaxation
patterns obtained with the model are in full agreement with the ab-initio ones. Regarding the atomic displacements, they are, in general, overestimated with respect
to the first-principles results; except for the first-layer atoms of the BaO surface,
which are considerably underestimated.
TABLE 4. Average surface energy for the unrelaxed
Esup(unrel) and relaxed cubic slabs Esup (relax), given in
eV per surface unit cell. ∆E is the surface relaxation
energy.
present calculation
Ref [4]
Ref [3]
Esup (unrel)
1.359
1.358
0.92
Esup (relax)
1.172
1.241
-
∆E
0.187
0.117
-
As was pointed out by Cohen [3], the average surface energy can be calculated
by adding the energies of the Ba-terminated and the Ti-terminated slabs, which
gives the energy of 7 BaTiO3 units. Then, we subtract 7 times the bulk energy per
cell and the result corresponds to 4 surfaces. In this way, we obtain the average
surface energy for the unrelaxed (Esup (unrel)) and the relaxed (Esup (relax)) cubic
slabs . The difference between these energies (∆E) gives the surface relaxation
energy. We compare our calculated values with those obtained previously by abinitio calculations in Table 4. The agreement is quite good. We note again that
∆E is many times larger than the bulk ferroelectric well depth, estimated to be of
the order of 0.03 eV.
Finally, in order to study the influence of surface relaxation effects upon the
ferroelectric distortion, we also determine the equilibrium atomic positions for the
tetragonal phase with FE polarization parallel to the surface in symmetric terminated slabs. The calculations were done at the theoretical unit cell parameters:
a(T) = 3.979(Å) and c(T)=4.055(Å), which are again larger than the ab-initio ones:
a(T) = 3.938(Å) and c(T)=3.993(Å). We do not show here the detailed results of
TABLE 5. Calculated interlayer relaxation (β) and rumpling (η) for the surface layer of the relaxed slabs in the cubic and tetragonal phases, given in percent of the repective unit cell parameters.
Numbers in parenthesis are from references [4](a) and [3](b).
slab
TiO2
BaO
-3.44
-0.90
β(C)
(-2.79)a
( -2.03)a
(-3.74)b
(-3.74)b
0.70
-0.20
η(C)
(1.27)a
(0.76)a
(1.0)b
(0.50)b
β(T)
-2.54 (-2.00)a
-0.42 (-2.0)a
η(T)
0.61 (1.25)a
-0.12 (0.75)a
the atomic relaxations, which are again in fairly good qualitative agreement with
the ab-initio calculations [4]. We prefer to present instead the calculated values of
three relevant quantities: interlayer relaxation, rumpling and ferroelectric distortion. So, we compute the interlayer relaxation (β) and rumpling (η) for the surface
layer of the relaxed slabs in the cubic and tetragonal phases. They were defined as
β = (δ z (O) + δ z (M))/2 and η = (δ z (O) - δ z (M))/2. The results are listed in Table
5. A comparison with Cohen’s results is also shown in the Table, although in his
calculations only the first layer atoms in an asymmetrically termined slab were reTABLE 6. Average layer-by-layer ferroelectric distortions δ F E
of the relaxed slabs, in percent of the theoretical lattice constant
c. Ab-initio results [4] are shown in parenthesis.
layer
1
2
3
4
Bulk
Ti-O terminated
(BaO) δ F E (TiO2 )
4.87 (4.38)
1.41 (1.44)
3.61 (3.44)
2.45 (1.65)
2.45 (1.50)
2.32 (3.20)
δ
FE
Ba-O terminated
δ F E (BaO) δ F E (TiO2 )
1.07 (1.56)
1.60 (1.82)
0.81 (1.31)
2.64 (3.32)
2.45 (1.50)
2.32 (3.20)
4
12Å
28Å
60Å
bulk
Energy (mRy/cell)
0
80Å
80Å
60Å
-4
28Å
-8
12Å
0.00
0.02
0.04
0.06
0.08
0.10
Ti-Ba displacement (Å)
FIGURE 1. Total energy as a function of the ferroelectric mode displacements along the [110]
direction for slabs of different widths. The in-plane lattice constant was a = 4.0Å. The energies are
referred to the unrelaxed cubic slab structures, and the displacements are represented through the
Ti displacement relative to Ba. The results are indicated by open symbols for the Ba-terminated
slabs, and by full symbols for the Ti-terminated slabs. The number near each curve indicates the
slab width in Å. For comparison, the total energy curve for the bulk material is shown in full
line.
laxed. While the agreement is quite good for the Ti-terminated slab, it is not
so good for the Ba-terminated one, because of the strong understimation of the
first-layer Ba relaxation. This could be related with the fact that the Ba p states
look quite different on the surface than in the bulk [3]. In fact, the surface Ba
atoms appear to be significantly less hybridized than the bulk ones. So, the BaO
suface layer is more ionic than the bulk layers, which suggest that the interatomic
potentials for the surface atoms should be modified in order to provide a more
accurate description of the BaO layer relaxation.
We compare, in Table 6, the average ferroelectric distortion (δ F E ) for each layer
of the tetragonal slabs, which is defined as δ F E = δ x(Ba) - δ x (OIII ) for BaO planes,
and δ F E = δ x (Ti) - [δ x(OI ) + δ x(OII )]/2 for TiO2 planes. The bulk values are given
for reference in the last row of the table. For the TiO2 -terminated surface, one can
see a clear increase in the average ferroelectric distortions δ F E when going from
the bulk values to the surface layer. Just the opposite behavior is found for the
BaO-terminated surface. So, the principal feature obtained by our calculation (in
agreement with the ab-initio result) is that the ferroelectricity is strongly enhanced
at the Ti-terminated surface and suppressed at the Ba-terminated surface.
We finally present a study of the ferroelectric instability (with polarization parallel to the surface) as a function of the slab width. To this end, we compute the total
energy of symmetrically terminated cubic slabs of different widths as a function of
the soft mode displacement pattern of the bulk material, along the [110] direction.
This will provide an idea of how the ferroelectric instability parallel to the surface
depends (”on average”) with the slab width. The results are shown in Figure 1.
For the Ti-terminated slab, the ferroelectric instability is stronger than the bulk
one, which corresponds to the above obtained result. The opposite is found for the
Ba-terminated slab. However, what it is interesting to remark is that the average
ferroelectric instability quickly converges to the bulk result when the slab width
increases: no significant deviations from the bulk result are observed for slab width
greater than ≈ 15 unit cells.
SUMMARY AND PERSPECTIVE
We have developed an atomistic model for BaTiO3 by mapping ab-initio potential energy surfaces for various configurations of some carefully selected atomic
displacements. The resulting model describes accurately several bulk properties,
such as Born effective charges, lattice dynamics, energy instabilities, phase transitions sequence, structural parameters, thermal expansivity, etc.
We have shown that the model developed for the bulk material proves also successful for describing surface properties, such as structural relaxations and surface
energies, which are found to be in quite good agreement with recent first-principles
total energy calculations.
In particular, a very good description is obtained for the TiO2 surface. For the
BaO surface, the description is not so accurate: the relaxation of the surface Ba
atoms is underestimated. In spite of that, the relaxation patterns obtained with
the model for both slabs are in full agreement with the ab-initio ones.
The average surface energy and the relaxation energy are in quite good agreement
with first-principles calculations. The energy gain on displacing the surface ions
is much greater than the bulk ferroelectric well depth, which indicates that ionic
motions on surfaces could indeed dominate the bulk energetics for thin slabs.
Regarding the ferroelectric instability with polarization parallel to the surface,
one can see a clear increase in the average ferroelectric distortions when going
from the bulk to the surface layer in the TiO2 -terminated slab. Just the opposite
behavior is found for the BaO-terminated surface. We have shown that the average
ferroelectric instability quickly converges to the bulk result when the slab width
increases.
Finally, we would like to point out that the presented comparative study constitutes a necessary initial step towards a finite-temperature simulation study, which
is in progress, of the ferroelectric transitions in BaTiO3 thin films.
Acknowledgments
Thanks to R.L.Migoni and M.Sepliarsky for helpful discussions. This work was
supported by Consejo Nacional de Investigaciones Cientı́ficas y Técnicas de la
República Argentina. M.G.S. thanks also support from CIUNR and FONCyT.
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