ISSN 10628738, Bulletin of the Russian Academy of Sciences. Physics, 2011, Vol. 75, No. 10, pp. 1388–1393. © Allerton Press, Inc., 2011.
Original Russian Text © R. Levitskii, I. Zachek, L. Korotkov, A. Vdovych, S. Sorokov, 2011, published in Izvestiya Rossiiskoi Akademii Nauk. Seriya Fizicheskaya, 2011, Vol. 75,
No. 10, pp. 1473–1478.
Features of the Behavior of Longitudinal Static
and Dynamic Dielectric, Piezoelectric, and Elastic Characteristics
of KH2PO4 and NH4H2PO4 Crystals
R. Levitskiia, I. Zachekb, L. Korotkovc, A. Vdovycha, and S. Sorokova
aInstitute
for Condensed Matter Physics, National Academy of Sciences of Ukraine, Lviv, 79011 Ukraine
b
Lviv Polytechnic National University, Lviv, 79013 Ukraine
cVoronezh State Technical University, Voronezh, 394026 Russia
email: vas@icmp.lviv.ua, sorok@mail.lviv.ua
Abstract—Along with their electromechanical coupling coefficients, the longitudinal dielectric, piezoelec
tric and elastic characteristics in ferroelectric KH2PO4 and antiferroelectric NH4H2PO4 crystals are calcu
lated using a modified proton ordering model that considers piezoelectric coupling and the fourparticle clus
ter approximation. The possibility of detecting piezoactivity in solid solutions of K1 – x(NH4)xH2PO4 is sub
stantiated.
DOI: 10.3103/S1062873811100212
INTRODUCTION
Ferroelectric crystals of the KH2PO4 family exhibit
piezoelectric behavior that affects their physical
behavior. This problem, however, has not been prop
erly studied in the past. Descriptions of the dielectric
features of the ferroelectric KH2PO4 family com
pounds using a common proton model is generally
limited to examining their static behavior and high
frequency relaxation [1–3]. Studies of piezoelectric
resonance on a model without piezoelectric coupling
are meaningless. It should be noted that the only way
to correct measurements of the highfrequency
dielectric behavior of the KH2PO4 family compounds
is to consider piezoelectric coupling. In contrast, the
classical proton model is unable to describe the effects
of unequal unclamped and clamped crystals, or the
phenomenon of a crystal clamped in a highfrequency
field.
A modified proton ordering model considering
piezoelectric coupling and fourparticle cluster
approximation was used to obtain static and dynamic
dielectric, piezoelectric and elastic behavior of the
KH2PO4 family crystals in [4–8]. A detailed numerical
analysis was made of the results, and the optimum sets
of microparameters were found in order to describe
well quantities of the experimental data.
In addition, a cluster theory of the thermody
namic and dynamic behavior of compounds of the
Rb1 – x(NH4)xH2PO4 type was developed in [9–13]. If
the parameters are selected properly, the theory
describes experimental data in terms of quantity fairly
well. Possible piezoelectric coupling in the crystal is
ignored in the proposed theory, however, though the
authors acknowledge its existence [14]. The results
obtained in [9–13] for ferroelectric (FE) KH2PO4
(KDP) and antiferroelectric (AFE) NH4H2PO4 (ADP)
are reviewed briefly in this work. The electromechani
cal coupling factors of the crystals are estimated
numerically. The possibility of detecting piezoelectric
coupling in compounds of K1 – x(NH4)xH2PO4 type is
discussed.
DIELECTRIC, PIEZOELECTRIC,
AND ELASTIC CHARACTERISTICS
Dynamic behavior of the FE KDP and AFE ADP
in the presence of piezoelectric coupling and deforma
tion ε 6 are studied using a dynamic model of ferro
electric orthophosphate using fourparticle cluster
approximation, which is based on the stochastic
Glauber model. The dynamic deformation process in
the crystals is modeled with classical motion equations
[7, 8]. On the basis of these techniques, we can write
the longitudinal dynamic susceptibility of mechani
cally unclamped KDP and ADP crystals:
σ
ε
χ33
(ω) = χ33
(ω) +
2
e36
(ω)
,
E
R6 ( ω) c66 ( ω)
1
(1)
where
1 = 2 tg k6l , k = ω ρ ,
6
E
R6 ( ω) k6l 2
c66
( ω)
(2)
and ω is a frequency of external field, ρ is a density of
the crystal, and l is the length of a thin square crystal
plate (l = 1 mm) made in the plane (001).
1388
FEATURES OF THE BEHAVIOR OF … CHARACTERISTICS … CRYSTALS
The longitudinal dynamic susceptibility of the
ε
mechanically clamped crystal χ33
(ω) in (1), and the
E
e36(ω) and c66(ω) for KDP, are determined as follows:
ε
ε0
χ 33
+
( ω) = χ 33
βμ 32 (1)
F ( ω) , β = 1 ,
k BT
2υ
(3)
βμ3
Ls ( ω) ,
υ
(4)
0
e36 ( ω) = e36
+
4βψ 6
2β
fs +
−δ s6 M s6
2(
υ Ds
υ Ds
4βψ 6
2
Ls ( ω)
+ δ16 M 16 + δ a6 M a6 ) +
υ
4ϕ f
2β ⎡ 2
− s s β Ls ( ω) −
δ s6 ch ( 2z 6 + βδ s6ε 6 )
υ Ds
υ Ds ⎣
E0
(5)
+ 4bδ16 ch ( z 6 − βδ16ε 6 ) + δ a6 2a ch βδ a6ε 6 ⎤⎦ ,
2
2
E
Along with the e36(ω) and c66
(ω) values for an ADP
crystal, the longitudinal dynamic susceptibility of a
ε
mechanically clamped crystal χ33
(ω) is
βμ 3 (1)
F ( ω) ,
υ
βμ
0
e36 ( ω) = e36
+ 3 La ( ω) ,
υ
ε
2
where kB is the Boltzmann constant; μ3 is the effective
dipole moment of a hydrogen bond; υ is the volume of
a primitive crystal cell consisting of two PO4 tetrahe
drons; ε6 is a spontaneous deformation; ψ6, δs6, δ16,
ε0
0
E0
and δa6 are deformation potentials; and χ33
are
, e36
, c66
the initial values of dielectric susceptibility, the piezo
electric coefficient, and the elastic constant, respec
tively:
( ω) −
δ a6Fa(1)
( ω) ,
while the expressions for F (1) ( ω) , Fs(1) ( ω) , Fa(1) ( ω) ,
F1(1) ( ω) , are listed in [15]:
f s = δ s6 cosh ( 2z 6 + βδ s6ε6 ) − 2bδ16 cosh ( z 6 − βδ16ε 6 )
+ ηs ( −δ s6M s6 + δa6M a6 + δ16M16 ) ;
M a6 = 2asinhβδa6ε6,
M s6 = sinh ( 2z 6 + βδ s6ε 6 ) , M16 = 4bsinh ( z 6 − βδ16ε 6 ) ,
ϕs =
(6)
(7)
4βψ 6
4ϕ f
4βψ 6
La ( ω) − a a β La ( ω) +
fa
υDa
υDa
υDa
(8)
2β ⎡ 2
2
2
⎤
−
δ s6a' + δ16 4b' + δ a6 (1 + cosh 2x )⎦ ,
υDa ⎣
where La ( ω) = −2ψ 6F (1) ( ω) + δ s6 Fs(1) ( ω) − δ a6Fa(1) ( ω) +
δ16F1(1) ( ω) , the expressions for F (1) ( ω), Fs(1) ( ω), Fa(1) ( ω),
F1(1) ( ω) are listed in [16].
f a = δ s6a' − δ16 2b' cosh x, ϕa =
1 + βν ,
c
1 − η2s
ηs = 1 ( sinh ( 2z 6 + βδ s6ε 6 ) + 2bsinh ( z 6 − βδ16ε 6 )) ,
Ds
Ds = cosh ( 2z 6 + βδ s6ε 6 ) + 4bcosh ( z 6 − βδ16ε 6 )
+ 2acoshβδ a6ε 6 + d,
1 + ηs
z 6 = 1 ln
+ βν cηs − βψ 6ε 6,
2 1 − ηs
a = e −βε, b = e −βw, d = e −βw1,
where ε, w, w1 are the energy parameters of the Slater–
Takagi model.
1 + βν (0);
c
1 − η2a
ηa = 1 ( sinh2x + 2b'sinhx ) ,
Da
Da = a' + cosh 2x + d ' + 4b' cosh x + 1,
1 + ηa
+ βν a k z ηa.
x = 1 ln
2 1 − ηa
( )
Ls ( ω) = −ψ 6 F (1) ( ω) + δ s6 Fs(1) ( ω)
+δ16F1(1)
2
ε0
χ 33 ( ω) = χ 33 +
E
c66
(ω) = c66E 0 +
c66 ( ω) = c66 +
E
1389
a' = e
−βε'
, b' = e
−β w'
, d' = e
−β w1'
,
where ε', w', w1' are the antiferroelectric energy values
of the Slater–Takagi model.
At the static limit (ω → 0 ), Eqs. (3)–(8) yield
expressions for the isothermal static dielectric suscep
tibility of clamped crystals:
μ2
2κ s
ε
ε0
KDP: χ 33
= χ 33
+ υ 32 1
, υ= υ,
kB
υ T D s − 2κ s ϕ s
(9)
μ2
2κ a
ε
ε0
ADP: χ 33
= χ 33
+ υ 32 1
,
υ T Da − 2κ a ϕ a
(10)
where
κ s = ch ( 2z6 + βδ s6ε6 ) + b ch ( z6 − βδ16ε6 ) − ηs Ds,
κa = a' + b' ch x.
Isothermal piezoelectric coefficients are found in a
similar manner:
In the case of KDP:
2μ β ( − 2κ s ψ 6 + f s )
0
(11)
+ 3
,
e36 = e36
υ Ds − 2ϕ s κ s
in the case of ADP:
μ β ( − 2κ a ψ 6 + f a )
0
(12)
+2 3
e36 = e36
,
υ Da − 2ϕa κ a
BULLETIN OF THE RUSSIAN ACADEMY OF SCIENCES. PHYSICS
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LEVITSKII et al.
Table 1. Collections of optimum model parameters for KDP crystal
Tc
ε/kB
w/kB
νc/kB
μ3–, 10–18
μ3+, 10–18
(K)
(K)
(K)
(K)
(CGSEq ⋅ cm)
(CGSEq ⋅ cm)
122.5
56.00
422.0
17.91
1.46
1.71
0.73
ψ6/kB
δs6/kB
δa6/kB
δ16/kB
0
c66
⋅ 10 −10
0
e36
(K)
(K)
(K)
(K)
(d/cm2)
(CGSEq/cm)
–150.00
82.00
–500.00
–400.00
7.10
1000.00
and the KDP and ADP crystals’ isothermal elastic ele
ments in a constant field are
c66 = c66 +
E
E0
(
8ψ 6 β −ψ 6 κ s + f s
υ Ds − 2ϕ s κ s
)
4βϕ s f s
2β ⎡ 2
−
δ s6cosh ( 2z 6 + βδ s6ε 6 )
(13)
Ds ⎣
υ
υ Ds (Ds − 2ϕ s κ s )
2
−
0
χ33
2
4bcosh ( z 6 − βδ16ε 6 )⎤⎦
+ δ 2a6 2acoshβδ a6ε 6 + δ16
2β
2
−δ s6 M s6 + δ a6 M a6 + δ16 M 16 ) ,
2(
υ Ds
+
E
E0
c66
= c66
+
−
8ψ 6 β ( −ψ 6 κ a + f a )
4βϕa f a2
−
υ Da − 2ϕa κ a
υDa Da − 2ϕa κ a
(
2β 2
δ16 4b'chx + δ 2s6a' + δ 2a6 2ch 2 x .
υDa
(
) (14)
)
Assuming that σ = const for KDP and ADP crys
tals, the isothermal elastic constant for fixed polariza
P
E
2
ε
tion c66
= c66
+ e36
χ33
;, the isothermal piezoelectric
E
strain coefficient d36 = e36 c66
, and isothermal dielec
ε
σ
tric susceptibility χ 33
= χ33
+ e36d36 can be found on
the basis of the wellknown relations between elastic,
dielectric and piezoelectric values [5, 6].
Table 2. Collections of optimum model parameters for ADP
crystal
0ε
TN
ε'/kB
w'/kB
νc(0)/kB
μ3+, 10–18
(K)
(K)
(K)
(K)
(CGSEq ⋅ cm)
148
20
490.0
–10.00
2.10
ψ6/kB, δs6/kB, δa6/kB, δ16/kB,
(K)
(K)
(K)
(K)
–160
1400
100
–300
0
c66,10
−10
χ33
0.23
0
e36
(d/cm2) (CGSEq/cm)
7.9
10000
COMPARING THE NUMERICAL RESULTS
AND EXPERIMENTAL DATA
Note that the theoretical results presented above
are, strictly speaking, valid for deuterated crystals of
DKDP and DADP. Considering the suppression of
tunneling by shortterm correlations [17], we shall
assume that these results are correct for both KDP and
ADP.
To calculate the physical characteristics of KDP
and ADP crystals, our theory proposes that the physi
cal characteristics of KDP and ADP crystals be calcu
lated on the basis of the predefined parameters
(Tables 1 and 2) found in [15, 16].
The volume of a primitive KDP crystal cell is taken to
be υ = 0.1936 × 10–21 cm3, and υ = 0.2110 × 10–21 cm3 for
ADP. Accordingly, the density of KDP and ADP crys
tals ρ = 2.31 g/cm3 and ρ = 1.804 g/cm3 [18].
The temperature dependence of calculated iso
thermal static dielectric permeability of unclamped
σ
ε
and clamped ε 33
crystals of KDP and ADP are pre
ε 33
sented in Fig. 1 together with experimental data. In the
σ
paraelectric phase, the value of ε 33
grows hyperboli
cally in the vicinity of Tc and reaches very high values
σ
(Т)
at T = Tc. Below Tc, when the temperature falls, ε 33
decays quickly. The CurieWeiss law is thus valid in the
temperature interval ΔT = T – Tc ≈ 50 K above the
σ
for ADP is approximately
value Tc. Permeability ε 33
ε
18% higher than ε 33 . This difference is insensitive to
temperature.
The calculated temperature dependence of piezo
electric strain coefficient d36 for KDP and ADP crys
E
tals is presented in Fig. 2. Elastic constant c66
of the
KDP crystal converges to zero when Т → Тс, while the
P
(Т) does not reveal any anomalies in the
function c66
vicinity of the phase transition. Unlike KDP, elastic
E
constant c66
for ADP reaches finite values at T = TN
and is insensitive to temperature.
As we can see in Figs. 1–3, the proposed theory
adequately describes experimental data for static
dielectric, piezoelectric, and elastic KDP and ADP
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FEATURES OF THE BEHAVIOR OF … CHARACTERISTICS … CRYSTALS
104
ε33
1391
d36, CGSE q d–1
10–3
10–4
103
1'
10–5
1
102
2
10–6
2'
1
2
1
10
100
140
180
10–7
220 T, K
100
Fig. 1. Temperature dependence of dielectric permeability
ε
ε33
ε
ε33
of clamped
(1) and unclamped
(1') KDP crystals,
of clamped (2) and unclamped (2') ADP crystals. Curves
correspond to the theoretical results; symbols mark exper
imental data adopted from referenced sources [18, 25].
E
c66
, 1010 d cm–2
8
1'
2'
6
2
ε'33
103
ε''33
600
4
100
400
2
0
100
300 T, K
Fig. 2. Temperature dependence of piezoelectric strain
coefficient d36 of KDP (1) and ADP (2) crystals. Curves
correspond to theoretical results; symbols mark experi
mental data: 䊊, 䊐 [18], 䉮 [19], 䉯 [20].
800
1
200
200
0
140
180
220 T, K
10–3
106 108 1010 1012
ν, Hz
106 108 1010 1012
ν, Hz
E
Fig. 3. Temperature dependence of elastic constants c66
P
and c66 of KDP (1, 1', respectively) and ADP (2, 2') crys
tals. Curves correspond to theoretical results, symbols
mark experimental data: 䊊 [18], 䊐 [21], 䉭 [22], 䉫 [18].
crystals. Note that studies of the temperature depen
2 E
σ
ε
= 4π e36d 36 = 4πd36
dency of the difference ε 33
− ε 33
c66
for KDP and ADP crystals is important in estimating
the values of piezoelectric coefficients and elastic con
stants of ferroelectric compounds.
Let us now examine the results from calculations of
the longitudinal dynamic characteristics of mechani
cally unclamped crystals of KDP and ADP. Diagonal
deformations εi are ignored for the sake of simplicity.
The frequency dependences of the real and imaginary
parts of the longitudinal dielectric permeability of
mechanically unclamped KDP crystals at ΔT = 5 K
Fig. 4. Frequency dependence of real and imaginary parts
of dielectric permeability of unclamped and clamped KDP
crystals at ΔT = 5 K: 䊊 [23], 䊐 [24]. Curves represent the
oretical results.
and ADP crystals at ΔT = 28 K are shown in Figs. 4
and 5. Resonancetype scattering is observed in the
range of 3 ⋅ 105–3 ⋅ 108 Hz. The resonance frequencies
are inversely proportional to the sample size.
Temperature dependency of electromechanical
σ
ε
σ
coupling coefficients k32 = ε 33
of KDP and
− ε 33
ε 33
(
)
ADP crystals are displayed in Fig. 6. The value k32 for
KDP reaches its maximum at transit point and decays
dramatically when pulled away from Tc point, espe
cially in ferroelectric phase. The ADP’s coefficient k32
reaches maximum of approximately 0.35, when T =
BULLETIN OF THE RUSSIAN ACADEMY OF SCIENCES. PHYSICS
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LEVITSKII et al.
ε'33
cluster density in these compounds would also be of
interest.
ε''33
40
101
(a)
100
10–1
10–2
10–3
10–4
10–5
10–6
20
0 4
10
108
1012
ν, Hz
104
(b)
108
1012
ν, Hz
Fig. 5. Frequency dependence of real and imaginary parts
of dielectric permeability of unclamped and clamped
(dashed line) ADP crystals at ΔT = 28 K: 䊐 [26]. Curves
represent theoretical results.
k23
1.0
0.8
KDP
0.6
REFERENCES
0.4
ADP
0.2
0
100
CONCLUSIONS
The expressions of permeability of unclamped and
clamped crystals, the piezoelectric strain coefficient,
the elastic constant, and the dynamic behavior of
KDP and ADP crystals were revealed by considering
piezoelectric coupling. Calculating the temperature
dependency of these characteristics demonstrated the
σ
ε
considerable difference between ε 33
and ε 33
measured
experimentally for KDP crystals, along with minor
differences for ADP. The optimum sets of parameters
and initial characteristics of the crystals under study
were found, and were helpful in describing the avail
able experimental data. The phenomena of clumping
and piezoelectric resonance in the studied crystals
have been described for the first time. Similar research
for compounds of type K1 – x(NH4)xH2PO4 will pro
duce important data whether or not they contain
piezoelectric coupling.
140
180
220 T, K
Fig. 6. Temperature dependence of the electromechanical
coupling coefficient of KDP and ADP crystals.
TN, and descends slowly with temperature growth.
Meanwhile, the ADP’s value of k32 in paraelectric
phase is substantially higher than that of KDP.
The results of this section indicate that piezoelec
tric coupling in crystals of the KH2PO4 family leads to
the difference between the dielectric permeability of
mechanically unclamped and clamped crystals, and
determines their piezoelectric modules and piezoelec
tric resonance. Unfortunately, similar theoretical and
experimental studies have not been undertaken on
irregular compounds such as K1 – x(NH4)xH2PO4. An
exception was described in [14]. In order to prove
there is piezoelectric coupling in a compound, it is
crucial to examine ε '33 ( ν,T ) and ε ''33 ( ν,T ) over wide
temperature and frequency ranges, and to estimate the
electromechanical coupling coefficient and difference
in permeability for unclamped and clamped crystals.
The latter indicates the presence of nonzero piezo
electric modules in a compound. A detailed study of
the listed characteristics as a function of NH4 ionic
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SPELL: 1. ok
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