# Electrocaloric and barocaloric effects in CsH2PO4 ferroelectric

A. S. Vdovych  $|$ ${}^{\textsf{\footnotesize{\mbox{?}}}}$, R. R.
Levitskii${}^{\textsf{\footnotesize{\mbox{?}}}}$, I. R.
Zachek${}^{\textsf{\footnotesize{\mbox{?}}}}$ Corresponding author:
<EMAIL_ADDRESS>
(Received June 30, 2022, in final form August 23, 2022)

###### Abstract

To investigate the caloric effects in the CsH2PO4 ferroelectric, a modified
pseudospin model of this crystal is used, which takes into account the
dependence of the parameters of interaction between pseudospins on lattice
strains. The model also takes into account the dependence of the effective
dipole moment of a pseudospin on the order parameter. In the two-particle
cluster approximation, the influence of the longitudinal electric field and
hydrostatic pressure on the molar entropy of the crystal was studied. The
electrocaloric and barocaloric effects were studied. The calculated
electrocaloric temperature change is about $1$ K; it can change its sign under
the influence of hydrostatic pressure. Barocaloric temperature change is about
$-0.5$ K; lattice anharmonicities were not taken into account in its
calculations.

Key words: ferroelectricity, phase transitions, dielectric permittivity,
mechanical deformation, hydrostatic pressure effect

###### Abstract

Äëÿ äîñëäæåííÿ êàëîðèíèõ åôåêòâ ó ñåãíåòîåëåêòðèêó CsH2PO4 âèêîðèñòàíî
ìîäèôêîâàíó ïñåâäîñïíîâó ìîäåëü öüîãî êðèñòàëà, ÿêà âðàõîâó çàëåæíñòü
ïàðàìåòðâ âçàìîä¿ ìæ ïñåâäîñïíàìè âä äåôîðìàöé ãðàòêè. Ìîäåëü òàêîæ âðàõîâó
çàëåæíñòü åôåêòèâíîãî äèïîëüíîãî ìîìåíòà íà âîäíåâîìó çâ’ÿçêó âä ïàðàìåòðà
âïîðÿäêóâàííÿ. Â íàáëèæåíí äâîàñòèíêîâîãî êëàñòåðà âèâåíî âïëèâ ïîçäîâæíüîãî
åëåêòðèíîãî ïîëÿ ãäðîñòàòèíîãî òèñêó íà ìîëÿðíó åíòðîïþ êðèñòàëà. Äîñëäæåíî
åëåêòðîêàëîðèíèé áàðîêàëîðèíèé åôåêòè. Ðîçðàõîâàíà åëåêòðîêàëîðèíà çìíà
òåìïåðàòóðè áëèçüêî 1 K; âîíà ìîæå ìíÿòè çíàê ïä äþ ãäðîñòàòèíîãî òèñêó.
Áàðîêàëîðèíà çìíà òåìïåðàòóðè áëèçüêî $-0.5$ K; ïðè ¿¿ ðîçðàõóíêàõ íå
âðàõîâóâàëèñÿ àíãàðìîíçìè ãðàòêè.

Ключов слова: ñåãíåòîåëåêòðèêè, ñåãíåòîåëåêòðèíèé ôàçîâèé ïåðåõä,
åëåêòðîêàëîðèíèé åôåêò, áàðîêàëîðèíèé åôåêò

##  1 Introduction

Currently, the greatest electrocaloric (EC) effect, as the change in
temperature of dielectric with an adiabatic change of electric field, is
observed in thin films of perovskite ferroelectrics and relaxors. In
particular, there was achieved a change in temperature $\Delta
T_{\text{ec}}=12$ K in the presence of strong electric field ($E=480$ kV/cm)
in crystal PbZr0.95Ti0.05O3 [1], 45.3 K at field strength $E=598$ kV/cm in
Pb0.8Ba0.2ZrO3 [2], $-42.5$ K at $E=1632$ kV/cm in
0.5(Ba0.8Ca0.2)TiO30.5Bi(Mg0.5Ti0.5)O3 [3], 40 K at $E=1200$ kV/cm in
Pb0.88La0.08Zr0.65Ti0.35O3 [4].

In bulk samples, the EC effect is an order of magnitude weaker due to a less
dielectric strength. In particular, there was achieved a temperature change
$\Delta T_{\text{ec}}=4.5$ K at $E=90$ kV/cm in
Pb0.88La0.12(Zr0.65Ti0.35)0.97O3 [5], 3.5 K at $E=197$ kV/cm in lead scandium
tantalate [6], 11 K at $E=29.7$ kV/cm in [(CH3)2CHCH2NH3]2PbCl4 [7].

In cheaper and more accessible KH2PO4 (KDP) type ferroelectrics with hydrogen
bonds, the EC effect has been investigated in relatively weak fields or not at
all. In particular, in the KDP crystal there was achieved $\Delta
T_{\text{ec}}\thickapprox$ 0.04 K at the field strength $E$ $\thickapprox 4$
kV/cm [8], $\Delta T_{\text{ec}}\thickapprox$ 1K at $E$ $\thickapprox 12$
kV/cm [9] and $\Delta T\thickapprox$ 0.25 K at temperature $T_{c}$ at $E$
$\thickapprox 1.2$ kV/cm [10]. Calculations carried out in [11] based on the
pseudospin model of a deformed KDP crystal show that the $\Delta
T_{\text{ec}}$ in this crystal can exceed 5 K.

Ferroelectrics, in which $\Delta T_{\text{ec}}$ is smaller than those
mentioned above, are also promising for electrocaloric cooling since, in order
to obtaine a given $\Delta T_{\text{ec}}$, electrocaloric devices can be
combined into a cascade of several links, in which the heater for the previous
link is at the same time the cooler for the next link [12].

In ferroelectric materials, the phase transition temperature depends on the
pressure. Therefore, they also exhibit a significant barocaloric (BC) effect,
which is a change in the crystal temperature during an adiabatic change in
hydrostatic pressure. The strongest BC effect was achieved in crystals with
hydrogen bonds NH4HSO4 [13] ($\Delta T_{\text{bc}}=-10$ K at pressure $p=0.15$
GPa) and (NH4)2SO4 [14] ($\Delta T_{\text{bc}}=-8$ K at pressure $p=0.1$ GPa).

Crystal CsH2PO4 (CDP) is another example of a hydrogen-bonded ferroelectric of
the KDP family. Neither EC nor BC effects in this crystal have been studied at
all. In the CDP crystal, there are two structurally non-equivalent types of
hydrogen bonds of different lengths (figure 1b). Longer bonds have one
equilibrium position for protons, while shorter bonds have two equilibrium
positions. They connect PO4 groups in chains along the $b$-axis (figure 1a);
therefore, the crystal is quasi-one-dimensional.

a b

Figure 1: (Colour online) Primitive cell of CDP crystal in the ferroelectric
phase [15].

At room temperature in the absence of pressure, the crystal is in the
paraelectric phase and has monoclinic symmetry (space group P21/m) [16, 17].
At the same time, protons on short bonds are in two equilibrium positions with
the same probability. Below $T_{c}=153$ K, the crystal passes to the
ferroelectric phase (space group P21) [18, 19] with spontaneous polarization
along the crystallographic $b$-axis, and protons with a higher probability
occupy the upper position (figure 1a). On the basis of dielectric studies [20,
21] it was established that at pressures $p=p_{c}=0.33$ GPa and
$T_{c}^{\text{cr}}=124.6$ K, double hysteresis loops appear, that is, a
transition to the antiferroelectric phase occurs. With the help of neutron
diffraction studies [22], it was established that in the antiferroelectric
phase, the unit cell of the CDP crystal doubles along the a-axis, as two
sublattices in the form of bc planes arise, which are polarized antiparallel
along $b$-axis and alternate along the a-axis. The symmetry remains monoclinic
(space group P21). Protons on hydrogen bonds are arranged in neighboring
sublattices in an antiparallel manner. At very high pressures, an
antiferroelectric phase of the second type (AF2) occurs, in which two
sublattices have the form of chains along the b-axis, and they are polarized
antiparallel along the $b$-axis and alternate in a checkerboard pattern. The
AF2 phase was predicted on the basis of NMR studies [23] and confirmed in [24]
on the basis of X-ray diffraction measurements and dielectric measurements
[25].

The effect of hydrostatic pressure on the phase transition temperature and
dielectric properties of Cs(H1-xD${}_{x})_{2}$PO4 ferroelectrics was studied
in [26, 20, 21, 27, 24, 28, 25]. The molar heat capacity of CDP was measured
in [29], and was also calculated based on the lattice dynamics simulations in
[30, 31]. Later, based on the ab-initio calculations [32] and using
calculations based on the quasi-one-dimensional model [33], the important role
of proton tunneling on the bonds was established. Piezoelectric coefficients,
elastic constants, and molar heat capacity of CDP [34, 35] were also
calculated on the basis of first-principle calculations.

A theoretical description of the dielectric properties of CDP at different
values of hydrostatic pressure was carried out in [36, 37] based on the
pseudospin model. However, in these works, the interaction parameters do not
depend on the lattice strains. As a result, it is impossible to obtain
piezoelectric and elastic characteristics of the crystal, and the critical
pressure does not depend on temperature.

In [38], temperature dependences of lattice strains $u_{1}$, $u_{2}$, $u_{3}$,
$u_{5}$ were measured. A quasi-one-dimensional Ising model for the CDP crystal
is also proposed there, in which the interaction parameters are linear
functions of these strains. Based on this model, the temperature behavior of
$u_{j}(T)$ was explained. However, this model does not consider the crystal as
two sublattices and does not allow describing the ferro-antiferroelectric
transition at high pressures.

In the papers [39, 40, 15, 41], a two-sublattice pseudospin model of a
deformed CDP crystal is proposed, in which the interactions between the
nearest pseudospins in the chain are taken into account in the two-particle
cluster approximation, and the long-range (including interchain) interactions
are taken into account in the mean field approximation. At the same time, the
interaction parameters are linear functions of $u_{j}$ strains. As a result,
the temperature dependences of spontaneous polarization, dielectric constant,
piezoelectric coefficients and elastic constants were calculated, and the
influence of hydrostatic and uniaxial pressures and longitudinal electric
field on these characteristics was studied. In [41], the behavior of the
thermodynamic characteristics of the CDP crystal under the action of
hydrostatic and uniaxial pressures and a longitudinal electric field, as well
as under the simultaneous action of pressures and the electric field, was
investigated.

In the present paper, the electrocaloric and barocaloric effects in CDP
crystal are calculated based on the model proposed in [15].

##  2 Model of CDP crystal

The [15] model was used to calculate the thermodynamic characteristics of CDP,
which considers the system of protons on O-H…O bonds with a two-minimum
potential as a system of pseudospins. The primitive cell contains one chain,
marked in figure 1 as “A”. To describe the transition to the antiferroelectric
phase at high pressures, in [15] an extended primitive cell formed by two
chains (“A” and “B”) is considered. All “A” chains form the “A” sublattice,
and all “B” chains form the “B” sublattice. Each chain in the primitive cell
contains two neighboring PO4 tetrahedra (of type “I” and “II”) together with
two short hydrogen bonds (“1” and “2”, respectively). The dipole moments
${\vec{d}}_{q1}^{A}$, ${\vec{d}}_{q2}^{A}$, ${\vec{d}}_{q1}^{B}$,
${\vec{d}}_{q2}^{B}$ are attributed to the protons on the bonds. Pseudospin
variables ${\sigma_{q1}^{A}}/{2}$, ${\sigma_{q2}^{A}}/{2}$,
${\sigma_{q1}^{B}}/{2}$, ${\sigma_{q2}^{B}}/{2}$ describe changes associated
with the rearrangement of the corresponding dipole moments of structural
units:
${\vec{d}}_{q1,2}^{A,B}=\vec{\mu}_{q1,2}^{A,B}\frac{\sigma_{q1,2}^{A,B}}{2}$.

Further, we use the notation “2” instead of “y” for the components of vectors
and tensors, for convenience. In the presence of mechanical stresses that do
not change the symmetry of the crystal $\sigma_{1}=\sigma_{xx}$,
$\sigma_{2}=\sigma_{yy}$, $\sigma_{3}=\sigma_{zz}$, $\sigma_{5}=\sigma_{xz}$
(X $\perp$ (b,c), Y $\parallel$ b, Z $\parallel$ c), as well as of the
electric field $E_{2}=E_{y}$, the Hamiltonian of the CDP model has the form
[15]:

$\displaystyle\hat{H}=NU_{\text{seed}}+\hat{H}_{\text{short}}+\hat{H}_{\text{long}}+\hat{H}_{E}+\hat{H}^{\prime}_{E},$
(2.1)

where $N$ is the total number of extended primitive cells.

The first term in (2.1) is the “seed” energy, which corresponds to the lattice
of heavy ions and does not explicitly depend on the configuration of the
proton subsystem. It includes elastic, piezoelectric and dielectric parts
expressed through electric field $E_{2}$ and strains that do not change the
lattice symmetry, $u_{1}=u_{xx}$, $u_{2}=u_{yy}$, $u_{3}=u_{zz}$,
$u_{5}=2u_{xz}$:

$\displaystyle
U_{\text{seed}}=v\left\\{\frac{1}{2}\sum\limits_{j,j^{\prime}}c_{jj^{\prime}}^{E0}u_{j}u_{j}^{\prime}-\sum\limits_{j}e_{2j}^{0}E_{2}u_{j}-\frac{1}{2}\varepsilon_{0}\chi_{22}^{u0}E_{2}^{2}\right\\},~{}~{}~{}~{}j,j^{\prime}=1,2,3,5,$
(2.2)

where $\varepsilon_{0}=8.8542\cdot 10^{-12}$ F/m is electric constant,
$c_{jj^{\prime}}^{E0}$, $e_{2j}^{0}$, $\chi_{22}^{u0}$ are “seed” elastic
constants, piezoelectric stress coefficients and dielectric susceptibility of
a mechanically clamped crystal. $v$ is the volume of the extended primitive
cell. In the paraelectric phase, all coefficients $e_{2j}^{0}\equiv 0$.

The other terms in (2.1) describe the pseudospin part of the Hamiltonian. In
particular, the second term in (2.1) is the Hamiltonian of short-range
interactions

$\hat{H}_{\text{short}}=-2w\sum\limits_{qq^{\prime}}\left(\frac{\sigma_{q1}^{A}}{2}\frac{\sigma_{q^{\prime}2}^{A}}{2}+\frac{\sigma_{q1}^{B}}{2}\frac{\sigma_{q^{\prime}2}^{B}}{2}\right)\bigl{(}\delta_{{\bf
R}_{q}{\bf R}_{q^{\prime}}}+\delta_{{\bf R}_{q}+{\bf R}_{b},{\bf
R}_{q^{\prime}}}\bigr{)}.$ (2.3)

In (2.3), $\sigma_{q1,2}^{A,B}$ are $z$-components of pseudospin operator,
that describe the state of the bond “1” or “2” of the chain “A” or “B”, in the
$q$-th cell, ${\vec{R}}_{b}$ is the lattice vector along $OY$-axis. The first
Kronecker delta corresponds to the interaction between neighboring pseudospins
in the chains near the tetrahedra PO4 of type “I”, where the second Kronecker
delta is near the tetrahedra PO4 of type “II”. Contributions to the energy of
interactions between pseudospins near tetrahedra of different types are
identical. Parameter $w$, which describes the short-range interactions within
the chains, is expanded linearly into a series with respect to strains
$u_{j}$:

$w=w_{0}+\sum\limits_{j}\delta_{j}u_{j},(j=1,2,3,5).$ (2.4)

The term $\hat{H}_{\text{long}}$ in (2.1) describes long-range dipole-dipole
interactions and indirect (through the lattice vibrations) interactions
between pseudospins which are taken into account in the mean field
approximation:

$\hat{H}_{\text{long}}=NH^{0}+\hat{H}_{2},$ (2.5)

where such notations are used:

$\displaystyle\hat{H}^{0}=\nu_{1}(\eta_{1}^{2}+\eta_{2}^{2})+2\nu_{2}\eta_{1}\eta_{2},$
(2.6)
$\displaystyle\hat{H}_{2}=\sum\limits_{q}\left\\{-(2\nu_{1}\eta_{1}+2\nu_{2}\eta_{2})\\!\left(\\!\frac{\sigma_{q1}^{A}}{2}+\frac{\sigma_{q2}^{A}}{2}\\!\right)\\!-(2\nu_{2}\eta_{1}+2\nu_{1}\eta_{2})\\!\left(\\!\frac{\sigma_{q1}^{B}}{2}+\frac{\sigma_{q2}^{B}}{2}\\!\right)\right\\}.$
(2.7)
$\displaystyle\nu_{1}=\nu_{1}^{0}+\sum\limits_{j}\psi_{j1}u_{j},~{}\nu_{2}=\nu_{2}^{0}+\sum\limits_{j}\psi_{j2}u_{j},~{}~{}~{}~{}\langle\sigma_{q1}^{A}\rangle=\langle\sigma_{q2}^{A}\rangle=\eta_{1},\quad\langle\sigma_{q1}^{B}\rangle=\langle\sigma_{q2}^{B}\rangle=\eta_{2}.$
(2.8)

The parameter $\nu_{1}$ describes the effective long-range interaction of the
pseudospin with the pseudospins within the same sublattice, and $\nu_{2}$ —
with the pseudospins of the other sublattice.

The fourth term in (2.1) describes the interactions of pseudospins with the
external electric field:

$\displaystyle\hat{H}_{E}=-\sum\limits_{q}\mu_{y}E_{2}\left(\frac{\sigma_{q1}^{A}}{2}+\frac{\sigma_{q2}^{A}}{2}+\frac{\sigma_{q1}^{B}}{2}+\frac{\sigma_{q2}^{B}}{2}\right),$
(2.9)

where $\mu_{y}$ is y-component of effective dipole moments per one pseudospin.

The term $\hat{H}^{\prime}_{E}$ in Hamiltonian (2.1) takes into account the
dependence of the effective dipole moment on the mean value of pseudospin
$s_{f}$:

$\displaystyle\hat{H}^{\prime}_{E}=-\sum\limits_{qf}s_{f}^{2}\mu^{\prime}E_{2}\frac{\sigma_{qf}}{2}=-\sum\limits_{qf}\left(\frac{1}{N}\sum\limits_{q^{\prime}}\sigma_{q^{\prime}f}\right)^{2}\mu^{\prime}E_{2}\frac{\sigma_{qf}}{2},$
(2.10)

where $\sigma_{qf}$ (f=1, 2, 3, 4) are a brief notation of pseudospins
$\sigma_{q1}^{A}$, $\sigma_{q2}^{A}$, $\sigma_{q1}^{B}$, $\sigma_{q2}^{B}$,
respectively. Here, we use corrections to dipole moments
$s_{f}^{2}\mu^{\prime}$ instead of $s_{f}\mu^{\prime}$ because of the symmetry
considerations and the energy should not change when the field and all
pseudospins change their sign.

The term $\hat{H}^{\prime}_{E}$, as well as long-range interactions, is taken
into account in the mean field approximation:

$\displaystyle\hat{H}^{\prime}_{E}=-3\sum\limits_{q}\mu^{\prime}E_{2}\left(\frac{\eta_{1}^{2}\sigma_{q1}^{A}}{2}+\frac{\eta_{1}^{2}\sigma_{q2}^{A}}{2}+\frac{\eta_{2}^{2}\sigma_{q1}^{B}}{2}+\frac{\eta_{2}^{2}\sigma_{q2}^{B}}{2}\right)+2N(\eta_{1}^{3}+\eta_{2}^{3})\mu^{\prime}E_{2}.$
(2.11)

In the two-particle cluster approximation for short-range interactions, the
thermodynamic potential per one extended primitive cell is as follows:

$\displaystyle g$ $\displaystyle=$ $\displaystyle
U_{\text{seed}}+H^{0}+2(\eta_{1}^{3}+\eta_{2}^{3})\mu^{\prime}E_{2}+2k_{\text{B}}T\ln
2-2w-v\sum\limits_{j}\sigma_{j}u_{j}$ (2.12) $\displaystyle-$ $\displaystyle
k_{\text{B}}T\ln(1-\eta_{1}^{2})-k_{\text{B}}T\ln(1-\eta_{2}^{2})-2k_{\text{B}}T\ln
D.$

Here, the following notations are used:

$\displaystyle D=\cosh(y_{1}+y_{2})+\cosh(y_{1}-y_{2})+2a\cosh y_{1}+2a\cosh
y_{2}+2a^{2},~{}~{}~{}~{}a=\mathrm{e}^{-\beta w}.$ $\displaystyle
y_{1}=\frac{1}{2}\ln\frac{1+\eta_{1}}{1-\eta_{1}}+\beta\nu_{1}\eta_{1}+\beta\nu_{2}\eta_{2}+\frac{1}{2}\beta(\mu_{y}E_{2}+3\eta_{1}^{2}\mu^{\prime}E_{2}),$
$\displaystyle
y_{2}=\frac{1}{2}\ln\frac{1+\eta_{2}}{1-\eta_{2}}+\beta\nu_{2}\eta_{1}+\beta\nu_{1}\eta_{2}+\frac{1}{2}\beta(\mu_{y}E_{2}+3\eta_{2}^{2}\mu^{\prime}E_{2}),$

where $\beta=\frac{1}{k_{\text{B}}T}$, $k_{\text{B}}$ is Boltzmann constant.

Minimizing the thermodynamic potential with respect to the order parameters
$\eta_{f}$ and strains $u_{j}$ in [15], we obtain a system of equations for
$\eta_{f}$ and $u_{j}$:

$\displaystyle\eta_{1}=\frac{1}{D}\left[\sinh(y_{1}+y_{2})+\sinh(y_{1}-y_{2})+2a\sinh
y_{1}\right],$ (2.13)
$\displaystyle\eta_{2}=\frac{1}{D}\left[\sinh(y_{1}+y_{2})-\sinh(y_{1}-y_{2})+2a\sinh
y_{2}\right],$
$\displaystyle\sigma_{j}=c_{j1}^{E0}u_{1}+c_{j2}^{E0}u_{2}+c_{j3}^{E0}u_{3}+c_{j5}^{E0}u_{5}-e_{2j}^{0}E_{2}-\frac{2\delta_{j}}{v}+\frac{4\delta_{j}}{vD}M-\frac{1}{v}\psi_{j1}(\eta_{1}^{2}+\eta_{2}^{2})-\frac{2}{v}\psi_{j2}\eta_{1}\eta_{2},$

where

$M=\bigl{[}a\cosh y_{1}+a\cosh y_{2}+2a^{2}\bigr{]}.$

In the presence of hydrostatic pressure $\sigma_{1}=\sigma_{2}=\sigma_{3}=-p$,
$\sigma_{4}=\sigma_{5}=\sigma_{6}=0$.

In [15], the expression for the longitudinal component of polarization $P_{2}$
was also obtained:

$\displaystyle P_{2}=-\left(\frac{\partial g}{\partial
E_{2}}\right)_{\sigma_{j}}=\sum\limits_{j}e_{2j}^{0}u_{j}+\chi_{22}^{u0}E_{2}+\frac{\mu_{y}}{v}\bigl{(}\eta_{1}+\eta_{2}\bigr{)}+\frac{\mu^{\prime}}{v}\bigl{(}\eta_{1}^{3}+\eta_{2}^{3}\bigr{)}.$
(2.14)

Based on the thermodynamic potential (2.12), we obtain an expression for the
entropy of the pseudospin subsystem:

$\displaystyle S=$ $\displaystyle-$
$\displaystyle\frac{N_{A}}{N_{m}}\left(\frac{\partial g}{\partial
T}\right)_{\eta,\varepsilon_{i}}=\frac{R}{N_{m}}\left\\{-2\ln
2+\sum\limits_{f=1}^{2}\ln\bigl{(}1-\eta_{f}^{2}\bigr{)}+2\ln D\right.$ (2.15)
$\displaystyle-$ $\displaystyle
2\eta_{1}\beta\left[\nu_{1}\eta_{1}+\nu_{2}\eta_{2}+\frac{1}{2}\left(\mu_{y}E_{2}+3\eta_{1}^{2}\mu^{\prime}E_{2}\right)\right]$
$\displaystyle-$
$\displaystyle\left.2\eta_{2}\beta\left[\nu_{2}\eta_{1}+\nu_{1}\eta_{2}+\frac{1}{2}\left(\mu_{y}E_{2}+3\eta_{2}^{2}\mu^{\prime}E_{2}\right)\right]+\frac{4M\beta
w}{D}\right\\}.$

Here, $N_{\text{A}}$ is Avogadro constant, $R$ is the universal gas constant,
$N_{m}=4$ is the number of CsH2PO4 molecules in the extended primitive cell.

The molar heat capacity of the pseudospin subsystem of the CDP crystal:

$\displaystyle
C=T\left(\frac{\mathrm{d}S}{\mathrm{d}T}\right)_{E_{2},\sigma_{j}}=T\left(S^{\prime}_{T}+\sum\limits_{f=1}^{2}S^{\prime}_{\eta_{f}}\eta^{\prime}_{Tf}+\\!\\!\sum\limits_{j=1,2,3,5}\\!\\!S^{\prime}_{u_{j}}u^{\prime}_{Tj}\right).$
(2.16)

The explicit expressions for derivatives $S^{\prime}_{T}$,
$S^{\prime}_{\eta_{f}}$, $S^{\prime}_{u_{j}}$, $\eta^{\prime}_{Tf}$,
$u^{\prime}_{Tj}$ are given in the appendix.

We consider the total heat capacity to be the sum of the pseudospin and
lattice components:

$C_{\text{total}}=C+C_{\text{lattice}}.$ (2.17)

The heat capacity of the lattice subsystem is considered to be the CDP heat
capacity, calculated on the basis of first-principle calculations [34]. Its
temperature dependence in the range of 80–350 K, in which the calculations
were carried out, is well approximated by a polynomial

$C_{\text{lattice}}=\sum\limits_{l=0}^{4}k_{l}T^{l},$ (2.18)

where the coefficients $k_{l}$: $k_{0}=17.62$ J/(mol K), $k_{1}=0.5955$ J/(mol
K2), $k_{2}=-0.001885$ J/(mol K3), $k_{3}=4.376\cdot$ 10-6 J/(mol K4),
$k_{4}=-4.034\cdot$ $10^{-9}$ J/(mol K5). The entropy of the lattice subsystem
near $T_{c}$:

$S_{\text{lattice}}=\int\frac{C_{\text{lattice}}}{T}\mathrm{d}T=k_{0}\ln(T)+\sum\limits_{l=1}^{4}\frac{k_{l}T^{l}}{l}+\text{const}.$
(2.19)

Total entropy as a function of temperature, field component $E_{2}$ and
hydrostatic pressure $p$:

$S_{\text{total}}(T,E_{2},p)=S+S_{\text{lattice}}.$ (2.20)

Solving (2.20) with respect to the temperature at
$S_{\text{total}}(T,E_{2},p)=\text{const}$ and two magnitudes of the field, it
is possible to calculate the electrocaloric temperature change (as shown in
figure 3b):

$\Delta
T_{\text{ec}}=T[S_{\text{total}},E_{2}(2),p]-T[S_{\text{total}},E_{2}(1),p].$
(2.21)

The change in temperature during the adiabatic change in the field $E_{2}$ can
also be calculated by the well-known formula

$\Delta
T_{\text{ec}}=-\int\limits_{0}^{E_{2}}\frac{TV}{C_{\text{total}}}\left(\frac{\partial
P_{2}}{\partial T}\right)_{E_{2}}\mathrm{d}E_{2},$ (2.22)

where pyroelectric coefficient

$\displaystyle\left(\frac{\partial P_{2}}{\partial
T}\right)_{E_{2}}=\sum\limits_{j}e_{2j}^{0}u^{\prime}_{jT}+\frac{\mu_{y}}{v}\bigl{(}\eta^{\prime}_{1T}+\eta^{\prime}_{2T}\bigr{)}+\frac{3\mu^{\prime}}{v}\bigl{(}\eta_{1}^{2}\eta^{\prime}_{1T}+\eta_{2}^{2}\eta^{\prime}_{2T}\bigr{)},$
(2.23)

and $V=vN_{A}/N_{m}$ is molar volume.

Similarly, solving (2.20) with respect to temperature at
$S_{\text{total}}(T,E_{2},p)=\text{const}$ and two pressure values, it is
possible to calculate the barocaloric temperature change (as shown in figure
3b):

$\Delta
T_{\text{bc}}=T[S_{\text{total}},E_{2},p(2)]-T[S_{\text{total}},E_{2},p(1)].$
(2.24)

The change in temperature under the adiabatic change in pressure $p$ can also
be calculated by the known formula

$\Delta
T_{\text{bc}}=\int\limits_{0}^{p}\frac{T}{C_{\text{total}}}\left(\frac{\partial
V}{\partial
T}\right)_{p}\mathrm{d}p=\int\limits_{0}^{p}\frac{N_{A}T}{N_{m}C_{\text{total}}}(u^{\prime}_{1T}+u^{\prime}_{2T}+u^{\prime}_{3T})\mathrm{d}p.$
(2.25)

##  3 Discussion of the obtained results

The theory parameters are determined in [15] from the condition of agreement
of calculated characteristics with experimental data for temperature
dependences of spontaneous polarization $P_{2}(T)$ and dielectric permittivity
$\varepsilon_{22}(T)$ at different values of hydrostatic pressure [21],
spontaneous strains $u_{j}$ [38], molar heat capacity [29] and elastic
constants [42]; as well as agreement with ab-initio calculations of the
lattice contributions into molar heat capacity [34] and dielectric
permittivity at zero temperature [35].

It should be noted that the temperature dependences of the dielectric constant
$\varepsilon_{22}$ at different values of hydrostatic pressure were also
measured in [25]. However, they do not agree with experimental data [21]. It
is possible that another crystal sample was used there, which was grown under
different conditions. In addition, in [25] there are no data for the
temperature dependences of spontaneous polarization at different pressures, as
well as no data for dielectric characteristics at zero pressure. Therefore, we
used experimental data [21] to determine the model parameters.

Parameters of short-range interactions $w_{0}$ and long-range interactions
$\nu_{1}^{0}$ (“intra-sublattice”), $\nu_{2}^{0}$ (“inter-sublattice”) mainly
fix the phase transition temperature from paraelectric to ferroelectric phase
at the absence of external pressure and field, the order of phase transition
and the shape of curve $P_{2}(T)$. Their optimal values are:
$w_{0}/k_{\text{B}}=650$ K, $\nu_{1}^{0}/k_{\text{B}}=1.50$ K,
$\nu_{2}^{0}/k_{\text{B}}=0.23$ K.

In order to determine the deformational potentials $\delta_{j}$ [see (2.4)]
and $\psi_{j1}$, $\psi_{j2}$ [see (2.8)], it is necessary to use experimental
data for the shift of the phase transition temperature under hydrostatic and
uniaxial pressures as well as the data for temperature dependences of
spontaneous strains $u_{j}$, piezoelectric coefficients and elastic constants.
Unfortunately, only the data for the spontaneous strains and hydrostatic
pressure effect on the dielectric characteristics are available. As a result,
the experimental data for strains and dielectric characteristics can be
described using a great number of combinations of parameters $\psi_{j1}$,
$\psi_{j2}$. Therefore, for the sake of simplicity, we chose $\psi_{j2}$ to be
proportional to $\psi_{j1}$. Optimal values of deformational potentials are:
$\delta_{1}/k_{\text{B}}=1214$ K, $\delta_{2}/k_{\text{B}}=454$ K,
$\delta_{3}/k_{\text{B}}=1728$ K, $\delta_{5}/k_{\text{B}}=-131$ K;
$\psi_{11}/k_{\text{B}}=92.2$ K, $\psi_{21}/k_{\text{B}}=23.2$ K,
$\psi_{31}/k_{\text{B}}=139.7$ K, $\psi_{51}/k_{\text{B}}=5.5$ K; $\psi_{j2}$
= $\frac{1}{3}\psi_{j1}$.

The effective dipole moment in the paraelectric phase is found from the
condition of agreement of the calculated curve $\varepsilon_{22}(T)$ with
experimental data. We consider it to be dependent on the value of hydrostatic
pressure $p$, that is $\mu_{y}=\mu_{y}^{0}(1-k_{p}p)$, where
$\mu_{y}^{0}=8.77\cdot 10^{-30}$ C$\cdot$m, $k_{p}=0.4\cdot 10^{-9}$ Pa-1. The
correction to the effective dipole moment $\mu^{\prime}=-1.43\cdot 10^{-30}$
C$\cdot$m is found from the condition of agreement of the calculated
saturation polarization with experimental data.

The “seed” dielectric susceptibility $\chi_{22}^{u0}$, coefficients of
piezoelectric stress $e_{2j}^{0}$ and elastic constants $c_{ij}^{E0}$ are
found from the condition of agreement of theory with experimental data in the
temperature regions far from the phase transition temperature $T_{c}$. Their
values are obtained as follows: $\chi_{22}^{u0}=5.57$; $e_{2j}^{0}=0$ C/m2;
$c_{jj^{\prime}}^{E0}$ ($10^{9}$N/m2): $c_{11}^{E0}=28.83$,
$c_{12}^{E0}=11.4$, $c_{13}^{E0}=42.87$, $c_{22}^{E0}=26.67$,
$c_{23}^{E0}=14.5$, $c_{33}^{E0}=65.45$, $c_{15}^{E0}=5.13$,
$c_{25}^{E0}=8.4$, $c_{35}^{E0}=7.50$, $c_{55}^{E0}=5.20$.

The volume of the extended primitive cell is $\upsilon=0.467\cdot 10^{-27}$ m3
[22].

In the paper [15], a phase diagram (figure 2) was calculated, which explains
the effect of hydrostatic pressure and longitudinal electric field on the
temperatures of phase transitions, in particular, the transition to the
antiferroelectric phase at pressures greater than the critical one.

Figure 2: Dependence on the hydrostatic pressure of the temperature of the
transition from the paraelectric to the ferroelectric phase $T_{c}$, from the
paraelectric to the antiferroelectric phase $T_{N}$, from the ferroelectric to
the antiferroelectric phase $T_{AF}$ at different values of the electric field
$E_{2}$ (MV/m): 0.0 –1 , 0.1 – 2, 0.2 – 3, 0.3 – 4, 0.4 – 5, 0.5 – 6 for the
CDP crystal. Symbols are experimental data [20], lines are theoretical
calculations [15]. Tricritical points $T_{N}^{\text{tr}}$ (marked as *)
separate the curves of the first-order phase transitions (dashed lines) and of
the second-order ones (solid lines).

As mentioned above, the EC effect is calculated as a change in the crystal
temperature $\Delta T_{\text{ec}}$ during adiabatic (at constant entropy)
application of an electric field, as shown in figure 3.

a b

Figure 3: (Colour online) Temperature dependences of the pseudospin
contribution to the molar entropy (a) and total entropy (b) of the CDP crystal
at different values of the field $E_{2}$ and of the hydrostatic pressure $p$.

At the pressures less than critical, longitudinal field $E_{2}$ decreases the
entropy of the crystal in the entire temperature range (figure 3), because it
puts the pseudospins in order in both sublattices, “A” and “B” (figure 1a).
Therefore, the $\Delta T_{\text{ec}}$ is positive. As we can see, the effect
of the field on the total entropy $S_{\text{total}}$ (figure 3b) is much
weaker than the effect on only the pseudospin contribution $S$ (figure 3a),
because the lattice heat capacity quite strongly stabilizes the temperature of
the crystal.

The calculated field and temperature dependences of $\Delta T_{\text{ec}}$ are
shown in figure 4.

a b

Figure 4: (Colour online) a) Field dependence of the electrocaloric
temperature change $\Delta T_{\text{ec}}$ at different values of temperature
$\Delta T=T-T_{c}$ and at zero hydrostatic pressure $p$. b) Temperature
dependence of $\Delta T_{\text{ec}}$ at different values of the longitudinal
electric field $E_{2}$ (MV/m): 1.0 – 1; 2.0 – 2; 5.0 – 3; 10.0 – 4; 20.0 – 5;
30.0 – 6; 40.0 – 7; 50.0 – 8 and at zero hydrostatic pressure $p$.

In the weak fields ($E_{2}<1$ MV/m) at the initial temperature $T=T_{c}$, the
change in temperature $\Delta T_{\text{ec}}\sim E_{2}^{2/3}$ (green curve in
figure 4a); at $T<T_{c}$, $\Delta T_{\text{ec}}\sim E_{2}$ (blue dashed curves
in figure 4); at T>$T_{c}$, $\Delta T_{\text{ec}}\sim E_{2}^{2}$ (red curves
in figure 4). At fields $E_{2}>1$ MV/m, the dependences of $\Delta
T_{\text{ec}}(E_{2})$ significantly deviate from the mentioned laws.

At high pressures, but less than the critical one, the field and temperature
dependences of $\Delta T_{\text{ec}}$ are qualitatively similar, as in the
absence of pressure (figure 5).

a b

Figure 5: (Colour online) a) Field dependence of electrocaloric temperature
change $\Delta T_{\text{ec}}$ at different temperature values $\Delta
T=T-T_{c}$ and at hydrostatic pressure $p=0.3$ GPa. b) Temperature dependence
of the electrocaloric temperature change $\Delta T_{\text{ec}}$ at different
values of the longitudinal electric field $E_{2}$ (MV/m): 1.0 – 1; 2.0 – 2;
5.0 – 3; 10.0 – 4; 20.0 – 5; 30.0 – 6; 40.0 – 7; 50.0 – 8 and at hydrostatic
pressure $p=0.3$ GPa.

At pressures greater than the critical one, at temperatures $T\geqslant
T_{N}$, EC effect is qualitatively similar to the case of subcritical
pressures in the paraelectric phase: at weak fields $\Delta T_{\text{ec}}\sim
E_{2}^{2}$ (green and red curves in figure 6a), at strong fields, the $\Delta
T_{\text{ec}}(E_{2})$ dependencies deviate from the quadratic law.

a b

Figure 6: (Colour online) a) Field dependence of electrocaloric change of
temperature $\Delta T_{\text{ec}}$ at different values of initial temperature
$\Delta T=T-T_{c}$ and at hydrostatic pressure $p=0.45$ GPa. b) Temperature
dependence of $\Delta T_{\text{ec}}$ at different values of the longitudinal
electric field $E_{2}$ (MV/m): 1.0 – 1; 2.0 – 2; 5.0 – 3; 10.0 – 4; 20.0 – 5;
30.0 – 6; 40.0 – 7; 50.0 – 8 and at hydrostatic pressure $p=0.5$ GPa.

At initial temperatures $T<T_{N}$ and weak fields $E_{2}$, the temperature of
the crystal decreases nonlinearly with the field (blue curves in figure 6a).
This is due to antiferroelectric ordering because the crystal passes into the
antiferroelectric phase at pressures higher than the critical one. The
ordering of pseudospins in sublattice “B” (which is oriented opposite to the
field) under the action of the field is stronger than the ordering of
pseudospins in sublattice “A”, which leads to the isothermal increase of
entropy and adiabatic (at constant entropy) lowering of temperature. With the
further strengthening of the field, the pseudospins in the “B” sublattice are
overturned and ordered in the direction of the field, which leads to the
isothermal decrease of entropy and to the isoentropic increase of temperature.

Hydrostatic pressure $p$ lowers the Curie temperature. This leads to the
isothermal increase of entropy and to the isentropic lowering of temperature,
as shown in figure 3. Therefore $\Delta T_{\text{bc}}$ is negative and at
$T\geqslant T_{c}^{0}$ it lowers almost linearly with increasing pressure
(figure 7a, green and red solid curves).

a b

Figure 7: (Colour online) a) Pressure dependence of the barocaloric
temperature change $\Delta T_{\text{bc}}$ at different values of temperature
$\Delta T=T-T_{c}^{0}$ and in the absence of a field. b) Temperature
dependence of barocaloric temperature change $\Delta T_{\text{bc}}$ at
different values of adiabatically applied pressure $p$ and in the absence of a
field.

At $T<T_{c}^{0}$ (ferroelectric phase) at low pressures, the BC effect is
stronger than in the paraelectric phase (in figure 7a these are the blue
dashed curves corresponding to $T-T_{c}^{0}=-10K,\,\,-20$ K). At a certain
value of pressure, the crystal passes to the paraelectric phase (see figure
2), in which the rate of cooling with pressure is less, and therefore a break
appears in the $\Delta T_{\text{bc}}(p)$ curve.

Figure 8: Temperature dependence of lattice strains $u_{j}$ under zero
pressure, calculated in [15].

As can be seen from figure 2, at $T-T_{c}^{0}=-40$ K there are two phase
transitions when increasing pressure: from ferroelectric to antiferroelectric
phase, and then from antiferroelectric to paraelectric phase. Accordingly, in
figure 7a two breaks appear on the curve $\Delta T_{\text{bc}}(p)$.

It should be noted that in this work, only the pseudospin (proton)
contribution to the BC effect was calculated, and lattice anharmonicities were
not taken into account. The interaction between pseudospins leads to the
occurrence of stretching strains due to the electrostrictive coupling of the
pseudospin and lattice subsystems, since after substitution of (2.4) into
(2.3) and also (2.8) into (2.6), there appear terms of the type
$\delta_{j}u_{j}\frac{\sigma_{q1}^{A}}{2}\frac{\sigma_{q^{\prime}2}^{A}}{2}$
and $\psi_{j1}u_{j}\eta_{1}^{2}$. The mean values of pseudospins decrease with
an increase of temperature. As a result, the electrostrictive coupling becomes
weaker and the diagonal strains $u_{1}$, $u_{2}$, $u_{3}$ decrease (figure 8).

The volume of the crystal decreases along with strains, $(\partial V/\partial
T)_{p}<0$. Therefore, $\Delta T_{\text{bc}}$ is negative, according to the
formula (2.25). We also note that it is possible to take a set of deformation
potentials $\delta_{j}$ [see (2.4)] and $\psi_{j1}$, $\psi_{j2}$ [see (2.8)],
which leads to an increase of the volume of the crystal with an increase of
temperature. However, this simultaneously leads to an increase in the Curie
temperature with an increase in pressure, which contradicts the experimental
data.

In contrast to electrostrictive coupling, lattice anharmonicities lead to
thermal expansion of the crystal and give a positive contribution to the BC
effect. This contribution competes with the pseudospin contribution, and, in a
certain temperature range, it can be larger than the pseudospin contribution.

##  4 Conclusions

In the case of a weak longitudinal field $E_{2}$, the electrocaloric change in
temperature $\Delta T_{\text{ec}}$ increases linearly with the field in the
ferroelectric phase, quadratically in the paraelectric phase, and according to
the law $\Delta T_{\text{ec}}\sim E_{2}^{2/3}$ at the initial temperature
$T=T_{c}$. In the strong field, the dependences $\Delta T_{\text{ec}}(E_{2})$
deviate from the mentioned laws. Applying the hydrostatic pressure, the EC
effect is qualitatively similar to the one at zero pressure. At pressures
greater than the critical one, the EC effect may be negative due to the
transition of the crystal into the antiferroelectric phase.

The barocaloric change in temperature $\Delta T_{\text{bc}}$ has a negative
sign and decreases almost linearly with pressure since the Curie temperature
decreases with pressure. The nonlinearity is strongly manifested at low
initial temperatures. In our calculations, only the pseudospin contribution to
the BC effect is taken into account. The electrostrictive coupling of the
pseudospin and lattice subsystem leads to a decrease in the volume of the
crystal with increasing temperature, and as a result the BC effect is
negative. To obtain $\Delta T_{\text{bc}}$, which can be compared with
experimental data, it is necessary to take into account the thermal expansion
associated with the lattice anharmonicities.

## Appendix. Notations in the expression for molar heat capacity

The notations introduced in expression (2.16) are as follows:

$\displaystyle S^{\prime}_{T}=\frac{R}{N_{m}}\left(\frac{4\beta
w}{D}\left[y_{1}^{T}a\sinh y_{1}+y_{2}^{T}a\sinh y_{2}+\frac{\beta
w}{T}aM^{a}\right]-\frac{4M\beta
w}{D}\left[y_{1}^{T}\eta_{1}+y_{2}^{T}\eta_{2}+\frac{\beta}{T}\frac{2Mw}{D}\right]\right),$
$\displaystyle
S^{\prime}_{\eta_{1}}=\frac{R}{N_{m}}\left(2Ty_{1}^{T}+\frac{4\beta
w}{D}\left(y_{1}^{\eta_{1}}a\sinh y_{1}+\beta\nu_{2}a\sinh
y_{2}\right)-\frac{4M\beta
w}{D}\left[\eta_{1}y_{1}^{\eta_{1}}+\eta_{2}\beta\nu_{2}\right]\right),$
$\displaystyle
S^{\prime}_{\eta_{2}}=\frac{R}{N_{m}}\left(2Ty_{2}^{T}+\frac{4\beta
w}{D}\left(\beta\nu_{2}a\sinh y_{1}+y_{2}^{\eta_{2}}a\sinh
y_{2}\right)-\frac{4M\beta
w}{D}\left[\eta_{2}y_{2}^{\eta_{2}}+\eta_{1}\beta\nu_{2}\right]\right),$
$\displaystyle S^{\prime}_{u_{j}}=\frac{R}{N_{m}}\left(\frac{4\beta
w}{D}\left[y_{1}^{u_{j}}a\sinh y_{1}+y_{2}^{u_{j}}a\sinh
y_{2}-\beta\delta_{j}aM^{a}\right]-\frac{4M\beta
w}{D}\left[\eta_{1}y_{1}^{u_{j}}+\eta_{2}y_{2}^{u_{j}}-\frac{2M\beta\delta_{j}}{D}\right]\right).$
(A.1)

Here are the notations:

$\displaystyle
y_{1}^{T}=-\frac{\beta}{T}\left[\nu_{1}\eta_{1}+\nu_{2}\eta_{2}+\frac{1}{2}\left(\mu_{y}E_{2}+3\eta_{1}^{2}\mu^{\prime}E_{2}\right)\right],~{}~{}y_{2}^{T}=-\frac{\beta}{T}\left[\nu_{2}\eta_{1}+\nu_{1}\eta_{2}+\frac{1}{2}\left(\mu_{y}E_{2}+3\eta_{2}^{2}\mu^{\prime}E_{2}\right)\right].$
$\displaystyle
y_{1}^{\eta_{1}}=\frac{1}{1-\eta_{1}^{2}}+\beta\nu_{1}+3\beta\eta_{1}\mu^{\prime}E_{2},~{}~{}~{}~{}y_{2}^{\eta_{2}}=\frac{1}{1-\eta_{2}^{2}}+\beta\nu_{1}+3\beta\eta_{2}\mu^{\prime}E_{2},$
$\displaystyle
y_{1}^{u_{j}}=\beta(\psi_{j1}\eta_{1}+\psi_{j2}\eta_{2}),~{}~{}~{}~{}y_{2}^{u_{j}}=\beta(\psi_{j2}\eta_{1}+\psi_{j1}\eta_{2}),$
$\displaystyle\phantom{\frac{1}{2}}M^{a}=\cosh y_{1}+\cosh y_{2}+4a.$

After differentiating the system of equations (2.13) with respect to the
temperature, we obtain a system of equations, from which we determine
$\eta^{\prime}_{Tf}$ and $u^{\prime}_{Tj}$:

$\displaystyle\left(\\!\begin{tabular}[]{cc}$\hat{A}^{\eta}-\hat{I}$&$\hat{A}^{u}$\\\
$\hat{B}^{\eta}$&$\hat{B}^{u}$\\\
\end{tabular}\\!\right)\left(\\!\begin{tabular}[]{c}$\vec{\eta}^{\prime}_{T}$\\\
$\vec{u}^{\prime}_{T}$\\\
\end{tabular}\\!\right)+\left(\\!\begin{tabular}[]{c}$\vec{\\!A}^{T}$\\\
$\vec{\\!B}^{T}$\\\
\end{tabular}\\!\right)=\vec{0}.~{}\Rightarrow~{}\left(\\!\begin{tabular}[]{c}$\vec{\eta}^{\prime}_{T}$\\\
$\vec{u}^{\prime}_{T}$\\\
\end{tabular}\\!\right)=-\left(\\!\begin{tabular}[]{cc}$\hat{A}^{\eta}-\hat{I}$&$\hat{A}^{u}$\\\
$\hat{B}^{\eta}$&$\hat{B}^{u}$\\\
\end{tabular}\\!\right)^{-1}\left(\\!\begin{tabular}[]{c}$\vec{\\!A}^{T}$\\\
$\vec{\\!B}^{T}$\\\ \end{tabular}\\!\right),$ (A.14)

where $\hat{I}$ is a 2$\times$2 identity matrix. Coefficients of the
$\hat{A}^{\eta}$ matrix are:

$\displaystyle\\!\\!\\!\\!A^{\eta}_{11}=\eta_{1}^{y_{1}}y_{1}^{\eta_{1}}+\eta_{1}^{y_{2}}\beta\nu_{2},~{}~{}~{}A^{\eta}_{12}=\eta_{1}^{y_{1}}\beta\nu_{2}+\eta_{1}^{y_{2}}y_{2}^{\eta_{2}},~{}~{}$
$\displaystyle\\!\\!\\!\\!A^{\eta}_{21}=\eta_{2}^{y_{1}}y_{1}^{\eta_{1}}+\eta_{2}^{y_{2}}\beta\nu_{2},~{}~{}~{}A^{\eta}_{22}=\eta_{2}^{y_{1}}\beta\nu_{2}+\eta_{2}^{y_{2}}y_{2}^{\eta_{2}},~{}~{}$

where the notations are entered:

$\displaystyle\eta_{1}^{y_{1}}=\frac{1}{D}\left[\cosh(y_{1}+y_{2})+\cosh(y_{1}-y_{2})+2a\cosh
y_{1}-\eta_{1}^{2}\right],$
$\displaystyle\eta_{1}^{y_{2}}=\eta_{2}^{y_{1}}=\frac{1}{D}\left[\cosh(y_{1}+y_{2})-\cosh(y_{1}-y_{2})-\eta_{1}\eta_{2}\right],~{}~{}$
$\displaystyle\eta_{2}^{y_{2}}=\frac{1}{D}\left[\cosh(y_{1}+y_{2})+\cosh(y_{1}-y_{2})+2a\cosh
y_{2}-\eta_{2}^{2}\right],~{}~{}$

coefficients of matrix $\hat{A}^{u}$:

$\displaystyle\\!\\!\\!\\!A^{u}_{1j}=\eta_{1}^{y_{1}}y_{1}^{u_{j}}+\eta_{1}^{y_{2}}y_{2}^{u_{j}}-\frac{\beta\delta_{j}}{D}\left[2a\sinh
y_{1}-2M\eta_{1}\right],$
$\displaystyle\\!\\!\\!\\!A^{u}_{2j}=\eta_{2}^{y_{1}}y_{1}^{u_{j}}+\eta_{2}^{y_{2}}y_{2}^{u_{j}}-\frac{\beta\delta_{j}}{D}\left[2a\sinh
y_{2}-2M\eta_{2}\right],~{}$

coefficients of matrix $\hat{B}^{\eta}$:

$\displaystyle
B^{\eta}_{j1}=-\frac{2}{v}(\psi_{j1}\eta_{1}+\psi_{j2}\eta_{2})+\frac{4\delta_{j}}{vD}(a\sinh
y_{1}y_{1}^{\eta_{1}}+a\sinh
y_{2}\beta\nu_{2})-\frac{4M\delta_{j}}{vD}\left(\eta_{1}y_{1}^{\eta_{1}}+\eta_{2}\beta\nu_{2}\right),$
$\displaystyle
B^{\eta}_{j2}=-\frac{2}{v}(\psi_{j1}\eta_{2}+\psi_{j2}\eta_{1})+\frac{4\delta_{j}}{vD}\left(a\sinh
y_{1}\beta\nu_{2}+a\sinh
y_{2}y_{2}^{\eta_{2}}\right)-\frac{4M\delta_{j}}{vD}(\eta_{1}\beta\nu_{2}+\eta_{2}y_{2}^{\eta_{2}}),$

coefficients of matrix $\hat{B}^{u}$:

$\displaystyle
B^{u}_{jj^{\prime}}=c_{jj^{\prime}}^{E0}+\frac{4\delta_{j}}{vD}\left[y_{1}^{u_{j^{\prime}}}a\sinh
y_{1}+y_{2}^{u_{j^{\prime}}}a\sinh
y_{2}-\beta\delta_{j^{\prime}}aM^{a}\right]-\frac{4M\delta_{j}}{vD}\left[\eta_{1}y_{1}^{u_{j^{\prime}}}+\eta_{2}y_{2}^{u_{j^{\prime}}}-\frac{2M\beta\delta_{j^{\prime}}}{D}\right],$

coefficients of vectors $\vec{\\!A}^{T}$ and $\vec{\\!B}^{T}$:

$\displaystyle
A^{T}_{1}=\eta_{1}^{y_{1}}y_{1}^{T}+\eta_{1}^{y_{2}}y_{2}^{T}+\frac{\beta
w}{DT}\left(2a\sinh y_{1}-2M\eta_{1}\right),$ $\displaystyle
A^{T}_{2}=\eta_{2}^{y_{1}}y_{1}^{T}+\eta_{2}^{y_{2}}y_{2}^{T}+\frac{\beta
w}{DT}\left(2a\sinh y_{2}-2M\eta_{2}\right),$ $\displaystyle
B^{T}_{j}=\frac{4\delta_{j}}{vD}\left[y_{1}^{T}a\sinh y_{1}+y_{2}^{T}a\sinh
y_{2}+\frac{aM^{a}\beta
w}{T}\right]-\frac{4\delta_{j}M}{vD}\left[y_{1}^{T}\eta_{1}+y_{2}^{T}\eta_{2}+\frac{2M\beta
w}{DT}\right].$

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