Recursive Asymptotic Hybrid Matrix Method for Acoustic Waves in Multilayered Piezoelectric Media

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1 Open Journa o Acoustics, 2,, doi:.4236/oja.2.4 Pubished Onine September 2 ( Recursive Asymptotic Hybrid Matrix Method or Acoustic Waves in Mutiayered Piezoeectric Media Abstract Eng Leong Tan Nanyang Technoogica University, Singapore E-mai: eetan@ntu.edu.sg Received June 3, 2; revised Juy 8, 2; accepted Juy 23, 2 This paper presents the recursive asymptotic hybrid matrix method or acoustic aves in mutiayered piezoeectric media. The hybrid matrix method preserves the numerica stabiity and accuracy across arge and sma thicknesses. For discussion and comparison, the scattering matrix method is aso presented in physics-based orm and coherent orm. The atter orm resembes cosey that o hybrid matrix method and heps to highight their reationship and distinction. For both scattering and hybrid matrix methods, their ormuations in terms o eigenaves soution are provided concisey. Making use o the hybrid matrix, the recursive asymptotic method ithout eigenaves soution is described and discussed. The method bypasses the intricacies o eigenvaue-eigenvector approach and requires ony eementary matrix operations aong ith thinayer asymptotic approximation. It can be used to determine Green s unction matrix readiy and aciitates the trade-o beteen computation eiciency and accuracy. Keyords: Hybrid Matrix Method, Recursive Asymptotic Method, Mutiayered Media, Piezoeectric Media. Introduction For many years there has been considerabe interest in the study o acoustic ave propagation in mutiayered piezoeectric media. Many techniques have been deveoped or anaysis o such media, incuding transer matrix method [], impedance/stiness matrix method [2-4], scattering/reection matrix method [5-7] and hybrid matrix method [8]. A comprehensive revie o these methods has been provided in [9] aong ith their variants, numerica stabiity, computationa eiciency, useuness and deiciency. Since the transer matrix method becomes unstabe toard arge thicknesses, hie the impedance matrix method is inaccurate toard sma thicknesses, they are not to be discussed urther beo. On the other hand, oing to their numerica stabiity and accuracy, both scattering and hybrid matrix methods deserve to be expoited urther as mentioned or demonstrated in some recent orks [-]. In particuar, the scattering matrix methods so ar have been presented more in physics-based orm (in terms o reections and transmissions), hich has motivated the uniied matrix ormaism in []. Hoever, ith the uniied ormaism therein, it is sti not cear about any reationship or distinction ith hybrid matrix method. Moreover, most matrix methods thus ar rey on the eigenaves soution in their basic buiding bocks. Since eigensover oten takes substantia computations, it is useu to consider other methods ithout the need or eigenaves soution [9]. In this paper, e present the recursive asymptotic hybrid matrix method or acoustic aves in mutiayered piezoeectric media. The method is extended rom the non-piezoeectric case [8] and expoits the hybrid matrix hich preserves the numerica stabiity and accuracy across arge and sma thicknesses (instead o stiness matrix [3] that may become inaccurate). For discussion and comparison, e aso present the scattering matrix method in physics-based orm and coherent orm. The atter orm resembes cosey that o hybrid matrix method and heps to highight their reationship and distinction. For both scattering and hybrid matrix methods, their ormuations in terms o eigenaves soution are provided concisey. Making use o the hybrid matrix, the recursive asymptotic method ithout eigenaves soution is described and discussed. The method bypasses the intricacies o eigenvaue-eigenvector approach and requires ony eementary matrix operations aong ith thin-ayer asymptotic approximation. It can be used to determine Green s unction matrix

2 28 E. L. TAN readiy and aciitates the trade-o beteen computation eiciency and accuracy. 2. Acoustic Waves in Mutiayered Piezoeectric Media 2.. Probem Formuation Figure shos a panar mutiayered structure comprising N piezoeectric ayers stratiied aong ẑ direction (ithin optiona externa ayers and N+). For each ayer o thickness h, its upper and oer interaces/ boundaries are denoted by Z and Z, respectivey. Let the ieds in each ayer be described by ied vector ormed by generaized stress vector (comprising norma stress τ and norma eectric dispacement D z ) and generaized veocity vector υ (comprising veocity v and the rate o change o eectric potentia j ), i.e., υ τ, v Dz υ (2) Assuming pane harmonic ave ith exp( j t) time dependence and transverse avenumber k t s t, the ied vector satisies a irst-order dierentia equation as d A (3) dz here the ayer system matrix is A j s I ttp pp s ( pp ) ( pp ) pt st 3 t tt tp( pp) pt st () (4) A consists o the materia parameters o ayer speciied in terms o mass density and various stiness constants, piezoeectric stress constants and permittivity via s (see []) Soution ith Eigenaves Equation (3) can be ritten as an eigenvaue probem A ( jkz ) (5) hose soutions represent the eigenaves ithin each ayer. For convenience, the norma avenumbers k z Figure. Geometry o mutiayered piezoeectric media. and their associated eigenvectors can be grouped into the ooing matrices: P P (6) P ψ (7) υ υ Here, P is a diagona matrix o exp( jkz z) that is partitioned into P and P. The superscripts > and < stand or upard-bounded and donard-bounded partitions, hich correspond to upardbounded and donard-bounded eigenaves respectivey (c. boundedness/radiation condition). In ine ith ied vector (), the eigenave matrix ψ is decomposed into, υ and, υ partitions. Each o these may be urther partitioned in accordance ith their compositions in (2) as τ v, υ (8) D z (Note that our convention in (7)-(8) is that the notation ithout superscript > or < represents ieds hie the same notation ith such superscript represents aves o upard-bounded or donard-bounded type.) Using the matrices above, the ied vector soution can be expressed as ψ P c ψ (9) c is the coeicient vector (to be determined), hie P c is the ave ampitude vector that umps the exponentia terms together. Fooing the upard-bounded and donard-bounded associations

3 E. L. TAN 29 above, these vectors can be partitioned into c c, () c Note that, P, (and their decompositions) are unctions o z, hie ψ, c (and their decompositions) are not. Furthermore, the ied vector is continuous across the interace o to dierent ayers, so e have ( Z ) ( Z ). Hoever, the ave ampitude vector is not continuous ith ( Z ) ( Z ). Thus, it is important to speciy exacty the z ocation o the interace to be ithin hich o the to adjacent ayers Scattering Matrix Method Using the eigenaves in each ayer, one can proceed to determine the soution or a stack o mutiayered media. To that end, e irst deine the oca interace scattering matrix that better describes the physics o ave scattering (reection/transmission) at the interace o ayers and + : Z r,, Z t Z,, Z t r () r, and t, denote the oca reection and transmission matrices or aves incident rom ayer to +, hie r, and t, denote those or incidence rom ayer + to. These matrices can be derived directy in terms o the eigenaves o both ayers as,,,, r t t r υ υ υ υ () Based on the oca interace scattering matrix, one can determine the scattering matrix or additiona ayers (one at a time) o a stack using certain recursive agorithm. In particuar, consider the donard-bounded aves incident rom ayer + toard ayer. The stack reection and transmission matrices, r, and t,, can be obtained rom the oca interace scattering matrix and the preceding r, and t, using the recursive agorithm (c. (23) and (25) o [9]): r, r, t, P ( h ) r, (3) IP ( h ) r, P ( h ) r, P ( h ) t, t t,, IP ( h ) r, P ( h ) r, P ( h ) t, (4) Likeise, the stack reection and transmission matrices r, and t, or incidence o upard-bounded aves rom ayer toard ayer + can aso determined via recursive agorithm: r, r, t, P ( h ) r, P ( h ) (5) Ir, P ( h ) r, P ( h ) t, t, t, P ( h ) (6) Ir, P ( h ) r, P ( h ) t, The orm o (3)-(6) aciitates the physics-based description o ave mutipe reections in the stack o mutiayered media. As an aternative, it is instructive to deine the matrix reating the ave ampitude vectors in the orm Z Z S S Z 22 Z S S (7) Such matrix has been denoted as ayer-interace scatterer [9], since it combines the ayer scatterers P ( h ) and P ( h ) ith interace scattering matrix as S S P ( h ) S S22 I (8) r, t, P ( h ) t, r, I To be in coherent orm, the stack scattering matrix [: ] S is aso deined in pace o rt,, hich embeds (ithin ayers and +) the stack rom ayer to (denoted by the superscript [:]), i.e. [: ] [: ] Z Z S S [: ] [: ] Z 22 Z S S [: ] [: ] S S r, t, [: ] [: ] S S22 t, r, (9) (2) In terms o such stack matrix, (3)-(6) can be reritten as S S S S ( IS S ) S () [: ] [: ] [: ] [: ] [: ] 22 S S ( IS S ) S (22) [: ] [: ] [: ] 22 S S ( IS S ) S (23) [: ] [: ] ) [: ] 22 S S S S ( IS S ) S (24) [: ] [: ] [: ] Equations ()-(24) constitute the recursive agorithm o the so-caed generaized tota scattering matrix o []. In essence, they represent a u matrix variant o agorithm A3 in Tabe o [9].

4 3 E. L. TAN 3. Recursive Asymptotic Hybrid Matrix Method 3.. Hybrid Matrix Method The scattering matrix method in the previous section invoves reations among ave ampitude vectors and. As mentioned earier, since these vectors are not continuous across interaces, it shoud be more convenient to ork directy ith ied variabes instead. In this aspect, a variety o deinitions and agorithms are possibe incuding the transer and impedance matrix methods. These methods are not unconditionay stabe since they may cause numerica instabiity or inaccuracy probem or very arge or very sma ayer thickness. Such probem can be overcome atogether by resorting to Z Z H H υ 25) Z 22 Z υ H H H is caed ayer hybrid matrix since it has a mixture o impedance, admittance and transer eements. Using the eigenaves in each ayer, the ayer hybrid matrix can be determined as H H P h H H22 υ P h υ (26) υ υp ( h ) P ( h ) It can be anayticay shon that H is sti numericay stabe even hen the ayer thickness tends to ininity or zero. Indeed, assuming at east sight oss as in practice, hen the ayer thickness tends to ininity, i.e. h, P ( h ) and P ( h ) tend to zero, the hybrid matrix in (26) reduces to H H υ H H22 υ h (27) On the other hand, hen the ayer thickness tends to zero, it is evident that P ( h ) P ( h ) I ith h, thus the hybrid matrix in (26) becomes H H I H I H22 h (28) Thereore, the hybrid matrix preserves the numerica stabiity and accuracy across arge and sma thicknesses. [: ] For mutiayered media, the stack hybrid matrix H or a stack rom ayer to (denoted by the superscript [:]) is deined by [: ] [: ] Z Z H H υ [: ] [: ] Z 22 Z υ H H (29) This stack matrix can be obtained by incorporating the ayer matrix H into the recursive agorithm: [: ] [: ] [: ] [: ] [: ] H H H H( IH22 H) H (3) [: ] [: ] [: ] H H ( IH H ) H (3) 22 H H ( IH H ) H (32) [: ] [: ] ) [: ] 22 H H H H ( IH H ) H (33) [: ] [: ] [: ] Notice that the orm o (3)-(33) resembes cosey that o ()-(24), hich heps to highight their reationship and distinction. In particuar, both scattering matrix and hybrid matrix do not dier much in their recursive agorithms or a stack o mutiayered media. Hoever, besides reating dierent entities (aves vs. ieds ), their basic matrices are distinct, i.e., S invoves eigenaves o to ayers in () and (8); hie H invoves eigenaves o individua ayer ony in (26) Soution ithout Eigenaves-Recursive Asymptotic Method Thus ar both scattering and hybrid matrix methods rey on the eigenaves (o to or one ayers) as the input (or S and H ) in each recursion to arrive at S [: ] and [: ] H. For such eigenaves soution, there exist various intricacies o soving the eigenvaues and eigenvectors incuding compex root searching, degeneracy treatment and upard/donard eigenvector sorting or seection. To obviate the need or eigenaves, e resort to the recursive asymptotic hybrid matrix method. The method bypasses the intricacies o eigenvaue-eigenvector approach and requires ony eementary matrix operations aong ith thin-ayer asymptotic approximation as described beo. For each individua ayer, e geometricay subdivide the ayer into n+ subayers having thicknesses d /2 i i h or i, 2,..., n and d /2 n n h as shon in Figure 2. For the thinnest subayer n+, its hybrid matrix is obtained directy by thin-ayer asymptotic approximation: ( n) H dn dn dn dn I A A 2 2 A I A 2 2 dn d dn dn A I A I A A 2 2 (34)

5 E. L. TAN 3 Tabe. Agorithms or scattering and hybrid matrix methods. z x Input With eigenaves (5) (7) A ψ Agorithm Scattering matrix method: Physics-based orm ψ () (3) (6), ψ r, t r, t Coherent orm ψ, ψ S S (),(8) () (24) [: ] Hybrid matrix method: ψ H H (26) (3) (33) [: ] Figure 2. Geometric subdivision o ayer into n + subayers. Starting ith this matrix, e impement se-recursions as H H H H ( IH H ) H (35) 22 H H ( IH H ) H (36) 22 H H ( IH H ) H (37) ( i) ( i) ( i) ( i) ( i) 22 H H H H ( IH H ) H (38) This recursive agorithm proceeds unti i = and the () ayer hybrid matrix is ound as H H. Throughout the procedure, there is no need to sove any eigenprobem and the hybrid matrix can be computed staby and accuratey even or very thick or very thin ayer. 4. Discussion and Numerica Resuts The previous sections have discussed some agorithms or scattering and hybrid matrix methods. For concise comparison, Tabe ists each o the agorithms and its pertaining equations invoved or each major step represented by an arro. In each major step, there is at east one (dense) matrix inversion to be deat ith, hich oten constitutes the most time-consuming operation. For the scattering matrix method, e ist the agorithms in physics-based orm as e as coherent orm. The atter orm heps to bring out the cose resembance ith the agorithm o hybrid matrix method. Aso isted in Tabe is the input required or each agorithm. Since the scattering matrix reates ave ampitude vectors across interaces, the input ought to be eigenaves o to ayers. As or the hybrid matrix that reates ied variabes, the input may need ony the eigenaves o individua ayer. Through the recursive asymptotic method, the input does not invoke any eigenaves at a. To highight the distinctions beteen the hybrid ma- Without eigenaves ( A directy) Recursive asymptotic hybrid matrix method: (34) (38) (3) (33) [: ] A H H trix method ith eigenaves and the recursive asymptotic method ithout eigenaves, e urther ist don beo the key steps in their respective procedure. In particuar, the procedure ith eigenaves is i) Sove the eigenvaue probem (5) or avenumbers and eigenvectors ii) Perorm upard/donard-bounded eigenvectors sorting or seection in (6)-(7), noting the boundedness/ radiation condition and degeneracy treatment i needed iii) Derive the ayer hybrid matrix using (26). Step i) is oten time-consuming, hie step ii) deserves much careu attention and coud be rather bothersome in practice. On the other hand, the procedure ithout eigenaves via the recursive asymptotic method is i') Initiaize the thin-ayer asymptotic approximation (34) directy rom A ii') Perorm se-recursions (35)-(38) unti i = iii') The ayer hybrid matrix is ound as A steps here are straightorard and invove eementary matrix operations ony. To assess the accuracy o recursive asymptotic hybrid matrix method, e investigate the reative error changes ith the number o geometric subdivisions n+ or equivaenty, the recursion number n. We arbitrariy take a ZnO ayer o μm thick at GHz as an exampe. Figure 3 shos the average reative error versus recursion number n. The reative error is measured by () H H. H -H H (39) a e e here H a and He represent the ayer hybrid matrix obtained rom the recursive asymptotic method and eigenaves soution, respectivey. The error is cacuated by taking the average over a range o transverse avenumbers. Notice that the error decreases initiay due to smaer truncation error or smaer initia subayer thickness d n. Ater certain minimum point, the error increases sighty and reaches a pateau ithout increasing urther.

6 32 E. L. TAN Figure 3. Average reative error vs. recursion number n. To iustrate the useuness o recursive asymptotic hybrid matrix method, et us consider a ZnO/ diamond/si structure at 2 GHz. The thicknesses o ZnO and diamond ayers are.2 and μm respectivey, hie Si substrate and vacuum are assumed semi-ininite. For anaysis o surace acoustic ave (SAW) on such structure, one can derive the generaized Green s unction matrix G deined by v N( ZN) τ N( ZN) G (4) N( Z N) s here s is the charge density on the surace. G can be ormuated using the scattering matrix ith eigenaves in a robust manner, see [5]. Aternativey, one can aso determine G using the stack hybrid matrix [: N ] H (or a stack rom ayer to N) as G IY s diag(,,, j s t ) (4) Ys diag(,,, ) here is the permittivity or vacuum (ayer N+), [: ] [: ] [: ] [: ] Y H H ( Z H ) H (42) s 22 sub and Z sub is the characteristic surace impedance or Si substrate (ayer ) Zsub ( υ ) (43) The stack hybrid matrix H [: N ] can be obtained ith or ithout eigenaves soution as mentioned earier. Figure 4 shos the Green s unction eement G 44 computed ith and ithout eigenaves. In the atter case, e appy the recursive asymptotic hybrid matrix method ith n=6. Athough this recursion number is rather sma, the resuts agree quite e and the pots are barey distinguishabe. Reerring to Figure 3, one can seect higher recursion number or better accu- Figure 4. Green unction eement computed ith and ithout eigenaves (via recursive asymptotic hybrid matrix method ith n = 6). racy, athough this may not be needed in many cases (e.g. hen materia data is not that accurate). In genera, the computation eiciency is improved or oer accuracy required and aso or thinner ayer ith eer geometric subdivisions. Thereore the method provides a very convenient ay that aciitates the trade-o beteen computation eiciency and accuracy. Note that the eiciency improvement here is meant or every ayer and one i gain substantia savings in the tota computation time hen there are many ayers in the stack or modeing inhomogeneous media. Moreover, the method is very useu or being simpe enough since it does not require any eigenaves or a ayers (even semi-ininite substrate). Thus, it may be appicabe even hen the eigensover package is not readiy accessibe, such as on ight-eight muti-thread processors (e.g. GPUs). 5. Concusions This paper has presented the recursive asymptotic hybrid matrix method or acoustic aves in mutiayered piezoeectric media. The hybrid matrix method preserves the numerica stabiity and accuracy across arge and sma thicknesses. For discussion and comparison, the scattering matrix method has aso been presented in physics-based orm and coherent orm. The atter orm resembes cosey that o hybrid matrix method and heps to highight their reationship and distinction. For both scattering and hybrid matrix methods, their ormuations in terms o eigenaves soution have been provided concisey. Making use o the hybrid matrix, the recursive asymptotic method ithout eigenaves

7 E. L. TAN 33 soution has been described and discussed. The method bypasses the intricacies o eigenvaue-eigenvector approach and requires ony eementary matrix operations aong ith thin-ayer asymptotic approximation. It can be used to determine Green s unction matrix readiy and aciitates the trade-o beteen computation eiciency and accuracy. 6. Reerences [] E. L. Ader, Matrix Methods Appied to Acoustic Waves in Mutiayers, IEEE Transactions on Utrasonics, Ferroeectrics and Frequency Contro, Vo. 37, No. 6, 99, pp doi:.9/ [2] B. Coet, Recursive Surace Impedance Matrix Methods or Utrasonic Wave Propagation in Piezoeectric Mutiayers, Utrasonics, Vo. 42, No. -9, 24, pp [3] L. Wang and S. I. Rokhin, A Compiance/Stiness Matrix Formuation o Genera Green s Function and Eective Permittivity or Piezoeectric Mutiayers, IEEE Transactions on Utrasonics, Ferroeectrics and Frequency Contro, Vo. 5, No. 4, 24, pp [4] E. L. Tan, Stiness Matrix Method ith Improved Eiciency or Eastic Wave Propagation in Layered Anisotropic Media, Journa o the Acoustica Society o America, Vo. 8, No. 6, 25, pp doi:./.8287 [5] E. L. Tan, A Robust Formuation o SAW Green s Functions or Arbitrariy Thick Mutiayers at High Frequencies, IEEE Transactions on Utrasonics, Ferroeectrics and Frequency Contro, Vo. 49, No. 7, 22, pp doi:.9/tuffc [6] E. L. Tan, A Concise and Eicient Scattering Matrix Formaism or Stabe Anaysis o Eastic Wave Propagation in Mutiayered Anisotropic Soids, Utrasonics, Vo. 4, No. 3, 23, pp doi:.6/s4-624x(2)447-x [7] T. Pastureaud, V. Laude, and S. Baandras, Stabe Scatteringmatrix Method or Surace Acoustic Waves in Piezoeectric Mutiayers, Appied Physics Letters, Vo. 8, No. 4, 22, pp doi:.63/ [8] E. L. Tan, Hybrid Compiance-Stiness Matrix Method or Stabe Anaysis o Eastic Wave Propagation in Mutiayered Anisotropic Media, Journa o the Acoustica Society o America, Vo. 9, No., 26, pp doi:./.3967 [9] E. L. Tan, Matrix Agorithms or Modeing Acoustic Waves in Piezoeectric Mutiayers, IEEE Transactions on Utrasonics, Ferroeectrics and Frequency Contro, Vo 54, No., 27, pp doi:.9/tuffc [] V. Y. Zhang and V. Laude, Uniied and Stabe Scattering Matrix Formaism or Acoustic Waves in Piezoeectric Stacks, Journa o Appied Physics, Vo. 4, 28, [] M. Lam, E. Le Cézio, H. Amorín, M. Agueró, Janez Hoc, Marija Kosec, A. C. Hadky-Hennion and G. Feuiard, Acoustic Wave Transmission through Piezoeectric Structured Materias, Utrasonics, Vo. 49, No. 4-5, 29, pp [] E. L. Tan, Generaized Eigenprobem o Hybrid Matrix or Foquet Wave Propagation in One-Dimensiona Phononic Crystas ith Soids and Fuids, Utrasonics, Vo. 5, No., 2, pp [3] L. Wang and S. I. Rokhin, Modeing o Wave Propagation in Layered Piezoeectric Media by a Recursive Asymptotic Matrix Method, IEEE Transactions on Utrasonics, Ferroeectrics and Frequency Contro, Vo. 5, No. 9, 24, pp doi:.9/tuffc