Effects of Conducting Liquid Loadings on Propagation Characteristics of Surface Acoustic Waves

Transcription

1 Proc. Natl. Sci. Counc. ROC(A) Vol. 25, No. 2, pp Effects of Conducting Liquid Loadings on Propagation Characteristics of Surface Acoustic Waves RUYEN RO *, SHIUH-KUANG YANG **, HUNG-YU LEE *, AND CHI-YEN SHEN * * Department of Electrical Engineering I-Shou University Kaohsiung, Taiwan, R.O.C. ** Department of Mechanical Engineering National Sun Yat-Sen University Kaohsiung, Taiwan, R.O.C. (Received May 10, 2000; Accepted July 7, 2000) ABSTRACT Propagation characteristics of surface acoustic waves (SAWs) at the boundary between a fluid medium and a piezoelectric substrate are functions of the material properties of the piezoelectric crystal and acoustoelectric properties of the fluid medium. Without using the perturbation method, characteristics of SAWs were determined in this study by directly solving Christoffel s equations subjected to appropriate boundary conditions at the interface. The effects of conductivities and dielectric constants of adjacent liquid on the phase velocities and attenuation constants of Rayleigh and shear horizontal leaky SAWs were then investigated numerically. Results obtained can be employed to design liquid sensors as well as to characterize the electrical properties of the fluid medium using SAW devices. Key Words: surface acoustic wave, conductivity, dielectric constant, phase velocity, attenuation constant, liquid sensor I. Introduction Surface acoustic wave (SAW) devices have been widely adopted for signal-processing and sensing applications in the microwave frequency range due to their compact size and integrated circuit (IC) compatibility as well as to remarkable recent progress in micromachining and microfabrication (Campbell, 1998; Ballantine et al., 1997). SAW devices, in general, consist of input and output paired interdigital transducers (IDTs), which are photolithographed on piezoelectric crystals. Propagation characteristics of SAWs, which can be obtained numerically either by using an exact method or by using a perturbation approach, are functions of the material properties of the piezoelectric substrate and the acoustoelectric properties of surface loadings (Auld, 1973; Kino, 1987; Matthews, 1977). An exact method was developed by Campbell and Jones (1970) for investigating characteristics of SAWs at the boundary between a piezoelectric crystal and a nonviscous and nonconducting fluid medium. They showed that propagating waves under the influence of the fluid medium must take the form of leaky waves, which will transfer acoustic energy to the fluid medium. The perturbation method, on the other hand, was employed to study the effects of the conductivity or viscosity of adjacent liquids on the characteristics of SAWs for sensing applications (Kondoh and Shiokawa, 1995; Josse and Shana, 1988, 1991). Based upon these research results, acoustic devices exploiting different propagation modes generated by IDTs, e.g., shear horizontal SAW (SH-SAW), acoustic plate mode (APM), and flexural plate wave (FPW), have been implemented for detecting the conductivity and/ or viscosity of liquid loadings (Shiokawa and Kondoh, 1996; Andle and Vetelino, 199; Josse, 199; Martin and Ricco, 1987). The measured oscillation frequencies or transmission characteristics have been employed to represent liquid loadings for pattern recognition using multivariate analysis and/ or neural networks (Kondoh and Shiokawa, 199; Kondoh et al., 1996; Ro et al., 1999a, 1999b). Identification results show that SAW devices can be applied effectively to discriminate between different liquid loadings. The perturbation method, as mentioned above, has been employed as a theoretical basis for designing liquid sensors using SAW devices. However, the perturbation method is essentially an approximation approach; hence, its application is limited to some extent. To explore SAW devices for further applications, e.g., measurement of the electrical and acoustic properties of the fluid medium, the characteristics of SAWs under the influence of the fluid medium must be understood first. Following the ideas that Campbell and Jones (1970) have proposed, the characteristics of SAWs at the boundary between a conducting fluid medium and a piezoelectric crystal are thoroughly discussed in this paper. Effects of conductivity and the dielectric constant on Rayleigh SAW and SH SAW are then illustrated numerically. Results obtained in this study can be extended to analyze the effect of viscosity on SAWs 131

2 R. Ro et al. and, furthermore, to measure the electrical and acoustic properties of the fluid medium. II. Theoretical Analysis The geometry associated with the problem considered in this study is depicted in Fig. 1. Let medium 1, which is a piezoelectric crystal, represent the z < 0 region, and let the half-space z > 0 region, which is a conducting fluid region, be represented by medium 2. The surface acoustic wave is assumed to propagate in the x direction, and no variation of the fields in either media is assumed in the y direction. Accordingly, the acoustic displacement fields and electric potential fields of SAW in medium 1 can be expressed as (Auld, 1973; Campbell and Jones, 1970) u j (1) = a j exp(ikbz)exp[ik(px vt)], j = 1, 2, 3, (1) φ (1) = a exp(ikbz)exp[ik(px vt)], (2) where u and φ are the acoustic displacement and electric potential fields, respectively, k and v are the wave number and phase velocity of SAW, respectively, P = 1 + iγ, γ is the attenuation coefficient, b is the wave number ratio, and a is the unknown constant. Substituting Eqs. (1) and (2) into stiffened Christoffel equations, which describe the acoustic and electric field behavior in a piezoelectric crystal, yields an eighth-order algebraic equation in the wave number ratio b (Auld, 1973; Campbell and Jones, 1970). Thus for each pair of values of (v, γ), there are eight real or complex values of b. For a semiinfinite piezoelectric crystal like that considered in this study, four complex roots with negative imaginary parts are selected for a Rayleigh SAW; meanwhile, in the case for a leaky SAW, one complex root, instead, has a positive imaginary part (Campbell, 1998; Tonami et al., 1995). Once the proper selection of b for a specific propagation mode is determined, the accompanying eigenvector, a, can also be obtained from the same equation. The resultant acoustic and electric fields in a piezoelectric crystal comprise four partial waves, given by u j (1) = C m a j (m) exp(ikb (m) z)exp[ik(px vt)], j = 1, 2, 3, (3) φ (1) = C m a (m) exp(ikb (m) z)exp[ik(px vt)], where C is the weighting factor still to be determined for the piezoelectric substrate. Similarly, the acoustic and electric fields in medium 2 take the form 3 u j (2) = X n α j (n) exp(ikβ (n) z)exp[ik(px vt)], n =1 () j = 1, 2, 3, (5) Fig. 1. Geometry of the propagation of surface acoustic waves at the boundary between a conducting fluid medium and a piezoelectric substrate. φ (2) = C m a (m) exp( kpz)exp[ik(px vt)], where X is the weighting factor for the fluid medium. The (m) coefficient C m a in Eq. (6) is obtained by applying the electrical boundary condition, the continuity of electric potential, at the interface, z = 0. For a fluid medium (an isotropic material), the wave number ratio β and its corresponding eigenvector α in Eq. (5) can be determined analytically as β (1) = ± ρv 2 /(λ +2µ) P 2, (6) (α 1 (1), α 2 (1), α 3 (1) )=(P,0,β (1) ), (7) β (2) = ± ρv 2 /µ P 2,(α 1 (2), α 2 (2), α 3 (2) )=(0,1,0), (8) β (3) = ± ρv 2 /µ P 2, (α 1 (3), α 2 (3), α 3 (3) )=( β (3),0,P), (9) in which ρ, λ and µ are the density and Lame constants of the fluid medium. In this paper, the effects of the electric properties of the fluid medium on the characteristics of SAWs will be elucidated, and the effects of viscosity ignored; hence, the fluid medium is described here by two real constants, λ and µ. The ambiguity in the sign of β in Eqs. (7) (9) is resolved by determining whether the imaginary part of β is positive or negative. When a positive imaginary part of β is selected, the solution is called a proper one since the displacement field is bound as z +. In other words, when a negative imaginary part of β is chosen, the solution is said to be improper. In all the simulations conducted in this study, the improper solution was selected to satisfy the condition that the surface acoustic wave is bound as x +. Selection of 132

3 Conducting Liquid Effects on SAWs the proper or improper solution for β will be explained in detail along with the assumed material properties of the piezoelectric substrate and the fluid medium. The boundary conditions required at the interface between a fluid medium and a piezoelectric substrate are the continuity of the acoustic displacement, the continuity of the stress field, and the continuity of the normal component of the electric displacement; that is, at the interface z = 0 u (1) x = u (2) x, u (1) y = u (2) y, u (1) z = u (2) z, T (1) zz = T (2) zz, T (1) zx = T (2) zx, T (1) zy = T (2) zy, and D (1) z = D (2) z, (10) where the stress field T and the electric displacement D are obtained by substituting displacement and potential fields into stiffened Christoffel equations. Application of Eq. (10) to Eqs. (3) (6) leads to a set of seven homogeneous equations: (M) 7 7 C 1 C 2 C 3 C X 1 X 2 X 3 =0. (11) The SAW phase velocity v and the attenuation coefficient γ can then be determined by vanishing the determinant of the matrix M, i.e., det M = 0. Consequently, the corresponding weighting coefficients can be evaluated from Eq. (11); in turn, the acoustic and electric fields in medium 1 and 2 can be obtained by inserting those values into Eqs. (3) (6). III. Simulation Results and Discussion A program was written to study the effects of conducting liquid loadings on the propagation characteristics of SAWs. Two piezoelectric crystals, 128 -rotated Y-cut X-propagation LiNbO 3 (128YX.LN) and 36 -rotated Y-cut X-propagation LiTaO 3 (36YX.LT), were used in this study to investigate those effects on the characteristics of Rayleigh and SH leaky SAWs, respectively. The input material properties for the fluid medium were the density ρ, Lame constants λ and µ, dielectric constant ε r, and conductivity σ. The density ρ was assumed to be 1000 kg/m 3, and the Lame constants λ and µ were chosen as and N/m 2, respectively; meanwhile, the values of the dielectric constant and conductivity were varied to examine their effects on the propagation characteristics of SAWs. The elastic, dielectric, and piezoelectric constants for LiNbO 3 and LiTaO 3 were taken from the book written by Auld (1973). With the given material properties, the longitudinal and transverse phase velocities of acoustic waves, [(λ + 2µ) Fig. 2. (a) Phase velocity and (b) attenuation constant of Rayleigh SAWs versus frequency. The piezoelectric crystal is 128YX.LN. The material properties of the fluid medium are ρ = 1000 kg/m 3, λ = N/m 2, µ = N/m 2, ε r = 80, and various conductivities. /ρ] 1/2 and [µ/ρ] 1/2, in the fluid medium were approximately 1530 and 220 m/s, respectively; the phase velocity of either the Rayleigh SAW for 128YX.LN or the SH leaky SAW for 36YX.LT was around 000 m/s, which was significantly greater than the longitudinal phase velocity in the fluid medium. Substituting these values into Eqs. (7) (9), one can easily show that if the real part of β is greater than zero, then the imaginary part of β must have a negative value, and vice versa. There is no possible solution for β with both positive real and imaginary parts. This demonstrates that we are searching for an improper solution instead of a proper one. The real part of β must be chosen to be positive so that surface waves will propagate in the proper direction, toward the fluid medium. Consequently, the imaginary part of β will have a negative value; the amplitudes of the displacement fields will increase as the SAW penetrates into the fluid medium. The phase velocities v and attenuation constants γ of SAWs for the fluid medium with the dielectric constant ε r = 80 and various conductivities are plotted against the frequency in Figs. 2 and 3. The selected piezoelectric substrates shown in Figs. 2 and 3 are 128YX.LN and 36YX.LT, respectively. Hence, the effects of the conductivity and operating frequency on the propagation characteristics of the Rayleigh SAW and SH leaky SAW are considered, respectively, in Figs. 2 and 3. At lower frequencies, v and γ for each conductivity curve 133

4 R. Ro et al. Fig. 3. (a) Phase velocity and (b) attenuation constant of SH leaky SAWs versus frequency. The piezoelectric crystal is 36YX.LT. The material properties of the fluid medium are ρ = 1000 kg/m 3, λ = N/m 2, µ = N/m 2, ε r = 80, and various conductivities. Fig.. (a) Phase velocity and (b) attenuation constant of Rayleigh SAWs versus frequency. The piezoelectric crystal is 128YX.LN. The material properties of the fluid medium are ρ = 1000 kg/m 3, λ = N/m 2, µ = N/m 2, σ = 1.0 S/m, and various dielectric constants. approach some specific values, e.g., 3887 m/s in Fig. 2(a) and in Fig. 2(b). These values can be obtained when vanishing of the electric potential at the interface is adopted as the only electric boundary condition. This is because the ratio of the conductivity to the product of the angular frequency and the permittivity, σ/ωε, tends to infinity such that the fluid medium behaves as an electrically inactive medium. As the frequency increases, v increases monotonically and converges to 3930 m/s in Fig. 2(a) and to 165 m/s in Fig. 3(a) while γ reaches a maximum value and then decreases gradually to in Fig. 2(b) and to in Fig. 3(b). The values of v and γ in the higher frequency limit can be obtained by simply assuming that σ = 0 since σ/ωε approaches zero. As a result, the electromechanical coupling coefficient K 2, which can be determined from the lower and upper bounds of v, does not vary with the conductivity. This illustrates that the conductivity of the fluid medium will not affect the transduction efficiency of the piezoelectric substrate. For each conductivity curve, it is clear that for a specific value of σ/ω, a maximum value appears in the γ curve; correspondingly, a largest slope is observed in the v curve. The critical values of (σ/ωε) c when the maximum values of γ occur in Figs. 2 and 3, are approximately 1.81 and 1.78, respectively. The effects of the dielectric constant ε r of the fluid medium with σ = 1.0 S/m on v and γ plotted against frequency are shown in Figs. and 5. At lower frequencies, the values of v approach 3887 m/s in Fig. (a) and 11 m/s in Fig. 5(a) while the values of γ reach in Fig. (b) and in Fig. 5(b). It is clear in Figs. and 5 that the conductivity does not affect the lower bound value of v and the corresponding value of γ. Meanwhile, the upper bounds of v converge to 398, 3930, and 3920 m/s in Fig. (a) for ε r = 0, 80, and 120 curves, respectively. It is evident that the upper bound of v decreases as ε r increases, and vice versa. The dynamic range of v, hence, apparently decreases as ε r increases, and so does the coupling coefficient, e.g., K 2 = and in Fig. (a) and K 2 = and in Fig. 5(a) for ε r = 0 and 120 curves, respectively. The maximum value of γ, which is proportional to K 2, also varies with the dielectric constant. The value of (σ/ωε) c at which the maximum value of γ occurs also varies with the dielectric constant; (σ/ωε) c increases as the dielectric constant decreases, e.g., (σ/ωε) c = 2.62 and 1.5 in Fig. for ε r = 0 and 120 curves, respectively. This illustrates that the dielectric constant of the fluid medium not only alters the characteristics of SAWs, but also significantly affects the transduction efficiency of the piezoelectric substrate. It is noted that γ is approximately in Fig. 2(b) and in Fig. 3(b) when σ equals zero. This indicates that when a nonconductive fluid medium is loaded, the loss of the 13

5 Conducting Liquid Effects on SAWs Fig. 6. Normalized displacement and potential fields of Rayleigh SAWs versus the normalized distance from the interface. The piezoelectric crystal is 128YX.LN. The material properties of the fluid medium are ρ = 1000 kg/m 3, λ = N/m 2, µ = N/m 2, ε r = 80, and σ = 0. The displacement and potential fields are both normalized with respect to u z evaluated at z = 0, but the potential fields are then divided by Fig. 5. (a) Phase velocity and (b) attenuation constant of SH leaky SAWs versus frequency. The piezoelectric crystal is 36YX.LT. The material properties of the fluid medium are ρ = 1000 kg/m 3, λ = N/m 2, µ = N/m 2, σ = 1.0 S/m, and various dielectric constants. Rayleigh SAW is significantly greater than that of the SH leaky SAW. To provide further information on this loss mechanism, the normalized acoustic displacement and electric potential fields of SAWs plotted against the normalized displacement are shown in Figs. 6 and 7. The selected piezoelectric crystals shown in Figs. 6 and 7 are 128YX.LN and 36YX.LT, respectively. Hence, the fields in Figs. 6 and 7 represent the corresponding behaviors of the Rayleigh SAW and SH leaky SAW, respectively. Consider the acoustic displacement fields at the interface z = 0. The magnitude of either u x or u z in Fig. 6 is significantly greater than that of u y while u y is the largest displacement component in Fig. 7. Multiplying these displacement fields by the corresponding stress fields, one can calculate the acoustic power propagating in the fluid medium. The result indicates that more acoustic energy will penetrate into the fluid medium for the Rayleigh SAW. This causes the attenuation coefficient of the Rayleigh SAW for 128YX. LN to be significantly greater than that of the SH leaky SAW for 36YX.LT. IV. Conclusions In this paper, an exact method for determining the propagation characteristics of SAWs at the boundary between a piezoelectric crystal and a conducting fluid medium has been Fig. 7. Normalized displacement and potential fields of SH leaky SAWs versus the normalized distance from the interface. The piezoelectric crystal is 36YX.LT. The material properties of the fluid medium are ρ = 1000 kg/m 3, λ = N/m 2, µ = N/m 2, ε r = 80, and σ = 0. The displacement and potential fields are both normalized with respect to u y evaluated at z = 0, but the potential fields are then divided by presented. The effects of the conductivity and dielectric constant of the fluid medium on the phase velocity and attenuation constant have been elucidated. Both the dielectric constant and conductivity can alter the characteristics of SAWs significantly. In addition, the dielectric constant can also affect the transduction efficiency; i.e., increasing the dielectric constant will cause the electromechanical coupling coefficient to decrease, and vice versa. The displacement and potential field have been plotted to examine the loss mechanism of SAWs. It has been found that the effects of the nonconducting fluid medium cause greater loss of the Rayleigh SAW for 128YX. LN than of the SH leaky SAW for 36YX.LT. This result can be used to design SAW devices that employ appropriate propagation modes for specific applications. Although the effect of viscosity has been ignored in this study, it can be 135

6 R. Ro et al. taken into account by assuming that λ and µ are complexvalued variables. A detailed discussion focusing on the selection of a proper or improper solution of β and related simulation results will be published in the near future. Acknowledgment The authors would like to thank the National Science Council of the Republic of China for financially supporting this research under contract NSC E References Andle, J. C. and J. F. Vetelino (199) Acoustic wave biosensors. Sensors and Actuators,, Auld, B. A. (1973) Acoustic Fields and Waves in Solids, Vol John Wiley and Sons, New York, NY, U.S.A. Ballantine, D. S., R. M. White, S. J. Martin, A. J. Ricco, E. T. Zellers, G. C. Frye, and H. Wohltjen (1997) Acoustic Wave Sensors: Theory, Design, and Physico-Chemical Applications. Academic Press, Inc., San Diego, CA, U.S.A. Campbell, C. (1998) Surface Acoustic Wave Devices for Mobile and Wireless Communications. Academic Press, Inc., San Diego, CA, U.S.A. Campbell, J. J. and W. R. Jones (1970) Propagation of surface waves at the boundary between a piezoelectric crystal and a fluid medium. IEEE Trans. SU, 17, Josse, F. (199) Acoustic wave liquid-phase-based microsensors. Sensors and Actuators,, Josse, F. and Z. Shana (1988) Analysis of shear horizontal surface waves at the boundary between a piezoelectric crystal and a viscous fluid medium. J. Acoust. Soc. Am., 8, Josse, F. and Z. A. Shana (1991) Acoustoionic interaction of SH surface waves with dilute ionic solutions. IEEE Trans. UFFC, 38, Kino, G. S. (1987) Acoustic Waves: Devices, Imaging, and Analog Signal Processing. Prentice-Hall, Inc., Englewood Cliffs, NJ, U.S.A. Kondoh, J. and S. Shiokawa (199) New application of shear horizontal surface acoustic wave sensors. Jpn. J. Appl. Phys., 33, Kondoh, J. and S. Shiokawa (1995) Shear surface acoustic wave liquid sensor based on acoustoelectric interaction. Electronics and Communications in Japan, 78, Kondoh, J., T. Imayama, Y. Matsui, and S. Shiokawa (1996) Enzyme biosensor based on surface acoustic wave device. Electronics and Communications in Japan, 79, Martin, S. J. and A. J. Ricco (1987) Acoustic wave viscosity sensor. Appl. Phys. Lett., 50, Matthews, H. (1977) Surface Wave Filters. John Wiley and Sons, New York, NY, U.S.A. Ro, R., S. Y. Chang, and D. H. Lee (1999a) Measurement of SAW liquid sensors. Proc. Natl. Sci. Counc. ROC(A), 23, Ro, R., S. Y. Chang, R. C. Hwang, and D. H. Lee (1999b) Identification of ionic solutions using a SAW liquid sensor. Proc. Natl. Sci. Counc. ROC(A), 23, Shiokawa, S. and J. Kondoh (1996) Surface acoustic wave microsensors. Electronics and Communications in Japan, 79, Tonami, S., A. Nishikata, and Y. Shimizu (1995) Characteristics of leaky surface waves propagating on LiNbO 3 and LiTaO 3 substrates. Jpn. J. Appl. Phys., 3, * ** * * Christoffel Rayleigh SH leaky 136