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Energy spectrum of piezoelectric resonators – IEEE IUS contribution   

By Gergely Simon 14 December 2020

We are excited to share the results of one of our papers that was presented at the virtual IEEE International Ultrasonic Symposium this year! This work focused on the design of high-quality factor resonators and the methods one can employ to investigate energy content using a coupled time domain simulation and frequency domain data extraction method. 

These resonator structures are usually considered a difficult simulation problem in 3D, the model size and fine meshing results in hundreds of millions of degrees of freedom. These would be impossible to attempt using traditional frequency domain solvers. We employ a time domain wave propagation approach combined with domain decomposition: the simulation problem is distributed across hundreds of compute nodes. 

However, frequency domain quantities, such as impedance or dissipation at various frequencies are required to analyze device operation. These can be extracted using rolling Fourier transforms, and implementing it we considered various optimization approaches to reduce computational overhead and memory requirement. 

Without further ado, let us look at the details of this paper! 

Device overview 

We used three example devices: a 2D thin film bulk acoustic resonator (TFBAR, Fig. 1), its 3D version (Fig. 2), and a 2D solidly mounted resonator (SMR, Fig. 3). The idea here was to show that the method is applicable to various device types, and to see how it scales with complexity. 

For all model setup, we used our brand-new Matlab Toolbox, check it out here. 


Figure 1. Schematic of a 2D TFBAR used in simulations. Zoomed in view at the bottom helps identifying thin layers


Figure 2. Schematic of a 3D TFBAR used in simulations. The pentagon shape suppresses unwanted modes



Figure 3. Schematic of a 2D SMR used in simulations. Zoomed in view at the bottom helps identifying thin layers


In all models, the size of the resonator area was selected to achieve 50  impedance at resonance. (This can be simply obtained assuming a parallel plate capacitor with relative permittivity of 9.5 for the aluminum nitride layer.) 

For the various material properties, please refer to the paper itself. We only highlight here that the quality factor of materials was 1,000 for the 2D models and 200 for the 3D model. 

The mesh size was 25 elements per wavelength (EPW), 10 EPW and 25 EPW, for the 2D FBAR, 3D FBAR and 2D SMR models, respectively. Runtimes were 2.5 times the Q value of the materials. The 2D models were run on 4-core machines in an hour, while the 3D problem was distributed across 128 compute nodes and finished in 7.5 hours. 


In all cases, the following plots were generated: an impedance spectrum, the S11 scattering parameter spectrum, input electrical energy vs mechanical losses spectrum and a plot detailing the contribution of various loss mechanisms to mechanical losses. 

Note here, that it is clearly expected that the input electrical energy equals the sum of acoustic losses and boundary losses. The model was run for long enough that all energies are dissipated and no stored energy stays, therefore energy conservation states the equivalence of input energy and all dissipated losses. See Fig. 4 for an example. 

Piezoelectric Resonator

Figure 4. Energy conservation between input electrical energy and the sum of all types of mechanical losses



Figure 5. Acoustic loss and boundary power flow for the 2D FBAR device


Piezoelectric Resonator

Figure 6. Acoustic loss and boundary power flow for the 3D FBAR device


Piezoelectric Resonator

Figure 7. Acoustic loss and boundary power flow for the 2D SMR device

These results can be used to draw a few very important conclusions: 

  • The FBAR device generally has less boundary loss through the bottom than the SMR device (Fig. 5 vs Fig. 7, faint orange curves). This is expected, as although the SMR has Bragg reflectors, there is a solid mechanical contact between the resonator structure and the substrate. However, the TFBAR device is mostly suspended, and only flanking transmission of mechanical energy occurs at the anchors. Also note the width of the SMR boundary loss frequencies is also considerably larger 
  • The 3D TFBAR has almost zero boundary loss. This is due to the lower quality factor. This illustrates, that having an accurate quality factor of all materials is very important to capture losses accurately 
  • The absolute magnitude of the energies varies between devices, even though their impedances are similar, and the exact same excitation signal was used. This draws attention to how to interpret the data and carry out normalization if necessary 

We see this method more applicable to large resonator and filter designs compared to pure frequency domain approaches. 

The paper is available on IEEE Explore as G. Simon, M. S. Patel, A. Tweedie and G. Harvey, “Energy Spectrum Analysis for Optimal Design of Ultra-High Frequency (UHF) Piezoelectric Resonators Leveraging 3D FEA Domain Decomposition Method with Cloud HPC,” 2020 IEEE International Ultrasonics Symposium (IUS), Las Vegas, NV, USA, 2020, pp. 1-4, doi: 10.1109/IUS46767.2020.9251423. 

IEEE Explore Paper – Energy Spectrum Analysis for Optimal Design of Ultra-High Frequency (UHF) Piezoelectric Resonators Leveraging 3D FEA Domain Decomposition Method with Cloud HPC


Gergely Simon
Gergely Simon

Gergely Simon is an Application Engineer at OnScale. He received his PhD in Smart Systems Integration from Heriot-Watt University. As part of our engineering team Gergely assists with developing applications, improving our existing software and providing technical support to our customers.