Extraction of Electrical Equivalent Circuit of One Port SAW Resonator

Extraction of Electrical Equivalent Circuit of One Port SAW

Resonator using FEM based Simulation

Ashish Kumar Namdeo and Harshal B. Nemade* Department of Electronics and Electrical Engineering, Indian Institute of Technology Guwahati *Corresponding author: Professor, Department of Electronics and Electrical Engineering, Indian Institute of Technology Guwahati, Guwahati, India, [email protected]

Abstract: The paper presents a method of made of piezoelectric materials [3]. An IDT is a extraction of electrical equivalent circuit of a one metallic comb-shaped periodic structure which port surface acoustic wave (SAW) resonator converts electrical energy into acoustic energy from the results of simulation based on finite and vice versa [3], [4]. The SAW devices are element method (FEM) using COMSOL generally operated in two different ways: Multiphysics. A one port SAW resonator resonator and delay line [5]. A SAW delay line consists of large number of periodic interdigital type device is a two port device, where two IDTs transducer (IDT) electrodes fabricated on the are fabricated at the two ends of the substrate surface of a piezoelectric substrate. A section of separated by a few wavelengths. One IDT is aluminium IDT structure patterned on YZ called as a transmitter IDT and the IDT at the lithium niobate piezoelectric substrate with other end is called as a receiver IDT. The periodic boundary condition is incorporated in resonator devices are mainly of two types: one the simulation. The equivalent circuit of a SAW port resonator and two port resonator. In one port resonator consists of motional resistance, SAW resonator, a bidirectional IDT is fabricated capacitance and inductance connected in series, with a set of reflectors on either side or a and static capacitance in parallel. The electrical bidirectional IDT with large number of resonance characteristics of SAW resonator electrodes is fabricated without reflectors. In obtained from frequency domain analysis case of two port resonator, two sets of reflectors provided by COMSOL Multiphysics are used to are fabricated on the outer sides of the two calculate the equivalent circuit parameters. The bidirectional IDTs. The function of IDT calculation of electrical equivalent circuit fabricated on the piezoelectric substrate in parameters is useful in designing matching the SAW device is described by several circuits for SAW devices. methods of modeling such as discrete source or delta function method, impulse response Keywords: Equivalent electrical circuit, FEM, method, piezoelectric permittivity method, IDT, MEMS, SAW device, SAW resonator. matrix representation, coupling of mode (COM) method, numerical techniques using 1. Introduction equations, and equivalent circuit method. The detailed description of the above methods is The surface acoustic wave (SAW) devices presented by many researchers [2], [3], [7]. such as delay line, resonator, filter, oscillator, The discrete source method or delta function convolver, and correlator are used in method is one of the earliest and simplest communication and signal processing methods and explains the shape of the IDT applications [1]–[4]. Other SAW devices such as frequency response [3], [7]. It does not sensor and actuator are used for industrial consider energy, capacitance, and applications [5], [6]. The SAW technology plays electromechanical coupling coefficient of the important role in MEMS industries due to low material used in the device [3]. The impulse cost, batch fabrication, light weight and inherent response method predicts the absolute properties of its device substrate [2], [4]. The amplitude, hence the power of SAW by development of mobile phones stirred up more considering the energy of the wave [3]. The interest in the investigation of SAW devices. The piezoelectric permittivity method describes the SAW device consists of interdigital transducer electric potential associated with SAW in terms (IDT) made of aluminum or gold material of actual charge density by considering the fabricated on the surface of the device substrate

Excerpt from the Proceedings of the 2015 COMSOL Conference in Pune surface piezoelectric permittivity and quasi-static are governed by the continuum equation of approximation [2], [3], [8]. COM model is motion, Maxwell’s equations under the quasi- widely used in designing of SAW devices [2], static assumptions, the strain-mechanical [3], [9], [10]. This method is based on local displacement relations and appropriate boundary analysis of way of two coupled waves evolve conditions [2], [8]. In a homogeneous piezo- when propagating in both directions opposite electric substrate, the stress component Tij at to each other. The Mason equivalent circuit for each point of substrate depends on the applied surface wave transducers is well explained by electric field E or electric displacement D. The smith et al. [11]. The equivalent electrical stress tensor component Tij can be expressed as circuit at frequency near the resonance of one [2] port SAW resonator can be presented by E Tij  c S e E motional resistance, capacitance and inductance ijkl kl  kij k k l k connected in series and static capacitance in E parallel [7], [8]. It is called as Butterworth Van- where, cijkl is the stiffness tensor for constant Dyke model of piezoelectric resonator [8]. electric field, S is the strain tensor, e is the In this paper, we present a method of kl kij extraction of the electrical equivalent circuit piezoelectric tensor, and Ek is the electric field. from the finite element simulation of one port The electric displacement D is determined by SAW resonator having large number of IDT electric field E and permittivity of the material electrodes fabricated on the surface of  ij . The electric displacement D in piezoelectric piezoelectric substrate. The electrical equivalent substrate is also related to strain. It can be circuit parameters are calculated using electrical expressed as [2] resonance characteristics obtained from the simulation of SAW resonator. The electrical S Di ij E j  eijk S jk equivalent circuit is useful in designing of j j k matching circuits for SAW devices and also understanding about the electrical and resonance where, Di represents the electric displacement S behavior of SAW devices in easy manner. The and  ij represents the permittivity tensor for piezoelectric constitutive equations, electrical constant stress. equivalent circuit parameters, simulation A quasi-static approximation is used to methodology, results and discussion, and express the electric field since the elastic conclusions are given in the following sections. vibration travels much slower than the electromagnetic waves [2], [3], [8]. 2. Piezoelectric Constitutive Equations 3. Electrical Equivalent Circuit The electromechanical constitutive equations Parameters for piezoelectric substrate used in the simulation

IDT

Lm Cm Rm

Piezoelectric substrate

(a) (b) Figure 1. (a) Schematic of a one port SAW resonator with large number of IDT electrodes without reflectors and (b) electrical equivalent circuit of a one port SAW resonator.

Excerpt from the Proceedings of the 2015 COMSOL Conference in Pune The basic resonator modeling requires theory Cm of acoustic wave propagation, material C0  2  1 properties, geometric dimensions and electric ar r  field approximations [2], [8]. Another successful GQ way, there is an electrical equivalent circuit L  rr m  model used to characterize the resonance r behavior of the SAW resonators. The conceptual where G is the peak value of the conductance, picture of a one port SAW resonator with large r number of IDT electrodes without reflectors is and  2 f is the angular frequency of the shown in figure 1 (a) [12]. The corresponding SAW resonator. electrical equivalent circuit at near resonance The quality factor Qr at resonance of SAW frequency is shown in figure 1 (b), where C0 is resonator is determined as [13] the static capacitance which represents the fixed f dielectric capacitance of the resonator, Lm and Cm Q  r r f are the motional inductance and capacitance, which correspond to the contribution of inertia and elasticity respectively, and Rm is the where f is the bandwidth at half of the peak- motional resistance which corresponds to the conductance. contribution of the damping [7]. The motional The resonance frequency fr, anti-resonance resistance carries the information about acoustic frequency far, conductance G and quality factor energy dissipation in the SAW devices [8]. Qr at resonance frequency are determined from These parameters are calculated from resonance the electrical resonance characteristics of SAW frequency fr, anti-resonance frequency far, resonator obtained from the frequency domain conductance G and quality factor Qr at resonance analysis provided by COMSOL Multiphysics. frequency of the SAW resonator [7]. The equations for the calculation of these parameters 4. Simulation Methodology are given as A one port SAW resonator with metallic IDT RGGmr ff r having large number of electrodes fabricated on piezoelectric substrate is considered for the 1 C  simulation. Due to periodic nature of IDT m  QG r r r structure, a section of IDT electrodes is incorporated and appropriate boundary

x3 IDT

x x 1 2 IDT electrode

p p/4

ΓL ΓR

≈ ≈ Piezoelectric substrate

(a) (b) (c) Figure 2. (a) 2D geometry of one port SAW resonator with one periodic section of IDT electrodes, (b) picture of mesh applied to one port SAW resonator and (c) displacement profile at resonance frequency of 212.88 MHz.

Excerpt from the Proceedings of the 2015 COMSOL Conference in Pune conditions are applied in the simulation [12], [14] [14]. The 2D geometry of the SAW resonator (,,V)(,,V)u v u v with one periodic section of IDT electrodes used LR for simulation is shown in figure 2 (a). The (  1)n ,na  2  dimensions are as follows: depth of the piezoelectric substrate 160 µm (10 λ) in –x3 where u, v and V are the x-displacement in x1 direction, width of the piezoelectric substrate 16 direction, y-displacement in x3 direction and µm (1 λ) in –x1 direction, period of IDT applied electric potential, respectively, and a is electrode (p) 8 µm, width of IDT electrode (p/2) the width of the piezoelectric substrate in 4 µm, electrode thickness 0.2 µm in x3 direction. multiples of λ. The YZ lithium niobate (YZ LiNbO3) is used for The electric potential of 1 V is applied to the piezoelectric substrate and aluminium metal is IDT electrode. Extremely fine mesh with used for IDT electrodes. The material properties minimum element quality of 0.5 is used. The such as values of elastic constants, permittivity number of degree of freedom solved is in the 5 constants, stress constants and density are used order of around 2 × 10 . The picture of mesh from Warner et al. [15]. applied to the SAW resonator in the simulation is The bottom boundary of the piezoelectric shown in figure 2 (b). substrate is fixed and all other boundaries are The simulation is carried out using MEMS traction free. The periodic boundary condition is module provided by COMSOL Multiphysics applied to left (ΓL) and right (ΓR) sides of the [16]. The frequency domain analysis is used for device as shown in figure 2 (a). The applied calculation of electrical resonance characteristics periodic boundary condition is given as [12], of SAW resonator.

1.E+8108 fr 14 Conductance 1.E+6106 12 Susceptance

1.E+44

10 S) 10 6 1.E+2102 8 Δf 1.E+0100 6 4 1.E10--22 Admittance (S) Admittance f Admittance (10 Admittance 2 r 1.E10--44 0 1.E10--66 0.9793460 0.9793462 0.9793464 far -2 1.E10--88 -4 0.970 0.976 0.982 0.988 0.994 1.000 -6 Normalized frequency Normalized frequency (a) (b)

16.16 pF

9.72 nH 16.16 pF 77.29 nΩ

(c) Figure 3. Results of simulation of one port SAW resonator: (a) plot of magnitude of harmonic admittance per period versus normalized frequency and (b) plot of harmonic admittance per period versus normalized frequency of one port SAW resonator, and (c) electrical equivalent circuit of a one port SAW resonator with parameters value.

Excerpt from the Proceedings of the 2015 COMSOL Conference in Pune 4. Results and Discussion motional resistance corresponding to damping is observed at 77.29 nΩ. The electrical equivalent The results of simulation of a one port SAW circuit is useful in understanding the resonance resonator are shown in figure 2 (c) and figure 3 behavior and designing of matching circuits for (a) and (b). Figure 2 (c) shows the displacement SAW devices. This simulation can be further profile at resonance frequency. Figure 3 (a) and extended to study the equivalent circuit (b) shows the electrical resonance characteristics parameters of SAW resonator with finite number of SAW resonator. The plot of harmonic of IDT electrodes and other SAW devices. admittance per period as a function of normalized frequency is obtained for an electric 6. References potential of 1 V applied to the IDT electrodes. The normalized frequency is expressed as 1. C. C. W. Ruppel and T. A. Fieldly, Advances

  2 p f v0 , where v0 = 3477.91 m/s is the free in Surface Acoustic Wave Technology, Systems surface velocity of the piezoelectric substrate and Applications (vol. I), World Scientific calculated from eigenmode analysis. Figure 3 (a) Publishing Co. Pte. Ltd., Singapore (2000) shows the plot of magnitude of harmonic 2. D. Morgan, Surface Acoustic Wave Filters admittance as a function of normalized with Applications to Electronic Communications frequency. From figure 3 (a), the resonance and and Signal Processing, Elsevier, UK (2007) anti-resonance frequencies of the SAW resonator 3. D. Royer and E. Dieulesaint, Elastic Waves are observed at 212.88 MHz and 216.64 MHz, in Solids II – Generation, Acoustic-optic respectively. The value of admittance is high due Interaction, Applications, Springer–Verlag, New to long aperture of IDT electrodes. Figure 3 (b) York (1999) shows the plot of harmonic admittance as a 4. J. W. Gardner, V. K. Varadan and O. O. function of normalized frequency. From figure 3 Awadelkarim, Microsensors MEMS and Smart Devices, John Wiley & Sons Ltd., England (b), the quality factor Qr is calculated which is 168.15 × 106. The electrical equivalent circuit (2002) parameters calculated from the results of 5. M. Thompson, Surface-Launched Acoustic simulation of one port SAW resonator are given Wave Sensors: Chemical Sensing and Thin Film in Table 1. Characterization. John Wiley & Sons, New York (1997) Table 1 Electrical equivalent circuit parameters 6. T. Shigematsu, M. K. Kurosawa, and K. Asai, Nanometer stepping drives of surface Parameter(s) Value acoustic wave motor, IEEE Transactions on Motional resistance 77.29 nΩ Ultrasonics, Ferroelectronics and Frequency Motional capacitance 16.16 pF Control, Volume 50, no. 4, 376–385 (2003) Motional inductance 9.72 nH 7. K.-Y. Hashimoto, Surface Acoustic Wave Static capacitance 16.16 pF Devices in Telecommunications: Modelling and Simulation. Springer–Verlag, New York The extracted electrical equivalent circuit of (2000) the simulated one port SAW resonator having 8. A. Arnau (Ed.), Piezoelectric Transducers large number of IDT electrodes is shown in and Applications. Springer–Verlag, New figure 3 (c). York (2004) 9. C. C. W. Ruppel and T. A. Fieldly, 5. Conclusions Advances in Surface Acoustic Wave Technology, Systems and Applications (vol. The paper has presented the extraction of II). World Scientific Publishing Co. Pte. Ltd., electrical equivalent circuit of one port SAW Singapore (2001) resonator with large number of IDT electrodes. 10. V. Plessky and J. Koskela, Coupling-of- The admittance value of admittance is high due modes analysis of SAW devices, to long aperture and no damping in the International Journal of High Speed simulation. As the damping is not applied in the Electronics and Systems, Volume 10, 867– simulation of the SAW resonator, the value of 947 (2000) 11. W. R. Smith, H. M. Gerard, J. H. Collins,

Excerpt from the Proceedings of the 2015 COMSOL Conference in Pune T. M. Reeder, and H. J. Shaw, Analysis of interdigital surface wave transducers by use of an equivalent circuit model, IEEE Transactions on Microwave Theory and Techniques, Volume 17, 856–864 (1969) 12. A. K. Namdeo and H. B. Nemade, Simulation on effects of electrical loading due to interdigital transducers in surface acoustic wave resonator, Journal of Procedia Engineering, Volume 64, 322–330 (2013) 13. J. Koskela, V. P. Plessky, and M. M. Salomaa, SAW/LSAW COM parameter extraction from computer experiments with harmonic admittance of a periodic array of electrodes, IEEE Transactions on Ultrasonics, Ferroelectronics and Frequency Control, Volume 46, no. 4, 806–816 (1999) 14. M. N. Hamidon, S. A. Mousavi, M. M. Isa, and M. A. Mahdi, Finite element method on mass loading effect for gallium phosphate surface acoustic wave resonators, in Proceedings of the World Congress Engineering, Volume 1 (2009) 15. A. W. Warner, M. Onoe, and G. A. Coquin, Determination of elastic and piezoelectric constants for crystals in class (3M), The Journal of the Acoustical Society of America, Volume 42, 1223–1231 (1968) 16. COMSOL Multiphysics Users Guide Version 3.4, COMSOL, Inc., Burligton (2007)

Excerpt from the Proceedings of the 2015 COMSOL Conference in Pune