1 analysis of nonlinear acoustoelastic effect of surface acoustic waves in laminated structures by transfer matrix method

The phase velocity equations for surface acoustic waves are obtained.. Keywords: Surface acoustic waves; Initial stress; Layered half-space; Phase velocity; Transfer matrix 1.. Residual

Analysis of nonlinear acoustoelastic effect of surface

acoustic waves in laminated structures by transfer

matrix method

H Liu a, J.J Lee a,*, Z.M Cai b

a Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology, 373-1,

Kusong-dong, Yusong-gu, Taejon 305-701, South Korea

b Department of Engineering Mechanics, School of Civil Engineering, Shanghai Jiaotong University, Shanghai 200240, PR China

Available online 17 May 2004

Abstract

The propagation of surface acoustic waves in a layered half-space is investigated in this paper, where a thin cubic Ge film is perfectly bonded to an isotropic elastic Si half-space Application of the transfer matrix and by solving the coupled field equations, solutions to the mechanical displacements are obtained for the film and elastic substrate, respectively The phase velocity equations for surface acoustic waves are obtained Effects of the homogeneous initial stresses induced by the mismatch of the film and substrate are discussed in detail The results are useful for the design of acoustic surface wave devices

Ó 2004 Elsevier Ltd All rights reserved

Keywords: Surface acoustic waves; Initial stress; Layered half-space; Phase velocity; Transfer matrix

1 Introduction

The nonlinear interactions between the stress in solids and acoustic waves have been investigated through the years Under the influence of initial stresses, the propagation velocity is slightly changed and different from the speed in an unstressed medium, which is referred as acoustoelastic effect (Hirao et al., 1984) The property of this effect is dependent on the propagation wave mode, propagation direction and material nonlinearity Acoustoelasticity could be used as a branch of nondestructive techniques for mea-suring residual stresses in bulk materials Residual stresses and externally applied variables such as a biasing electric field, stresses, stains, pressure and temperature can all lead to a change in the propagation velocity for the bulk waves (Baryshnikova et al., 1981), for surface acoustic waves (Palmieri et al., 1986) and for Lamb waves (Palma et al., 1985) The fabrications of acoustic devices such as voltage sensors and elec-trically controlled delay lines are stimulated

Mechanics Research Communications 31 (2004) 667–675

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*

Corresponding author Address: Department of Electrical Engineering, Korea Advanced Institute of Science and Technology, 373-1, Kusong-dong, Yusong-gu, Taejon 305-701, South Korea Tel.: +82-42-869-3033/3432/5432; fax: +82-42-869-3210/3410/5210 E-mail addresses: jjlee@mail.kaist.ac.kr, jjlee@ee.kaist.ac.kr (J.J Lee).

0093-6413/$ - see front matter Ó 2004 Elsevier Ltd All rights reserved.

doi:10.1016/j.mechrescom.2004.03.004

Acoustoelastic effect can be interpreted in terms of the change in the second order material constants, which are linear combinations of the second and third order material constants relevant to the initial

the maximal compressive strains those materials can stand, the magnitude of the initial stress is less than

100 MPa The acoustoelastic effect is very small, high precision techniques for the measurement of acoustic velocity are needed In a recent report by Wedler et al (1998) it is found that considerably larger com-pressive stresses can occur and have been measured in Ge/Si(0 0 1) structures The values of the initial

effect of Love modes propagating in the layered Ge/Si(0 0 1) system on the basis of the state space approach suggested by Fahmy and Adler

In this paper, we attempt to investigate the acoustoelastic effect of surface acoustic waves in Ge films on

Si substrates It is well known that for the layered structure, due to the mismatch of material properties of the film and substrate, all films are in a state of internal stress by whatever means they are produced (Sinha and Locke, 1989) Two major causes of thin film stress are intrinsic stress and thermal stress Excessive stress in the film will lead to film cracking, loss of adhesion, etc The measured residual stress is an equi-librium of surface stresses The propagation characteristics of waves can be influenced by the prestress distributions, and a knowledge of the magnitude of such effect plays an important role in improving the performance and long term aging characteristics of surface acoustic wave devices (Sinha et al., 1985) Based

on the transfer matrix, the solutions for each layer of the plate are derived and expressed in terms of wave amplitudes By satisfying appropriate interfacial continuity conditions, a global transfer matrix that relates the displacements and stresses on the bottom of the plate to those on the top is constructed Introducing the external boundary conditions on the upper surface of the plate, the solutions to the wave propagation problem are obtained

2 Wave motion in a prestressed medium

The nonlinear acoustoelastic equations for small fields superposed on a bias may be established by the nonlinear continuum mechanics (Pao et al., 1984) A body in the natural state is free of stress and strain In the initial state, the body is deformed due to residual stresses or applied loading The deformation from the natural to initial state is static When a wave motion is superposed on the initial state, the body is further deformed to the final state A physical variable in the initial and final state can be designated by a superscript label ‘‘0’’ and ‘‘f ’’, respectively Thus the increments of the second Piola–Kirchhoff stress tensor

ij; ui¼ ufi
u0

The equations of motion for the incremental displacements can be established either referring to the natural state or the initial state, the natural state is unknown for a genuine problem of residual stresses Generally the equations of motion for the incremental displacement are referred to the initial coordinates, which are written as

where the indices preceded by a comma denote space-coordinate differentiation with respect to the

in the initial state For small deformation, it is approximated to the mass density q of the unstressed material by

668 H Liu et al / Mechanics Research Communications 31 (2004) 667–675

q0ffi q 1

mm



where the repeated index in the subscript implies summation with respect to that index,

mm¼ u0

1;1þ u0

2;2þ u0

3;3, e0

The constitutive equation for the incremental stress r is

mn¼1

2ðu0 m;nþ u0

material The initial fields can be obtained from the field equations in the initial state

3 Surface acoustic waves in Ge films on Si substrates

The global rectangular Cartesian coordinate system is illustrated in Fig 1 The structure is made up of a

stress The thickness of the substrate is considerably larger than h and can be treated as a half space The half space noted with ‘‘0’’ is the Si substrate, ‘‘2’’ is vacuum The layer labeled with ‘‘1’’ is the Ge film The basis for the transfer matrix method is to develop a transfer matrix for each layer s which maps displacements and stress tractions from the lower surface of the layer s to it’s upper surface (Stewart and Yong, 1994; Nayfeh, 1991) For each layer, it is assumed that the initial stresses are space independent

to

Here the body force is not considered

Generally the wavelengths are considerably smaller than the width of the plate, the plain strain analysis

2

Si substrate

0

x3

x

Fig 1 The geometry of the problem and corresponding coordinate system.

uj¼ Bjexpðijbx3Þ exp½ijðx1
ctÞ; ð7Þ

1

p , c is the phase velocity of wave propagation, b is unknown parameter

For cubic media, substituting Eqs (4) and (7) into Eq (6) leads to the three equations:

11þ r0

11
q0c2þ c

55b2;

13þ c

55Þb;

11
q0c2;

55þ r0

11
q0c2þ c

33b2;

ð9Þ

where the contracting subscript notations are used

acoustic waves with particle motions entirely in the sagittal plane are usually applied in ultrasonics, thus in this paper we concentrate on the Rayleigh waves and exclude Love waves For Rayleigh waves, the

a fourth order algebraic equation in b with velocity c as the parameter, i.e.,

where

55c33;

33
c11

11
q0c2

55ðc

55þ r0

13

55

;

11

11
q0c2

11
q0c2

:

ð11Þ



13þ c 55

55þ r0

11
q0c2þ c

Then the mechanical displacements may be expressed as

g¼1

g¼1

ð13Þ

From Eq (13) and the constitutive relations (4), one can write the formal solutions for the displacements and stresses in the matrix form

670 H Liu et al / Mechanics Research Communications 31 (2004) 667–675

13þ c

33b1b1

13þ c

33b2b2

13þ c

33b3b3

13þ c

33b4b4

55ðb1þ b1Þ ijc

55ðb2þ b2Þ ijc

55ðb3þ b3Þ ijc

55ðb4þ b4Þ

2

6

4

3 7 5;

Then the column vectors specified to the lower and upper surfaces of the layer s can be respectively written as

and

lower surface of the sth layer to those at the upper surface, i.e.,

and

this leads to the displacement and stress vectors at the top of the film in the form

and

It is well known that the major disturbance of surface acoustic wave motion is confined to the region

corresponding quantities in the substrate Thus the displacement and stress vectors of the substrate can be found as

0

2x3Þ 0

4x3Þ

8

>

>

9

>

Due to the fact that the thickness of the substrate is significantly larger than the film thickness, the initial

11, c0

v0ð0Þ ¼ D0ð0Þ 0; B
012;0; B014T

The displacements and stresses must be continuous across the interface between two layers, therefore,

From Eqs (20), (22), (23) and (24), the displacement and stress vectors at the top of the layer can be written as

12;0; B014

Let

Now introduce the mechanical boundary conditions at the surface of the film, i.e.,

Substituting of Eqs (25) and (26) into the boundary conditions (27), we obtain a set of two algebraic equations in the form

12

14

0

 

vanish The phase velocity is thus found by searching for values of c that make the determinant of the

Eq (29) is the phase velocity equation for the Ge–Si system in case that the film is in presence of homo-geneous initial stresses

4 Discussion

According to the equations in the preceding section, calculations are performed for a Ge film deposited

on a silicon substrate The material constants of Ge and Si are listed in Table 1 Fig 2 displays the dis-persion relations for the first four modes of surface acoustic waves in the case that the Ge/Si(0 0 1) system is

in absence of initial stresses It is noted that the phase velocity for the first mode of the surface acoustic waves is asymptotic to the Rayleigh wave velocity of the Si substrate as the product of jh approaches zero and decreases to the Rayleigh velocity of the Ge layer as the product of jh increases to infinity For higher modes, there exist cut-off frequencies The phase velocities are asymptotic to the transverse velocity of the Si

Table 1

Material constants of unstressed media (c ij , c ijk , 10 10 Pa; q, kg/m 3 )

c 11 c 12 c 44 q c 111 c 112 c 123 c 144 c 155 c 456

Si 16.5 6.4 7.92 2329

Ge 12.9 4.8 6.71 5323.4 )72 )38 )3 )1 )30.5 )4.5

672 H Liu et al / Mechanics Research Communications 31 (2004) 667–675

substrate as the product of jh approaches the cut-off frequency and decrease to the transverse velocity of the Ge layer as the product of jh increases to infinity

The phase velocity shifts for the first mode due to the presence of initial stress in the Ge films are plotted

presence and absence of initial stresses, respectively It is assumed that the value of initial stress component

wave For small values of jh, the wavelength is larger than the film thickness, the phase velocity shift is very small With increasing the values of jh, the wavelength is considerably smaller than the film thickness and the wave is completely confined to the stressed Ge film, the acoustoelastic effect is obviously found The maximal change in phase velocity is 14.5 m/s as the product of jh is greater than 5.5

Fig 4 shows the variations of the phase velocity shift with the initial stress for fixed values of jh An almost linear behavior of the phase velocity shift versus the initial stress is obtained in our calculated range This feature is useful for estimating the magnitude of intrinsic, surface stresses and thereby characterizing the fabrication process by the measurement of the stress-induced frequency shifts (Sinha et al., 1985)

0 2 4 6 8 10 12 14 16 3.0

3.5 4.0 4.5 5.0 5.5 6.0

h

1st mode 2nd mode 3rd mode 4th mode

κ

Fig 2 The dispersion curves for the first four modes of surface acoustic waves in Ge/Si(0 0 1) system in the absence of initial stresses.

-4 -2 0 2 4 6 8 10 12 14 16

κ h

Fig 3 Relations between the phase velocity shift for the first mode and the product of wave number and film thickness.

5 Conclusions

The propagation of surface acoustic waves in laminated structure is studied in this paper and numerical calculations are performed for the Ge films on Si substrates It is found that the second order elastic constants are linear combinations of the second and third order material constants relevant to the initial biasing state The residual stresses in the Ge film modify the phase velocity Linear behavior of the phase velocity shift versus the initial stress is obtained The change in phase velocity is dependent on the product

of wave number and film thickness For surface acoustic waves in layered Ge/Si(0 0 1) system, the maximal phase velocity change is found for larger values of the product of wave number and film thickness in case that the wave propagates along the inplane axis of symmetry of the film

Acknowledgements

This study was supported by KOSEF (Korea Science and Engineering Foundation) research fund through HWRS–ERC of KAIST We would like to thank for this support

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