XX Session of the Russian Acoustical Society
Moscow, October 27-31, 2008
A.S. Kuznetsova, I.E. Kuznetsova, B.D. Zaitsev
INFLUENCE OF TEMPERATURE ON ACOUSTIC WAVE CHARACTERISTICS IN
STRUCTURE “PIEZOELECTRIC PLATE - NANOCOMPOSITE POLYMERIC
FILM”
Institute of Radio Engineering and Electronics of RAS, Saratov Branch
Zelyonaya str, 38, Saratov, 410019 Russia
Phone: (8452) 27-7323; Fax: (8452) 27-2401
E-mail: tigrnasya@yandex.ru
As is well known that the development of high effective and thermostable acoustic devices assumes using the
acoustic waves with high coefficient of electromechanical coupling (K2) and low temperature coefficient of delay
(TCD) is very important. At present there exist the papers showing that fundamental shear-horizontal (SH0)
acoustic waves in thin (in comparison with wavelength) piezoelectric plates possess significantly more
electromechanical coupling coefficient than surface acoustic waves (SAW). However, although the value of TCD
of such waves is equal 66 ppm/C that is some less than for SAW but this is not enough for development of
temperature stable devices. In this paper the theoretical analysis of temperature influence on velocity of SH0
wave in YX lithium niobate plate, which contacted by one side with nanocomosite polymeric films with various
concentrations of Fe nanoparticles has been carried out. Calculation showed that there exist such
concentrations of Fe nanoparticles when the TCD of SH0 wave in aforementioned structure is equal 15 ppm/C.
In whole the obtained results open the real prospective of use of SH0 wave in structures based on thin
piezoelectric plates for development of temperature stable acoustic devices for signal processing and high
sensitive sensors.
As known on development of acoustoelectronic devices the electro-mechanical coupling
coefficient (K2) and temperature coefficient of delay (TCD) of acoustic wave have a great importance.
In this connection one of the modern lines of investigation in acoustics is the search of such wave
types and crystallographic orientations of wave-guides for which the value of K2 is maximal and value
of TCD is minimal. Such combination of parameters is necessary to development of high effective
thermostable devices. At present there exist the papers showing that fundamental shear-horizontal
(SH0) acoustic waves in thin (in comparison with wavelength) piezoelectric plates possess
significantly more electromechanical coupling coefficient than surface acoustic waves (SAW) in the
same material [1-3]. For example, it was shown that SH0 wave has K2 = 34% for hf = 500 m/s in YX
LiNbO3 plate (h = plate thickness, f = wave frequency). In this case although the value of TCD (66
ppm/C) [4] is slightly less than TCD of SAW (88 ppm/C) this is not enough for development of
thermo-stable devices. Thus the problem of decreasing TCD with high value of K2 has the practical
significance.
It is known that velocity of SH0 wave in piezoelectric plates always decreases when the
temperature increases. From other hand it has been found that the velocity of SH_0 wave in plate
always increases when dielectric permittivity of contacting medium decreases. Thus if we have found
the material whose dielectric permittivity decreases when temperature increases this opens the
possibility to develop the structure where increasing the temperature leads to increasing of SH0 wave
velocity. This is allowed significantly to reduce TCD SH0 wave propagating in such structure. The
possibility of reducing TCD SH0 wave by using as contacting media the specific liquid (butyl acetate
or chlorbenzol) is theoretically and experimentally shown in [5]. However the making of such
structure and maintenance of devices developed on their basis is very difficult from technologic point
of view. In this connection the search of material characterized by specific dependence of dielectric
permittivity versus temperature has been carried out. Investigations showed that as such material can
be used the nanocomposite polymeric films with various contents of Fe nanoparticles.
Thus, in this paper the theoretical analysis of influence of temperature on velocity of SH0 wave
in structure consisting of thin piezoelectric plate and aforementioned nanocomposite film has been
carried out.
The geometry of the problem under consideration is shown in Fig.1. Wave propagates along the
x1 direction of a piezoelectric plate bounded by planes x3=0 and x3=-h. The nanocomposite polymeric
film is bounded by planes x3 = 0 and x3 = d. We consider that this film is viscoelastic, nonconductive
and isotropic. The regions x3 < -d and x3 > h correspond to vacuum. We consider a two dimensional
problem in which all field components are assumed to be constant in the x2 direction. For analysis of
290
XX Session of the Russian Acoustical Society
Moscow, October 27-31, 2008
wave propagation we used the motion equation, Laplace’s equation, and constitutive equations for
piezoelectric medium and polymeric film [5,6]:
Vacuum I
-h
Piezoel. plate
x1
0
film
d
Vacuum II
x3
ρ ' ∂ 2U i ∂t 2 = ∂Tij ∂x j , ∂D j ∂x j = 0 ,
(1)
'
'
Tij = C ijkl
∂U l ∂x k + e kij
∂Φ ∂x k ,
(2)
D j = −ε 'jk ∂Φ ∂x k + e 'jlk ∂U l ∂x k ,
(3)
ρ f ∂ 2U i f ∂t 2 = ∂Tijf ∂x j , ∂D jf ∂x j = 0 ,
(4)
f
f
Tijf = C ijkl
∂U l f ∂x k + η ijkl
∂U l f ∂U l f ∂t∂x k ,
(5)
D jf = −ε jkf ∂Φ f ∂x k .
Fig.1. Geometry of the problem
(6)
Here Ui is the component of mechanical displacement of particles, t is time, Tij is the component
of the mechanical stress, xj is the coordinate, Dj is the component of the electrical displacement, ηijkl is
component of viscosity, Φ is the electrical potential. The index f denotes that the appropriate variable
refers to polymeric nanocomposite film. Because the considered nanocomposite film is viscoelastic
the equation (5) shows that the effective elastic constants are complex and their imaginary part is equal
ωηijkl for harmonic waves.
The influence of temperature was considered in the following way. It is known [7] that
'
'
, piezoelectric eikl
, and dielectric ε 'jk constants and density ρ' of piezoelectric material at
elastic C ijkl
temperature T may be respectively expressed as:
Here C
E
ijkl
, e kij , ε
S
jk
'
E
T
C ijkl
= C ijkl
[1 + C ijkl
(T − TR )] ,
(7)
T
e'kij = e kij [ 1 + e kij
( T − TR )] ,
(8)
ε 'jk = ε Sjk [ 1 + ε Tjk ( T − TR )] ,
(9)
ρ ' = ρ[1 − (2α 11 + α 33 )(T − TR )] .
(10)
and ρ are elastic, piezoelectric, and dielectric constants and density of
T
T
, e kij
and ε Tjk are the temperature
piezoelectric media at room temperature TR=20C, respectively; C ijkl
coefficients of material constants, αij is the temperature coefficient of expansion. As for
nanocomposite film it was measured the dependence of their dielectric permittivity versus
temperature, which is presented in Table 1.
Outside of the structure, in regions I and II, the electrical displacement must satisfy the
Laplace’s equation:
∂D Ij ∂x j = 0,
∂D IIj ∂x j = 0,
(11)
where D Ij = −ε 0 ∂Φ I ∂x j and D IIj = −ε 0 ∂Φ II ∂x j . Here, ε0 is the permittivity of vacuum, indices
I and II denote that the values refer to regions x3 < -h and x3 > d, respectively.
Acoustic waves propagating in aforementioned structure must also satisfy the mechanical and
electrical boundary conditions:
x3=d: Ti 3f = 0 ; Φ f = Φ I , D3f = D3I ;
(12)
x3=0: U i f = U i ; Ti 3f = Ti 3 ; Φ f = Φ ; D3f = D3 ;
(13)
x3=-h: Ti 3 = 0 ; Φ = Φ II ; D3 = D3II .
(14)
Here i=1÷3, d and h are the thickness of nanocomposite film and piezoelectric plate,
respectively.
Substituting the solution as the plane inhomogeneous wave in equations system (1) - (3), (4)-(6)
and (11) yields systems of eight linear algebraic equations for piezoelectric plate, of six and two linear
algebraic equations for nanocomposite film and of two linear algebraic equations for vacuum in
291
XX Session of the Russian Acoustical Society
Moscow, October 27-31, 2008
regions x3 < -h and x3 > d, respectively [5]. At first the phase velocity (V) is considered as the
unknown parameter of these systems. This allows finding the eigenvalues and eigenvectors as the
functions of unknown velocity and writing the matrix of boundary conditions by using (12), (13) and
(14). The sought value of phase velocity is that value for which the determinant of this matrix is equal
zero.
When we have defined the value of velocity it can be found the temperature coefficient of phase
velocity (TCV) and delay (TCD). The expression for TCV and TCD take the form [7]:
v ph ( T ) − v ph ( TR )
TCV =
,
(15)
v ph ( TR )( T − TR )
TCD = α 11 − TCPV
(16)
We used the material constants of lithium niobate and their temperature coefficients from [8]
and [7], respectively. Material constants of nanocomposite polymeric films containing Fe
nanoparticles have been measured by authors using way [9]. These constants are presented in Table 1.
Moreover authors measured the temperature dependence of dielectric permittivity of used
nanocomposite films. These data are also presented in Table 1.
Table 1. Параметры нанокомпозитной полимерной железосодержащей пленки
Fe, %
2
5
7
12
15
17
20
T, C
5
20
5
20
5
20
5
20
5
20
5
20
5
20
L, mkm
60
ρ, kg/m3
884.1
C11, 108 Pa
23.2
η11, Pa*s
24.6
C66, 108 Pa
2.6
η66, Pa*s
2.2
63
918.6
20.1
6.2
2.8
2.3
64
972.6
17.8
4
3.4
2.4
55
993.7
15.1
7.6
3.5
1.1
60
991.3
14
4
3.4
0.7
50
986.6
12.8
11.3
3.0
1.3
36
1052.0
10.2
17.6
3.3
1.1
εii/ε0
2.1175
2.0938
2.3687
2.3488
2.3775
2.3526
2.2814
2.2582
2.3442
2.3192
3.0767
3.0551
2.5277
2.5002
As a result of conducted calculation TCD have been found for SH0 wave propagating in
structure “piezoelectric plate – nanocomposite polymeric film”.
Fig. 2 shows the dependence of TCD of SH0 wave propagating in structure “Y-X plate of
lithium niobate - nanocomposite polymeric film with 12% of Fe nanoparticles” on parameter d/h. The
analysis of obtained results have shown that there exist such concentrations of Fe nanoparticles at that
the increase of film thickness leads to significant decreasing of TCD SH0 wave in investigated
structure from 65 ppm/C to 15 ppm/C.
TCD, ppm/C
80
60
40
20
0
0
0.2
0.4
0.6
0.8
d/h
Fig.2. The dependence of temperature coefficient of delay (TCD) of SH0 wave in structure “Y-X plate of lithium
niobate - nanocomposite polymeric film with 12% of Fe nanoparticles” on thickness relation d/h.
292
XX Session of the Russian Acoustical Society
Moscow, October 27-31, 2008
This work is supported by the grant of Ministry of Education and Science of Russia (grant RNP
2.1.1.8014), grant RFBR 06-08-01011, by the grant f Ministry of Rosnauka (contract№2007-3-1.3-0715-036). Dr. Iren Kuznetsova thanks the Russian Science Support of Foundation.
REFERENCES
1.
2.
3.
4.
5.
6.
7.
8.
9.
Zaitsev B. D., Joshi S. G., and Kuznetsova I. E. Propagation of QSH (quasi shear horizontal) acoustic
waves in piezoelectric plates // IEEE Trans. Ultrason. Ferroel. and Freq. Control.1999.V.46.P.12981302.
Kuznetsova I. E., Zaitsev B. D., Joshi S. G., and Borodina I. A. Investigation of acoustic waves in thin
plates of lithium niobate and lithium tantalate// IEEE Trans. Ultrason. Ferroel. and Freq. Control. 2001.
V.48. P.322 – 328.
Kuznetsova I. E., Zaitsev B. D., Joshi S. G., and Borodina I. A. Acoustic plate waves in potassium
niobate single crystal // Electronic Letters. 1998. V.34. N23. P.2280 – 2281.
Kuznetsova I. E., Zaitsev B. D., Joshi S. G. Temperature characteristics of acoustic waves propagating
in thin piezoelectric plates // Proc. of IEEE Ultrason. Symp. 2001. V.1. P.157-160.
Zaitsev B. D., Kuznetsova I. E., Joshi S. G., Kuznetsova A.S. New method of change in temperature
coefficient delay of acoustic waves in thin piezoelectric plates // IEEE Trans. Ultrason. Ferroel. and
Freq. Control. 2001. V.48. N1. P.322 – 328.
Kuznetsova I. E., Zaitsev B. D., Ushakov N.M., Kuznetsova A.S. Acoustic waves in structure
“piezoelectric plate – polymeric nanocomposite film with CdS nanoparticles” // Proceed. XIX sessia
RAO, Sept. 24-28, 2007, Nizhnii Novgorod. 2007. V.2. P.69-72. (in Russian).
Slobodnik A. J. The Temperature Coefficients of Acoustic Surface Wave Velocity and Delay on
Lithium Niobate, Lithium Tantalate, Quartz, and Tellurium Dioxide //Phys. Sci. Res. Pap. 1972.No.477.
Kovacs G., Anhorn M., Engan H.E., Visintini G., and Ruppel C.C.W. Improved material constants for
LiNbO3 and LiTaO3// Proc. IEEE Ultrasonics Symp. 1990. V.1. P.435-438.
Kuznetsova I.E., Ulzutuev A.N., Zaitsev B.D., Ushakov N.M., Kosobudskii I.D. Acoustic
characteristics of polymeric nanocomposite films // Proceed. of XVIII sessii RAO, Sept. 11-15 2006 г.,
Taganrog. 2006. V.1. P.15-19. (in Russian).
293