ENERGY CONSIDERATIONS ON THE PROPAGATION AND THE GENERATION OF ACOUSTIC SURFACE WAVES IN PIEZOELECTRIC MEDIA

Transcription

1 RB03 Philips Res. Repts 27, , 1972 ENERGY CONSDERATONS ON THE PROPAGATON AND THE GENERATON OF ACOUSTC SURFACE WAVES N PEZOELECTRC MEDA by P. A. van DALEN and C. A. A. J. GREEBE Abstract A derivation based on energy considerations - which the authors regard to have considerable heuristic value - of the series resistance at resonance of interdigital acoustic-surface-wave transducers is given. The result agrees with previously known results, and rests on the assumption that the free wave propagation can be described by ngebrigtsen's well-known approximation. A confrontation with experimental results obtained from interdigital transducers on one face of thin piezoceramic plates having the Bleustein-Gulyaev geometry is presented. This example is particularly convenient because the various plate modes can be separately described by ngebrigtsen's approximation. Metallization of the opposite face of the plate has drastic influence and thus constitutes a useful experimental degree offreedom. Some comments are given on the behaviour of meander transducers on metallized plate faces. The influence of plate thickness on the series resistance of the transducers indicates that Bleustein-Gulyaev waves in a half-space will be difficult to generate. 1. ntroduetion The primary purpose of this paper is to provide a heuristic derivation of the magnitude at resonance of the series resistance R, of an interdigital transducer of acoustic surface waves (a.s.w.). Apart from some preliminary remarks (sec. 2) the reasoning chiefly consists of energy considerations in connection with ngebrigtsen's approximate description of the electrical properties of an a.s.w. (sec. 3), together with some elementary reflections on the currents and voltages in an interdigital transducer (sec. 4). We subsequently apply our results to the specific case of the excitation of Bleustein-Gulyaev plate modes by means of an interdigital transducer on one surface of piezoceramic plates. This turns out to be a very convenient example because if the plates are sufficiently thin it gives rise to various separate peaks in R., which can be altered drastically by metallizing the other surface of the plate (sec. 5). Experimental results obtained with such plates are then shown to agree reasonably well with our heuristic results for normal interdigital transducers. From this an interesting conclusion can be drawn as to the behaviour of transducers consisting of an insulating meander on a metallized surface of a thin piezoceramic plate. Finally we discuss measurements on the influence of the plate thickness on R., which indicate in accordance with expectation that for increasing plate thickness Bleustein-Gulyaev waves are generated with decreasing efficiency.

2 ENERGY CONSDERATONS ON ACOUSTC SURFACE WAVES Theoretical preliminaries t is well known 1) that the electric coupling between an a.s.w. propagating on the surface of a piezoelectric medium and the adjacent substance can be adequately described by characterizing each medium by an effective dielectric constant Beff' We shall consider the a.s.w. to propagate in the x-direction along a surface perpendicular to the y-axis, the positive y-axis pointing into the pertinent halfspace. We write all variables involved as where w is the angular frequency (which is assumed to be real), kx = kxr +j kxi the wavevector component in the propagation direction, and k; = k yr +j k Yi the wavevector component perpendicular to the surface. The quantity Beff is for positive real k; defined by 1) (1) Dy Beff(W, k x ) = --. kx cp (2) n (2) Dy is the normal component of the dielectric displacement and cp the electric potential, both taken at the surface. The dispersion relation of a surface wave travelling along the interface between two substances (a and b) which are only electrically coupled, reads 1) Beff (a) (w, k x ) + Beff (b) (W, k x ) = O. (3) t follows from ngebrigtsen's treatment 2) that in the case of small coupling the free propagation of a Rayleigh wave on a piezoelectric substance (a) can be quite well described by the approximate equation 1) wjk x - Voo Beff (al (W, k x ) = Bp, w/k x - Vo (4) where V = w/kx is the phase velocity ofthe wave. n eq. (4) Bp is the appropriate dielectric constant of the piezoelectric, and Voo and Vo are the a.s.w. phase velocities in the case of an adjacent substance with zero and infinite dielectric constant respectively. The relative difference (voo - vo)/vo is a direct measure for the piezoelectric coupling 3). f the adjacent medium has an effective dielectric constant eq. (3), together with (4) and (5) yields a dispersion relation (5) (6)

3 342 P. A. van DALEN and C. A. A. J. GREEBE From (6) one obtains in the approximation of small coupling, already implicit in ref. 4, i.e. in the approximation w/kxr ~ Va ~ v"" and kxdkxr «1, for the growth/decay constant k X of the a.s.w.: (7) Equation (7) is known to describe quite well the acoustoelectric interaction in the separate medium a.s.w. amplifier 4.5). ' 3. Acoustic surface waves on piezoelectrics: energy considerations n this section we derive a general relationship between the total power flow in an a.s.w. and the amplitude at the surface of the electric potential which it carries along. The derivation initially assumes an adjacent medium with a complex effective dielectric constant as given by (5). The ultimate expression ensues by taking the limit 81 ~ O. Adopting the same coordinate system as in sec. 2 we obtain for the power flow from the adjacent medium into the piezoelectric, which is necessarily electromagnetic, an averaged value where the y-axis points into the adjacent medium. n the quasi-static approximation the electric-field components Ex and E; follow directly from the electric potential cp. Using (1) we find (8) ~cp ~cp Ex = - - = j le; cp; Ez = - - = O. öx öz (9) The relevant component Hz of the magnetic field is found from Maxwell's equation VxH=D. (10) Using (1), (2) and (5) we find from the y-component of (10) Hz = w cp (s, +j S)' (11) nsertion of (9) and (11) into (8) gives <- Py) = <Ex Hz) = t Re [j kx cp w cp* (s, - j 81)] = t kx W 81 cp tp", (12) Here we assumed kxl «kxr>which on account of (7) is certainly justified in the limit 81 ~ 0, which we shall ultimately take. Since (12) describes the power fed into, or extracted from, the piezoelectric, it

4 ENERGY CONSDERATONS ON ACOUSTC SURFACE WAVES 343 must equal the divergence of the total power flow P of the a.s.w. which can bewritten as if we assume P to be parallel to the surface ( P ~ Px). Consequently we derive from (12) and (13) (13) t "x W S cp cp* = 2 "xl P.,(14) f we now substitute (7) into (14), we obtain in the limit S ~ 0: Voo- Vo Sp t w cp cp* = 2 Vo (s, + sr)2 P. (15) For our present purpose, i.e. deriving the series resistance R, at resonance of an interdigital transducer on a piezoelectric, we can simplify eq. (15) further by using the fact that the materials which are relevant in this context have dielectric constants which are large compared with that of the adjacent medium. This then leads to the fairly general equation ' Voo- Vo P cp cp* = 4--- Vo Sp w (16) A similar result has been obtained by Bers 6) in a different way. t should be noted in connection with what will be said below that the only assumptions about the a.s.w. made in the derivation of (16) have been (1) (voo -vo)/vo «1, (2) the validity of eq. (4), (3) the acoustic powerflow is parallel to the surface. 4. Considerations on voltage and current in an interdigital transducer of acoustic surface waves at resonance We consider a periodic interdigital transducer of the usual type, on the surface of a piezoelectric medium. No specific assumptions regarding the finger width, the finger spacing and the number of finger pairs shall be made. End effects will be ignored, and accordingly the currents in each finger pair may be assumed to have the same magnitude and phase. At fixed current amplitude and at fixed frequency the voltage difference between two adjacent fingers is due partly to the charges on the fingers and partly to the acoustic surface waves which are being generated. Since the emission of acoustic surface waves constitutes the only possibility of net energy transfer from the electric power source *), only the latter part of the voltage can have a component that is in phase with the current. *) t is assumed that no losses occur.

5 344 P. A. van DALEN and C. A. A. J. GREEBE n fact, a standard calculation on a spatially periodic transducer of acoustic volume waves in a piezoelectric medium having a simple symmetry shows that for that case the entire voltage caused by the waves is in phase with the current. f we extrapolate this result to the case of an interdigital transducer of acoustic surface waves, thus making an assumption about phase only and not about magnitudes, we may write down the relation Vn = 2 cp, (17) where V n is the voltage across the series resistance R, in the equivalent circuit of the transducer (fig. 1) 7). The factor 2 arises because in a wave the voltage o Fig. 1. Series representation of the transducer electrical equivalent circuit. difference between two points which are half a wavelength apart equals at most twice the amplitude of the electric potential carried by the wave. The power transmitted by the transducer equals VnVn*/2Rs = 2 cp cp*/rs Since acoustic surface waves are radiated at each end of the transducer, we thus obtain for the power carried by one a.s.w. P = cp cp*/rs (18) Combination of (6), (17) and (8) yields Voo - Vo 1 Rs=4 -- Vo sp co (19) Theexpression is in accordance with previously known results 2,7,8), but has been derived in a more general way. n deriving eq. (19) no assumptions or calculations on the exact field pattern in the piezoelectric medium have been made. nstead of this the conditions mentioned at the end of the previous section must be fulfilled for (19) to be valid, and of course we implicitly assumed that the individual waves do not drastically change their field pattern when leaving the region underneath the transducer.

6 ENERGY CONSDERATONS ON ACOUSTC SURFACE WAVES Bleustein-Gulyaev waves in piezoelectric plates For Bleustein-Gulyaev waves on the surface of a piezoceramic half-space one derives an expression 1) for Beff(W, k x ) which is quite different from (4). n fig. 2 this Berr is depicted as a function of v/vs, where Vs is the velocity of transverse bulk sound. However, it has been shown numerically 9) that for transverse electroacoustic waves propagating in piezoelectric plates of the Beustein- Gulyaev geometry a dependence of Beff on v ensues which for each plate mode may very well be approximated by an expression of the form (4). n fig. 3 the loef )~0 2S o.. v",/vs /' vo/vs _v/vs -St; -lot; Fig.2. eeer as a function of the phase velocity for Bleustein-Gulyaev waves. Drawn curve: positive penetration depth; dashed curve: negative penetration depth., y2 and ells are the appropriate electromechanical coupling factor and dielectric constant of the piezoelectric respectively. Beff pertaining to a specific piezoelectric plate is shown as a function of v/vs. f one takes for each mode the relevant zero of Beff as voo, the relevant pole of Berf as Vo, and if one derives Bp from the slope of the Beff vs v curve in Voo - henceforth denoted by BP' - then the Beff of each plate is approximated in the relevant region near v = Voo by an expression of the form (4). Since our main result (19) was derived by using only the form of (4), we now expect the series resistance R, ofinterdigital transducers on piezoceramic plates of the Bleustein- Gulyaev geometry for each mode to show a peak, its magnitude being described by (19), if we make the substitutions just mentioned. Furthe-more we have to take the following points into consideration:

7 346 P. A. van DALEN and C. A. A. J. GREEBE kxd=:tr open y2=0'2s Er,=TOOE O ~ -Scf o /."" ~ r:- 0 5 T T S ~'vs -Totf a) -Scf kxd=:tr shorted Y2=0'2S tft=tooeo _) 1,.1 ~ T T S _v/vs -Toef b) Fig. 3. Beff at the upper surface of a piezoelectric plate of the Bleustein-Gulyaev symmetry as a function ofthe phase velocity; (a) lower surface open, (b) lower surface shorted. d is the plate thickness, y2 and B11 S are the appropriate electromechanical coupling factor and dielectric constant of the piezoelectric respectively. (a) the peaks in R, should be sufficiently separated, which implies that the plate should be thin enough; (b) (voo - vo)/vo «1, which is borne out by fig. 3; (c) the power flow is parallel to the surface. The effect on Rs of metallizing the opposite surface of the plate should be fully reflected by the influence of this operation on 8 e rr(w, k x ) which can be obtained from comparing figs 3a and 3b. t should be noted that this metallization drastically changes the electric-field distribution underneath the transducer. A comparison between experiment and calculated resuls obtained with the procedure described in this paper is given in fig. 4. Here only the height and the position of the resonance peak in R, are given. The experimental results are taken from fig. 4 of ref. 9, and the theoretical results are calculated from (19). We have also investigated the contribution of diffraction losses to the value of R., by measuring R, of interdigital transducers with different finger lengths. t

8 ENERGY CONSDERATONS ON ACOUSTC SURFACE WAVES 347 Rmox (.a) 300K open 77K open a) o o _Freq. (104Hz) Rmox T b) o 300K shorted o Freq. (14Hz) 77K shorted Fig. 4. R, of an interdigital transducer deposited on the upper surface of a plate of PXE5 at 300 K and 77 K for two cases: (a) lower surface open, (b) lower surface shorted. Drawn line: experimental result, dashed line: theoretical result.

9 348 P. A. van DALEN and C. A. A. J. GREEBE turned out that the contribution of diffraction losses to R, was only a few per cent. We have also checked whether taking in (19) the actual value of Bp instead of B/ had influence on the overall agreement between theoretical and experimental results. However, it was found that a better fit between experiment and theory was obtained by taking the value of B/ instead oftaking the value of Bp' From fig. 4 we may conclude that agreement between theory and experiment is present notwithstanding the fact that the various peaks mutually show large relative differences. n particular it should be noted that the drastic change in the relative magnitude of the two peaks of lowest frequency, which is experimentally seen to result from the metallizing process described above, is represented quite well by the picture we propose. Another type of transducer described in an earlier paper 10) is an insulating meander on a metallized surface as displayed in fig. 5. We also performed experi- Fig. 5. Meander-type interdigital transducer. ments on these meander transducers for the case where the opposite surface of the plate was unmetall zed, and found that these show peaks in R, which do not differ s gnificantly from those of normal interdigital transducers, although clearly the potential at the surface carred by the emitted waves must be zero in this case. So here our result (19) seems to be correct, although outside the transducer the surface is shorted. Therefore, we conclude that the waves change their anatomy when leaving the transducer area so as to reduce their surface potential to zero with little reflection against the ends of the meander transducer. This is in accordance with theoretical expectations 11) that these plate modes show very little reflection when passing from a non-metallized to a metallized area. n conclusion we mention an experiment where we performed measurements on.piezoceramic plates in the Bleustein-Gulyaev geometry of increasing thickness. n the limit of infinite thickness the curve representing Berr as a function of v = calk; has the form 1) shown in fig. 2 9), where Bp' as determined from the slope at the point v = Voo clearly is infinite. This suggests, according to (19), that R, should decrease as the thickness of the plate increases, n fig. 6 this

10 ENERGY CONSDERATONS ON ACOUSTC SURFACE WAVES 349 n=o open R TOOOl--~' l 100~ ~ o _ d (um) Fig. 6. R; of an interdigital transducer deposited on the upper surface of a plate of PXE5 at 300 K as a function of plate thickness. Lower surface open. Dots: experimental results; drawn curve: theoretical result obtained by using the value of EP'; dashed curve: theoretical result obtained by using the value of Ep' expectation (drawn curve) is seen to be borne out by the experimental results. The dashed curve represents an alternative expression of the form (15), where the actual dielectric constant sp of the piezoceramic was substituted. This shows that to a large extent the observed decrease in R, with increasing plate thickness is due to the increasing slope of Seff at the point v = Voo- The latter result indicates that a Bleustein-Gulyaev wave in a half-space - which wave corresponds to the two lowest plate modes for plates offinite thickness 9) - cannot be generated by means of an interdigital transducer. REFERENCES Eindhoven, June ) C. A. A. J. Greebe, P. A. van Dalen, T. J. B. Swanenburg and J. Wolter, Physics Reports C, 235, ) K. A. ngebrigtsen, J. appl. Phys. 40, 2681, ) J. J. Campbell and W. R. Jones, EEE Trans. Sonics and Ultrasonics SU-S, 209, ) W. L. Bond, J. H. Collins, H. M. Gerard, T. M. Reeder and H. J. Sh aw, Appl. Phys. Letters 14, 122, ) J. Wolter, Phys. Letters 34A, 87, ) A. Bers, "nvited" Proceedings 1970 Ultrasonics Symposium EEE 70C69SU, 138, ) W. R. Smith, H. M. Gerard, J. H. Co llins, T. M. Reeder and H. J. Shaw, EEE Trans. Microwave Theory Tech. MTT-17, 856, ) B. A. A uld and G. S. Kino, EEE Trans. Electron Devices EO-S, 898, ) P. A. van Dalen, Philips Res. Repts 27, , ) P. A. van Dalen and C. A. A. J. G reebe, Physics Letters 33A, 93, ) J. Wol ter, private communication.