New Approach for Piezoelectric Resonances and
Phase Transitions in KDP and ADP crystals.
D P Pereira1, P C de Oliveira2, J Del Nero3, 4, P Alcantara Jr3,
C M R Remédios3, S G C Moreira3.
1Centro
Federal de Educação Tecnológica do Pará,66090-100, Belém, Pará, Brazil.
2Departamento
de Física, Universidade Federal da Paraíba, 58051-970,
João Pessoa, Paraíba, Brazil.
3Departamento
de Física, Universidade Federal do Pará, 66075-110,
Belém, Pará, Brazil.
4 Instituto
de Física, Universidade Federal do Rio de Janeiro, 21941-972,
Rio de Janeiro, RJ, Brazil
Abstract
In this work we introduce a new experimental setup, here called ThreeElectrode System (TES), which is very useful to analyze and compare
piezoelectric crystals, as well as to study their phase transitions and measure
their piezoelectric coefficients. ADP and KDP, that are very well known crystals,
were used in this work just as good piezo-crystal examples. The principal TES
characteristic is the application of an electric field on a small region of the
crystal and the observation of the electric field at another region, usually located
at the extremes of the crystal. In this way, we can see several interesting
effects, like piezoelectric resonances, phase transitions and harmonic
generation, which are dependent on voltage and frequency conditions. We
consider the piezoelectric concept for developing a mathematical model for the
generated elastic wave, starting at the excitation electrode and traveling to the
second electrode, where the signal depends on the piezoelectric coefficient.
Our results have shown that this technique is very efficient in the determination
of phase transitions.
1. Introduction
Piezoelectric crystals have been deeply investigated for more than 120
years, mainly due to their actuator and sensor applications. At room
temperature both KDP and ADP crystals are piezoelectric and their structure is
tetragonal with 42m symmetry. The field of applicability is really very vast for
this material class [1]. The propagation signal studies through piezoelectric
crystals have been developed under different experimental configurations [2,3].
Some of these studies consider only two electrodes and other ones three
electrodes [4,5].
Mason [6], in 1945, measured elastic, piezoelectric, and dielectric properties
of KDP and ADP at temperatures above the Curie point (122 K or -151oC). The
piezoelectric properties were explained by a phenomenological theory by
Mueller [7]. Bantle and Caflisch, according to citation of Lang [8], performed a
detailed description of the direct piezoelectric effect measurements for the KDP
crystal, from room temperature up to the Curie point. Still according to Lang [8],
Von Arx and Bantle have used an interferometric technique to find the inverse
piezoelectric effect, under the same temperature interval, which agrees with the
direct piezoelectric effect, showing that the piezoelectric modulus has the same
value for both effects, as described by Nye [9]. Nowadays, new techniques
have being employed to determine the piezoelectric moduli of piezoelectric
crystals, such as the multiple X-ray diffraction [10,11].
Three decades ago the first piezoelectric voltage transformer [12] was built,
which can be applied in small electronic circuits where the traditional inductive
transformer can not be used. This kind of device was built using a setup similar
to the one presented in this paper. The theoretical studies of electromagnetic
wave propagation in piezoelectric crystals have already been developed
[13,14].
Also, classical works showing frequency filter crystals could be explained in
terms of existing elastic wave theory and a cut-off frequency, where this effect
confines the vibratory energy to a limited region surrounding the electrode
portion. For values lower than the cut-off frequency, the vibratory energy
presents exponential behavior with distance of the electrode [15,16,17].
Generally, the piezoelectric crystals properties are defined by their
piezoelectric coefficients, where the crystal symmetry is one of the most
important aspects [9]. These coefficients are specific of each crystal and are
temperature [18] and pressure (uniaxial or hydrostatic) dependent.
Many
authors have characterized phase transitions by piezoelectric properties, since
these properties may present anomalous behavior or discontinuities related to
the phase transitions [19-22]. In this work we report on the characterization of
the ADP and KDP crystals phase transitions by the piezoelectric resonances
using the Three-Electrode System (TES).
The piezoelectric properties of these crystals was largely investigated [8,9]
and it produces a good example in connection with our measurements, as well
as, the proposed model.
2. Experimental
The crystals used in this work were grown by low evaporation method, starting
from a supersaturated solution at constant temperature (35  0.3 oC). After the
crystals are formed the samples were cut and polished to shape a
parallelepiped plate with 5 x 3 x 0.4 mm 3, and the electrodes were painted with
silver tin on [001] on both faces according to Figure 1. In this way we have an
input electrode (electrode 1), an output electrode (electrode 2) and a ground
electrode (electrode 3).
Figure 1. Three electrodes system pictures. The dashed areas are the silvered
regions.
We excited the piezoelectric sample using a Lock-in amplifier (EG&G
model 5302) and the frequency was scanned from 10 kHz up to 500 kHz with 5
Volts pp amplitude. The generated signal at output electrode 2 was connected
to the lock-in input channel.
We perform two kinds of experiments: one at constant temperature,
scanning the frequency and the other one at constant frequency while the
temperature is varied. For the experiments at constant frequency the sample
was placed into a cryostat and the temperature was controlled from -196oC up
to 27oC.
3. Theoretical Modeling
The propagation of a mechanical wave in a crystal follows specific rules that
depend on elastic and piezoelectric properties of the crystal and such wave is
accompanied by an electric field through piezoelectric coupling. Due to the
electric field to be associated with the sound it is necessary to take electrical
conditions into account. The elastic equation of motion is:

It
is
 2 i  ij

.
t 2
x j
convenient
to
take
(1)
the
boundary
conditions:
 (0,0)   (0, L)  0 and  (0,0)   (0, L)  0 , where L is sample length.
Where i are the components of the displacement vector and  is the mass
density. This set of equations can assume different forms, depending on what
parameters are fixed or controlled and also on the symmetries of the crystal. In
our case, having strain  and electric field E as controlled variables, we can
write [24]:
 ij  Cijkl kl  Cijkl d mij E m
Dn  d nkl  kl  nm E m
(2)
where  kl is the strain tensor component, d mij is the piezoelectric coefficient, nm is
the dielectric constant,  ij is the stress tensor component, Cijkl is the elastic
constant, Em is the electric field amplitude, and Dn is the electric displacement.
A solution for equation (2) can be written as a plane wave
  0 .ei (t k .r ) , where
and 0 is the wave amplitude.
k is the wave vector,  is the angular frequency,
Considering a 45o cut and symmetry specific conditions for KDP and
ADP crystals, with an electric field applied on [001] direction, we have:
 1  C11 1  C12 2  C13 3  0
 2  C 22 2  C 23 3  0
 3  C33 3  0
 4  C 44 4  0
 5  C55 5  0
 1  C 66 6  C 66 d 36 E3
(3)
Consequently,
 2  6 C 66  6
S

 C 66 d 36 E3
L
t 2
where
(4)
 S  L .
This equation completely describes the wave propagation of our system.
Considering the experimental setup, we can divide the samples in two
regions, as shown in Figure 2. The region 1 is constituted by all crystal parts
under the electrode 1. This region of the crystal is excited by an electric field,
given by E3  E1 cos(t ) , in the [001] direction and will vibrate with an
0
amplitude 1 . The remaining region of the crystal, despite there is no direct field
action on it, is driven to vibrate because of the mechanical perturbation of the
crystal lattice in addition to the piezoelectric effect. However, the amplitude in
this part will be damped as a function of the distance of propagation. Then,
considering  6 (1)  1 and  6 ( 2 )   2 as the deformations in the two regions,
respectively, we have:

1   .e 
 
0 i t  k . 
   .e
0
1
1
2
 
i t k .
(5)
(6)
In Figure 2, each solution (5) and (6) is indicated in the two different regions.
The equation (5) is the solution for the region 1 while (6) is the solution for the
region 2.
Figure 2. Presents two regions: region 1 droved to an applied electric field with
solution (5) and the region 2 without electric field with solution (6).
Considering the equations (5) and (6) the equation (4) becomes:
 2 1 C 66 1 C 66  2
S


 C 66 d 36 E 3
L
L
t 2
C 66 1 C 66  2
 2 1
S



2
L
L
t
The
solutions
for
the
two
regions
have
(7)
amplitudes 1 and 2 ,
0
0
respectively, given by:
  2 L  C66 
0

  
C
d
E
66
36
1
2 4 3

   L 
0
1
 
0
2
2
C 66
  L
2
4
3
e l d 36 E10
(8a)
(8b)
where  is mechanical wave attenuation coefficient and l is distance between
electrodes. We need to introduce a dissipation factor that is observed in
experiments. This dissipative term arises when we consider the elastic constant
C66 and the dielectric constant 3 as complex numbers. In this case, a
exponential decay as exp(-l) for amplitude (equation 8b) naturally appears.
The electric field amplitude in electrode 2 corresponding to the second
electrode given by [25]:
E20  4P3
(CGS units)
(9)
where P3 is polarization in region 2 with [001] direction. Using the direct
piezoelectric relation, we have
P3  d36 6
(10)
Take into account also the Hooke’s law and equation (8b), we obtain the
voltage in electrode 2:
V20 
4
 L
2
4 4
Introducing the relation
V20 
4
 L
2
4 4
3
2 l 0
C66
d36
e V1
 3  d 362 C66
 3C662 e lV10
(11)
[8]:
(12)
The results (11) and (12) show that the voltage in output electrode (V 20)
carries the dependence with the following parameters: the voltage in input
electrode (V10), the piezoelectric coefficient (d36), the elastic constant (C66) or of
electric susceptibility (3) and constant (C66) and decay exponentially with the
propagation distance (). All these dependences were experimentally observed
in our measurements.
For a realistic case we need to introduce a dissipation factor that is
observed in experiments. This dissipative term arises when we consider the
elastic constant C66 and the dielectric constant ij as complex numbers. In this
case, a exponential decay as exp(-) for the 2 amplitude (equation 6)
naturally appears.
For ADP and KDP, the elastic wave at room temperature, and taking into
account an electric field E // z direction, may be obtained from
 2 6
 2  C66 6  d36E3
t
This equation completely describes the wave propagation of our system.
4. Results and discussions
4.1 ADP crystal
Figure 3 shows the three dimensional experimental data set of the
piezoelectric resonance spectra (PRS) from –80 oC to 5 oC. These spectra were
generated when we collect the output signal as a function of the excitation
frequency. Figure 3 also shows that when the temperature is lowered the output
signal becomes stronger for all frequencies.
20
UDE (1
AMPLIT
10
-3
0 V)
0
300
-80
-60
o
NC
UE
EQ
FR
400
Y
(
-40
RE
U
T
RA
E
P
Hz
(k
-20
500
C)
)
M
TE
0
Figure 3. 3-D graphic contained resonance spectra taken in different
temperatures for the ADP crystal, where there isn’t evidence of phase transition.
This behavior for ADP crystal had already been reported by W. P. Mason
[6], and it is a direct consequence of the decrease of d36 with the temperature.
According equation (11) the output amplitude V20 is directly dependent on (d36)2.
We can also see, around 320 Hz and 400 Hz, that some frequency splitting
arise. Such splitting can be explained by the breaking of the degenerated
acoustic mode, therefore ADP crystal belongs to the 42m symmetry group with
E (double degenerated mode). Other techniques, like Raman spectroscopy,
show the same splitting behavior as a function of the temperature [26].
Furthermore, the temperature effect always leads to a linewidth narrowing and
an increasing of the intensity. Another interesting observation is the resonance
frequency shift as a function of the temperature [27,28]. This resonance
frequency shift is more evident at high temperatures.
The strongest resonance is at f = 260.2 kHz. This fundamental resonance
frequency was used to evaluate the elastic compliance S66 through equation
[29]:
f 
1
2l  S66
Where l is the length of the sample and S66 is the elastic compliance
constant (the elastic constant reverse S66 = 1/C66).
The found value was: S66 
1
4l  f
2
2
 16.9 x 1012 cm2 / dyn
This value is in completely agreement with the literature value [30].
4.2 KDP Results
Figure 4 shows a three dimensional experimental data set of the PRS
from –200 oC to –120 oC, for KDP crystal using TES. In this figure, we can see
the PRS evolution as a function of the temperature. The PRS shows different
qualitative and quantitative characteristics for each phase before and after a
phase transition (at – 151
oC).
Therefore, the PRS obtained with the
experimental setup presented in this article may be used to monitor a phase
transition. Similarly to other spectroscopy techniques, the PRS is related to the
vibrational acoustic modes because the obtained spectra is generated basically
by an electric field applied on piezo material generating a mechanical wave
propagated from electrode 1 to electrode 2. Evidently, the produced strain and
stress inside the sample are symmetry dependent, and any modification on the
elastic parameter, like a phase transition, must reflect on the output signal (and
consequently on PRS). For the case of KDP crystal the well known ferroelectric
phase transition, at –151 oC, can be seeing by the modification presented on
PRS showed in Figure 3. The principal high light in the figure are: a) spectra
intensity increase around Tc = –151 oC, b) there is a characteristic piezoelectric
resonance spectra for each phase of the crystal (paraelectric or ferroelectric).
1400
1200
(mV)
ITUDE
AMPL
1000
800
600
400
200
0
100
z)
kH
Y(
NC
UE
EQ
FR
200
-180
300
-160
o )
C
E(
R
U
T
-140
RA
PE
M
TE
400
500
-120
Figure 4. 3-D graphic contained resonance spectra taken in different
temperatures for the KDP crystal, where is showed the phase transition to –
151oC.
For our experimental setup and taking into account the resonant
frequency at f = 171 kHz and density  = 2.332 g/cm3 for KDP, we obtain the
elastic compliance S66.
S66 
1
4l  f
2
2
 1.66 x 1011 cm2 / dyn
This result is in complete agreement with the literature [30,31,32].
The Table 1 resume the (C66 and d36) evaluated data by equation (11), at
room temperature. Those values have been compared with the literature
values.
Table 1. Obtained values of C66 and d36 constants to ADP and KDP crystals.
Present paper
Other authors
crystal
C66(dyn/cm2)
d36(pC/N) C66(dyn/cm2) [ 30]
d36(pC/N) [6]
ADP
5,9x1010
52
6,02x1010
51,7
KDP
6,02x1010
20,5
6,25x1010
21
Another important result is that there is a temperature and frequency
where the observed output signal is stronger than the input signal. For example,
at –151 oC it was observed an output of 1,400 mV when the input signal was
just 1,000 mV. This represents a 40% increase in the input signal. Other
authors [33,34] already showed the piezo-transformer effect but they use a
different setup to observe such effect.
In a further exploration of the tree-electrode system we also maintained the
frequency constant at f = 80 Hz and scanned the temperature between T 1 = –
200 oC and T2 = 40 oC to observe the output signal behavior. The Figure 5
shows the curves for the studied crystals. Although it is a very simple
experiment, it presents an interesting result because it contains the thermalelastic behavior of the crystal. And more, it is possible to see a phase transition
happening because the elastic and piezo thermal properties affect the output
signal, as we demonstrated in equations (11) and (12).
KDP
ADP
280
AMPLITUDE (mV)
240
200
Frequency = 80 kHz
160
120
80
40
0
-200
-160
-120
-80
-40
0
40
o
TEMPERATURE ( C)
Figure 5. Amplitude of output signal versus temperature. The frequency was
constant at 80 kHz. ■ (KDP) and ▲ (ADP).
5. Conclusion
We introduced a new experimental setup, here named the three-electrode
system (TES). Such system has permitted us to analyze and measure the
piezoelectric resonances on piezoelectric crystals such as ADP, KDP and
Rochelle salt. Through such simple system we observed the phase transitions
undergone by crystals. The experimental setup is relatively simple, easy to work
and provides a new way to test the piezoelectricity in crystals, ceramics or on
any piezoelectric materials. Furthermore, the experimental setup was very
sensitive to verify the occurrence of the phase transitions in the tested samples.
We developed a theoretical model to evaluate, through TES, the piezoelectric
moduli for ADP and KDP crystals using a very simple methodology.
6. Acknowledgments
PAJr, and CMRR are grateful to CNPq and FAPESPA. JDN would like to
thank FAPERJ, FAPESPA and CAPES-DAAD for financial support. SGCM
would like to thank CNPq and CAPES for financial support.
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