## Abstract

Solid-state electrothermal energy interconversion utilising the electrocaloric
effect is currently being considered as a viable source of applications
alternative to contemporary cooling and heating technologies. Electrocaloric
performance of a dielectric system is critically dependent on the number of
uncorrelated polar states, or ‘entropy channels’ present
within the system phase space. Exact physical origins of these states are
currently unclear and practical methodologies for controlling their number and
creating additional ones are not firmly established. Here we employ a multiscale
computational approach to investigate the electrocaloric response of an
artificial layered-oxide material that exhibits Goldstone-like polar
excitations. We demonstrate that in the low-electric-field poling regime, the
number of independent polar states in this system is proportional to the number
of grown layers, and that the resulting electrocaloric properties are tuneable
in the whole range of temperatures below
*T*_{C} by application of
electric fields and elastic strain.

## Introduction

Electrocaloric (EC) effect is defined as the variation of the dielectric material
entropy as a function of the electric field at a given temperature, which results in
an adiabatic temperature change. Recently, there has been significant
progress^{1,2,3} in the development of polar (i.e., possessing
spontaneous polarisation) dielectrics that display large EC temperature shifts
Δ*T* under electric-field poling. These systems
include a variety of ferroelectric ceramics,^{4,5,6} polymers^{7,8,9,10} and liquid
crystals.^{11} A few relaxor
and antiferroelectric materials may exhibit negative EC effect, which is especially
advantageous for solid-state refrigeration.^{12} The best values of positive EC
Δ*T* in modern nanoengineered materials range from 20
to 45 K, for electric-field sweeps
Δ*E*⩾500 kV/cm,^{1,2,3,4,5,6,13,14} whereas negative EC Δ*T* remain
below −10 K for much smaller
Δ*E*.^{15,16,17,18}

The magnitude of the EC Δ*T* is proportional to a
logarithm of the number of possible polar states, or independent
‘entropy channels’ in the system.^{3,19} Therefore,
it is highly desirable to acquire precise understanding of physical phenomena
underpinning the emergence of these states and develop practical methods for
engineering additional entropy channels into dielectrics. Moreover, since entropic
changes involving evolution of other ferroic order parameters can be cumulative with
the EC effect, more advanced multicaloric materials concepts blending polar,
magnetic and elastic energy-interconversion functionalities, are also being
considered.^{20,21,22,23,24}

Here we use a multiscale computational approach that combines *ab
initio* quantum mechanical simulations, phenomenological Landau theory
and thermodynamical evaluations to investigate the EC response of a
‘material template’ based on a quasi-two-dimensional system
that exhibits polar Goldstone-like^{25,26,27} excitations. This template system is an
‘*n*=2’ Ruddlesden-Popper (RP)
type^{28}
PbSr_{2}Ti_{2}O_{7} (PSTO) layered-oxide superlattice
shown in Figure 1a. Such or similar materials
have been successfully grown, e.g., by utilising molecular-beam epitaxy synthetic
techniques.^{29,30,31}

The most interesting predicted feature of this system is that under biaxial
basal-plane strains ranging from slight tensions (up to 0.2%) to modest compressions
(up to 2%) it loses crystalline anisotropy with respect to the direction of its
spontaneous polarisation
**P**=(*P*_{x},
*P*_{y}),^{32} whereas rocksalt-type SrO–SrO inserts
in between perovskite slabs prevent any polarisation in the out-of-plane direction
*z*. The system free-energy density landscape
*f*(*P*_{x},
*P*_{y}) is shown in Figure 1b and has a distinctive sombrero-hat
shape characteristic of a presence of a Goldstone-like or phason
excitation.^{25,26,27} In the case of PSTO, that excitation
manifests itself as an easy—i.e., requiring almost no consumption of
energy—rotation of **P** along the minimum-energy path within
the basal plane, whereas its amplitude is kept constant.

This path, shown in greater detail in Figure 1c
for zero biaxial strain, is only approximately circular—hence the
‘-like’ suffix above, as for the true Goldstone excitation
it has to be a perfect circle. Here the absolute energy minima (at
−6.26 meV per structural unit, or s.u.) are located along
the [100], *θ*=0°, and three symmetrically
equivalent crystallographic directions, and separated by saddle points (at
−6.01 meV/s.u.) located along [110],
*θ*=45°, and symmetrically equivalent
directions. The competition between the [100]- and [110]-oriented sets of energy
extrema can be controlled by applied strain; e.g., a small amount of tension can
shift the minima to the [110] set of directions and saddle points to the [100]
set.

## Results

We have developed Landau-type polynomial expansions for *f* from
*ab initio* calculations,^{32} including couplings to elastic strain and applied electric
field **E**. Since RP structure does not support out-of-plane polar
distortions,^{32,33} only the in-plane polarisation
components were included. Also, only planar biaxial strain–tensor
components ${\mathit{\epsilon}}_{xx}={\mathit{\epsilon}}_{yy}\equiv \mathit{\epsilon}$,
accounting for the epitaxial misfit strain on a square substrate, were considered in
the expansion. Both linear and quadratic couplings involving
*ε* were included to reproduce polarisation
locking–unlocking phase transition under changing strain. With all the
simplifications, this quasi-two-dimensional free-energy function is $$\begin{array}{}\text{(1)}& f={\mathit{\alpha}}_{1}(T-{T}_{C})\left({P}_{x}^{2}+{P}_{y}^{2}\right)+{\mathit{\alpha}}_{2}\left({P}_{x}^{4}+{P}_{y}^{4}\right)+{\mathit{\alpha}}_{3}{P}_{x}^{2}{P}_{y}^{2}+{\mathit{\alpha}}_{4}\left({P}_{x}^{6}+{P}_{y}^{6}\right)+{\mathit{\alpha}}_{5}\left({P}_{x}^{4}{P}_{y}^{2}+{P}_{x}^{2}{P}_{y}^{4}\right)+\left({\mathit{\chi}}_{1}\left({P}_{x}^{2}+{P}_{y}^{2}\right)+{\mathit{\chi}}_{2}\left({P}_{x}^{4}+{P}_{y}^{4}\right)+{\mathit{\chi}}_{3}{P}_{x}^{2}{P}_{y}^{2}\right)\epsilon +\left({\mathit{\chi}}_{4}\left({P}_{x}^{2}+{P}_{y}^{2}\right)+{\mathit{\chi}}_{5}\left({P}_{x}^{4}+{P}_{y}^{4}\right)+{\mathit{\chi}}_{6}{P}_{x}^{2}{P}_{y}^{2}\right){\epsilon}^{2}-{P}_{x}{E}_{x}-{P}_{y}{E}_{y}.\end{array}$$

The estimated system
*T*_{C}≃120 K, while
the rest of the expression coefficients are given in Supplementary Information.

The equilibrium state of the system at a given temperature *T* and
strain *ε* is determined by minimising *f*
with respect to the polarisation vector components
*P*_{x},
*P*_{y}. System excess heat
capacity^{34}
Δ*C* and pyroelectric coefficients
${p}_{\mathit{\gamma}}$,
$\mathit{\gamma}=x,y$,
can be obtained from the equilibrium values of polarisation
(**P**^{0}) and energy density
*f*^{0}≡*f*(**P**^{0})
as $$\begin{array}{}\text{(2)}& \Delta C=-T{\left({\partial}^{2}{f}^{0}/\partial {T}^{2}\right)}_{\mathbf{E},\mathit{\epsilon}},\end{array}$$
$$\begin{array}{}\text{(3)}& {p}_{\mathit{\gamma}}={\left(\partial S/\partial {E}_{\mathit{\gamma}}\right)}_{\mathit{\epsilon},T}={\left(\partial {P}_{\mathit{\gamma}}^{0}/\partial T\right)}_{\mathit{\epsilon},\mathbf{E}}.\end{array}$$

Then the adiabatic change in the system temperature Δ*T*
under the influence of varying applied electric field **E** is^{35}
$$\begin{array}{}\text{(4)}& \Delta T=-T\sum _{\mathit{\gamma}=x,y}{\int}_{{E}_{\mathit{\gamma},a}}^{{E}_{\mathit{\gamma}\mathit{,}b}}d{E}_{\mathit{\gamma}}{\left[\frac{{p}_{\mathit{\gamma}}}{C+\Delta C}\phantom{\rule{.03em}{0ex}}\right]}_{\mathit{\epsilon},\mathbf{E},T},\phantom{\rule{.03em}{0ex}}\end{array}$$
where
${E}_{\mathit{\gamma},b}-{E}_{\mathit{\gamma},a}\equiv \Delta {E}_{\mathit{\gamma}}$,
${E}_{\mathit{\gamma},a}\u2a7d{E}_{\mathit{\gamma},b}$,
are changes in the *x* and *y* components of
**E** during the poling procedure, and
*C*(*T*) is the system heat capacity. Equations (1)–(4) were then
used to study polarisation switching and the resulting temperature change
Δ*T* in PSTO under applied field **E**. More
details on the evaluation of both heat capacity terms and their influence on the
magnitude of Δ*T* are provided in the Materials and
Methods section below and in Supplementary Information.

Remarkably different types of the system polarisation behaviour are observed under
conditions of either high-, or low-field poling. In the former case, i.e., for
$\Delta {E}_{\mathit{\gamma}}\u2a7e40-50\phantom{\rule{.25em}{0ex}}\mathrm{kV}/\mathrm{cm}$ (depending on the value of
*ε*), energy landscape shown in Figure 1 is deformed away from its original sombrero-hat shape.
Application of a large electric field
(*E*_{x,a},
*E*_{y,b}) creates a deep energy
minimum along its direction, locking the polar vector **P**^{0}
inside this minimum and destroying the Goldstone-like excitation. Poling field
(Δ*E*_{x},
Δ*E*_{y}) then preserves the
locked state of the polarisation. If the direction of **E** remains fixed
during the poling, polarisation azimuthal angle *θ* stays
constant, while only the amplitude $\left|{\mathbf{P}}^{0}\right|$
changes value. Such amplitudon mode of polarisation switching is usually seen in
conventional EC materials,^{5,6,34,35} where the largest
variations in $\left|{\mathbf{P}}^{0}\right|$ and
thus the largest changes in
${p}_{\mathit{\gamma}}$
and Δ*T* occur near
*T*_{C}.

In Figure 2a, we present the
$\left|{\mathbf{P}}^{0}\right|$
versus *T* dependence at different values of
*ε* (see Supplementary Figure 1 for more details). This includes
$\mathit{\epsilon}=\mathrm{1\%}$,
where even at zero field the Goldstone-like excitation disappears in favor of a
localised minimum along [110]. These results display transitional behaviour typical
for a generic perovskite ferroelectric around
*T*_{C}.^{36} Curves for a number of different applied fields
*E*_{x,a}=*E*_{y,a}
are also shown for $\mathit{\epsilon}=\mathrm{1\%}$
and, as expected, show a persistence of the polar phase beyond the zero-field
*T*_{C}. For the poling field
$\Delta {E}_{\mathit{\gamma}}\sim 200\phantom{\rule{0.5mm}{0ex}}\mathrm{kV/cm}$, we
obtain the values of Δ*T* in the range of
1–2 K (Supplementary Figure 2), which is similar to the performance of
conventional EC materials undergoing amplitudon-polarisation switching.^{5,6,34,35}

In the case of low-field poling, i.e., for
${E}_{\mathit{\gamma},a}\sim 1-5\phantom{\rule{0.5mm}{0ex}}\mathrm{kV/cm}$ and
$\Delta {E}_{\mathit{\gamma}}\le 10\u201330\phantom{\rule{.25em}{0ex}}\mathrm{kV}/\mathrm{cm}$, the *sombrero-hat*
energy landscape is only slightly perturbed. If the
$\mathbf{P}\left|\right|$
[100] state of Figures 1b and c is taken as a
starting point, applying noncollinear **E** induces a
*rotation* of the polar vector along the minimum-energy
*groove* until $\mathbf{P}\left|\right|\mathbf{E}$. During this
rotation process, which we refer to as *phason* polarisation
switching, its azimuthal angle *θ* changes while its
amplitude $\left|{\mathbf{P}}^{0}\right|$
remains approximately constant.

The *θ*(*T*) dependence for a number of
different magnitudes of **E** is shown in Figure 2b and Figure 2c for
symmetric,
*E*_{x,a}≡*E*_{y,a},
and asymmetric,
*E*_{x,a}≠*E*_{y,a},
*static* applied fields, respectively. Here, the polarisation is
initially pointing towards the nearest energy minimum, so that, e.g.,
*θ*=0°, while the
$\widehat{\mathbf{P},\mathbf{E}}$
angle is <45°. In the both cases, it is observed that
**P** can align with **E** only if sufficient energy is
provided to the system in the form of heat. The temperature value at which the
alignment happens (*T*_{lock}) can be adjusted by changing
the magnitude of **E** throughout the whole temperature interval
(0→*T*_{C}), similarly to
how the value of *T*_{C} can be attuned by
the applied strain *ε* during the amplitudon switching.
However, for asymmetrically applied fields, the alignment of **P** with
**E** always happens smoothly (see Figure
2c), not resulting in an emergence of large
${p}_{\mathit{\gamma}}$
in the vicinity of *T*_{lock}. On the other hand, for
symmetrically applied fields, polarisation alignment occurs abruptly (see Figure 2b; Supplementary Figure 3) and
produces large pyroelectric response, which is strikingly similar to the behaviour
of $\left|{\mathbf{P}}^{0}\right|$ near
*T*_{C} during the high-field poling.
The application of sufficiently large **E**—which is dependent
on *T*—along [110] lowers the energy of the associated
saddle point until it becomes the new energy minimum, while simultaneously raising
the energy of the original minima along [100] and [010]. This creates a strong bias
for locking **P** along the saddle-point direction, while such an incentive
would be missing in the case of asymmetrically applied fields.

Since passage of the applied field through the saddle-point direction is accompanied
by an emergence of large pyroelectric coefficients, which should in turn result in
large EC Δ*T*, the following simple poling scenario can
be considered. Starting with the polarisation along the *x*
direction, *θ*=0°, a *static*
field
${E}_{x}\equiv \tilde{E}$
is applied, i.e., Δ*E*_{x}=0
which eliminates one of the two integrals in Equation (4). Then, the field along the *y* direction is
changed from
${E}_{y,a}<\tilde{E}$
to
${E}_{y,b}>\tilde{E}$.
This situation is illustrated step by step in Figure
3, which suggests that even for modest poling-field ramps
Δ*E*_{y}, bracketing the
value of $\tilde{E}$,
it is possible to trigger rotation of **P** by an angle close to
90°, e.g., from the [100] energy minimum, through the saddle point along
[110] and into the [010] energy minimum. The
*θ*(*T*) plot in Figure 3b shows that
${E}_{y,b}<\tilde{E}$
initiates a smooth alignment of **P** with **E**, while only
${E}_{y,b}>\tilde{E}$
results in an abrupt switching of **P** between the two neighbouring basins
within the energy landscape, after which it again smoothly aligns with
**E** as *T* is raised above
*T*_{lock}.

Figure 3c shows that abrupt changes in
*θ* translate into large negative values of EC
Δ*T*. As illustrated by the two different switching
cases, the position of the phason EC peak on the temperature axis, as well as its
width, can both be controlled purely by means of applied electric
field—specifically by setting the values of
$\tilde{E}$
and Δ*E*_{y}, respectively. For
example, changing $\tilde{E}$
from 5 to 13 kV/cm moves the center of the phason EC peak down from 60
to 30 K. Such precise tuning of the shape and location of the
phason-induced EC response can be accomplished for all temperatures below
*T*_{C}. Typical maximum entropy changes
achieved in this poling scenario are 0.5–1 J/kg/K (Supplementary Figure 4),
similar to values observed by others for low-field switching.^{37}

The negative sign of the phason-switching induced Δ*T*
originates from the following consideration: as
*T*→*T*_{lock} and
polarisation rotation occurs, the $\widehat{\mathbf{P},\mathbf{E}}$
angle is always diminished. Naturally, polarisation component
${P}_{\mathit{\gamma},b}$
after the rotation is always larger than component
${P}_{\mathit{\gamma},a}$
before the rotation, where *γ* is the direction along the
field. Thus, the associated
${p}_{\mathit{\gamma}}\sim ({P}_{\mathit{\gamma},b}-{P}_{\mathit{\gamma},a})/({T}_{\mathrm{lock}}-T)$ is positive,
which results in negative Δ*T* according to Equation (4).

As shown in Figure 3c, abrupt changes of
*θ*, each occurring under a specific fixed value of
${E}_{y}>\tilde{E}$,
result in an emergence of sharp bumps in the EC Δ*T*
response. Furthermore, by choosing the values of the stationary
$\tilde{E}$
field and the bracketing
Δ*E*_{y} poling interval,
these peaks can be shifted to lower or higher temperatures on demand. When during
the poling procedure *E*_{y} is swept
continuously from
${E}_{y,a}<\tilde{E}$
to
${E}_{y,b}>\tilde{E}$,
individual peaks in the Δ*T* curve merge into an envelope
that is presented in Figure 4 for different
values of *ε* (see also Supplementary Figure 5).
Remarkably, the variation of *ε* by ⩽1% can
result in the system EC response changing from cooling (phason contribution) to
heating (amplitudon contribution) at the same operating temperature.

## Discussion

The EC temperature changes in PSTO in the low-field poling regime range from
−50 to −100 mK, i.e., they are two orders of
magnitude smaller compared to the (positive) Δ*T* values
produced during the conventional, high-field poling. However, since, without the
loss of generality, the quasi-two-dimensional form of *f* can
describe the behaviour of a single slab whose polarisation is decoupled from those
of its neighbours, cumulative EC Δ*T* in a multi-slab
system should look very different for the high- and low-field poling cases. It is
noteworthy that the approximation of electrically decoupled polar slabs is already
quite good for bulk PSTO, as illustrated by an absence of phonon-band dispersion for
polar modes in the direction perpendicular to the slab planes; see Figure 2 in reference 32.

In the high-field regime, individual slab polarisations are switched ‘all
at once,’ being forcefully correlated by the applied field. That is, the
multi-slab system possesses only one entropy channel and the value of
Δ*T*≃1–2 K quoted
above for PSTO should represent its aggregate EC response. In the low-field regime,
applied fields are insufficient to correlate the directions of (disordered) polar
vectors in different slabs and, therefore, polarisation rotations under the cycling
of the field should occur independently in each slab. In such a case, each slab acts
as a separate entropy channel and the aggregate EC response of the whole system
should be proportional to the logarithm of the number of slabs. Then, even with
individual slab contributions being low (~100 mK for PSTO),
a system comprised of a large number of slabs should possess an aggregate
Δ*T* that is at least comparable with other
state-of-the-art *negative* EC materials.^{15,16,17,18}

The estimate of one entropy channel per slab is conservative, as in the model
presented here any influence of polar domain-wall dynamics on the EC response is not
taken into account, i.e., each slab is considered to be in a monodomain state. An
investigation of polarisation-closure patterns in PSTO nanostructures predicted that
these patterns are likely to form and their behaviour sensitively depends on the
applied strain,^{38} suggesting that
multiple entropy channels may be created (or destroyed, if needed) in each slab by
distorting its shape. Alternatively, a nano-island device geometry, where polar
states of individual islands uncorrelated from each other, can be used instead of a
continuous-film one.

In summary, we have shown that layered polar systems with Goldstone-like
instabilities should exhibit attractive EC properties that are highly tunable by
applying electric fields and elastic distortions over a wide range of operating
temperatures below *T*_{C}, which may even
include on-demand switching between cooling and heating. By virtue of operating at
low electric fields, device applications of actual materials will require modest
power consumption, compared with most other known EC compounds, where fields of
upwards of 500 kV/cm are needed for best performance. Furthermore, in
such model systems the mechanisms behind the emergence of independent entropy
channels can be clearly established allowing for an easy up- or down-scaling of the
system entropy-flow and EC characteristics by growing an appropriate number of
layers. Although for the specific example of PSTO, a single monodomain-slab EC
response in the low-field operation is only around 0.1 K, it should be
possible to employ ‘materials by design’ principles to
develop new compounds with improved EC properties. The competition between the
[100]- and [110]-oriented sets of energy minima in polar-perovskite layers, leading
to Goldstone-like excitations, appears to be a generic geometrical feature of such
systems, rather than a peculiar trait distinctive to PSTO. We have recently
identified other layered-oxide structures that should possess similar
properties.^{39}

## Materials and methods

*Ab initio* calculations were performed with the density-functional
theory code Quantum Espresso^{40}
utilising local-density approximation in the Perdew–Zunger
parametrisation^{41} and
Vanderbilt ultrasoft pseudopotentials.^{42} Vibrational frequencies, ionic displacement patterns and system
vibrational density of states, were obtained using density-functional perturbation
theory approach within Quantum Espresso.^{43} The system total heat capacity
*C*(*T*), used in Equation (4) above, was evaluated from the vibrational density
of states^{44,45} computed for the non-polar system configuration
(see Supplementary
Information and Supplementary Figure 6 for more details).

Ionic forces were relaxed to less than
0.2×10^{−5} Ry/bohr
(~0.5×10^{−4} eV/Å)
and the appropriate stress tensor components
${\sigma}_{\mathit{\alpha}\mathit{\beta}}$
(*α* and *β* are Cartesian
directions *x*, *y* and *z*) were
converged to values below 0.2 kbar. Epitaxial thin-film constraint on a
cubic (001)-oriented substrate was simulated by varying the in-plane lattice
constant *a* of the tetragonal cell and allowing the out-of-plane
lattice constant *c* to relax (by converging the normal stress in
this direction to a small value), while preserving the imposed tetragonal unit-cell
symmetry. The biaxial misfit strain *ε* is defined as
(*a−a*_{0})/*a*_{0},
where *a*_{0} is the in-plane lattice parameter of the free
standing PSTO structure with all the normal stress tensor components relaxed to
<0.2 kbar.

Ionic Born effective-charge tensors
${Z}_{i,\mathit{\alpha}\mathit{\beta}}^{*}$,
where *i* is the ion number, and high-frequency dielectric
permittivity tensor
${\epsilon}_{\mathit{\alpha}\mathit{\beta}}^{\infty}$
were calculated by utilising the density-functional perturbation theory approach.
The system polarisation was evaluated with a linearised approximation^{46} involving products of
${Z}_{i,\mathit{\alpha}\mathit{\beta}}^{*}$
and ionic displacements away from centrosymmetric positions. The value of
*T*_{C} (~120 K)
was estimated from energy differences between the non-polar
*I*4/*mmm* and polarised
$\mathbf{P}\left|\right|$
[100], and $\mathbf{P}\left|\right|$
[110] states of the system.

We should point out that due to the purely analytical nature of our approach it does
not account for spatial inhomogeneities of the polarisation **P** and
elastic-field ${\mathit{\epsilon}}_{\mathit{\alpha}\mathit{\beta}}$
variables. Such inhomogeneities can be present in epitaxially clamped thin films
producing additional contributions to the aggregate Δ*T*
that stem from, e.g., elastocaloric effects and polydomain behaviour. Although we
have not yet studied piezoelectric response of PSTO in detail, we expect most of its
piezoelectric coefficients to be low—with the same being true about the
magnitude of the intrinsic elastocaloric effect—as polarisation rotation
in this material is not accompanied by large elastic distortions. Instead of
utilising the Maxwell relation (3), a number of approaches involving combinations of
effective-Hamiltonian techniques with molecular dynamics or Monte Carlo simulations
have been used to directly compute field-induced temperature and entropy changes in
ferroelectrics.^{47,48,49} (Such techniques can indeed handle polar and
elastic spatial inhomogeneities when appropriately parameterised.) Although with no
first-order phase transitions present both types of approaches should produce the
same results,^{49} the accuracy of
the type adopted here may depend on precision of numerical integration of the
Maxwell relations.^{47} We have
investigated convergence of the integral in Equation (4) for the poling schemes described above, with variations
⩽15–20% found for the resulting
Δ*T*, as long as step of numerical sweeping of the
poling field
$\Delta {E}_{\mathit{\gamma}}$
was not too small (typical step values ranged from 0.5 to
0.05 kV/cm).

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## Acknowledgements

S.N., K.C.P. and J.M. are grateful to the National Science Foundation (DMR 1309114) for partial funding of this project. S.N. also thanks Olle Heinonen and Joseph Mantese for illuminating discussions.

## Author information

## Affiliations

### Department of Physics, University of Connecticut, Storrs, CT, USA

- John Mangeri
- , S Pamir Alpay
- & Serge Nakhmanson

### Department of Materials Science & Engineering and Institute of Materials Science, University of Connecticut, Storrs, CT, USA

- Krishna C Pitike
- , S Pamir Alpay
- & Serge Nakhmanson

## Authors

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### Contributions

S.N. and S.P.A. developed the multiscale theoretical framework and supervised the project. S.N. and K.C.P. performed all the DFT simulations. J.M. fitted the Landau-type energy expressions from the DFT results and conducted thermodynamical modeling. J.M., S.N. and S.P.A. co-wrote the manuscript.

### Competing interests

The authors declare no conflict of interest.

## Corresponding author

Correspondence to Serge Nakhmanson.

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