Understanding the morphotropic phase boundary of
perovskite solid solutions as a frustrated state
Ying Shi Teh
1
, Jiangyu Li
2
, and Kaushik Bhattacharya
∗
1
1
California Institute of Technology, Pasadena CA 91125, USA
2
Southern University of Science and Technology, Shenzhen, PRC
November 3, 2020
Perovskite solid solutions that have a chemical composition A(C
x
D
1
−
x
)
O
3
with tran-
sition metals C and D substitutionally occupying the B site of a perovskite lattice are
attractive in various applications for their dielectric, piezoelectric and other proper-
ties. A remarkable feature of these solid solutions is the
morphotropic phase boundary
(MPB), the composition across which the crystal symmetry changes. Critically, it
has long been observed that the dielectric and piezoelectric as well as the ability to
pole a ceramic increases dramatically at the MPB. While this has motivated much
study of perovskite MPBs, a number of important questions about the role of dis-
order remain unanswered. We address these questions using a new approach based
on the random-field Ising model with long-range interactions that incorporates the
basic elements of the physics at the meso-scale. We show that the MPB emerges
naturally in this approach as a frustrated state where stability is exchanged between
two well-defined phases. Specifically, long-range interactions suppress the disorder at
compositions away from MPB but are unable to do so when there is an exchange
of stability. Further, the approach also predicts a number of experimentally observed
features like the fragmented domain patterns and superior ability to pole at the MPB.
The insights from this model also suggest the possibility of entirely new materials with
strong ferroelectric-ferromagnetic coupling using an MPB.
Significance
Perovskites are widely used in capacitor, ultrasonic, photonic, sensor and actuator
applications for their dielectric, piezoelectric and optical properties. Many of these properties are
enhanced in perovskite solid solutions at compositions close to the morphotropic phase boundary
(MPB). This observation drives the search for new materials including lead-free piezoelectrics; yet
the physics of MPB are incompletely understood. We present a new approach which shows that the
MPB arises as a frustrated state in a competition between local chemical disorder and long-range
interactions. It explains various experimental observations and the insights are useful in the search
for new piezoelectrics. Further, the model also suggests the possibility of entirely new phenomena
by exploiting MPBs.
Perovskites are a class of materials with a chemical composition of ABO
3
, where A and B are
typically transition metals, and a crystal structure similar to that of the mineral perovskite CaTiO
3
∗
Corresponding Author: bhatta@caltech.edu
1
arXiv:2011.00752v1 [cond-mat.mes-hall] 2 Nov 2020
500
Temperature
(
°
C
)
400
300
200
100
100% PT
% composition
100% PZ
Ferroelectric
rhombohedral
Paraelectric
Ferroelectric
tetragonal
MPB
A
B
O
(
a
)
(
b
)
(
c
)
Figure 1: (a) Perovskite structure. (b) Phase diagram of PZT adapted from Cross [3]. (c) Schematic
illustration of the lattice model proposed in this work.
shown in Figure 1(a) (e.g. [1, 2]). It is common that these materials undergo a series of displacive
phase transitions from a high temperature cubic ideal perovskite (Pm
3m) structure to distorted
tetragonal (P4mm), rhombohedral (R3m), orthorhombic (Amm2) and other structures. These
lower symmetry structures may be non-centro-symmetric and therefore can become electrically
polarized or magnetized. Therefore these materials are widely used in capacitor, ultrasonic, optical,
sensor and actuator applications for their dielectric and piezoelectric properties.
The basic structure is extremely stable and it is possible to have solid solutions A(C
x
D
1
−
x
)O
3
where two metallic species C and D substitutionally occupy the B site of the lattice. The low
temperature structure in these compounds depends on composition. A remarkable feature that
is observed in a number of such solid solutions is the
morphotropic phase boundary
(MPB), the
composition across which the symmetry changes, see Figure 1(b). This composition is largely
independent of temperature (up until the temperature where it transforms to the cubic structure).
Critically, it has long been observed that the dielectric and piezoelectric as well as the ability to pole
a ceramic increases dramatically at the MPB [3, 4, 5]. This has been central to the widespread use
of lead zirconate titanate (PbZr
x
Ti
1
−
x
O
3
or PZT) that has a MPB at
x
= 0
.
52 with a ferroelectric
rhombohedral (R3m) structure in the Zr-rich compositions
1
and ferroelectric tetragonal (P4mm)
structure in the Ti-rich phases. The search for lead-free dielectric and piezoelectric materials has
also focussed on solid solutions with MPBs (e.g. [6] for a recent review).
Given the importance of MPBs, it has been and continues to be the subject of intense study.
Classically, it was believed that the tetragonal and rhombohedral phases coexist at the MPB. This
was challenged by the discovery of a low-symmetry monoclinic phase (Cm) at the MPB by Noheda
et al.
[7] using x-ray powder diffraction. This was supported by first principles calculations that
developed a composition-dependent hybrid pseudopotential [8]. Importantly, it was recognized that
the presence of a bridging phase enable a larger intrinsic piezoelectric effect at the MPB [9, 4, 10].
Further, either the coexistence or the availability of a low symmetry bridging phase enabled a high
extrinsic piezoelectric effect at the MPB [5].
Since then, there have been a number of studies of the crystal structure of MPB-PZT, and
there are observations consistent with various structures. Examples include the combination of two
monoclinic phases (Cm and Ic, [11] or Cm and Pm [12]), combination of tetragonal (P4mm) and
monoclinic (Cm) [13] and combination of rhombohedral (R3m) and monoclinic (Cm) [14]. This
1
PZT shows a second rhombohedral (R3c) phase at low temperature at high Zr compositions, but we focus on the
compositions near the MBP
2
uncertainty has been attributed to the disorder in the composition resulting in a disorder in the
structure, and the difficulty of resolving local structures [15]. This role of disorder is also supported
by first principles calculations [16, 17]. This however raises the question as to why the disorder
does not affect the structure away from the MPB. Another interesting observation concerns the
domain patterns. Classical well-defined domain patterns are observed away from the MPB, but
highly fragmented domain patterns are observed near the MPB [18]. It has been argued that this
fragmented domain pattern also contributes to the high piezoelectric response near the MPB [19].
In short, critical questions remains open. Why is the effect of compositional disorder suppressed
to form an unambiguous structure away from the MPB, but suddenly revealed at the MPB? Is there
a definitive crystal structure at the MPB? Why do domain patterns become fragmented near the
MPB? Does compositional disorder play a role in the ease of poling at the MPB? Can the MPB be
exploited to create new phenomena? These questions are important because the MPB is the focus
of the development of new materials. However, they have proved to be challenging. The disordered
nature of the solid solution requires a large ensemble that takes it beyond the scope of direct
first principles calculations without the introduction of an averaged pseudopotential. On the other
hand, phase-field Landau-Ginzburg methods can provide insight into domain patterns. However,
they are too coarse-grained to account for atomic-scale interactions and instead incorporate the
MPB phenomenologically.
We address these questions using a new approach based on the random-field Ising model with
long-range interactions that incorporates the basic elements of the physics at the meso-scale. First,
the B sites of a perovskite form a reference cubic lattice that is occupied randomly by atoms of
either C or D species. Second, the local quantum mechanical interactions create a propensity for
the unit cell to break cubic symmetry depending on the species at the B site. Finally, there are
long-range interactions due to ferroelectric, ferromagnetic and ferroelastic polarizations. We create
an effective Hamiltonian with these physics and study the ground states using the Markov chain
Monte-Carlo (MCMC) method with cooling.
In the first part of the paper, we show that this simple model provides new insights to the
the questions concerning the MPB of ferroelectric solid solutions like PZT. In particular, the long-
range interactions which have an ordering feature overwhelm the local disorder in the C-rich and
D-rich compositions with the exchange of stability taking place at a specific composition where
the material is frustrated. This frustration manifests itself as the MPB. The frustrated state also
enables easy poling as observed. In the second part, we use the model to explore the possibility
of obtaining materials with strong ferroelectric-ferromagnetic coupling using the insights obtained
in the first part. Such multi-ferroic coupling is limited in single materials [20], and is thus realized
using composite media.
1 Ferroelectric solid solution
Model
Consider a
d
-dimensional periodic lattice (
d
= 2 or 3) with
N
lattice points as shown in
Figure 1(c). Each lattice point
i
is characterized by fixed (quenched) chemical composition (
c
i
) of
either type
C
(
c
i
= 0 indicated by a red open circle in Figure 1(c)) or type
B
(
c
i
= 1 indicated by
a blue closed circle). Each lattice point carries a dipole state (
p
i
indicated by the arrows in Figure
1(c)) that can take one of a number of orientations determined by the Hamiltonian
W
tot
(
{
p
i
}
;
{
c
i
}
) =
N
∑
i
=1
h
loc
(
p
i
;
c
i
)
−
J
e
2
∑
<i,j>
p
i
·
p
j
+
D
e
W
dip
(
{
p
i
}
)
−
E
ext
·
N
∑
i
p
i
.
(1)
3
The first term encodes the information that lattice site of type
C
(respectively
D
) energetically
prefers the set of dipole states
C
indicated by the red arrows (respectively
D
indicated by the blue
arrows), though they can take states in
D
(respectively
C
) with an energetic cost
h >
0:
h
loc
(
p
i
;
c
i
) =
{
0 if
c
i
= 0 and
p
i
∈C
,
or,
c
i
= 1 and
p
i
∈D
,
h
otherwise.
(2)
The second term is the exchange energy with
J
e
>
0 (and the sum is limited to nearest neighbors)
that promotes like neighbors. The third term is the long-range electrostatic dipole-dipole interaction
where
W
dip
(
{
p
i
}
) =
1
(
d
−
1)
N
∑
i,j
=1
,
∑
R
1
r
d
ij
[
p
i
·
p
j
−
d
(
p
i
·
r
ij
)(
p
j
·
r
ij
)
r
2
ij
]
+
2
π
d
N
∑
i
=1
|
p
i
|
2
(3)
with strength
D
e
which incorporates the dipole strength, lattice constant and electro-magnetic
constants. The final term is the influence of the applied external electric field
E
ext
.
Given a lattice where the composition of each site is randomly assigned subject to a fixed aver-
age, we use a Markov chain Monte-Carlo (MCMC) method with cooling to obtain the equilibrium
distribution at a given temperature. The state is initialized by randomly assigning a polarization
from
C∪D
. Adapting the Metropolis-Hastings algorithm to our multi-state setting, a site is chosen
at random and its dipole state is updated to one of the
N
states
states according to the transition
probability
P
s
=
exp(
−
βW
(
s
)
tot
)
∑
N
states
r
=1
exp(
−
βW
(
r
)
tot
)
, s
= 1
,
2
,...,N
states
(4)
where
β
is the inverse temperature and
N
states
is the cardinality of
C ∪D
. We avoid the system
getting trapped in local minima at low temperatures by starting at a high temperature (
β
= 0)
and slowly cooling (increasing
β
) to the temperature of interest, while performing enough MCMC
steps to reach equilibrium at each temperature. The details are provided in Methods.
Results
We study an example motivated by PZT though the results are generic. Here the C
lattice points represent unit cells containing Zr atoms while the D lattice points represent unit
cells containing Ti atoms. Recall that the former prefers rhombohedral or
〈
111
〉
polarization states
while the latter prefer tetragonal or
〈
100
〉
polarization states. We begin in two dimensions
d
= 2
so that the
C
= 1
/
√
2
{
[1
,
1]
,
[1
,
−
1]
,
[
−
1
,
1]
,
[
−
1
,
−
1]
}
while
D
=
{
[1
,
0]
,
[0
,
1]
,
[
−
1
,
0]
,
[0
,
−
1]
}
. We
set
h
=
J
e
=
D
e
= 1, Ewald parameters
σ
= 0
.
157 and
M
cut
= 16, and conduct our simulations on
a 256
2
lattice. In each simulation, we begin with an inverse temperature of
β
= 0, and repeatedly
increase its value with a small step size of ∆
β
= 0
.
05 until we reach
β
= 5. At each temperature
value, at least 2000 Monte Carlo (MC) sweeps (each sweep consists of
N
= 256
2
steps) are performed
with a total of
≈
2
×
10
5
sweeps.
Figure 2 shows the results of these simulations. Figure 2(a) shows the evolution of the order
parameter (
ξ
=
1
κ
max
∑
κ
max
κ
=1
κC
(
κ
) where
C
(
κ
) =
〈
p
i
·
p
j
〉
is the correlation function over any
two sites
i
and
j
that satisfy
κ
−
1
<
|
r
i
−
r
j
| ≤
κ
) in a series of simulations with varying
average composition. The material is disordered at high temperature, but becomes ordered at low
temperatures. The phase transition is somewhat diffuse due to the disorder. Figure 2(b) shows
the nature of the ordered phase. Remarkably, we find that all dipoles are in the rhombohedral
(
C
) states till a critical composition of about 50% beyond which all dipoles are in the tetragonal
(
D
) states. Indeed, at a composition of 33
.
3%, a third of the sites would prefer tetragonal dipoles.
However, the exchange and electrostatic interaction with the neighbors overwhelm this preference
4