Longitudinal relaxation of ND₄D₂PO₄ type antiferroelectrics. Piezoelectric resonance and sound attenuation
Within the framework of the modi ed proton model with taking into account the interaction with the shear strain ε6, a dynamic dielectric response of ND₄D₂PO₄ type antiferroelectrics is considered. Dynamics of the piezoelectric strain is taken into account. Experimentally observed phenomena of crys...
Збережено в:
| Дата: | 2009 |
|---|---|
| Автори: | , , , |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Інститут фізики конденсованих систем НАН України
2009
|
| Назва видання: | Condensed Matter Physics |
| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/119986 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Longitudinal relaxation of ND₄D₂PO₄ type antiferroelectrics. Piezoelectric resonance and sound attenuation / R.R. Levitskii, I.R. Zachek, A.P. Moina, A.S. Vdovych // Condensed Matter Physics. — 2009. — Т. 12, № 2. — С. 275-294. — Бібліогр.: 35 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
nasplib_isofts_kiev_ua-123456789-119986 |
|---|---|
| record_format |
dspace |
| spelling |
nasplib_isofts_kiev_ua-123456789-1199862025-02-09T22:26:18Z Longitudinal relaxation of ND₄D₂PO₄ type antiferroelectrics. Piezoelectric resonance and sound attenuation Поздовжня релаксацiя антисегнетоелектрикiв типу ND₄D₂PO₄. П’єзоелектричний резонанс та поглинання звуку Levitskii, R.R. Zachek, I.R. Moina, A.P. Vdovych, A.S. Within the framework of the modi ed proton model with taking into account the interaction with the shear strain ε6, a dynamic dielectric response of ND₄D₂PO₄ type antiferroelectrics is considered. Dynamics of the piezoelectric strain is taken into account. Experimentally observed phenomena of crystal clamping by high frequency electric field, piezoelectric resonance and microwave dispersion are described. Ultrasound velocity and attenuation are calculated. Character of behaviour of attenuation in the paraelectric phase and the existence of a cut-off frequency in the frequency dependence of attenuation are predicted. At the proper choice of the parameters, a good quantitative description of experimental data for longitudinal static dielectric, piezoelectric and elastic characteristics and sound velocity for ND₄D₂PO₄ and NH₄H₂PO₄ is obtained in the paraelectric phase. В рамках модифiкованої протонної моделi з врахуванням взаємодiї зi зсувною деформацiєю ε₆ розглянуто динамiчний дiелектричний вiдгук антисегнетоелектрикiв типу ND₄D₂PO₄. Враховано динамiку п’єзоелектричної деформацiї. Явно описано явища затискання кристалу високочастотним електричним полем, п’єзоелектричного резонансу i НВЧ дисперсiї, що спостерiгаються на експериментi. Розраховано швидкiсть та коефiцiєнт поглинання звуку. Передбачено характер поведiнки коефiцiєнта поглинання в парафазi та наявнiсть обрiзаючої частоти у частотнiй залежностi коефiцiєнта поглинання звуку. При належному виборi мiкропараметрiв в параелектричнiй фазi отримано добрий кiлькiсний опис експериментальних даних для поздовжнiх статичних дiелектричних, п’єзоелектричних i пружних характеристик та швидкостi звуку для ND₄D₂PO₄ i NH₄H₂PO₄. 2009 Article Longitudinal relaxation of ND₄D₂PO₄ type antiferroelectrics. Piezoelectric resonance and sound attenuation / R.R. Levitskii, I.R. Zachek, A.P. Moina, A.S. Vdovych // Condensed Matter Physics. — 2009. — Т. 12, № 2. — С. 275-294. — Бібліогр.: 35 назв. — англ. 1607-324X PACS: 77.22.Ch, 77.22.Gm, 77.65.Bn 77.84.Fa, 77.65.Fs DOI:10.5488/CMP.12.2.275 https://nasplib.isofts.kiev.ua/handle/123456789/119986 en Condensed Matter Physics application/pdf Інститут фізики конденсованих систем НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
Within the framework of the modi ed proton model with taking into account the interaction with the shear
strain ε6, a dynamic dielectric response of ND₄D₂PO₄ type antiferroelectrics is considered. Dynamics of
the piezoelectric strain is taken into account. Experimentally observed phenomena of crystal clamping by
high frequency electric field, piezoelectric resonance and microwave dispersion are described. Ultrasound
velocity and attenuation are calculated. Character of behaviour of attenuation in the paraelectric phase and
the existence of a cut-off frequency in the frequency dependence of attenuation are predicted. At the proper
choice of the parameters, a good quantitative description of experimental data for longitudinal static dielectric,
piezoelectric and elastic characteristics and sound velocity for ND₄D₂PO₄ and NH₄H₂PO₄ is obtained in the
paraelectric phase. |
| format |
Article |
| author |
Levitskii, R.R. Zachek, I.R. Moina, A.P. Vdovych, A.S. |
| spellingShingle |
Levitskii, R.R. Zachek, I.R. Moina, A.P. Vdovych, A.S. Longitudinal relaxation of ND₄D₂PO₄ type antiferroelectrics. Piezoelectric resonance and sound attenuation Condensed Matter Physics |
| author_facet |
Levitskii, R.R. Zachek, I.R. Moina, A.P. Vdovych, A.S. |
| author_sort |
Levitskii, R.R. |
| title |
Longitudinal relaxation of ND₄D₂PO₄ type antiferroelectrics. Piezoelectric resonance and sound attenuation |
| title_short |
Longitudinal relaxation of ND₄D₂PO₄ type antiferroelectrics. Piezoelectric resonance and sound attenuation |
| title_full |
Longitudinal relaxation of ND₄D₂PO₄ type antiferroelectrics. Piezoelectric resonance and sound attenuation |
| title_fullStr |
Longitudinal relaxation of ND₄D₂PO₄ type antiferroelectrics. Piezoelectric resonance and sound attenuation |
| title_full_unstemmed |
Longitudinal relaxation of ND₄D₂PO₄ type antiferroelectrics. Piezoelectric resonance and sound attenuation |
| title_sort |
longitudinal relaxation of nd₄d₂po₄ type antiferroelectrics. piezoelectric resonance and sound attenuation |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2009 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/119986 |
| citation_txt |
Longitudinal relaxation of ND₄D₂PO₄ type antiferroelectrics. Piezoelectric resonance and sound attenuation / R.R. Levitskii, I.R. Zachek, A.P. Moina, A.S. Vdovych // Condensed Matter Physics. — 2009. — Т. 12, № 2. — С. 275-294. — Бібліогр.: 35 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT levitskiirr longitudinalrelaxationofnd4d2po4typeantiferroelectricspiezoelectricresonanceandsoundattenuation AT zachekir longitudinalrelaxationofnd4d2po4typeantiferroelectricspiezoelectricresonanceandsoundattenuation AT moinaap longitudinalrelaxationofnd4d2po4typeantiferroelectricspiezoelectricresonanceandsoundattenuation AT vdovychas longitudinalrelaxationofnd4d2po4typeantiferroelectricspiezoelectricresonanceandsoundattenuation AT levitskiirr pozdovžnârelaksaciâantisegnetoelektrikivtipund4d2po4pêzoelektričniirezonanstapoglinannâzvuku AT zachekir pozdovžnârelaksaciâantisegnetoelektrikivtipund4d2po4pêzoelektričniirezonanstapoglinannâzvuku AT moinaap pozdovžnârelaksaciâantisegnetoelektrikivtipund4d2po4pêzoelektričniirezonanstapoglinannâzvuku AT vdovychas pozdovžnârelaksaciâantisegnetoelektrikivtipund4d2po4pêzoelektričniirezonanstapoglinannâzvuku |
| first_indexed |
2025-12-01T10:02:48Z |
| last_indexed |
2025-12-01T10:02:48Z |
| _version_ |
1850299775356764160 |
| fulltext |
Condensed Matter Physics 2009, Vol. 12, No 2, pp. 275–294
Longitudinal relaxation of ND4D2PO4 type
antiferroelectrics. Piezoelectric resonance and sound
attenuation
R.R.Levitskii1, I.R.Zachek2, A.P.Moina1, A.S.Vdovych1
1 Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine, 1 Svientsitskii str,
79011 Lviv, Ukraine
2 Lviv Polytechnic National University, 12 Bandera Str., 79013 Lviv, Ukraine
Received April 2, 2009, in final form May 18, 2009
Within the framework of the modified proton model with taking into account the interaction with the shear
strain ε6, a dynamic dielectric response of ND4D2PO4 type antiferroelectrics is considered. Dynamics of
the piezoelectric strain is taken into account. Experimentally observed phenomena of crystal clamping by
high frequency electric field, piezoelectric resonance and microwave dispersion are described. Ultrasound
velocity and attenuation are calculated. Character of behaviour of attenuation in the paraelectric phase and
the existence of a cut-off frequency in the frequency dependence of attenuation are predicted. At the proper
choice of the parameters, a good quantitative description of experimental data for longitudinal static dielectric,
piezoelectric and elastic characteristics and sound velocity for ND4D2PO4 and NH4H2PO4 is obtained in the
paraelectric phase.
Key words: antiferroelectrics, dielectric permittivity, piezoelectric resonance
PACS: 77.22.Ch, 77.22.Gm, 77.65.Bn 77.84.Fa, 77.65.Fs
1. Introduction
Ferroelectric compounds of the MD2XO4 (M=K, D4; x=P, As) type crystallize in the 4̄ ·m class
of the tetragonal syngony (space group I 4̄2d with non-centrosymmetric point group D2d) in the
paraelectric phase and possess piezoelectric properties. When appropriate electric fields and shear
stresses are applied, one can explore the role of piezoelectric coupling in the phase transition and
in the formation of physical characteristics of the crystals. Theoretical investigations of the role of
piezoelectricity in the KH2PO4 type ferroelectricity were initiated in [1], where the Slater theory
[2] was modified by taking into account the splitting of the lowest ferroelectric energy level of the
proton subsystem due to the strain ε6.
Important results for strained ferroelectric compounds of the KH2PO4 type were obtained in
[3–11]. In [3,4] the proton ordering model was modified by taking into account the ε6 contributions
to the proton subsystem energy linear in strain. The obtained Hamiltonian contains a deformati-
onal molecular field and takes splitting of lateral proton configurations into account. Later [5–7]
all possible splittings of proton configuration energies by the strain ε6 were taken into account.
In [5] a phase transition in the strained K(H0,12D0,88)2PO4 crystal was explored; its thermody-
namic, longitudinal dielectric, piezoelectric, and elastic characteristics were calculated; the effect
of the stress σ6 on the calculated quantities was studied. Similar calculations of thermodynamic,
longitudinal and transverse dielectric, piezoelectric, and elastic characteristics of KH2PO4 type
ferroelectrics were performed in [6–8] with tunneling taken into account. A good description of ex-
perimental data for the KH2PO4 ferroelectrics and NH4H2PO4 antiferroelectrics in the paraelectric
phase was obtained. In [9–11], the effect of longitudinal electric field on the physical characteristics
of K(H0,12D0,88)2PO4 and KH2PO4 was studied; a satisfactory quantitative agreement with the
available experimental data was obtained.
c© R.R.Levitskii, I.R.Zachek, A.P.Moina, A.S.Vdovych 275
R.R.Levitskii et al.
We should also mention the paper [12], where the mechanism of spontaneous strain ε6 formation
in the KH2PO4 type ferroelectrics and the role of proton interactions with acoustic lattice vibrations
in this process were explored.
In [5–11], the dynamic properties of KH2PO4 type ferroelectrics were not studied with taking
into account the piezoelectric coupling. Such a problem, however, is very important. Due to the
effect of tunneling suppression in KH2PO4 family crystals found in [13–15], and due to the principal
difficulties arising at calculations of dynamic characteristics in the presence of tunneling, this
problem should be approached by neglecting tunneling. In [16–19], within the framework of the
modified proton ordering models, the thermal, longitudinal and transverse dielectric, piezoelectric,
and elastic characteristics of the KH2PO4 family ferroelectrics were calculated. The relaxational
phenomena in these crystals were explored; sound velocity and attenuation were obtained. It was
shown that for a proper choice of the theory parameters, the experimental data for longitudinal
dynamic characteristics of these crystals should be taken into account.
Description of dynamic dielectric characteristics of the ND4D2PO4 type antiferroelectrics [20–
22] was restricted to the static limit and high-frequency relaxation. The attempts to explore the
piezoelectric resonance within a model that does not take into account the piezoelectric coupling
are pointless. The traditional proton ordering model for the ND4D2PO4 type antiferroelectrics
does not allow one to describe the difference of the behaivor of free and clamped crystals in the
static limit or the effect of crystal clamping produced by high-frequency field. It seems natural to
calculate the dynamic characteristics of the ND4D2PO4 type antiferroelectrics using the proton
ordering model proposed in [5,6,18] in a wide frequency range from 103 kHz up to 1012 Hz, including
the piezoelectric resonance region as well.
In the present paper, following the approach developed in [23,24], within the framework of
the modified proton ordering model with taking into account the coupling with shear strain ε6,
we calculate the longitudinal dynamic dielectric, piezoelectric, and elastic characteristics of the
ND4D2PO4 type antiferroelectrics and explore their temperature and frequency dependences. The
effect of crystal clamping produced by a high-frequency longitudinal electric field is studied. Sound
velocity and attenuation in these crystals are also calculated.
2. Hamiltonian of proton ordering model
We shall consider a system of deuterons moving on the O–D. . . O bonds in deuterated
ND4D2PO4 type crystals. The primitive cell of the Bravais lattice of these crystals consists of
two neighboring tetrahedra PO4 along with four hydrogen bonds attached to one of them (the “A”
type tetrahedron). The hydrogen bonds attached to the other tetrahedron (“B” type) belong to
the four structural elements surrounding it. Spontaneous polarization in these crystals is zero due
to antipolar ordering of dipole moments of hydrogen bonds. External fields applied along a, b, and
c axes induce non-zero net polarization.
The model Hamiltonian, with taking into account the short-range and long-range interactions,
in the presence of mechanical stress σ6 = σxy and external electric field E3 directed along the
crystallographic axis c, consists of the “seed” and pseudospin parts. The “seed” energy of a primitive
cell corresponds to the lattice of heavy ions and is explicitly independent of the configurations of
hydrogen bonds. The pseudospin part of the Hamiltonian includes long-range (Ĥlong) and short-
range (Ĥshort) deuteron interactions as well as the effective interactions of deuterons with the
electric field E3. Hence,
Ĥ = NUseed + Ĥlong + Ĥshort −
∑
qf
µf3E3
σqf
2
, (2.1)
where N is the number of primitive cells; σqf is the operator of the z-component of a pseudospin
describing the state of a deuteron in the q-th cell on the f-th bond. Eigenvalues of the operator
σqf = ±1 correspond to the two possible equilibrium positions of the deuteron on the bond.
Symmetry of the effective dipole moments of the primitive cells along the c-axis per one hydrogen
276
Longitudinal relaxation of ND4D2PO4 type antiferroelectrics
bond is as follows:
µ3 = µ13 = µ23 = µ33 = µ43 .
The “seed” energy Useed is expressed in terms of the electric field E3 and strain ε6. It consists
of the elastic, piezoelectric, and dielectric parts
Useed = v̄
(
1
2
cE0
66 ε
2
6 − e036ε6E3 −
1
2
χε033E
2
3
)
, (2.2)
where v̄ = v
kB
, v is the primitive cell volume; kB is the Boltzmann constant; cE0
66 , e036, χ
ε0
33 are the
“seed” elastic constant, coefficient of piezoelectric stress, and dielectric susceptibility, respectively.
The “seed” quantities determine the temperature behavior of the corresponding characteristics at
temperatures far from the transition point TN.
The Hamiltonian Ĥlong includes the long-range interactions between deuterons and an indirect
lattice-mediated deuteron interactions taken into account within the mean field approximation, as
well as the linear in the strain ε6 molecular field [3,4], induced by piezoelectric coupling
Ĥlong =
1
2
∑
qq′
ff ′
Jff ′(qq′)
〈σqf 〉
2
〈σq′f ′〉
2
−
∑
qf
2µFqf
σqf
2
. (2.3)
Here
2µFq13 = ∓2νa(k
z)η(1)eikz
aq + 2νc(0)η(1)z − 2ψ6ε6,
2µFq24 = ±2νa(k
z)η(1)eikz
aq + 2νc(0)η(1)z − 2ψ6ε6, (2.4)
and we took into account the fact that the single-particle deuteron distribution functions can be
presented as a sum of a modulated part and uniform terms induced by the longitudinal electric
field
〈σq13 〉 = ∓η(1)eikz
aq + η(1)z , 〈σq24 〉 = ±η(1)eikz
aq + η(1)z .
In (2.4) we use the following notations
4νa(k
z) = J11(k
z) − J13(k
z), 4νc(0) = J11(0) + 2J12(0) + J13(0),
Jff ′(kz) =
∑
aq−aq′
Jff ′(qq′)e−ikz(aq−aq′ );
k
z = 1/2(b1 + b2 + b3), b1, b2, b3 are vectors of the reciprocal lattice; eikz
aq = ±1, ψ6 is the
deformational potential.
The Hamiltonian Ĥshort reads [18]:
Ĥshort =
∑
q
{(
−δs6
4
+
δ16
2
)
ε6
(σq1
2
+
σq2
2
+
σq3
2
+
σq4
2
)
+ (−δs6 − 2δ16) ε6
(σq1
2
σq2
2
σq3
2
+
σq1
2
σq2
2
σq4
2
+
σq1
2
σq3
2
σq4
2
+
σq2
2
σq3
2
σq4
2
)
+(Va + δa6ε6)
(σq1
2
σq2
2
+
σq3
2
σq4
2
)
+ (Va − δa6ε6)
(σq2
2
σq3
2
+
σq4
2
σq1
2
)
+Ua
(σq1
2
σq3
2
+
σq2
2
σq4
2
)
+ Φa
σq1
2
σq2
2
σq3
2
σq4
2
}
, (2.5)
where we use the notations
Va =
1
2
ε′ − 1
2
w′
1, Ua =
1
2
ε′ +
1
2
w′
1, Φa = 2ε′ − 8w′ + 2w′
1.
277
R.R.Levitskii et al.
Here
ε′ = εs − εa; w′ = ε1 − εa; w′
1 = ε0 − εa ,
where εs, εa, ε1, ε0 are the configurational energies of deuterons, and ε′, w′, w′
1 are the antiferro-
electric energies of the extended Slater-Takagi model.
Considering the peculiarities of the crystal structure of ND4D2PO4 type crystals, we shall use
the four-particle cluster approximation [25]. The longitudinal static dielectric and elastic charac-
teristics can be calculated using the thermodynamic potential, which in the cluster approximation
reads [18]:
G = NUseed +
1
2
∑
qq′
ff ′
Jff ′(qq′)
〈σqf 〉
2
〈σq′f ′〉
2
− 1
2
T
∑
q
4
∑
f=1
lnZq1f − T
∑
q
lnZq4 −Nv̄σ6ε6 , (2.6)
where Zq1f = Sp e−βĤ
(1)
qf , Zq4 = Sp e−βĤ
(4)
q are the single-particle and four-particle partition
functions. The single-particle Ĥ
(1)
qf and four-particle Ĥ
(4)
q deuteron Hamiltonians read
Ĥ
(1)
q13
= ∓ 1
β
x̄q
σq13
2
+
1
β
z̄
σq13
2
, Ĥ
(1)
q24
= ± 1
β
x̄q
σq24
2
+
1
β
z̄
σq24
2
, (2.7)
Ĥ(4)
q =
(
−δs6
4
+
δ16
2
)
ε6
(σq1
2
+
σq2
2
+
σq3
2
+
σq4
2
)
+ (−δs6 − 2δ16) ε6
(σq1
2
σq2
2
σq3
2
+
σq1
2
σq2
2
σq4
2
+
σq1
2
σq3
2
σq4
2
+
σq2
2
σq3
2
σq4
2
)
+ (Va + δa6ε6)
(σq1
2
σq2
2
+
σq3
2
σq4
2
)
+ (Va − δa6ε6)
(σq2
2
σq3
2
+
σq4
2
σq1
2
)
+ Ua
(σq1
2
σq3
2
+
σq2
2
σq4
2
)
+ Φa
σq1
2
σq2
2
σq3
2
σq4
2
− 1
β
xq
(
−σq1
2
+
σq2
2
+
σq3
2
− σq4
2
)
− 1
β
z
(σq1
2
+
σq2
2
+
σq3
2
+
σq4
2
)
. (2.8)
Here we use the notations
xq = β(−∆ae
ikz
aq + 2νa(k
z)η(1)eikz
aq ), z = β(−∆c + 2νc(0)η(1)z − 2ψ6ε6 + µ3E3),
x̄q = −β∆ae
ikz
aq + xq , z̄ = −β∆c + z,
and ∆a, ∆c are the effective fields exerted by the neighboring hydrogen bonds O–D. . . O from
outside the cluster.
Having calculated the eigenvalues of the single-particle and four-particle Hamiltonians, we
present the thermodynamic potential per unit cell in the form [18]:
g =
v̄
2
cE0
66 ε
2
6 − v̄e036ε6E3 +
v̄
2
χε033E
2
3 + 2T ln 22w̃′ + ε̃′ + 2νa(k
z)η(1)2 + 2νc(0)(η(1)z)2
− T ln[1 − (η(1) − η(1)z)2] − T ln[1 − (η(1) + η(1)z)2] − 2T lnD6 − v̄σ6ε6. (2.9)
Here and further we note ε̃′ = ε′
kB
, w̃′ = w′
kB
.
From the conditions of thermodynamic equilibrium
1
v̄
(
∂g
∂ε6
)
E3
= 0,
1
v
(
∂g
∂E3
)
σ6
= −P3 (2.10)
we obtain (in the limit w′
1 → ∞) an equation for the strain ε6 and polarization P3:
σ6 = cE0
66 ε6 − e036E3 −
2
v
δs6
Ns6
D6
+
2
v
δ16
N16chx
D6
+
2
v̄
δa6
Na6
D6
+
4
v
ψ6η
(1)z ,
P3 = e036ε6 + χε033E3 + 2
µ3
v
η(1)z. (2.11)
278
Longitudinal relaxation of ND4D2PO4 type antiferroelectrics
Here we use the notations
Ns = ach(2z + βδs6ε6), N1 = bch(z − βδ16ε6),
Ns6 = ash(2z + βδs6ε6), N16 = 4bsh(z − βδ16ε6), Na6 = a6 −
ch2x
a6
,
D6 = ach(2z + βδs6ε6) +
1
a6
ch2x+ a6 + d
+ 2b[ch(x+ z − βδ16ε6) + ch(x− z + βδ16ε6)],
a = e−βε
′
, b = e−βw
′
, d = e−βw
′
1 , a6 = e−βδa6ε6 .
3. Longitudinal dynamic permittivity of ND4D2PO4 type crystals
The dynamic characteristics of the ND4D2PO4 type crystals will be explored within the frame-
work of the dynamic model of these crystals based on the stochastic Glauber approach [26], where
the time dependence of the deuteron distribution functions is described by the following equation
−α d
dt
〈
∏
f
σqf
〉
=
∑
f ′
〈
∏
f
σqf
[
1 − σqf ′ tanh
β
2
εzqf
]
〉
, (3.1)
where α is the time constant that effectively determines the time scale of the dynamic processes
in the system; εzqf is the local field acting on the f -th bond in the q-th cell in the presence of the
field E3. The fields can be determined from the Hamiltonian (2.8)
tanh
β
2
εzq1 = tanh
{
−β
4
(Va + δa6ε6)σq2 −
β
4
(Va − δa6ε6)σq4 −
β
4
Uaσq3 −
β
16
Φaσq2σq3σq4
−β
4
(
−δs6ε6
2
− δ16ε6
)
(σq2σq3 + σq3σq4 + σq2σq4) −
β
4
(
−δs6ε6
2
+ δ16ε6
)
− 1
2
zq14
}
,
tanh
β
2
εzq2 = tanh
{
−β
4
(Va + δa6ε6)σq1 −
β
4
(Va − δa6ε6)σq3 −
β
4
Uaσq4 −
β
16
Φaσq1σq3σq4
−β
4
(
−δs6ε6
2
− δ16ε6
)
(σq1σq4 + σq3σq4 + σq1σq3) −
β
4
(
−δs6ε6
2
+ δ16ε6
)
+
1
2
zq23
}
,
tanh
β
2
εzq3 = tanh
{
−β
4
(Va + δa6ε6)σq4 −
β
4
(Va − δa6ε6)σq2 −
β
4
Uaσq1 −
β
16
Φaσq1σq2σq4
−β
4
(
−δs6ε6
2
− δ16ε6
)
(σq1σq2 + σq1σq4 + σq2σq4) −
β
4
(
−δs6ε6
2
+ δ16ε6
)
+
1
2
zq23
}
,
tanh
β
2
εzq4 = tanh
{
−β
4
(Va + δa6ε6)σq3 −
β
4
(Va − δa6ε6)σq1 −
β
4
Uaσq2 −
β
16
Φaσq1σq2σq4
−β
4
(
−δs6ε6
2
− δ16ε6
)
(σq1σq2 + σq2σq3 + σq1σq3) −
β
4
(
−δs6ε6
2
+ δ16ε6
)
− 1
2
zq14
}
, (3.2)
where
zq14 = −xq + z, zq23 = xq + z.
The right hand sides in (3.2) can be written as
tanh
β
2
εzq1 = P zq14σq3 +Qzq141σq2 +Qzq142σq4 +Rzq14σq2σq3σq4
+Mz
q141σq2σq3 +Mz
q142σq3σq4 +Nz
q14σq2σq4 + Lzq14 ,
. . .
tanh
β
2
εzq4 = P zq14σq2 +Qzq241σq3 +Qzq142σq1 +Rzq14σq1σq2σq4
+Mz
q141σq2σq3 +Mz
q142σq1σq2 +Nz
q14σq1σq3 + Lzq14 . (3.3)
279
R.R.Levitskii et al.
Equating the right hand sides of (3.2) and (3.3) and taking into account the fact that σqf = ±1,
we find
P z
q
14
23
=
1
8
(
lz
q1 14
23
− lz
q2 14
23
+ nz
q1 14
23
− nz
q2 14
23
+mz
q1 14
23
−mz
q2 14
23
+mz
q3 14
23
−mz
q4 14
23
)
,
Qz
q
14
23 1
=
1
8
(
lz
q1 14
23
− lz
q2 14
23
− nz
q1 14
23
+ nz
q2 14
23
+mz
q1 14
23
+mz
q2 14
23
−mz
q3 14
23
−mz
q4 14
23
)
,
Qz
q
14
23 2
=
1
8
(
lz
q1 14
23
− lz
q2 14
23
− nz
q1 14
23
+ nz
q2 14
23
−mz
q1 14
23
−mz
q2 14
23
+mz
q3 14
23
+mz
q4 14
23
)
,
Rz
q
14
23
=
1
8
(
lz
q1 14
23
− lz
q2 14
23
+ nz
q1 14
23
+ nz
q2 14
23
−mz
q1 14
23
+mz
q2 14
23
−mz
q3 14
23
+mz
q4 14
23
)
,
Mz
q
14
23 1
=
1
8
(
lz
q1 14
23
+ lz
q2 14
23
− nz
q1 14
23
− nz
q2 14
23
+mz
q1 14
23
−mz
q2 14
23
−mz
q3 14
23
+mz
q4 14
23
)
,
Mz
q
14
23 2
=
1
8
(
lz
q1 14
23
+ lz
q2 14
23
− nz
q1 14
23
− nz
q2 14
23
−mz
q1 14
23
+mz
q2 14
23
+mz
q3 14
23
−mz
q4 14
23
)
,
Nz
q
14
23
=
1
8
(
lz
q1 14
23
+ lz
q2 14
23
+ nz
q1 14
23
+ nz
q2 14
23
−mz
q1 14
23
−mz
q2 14
23
−mz
q3 14
23
−mz
q4 14
23
)
,
Lz
q
14
23
=
1
8
(
lz
q1 14
23
+ lz
q2 14
23
+ nz
q1 14
23
+ nz
q2 14
23
+mz
q1 14
23
+mz
q2 14
23
+mz
q3 14
23
+mz
q4 14
23
)
, (3.4)
where
lz
q
1
2
14
23
= tanh
β
2
[
∓(ε′ − ω′) + (δs6 + δ16)ε6 +
1
β
zq 14
23
]
,
nz
q
1
2
14
23
= tanh
β
2
[
∓(ω′ − ω′
1) − δ16ε6 +
1
β
zq 14
23
]
,
mz
q
1
4
14
23
= tanh
β
2
[
∓ω′ − (±δa6 + δ16)ε6 +
1
β
zq 14
23
]
,
mz
q
3
2
14
23
= tanh
β
2
[
∓ω′ − (∓δa6 + δ16)ε6 +
1
β
zq 14
23
]
. (3.5)
When an electric field E3 along the c-axis is applied, the deuteron distribution functions possess
the following symmetry
η
(1)z
q14 = 〈σq1〉 = 〈σq4〉, η
(1)z
q23 = 〈σq2〉 = 〈σq3〉,
η
(3)z
q14 = 〈σq2σq3σq4〉 = 〈σq1σq2σq3〉, η
(3)z
q23 = 〈σq1σq3σq4〉 = 〈σq1σq2σq4〉,
η
(2)z
q14 = 〈σq1σq4〉, η
(2)z
q23 = 〈σq2σq3〉,
η
(2)z
q2 = −〈σq1σq2〉 = −〈σq3σq4〉, η
(2)z
q3 = −〈σq1σq3〉 = −〈σq2σq4〉. (3.6)
Substituting (3.3) into the system (3.1) and taking into account the symmetry of the distribu-
tion functions (3.6), we obtain the following system of equations for the time-dependent deuteron
distribution functions in the presence of the field E3:
d
dt
η
(1)z
q14
η
(1)z
q23
η
(3)z
q14
η
(3)z
q23
η
(2)z
q14
η
(2)z
q23
η
(2)z
q2
η
(2)z
q3
=
c̄q11 c̄q12 . . . c̄q18
c̄q21 c̄q22 . . . c̄q28
c̄q31 c̄q32 . . . c̄q38
c̄q41 c̄q42 . . . c̄q48
c̄q51 c̄q52 . . . c̄q58
c̄q61 c̄q62 . . . c̄q68
c̄q71 c̄q72 . . . c̄q78
c̄q81 c̄q82 . . . c̄q88
η
(1)z
q14
η
(1)z
q23
η
(3)z
q14
η
(3)z
q24
η
(2)z
q14
η
(2)z
q23
η
(2)z
q2
η
(2)z
q3
+
c̄q1
c̄q2
c̄q3
c̄q4
c̄q5
c̄q6
c̄q7
c̄q8
. (3.7)
280
Longitudinal relaxation of ND4D2PO4 type antiferroelectrics
Expressions for the coefficients c̄q11, . . . , c̄q88 are given in [18]. In the one-particle approximation,
we obtain the following system of equations
d
dt
η
(1)z
q14 = − 1
α
η
(1)z
q14 +
1
α
tanh
1
2
z̄q14 ,
d
dt
η
(1)z
q23 = − 1
α
η
(1)z
q23 +
1
α
tanh
1
2
z̄q23 . (3.8)
We shall consider the vibrations of a thin square plate with sides l of a ND4D2PO4 type crystal
cut in the [001] plane, produced by an external time-dependent electric field E3t = E3e
iωt. For
the sake of simplicity we shall neglect the diagonal strains εi (i = 1, 2, 3), which, in fact, are also
created in the crystal.
The shear strain ε6 is determined by the displacements ux = u1 and uy = u2, namely
ε6 = εxy =
∂u1
∂y
+
∂u2
∂x
.
The classical equations of motion of an elementary volume, describing the dynamics of deformati-
onal processes in ND4D2PO4 type crystals, read
ρ
∂2u1
∂t2
=
∂σ6
∂y
, ρ
∂2u2
∂t2
=
∂σ6
∂x
, (3.9)
where ρ is the crystal density.
Taking into account (2.11) and (3.9), we obtain
ρ
∂2u1
∂t2
= cE0
66
∂ε6
∂y
+
4ψ6
v
∂η
(1)z
t
∂y
+
2δa6
v
∂
∂y
(
Ma6
D6
)
− 2δs6
v
∂
∂y
(
Ns6
D6
)
+
2δ16
v
∂
∂y
(
N16chxq
D6
)
,
ρ
∂2u2
∂t2
= cE0
66
∂ε6
∂x
+
4ψ6
v
∂η
(1)z
t
∂x
+
2δa6
v
∂
∂x
(
Na6
D6
)
− 2δs6
v
∂
∂x
(
Ns6
D6
)
+
2δ16
v
∂
∂x
(
N16chxq
D6
)
. (3.10)
Assuming that the crystal is mechanically free, we present the distribution functions, effective
fields, and the strain ε6 as sums of two terms: the equilibrium functions and their fluctuations.
Hence
η
(1)z
q
14
23
= ∓η(1)
q + η
(1)z
t , η
(3)z
q
14
23
= ∓η(3)
q + η
(3)z
t ,
η
(2)z
q14 = η
(2)
1 − η
(2)z
qt , η
(2)z
q12 = η
(2)
1 + η
(2)z
qt , η
(2)z
q2 = −η(2)
2 , η
(2)z
q3 = −η(2)
3 ,
ε6 = ε6t , E3 = E3t , zq14 = −xq + zt − 2βψ6ε6t , zq23 = xq + zt − 2βψ6ε6t , (3.11)
where
xq = −β∆qa + 2βνa(k
z)η(1)
q , zt = −β∆ct + 2βνc(0)η
(1)z
t + βµ3E3t .
The calculated statistical distribution functions in the ND4D2PO4 type crystal in the particular
case at E3 = 0 and σ6 = 0 have the following form
η(1) =
1
D
(sinh 2x+ 2b sinhx), η(3) =
1
D
(sinh 2x− 2b sinhx),
η
(2)
1 =
1
D
(cosh 2x− 1 + a+ d), η
(2)
2 =
1
D
(cosh 2x−1−a+d), η
(2)
3 =
1
D
(cosh 2x− 1 + a− d).
D = a+ ch2x+ d+ 4bchx+ 1, x =
1
2
ln
1 + η(1)
1− η(1)
+ βνa(k
z)η(1).
Let us expand the coefficients (3.4) in series over the time-dependent terms. Taking into account
(3.11) and eliminating ∆ct from the system (3.7)–(3.8), we obtain a system of equations for the
281
R.R.Levitskii et al.
time-dependent distribution functions for a mechanically free crystal
d
dt
η
(1)z
t
η
(3)z
t
η
(2)z
qt
=
c011 c012 cq13
c021 c022 cq23
cq31 cq32 c033
η
(1)z
t
η
(3)z
t
η
(2)z
qt
− βµ3
2
E3t
c01
c02
cq3
+βψ6ε6t
c01
c02
cq3
− βδs6ε6t
c0s1
c0s2
cqs3
+ βδa6ε6t
c0a1
c0a2
cqa3
− βδ16ε6t
c061
c062
cq63
. (3.12)
The expressions for coefficients of this system are given in [18].
Taking into account (3.10) and (3.11), we get
ρ
∂2u1t
∂t2
= c16
∂ε6t
∂y
+ c26
∂η
(1)z
t
∂y
, ρ
∂2u2t
∂t2
= c16
∂ε6t
∂x
+ c26
∂η
(1)z
t
∂x
, (3.13)
where
c16 = cE0
66 +
4βψ6
vD
f6 −
2β
vD
[
δ2s6a+ δ2164b+ δ2a6(1 + cosh 2x)
]
,
c26 =
4
v
(
ψ6 −
ϕηc
D
f6
)
. (3.14)
f6 = δs6a− δ162bchx, ϕηc =
1
1 − η(1)2
+ βνc(0).
We look for the solutions of the systems (3.12) and (3.13) in the form of harmonic waves
η
(1)z
t = η
(1)
E (x, y)eiωt, η
(3)z
t = η
(3)
E (x, y)eiωt, η
(2)z
t = η
(2)
E (x, y)eiωt,
ε6t = ε6E(x, y)eiωt, u1t = u1E(y)eiωt, u2t = u2E(x)eiωt. (3.15)
Solving the system (3.12) with taking into account (3.15), we find that
η
(1)
E (x, y) =
βµ3
2
F (1)(ω)E3 +
[
−βψ6F
(1)(ω) − βδs6F
(1)
s (ω)
− βδa6F
(1)
a6 (ω) + βδ16F
(1)
1 (ω)
]
ε6E(x, y), (3.16)
where
F (1)(ω) =
(iω)2r(2) + (iω)r(1) + r(0)
(iω)3 + (iω)2r2 + (iω)r1 + r0
, F (1)
s (ω) =
(iω)2r
(2)
s + (iω)r
(1)
s + r
(0)
s
(iω)3 + (iω)2r2 + (iω)r1 + r0
,
F (1)
a (ω) =
(iω)2r
(2)
a + (iω)r
(1)
a + r
(0)
a
(iω)3 + (iω)2r2 + (iω)r1 + r0
, F
(1)
1 (ω) =
(iω)2r
(2)
1 + (iω)r
(1)
1 + r
(0)
1
(iω)3 + (iω)2r2 + (iω)r1 + r0
, (3.17)
and the expressions for r2, . . . , r
(0)
1 are presented in [18].
Taking into account (3.13) and (3.16), we obtain the following wave equations for u1E and u2E:
∂2u1E
∂y2
+ k6u1E = 0,
∂2u2E
∂x2
+ k6u2E = 0, (3.18)
where the wavenumber is
k6 =
ω
√
ρ
√
cE66(ω)
,
whereas
cE66(ω) = cE0
66 +
4βψ6
vD
[
−2ψ6F
(1)(ω) + δs6F
(1)
s (ω) + δ16F
(1)
1 (ω) − δa6F
(1)
a (ω)
]
− 4ϕηcf6
vD
β
[
−2ψ6F
(1)(ω) + δs6F
(1)
s (ω) + δ16F
(1)
1 (ω) − δa6F
(1)
a (ω)
]
+
4βψ6
vD
f6 −
2β
vD
[
δ2s6a+ δ2164b+ δ2a6(1 + cosh 2x)
]
. (3.19)
282
Longitudinal relaxation of ND4D2PO4 type antiferroelectrics
We look for the solutions of (3.18) in the form
u1E = A1 cos k6y +B1 sin k6y, u2E = A2 cos k6x+B2 sin k6x.
As a result,
ε6E(x, y) = k6[−(A1 cos k6y +A2 cos k6x) + (B1 sin k6y +B2 sin k6x)]. (3.20)
We set the boundary conditions in the following form
ε6E(0, 0) = ε6E(l, l) = ε6E(0, l) = ε6E(l, 0) = ε06 . (3.21)
Using expressions (2.11) and (3.17), we find that
ε06 =
e36(ω)
cE66(ω)
E3 , (3.22)
where
e36(ω) = e036+
βµ3
v
[
−2ψ6F
(1)(ω)+δs6F
(1)
s (ω)−δa6F (1)
a (ω)+δ16F
(1)
1 (ω)
]
. (3.23)
Taking into account the boundary conditions (3.22) and (3.20), we get
ε6E(x, y) =
ε06
2
[
−cos k6l − 1
sin k6l
(sin k6y + sin k6x) + (cos k6y + cos k6x)
]
. (3.24)
Using the relation between polarization P3 and the order parameter η(1) and strain ε6 (2.11),
as well as (3.17), we find
P3(x, y, t) = P3E(x, y)eiωt, (3.25)
where
P3E(x, y) = e36(ω)ε6E(x, y) + χε33(ω)E3 ,
and
χε33(ω) = χε033 +
βµ2
3
v
F (1)(ω), ω = 2πν. (3.26)
The longitudinal dielectric dynamic permittivity of a ND4D2PO4 type crystal can be calculated
using the relation
χσ33(ω) =
1
l2
∂
∂E3
l
∫
0
l
∫
0
P3E(x, y)dxdy. (3.27)
Since
1
l2
l
∫
0
l
∫
0
dxdyε6(x, y) =
2ε06
k6
tanh
k6l
2
=
ε06
R(ω)
, (3.28)
where
R6(ω) =
2
k6l
tanh
k6l
2
,
then from (3.27) we find that
χσ33(ω) = χε33(ω) +
1
R6(ω)
e236(ω)
cE66(ω)
. (3.29)
Thereafter, longitudinal dynamic dielectric permittivity of the ND4D2PO4 type crystals is
εσ33(ω) = 1 + 4πχσ33(ω). (3.30)
It should be noted that at ω → ∞ R6(ω) → ∞ and χσ33(ω) → χε33(ω).
283
R.R.Levitskii et al.
4. Sound attenuation and velocity in ND4D2PO4 type crystals
We consider propagation through the ND4D2PO4 type crystals of a sound wave, whose length is
much smaller than sample dimensions. Then, all the dynamic variables, namely, the order parameter
and elementary displacements depend only on the spatial coordinate which is the direction of sound
propagation. For the thin bars cut along [001] we should consider a transverse ultrasound wave
polarized along [010]. Among the derivatives ∂ui
∂xj
only ∂u2
∂x
is different from zero; therefore, instead
of (3.12) and (3.13) we can write
d
dt
η
(1)z
t
η
(3)z
t
η
(2)z
qt
=
c011 c012 cq13
c021 c022 cq23
cq31 cq32 c033
η
(1)z
t
η
(3)z
t
η
(2)z
qt
+βψ6ε6t
c01
c02
cq3
− βδs6ε6t
c0s1
c0s2
cqs3
+ βδa6ε6t
c0a1
c0a2
cqa3
− βδ16ε6t
c011
c012
cq13
,
ρ
∂2u2t
∂t2
= c16
∂ε6t
∂x
+ c26
∂η
(1)z
t
∂x
. (4.1)
Solving the system (4.1), we obtain the wavenumber that coincides with the one found above
k6 =
ω
√
ρ
√
cE66(ω)
. (4.2)
Using (4.2), we can calculate the ultrasound velocity
v6(ω) =
ω
Re(k6)
= Re
√
cE66(ω)√
ρ
(4.3)
and attenuation
α6(ω) = α06 − Im(k6) = α06 − Im
(
(ω)
√
ρ
√
cE66(ω)
)
, (4.4)
where α06 is the constant frequency and temperature independent term, describing contributions
of other mechanisms to the observed attenuation.
5. Longitudinal static dielectric, piezoelectric, and elastic characteristics of
ND4D2PO4 type crystals
In the static limit ω → 0 in (3.26), (3.23), and (3.19), we obtain the isothermal static dielec-
tric susceptibility of a mechanically clamped crystal, coefficient of piezoelectric stress, and elastic
constant and constant field in the antiferroelectric phase in the following form
χε33 = χε033 +
µ2
3
v
β
2κ6
D − 2κ6ϕ
η
c
, (5.1)
e36 = e036 + 2
µ3
v
β
−2κ6 + f6
D − 2κ6ϕ
η
c
, (5.2)
cE66 = cE0
66 +
8ψ6
v
β(−ψ6κ6 + f6)
D − 2κ6ϕ
η
c
− 4βϕηcf
2
6
vD(D − 2κ6ϕ
η
c )
− 2β
vD
(δ2164bchx+ δ2s6a+ δ2a62ch2x). (5.3)
Here we use the notation
κ6 = a+ bchx.
284
Longitudinal relaxation of ND4D2PO4 type antiferroelectrics
In the paraelectric phase, from (5.1)–(5.3) one easily obtains
χε33 = χε033 +
µ2
3
v
β
2(a+ b)
2 − a+ 2b− 2βνc(0)(a+ b)
. (5.4)
e36 = e036 + 2
µ3
v
β
−2ψ6(a+ b) + δs6a− 2δ16b
−a+ 2 + 2b− 2βνc(0)(a+ b)
. (5.5)
cE66 = cE0
66 +
8ψ6
v
β
−2ψ6(a+ b) + δs6a− 2δ16b
−a+ 2 + 2b− 2βνc(0)(a+ b)
− 4β
v
[1 + βνc(0)](δs6a− 2δ16b)
2
(2 + a+ 4b)[−a+ 2 + 2b− 2βνc(0)(a+ b)]
− 2β
v
δ2s6a+ δ2164b+ 2δ2a6
2 + a+ 4b
. (5.6)
Using the known relations between elastic, dielectric, and piezoelectric characteristics, we find
the isothermal constant of piezoelectric stress h36:
h36 =
e36
χε33
; (5.7)
isothermal elastic constant at constant polarization cP66:
cP66 = cE66 + e36h36; (5.8)
isothermal coefficient of piezoelectric strain d36:
d36 =
e36
cE66
; (5.9)
isothermal constant of piezoelectric strain g36:
g36 =
h36
cP66
; (5.10)
isothermal dielectric susceptibility at σ = const:
χσ33 = χε33 + e36d36 . (5.11)
6. Comparison of numerical calculations with experimental data
Let us now evaluate the found above longitudinal dielectric, piezoelectric, and elastic charac-
teristics of the NH4H2PO4 (ADP) and ND4D2PO4 (DADP) crystals and compare them with the
corresponding experimental data. It should be noted that the developed theory is valid, strictly
speaking, only for highly deuterated ND4D2PO4 type crystals. The experimentally established re-
laxational character of ε∗33(ω, T ) dispersion [27–29] in these crystals, according to [13–15] is most
likely related to suppression of tunneling by the short-range interactions. Therefore, proton tunnel-
ing for the NH4H2PO4 type crystals will be neglected. Since the majority of experimental studies
were performed for the paraelectric phase, we shall also restrict our calculations to temperatures
T > TN.
To calculate the paraelectric temperature and frequency dependences of the physical charac-
teristics of the NH4H2PO4 and ND4D2PO4 crystals we need to set the values of the following
parameters:
– energies of proton and deuteron configurations ε′H, w′
H, w′
1H, ε′D, w′
D, w′
1D;
– the long–range interaction parameters νcH(0) and νcD(0);
– deformational potentials ψ6, δs6, δ16, δa6, δ1i;
– effective dipole moments µ3H and µ3D;
– “seed” static dielectric susceptibility χε033, coefficient of piezoelectric stress e036, elastic constants
cE0
66 , cE0
ij ;
– parameters αH, αD.
285
R.R.Levitskii et al.
The volumes of the primitive cell v were taken to be equal to 0,2110·10−21 cm3 for NH4H2PO4
[30], and 0,213·10−21 cm3 for ND4D2PO4 [31]; whereas the crystal density is ρ = 1, 804 g/cm3 [32]
both for NH4H2PO4 and ND4D2PO4.
To determine the mentioned parameters we use the experimental temperature dependences
of the physical characteristics of ADP and DADP crystals. Thus, for ADP we used the data for
εσ33(0, T ) [32,33], ε∗33(ω, T ) [29], d36(T ) [32], sE,P66 (T ) [32], sEij(T ) [32], whereas for DADP we use
εσ33(0, T ) [34], ε∗33(ω, T ) [29], d36(T ) [34], sE66(T ) [34], sEij [34]. Also, using the known relations for
dielectric, piezoelectric, and elastic characteristics of ADP and DADP, we calculated, using the
experimental data of [32,34], the “experimental” temperature dependences of cE66 = 1
sE
66
, e36 = d36
sE
66
,
εε33 = εσ33 − 4π
d236
sE
66
, h36 = d36
χσ
33s
E
66−d
2
36
, cP66 = cE66 + e36h36, g36 = h36
cP
66
.
Using the experimental data for εσ33(0, T )–ε0σ33 , ε
′ε
33(ω, T )–ε0ε33 and TN, we determined the pa-
rameters ε′, w′, νc(0), at which the value µ3 is weakly temperature dependent. Then, using the
experimental data for ε∗33(ω, T ), we determine the value of α, which turns out to be also weakly
temperature dependent: α = [P + R(∆T )] · 10−14 (∆T = T − TN). The energy w′
1 of the proton
configurations without any proton and with four protons next to the PO4 group is much larger
than ε′ or w′. Hereafter we take w′
1 = ∞ (d = 0).
The “seed” quantities χε033, e
0
36, c
E0
66 = 1
sE0
66
are determined by fitting the theoretical curves of
the characteristics to the experimental points at temperatures far from the transition point TN.
To determine the deformational parameters ψ6, δs6, δa6, δ16 we explore their effect on the
temperature curves of the calculated piezoelectric characteristics d36, e36, h36, g36 and of the elastic
constant cE66 and find such a set of the parameters, yielding a good agreement with experimental
data [32,34].
The obtained optimum set of the model parameters for ADP and DADP is given in table 1.
Table 1. Optimum sets of the model parameters for ADP and DADP crystals.
TN,
ε′
kB
, w′
kB
, νc(0)
kB
, µ3, 10−18, χ0ε
33 P , R,
(K) (K) (K) (K) (esu· cm) (s) (s/k)
ADP 148 20 490,0 -10,00 2,10 0,23 0,38 0,0090
DADP 240 78,8 715,4 -17,35 2,75 0,34 6,72 0,0090
ψ6
kB
, δs6
kB
, δa6
kB
, δ16
kB
, c066 · 10−10 e036
(K) (K) (K) (K) (dyn/cm2) (esu/cm2)
ADP -160 1400 100 -300 7.9 10000
DADP -200 2000 200 -100 7.6 28000
Let us note that using the relations ε = −ε′ and w = w′ − ε, we obtain practically the same
values of the proton and deuteron configuration energies of ADP and DADP crystals, as in [21]. In
figures 1a and 1b we show the temperature curves of the calculated longitudinal static dielectric
permittivities of mechanically free and clamped ADP and DADP crystals along with the available
experimental data. Hereafter, in figures for the ADP crystal the dashed lines denote the theoretical
temperature curves calculated within the theory that takes tunneling into account [6]. As one can
see in figure 1, a satisfactory quantitative description of the experimental data is obtained. The
static dielectric permittivities of free and clamped ADP and DADP crystals have finite values at
the transition points and are weakly decreasing functions of temperature. The permittivity εσ33 of
a free crystal is about ∼ 18% larger than the permittivity εε33 of a clamped crystal; this difference
is practically temperature independent. Let us note (see [16]) that in the case of KH2PO4 the
values of εσ33(0) increase by the hyperbolic law at approaching Tc in the paraelectric phase and are
very large at T = Tc. The difference between εσ33(0) and εε33(0) rapidly decreases with temperature
increasing.
The calculated temperature dependences of the coefficients of piezoelectric strain d36 and stress
e36 of ADP and DADP crystals along with the experimental points are given in figures 2, 3. A
286
Longitudinal relaxation of ND4D2PO4 type antiferroelectrics
0 50 100 150
10
15
20
25
30
ε
33
∆T, K
ε
33
ε
ε
33
σ
0 20 40 60 80
10
15
20
25
30
ε
33
∆T, K
ε
33
ε
ε
33
σ
(a) (b)
Figure 1. The temperature dependence of static dielectric permittivities of a clamped εε
33 •, [32]
and free εσ
33 ◦ [32], � [33] NH4H2PO4 crystal (a), as well as clamped �, [34] and free �, [34]
N(H0.02D0.98)4(H0.02D0.98)2PO4 crystal (b).
0 50 100 150
1
1.5
2
2.5
3
x 10−6d
36
, esu/dyn
∆T, K 0 20 40 60 80
1
1.5
2
2.5
3
x 10−6d
36
, esu/dyn
∆T, K
(a) (b)
Figure 2. The temperature dependence of the coefficient of piezoelectric strain d36 of NH4H2PO4
◦, [32]; N(H0.02D0.98)4(H0.02D0.98)2PO4 �, [34].
good quantitative description of the experimental points is obtained. At T = TN the coefficients
d36 and e36 are finite and decrease with temperature increasing. The coefficients d36 and e36 of
KH2PO4 at T = Tc are about one order of magnitude larger than the corresponding values in the
ADP crystal and decrease with temperature increasing much faster than the coefficients d36 and
e36 of ADP [16].
In figures 4 and 5 we plot the temperature dependences of the constants of piezoelectric stress
h36 and piezoelectric strain g36 of ADP and DADP crystals. The experimental data are well de-
scribed by the proposed theory. The constants h36 and g36 are practically temperature independent.
The temperature dependences of the h36 and g36 constants of KH2PO4 are also weak, with their
values being nearly three times smaller than the values of h36 and g36 of ADP. Even though the
dielectric permittivities of ADP and DADP along the c-axis are relatively small, the values of the
constants of piezoelectric strain and piezoelectric stress in this direction are rather significant.
The temperature dependences of the calculated isothermal elastic constants cE66 and cP66 of ADP
(a) and DADP (b) well agree with the corresponding experimental data (see figure 6). The elastic
constants cE66 of ADP and DADP, in contrast to those of KH2PO4, are finite at T = TN and hardly
depend on temperature.
Let us analyse now the temperature and frequency dependences of the calculated dynamic
287
R.R.Levitskii et al.
0 50 100 150
0.6
0.8
1
1.2
1.4
1.6
1.8
x 105e
36
, esu/cm2
∆T, K 0 20 40 60 80
0.6
0.8
1
1.2
1.4
1.6
1.8
x 105 e
36
, esu/cm2
∆T, K
(a) (b)
Figure 3. The temperature dependence of the coefficient of piezoelectric stress e36 of NH4H2PO4
•, [32]; N(H0.02D0.98)4(H0.02D0.98)2PO4 �, [34].
0 50 100 150
0.8
0.85
0.9
0.95
1
1.05
1.1
1.15
1.2
x 105
∆T, K
h
36
, dyn/esu
0 20 40 60 80
0.8
0.85
0.9
0.95
1
1.05
1.1
1.15
1.2
x 105
∆T, K
h
36
, dyn/esu
(a) (b)
Figure 4. The temperature dependences of the constant of piezoelectric stress h36 of NH4H2PO4
•, [32]; N(H0.02D0.98)4(H0.02D0.98)2PO4 �, [34].
0 50 100 150
1
1.1
1.2
1.3
1.4
1.5
x 10−6g
36
, cm2/esu
∆T, K 0 20 40 60 80
1
1.1
1.2
1.3
1.4
1.5
x 10−6g
36
, cm2/esu
∆T, K
(a) (b)
Figure 5. The temperature dependences of the constant of piezoelectric strain g36 of NH4H2PO4
•, [32]; N(H0.02D0.98)4(H0.02D0.98)2PO4 �, [34].
288
Longitudinal relaxation of ND4D2PO4 type antiferroelectrics
0 50 100 150
5.5
6
6.5
7
7.5
8
x 1010
∆T, K
c
66
, dyn/cm2
c
66
P
c
66
E
0 20 40 60 80
6
6.5
7
7.5
8
x 1010
∆T, K
c
66
, dyn/cm2
c
66
P
c
66
E
(a) (b)
Figure 6. The temperature dependences of the elastic constants cE
66 •, [32] and cP
66 ◦ [32] of
NH4H2PO4; cE
66 �, [34] and cP
66 �, [34] of N(H0.02D0.98)4(H0.02D0.98)2PO4.
characteristics of mechanically free ADP and DADP crystals cut in the [001] plane as thin square
plates with sides l = 1 mm long. Unfortunately, we are not aware of a corresponding experimental
measurement. From the equation for resonance frequencies
νn =
2n+ 1
2l
√
cE66
ρ
for NH4H2PO4 and n = 1 we obtain the value of the first resonance frequency ν1 ≈ 0.92793 MHz
at ∆T = 28 K. Depending on frequency ν (in the resonance region) and temperature ∆T , the
temperature curves of real and imaginary parts of dielectric permittivity of mechanically free ADP
and DADP crystals exhibit one, two, or more resonance peaks.
The calculated frequency curves of real and imaginary parts of dielectric permittivity ε∗33(ω, T )
and experimental points of [29] are presented in figure 7 for ADP at ∆T = 28 K and in figure 8 for
DADP at ∆T = 64 K. In the frequency range of 106 − 108 Hz a resonance dispersion is observed.
104 106 108 1010 1012
0
10
20
30
40
104 106 108 1010 1012
10−6
10−5
10−4
10−3
10−2
10−1
100
101ε’
33
ε’’
33
ν, Hz ν, Hz
Figure 7. Frequency curves of real and imaginary part of dielectric permittivity of free and
clamped (dashed line) NH4H2PO4 crystals at ∆T = 28 K, � – [29].
At ω → 0 we obtain a static dielectric permittivity of a free crystal. The dashed line corresponds
to the low-frequency part of the clamped permittivity. Above the resonances, the permittivity
corresponds to a clamped crystal and has a relaxational character.
Theoretical results and experimental points for the temperature dependences of real and imag-
inary parts of complex dielectric permittivity ε∗33(ω, T ) of ADP and DADP at frequencies where
289
R.R.Levitskii et al.
104 106 108 1010 1012
0
10
20
30
40
104 106 108 1010 1012
10−6
10−5
10−4
10−3
10−2
10−1
100
101ε’
33
ε’’
33
ν, Hz ν, Hz
Figure 8. Frequency curves of real and imaginary part of dielectric permittivity of free and
clamped (dashed line) N(H0.02D0.98)4(H0.02D0.98)2PO4 crystals at ∆T = 64 K, M – [27,29].
0 40 80 120
0
5
10
15
20
0 40 80 120
0
1
2
3
4
5
6
7
8
9
ε’
33
ε’’
33
∆T,K ∆T,K
1
1
2
3
4
5
6
7
8
9
7
8
9
Figure 9. The temperature dependence of ε′33 and ε′′33 of NH4H2PO4 at different frequencies ν
(GHz): 9.2 – 1, ◦[35]; 180.0 – 2, M[29]; 249.9 – 3, .[29]; 320.1 – 4, O[29]; 390.0 – 5, /[29]; 600.0
– 6; 1000.0 – 7; 2000.0 – 8; 5000.0 – 9. Symbols are experimental points; lines are theoretical
results.
0 40 80 120
0
5
10
15
20
0 40 80 120
0
1
2
3
4
5
6
7
8
9
ε’
33
ε’’
33
∆T,K ∆T,K
1
4
2
3
5
2
3
4
5
6
7
6
7
1
Figure 10. The temperature dependence of ε′33 and ε′′33 of N(H0.02D0.98)4(H0.02D0.98)2PO4 at
different frequencies ν (GHz): 9.2 – 1; 80.0 – 2; 150.0 – 3; 262.0 – 4, �[27,29]; 330.0 – 5 M[27,29];
437.0 – 6 ◦[27,29]; 540.0 – 7 ♦[27,29]. Symbols are experimental points; lines are theoretical
results.
290
Longitudinal relaxation of ND4D2PO4 type antiferroelectrics
the effect of crystal clamping by a high-frequency field takes place are given in figures 9, 10, re-
spectively. As one can see, the experimental data of [27,29] are quantitatively well described by
the proposed theory. At the transition temperature the real and imaginary parts of permittivity
ε∗33(ω, T ) of ADP have finite maxima at all frequencies. With ∆T increasing the values of ε
′
33(ω, T )
and ε
′′
33(ω, T ) slightly decrease at all frequencies.
In the temperature curves of ε
′
33(ω, T ) and ε
′′
33(ω, T ) of DADP a maximum is observed at
T = TN at frequencies below the dispersion frequency and there is a shallow minimum at higher
frequencies. With ∆T increasing at dispersion frequencies the values of ε
′
33(ω, T ) and ε
′′
33(ω, T )
increase, reaching a maximum, which shifts to higher ∆T with frequency increasing.
The calculated frequency dependences of ε∗33(ω, T ) along with the experimental points are
presented in figure 11 for ADP and in figure 12 for DADP. A good quantitative description of
1010 1011 1012 1013
0
5
10
15
20
1010 1011 1012 1013
0
2
4
6
8
10
ε’
33
ε’’
33
ν, Hz ν, Hz
1
2
3
1
2
3 4
4
Figure 11. Frequency dependence of ε′33 and ε′′33 of NH4H2PO4 at different temperatures ∆T(K)
[29]: 0.0 – 1; 5.0 – 2, ◦; 28.0 – 3, �; 82.0 – 4, M. Symbols are experimental points; lines are
theoretical results.
1010 1011 1012
0
5
10
15
20
1010 1011 1012
0
2
4
6
8
10
ε’
33
ε’’
33
ν, Hz ν, Hz
1
5
1
5
Figure 12. Frequency dependence of ε′33 and ε′′33 of the N(H0.02D0.98)4(H0.02D0.98)2PO4 crystal
at different temperatures ∆T(K) [27,29]: 0.0 – 1; 19.0 – 2, ◦; 41.0 – 3, �; 64.0 – 4, M; 108.0 – 5,
♦. Symbols are experimental points; lines are theoretical results.
experimental data is obtained. The experimental frequency dependences of ε∗33(ω, T ) for DADP
are for the dispersion region (1011−−1013Hz), whereas for ADP they are below the dispersion. At
∆T = 0 K the dispersion frequency for ADP equals 2062 GHz, whereas for DADP it is 228.5 GHz.
With temperature ∆T increasing the dispersion frequency of ε∗33(ω, T ) slightly increases in DADP
and does not change in ADP.
The temperature and frequency dependences of sound attenuation α6 of ADP and DADP
291
R.R.Levitskii et al.
0 50 100 150 200 250
10−2
100
102
104
106
108 α
6
, cm−1
∆T, K
1
2
3
1’
2’
3’
4’
4
104 106 108 1010 1012
10−2
100
102
104
106
108
ν, Hz
α
6
, cm−1
1 2
Figure 13. Temperature dependence of sound
attenuation α6 of NH4H2PO4 (1,2,3,4),
N(H0.02D0.98)4(H0.02D0.98)2PO4 (1’,2’,3’,4’)
crystals at different frequencies ν, Hz: 1,1’ –
106, 2,2’ – 109, 3,3’ – 1011, 4,4’ – 1013, and
DADP at the same frequencies.
Figure 14. Frequency dependence of sound
attenuation α6 of NH4H2PO4 (1) and
N(H0.02D0.98)4(H0.02D0.98)2PO4 (2) crystals
at ∆T=28K and 64K, respectively.
crystals are shown in figure 13, 14, respectively. At T = TN the attenuation α6 is finite and slightly
decreases with temperature increasing. Below 108 Hz attenuation α6 is small, whereas at further
increase of frequency up to 1011 Hz α6 it rapidly increases and saturates. Such high values of α6
at saturation mean that sound does not propagate in the crystal. In contrast, in the KH2PO4 type
crystals, the attenuation rapidly increases at temperatures close to T = Tc.
In figure 15 we plot the calculated temperature dependence of the sound velocity v6 for ADP(a)
and DADP(b) crystals. The sound velocity is practically independent of temperature and frequency,
0 50 100 150
0
0.5
1
1.5
2
x 105 v
6
, cm/c
∆T, K 0 20 40 60 80
0
0.5
1
1.5
2
x 105 v
6
, cm/c
∆T, K
a b
Figure 15. The temperature dependence of sound velocity in the NH4H2PO4 (a) and
N(H0.02D0.98)4(H0.02D0.98)2PO4 (b) crystals. •, � are calculated as v6 =
√
cE
44√
ρ
[32,34].
except for the frequency region where the dispersion of the clamped dielectric permittivity is
observed; in this region the sound velocity v66 rapidly increases and saturates.
7. Concluding remarks
In this paper, using the modified proton ordering model for the KH2PO4 family crystals, with
taking into account the linear in the strain ε6 contribution to the proton system energy, without
292
Longitudinal relaxation of ND4D2PO4 type antiferroelectrics
tunneling, within the framework of the four-particle cluster approximation, we develop a theory
of dynamic longitudinal dielectric, piezoelectric, and elastic properties of the ND4D2PO4 type
antiferroelectrics. Sound velocity and attenuation in these crystals are also calculated. Numerical
analysis of the dependences of the found characteristics on the values of the theory parameters
is performed. Optimum sets of the model parameters and “seed” quantities for ND4D2PO4 and
NH4H2PO4 crystals are found. They permit a satisfactory description of the available experimental
data.
The piezoelectric coupling (ψ6 6= 0) being taken into account gave rise to understandable
differences between static dielectric permittivities of mechanically free εσ33 and clamped εε33 crystals.
In the ADP type crystals, the permittivity εσ33 is ≈ 18% larger than εε33, and this difference is
practically temperature independent. The isothermal elastic constants cP66 and cE66 in ADP and
DADP crystals are different, just like in the KH2PO4 type crystals, but they have no peculiarities
at T = TN. The sound attenuation coefficient α6 in the ADP type antiferroelectrics is finite and
has a weak temperature dependence, whereas in the KDP type ferroelectrics it has an anomalous
behavior in the phase transition region.
The obtained results for the ADP crystals are compared with the calculations performed in
[6,7]. It is established that tunneling practically does not affect the static dielectric, piezoelectric,
and elastic characteristics of ADP.
References
1. Yomosa Sh., Nagamiya T., Progr. Theor. Phys., 1949, 4, No. 3, 263.
2. Slater J.C., J. Chem. Phys., 1941, 9, No. 1, 16.
3. Stasyuk I.V., Biletskii I.N., Izv. AN SSSR, ser. fiz., 1983, 47, 705 (in Russian).
4. Stasyuk I.V., Biletskii I.N., Styagar O.N., Ukr. J. Phys., 1986, 31, No. 4, 567.
5. Stasyuk I.V., Levitskii R.R., Zachek I.R., Moina A.P., Phys. Rev. B, 2000, 62, No. 10, 6198.
6. Levitskii R.R., Lisnii B.M., J. phys. study, 2003, 7, No. 4, 431 (in Ukrainian).
7. Levitskii R.R., Lisnii B.M., phys. stat. sol. (b), 2004, 241, No. 6, 1350.
8. Lisnii B.M., Levitskii R.R., Ukr. J. Phys., 2004, 49, No. 7, 701.
9. Stasyuk I.V., Levitskii R.R., Moina A.P., Lisnii B.M., Ferroelectrics, 2001, 254, 213.
10. Lisnii B.M., Levitskii R.R., Baran O.R., Phase Transitions, 2007, 80, No. 1-2, 25.
11. Stasyuk I.V., Levitskii R.R., Moina A.P., Velychko O.V., Ukr. J. Phys.: Reviews, 2008, No. 1, 3 (in
Ukrainian).
12. Stasyuk I.V., Kaminska N.M., Ukr. J. Phys., 1974, 19, No. 2, 237 (in Ukrainian).
13. Stasyuk I.V., Levitskii R.R., Korinevskii N.A., Phys. Stat. Sol. (b), 1979, 91, No. 2, 541.
14. Levitskii R.R., Stasyuk I.V., Korinevsky H.A., Ferroelectrics, 1978, 21, 481.
15. Korinevskii N.A., Levitskii R.R., Teor. Mat. Fiz., 1980, 42, No. 3, 416 (in Russian).
16. Levitskii R.R., Zachek I.R., Vdovych A.S. Preprint of the Institute for Condensed Matter Physics,
ICMP-06-08U, Lviv, 2006 (in Ukrainian).
17. Levitskii R.R., Zachek I.R., Vdovych A.S. Preprint of the Institute for Condensed Matter Physics,
ICMP-07-24U, Lviv, 2007 (in Ukrainian).
18. Levitskii R.R., Zachek I.R., Vdovych A.S. Preprint of the Institute for Condensed Matter Physics,
ICMP-08-19U, Lviv, 2008 (in Ukrainian).
19. Levitskii R.R., Zachek I.R., Vdovych A.S. Preprint of the Institute for Condensed Matter Physics,
ICMP-08-20U, Lviv, 2008 (in Ukrainian).
20. Levitsky R.R., Zachek I.R., Mits Ye.V., Moina A.P. Physical collection. Lviv: NTS, 1998, 3, 417 (in
Ukrainian).
21. Levitsky R.R., Zachek I.R., Vdovych A.S., Sorokov S.I. Condens. Matter Phys., 2009, 12, No. 1, 603.
22. Levitsky R.R., Vdovych A.S., Zachek I.R., Visnyk NULP, fiz. mat. nauky, 2008, 601, No. 625, 65 (in
Ukrainian).
23. Moina A.P. Levitskii R.R., Zachek I.R., Phys. Rev. B, 2005, 71, 134108.
24. Levitsky R.R., Zachek I.R., Moina A.P., Vdovych A.S., Condens. Matter Phys., 2008, 11, No. 3(55),
555.
25. Levitskii R.R., Korinevskii N.A., Stasyuk I.V., Ukr. J. Phys., 1974, 19, No. 8, 1289 (in Russian).
26. Glauber J., J. Math. Phys., 1963, 4, No. 2, 294.
293
R.R.Levitskii et al.
27. Levitsky R.R., Zachek I.R., Mits Ye.V., Volkov. A.A., Kozlov G.V., Lebedev S.P., Preprint of the
Bogolyubov Institute for Theoretical Physics, ITP-82-2R, Kyiv, 1982 (in Russian).
28. Volkov A.A., Kozlov G.V., Lebedev S.P., Fiz. Tverd. Tela, 1980, 22, No. 10, 3064 (in Russian).
29. Kozlov G.V., Lebedev S.P., Prokhorov A.M., Volkov A.A., J. Phys. Soc. Japan, 1980, 49, 188.
30. Fukami T., J. Phys. Soc. Jpn., 1988, 57, No. 4, 1287.
31. Kentsig V. Ferroelectrics and antiferroelectrics, Moscow, 1960 (in Russian).
32. Mason W. Piezoelectric crystals and their application in ultraacoustics, Moscow, 1952 (in Russian).
33. Matthias B., Merz W., Scherrer P., Helv. Phys. Acta, 1947, 20, 273.
34. Mason W.P., Mattias B.T., Phys. Rev., 1952, 88, No. 3, 477.
35. Kaminov I.P., Phys. Rev., 1965, 138, No. 5A, 1539.
Поздовжня релаксацiя антисегнетоелектрикiв типу
ND4D2PO4. П’єзоелектричний резонанс та поглинання звуку
Р.Р.Левицький1 , I.Р.Зачек2, А.P.Моїна1, А.С.Вдович1
1 Iнститут фiзики конденсованих систем НАН України, 79011 Львiв, вул. Свєнцiцького, 1, Україна
2 Нацiональний унiверситет “Львiвська полiтехнiка”, 79013 Львiв, вул. С. Бандери, 12, Україна
Отримано 2 квiтня 2009 р., в остаточному виглядi – 18 травня 2009 р.
В рамках модифiкованої протонної моделi з врахуванням взаємодiї зi зсувною деформацiєю ε6
розглянуто динамiчний дiелектричний вiдгук антисегнетоелектрикiв типу ND4D2PO4. Враховано ди-
намiку п’єзоелектричної деформацiї. Явно описано явища затискання кристалу високочастотним
електричним полем, п’єзоелектричного резонансу i НВЧ дисперсiї, що спостерiгаються на експе-
риментi. Розраховано швидкiсть та коефiцiєнт поглинання звуку. Передбачено характер поведiнки
коефiцiєнта поглинання в парафазi та наявнiсть обрiзаючої частоти у частотнiй залежностi коефi-
цiєнта поглинання звуку. При належному виборi мiкропараметрiв в параелектричнiй фазi отримано
добрий кiлькiсний опис експериментальних даних для поздовжнiх статичних дiелектричних, п’єзо-
електричних i пружних характеристик та швидкостi звуку для ND4D2PO4 i NH4H2PO4.
Ключовi слова: антисегнетоелектрики, дiелектрична проникнiсть, п’єзоелектричний резонанс
PACS: 77.22.Ch, 77.22.Gm, 77.65.Bn 77.84.Fa, 77.65.Fs
294
|