Surface Acoustic Wave Energy in Piezoelectric Material

Available online at www.sciencedirect.com

Physics Procedia 00 (2008) 000–000

www.elsevier.com/locate/XXX

Proceedings of the JMSM 2008 conference

Surface Acoustic Wave Energy in Piezoelectric Material

F. Takali*, A. Njeh, M.H. Ben Ghozlen

Laboratoire de Physique des Matériaux, Faculté des Sciences de Sfax, B.P.1171, 3000 Sfax, Tunisia

Elsevier use only: Received date here; revised date here; accepted date here

Abstract

The surface acoustic waves in semi infinite piezoelectric media transports energy either under electrical or mechanical forms.The identification of piezoactive and non-piezoactive mode is performed by use of "Ordinary Differential Equation ", suchapproach permits calculation of energy densities under different forms. Poynting vector associated to the whole energy transporthas been established. Acoustic modes have been pointed out from Poynting variations in term of incidence angle. These resultsare confirmed from the reflection coefficient analysis. All numerical developments are realized in the scheme of plane waves.© 2009 Elsevier B.V. All rights reserved

PACS: Type pacs here, separated by semicolons ;

Keywords:Surface waves, Piezoelectricity, Poynting vector, Energy density.

1. Introduction

The energy characteristics of acoustic waves propagating in piezoelectric material have a significant practicalimportance for development of various acoustic devices. They are based on the use of bulk acoustic waves [l] andsurface acoustic waves [2,3]. Much attention has been given to the study of the energy characterization of acousticwaves [3]-[8], because the attenuation of the energy by these waves is not big. As a result, the fundamental acousticenergy-power relations have been formulated [5,6], which allow to the calculation of energy-power characteristicsof acoustic waves. Particular emphasis should be placed on acoustic waves propagating in piezoelectric materialsbecause such waves produce not only mechanical energy but also electrical energy, electromechanical energy andmechanoelectrical energy [3], [5], [6].

We apply in this work a new method to identify the surface acoustic waves. It is based on the calculation of thepower flow as well as different energy densities implied in this process. The obtained results are compared to thosebased on the calculation of the reflection coefficient.

* Corresponding author. Tel.: +21674676607; fax: +21674676607.E-mail address: [email protected].

Received 1 January 2009; received in revised form 31 July 2009; accepted 31 August 2009

Physics Procedia 2 (2009) 1369–1375

www.elsevier.com/locate/procedia

doi:10.1016/j.phpro.2009.11.104

http://www.elsevier.com/locate/procedia

http://dx.doi.org/10.1016/j.phpro.2009.11.104

2 F. Takali/ Physics Procedia 00 (2009) 000–000

2. Matrix method

The matrix method approach is based on the development of the first-order matrix ordinary differential equation(ODE) [9]. Let us consider generally anisotropic piezoelectric media as shown in Fig. 1. We define the generaldisplacement vector, the general stress vector and the state vector by U = [u, �]T, T = [�, D3]

T and ξ=[U,T]T.respectively [9],[10].We note that u= [u1, u2, u3] and �= [T13, T23, T33] are the particle displacement vector and the normal of the stressvectors, respectively. � and D3 are the electrical potential and the normal component of the electrical displacementvector, respectively. We have chosen, without loss of generality, a coordinate system (x1, x2, x3) with (x3) is normalto the layer surface, as shown in Fig. 1, we remind that (x1, x3) is the wave propagation plane. The general solutionfor the state vector can be written in this form:

(1)

where � is the angular frequency and k1 is the projection on x1 axis of the wave vector. The governing equation forthe state vector �(x3) is given by a system of differential equations [9,10]:

(2)

in which A is the fundamental acoustic tensor defined as [5,6]:

(3)

We note that ρ is the density of the considered material, and the 4 × 4 matrices �ik are related to the elastic constantsCijkl, piezoelectric stress constants eijk, and dielectric permittivity constants �ik by:

(4)

X is the inverse matrix of �33 and I’ is the 4×4 identity matrix with zero (4, 4) element.

Fig .1: Geometry of the problem.

T33T13T23

D3

Reflected wave

Vacuum

ξξAi

x3

=∂∂

( ) ��

��

�′+−−

−=

XkIikXi

XiXkA

13x22

x311311

31x

ΓωρΓΓΓΓ

( )[ ]txkiexp)x()t,x,x( 11331 ωξξ −=

��

�

�

��

�

�

−

=

ikk3ik2ik1i

i3kk3i3k2i3k1i3

i2kk3i2k2i2k1i2

i1kk3lik2lik1i1

ik

eee

eCCC

eCCC

eCCC

ε

Γ

Incident wave

X3

X1

ZnO

1370 F. Takali et al. / Physics Procedia 2 (2009) 1369–1375

F. Takali / Physics Procedia 00 (2009) 000–000 3

3. Boundary conditions

Here we consider the propagation of piezoactive acoustic wave along x1 axis in semi-infinite piezoelectricmedium as illustrated in Fig. 1. For x3 > 0, the displacement vector uj (j = 1, 2, 3) and the electrical potential � areexpressed as linear combination of four plane waves as follows:

(5)

and

(6)

where A(k), pj(k) and are the amplitude, the unit displacement polarization vector and the slowness vector of the

partial plane wave, and ω is the angular frequency.In the free space (x3 < 0), only the incident displacement vector uIj, the reflected displacement uRj and the electricalpotential:

(7)

(8)

and

(9)

R design the reflection coefficient, pIj and pRj design the unit displacement polarization vectors for the incident andthe reflected waves respectively.The conventional analysis requires the choice of four out of eight plane waves, in order to fulfill the boundarycondition on the free surface.The continuity of the normal of the stress tensor is written as:

(10)

The continuity of the electric potential � and of the normal of the electrical displacement vector is defined by:

(11)and

(12)

4. Analysis of Power Flow

We are interested in finding the incidence angle for which the surface acoustic wave can be generated in thepiezoelectric material.For an acoustic wave propagating along x1 axis in semi-infinite piezoelectric media, which is characterized by thedisplacement vector and the electrical potential:

(13)

(14)

The time-averaged generalized power flow of acoustic wave through the Poynting vector is defined as.

�=

−=4

1k

)k()k(j

)k(j ))txm(i(exppAu ω

�=

−=4

1k

)k()k(4

)k( ))txm(i(exppA ωΦ

))txm(iexp(pu IIjIj −= ω

��

−−−=circuitedshort0

circuitedopen))txm(i(exp'' IωφΦ

0)0(T)0(T 1333 ==

)0(D)0(D 33 −=+

( ))txm(iexp.Uu jj −= ω

( ))txm(iexp.0 −= ωΦΦ

))txm(iexp(pRu RRjRj −= ω

)0()0( −=+ ΦΦ

(k)m

F. Takali et al. / Physics Procedia 2 (2009) 1369–1375 1371


(15)

Tij is the mechanical stress and Di is the components of the electrical displacement vector which are defined as [3]:

(16)

(17)

Eq. 15 contains two terms: the generalized mechanical power (PiGM) and the generalized electrical power (Pi

GE),which are defined as [6]:

(18)

(19)

The direction of the Poynting vector describes the direction of energy transport, and its magnitude is equal to theamount of energy across a unit area per unit time. For pure surface wave modes without attenuation and energyalways travels parallel to the surface, the power flow vector can be defined [15]:

(20)

We are interested now in the variation of the flow of the Poynting vector versus the incidence angle, the observedmaxima are related to excited acoustic modes.

(a) (b)Fig .2: Flow of the Poynting vector as a function as the incidence angle: (a) open-circuited ( b) short-circuited.

The calculations are performed at frequency of the incidental wave 20 MHz in semi-infinite piezoelectricmaterial ZnO, with C axis parallel to x1. Fig. 2.a shows Rayleigh mode at 7,39° for open-circuited, Fig. 2.b Rayleighmode at 7,45° for short-circuited. Both figures reveal narrow maximum, which can be related to the high selectivityof ZnO material.

�

�

��

�

�

∂∂

+ �

��

�

∂∂

−= *i

*j

iji t

D

t

uRe

2

1P ΦΤ

kkij

k

lijklij x

ex

uCT

∂∂+

∂∂

= φ

jij

j

kijki xx

ueD

∂∂−

∂∂

= φε

�

�

��

�

�

�

��

�

∂∂

−=*

jij

GMi t

uRe

2

1P Τ

�

��

�∂

∂= *iGE

i t

DRe

2

1P φ

3

0

i dxP2

1�∞−

=ϕ

7 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 80

5

10

15x 10

16

Incident angle (degree)

Flo

wof

Poy

ntin

gve

ctor

follo

win

gX

1(a

u)

Rayleigh Mode 7.45°"short-circuited"

7 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 80

1

2

3

4

5

6

7

8

9

10x 10

16


Flo

wof

Poy

ntin

gve

ctor

follo

win

gx1

(au)

Rayleigh Mode 7.39°"open-circuited"



5. Analysis of Energy Density

We consider below the propagation of piezoactive acoustic wave in piezoelectric media. In this case, the time-averaged generalized of density of kinetic energy WK may be written as follows [11]:

(21)

where � is the density of the material, ui is the instantaneous value of mechanical displacement, t is time and *denotes complex conjugate. A repeated index in the subscript implies summation with respect to that index.

As for the time-averaged generalized potential energy WP, it is equal to the sum of densities of generalizedmechanical WGM and generalized electrical WGE energies [5]:

(22)

where WGM and WGE may be presented as [11]:

(23)

and

(24)

Tij is the mechanical stress, Sij is the mechanical strain, Dj is the electrical displacement and Ej is the electric fieldintensity.Here we used the following expression for mechanical strain [5]:

(25)

and the quasistatic approximation [5]:

(26)

Substituting (16) and (17) into (23) and (24) shows that generalized mechanical energy WGM includes puremechanical WM and electromechanical WME contributions. In the same time, the generalized electrical energy WGE

includes pure electrical energy WE and electromechanical energy WEM [5]:

(27)

(28)

The development shows that WME = −WEM, i.e., mechanoelectrical and electromechanical energies compensate eachother and their total contribution is close to zero [5], [12], [13].

)t

u*

t

u(Re

2

1W

*iik

∂∂

∂∂

= ρ

GEGMP WWW +=

)ST(Re2

1W *

ijijGM =

)ED(Re2

1W *

jjGE =

)t

u

t

u(

2

1S

jiij ∂

∂+

∂∂

=

kk x

E∂∂−= Φ

MEMGM WWW +=

EMEGE WWW +=



Fig .3: All contribution of energy density

All contribution of energy density shown in Fig. 3 has been calculated as a function of the incident angle for asemi-infinite piezoelectric media of ZnO at a frequency of 20 MHz. These poles corresponding to Rayleigh modesat an incidence angle of 7,39° for open-circuited.

6. Reflection coefficient

Once the displacements and the stresses in the fluid and in the material are determined, we can calculate not onlythe phase velocity, but also the wave displacements amplitudes for each partial wave and the reflection coefficients[5], [6], [14].

(a) (b)Fig .4: Reflection coefficient, for open-circuited (a), and short-circuited (b)

The reflection coefficient has been calculated at a frequency of 20 MHz. The c-axis of the semi-infinitepiezoelectric media (ZnO) has been considered to be parallel to x1 axis. Fig. 4a and 4b show that Rayleigh mode canbe generated for an incidence angle of 7,39° for open-circuited system and for an incidence angle of 7,45° for short-circuited media. One can note the perfect agreements of these tow methods.

7. Conclusion

We have presented in this paper a new method to detect the Rayleigh modes. This method is based on thecalculation of the flow of the Poynting vector and of the density of energy. The obtained results are confirmed by another method related to the reflection coefficient, and agree very well with results previously reported in theliterature. The use of the elaborated programs can be extended to configurations of higher complexity asmultilayered systems.

6 6.2 6.4 6.6 6.8 7 7.2 7.4 7.6 7.8 80.995

1

1.005

1.01

1.015

1.02

1.025

1.03

1.035

1.04

1.045


Ref

lect

ion

coef

ficie

nt(a

u)

Rayleigh Mode 7.39°" open-circuited"

6 6.2 6.4 6.6 6.8 7 7.2 7.4 7.6 7.8 80.95

0.96

0.97

0.98

0.99

1

1.01


Ref

lect

ion

coef

ficie

nt(a

u)

Rayleigh Mode 7.45°" short-circuited"

7.3 7.35 7.4 7.45 7.5

0

0.5

1

1.5

2

2.5

x 1017


Ene

rgy

dens

ity(a

u)

WEWMWEMWME



References

[l] J. Tucker and V. Rampton, Microwave Ultmsonies in Solid State Physics. Amsterdam: North-Holland Pub., 1972.[2] D.P. Morgan, Surface- Wnue Devices for Signal Processing. Amsterdam: Elsevier, 1985.[3] H. Matthews, Surface Wove Filters: Design, Construction, and Use. New York Wiley, 1977, pp. 54.[4] G.A. Coquin, H.F. Tiersten, J. Acoust. Soc. 41 (1967) 921-939[5] B.A. Auld, Acoustic Fields and Waves in Solids, vol. 1. Malabar: Krieger, 1990, pp. 135-161.[6] G.W. Farnell, “Acoustic surface waves: in Topics in Applied Physics. vol. 24, Berlin: Springer-Verlag. 1978, ch. 2, pp. 26-81.[7] W.H. Chen and F.C. Fu, J. Appl. Phys. 59 (1986) 49-54[8]V.M. Bright, W.D. Hunt, J. Appl. Phys. 66 (1989) 15561564[9] E.L. Adler, IEEE Trans. Ultrason., Ferroelect., Freq. Contr., vol. 41 (1994) 876–882[10] A.H. Fahmy, E.L. Adler, Appl. Phys. Lett. 20 (1973) 495–497[11] B.A. Auld, Acoustic Fields and Waves in Solids. vol. 1, New-York: Wiley, 1973.[12] B.D. Zaitsev, I.E. Kuznetsova, IEEE Trans. Ultrason., Ferroelect., Freq. Contr., vol. 50 (2003) 1762–1765[13] V. Laude, A. Reinhardt, A. Khelif, IEEE Trans. Ultrason., Ferroelect., Freq. Contr., vol. 52 (2005) 1869–1871[14] S.G. Joshi, Y. Jin, J. Appl. Phys. 70 (1991) 4113–4120[15] E. Dieulesaint, D. Royer, Elastic Waves in Solids (Wiley, Chichester, 1980).


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Surface Acoustic Wave Energy in Piezoelectric Material