Electrocaloric effect in KH₂PO₄ family crystals
The proton ordering model for the KH₂PO₄ type ferroelectrics is modified by taking into account the dependence of the effective dipole moments on the proton ordering parameter. Within the four-particle cluster approximation we calculate the crystal polarization and explore the electrocaloric effect...
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Vdovych, A.S. Moina, A.P. Levitskii, R.R. Zachek, I.R. 2019-06-13T09:50:42Z 2019-06-13T09:50:42Z 2014 Electrocaloric effect in KH₂PO₄ family crystals / A.S. Vdovych, A.P. Moina, R.R. Levitskii, I.R. Zachek // Condensed Matter Physics. — 2014. — Т. 17, № 4. — С. 43703: 1–10 — Бібліогр.: 23 назв. — англ. 1607-324X arXiv:1502.02399 DOI:10.5488/CMP.17.43703 PACS: 77.84.Fa, 77.70.+a https://nasplib.isofts.kiev.ua/handle/123456789/152892 The proton ordering model for the KH₂PO₄ type ferroelectrics is modified by taking into account the dependence of the effective dipole moments on the proton ordering parameter. Within the four-particle cluster approximation we calculate the crystal polarization and explore the electrocaloric effect. Smearing of the ferroelectric phase transition by a longitudinal electric field is described. A good agreement with experiment is obtained. В моделi протонного впорядкування для кристалiв типу KH₂PO₄ враховано залежнiсть ефективних дипольних моментiв вiд параметра протонного впорядкування. В наближеннi чотиричастинкового кластера розраховано поляризацiю кристалiв та дослiджено електрокалоричний ефект у них. Описано розмивання сегнетоелектричного фазового переходу поздовжним електричним полем. Отримано добре узгодження з експериментальними даними. en Інститут фізики конденсованих систем НАН України Condensed Matter Physics Electrocaloric effect in KH₂PO₄ family crystals Електрокалоричний ефект у кристалах типу KH₂PO₄ Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| title |
Electrocaloric effect in KH₂PO₄ family crystals |
| spellingShingle |
Electrocaloric effect in KH₂PO₄ family crystals Vdovych, A.S. Moina, A.P. Levitskii, R.R. Zachek, I.R. |
| title_short |
Electrocaloric effect in KH₂PO₄ family crystals |
| title_full |
Electrocaloric effect in KH₂PO₄ family crystals |
| title_fullStr |
Electrocaloric effect in KH₂PO₄ family crystals |
| title_full_unstemmed |
Electrocaloric effect in KH₂PO₄ family crystals |
| title_sort |
electrocaloric effect in kh₂po₄ family crystals |
| author |
Vdovych, A.S. Moina, A.P. Levitskii, R.R. Zachek, I.R. |
| author_facet |
Vdovych, A.S. Moina, A.P. Levitskii, R.R. Zachek, I.R. |
| publishDate |
2014 |
| language |
English |
| container_title |
Condensed Matter Physics |
| publisher |
Інститут фізики конденсованих систем НАН України |
| format |
Article |
| title_alt |
Електрокалоричний ефект у кристалах типу KH₂PO₄ |
| description |
The proton ordering model for the KH₂PO₄ type ferroelectrics is modified by taking into account the dependence of the effective dipole moments on the proton ordering parameter. Within the four-particle cluster approximation we calculate the crystal polarization and explore the electrocaloric effect. Smearing of the ferroelectric phase transition by a longitudinal electric field is described. A good agreement with experiment is
obtained.
В моделi протонного впорядкування для кристалiв типу KH₂PO₄ враховано залежнiсть ефективних дипольних моментiв вiд параметра протонного впорядкування. В наближеннi чотиричастинкового кластера розраховано поляризацiю кристалiв та дослiджено електрокалоричний ефект у них. Описано розмивання сегнетоелектричного фазового переходу поздовжним електричним полем. Отримано добре узгодження з експериментальними даними.
|
| issn |
1607-324X |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/152892 |
| citation_txt |
Electrocaloric effect in KH₂PO₄ family crystals / A.S. Vdovych, A.P. Moina, R.R. Levitskii, I.R. Zachek // Condensed Matter Physics. — 2014. — Т. 17, № 4. — С. 43703: 1–10 — Бібліогр.: 23 назв. — англ. |
| work_keys_str_mv |
AT vdovychas electrocaloriceffectinkh2po4familycrystals AT moinaap electrocaloriceffectinkh2po4familycrystals AT levitskiirr electrocaloriceffectinkh2po4familycrystals AT zachekir electrocaloriceffectinkh2po4familycrystals AT vdovychas elektrokaloričniiefektukristalahtipukh2po4 AT moinaap elektrokaloričniiefektukristalahtipukh2po4 AT levitskiirr elektrokaloričniiefektukristalahtipukh2po4 AT zachekir elektrokaloričniiefektukristalahtipukh2po4 |
| first_indexed |
2025-11-25T23:08:43Z |
| last_indexed |
2025-11-25T23:08:43Z |
| _version_ |
1850578867210682368 |
| fulltext |
Condensed Matter Physics, 2014, Vol. 17, No 4, 43703: 1–10
DOI: 10.5488/CMP.17.43703
http://www.icmp.lviv.ua/journal
Electrocaloric effect in KH2PO4 family crystals
A.S. Vdovych1, A.P. Moina1, R.R. Levitskii1, I.R. Zachek2
1 Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine, 1 Svientsitskii St.,
79011 Lviv, Ukraine
2 National University “Lviv Polytechnic”, 12 Bandera St., 79013 Lviv, Ukraine
Received May 31, 2014, in final form October 7, 2014
The proton ordering model for the KH2PO4 type ferroelectrics is modified by taking into account the depen-
dence of the effective dipole moments on the proton ordering parameter. Within the four-particle cluster ap-
proximation we calculate the crystal polarization and explore the electrocaloric effect. Smearing of the fer-
roelectric phase transition by a longitudinal electric field is described. A good agreement with experiment is
obtained.
Key words: electrocaloric effect, KDP, cluster approximation, polarization
PACS: 77.84.Fa, 77.70.+a
1. Introduction
The electrocaloric (EC) effect is the change of temperature of a dielectric at an adiabatic change of the
applied electric field. Research in this field is driven by a quest for materials that can be used for efficient,
environment-friendly, and compact (on-chip) solid-state cooling devices.
The current state of the art on the electrocaloric effect research for ferroelectrics is well summarized
in [1, 2]. At the moment, the largest effect is observed in perovskite ferroelectrics. Thus, in [3] in the
PbZr0.95Ti0.05O3 thin film with a thickness of 350 nm in a strong electric field (480 kV/cm) the obtained
electrocaloric temperature change is ∆T = 12 K. Ab initio molecular dynamics calculations [4] predict
∆T ≈ 20 K in LiNbO3. In the hydrogen bonded ferroelectrics of the KH2PO4 (KDP) type, the electrocaloric
effect was studied for relatively low fields only. Thus, it has been obtained that∆T ≈ 0.04 K at E ≈ 4 kV/cm
[5], ∆T ≈ 1 K at E ≈ 12 kV/cm [6], and ∆T ≈ 0.25 K at Tc and E ≈ 1.2 kV/cm [7].
Theoretical calculations of the electrocaloric effect in KDP have been made in [8] within the Slater
model [9] and in the paraelectric phase only. It is also known that the Slater model gives incorrect results
in the ferroelectric phase, and more complicated versions of the proton ordering model are required for
an adequate description of these crystals. Thus, the effect of electric field on the physical characteristics
of the KDP type crystals, such as polarization, dielectric permittivity, piezoelectric coefficients, elastic con-
stants, has been described within the proton ordering model with the piezoelectric coupling to the shear
strain ε6 [10–12] and with proton tunneling [13] taken into account. However, these theories required,
in particular, invoking two different values of the effective dipole moments for the paraelectric and fer-
roelectric phase [10, 12]. This made impossible a correct description of the system behavior in the fields
high enough to smear out the first order phase transition. There is an inner logical contradiction in the
model: while no physical characteristic of a crystal should exhibit any discontinuity in the fields above
the critical one, there is no smooth transition between the values of model parameters, rigidly set to be
different for the two phases.
In the present paper we suggest a way to remove this contradiction. Assuming that the difference
between the dipole moments is caused by non-zero values of the order parameter, we modify the proton
ordering model accordingly. The field dependences of polarization, smearing of the first order phase
transition, and the electrocaloric effect are described.
© A.S. Vdovych, A.P. Moina, R.R. Levitskii, I.R. Zachek, 2014 43703-1
http://dx.doi.org/10.5488/CMP.17.43703
http://www.icmp.lviv.ua/journal
A.S. Vdovych et al.
2. Thermodynamic characteristics
We consider the KDP type ferroelectrics in the presence of an external electric field E3 applied along
the crystallographic axis c, inducing the strain ε6 and polarization P3. The total model Hamiltonian reads
Ĥ = N Ĥ0 + Ĥs , (1)
where N is the total number of primitive cells. The “seed” energy H0 corresponds to the sublattice of
heavy ions and does not explicitly depend on the proton subsystem configuration. It is expressed in terms
of the strain ε6 and electric field E3 and includes the elastic, piezoelectric, and dielectric contributions [11]
Ĥ0 = v
(
1
2
cE0
66 ε
2
6 −e0
36E3ε6 −
1
2
χε0
33E 2
3
)
, (2)
where v is the primitive cell volume; cE0
44
, e0
36
, χε0
33
are the “seed” elastic constant, piezoelectric coefficient,
and dielectric susceptibility, respectively.
The pseudospin part of the Hamiltonian reads
Ĥs =
1
2
∑
q f ,q ′ f ′
J f f ′ (qq ′
)
σq f
2
σq ′ f ′
2
+ Ĥsh+
∑
q f
2ψ6ε6
σq f
2
−
∑
q f
µ f E3
σq f
2
+ ĤE . (3)
Here, the first term describes the effective long-range interactions between protons, including also in-
direct lattice-mediated interactions [14, 15]; σq f is the operator of the z-component of a pseudospin,
corresponding to the proton on the f -th hydrogen bond ( f = 1, 2, 3, 4) in the q-th cell. Its eigenvalues
σq f =±1 are assigned to two equilibrium positions of a proton on this bond.
In (3), Ĥsh is the Hamiltonian of short-range interactions between protons, which includes terms lin-
ear over the strain [11]
Ĥsh =
∑
q
{
(
δs
8
ε6 +
δ1
4
ε6
)
(σq1 +σq2 +σq3 +σq4)
+
(
δs
8
ε6 −
δ1
4
ε6
)
(σq1σq2σq3 +σq1σq2σq4 +σq1σq3σq4 +σq2σq3σq4)
+
1
4
(V +δaε6)(σq1σq2 +σq3σq4)+
1
4
(V −δaε6)(σq2σq3 +σq4σq1)
+
U
4
(σq1σq3 +σq2σq4)+
Φ
16
σq1σq2σq3σq4
}
. (4)
Here,
V =−
1
2
w1 , U =
1
2
w1 −ε , Φ= 4ε−8w +2w1 ,
and ε, w , w1 are the energies of proton configurations.
The third term in (3) is a linear over the shear strain ε6 field due to the piezoelectric coupling;ψ6 is the
deformational potential. The fourth term effectively describes the system interaction with the external
electric field E3. Here, µ f is the effective dipole moment of the f -the hydrogen bond, and
µ1 =µ2 = µ3 =µ4 =µ.
The fifth term in (3) is introduced in the present paper for the first time. It takes into account the
assumed dependence of the effective dipole moment on the order parameter (pseudospin mean value)
ĤE =−
( 1
N
∑
q ′ f ′
σq ′ f ′
2
)2
µ′E3
∑
q f
σq f
2
. (5)
It is equivalent to a term proportional to P 3
3
E3 in a phenomenological thermodynamic potential. Note that
the terms like P 2
3
E3 are not allowed because of the symmetry considerations, and we keep the Hamilto-
nian to be linear in the field E3.
43703-2
Electrocaloric effect in KH2PO4 family crystals
In view of the crystal structure of the KDP type ferroelectrics, the four-particle cluster approximation
is most suitable for short-range interactions [15, 16]. Long-range interactions and the term ĤE are taken
into account in the mean field approximation. Thus,
ĤE ≈−12Nµ′E3η
2
4
∑
f =1
σq f
2
+16Nµ′E3η
3
. (6)
Combining the fourth term in (3) and the first term in (6), we obtain the following term in the Hamiltonian
−(µ+12µ′η2)E3
∑
q f σq f /2. Effectively, the term 12µ′η2 in (µ+12µ′η2) describes the jump of the dipole
moment at the first order phase transition, its different values for the paraelectric and ferroelectric phase,
and its smooth behavior in the fields above the critical one, when there is no jump of η. We can now use
a single value of µ for both phases and remove the logical contradiction of the earlier theories, described
in Introduction.
Proceeding with the standard calculations of the cluster approximation [10, 12, 16], we obtain the
following expression for the proton ordering parameter
η= 〈σq1〉 = 〈σq2〉 = 〈σq3〉 = 〈σq4〉 =
m
D
,
where
m = sinh(2z +βδsε6)+2b sinh(z −βδ1ε6),
D = cosh(2z +βδsε6)+4b cosh(z −βδ1ε6)+2a coshβδaε6 +d ,
z =
1
2
ln
1+η
1−η
+βνcη−βψ6ε6 +
βµ
2
E3 +6βµ′η2E3 ,
a = e
−βε
, b = e
−βw
, d = e
−βw1 ;
4νc = J11(0)+2J12(0)+ J13(0) is the eigenvalue of the long-range interactions matrix Fourier transform
J f f ′ =
∑
Rq−Rq′
J f f ′(qq ′); β= 1/kBT .
The thermodynamic potential is then obtained in the following form
G =
v
2
cE0
66 ε
2
6 − ve0
36ε6E3 −
v
2
χε0
33E 2
3 +2νcη
2
+16µ′E3η
3 (7)
+
2
β
ln 2−
2
β
ln
(
1−η2
)
−
2
β
ln D − vσ6ε6 .
Here, σ6 is the formally introduced shear stress conjugate to the strain ε6. In numerical calculations we
put σ6 = 0. The condition of the thermodynamic potential minimum
(
∂G
∂ε6
)
T,E3 ,σ6
= 0
yields an equation for the strain ε6
σ6 = cE0
66 ε6 −e0
36E3 +
4ψ6
v
η+
2r
vD
. (8)
In the same way, we derive the expressions for polarization P3 and molar entropy of the proton subsys-
tem
P3 = −
1
v
(
∂G
∂E3
)
T,σ6
= e0
36ε6 +χε0
33E3 +2
µ
v
η+8
µ′
v
η3
, (9)
S = −
NA
2
(
∂G
∂T
)
E3,σ6
= R
[
− ln 2+ ln(1−η2
)+ lnD +2T zT η+
M
D
]
. (10)
Here, NA is the Avogadro number; R is the gas constant. The following notations are used:
r =−δs Ms −δa Ma +δ1M1 ,
zT =−
1
kBT 2
(νcη−ψ6ε6 +6µ′η2E3),
43703-3
A.S. Vdovych et al.
M = 4bβw cosh(z −βδ1ε6)+βw1d +2aβεcoshβδaε6 +βε6r,
Ma = 2a sinhβδaε6, Ms = sinh(2z +βδsε6), M1 = 4b sinh(z −βδ1ε6).
Expressions for dielectric susceptibilities, piezoelectric coefficients, and elastic constants derived [17]
from equations (8), (9) are slightly different from the previous ones [10], where the dependence of the
effective dipole moment on the order parameter was not taken into account. Numerical calculations,
however, showed [17] that in zero electric field the difference is minor.
The molar specific heat of the subsystem described by the Hamiltonian (1) is
∆Cσ
= T
(
∂S
∂T
)
σ
= T (ST +SηηT +SεεT ). (11)
Here,
ST =
(
∂S
∂T
)
P3,ε6
=
R
DT
[
2T zT (q6 −ηM)+N6 −
M2
D
]
,
Sη =
(
∂S
∂η
)
ε6 ,T
=
2R
D
[
DT zT + (q6 −ηM)zη
]
,
Sε =
(
∂S
∂ε6
)
η,T
=
R
DT
[
−2
(
q6 −ηM
)
ψ6 −λ+
M
D
r
]
. (12)
Notations introduced here are described in appendix.
Then, the total specific heat is
C =∆Cσ
+Cregular . (13)
Here, ∆Cσ is assumed to describe all the anomalies of the specific heat at the phase transition, whereas
the regular background contribution to the specific heat, mostly from the lattice of heavy ions, is approx-
imated by a linear temperature dependence
Cregular =C0 +C1(T −Tc). (14)
As will be discussed later, this linear approximation agrees with the experimental data.
Finally, the electrocaloric temperature change is calculated using the known formula
∆T =
E3
∫
0
T V
C
(
∂P3
∂T
)
E
dE3 , (15)
where the pyroelectric coefficient is
(
∂P3
∂T
)
E
= e0
36εT +
2(µ+12µ′η2)
v
ηT , (16)
V = v NA/2 is the molar volume.
3. Numerical calculations
To perform the numerical calculations we need to set the values of the following theory parameters:
— the Slater energies ε, w , w1;
— the parameter of the long-range interactions νc;
— the effective dipole moment µ and the correction is due to proton ordering µ′;
— the deformation potentials ψ6, δs , δa , δ1;
43703-4
Electrocaloric effect in KH2PO4 family crystals
— the “seed” dielectric susceptibility χε0
33
, elastic constant cE0
66
, piezoelectric coefficient e0
36
;
— the parameters of the lattice specific heat C0 and C1.
They are chosen, obviously, by fitting the theoretical thermodynamic characteristics to the experimental
data, as described in [12]. The obtained optimum sets of the model parameters are given in table 1.
To describe crystals with different deuteration levels, we use the mean crystal approximation, where
the theory parameters are assumed to be linearly dependent on deuteron concentration (except for the
parameter νc, for which a small deviation from the linear dependence is assumed, as it is chosen from
the condition that the calculated transition temperature coincides with the experimental one, which is
also slightly non-linear). The dependence of the energy levels and interparticle interaction constants on
deuteration is caused by the corresponding geometrical changes in the crystal structure with deuteration
(elongation of the hydrogen bonds, changes in the distance between the equilibrium positions of H or D
on the bonds, changes in the lattice constants, etc).
Table 1. The optimum sets of the model parameters for different crystals. As KD2PO4 we denoted
K(H1−xDx )2PO4 with x = 0.89.
T 0
c ε/kB w/kB νc/kB µ µ′ χ0
33
(K) (K) (K) (K) (10−30 C·m) (10−30 C·m)
KH2PO4 122.22 56.00 430.0 17.55 5.6 −0.217 0.75
KD2PO4 211.73 85.33 730.4 39.26 6.8 −0.217 0.39
KH2AsO4 97 35.50 385.0 17.43 5.5 −0.033 0.7
KD2AsO4 162 56.00 690.0 31.72 7.3 −0.000 0.5
ψ6/kB δs /kB δa/kB δ1/kB cE0
66
e0
36
C0 C1
(K) (K) (K) (K) (109 N/m2) (C/m2) J/(mol K) J/(mol K2)
KH2PO4 −150.00 82.00 −500.00 −400.0 7.00 0.0033 60 0.32
KD2PO4 −139.89 48.64 −1005.68 −400.0 6.39 0.0033 93 0.32
KH2AsO4 −170.00 130.00 −500.0 −500.0 7.50 0.01 60 0.32
KD2AsO4 −160.00 120.00 −800.0 −500.0 6.95 0.01 98 0.40
The primitive cell volume is taken to be v = 0.1946 · 10−21 cm3 for K(H1−xDx )2PO4 and v = 0.202 ·
10−21 cm3 for K(H1−xDx )2AsO4, irrespectively of the deuteration. The energy w1 of proton configurations
with four or zero protons near the given oxygen tetrahedron should be much higher than ε and w .
Therefore, we take w1 =∞ (d = 0).
As we have already mentioned, when the dependence of the effective dipole moment on the order
parameter is taken into account, the agreement between the theory and experiment for most of the cal-
culated dielectric, piezoelectric, elastic characteristics, and specific heat of the studied crystals in the
absence of an external electric field is neither improved nor worsened (see [17]). However, the present
model allows us to describe more consistently the smearing of the first order phase in high electric fields.
The temperature dependence of the specific heat of KH2PO4 and KD2PO4 is shown in figure 1. The con-
tribution ∆Cσ is essential in the transition region and satisfactorily describes the experimental anoma-
lies. As one can see, the total specific heat above Tc can be well approximated by a linear temperature
dependence, thus justifying the linear dependence of Cregular , given by equation (14).
In figures 2 and 3 we plotted the temperature variation of polarization of K(H1−xDx )2PO4 in different
fields. The agreement with experiment is better at x = 0.89 (and 0.84, see [17]) than at x = 0. We believe
this is due to proton tunnelling, essential in non-deuterated samples, which is not included in our model.
The field E3, which in these crystals is the field conjugate to the order parameter, induces non-zero
polarization P3 above the transition point. Polarization has a jump at Tc, indicating the first order phase
transition. With an increasing field, the polarization jump decreases, whereas the transition temperature
Tc increases almost linearly. The corresponding ∂Tc/∂E3 slopes are 0.192 and 0.115 K cm/kV for x = 0
43703-5
A.S. Vdovych et al.
100 120 140 160 180 200 220
50
100
150
200
C
p
, J/(mol⋅K)
T, K
1 1’
2
2’
Figure 1. The temperature dependence of the molar specific heat of K(H1−xDx )2PO4 at x = 0.0 — ◦ [18],
ä [19]; at x = 0.86 — △ [19]. Dashed lines 1’ and 2’: the theoretical results of [12].
and x = 0.89, respectively (c.f. 0.22 and 0.13 K cm/kV from our earlier calculations [10] and experimen-
tal 0.125 K cm/kV of [23] for x = 0.89). At some critical field E∗, the jump vanishes, and the transition
smears out. The calculated coordinates of the critical point are E∗ = 125 V/cm, T ∗
c =122.244 K for x = 0
and 7.1 kV/cm, 212.55 K for x = 0.89, which agrees well with the experiment [22, 23]. It should be noted
that in our previous calculations [12] it was impossible to obtain a correct description of the polariza-
tion behavior in the fields above the critical one, due to the necessity of using two different values of the
effective dipole moment µ in calculations.
The calculated electrocaloric changes of temperature ∆T of the K(H1−xDx )2PO4 and K(H1−xDx )2AsO4
crystals with the adiabatically applied electric field are shown in figures 4 and 5. The experimental data
of [7] were obtained at T = 121 K, which was very close to the transition temperature of the sample used
in the measurements.
90 100 110 120 130
0
0.01
0.02
0.03
0.04
0.05
0.06
P
3
, C/m2
T, K
1 2
3
4
E=0.000 MV/m (1)
0.581 MV/m (2)
1.250 MV/m (2)
2.031 MV/m (3)
211 212 213 214
0
0.01
0.02
0.03
0.04
0.05
P
3
, C/m2
T, K
1 2 3 4
E=0.000 MV/m (1)
0.282 MV/m (2)
0.564 MV/m (3)
0.710 MV/m (4)
0.846 MV/m (5)
1.128 MV/m (6)
5 6
Figure 2. The temperature dependence of polariza-
tion of KH2PO4 at different E3(MV/m): 0.0 — 1, △
[5]; 0.581 — 2, ◦ [20]; 1.250 — 3, ä [20]; 2.031 — 4,
♦ [20]. Symbols are experimental points; solid lines:
the present theory; dashed lines: the theoretical re-
sults of [12].
Figure 3. The temperature dependence of polariza-
tion of K(H1−xDx )2PO4 at x = 0.89 and at different
E3 (MV/m): 0.0— 1; 0.282— 2, ◦; 0.564— 3,ä; 0.71—
4; 0.846— 5, ♦; 1.128— 6,△. Symbols are experimen-
tal points taken from [21]; lines: the present theory.
43703-6
Electrocaloric effect in KH2PO4 family crystals
0 0.1 0.2 0.3 0.4
0
0.05
0.1
0.15
0.2
0.25
E, MV/ m
∆T, K
2
2’
1’
1
3’
3
T−T
c
0=−2.04K (1,1’)
0.00K (2,2’)
3.28K (3,3’)
0 10 20 30 40
0
1
2
3
4
5
6
7
8
E, MV/ m
∆T, K
1’3
1
3’
T−T
c
0=−2.04K (1,1’)
0.00K (2,2’)
3.28K (3,3’)
2’
2
Figure 4. The field dependence of the electrocaloric temperature change of K(H1−xDx )2PO4 for x = 0.0
(solid lines) and x = 0.89 (dashed lines) at T −T 0
c =−2.04 K— 1, 1′ ,ä; T = T 0
c — 2, 2′, ◦; T −T 0
c = 3.2 K —
3, 3′, ⋄. Experimental points are taken from [5] — ◦, ä and [7]— ⋄.
As one can see, at small fields (figures 4, 5, left-hand) the calculated electrocaloric temperature change
is a linear function of the field below T 0
c (curves 1, 1′) and a quadratic function above T 0
c (at 2, 2′). The
experimental behavior below T 0
c is not linear at E3 < 2 kV/cm due to the domains: The domains, whose
polarization is oriented along the field, are heated, whereas the domains, polarized in the opposite di-
rection are cooled, thus the resulting net change of the sample temperature is close to zero. The experi-
mental data for the electrocaloric temperature change at and above T 0
c available for KH2PO4, as well as
the ∆T /∆E ratio below T 0
c at fields above 2 kV/cm (when the sample is in a single-domain state), are well
reproduced by the theory.
At higher fields (figures 4, 5, right-hand) the calculated electrocaloric temperature changes at tem-
peratures above T 0
c are larger than below T 0
c . The obtained curves deviate from linear and quadratic
behavior and reach saturation at E ≫ 500 kV/cm. It should be mentioned, however, that these curves are
calculated with the linear over the field E3 pseudospin Hamiltonian (3). It would be very interesting to
compare our results at high fields with experiment, for instance, to find out when non-linear contribu-
tions to the Hamiltonian cannot be omitted any longer. Unfortunately, no experimental data for ∆T in
0 0.1 0.2 0.3 0.4
0
0.02
0.04
0.06
0.08
0.1
0.12
E, MV/ m
∆T, K
2
2’
1’
1
3’
3
T−T
c
0=−2.04K (1,1’)
0.00K (2,2’)
3.28K (3,3’)
0 10 20 30 40
0
1
2
3
4
5
6
7
8
E, MV/ m
∆T, K
1’
3
1
3’
T−T
c
0=−2.04K (1,1’)
0.00K (2,2’)
3.28K (3,3’)
2’ 2
Figure 5. The field dependence of the electrocaloric temperature change of KH2AsO4 (solid lines) and
KD2AsO4 (dashed lines) at T −T 0
c =−2.04 K— 1, 1′ ; T = T 0
c — 2, 2′ ; T −T 0
c = 3.2 K — 3, 3′.
43703-7
A.S. Vdovych et al.
100 110 120 130 140
0
1
2
3
4
5
6
7
8
∆ T, K
T, K
1
2
3
4
E=2 MV/m (1)
5 MV/m (2)
10 MV/m (3)
20 MV/m (4)
50 MV/m (5)
5
190 200 210 220 230
0
2
4
6
8
10
∆ T, K
T, K
1
2
3
4
E=2 MV/m (1)
5 MV/m (2)
10 MV/m (3)
20 MV/m (4)
50 MV/m (5)
5
Figure 6. The temperature dependence of the electrocaloric temperature change of K(H1−xDx )2PO4 for
x = 0.0 (left-hand) and x = 0.89 (right-hand) in different fields.
the fields above 1 kV/cm are available. And, of course, possibilities for experimental measurements are
limited by the dielectric strength of the samples.
As one can see from the temperature dependence of ∆T (figure 6) for K(H1−xDx )2PO4 crystals, the
calculated electrocaloric temperature change is the largest at temperatures below T 0
c but close Tc and
can exceed 6 K; however, the fields required to reach ∆T that high are not accessible in reality, because
most likely they exceed the dielectric strength of the crystals.
4. Conclusions
Taking into account the dependence of the effective dipole moment on the order parameter within
the framework of the proton orderingmodel allows us to correctly describe the smearing of the ferroelec-
tric phase transition in high electric fields as well as the electrocaloric effect in the KDP family crystals.
The theory predicts the values of the electrocaloric temperature change of a few Kelvins in high fields.
Additional experimental measurements of ∆T in the fields above 2 kV/cm are necessary.
Appendix
The notations introduced in equations (11)–(12) are as follows:
N6 = 2a(βε)
2
coshβδaε6 +4b(βw)
2
cosh(z −βδ1ε6)+ (βw1)
2d +2ε6β
2
(−εδa Ma +wδ1M1)
+ε2
6
[
2a(βδa )
2
coshβδaε6 + (βδs )
2
cosh(2z +βδsε6)+4b(βδ1)
2
cosh(z −βδ1ε6)
]
,
q6 = 2bβw sinh(z −βδ1ε6)+ε6β
[
−δs cosh(2z +βδsε6)+2bδ1 cosh(z −βδ1ε6)
]
,
λ=−βεδa Ma +βwδ1M1 +ε6β
[
δ2
s cosh(2z +βδsε6)+2aδ2
a coshβδaε6 +4bδ2
1 cosh(z −βδ1ε6)
]
,
ηT = pε
6 +
v
2(µ+12µ′η2)
(e36 −e2
36)εT ,
εT =
[
2β
vD
(
2T zT f6 −λ+
Mr
D
)
−
4pε
6
v
(
ψ6 −
zη f6
D
)]
/
cE
66 .
In turn,
pε
6 =
1
T
2κT zT + [q6 −ηM]
D −2κzη
,
43703-8
Electrocaloric effect in KH2PO4 family crystals
cE
66
is the isothermal elastic constant at a constant field
cE
66 = cE0
66 +
8ψ6
v
β(−ψ6κ+ f6)
D −2zηκ
−
4βzη f 2
6
vD(D −2zηκ)
−
2β
vD
[
δ2
s cosh(2z +βδsε6)+2aδ2
a coshβδaε6 +4bδ2
1 cosh(z −βδ1ε6)
]
+
2βr 2
vD2
,
and e36 is the isothermal piezoelectric coefficient
e36 =−
(
∂σ6
∂E3
)
T,ε6
=
(
∂P3
∂ε6
)
T,E3
= e0
36 +
2(µ+12µ′η2)
v
βθ6
D −2zηκ
,
with
θ6 =−2κψ6 + f6 , f6 = δs cosh(2z +βδsε6)−2bδ1 cosh(z −βδ1ε6)+ηr,
zη =
1
1−η2
+βνc+12βµ′ηE3 .
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A.S. Vdovych et al.
Електрокалоричний ефект у кристалах типу KH2PO4
А.С. Вдович1, А.П. Моїна1, Р.Р. Левицький1, I.Р. Зачек2
1 Iнститут фiзики конденсованих систем НАН України, вул. I. Свєнцiцького, 1, 79011 Львiв, Україна
2 Нацiональний унiверситет “Львiвська полiтехнiка”, вул. С. Бандери, 12, 79013 Львiв, Україна
В моделi протонного впорядкування для кристалiв типу KH2PO4 враховано залежнiсть ефективних ди-
польних моментiв вiд параметра протонного впорядкування. В наближеннi чотиричастинкового класте-
ра розраховано поляризацiю кристалiв та дослiджено електрокалоричний ефект у них. Описано розми-
вання сегнетоелектричного фазового переходу поздовжним електричним полем. Отримано добре узго-
дження з експериментальними даними.
Ключовi слова: електрокалоричний ефект, KDP, кластерне наближення, поляризацiя
43703-10
Introduction
Thermodynamic characteristics
Numerical calculations
Conclusions
|