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 www.elsevier.com/locate/mechrescom MECHANICS RESEARCH COMMUNICATIONS * 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 biasing state. In the investigations of the acoustoelastic effect on LiNbO 3 (Kuznetsova et al., 1998; Liu et al., 2003), SrTiO 3 (Zaitsev and Kuznetsova, 1996) plates, the initial stress in the bulk is usually limited by the maximal compres sive 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 stresses in the Ge films are in the range of )2.8 to )5.8 GPa. Osetrov et al. (2002) studied the acoustoelastic 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 struc ture, 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 magni tude 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 r and displacement u due to the dynamic disturbance are respectively denoted by r ij ¼ r f ij À r 0 ij ; u i ¼ u f i À u 0 i ; i; j ¼ 1; 2; 3: ð1Þ The equations of motion for the incremental displacements can be established either refer ring 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 r ij À þ u j;k r 0 ik Á ; i þq 0 f j ¼ q 0 € u j ; i; j; k ¼ 1; 2; 3; ð2Þ where the indices preceded by a comma denote space-coordinate differentiation with respect to the coor- dinate variables x i and the dot denotes time differentiation. f j is a force per unit mass. q 0 is the mass density in the initial state. For small deform ation, 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 q 0 ffi q 1 À À e 0 mm Á ; m ¼ 1; 2; 3; ð3Þ where the repeated index in the subscript implies summation with respect to that index, e 0 mm ¼ u 0 1;1 þ u 0 2;2 þ u 0 3;3 , e 0 mm is the cubic dilatation. The constitutive equation for the incremental stress r is r ij ¼ c à ijkl u k;l ; ð4Þ where c à ijkl are effective elastic constants, and c à ijkl ¼ c ijkl ð1 À e 0 mm Þþc ijklmn e 0 mn þ c mjkl u 0 i;m þ c imkl u 0 j;m þ c ijml u 0 k;m þ c ijkm u 0 l;m ; ð5Þ where i; j; k; l; m; n ¼ 1; 2; 3, c ijkl are elastic constants in the unstressed material, e 0 mn is the Euler strains in the initial state, and e 0 mn ¼ 1 2 ðu 0 m;n þ u 0 n;m Þ. It is noted that the coefficients c à ijkl possess the same symmetry with c ijkl . The values of c à ijkl depend on the initial displacement gradient fields and the elastic constants of the 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 film with thickness of h, and the substrate is rigidly bonded at their interface and stacked normal to the x 3 axis. The Ge film is in the region Àh < x 3 < 0 and the Si substrate x 3 > 0. The surface of the film is free of 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. Because the layer is very thin, the components of stress r 0 3j (j ¼ 1; 2; 3) between laminas are small compared to others and can be negligible. Then the equations of motion for the prestressed layer s (s ¼ 0; 1) is reduced to r ij þ u j;ca r 0 ac ¼ q 0 € u j ; i; j ¼ 1; 2; 3; a; c ¼ 1; 2: ð6Þ Here the body force is not considered. Generally the wavelengths are considerably smaller than the width of the plate, the plain strain analysis is valid (Lowe, 1995), i.e., there is no variation of any quantity along the x 2 axis. When the wave propagates along the x 1 axis, the displacements u j can be expressed in the form 2 Ge film 1 h o 1 Si substrate 0 x 3 x Fig. 1. The geometry of the problem and corresponding coordinate system. H. Liu et al. / Mechanics Research Communications 31 (2004) 667–675 669 u j ¼ B j expðijbx 3 Þexp½ijðx 1 À ctÞ; ð7Þ where B j is wave amplitude, jð¼ 2p=kÞ is the wave number with k being the wavelength, i ¼ ffiffiffiffiffiffiffi À1 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: C ij B j ¼ 0; ð8Þ C ij is symmetric, i.e., C ij ¼ C ji ,and C 11 ¼ c à 11 þ r 0 11 À q 0 c 2 þ c à 55 b 2 ; C 12 ¼ 0; C 13 ¼ðc à 13 þ c à 55 Þb; C 22 ¼ c à 44 b 2 þ r 0 11 À q 0 c 2 ; C 23 ¼ 0; C 33 ¼ c à 55 þ r 0 11 À q 0 c 2 þ c à 33 b 2 ; ð9Þ where the contracting subscript notations are used. It is noted that for cubic media, the transverse displacement u 2 is decoupled from u 1 and u 3 . The surface 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 exis- tence condition of nontrivial solutions for B 1 and B 3 is that the determinant in Eq. (8) vanishes, which yields a fourth order algebraic equation in b with velocity c as the parameter, i.e., A 1 b 4 þ A 2 b 2 þ A 3 ¼ 0; ð10Þ where A 1 ¼ c à 55 c à 33 ; A 2 ¼ c à 33 c à 11 À þ r 0 11 À q 0 c 2 Á þ c à 55 ðc à 55 þ r 0 11 À q 0 c 2 ÞÀ c à 13 À þ c à 55 Á 2 ; A 3 ¼ c à 11 À þ r 0 11 À q 0 c 2 Á c à 55 À þ r 0 11 À q 0 c 2 Á : ð11Þ From the above equation, the four roots b g (g ¼ 1–4) of b can be solved, as long as the material constants, film thickne ss and the values of initial stresses are given. In addition, from Eq. (8), each b g (g ¼ 1–4) yields the amplitude ratio b g , i.e., b g ¼ B 3g B 1g ¼À c à 13 þ c à 55 ÀÁ b g c à 55 þ r 0 11 À q 0 c 2 þ c à 33 b 2 g : ð12Þ Then the mechanical displacements may be expressed as u 1 ¼ X 4 g¼1 B 1g expðijb g x 3 Þexp½ijðx 1 À ctÞ; u 3 ¼ X 4 g¼1 b g B 1g expðijb g x 3 Þexp½ijðx 1 À ctÞ: ð13Þ From Eq. (13) and the constitutive relations (4), one can write the formal solutions for the displacements and stresses in the matrix form vðx 3 Þ¼DRðx 3 ÞB; ð14Þ 670 H. Liu et al. / Mechanics Research Communications 31 (2004) 667–675 where v ¼½u 1 u 3 r 33 r 13 T ; D ¼ 1111 b 1 b 2 b 3 b 4 ij c à 13 þ c à 33 b 1 b 1 ÀÁ ij c à 13 þ c à 33 b 2 b 2 ÀÁ ij c à 13 þ c à 33 b 3 b 3 ÀÁ ij c à 13 þ c à 33 b 4 b 4 ÀÁ ijc à 55 ðb 1 þ b 1 Þ ijc à 55 ðb 2 þ b 2 Þ ijc à 55 ðb 3 þ b 3 Þ ijc à 55 ðb 4 þ b 4 Þ 2 6 6 4 3 7 7 5 ; Rðx 3 Þ¼diag½expðijb 1 x 3 Þ expðijb 2 x 3 Þ expðijb 3 x 3 Þ expðijb 4 x 3 Þ; B ¼ðB 11 B 12 B 13 B 14 Þ T : ð15Þ Then the column vectors specified to the lower and upper surfaces of the layer s can be respectively written as v s ðx 3s Þ¼D s R s ðx 3s ÞB s ð16Þ and v s ðx 3s À d s Þ¼D s R s ðx 3s À d s ÞB s ; ð17Þ where the subscript ‘‘s’’ indicates the quantities in the sth layer, x 3s is located in the lower surface of the layer s, d s is the thickness of the layer s, x 3s À d s is located in the upper surface of the layer. From Eqs. (16) and (17), one can establish the transfer matrix P s ðx 3s À d s ; x 3s Þ to relate the displacements and stresses at the lower surface of the sth layer to those at the upper surface, i.e., v s ðx 3s À d s Þ¼P s ðx 3s À d s ; x 3s Þv s ðx 3s Þð18Þ and P s ðx 3s À d s ; x 3s Þ¼D s R s ðÀd s ÞD À1 s ð19Þ this leads to the displacement and stress vectors at the top of the film in the form v 1 ðÀhÞ¼P 1 ðÀh; 0Þv 1 ð0Þð20Þ and P 1 ðÀh; 0Þ¼D 1 R 1 ðÀhÞD À1 1 ; ð21Þ where v 1 ðÀhÞ and v 1 ð0Þ are respectively the vectors at the upper and lower surfaces of the film. It is well known that the major disturbance of surface acoustic wave motion is con fined to the region near the surface, this requires that the components of displacement tend to zero when x 3 tends to 1, only two roots of b 0 with positive imaginary parts can be selected. The superscript label ‘‘ 0 ’’ is designa ted for the corresponding quantities in the substrate. Thus the displacement and stress vectors of the substrate can be found as v 0 ðx 3 Þ¼D 0 0 B 0 12 expðijb 0 2 x 3 Þ 0 B 0 14 expðijb 0 4 x 3 Þ 8 > > < > > : 9 > > = > > ; : ð22Þ Due to the fact that the thickness of the substrate is significantly larger than the film thickness, the initial stress in the substrate is negligible. D 0 can obtained by substitution of the elastic constants c 0 11 , c 0 44 and mass density q 0 of the isotropic Si substrate into Eq. (15). The vector at x 3 ¼ 0 becomes H. Liu et al. / Mechanics Research Communications 31 (2004) 667–675 671 v 0 ð0Þ¼D 0 ð0Þ 0; B 0 12 ; 0; B 0 14 ÀÁ T : ð23Þ The displacements and stresses must be continuous across the interface between two layers, therefore, v 1 ð0Þ¼v 0 ð0Þ: ð24Þ From Eqs. (20), (22), (23) and (24), the displacement and stress vectors at the top of the layer can be written as v 1 ðÀhÞ¼P 1 ðÀh; 0Þv 0 ð0Þ¼P 1 ðÀh; 0ÞD 0 ð0Þ 0; B 0 12 ; 0; B 0 14 ÀÁ T : ð25Þ Let E ¼ P 1 ðÀh; 0ÞD 0 ð0Þ: ð26Þ Now introduce the mechanical boundary conditions at the surface of the film, i.e., r 33 ðx 1 ; ÀhÞ¼0; r 13 ðx 1 ; ÀhÞ¼0: ð27Þ Substituting of Eqs. (25) and (26) into the boundary conditions (27), we obtain a set of two algebraic equations in the form r 33 r 13 &' ¼ E 32 E 34 E 42 E 44 ! B 0 12 B 0 14 &' ¼ 0 0 &' : ð28Þ To obtain the nontrivial solutions for B 0 12 and B 0 14 , the determinant of the boundary condition matrix should vanish. The phase velocity is thus found by searching for values of c that make the determinant of the coefficient matrix equal ½E zero, i.e., E 32 E 44 À E 34 E 42 ¼ 0: ð29Þ 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 in Fig. 3. The phase velocity shift Dc is defined by Dc ¼ c À c 0 , where c and c 0 are the phase velocities in the presence and absence of initial stresses, respectively. It is assum ed that the value of initial stress component r 0 11 ¼À1 GPa. It is clearly seen that the acoustoelastic effect is de pendent on the penetration depth of the 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 phase velocity (km/s) 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. 01234567 -4 -2 0 2 4 6 8 10 12 14 16 phase velocity shift (m/s) κ h Fig. 3. Relations between the phase velocity shift for the first mode and the product of wave number and film thickness. H. Liu et al. / Mechanics Research Communications 31 (2004) 667–675 673 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 resid ual 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. References Baryshnikova, L.F., Grachev, G.S., Ermilin, K.K., Lyamov, V.E., Prokhokov, V.M., 1981. Elliptical polarization of elastic shear waves and polarization effects in crystals. IEEE Trans. Sonics Ultrason. SU-28, 2–7. Hirao, M., Tomizawa, A., Fukuoka, H., 1984. Nonlinear resonance interaction of ultrasonic waves under applied stress. J. Appl. Phys. 56, 235–237. Kuznetsova, I.E., Zaitsev, B.D., Polyakov, P.V., Mysenko, M.B., 1998. External electric field effect on the properties of Bleustein– Gulyaev surface acoustic waves in lithium niobate and strontium titanate. Ultrasonics 36, 431–434. Liu, H., Kuang, Z.B., Cai, Z.M., 2003. Propagation of Bleustein–Gulyaev waves in a prestressed layered piezoelectric structure. Ultrasonics 41, 397–405. Lowe, M.J.S., 1995. Matrix techniques for modeling ultrasonic waves in multilayered media. IEEE Trans. Ultrason. Ferroelec. Freq. Contr. 42, 525–542. Nayfeh, A.H., 1991. The general problem of elastic wave propagation in multilayered anisotropic media. J. Acoust. Soc. Am. 89, 1521– 1531. Osetrov, A.V., Frohlich, H.J., Koch, R., Chilla, E., 2002. Acoustoelastic effect in stressed heterostructures. IEEE Trans. Ultrason. Ferroelec. Freq. Contr. 49, 94–98. 0.2 0.4 0.6 0.8 1.0 -4 -2 0 2 4 6 8 10 12 14 16 phase velocity shift (m/s) -σ 0 11 (GPa) κ κ κ h = 1.17 h = 2.50 h = 5.50 Fig. 4. Variations of the phase velocity shift with initial stress. 674 H. Liu et al. / Mechanics Research Communications 31 (2004) 667–675 Palmieri, L., Socino, G., Verona, E., 1986. Electroelastic effect in layer acoustic mode propagation along ZnO films on Si substrates. Appl. Phys. Lett. 49, 1581–1583. Palma, A., Palmieri, L., Socino, G., Verona, E., 1985. Acoustic Lamb wave–electric field nonlinear interaction in YZ LiNbO 3 plates. Appl. Phys. Lett. 46, 25–27. Pao, Y.H., Sachse, W., Fukuoka, H., 1984. In: Physical Acoustics, vol. XVII. Academic Press, New York, pp. 62–144. Sinha, B.K., Locke, S., 1989. Thin-film induced effects on the stability of SAW devices. IEEE Trans. Ultrason. Ferroelec. Freq. Contr. 36, 231–241. Sinha, B.K., Tanski, W.J., Lukaszek, T., Ballato, A., 1985. Influence of biasing stresses on the propagation of surface waves. J. Appl. Phys. 57, 767–776. Stewart, J.T., Yong, Y.K., 1994. Exact analysis of the propagation of acoustic waves in multilayered anisotropic piezoelectric plates. IEEE Trans. Ultrason. Ferroelect. Freq. Contr. 41, 375–390. Wedler, G., Walz, J., Hesjedal, T., Chilla, E., Koch, R., 1998. Stress and relief of misfit strain of Ge/Si(0 0 1). Phys. Rev. Lett. 80, 2382– 2385. Zaitsev, B.D., Kuznetsova, L.E., 1996. Electroacoustic SAW interaction in strontium titanate. IEEE Trans. Ultrason. Ferroelec. Freq. Contr. 43, 708–711. H. Liu et al. / Mechanics Research Communications 31 (2004) 667–675 675 . 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,. unstressed media (c ij , c ijk ,10 10 Pa; q, kg/m 3 ) c 11 c 12 c 44 q c 11 1 c 11 2 c 12 3 c 14 4 c 15 5 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 c à 55 c à 33 ; A 2 ¼ c à 33 c à 11 À þ r 0 11 À q 0 c 2 Á þ c à 55 ðc à 55 þ r 0 11 À q 0 c 2 ÞÀ c à 13 À þ c à 55 Á 2 ; A 3 ¼ c à 11 À þ r 0 11 À q 0 c 2 Á c à 55 À þ r 0 11 À q 0 c 2 Á : 11 Þ From the above