Also at ]ICAMS and ZGH, Ruhr-University Bochum.

# Field direction dependence of the electrocaloric effect in single crystal
BaTiO3

Lan-Tien Hsu<EMAIL_ADDRESS>Interdisciplinary Centre for
Advanced Materials Simulation (ICAMS) and Center for Interface-Dominated High
Performance Materials (ZGH), Ruhr-University Bochum, Universitätsstr. 150,
44801 Bochum, Germany  Frank Wendler<EMAIL_ADDRESS>Institute of
Materials Simulation, Department of Materials Science, Friedrich-Alexander
University of Erlangen-Nürnberg, Dr.-Mack-Strasse 77, 90762 Fürth, Germany
Anna Grünebohm<EMAIL_ADDRESS>[

###### Abstract

Single crystalline ferroelectric perovskites show a large electrocaloric
effect (ECE) at field-induced phase transitions, raising attention to solid-
state cooling technology. However, paraelectric-ferroelectric transition
temperatures are often too high for practical applications and lower
transitions are so far underrepresented in literature. Particularly, the role
of thermal hysteresis and field direction on the responses is critical, but
not yet fully understood. Using ab initio based coarse-grained molecular
dynamics, we show that the polarization changes at zero fields are good
descriptors for the field direction dependence of transition temperatures both
for BaTiO3 and PbTiO3 even though both materials show different microscopic
field coupling. Furthermore, we reveal that the choice of field direction can
reduce irreversibilities related to thermal hysteresis and can adjust the
temperature windows of large caloric responses. Based on our results we
identify favorable temperature- and texturing conditions for polycrystals.

keyword: electrocaloric effect, solid-state cooling, ferroelectrics,
structural phase transition, molecular dynamics, field direction dependence,
symmetry, perovskite oxides, thermal hysteresis

##  I Introduction

The electrocaloric effect (ECE) is promising for solid-state cooling
technologies.Ožbolt  _et al._ (2014); Aprea  _et al._ (2017); Shi  _et al._
(2019); Wang  _et al._ (2020); Greco and Masselli (2020); Torelló and Defay
(2022) It is the electrical field-induced adiabatic temperature change
($\Delta T_{ad.}$) by the reallocation of entropy between lattice vibrations
and the dipole configuration. Maximal caloric responses are observed at
ferroic phase transitions with large changes in polarization ($\vec{P}$) with
temperature and/or latent heat.Marathe  _et al._ (2017); Moya  _et al._
(2014); Mischenko  _et al._ (2006); Neese  _et al._ (2008) Conventional and
inverse ECEs appear when the field stabilizes the low- and high-temperature
phases, respectively. Both can achieve adiabatic cooling: the former under
field removal and the latter under field application.Wu and Cohen (2017a);
Grünebohm  _et al._ (2018)

Challenges in material design include tuning the temperature window with large
caloric responses to room temperature, maintaining a large response when going
from single crystals to technically more relevant ceramics, and realizing a
large reversible response, i.e., reducing or bypassing hysteresis. Firstly,
the largest caloric responses are found in small temperature windows at the
paraelectric-ferroelectric (PE-FE) phase transitions.Moya  _et al._ (2013);
Rose and Cohen (2012) These windows are commonly above room temperature, and
thus not suitable for most applications. It has been shown that the
temperature window with maximal response can be tuned by substitutionLisenkov
and Ponomareva (2018); Wu and Cohen (2017b); Grünebohm and Nishimatsu (2018)
or by utilizing the FE-FE transitions at lower temperatures.Marathe  _et al._
(2017); Taxil  _et al._ (2022) Particularly for the latter, it has furthermore
been reported that transition temperatures and temperature windows with large
$\Delta T_{ad.}$ depend on the direction of the applied field.Marathe  _et
al._ (2017); Li  _et al._ (2020)

Secondly, as field direction can change the sign of the caloric
response,Marathe  _et al._ (2017); Li  _et al._ (2020) it is obvious that this
direction dependence may reduce the ECE in polycrystalline materials. However,
a systematic analysis of all relative directions between the field and crystal
structure is missing. Finally, the response is reduced if the system does not
(fully) cross the phase transition during the field variation. Within the
coexistence range of two phases, the ECE depends on the thermal history and is
thus not reversible. Li  _et al._ (2022); Marathe  _et al._ (2018); Moya  _et
al._ (2014) Studies on the irreversibilities due to field and thermal
hysteresis are so far underrepresented.Li  _et al._ (2022); Ma  _et al._
(2018); Liu  _et al._ (2016a); Grünebohm  _et al._ (2021); Marathe  _et al._
(2018); Bai  _et al._ (2012); Moya  _et al._ (2014) Promising concepts to
bypass thermal hysteresis discussed in the literature include the weakening of
the first-order character by large fields,Novak  _et al._ (2013); Marathe  _et
al._ (2017); Liu  _et al._ (2016b); Li  _et al._ (2022) the reduction of the
energy barrier for nucleation and growth of the new phase by domain and defect
engineering, and the combination of different fields.Grünebohm  _et al._
(2021); Gutfleisch  _et al._ (2016) Under a strong field, a reversible ECE is
observed experimentally in $\langle 011\rangle$\- and $\langle
111\rangle$-oriented BaTiO3 single crystal.Li  _et al._ (2021)

The fundamental understanding and optimization of the ECE, especially in
ceramics, asks for a better understanding of the relation between field
direction and transition temperatures while good descriptors for that are so
far missing. To the best of our knowledge, also the impact of the field
direction on thermal hysteresis has only been studied for high symmetric
directions (i.e., $[100]$, $[110]$, and $[111]$), and while the ECE has been
analyzed also on the $(001)$, ($\bar{1}10$), and (0$\bar{1}$1) planes,Novak
_et al._ (2013); Marathe  _et al._ (2017); Li  _et al._ (2020, 2022) the
knowledge on the complete direction dependence is still missing for both
quantities.

In this paper, we discuss the influence of field direction on transition
temperatures of two prototypical ferroelectric perovskite oxides – BaTiO3 and
PbTiO3. The latter shows superior $|\Delta T_{\text{ad.}}|$Mikhaleva  _et al._
(2012); Moya  _et al._ (2013), while the former is non-toxic and experiences
three transitions. However, so far, there is no systematic comparison between
them. Furthermore, we discuss the thermal hysteresis and ECE of BaTiO3 for all
possible field directions. We show that a phenomenological polarization-field
coupling can qualitatively describe the field direction dependence of
transition temperatures and thus the temperature window with large responses
at all transitions in BaTiO3 and PbTiO3. Although low-symmetric field
directions cannot enhance the maximal ECE, they can reduce thermal hysteresis
and irreversibilities.

Figure 1: A sketch of sampled field directions. In addition to the field
directions parallel to the spontaneous polarization in phases T, O, and R
phases ($[100]$, $[110]$, and $[111]$, respectively), we sample low symmetric
field directions on the $(001)$, $(\bar{1}10)$, and $(0\bar{1}1)$ planes, and
the low symmetric directions $\theta_{1-3}$ between $[110]$ and $\theta_{4}$
(unit vector: $(0.82,0.41,0.41)$; on the $(0\bar{1}1)$ plane). The unit
vectors for $\theta_{1-3}$ are $(0.75,0.67,0.083)$, $(0.78,0.60,0.18)$, and
$(0.81,0.48,0.33)$, respectively.

##  II Method

As first step, we set up a simple phenomenological model to efficiently
predict the direction dependence of ferroelectric phase transition
temperatures. In Landau theory, the coupling between polarization and an
external field ($\vec{E}$) is given as $\vec{E}\cdot\vec{P}$. As the
transition temperature ($T_{C}$) between two phases depends on their relative
energies, one may thus expect that the transition temperature depends on
$\vec{E}$ as

$T_{C}(\vec{E})\sim\vec{E}\cdot\Delta\vec{P}\;,$ (1)

with $\Delta\vec{P}$ being the polarization difference between both phases in
a field, which is further approximated to the spontaneous polarization
difference in zero field cooling (ZFC).

Our main method is the effective Hamiltonian by Zhong et al.Zhong  _et al._
(1994, 1995) parametrized with density functional theory (DFT) calculations
for BaTiO3Nishimatsu  _et al._ (2010) and PbTiO3Nishimatsu  _et al._ (2012)
which has been successfully used to study ferroelectric phase diagrams and
electric-field-driven responses of ferroelectrics.Grünebohm  _et al._ (2023);
Walizer  _et al._ (2006) With this method, we characterize phase transitions
and caloric response and check the validity of Eqn. (1). For these purposes,
we perform coarse-grained molecular dynamics simulations (MD) using the open-
source feram code.Nishimatsu  _et al._ (2008) The energy surface is expressed
as a function of local optical displacement vectors (or dipole moments) and
local acoustic displacement vectors (or local strain), and global homogeneous
strain. As strain is internally optimized, the number of degrees of freedom
explicitly involved in time for each unit cell is reduced from 15 (five atoms
in three Cartesian directions) to 3.

To model ZFC and field cooling (FC), we initialize the dipoles in the
paraelectric phase with a random distribution either without field or under an
external field of 100 kV/cm. The system is cooled down from 380 K to 25 K.
Starting from these preconverged configurations at 25 K, zero field heating
(ZFH) and field heating (FH) simulations are performed. In all cases,
temperature steps of $\pm 5$ K ($\pm 1$ K around the second transition) are
used together with the Nosé-Poincaré thermostat,Bond  _et al._ (1999) a time
step of 1 fs, and a large simulation box with 36$\times$36$\times$36 unit
cells (side length: 14.35 nm) together with periodic boundary conditions. At
each temperature, we thermalize the system for 60 ps and average for 20 ps. We
determine the transition temperatures by the maximum change of one chosen
polarization component with temperature. Two repetitive simulations are done
for the three high symmetric field directions and the repetitive error of
transition temperature is within 5 K.

Due to the symmetry of the materials of interest, the full field space can be
sampled by the directions illustrated in Fig. 1:Vanderbilt and Cohen (2001)
the $(001)$ plane, i.e., between $[100]$ and $[110]$, the ($\bar{1}10$) plane,
i.e., between $[110]$ and $[111]$, the (0$\bar{1}$1) plane, i.e., between
$[100]$ and $[111]$, and the low symmetric directions in the region bounded by
these planes. For the latter, we sample three directions $\theta_{1-3}$ on the
connection line between $[110]$ and $\theta_{4}$ (a low symmetric direction on
the (0$\bar{1}$1) plane).

To classify the symmetry of the FE phases in the applied field, the notation
by Vanderbilt and CohenVanderbilt and Cohen (2001) based on the three
Cartesian components of $\vec{P}$, with $a>b>c>0$ is used:
$\langle$a,0,0$\rangle$ for tetragonal phases (T); $\langle$a,a,0$\rangle$ for
orthorhombic phases (O); $\langle$a,a,a$\rangle$ for rhombohedral phases (R);
$\langle$b,b,a$\rangle$, $\langle$a,a,b$\rangle$, $\langle$a,b,0$\rangle$ for
monoclinic phases (M${}_{\text{A}}$, M${}_{\text{B}}$, M${}_{\text{C}}$
respectively), and $\langle$a,b,c$\rangle$ for triclinic phases (Tri).

The ECE under field removal is directly determined by ramping off the field
under adiabatic conditions. Firstly, we start with the aforementioned
thermalized FC configurations, equilibrate them further for 80 ps in a
microcanonical ensemble, and measure the actual initial temperature
(${T_{in}}$) by averaging over another 40 ps. Secondly, we slowly ramp down
the field adiabatically with a ramping rate of 1 kV/(cm$\cdot$ps). Lastly, we
equilibrate the system for 80 ps without an external field and measure the
final temperature by averaging over 100 ps. Due to the coarse-graining (the
number of degrees of freedom goes from 15 to 3), the specific heat in our
model is underestimated by a factor of 5, and thus we correspondingly rescale
the $\Delta T_{ad.}$ by 1/5.Nishimatsu  _et al._ (2013) As entropy change and
heat and thus the ECE depend linearly on temperature, it is convenient to also
compare the normalized ECE ($\Delta T_{ad.}/T_{{in}}$) for different field
directions.

Figure 2: Field direction dependence of the field coupling (as in Eqn. (1)) at
the (a) first, (b) second, and (c) third phase transitions of BaTiO3. Figure
3:  Temperature-field direction phase diagram of (a) BaTiO3 and (b) PbTiO3
under a total field strength of 100 kV/cm. The phases are annotated for each
high symmetric direction and each plane of low symmetric direction. Thick blue
and red curves mark the transition temperatures $T_{C}^{\mathrm{FC}}$ and
$T_{C}^{\mathrm{FH}}$, and horizontal lines give corresponding ZF values as
reference. In (a) for chosen FC configurations, the color of the dots marks
the normalized temperature change ($\Delta T_{ad.}/T_{in}$) by field removal.
In (b) dark and light blue curves mark the transition temperatures
$T_{C}^{\mathrm{FC}}$ to single- and multi-domain states, respectively.
Possible macroscopic symmetries of the observed multidomain structures are
shown in parenthesis. (c) Exemplary multidomain structure induced in PbTiO3 by
a field along $[110]$ at 640 K. Each arrow represents the time-averaged (20
ps) local polarization of each unit cell and is color-coded according to its
component along the x-axis ($P_{x}$). The time-averaged $P_{z}$ is zero.

##  III Results

###  III.1 Phase diagrams

BaTiO3 shows three first-order phase transitions
(C$\leftrightarrow$T$\leftrightarrow$O$\leftrightarrow$R) while PbTiO3 only
shows the C$\leftrightarrow$T transition. The zero-field cooling (heating)
transition temperatures, $T_{C}^{\mathrm{ZFC}}$ ($T_{C}^{\mathrm{ZFH}}$),
predicted by the effective Hamiltionan are 280 K (295 K), 140 K (185 K), and
75 K (130 K) for BaTiO3; and $595$ K for PbTiO3. Note that the used method
underestimates experimental values of 393 K, 278 K, and 183 K due to the
underlying DFT calculations and the neglect of anharmonicities and other
phonon modes in the potential.Nishimatsu  _et al._ (2010) If a field is
applied parallel to the spontaneous polarization of a phase, it changes the
ordering and the magnitude of the dipoles. For all other field directions,
also the symmetry of the phases changes. Thus, for simplicity, we refer to the
transitions by their temperature sequence as the first (PE-FE), second, and
third transitions.

To predict the field direction dependence of these phase transitions, the
phenomenological field-coupling ($\vec{E}\cdot\Delta\vec{P}$) is shown in Fig.
2. The $\Delta\vec{P}$ for the first, second, and third phase transitions of
BaTiO3 is set to ($+28$,$0$,$0$), ($-8$,$+28$,$0$), and ($-5$,$-5$,$+25$) in
units of $\mu\text{C/cm}^{2}$, i.e. the polarization changes we find during
ZFC. The field coupling for the first transition shown in Fig. 2 (a) is always
positive and shows a maximum, a local minimum, and the global minimum for
fields along $[100]$, $[110]$, and $[111]$, respectively. Thus,
$T_{C}^{\mathrm{FC}}>T_{C}^{\mathrm{ZFC}}$ is expected for all field
directions with a maximal increase in $T_{C}^{\mathrm{FC}}$ for the field
along $[100]$, i.e., parallel to the spontaneous polarization of the
tetragonal phase, and with smaller increases for the other high symmetric
field directions. Compared with BaTiO3, the larger $\Delta\vec{P}$ for
PbTiO3111We find $\Delta\vec{P}$ for PbTiO3 being ($+35$,$0$,$0$) in units of
$\mu\text{C/cm}^{2}$. suggests a larger increase in the PE-FE transition
temperature.

At the second transition of BaTiO3, the field coupling changes can also be
negative. It is positive for most directions, see Fig. 2 (b), but close to
$[100]$, the projection of polarization on the field direction is larger in
the high-temperature phase, and a decrease in the transition temperature is
predicted. At the third transition shown in Fig. 2 (c), the field coupling is
positive if the field points close enough to $[111]$, i.e., if the field
stabilizes the R phase. For other field directions, particularly on the
$(001)$-plane, the high-temperature phase is stabilized.

Qualitatively, all these predictions are in good agreement with the
$T_{C}^{\mathrm{FC}}$ found in MD simulations as shown by thick blue curves in
Fig. 3 for a field strength of 100 kV/cm and in Fig. 6 for 10 kV/cm. Firstly,
indeed, $T_{C}^{\mathrm{FC}}>T_{C}^{\mathrm{ZFC}}$ for all field directions at
the first transition, for fields not pointing close to $[100]$ at the second
transition, and for fields on the $(001)$ plane at the third transition.
Secondly, it is also true that PbTiO3 shows a larger field-induced increase in
$T_{C}^{\mathrm{FC}}$ compared to BaTiO3. However, the simple estimate given
by Eqn. (1) does not include field-induced changes in polarization, domain
structure, and the dynamics of the transitions, and thus fails to
quantitatively predict the crossover field angle from enhanced to reduced
$T_{C}^{\mathrm{FC}}$. For example, the field direction range with
$T_{C}^{\mathrm{FC}}<T_{C}^{\mathrm{ZFC}}$ at the second transition around
$[100]$ is overestimated by about 33%, and the range with
$T_{C}^{\mathrm{FC}}>T_{C}^{\mathrm{ZFC}}$ at the third transition around
$[111]$ is underestimated by about 68%. 222At the second transition of BaTiO3
Eqn. (1) predicts that $T_{C}^{\mathrm{FC}}<T_{C}^{\mathrm{ZFC}}$ within 22∘
away from $[100]$ on the $(0\bar{1}1)$ plane and within 16∘ away from $[100]$
on the $(001)$ plane, while MD predicts a smaller angle range (8∘ and 5∘). At
the third transition, it predicts that
$T_{C}^{\mathrm{FC}}>T_{C}^{\mathrm{ZFC}}$ within 19∘ away from $[111]$ on the
$(\bar{1}10)$ plane and within 35∘ away from $[111]$ on the $(0\bar{1}1)$
plane, while MD predicts a larger angle range (28∘ and 51∘, respectively).
Thirdly, the quantitative differences between BaTiO3 and PbTiO3 are not fully
covered by the difference in $\Delta\vec{P}$. While $\Delta\vec{P}$ differs by
25%, the MD predicted increase in $T_{C}^{\mathrm{FC}}$ is considerably larger
for all field directions, e.g., 116% for the field along $[111]$. Fourthly, MD
simulations show that a field direction that favors the high-temperature phase
induces a larger reduction in $T_{C}^{\mathrm{FH}}$ than in
$T_{C}^{\mathrm{FC}}$, whereas a direction that favors the low-temperature
phase induces a larger increase in $T_{C}^{\mathrm{FC}}$ than in
$T_{C}^{\mathrm{FH}}$. This is, however, not reflected by this simple estimate
even with $\Delta\vec{P}$ from the ZFH simulation considered.

To conclude, using the spontaneous polarization difference in ZFC together
with Eqn. (1) results in a good descriptor for the qualitative change of
transition temperature with field direction, but does not suffice to
quantitatively predict the field-direction dependence of the transitions.

Figure 4: Field direction dependence of thermal hysteresis in BaTiO3 under
fields of 100 kV/cm in sampled directions at the (a) second and (b) third
transitions. For comparison, the thermal hysteresis in ZF are 45 K and 55 K,
respectively.

The MD simulations give us access to the change in the macroscopic
polarization, its spatial distribution, and the symmetry under fields, see
Fig. 3. At high temperatures where the phase is paraelectric in ZF, the field
induces polarization and reduces the symmetry correspondingly, e.g., to MB on
the $(\bar{1}10)$ plane. Within a ferroelectric phase, rotating the field away
from the direction of the spontaneous polarization reduces the symmetry to
monoclinic or triclinic. In all cases, a homogeneous polarization rotation is
induced in BaTiO3, i.e., the system stays in a single domain state and the
distribution of the local dipoles gradually rotates with the field direction.
This polarization rotation is small under a field strength of 100 kV/cm, cf.
Fig. 9.

For PbTiO3 one has to distinguish two regimes: Close to [100] (blue in Fig. 3
(b)) and all other field directions (light blue). For the former, the system
remains single domain with the same symmetry as BaTiO3, but the polarization
rotation is smaller (cf. Fig. 9). For the latter, the system decomposes to a
multidomain state: tetragonal domains separated by charge-neutral T 90∘ walls,
e.g., tetragonal domains with polarization pointing along $[100]$ and $[010]$
are observed for field direction $[110]$, see Fig. 3 (c). The macroscopic
symmetry depends on the multidomain states and can be monoclinic or triclinic.
Note that even though a field along $[111]$ would equally stabilize all three
types of $\langle 100\rangle$-domains, we always only find two types in our
simulations. Replicate simulations show that both the width and polarization
direction of the domains relate to random nucleation. After two types of
domains have nucleated, the formation of the third direction would result in
wall crossings and is too high in energy in our simulations. In agreement with
our findings, a larger tendency to form domains in PbTiO3 than in BaTiO3 has
also been found in ZFC and has been related to the larger polarization-strain
coupling of PbTiO3.Nishimatsu  _et al._ (2012) Importantly, neither the
appearance of the multidomain structures, nor wall direction and spacing
influence the field dependence of $T_{C}^{\mathrm{FC}}$ at the given
temperature resolution. In addition, the differences between
$T_{C}^{\mathrm{FC}}$ for fields along $[100]$ and $[111]$ are similar for
PbTiO3 (35 K) and BaTiO3 (33 K).

Field direction dependence of thermal hysteresis cannot be simply estimated by
Eqn. (1), even if $\Delta\vec{P}$ from the ZFH simulation is used. In the
following, we restrict our discussion to MD results on the thermal hysteresis
in BaTiO3. In general, thermal hysteresis decreases with the magnitude of the
applied field.Li  _et al._ (2022); Marathe  _et al._ (2017) For the first
transition, the used field is stronger than the critical field strengths for
fields along $[100]$ (about 40 kV/cmMarathe  _et al._ (2017)) and $[110]$
(about 50 kV/cmLi  _et al._ (2022)) reported for the used effective
Hamiltonian. Furthermore, fields along most directions induce isostructural
transition without clear jumps in polarization. Only for the field exactly
along $[111]$ (R-MA transition) does the thermal hysteresis exceed the
temperature resolution of 5 K.

The thermal hysteresis for both FE-FE transitions under fields of 100 kV/cm is
shown in Figure 4. For all high symmetric directions, a finite hysteresis
remains, in agreement with the predicted larger critical field strengths
Marathe  _et al._ (2017). The field exactly along $[100]$ induces the maximal
hysteresis for both FE-FE transitions. Particularly at the second transition,
the field along $[100]$ favors the high-temperature phase and thus reduces
$T_{C}^{\mathrm{FC}}$. On the other hand, the stability range of the low-
temperature phase is less affected, i.e., $T_{C}^{\mathrm{FH}}$ is less
sensitive to the direction. Small rotations away from this direction result in
the abrupt reduction of hysteresis at the second transition, and the third
transition (particularly with a rotation on the $(0\bar{1}1)$ plane). These
observations can be related to the stabilization of the initial phase exactly
by this field direction.

Generally, a rotation of field direction that changes only the symmetry of the
high-temperature (low-temperature) phase causes an abrupt change in
$T_{C}^{\mathrm{FC}}$ ($T_{C}^{\mathrm{FH}}$), while $T_{C}^{\mathrm{FH}}$
($T_{C}^{\mathrm{FC}}$) changes linearly with the field rotation. Therefore,
when rotating the field away from $[100]$ on the $(001)$ plane,
$T_{C}^{\mathrm{FC}}$ at the second transition changes abruptly while the
transition temperature under heating changes gradually, resulting in a huge
decrease in thermal hysteresis. With the same field rotation at the third
transition, the symmetry of the high-temperature phase remains unchanged,
resulting in an abrupt and gradual change in $T_{C}^{\mathrm{FH}}$ and
$T_{C}^{\mathrm{FC}}$, respectively, and the thermal hysteresis is related to
the change in $T_{C}^{\mathrm{FH}}$.

Low symmetric directions reduce thermal hysteresis, particularly if they
induce isostructural transitions. At the second transition, thermal hysteresis
is reduced to around 5 K for fields on the $(\bar{1}10)$ and $(0\bar{1}1)$
planes, and can be further reduced to close to 0 K when the transition becomes
isostructural, e.g., for the field along $\theta_{2}$
(Tri$\leftrightarrow$Tri) or on the $(001)$ plane
(M${}_{\text{C}}$$\leftrightarrow$M${}_{\text{C}}$). At the third transition,
thermal hysteresis can be reduced to around 15 K and 10 K for fields on the
$(001)$ and $(0\bar{1}1)$ planes, and can be further reduced to around 5 K or
lower for fields on the $(\bar{1}10)$ plane
(M${}_{\text{B}}$$\leftrightarrow$M${}_{\text{B}}$) or along $\theta_{i}$
(Tri$\leftrightarrow$Tri). In general, we can expect to find the lowest
thermal hysteresis at isostructural transitions between phase Tri for low
symmetric field directions.

###  III.2 Electrocaloric response

Which field direction yields a large reversible ECE in a broad temperature
window and is thus optimal for application? To answer this question, we first
analyze the adiabatic temperature change for representative field directions.
The magnitudes of the ECE (of pre-cooled systems) are shown as colored dots at
the sampled temperatures and field directions in Fig. 3 (a) and the
temperature dependence of the ECE for representative field directions are
illustrated in Figure 5.

At all transitions, a large conventional ECE (shown in red dots in Fig. 3 (a))
is induced only when the system undergoes a phase transition from the low- to
the high-temperature phase during field removal, i.e., only in the temperature
intervals from $T_{C}^{\mathrm{ZFH}}$ to $T_{C}^{\mathrm{FC}}$. Analogously, a
large conventional response has to be expected between $T_{C}^{\mathrm{ZFH}}$
and $T_{C}^{\mathrm{FH}}$ for pre-heated samples. The maximal $\Delta
T_{{ad.}}$ ($\Delta T_{{ad.}}/T_{in}$) for our pre-cooled samples at the
first, second, and third transitions are $-$4 K ($-$0.013), $-$1.2 K
($-$0.006), and $-$0.5 K ($-$0.004), respectively, see Fig. 5. The
conventional ECE is maximal at the first (PE-FE) transition, even if rescaled
with temperature. Note that the MD simulated transition temperatures are
systematically underestimated, and the conventional ECE at the second
transition under a field along the $[110]$ direction is supposed to take place
near room temperature as reported in experimental work Li  _et al._ (2021).

For all other temperatures, the conventional response is reduced to $|\Delta
T_{{ad.}}/{T_{in}}|\leq$ 0.005 above the PE-FE transition, and $\leq$ 0.002
near the FE-FE transitions, see yellow dots in Fig. 3 (a). Furthermore, at the
FE-FE transitions, field removal can induce an inverse ECE (blue dots) between
$T_{C}^{\mathrm{FC}}$ and $T_{C}^{\mathrm{ZFC}}$ for pre-cooled samples, or
between $T_{C}^{\mathrm{FH}}$ and $T_{C}^{\mathrm{ZFC}}$ for pre-heated
samples. For the chosen field strength, the maximal inverse ECE ($\Delta
T_{{ad.}}/T_{in}$) of $+$1.2 K ($+$0.009) and $+0.5$ K ($+$0.007) appear at
the second and third transitions, respectively, see Fig. 5 (b).

Figure 5: Temperature dependence of the ECE for representative field
directions at (a) the first and (b) second phase transitions. The color code
of directions is consistent with Fig. 1. The triangles mark the
$T_{C}^{\mathrm{FC}}$ under fields along the corresponding directions. Gray
regions are the ZF coexistence ranges. The small conventional ECE between
220-285 K is not shown and the small ECE at the third transition is shown in
Fig. 7.

As shown in Fig. 5, at each boundary to an FE phase, the ECE peaks show sharp
shoulders as the response within one phase is one order of magnitude smaller.
Only at the high-temperature side of the ECE peaks at the first transition
(where it is in PE phase in ZF) does the field-induced entropy, and thus the
ECE, change continuously with temperature, see Fig. 5 (a). While ECEs are not
too sensitive to temperature at the FE-FE transitions, they are at the PE-FE
transition: The caloric responses are maximal at the lowest temperature where
the system still undergoes a phase transition, i.e., at 310 K, consistent with
simulation work Li  _et al._ (2020); Marathe  _et al._ (2017); and decrease
continuously with increasing temperature. Only for the $[111]$ direction can
one distinguish between the continuous entropy change at high temperatures
(maximally 2 K) and the ECE peak related to the first-order phase transition
below T${}_{C}^{\mathrm{FC}}$ of 3 K.

For the PE-FE transition, the maximal ECE is the largest under fields along
the $[100]$ direction. Any rotation of the field causes a gradual reduction in
the field coupling, $T_{C}^{\mathrm{FC}}$, and also $\Delta T_{ad.}$. At the
FE-FE transitions the direction dependence of the maximal magnitude of ECE is
small. In our data, the largest direction dependence is found for the inverse
ECE at the third transition, which decreases by about 10% if the field rotates
from $[100]$ to $[110]$, cf. Fig. 7. This can be understood by a small
conventional ECE induced in the initial phase. The total contribution of this
conventional ECE increases with increasing field variation interval. For
example (cf. Fig. 8), during field removal, the system for field direction
$[100]$ already transits (MC$\leftrightarrow$MA) at a field strength of around
40 kV/cm; while for the other two directions on the $(001)$ plane, the
transitions (MC$\leftrightarrow$Tri) take place at a smaller field strength
(around 10 kV/cm), resulting in a larger temperature decrease.

Most importantly, the discussed direction dependence of
T${}_{C}^{\mathrm{FC}}$ at the FE-FE transitions results in large changes in
ECE. On one hand, fields along $[110]$ yield the largest temperature window
for the conventional ECE at the second transition and this window is robust to
a field rotation towards $[111]$; and an inverse response at the third
transition is possible for all field directions on the $(001)$ plane. On the
other hand, these responses vanish for small rotations toward other
directions. The strongest dependence on the field direction occurs for the
inverse ECE at the second transition for the field along $[100]$: A field
rotation of merely 8∘ can turn the inverse response into a small conventional
one.

What change in caloric responses can be expected for misaligned fields or
(textured) ceramics? At the first transition or at the second transition under
fields close to $[110]$, the change in the conventional ECE with field
rotation is moderate. Thus, no large changes are expected for misaligned
fields or in $\\{\bar{1}10\\}$-textured ceramics. Analogously, an inverse ECE
at the third transition is likely to be found in $\\{001\\}$-textured
ceramics. On the other hand, as the ECEs are very field direction sensitive at
the second transition for the field along $[100]$ and at the third transition
for the field along $[111]$, ceramics are required to be strictly textured to
the corresponding directions to observe the caloric responses. Otherwise,
small responses are expected.

So far, we have discussed the ECE of cooled samples during field removal. A
practically useful caloric response needs to be large and reversible in a
cycling field. Reversibility in single crystals is only possible for
temperatures outside the coexistence ranges with and without the field. That
is, for the chosen field strength, a reversible, conventional ECE can be
expected at temperatures larger than 305 K and temperatures between 185-206 K
for fields on the $(\bar{1}10)$ plane as well as those between $[110]$ and
$\theta_{2}$. Note that the fields close to the direction $\theta_{2}$ yield
large conventional ECEs with almost no thermal hysteresis at the second
transition. The inverse ECE is, however, irreversible for all field directions
at our applied field strength. Once the field is strong enough to drive
T${}_{C}^{\mathrm{FH}}$ lower than T${}_{C}^{\mathrm{ZFC}}$, reversible,
inverse ECEs are expectedNishimatsu  _et al._ (2013); Marathe  _et al._
(2017), e.g., at the third transition under fields on the $(001)$ plane or
between $[110]$ and $\theta_{i}$.

##  IV Summary and Conclusions

Using coarse-grained molecular dynamics, we study the field direction
dependence of ferroelectric transitions for two perovskite oxides – BaTiO3 and
PbTiO3. While BaTiO3 is susceptible to polarization rotation, non-collinear
fields induce multidomain structures in PbTiO3. Surprisingly, for both
materials, the changes in the transition temperatures with field direction can
be predicted based on their spontaneous polarizations. This finding can be
used for future screening and optimization of transition temperatures.

We find that the transition temperatures are strongly influenced by the field
direction. For example, the transition temperatures at the first (PE-FE)
transition for fields close to $[100]$ are higher than those for fields close
to $[111]$. Corresponding differences in transition temperatures are,
therefore, expected for differently textured ceramics. This could explain the
broad range of their experimentally reported values for (un)textured ceramics.
At the FE-FE transitions, the transition temperature of single crystals and
textured ceramics can even be above and below the zero-field coexistence
range, depending on field directions.

Field direction also impacts thermal hysteresis and is a promising path to
bypass thermal hysteresis without increasing the strength of the external
field. For the chosen field strength at the PE-FE phase transition, only for
the field along $[111]$ does a first-order phase transition with thermal
hysteresis remain. For the FE-FE transitions, the maximal hysteresis is found
under fields along the high symmetric directions, particularly along $[100]$.
On the other hand, hysteresis is reduced in low symmetric field directions,
especially in those where isostructural transitions are induced.

Most importantly, the direction of the applied field influences the sign and
magnitude of the ECE and the temperature windows with large caloric responses.
Generally, large ECEs ask for phase transitions. In ideal single crystals,
there is, thus, an abrupt drop of ECE at the boundary of FE phases, which
needs to be considered in application. At the PE-FE transition, fields close
to $[100]$ are favorable, as the temperature window and magnitude of the
conventional ECE are the largest. The same direction is also exceptional at
the second transition: it induces an inverse ECE. However, this is related to
the maximal thermal hysteresis and the inverse response vanishes for a small
field rotation. A more stable inverse ECE is possible at the third transition
for fields on the $(001)$ plane. However, a stronger field is needed to make
this response reversible. In addition, a large temperature window with a
reversible conventional response exists for fields on the whole $(\bar{1}10)$
plane and for lower symmetric directions. What’s more, this particular
response is close to room temperature and is thus promising for application.

The knowledge of field direction dependence of ECE is valuable for ceramics,
as the responses there normally suffer from partial cancellation and averaging
out. We expect to see a large conventional ECE at the PE-FE transition in
untextured ceramics, at the second transition in $\\{\bar{1}10\\}$-textured
ceramics, and at the third transition in strictly $\langle
111\rangle$-textured ones. Inverse ECEs are expected at the second transition
in strictly $\langle 100\rangle$-textured ceramics and at the third transition
in $\\{001\\}$-textured ones. Particularly, the conventional ECE at the second
transition of the $\\{\bar{1}10\\}$-textured ceramics is expected to take
place close to room temperature.

###### Acknowledgements.

All authors acknowledge support from the Deutsche Forschungsgemeinschaft
(DFG), Germany. F.W. is grateful for the financial support under grant
GRK2495/K, L.H. and A.G. under grant GR 4792/3.

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##  V Appendix

In this appendix, we provide information on the field direction dependence
under a weaker field (Fig. 6), on the field direction dependence of the ECE at
the third transition (Fig. 7), on the time evolution of temperature changes
during field removal (Fig. 8), and on the local dipole distribution for BaTiO3
and PbTiO3 (Fig. 9).

Figure 6: Temperature field direction phase diagram of BaTiO3 in 10 kV/cm.
Thick blue and red curves mark the phase transition temperatures for FC and
FH, and horizontal lines give ZFC and ZFH values as reference. The change in
$T_{C}$ and the reduction in thermal hysteresis are not as large as in field
strength 100 kV/cm. For example, at the third transition for field directions
between $[100]$ and $[110]$, the change in $T_{C}^{\mathrm{FC}}$ is below the
temperature resolution of 5 K. Note that the discontinuity of $T_{C}$, e.g.,
at the second transition along field direction $[111]$, is an artifact of the
5 K temperature resolution.  Figure 7: Temperature dependence of the ECE for
representative field directions at the third phase transition. The color code
of directions is consistent with Fig. 1. The triangles mark the
$T_{C}^{\mathrm{FC}}$ under fields along the corresponding directions. Gray
regions are the coexistence ranges at T$\leftrightarrow$O and
O$\leftrightarrow$R transitions in ZF simulations. Figure 8: Time evolution of
temperature changes relative to the initial temperature of about 70 K, i.e.,
at the third transition under field removal using a moving average over 100
data points. Between 120 and 220 ps, the field is slowly ramped down from 100
to 0 kV/cm. Colors encode angles of the applied fields away from $[100]$ on
the $(001)$ plane. During the field removal, the temperature, first, decreases
due to the conventional ECE before the transitions (e.g., 120 to 200 ps for
30∘) and then increases due to the inverse ECE at the transitions. The larger
the critical field strength for transition, the smaller the conventional
contribution, the inverse ECE at the transition depends on the initial and
final state.  Figure 9: Polarization distribution of BaTiO3 (at 300 K) and
PbTiO3 (at 640 K) below T${}_{C}^{\mathrm{FC}}$ on the $P_{x}-P_{y}$ plane for
field directions on the $(001)$ plane. The macroscopic polarizations (i.e.,
the average over the whole system) are annotated in units of
$\mu\text{C/cm}^{2}$. BaTiO3 remains single-domain for all directions. For the
$(100)$ direction, $P_{x}$ fluctuates between $-20$ and
$60~{}\mu\text{C/cm}^{2}$, and $P_{y}$ and $P_{z}$ fluctuate between $\pm
40\mu\text{C/cm}^{2}$. When the field rotates away from $[100]$ to $[110]$,
the field-induced polarization in $P_{y}$ increases. As the total polarization
is the sum of the spontaneous polarization (along $[100]$) and the field-
induced polarization, the polarization peak does not coincide exactly with the
applied field direction. PbTiO3 remains single-domain for fields deviating
within 30∘ away from $[100]$ and the field-induced polarization is smaller
compared to BaTiO3 due to a stronger polarization-strain couplingNishimatsu
_et al._ (2012), and the width of the peaks is larger due to the higher
temperature. For $[100]$ direction, $P_{x}$ fluctuates between $-$30 and 100
$\mu\text{C/cm}^{2}$, and $P_{y}$ and $P_{z}$ fluctuate between $\pm-75$
$\mu\text{C/cm}^{2}$. For the field direction $[110]$, PbTiO3 decomposes into
a multidomain state whose macroscopic symmetry is MC.