Hybrid Full-Wave Analysis of Surface Acoustic Wave Devices for Accuracy and Fast Performance Prediction

micromachines

Article Hybrid Full-Wave Analysis of Surface Acoustic Wave Devices for Accuracy and Fast Performance Prediction

Zhenglin Chen 1 , Qiaozhen Zhang 2,* , Sulei Fu 3, Xiaoyu Wang 4, Xiaojun Qiu 1 and Haodong Wu 4,*

1 School of Electronic Science and Engineering, Nanjing University, Nanjing 210093, China; [email protected] (Z.C.); [email protected] (X.Q.) 2 Mechanical and Electrical Engineering, College of Information, Shanghai Normal University, Shanghai 200234, China 3 Key Laboratory of Advanced Materials (MOE), School of Materials Science and Engineering, Tsinghua University, Beijing 100084, China; [email protected] 4 Key Laboratory of Modern Acoustics, Ministry of Education, Department of Acoustic Science and Engineering, School of Physics, Nanjing University, Nanjing 210093, China; [email protected] * Correspondence: [email protected] (Q.Z.); [email protected] (H.W.); Tel.: +86-186-5293-0171 (H.W.)

Abstract: In this paper, a hybrid full-wave analysis of surface acoustic wave (SAW) devices is proposed to achieve accurate and fast simulation. The partial differential equation (PDE) models of the physical system in question and graphics processing unit (GPU)-assisted hierarchical cascading technology (HCT) are used to calculate acoustic-electric characteristics of a SAW filter. The practical solid model of the radio frequency (RF) filter package is constructed in High Frequency Structure Simulator (HFSS) software and the parasitic electromagnetics of the entire package is considered in the design process. The PDE-based models of the two-dimensional finite element method (2D-FEM) are derived in detail and solved by the PDE module embedded in COMSOL Multiphysics. Due to the advantages of PDE-based 2D-FEM, it is universal, efficient and not restricted to handling arbitrary materials and crystal cuts, electrode shapes, and multi-layered substrate. Combining COMSOL

Multiphysics with a user-friendly interface, a ﬂexible way of modeling and mesh generation, it can greatly reduce the complicated process of modeling and physical properties deﬁnition. Based on a

Citation: Chen, Z.; Zhang, Q.; Fu, S.; hybrid full-wave analysis, we present an example application of this approach on a TC-SAW ladder ◦ Wang, X.; Qiu, X.; Wu, H. Hybrid Full- filter with 5 YX-cut LiNbO3 substrate. Numerical results and measurements were calculated for Wave Analysis of Surface Acoustic comparison, and the accuracy and efficiency of the proposed method were verified. Wave Devices for Accuracy and Fast Performance Prediction. Micromachines Keywords: partial differential equations; graphics processing unit (GPU); hierarchical cascading tech- 2021, 12, 5. https://dx.doi.org/ nology; parasitic electromagnetics; finite element method; perfect match layer; COMSOL Multiphysics; 10.3390/mi12010005 surface acoustic wave (SAW) devices

Received: 26 November 2020 Accepted: 19 December 2020 Published: 22 December 2020 1. Introduction

Publisher’s Note: MDPI stays neu- Because of their high performance, small size and low cost, surface acoustic wave tral with regard to jurisdictional claims (SAW) devices have become key radio frequency (RF) micro electro-mechanical components in published maps and institutional in the wireless communication field. With the coming of the 5G era, SAW devices have affiliations. gradually become stringent requirements on RF devices [1,2]. It promotes the emergence of a large number of new-type SAW devices, such as temperature compensated SAW (TC-SAW) devices [3–5], incredible high-performance SAW (I.H.P. SAW) devices [6–9], and laterally-excited bulk-wave resonators (XBARs) [10,11]. However, for such SAW Copyright: © 2020 by the authors. Li- devices with complicated structures, a universal method used for the accurate and fast censee MDPI, Basel, Switzerland. This simulation of the SAW devices calls for urgent demand. article is an open access article distributed As for practical SAW devices, especially SAW filters, hybrid full-wave analysis, under the terms and conditions of the including piezoelectric and electromagnetic analysis, is expected to be an effective way to Creative Commons Attribution (CC BY) simulate full-scale SAW devices [12]. Due to the consideration of multiphysics fields in the license (https://creativecommons.org/ analysis process, including acoustic, electrostatic and electromagnetic fields, the hybrid licenses/by/4.0/).

Micromachines 2021, 12, 5. https://dx.doi.org/10.3390/mi12010005 https://www.mdpi.com/journal/micromachines Micromachines 2021, 12, 5 2 of 13

full-wave analysis for SAW filter is expected to achieve higher accuracy. Since the filter is comprised of series and parallel resonators in which the frequency characteristics determine the response of the filter, the difficulty in simulation accuracy mainly depends on the simulation accuracy of the resonators. As for numerical simulation tools, the finite element method/boundary element method (FEM/ BEM) is an effective way to calculate a regular SAW resonator with finite- length structures [13]. Unfortunately, this method requires a long time to compute the Green function, especially an SAW resonator with a multi-layered and intricate structure, and the FEM/BEM method cannot effectively work. The pure FEM method is an effective and universal method to deal with a multi-layered and intricate structure, moreover, FEM modeling in COMSOL Multiphysics offers a more convenient and flexible way to build arbitrary structures [14]. However, the pure FEM method has difficulties in rapid calculation of the behavior of SAW devices due to amount of degrees of freedom (DOFs). The hierarchical cascading technique (HCT) is an innovative method to reduce the calculation of DOFs by cascading a two-dimensional (2D)-FEM model of unit blocks, as reported by Koskela et al. [15,16], which significantly reduces time consumption and memory consumption on full-scale SAW devices. However, it still takes a lot of time for the researcher to build 2D-FEM models and define physical properties. Subsequently, A. Shimko et al. [17] used COMSOL software to establish a quasi-three dimensional (3D) FEM model as a unit block, and as the COMSOL software has a user-friendly interface, flexible way of modeling and mesh generation, it significantly improved the efficiency of the FEM model and physical definition. However, the computation domain of the built-in quasi-3D FEM model is a 3D block, which has much more DOFs than that of the 2D-FEM model. More recently, the GPU-assisted HCT was proposed to accelerate calculation for SAW devices with finite-length structure [18–20]. Although GPUs have faster data processing capabilities than CPUs and the calculation speed is further improved, the difficulty with the 2D-FEM model established by the researcher himself or the quasi- 3D FEM model with more DOFs still exists, which led to a reduction of the simulation efficiency and calculation speed. In this work, considering the finite element method model is advantageous in versatil- ity for multi-layered and intricate structure, the partial differential equation (PDE)-based 2D-FEM model is proposed to calculate the acoustic-electric characteristics of full-scale resonators with arbitrary structure. In order to ensure its accuracy, the proposed model fully takes into account the propagation loss, dielectric loss, electrode resistance loss and electrode shape, making it consider practical factors as much as possible, and the angle of the trapezoid shape electrode was considered according to the process level. First, the PDE-based 2D-FEM model is solved by a solver of the PDE module embedded in COM- SOL, and due to its user-friendly interface, flexible definition, the simulation efficiency for the researcher, it is significantly improved. Then, FEM matrices derived from COMSOL were transferred to MATLAB via LiveLink for the remaining calculation, and because the computation domain of the proposed model is a 2D plane, the DOFs are significantly less than that of the quasi-3D FEM model. It is noted that the adequate number of mesh grids and DOFs is required to ensure the accuracy of the SAW simulation. In general, a wavelength must contain at least four grids or eight nodes, which can characterize accurately the behavior of SAW devices. Considering the number of DOFs, the PDE-based 2D-FEM model has remarkable advantages over a quasi-3D or 3D FEM model. Due to the fact that the calculation speed of the FEM largely depends on the number of DOFs, with the help of GPU-assisted HCT, the PDE-based 2D-FEM model can achieve simulation effectively and efficiently for full-scale SAW devices of large size and arbitrary structure. In addition, because the packaging structure has an increasing influence on the electrical performance of the SAW device by creating parasitic capacitance and inductance [21], the electromagnetic effect of the SAW filter is considered in the calculation process. By combining the acoustic field, electric field and electromagnetic field, hybrid full-wave simulation technology has been proven to be an efficient and powerful method to simulate the Micromachines 2021, 12, 5 3 of 13

By combining the acoustic field, electric field and electromagnetic field, hybrid full-wave simulation technology has been proven to be an efficient and powerful method to simulate the behavior of a SAW filter with higher accuracy. Firstly, the practical solid model of the SAW filter package is constructed and then the electromagnetic effect of the entire package is calculated in the High Frequency Structure Simulator (HFSS) software. Based on hybrid full-wave analysis, the frequency response of the TC-SAW filter on 5° YX-cut

Micromachines 2021, 12, 5 LiNbO3 substrate is investigated, and the simulated insertion loss curve is in3 ofgood 13 agreement with the measurement. The validity and accuracy of the proposed method were confirmed. The remainder of this paper is divided into three sections. Section 2 is devoted to the behavior of a SAW filter with higher accuracy. Firstly, the practical solid model of the SAW PDE-based 2D-FEM model for a full-scaled resonator and the electromagnetic model for filter package is constructed and then the electromagnetic effect of the entire package is calculateda filter. In in Section the High 3, Frequencythe admittance Structure curve Simulator and the (HFSS) relative software. bandwidth Based of on the hybrid TC-SAW ladder resonator and the insertion loss curve of the TC-SAW ladder◦ filter are calculated and full-wave analysis, the frequency response of the TC-SAW filter on 5 YX-cut LiNbO3 substratecompared is investigated,with the experimental and the simulated results. insertion Finally, loss conclusions curve is in goodare discussed agreement in with detail in the thelast measurement. section of the The paper. validity and accuracy of the proposed method were confirmed. The remainder of this paper is divided into three sections. Section2 is devoted to the2. Simulation PDE-based 2D-FEM Techniques model for a full-scaled resonator and the electromagnetic model for a filter. In Section3, the admittance curve and the relative bandwidth of the TC-SAW 2.1. Calculation Procedure ladder resonator and the insertion loss curve of the TC-SAW ladder filter are calculated and comparedFigure 1 with shows the the experimental overall flow results. chart Finally, of the conclusions calculation are procedure discussed infor detail the SAW in filter. theTo lastobtain section the of acoustic-electric the paper. characteristic of a SAW resonator, a PDE-based 2D-FEM model of the resonators in the SAW filter was solved by a solver of the PDE module em- 2. Simulation Techniques bedded in the FEM software COMSOL. GPU-assisted hierarchical cascading technology 2.1. Calculation Procedure (GPU-HCT) was used to achieve calculation acceleration. The calculation of the finite- Figure1 shows the overall flow chart of the calculation procedure for the SAW filter. length resonator was implemented on a workstation (Intel i9-9900K CPU @ 3.60 GHz, 96 To obtain the acoustic-electric characteristic of a SAW resonator, a PDE-based 2D-FEM modelGB RAM) of the with resonators NVIDIA in TITAN the SAW RTX filter GPUs was solved(24 GB by HBM2 a solver memory, of the PDECUDA module Compute Ca- embeddedpability 10.0 in the and FEM total software of 4608 COMSOL. CUDA cores). GPU-assisted hierarchical cascading technology (GPU-HCT)Meanwhile, was used the to practical achieve calculationsolid model acceleration. of the RF Thefilter calculation package ofwas the constructed finite- in lengthHFSS resonatorsoftware wasand implementedthe parasitic onelectromagneti a workstationcs (Intel were i9-9900K calculated CPU in @ the 3.60 design GHz, process. 96Finally, GB RAM) simulation with NVIDIA for the TITAN SAW RTX filter GPUs with (24 higher GB HBM2 accuracy memory, and CUDA speed Computewas achieved by Capabilitycombining 10.0 the and acoustic total of behavior, 4608 CUDA electric cores). behavior and electromagnetic behavior.

Input: Materials and crystal cuts, electrode shapes, multi-layered substrate

Geometry building and mesh generation of 2D-FEM

Package construction of SAW filter in Solved by PDE-based 2D-FEM model HFSS software

Derived FEM matrices from Load the resonator layout of SAW COMSOL to MATLAB via LiveLink filter in HFSS software

S parameters calculation for resonator Electromagnetic effect calculation of of SAW filter based on GPU-assisted SAW filter in HFSS software HCT

Hybrid full-wave analysis of SAW filter

Analyzed the behavior of SAW filter FigureFigure 1. 1.The The overall overall ﬂow flow chart chart of calculation of calculation for the for surface the surface acoustic acoustic wave (SAW) wave ﬁlter. (SAW) filter.

Meanwhile, the practical solid model of the RF ﬁlter package was constructed in HFSS software and the parasitic electromagnetics were calculated in the design process. Finally, simulation for the SAW ﬁlter with higher accuracy and speed was achieved by combining the acoustic behavior, electric behavior and electromagnetic behavior. Micromachines 2021, 12, 5 4 of 13

Micromachines 2021, 12, 5 4 of 13 2.2. A PDE-based 2D-FEM Model for Resonator Without loss of generality, the acoustic surface wave was assumed to propagate in the direction and2.2. Athe PDE-based partial 2D-FEM derivative Model of for the Resonator physical field along the direction ∂/∂ = 0. It means thatWithout the behavior loss of generality, of SAW devices the acoustic can surface be characterized wave was assumed by the to 2D-FEM propagate in model, which is equivalentthe x1 direction to the and previously the partial derivativereported ofquasi-3D the physical FEM field [17]. along As theshownx2 direction in ∂/∂x2 = 0. It means that the behavior of SAW devices can be characterized by the 2D-FEM Figure 2a, the finite-lengthmodel, which structure is equivalent of the to TC-SAW the previously resonator, reported with quasi-3D the FEMtrapezoid [17]. As shape shown in electrode, is surroundedFigure2 a,with the finite-lengtha perfectly structure matched of thelayer TC-SAW (PML) resonator, for absorbing with the thoroughly trapezoid shape unwanted reflectiveelectrode, waves isat surrounded any incident with an a perfectlygle, and matched the trapezoid layer (PML) angle for of absorbing the electrode thoroughly is set according to unwantedthe practical reflective processing waves at anytechnology. incident angle, Figure and 2b the shows trapezoid the angle unit of block the electrode of TC-SAW devices. is set according to the practical processing technology. Figure2b shows the unit block of TC-SAW devices.

FigureFigure 2. Schematic 2. Schematic diagram diagram of finite-length of finite-length temperature temper compensatedature compensated (TC)-SAW resonator. (TC)-SAW (a) The resonator. whole TC-SAW (a) resonator; (b) the unit block of the TC-SAW resonator. The whole TC-SAW resonator; (b) the unit block of the TC-SAW resonator. For piezoelectric material, the relation of the mechanical displacement and electric For piezoelectricfield material, is described the through relation the coupled of the constitutivemechanical equations displacement [22] and electric field is described through the coupled constitutive equations [22] TI = cIJSJ − eIjEj (1) = − (1) Di = eiJSJ + εijEj (2) = + where TI and SJ are stress and strain tensors, respectively. cIJ, eiJ and εij are stiffness(2) constant, piezoelectric stress constant and dielectric permittivity constant, respectively. where and are stress and strain tensors, respectively. , and are stiffness Di and Ej are the electric displacement vector and electric field, respectively. constant, piezoelectricThe stress relation constant of strain and and mechanicaldielectric displacementpermittivity can constant, be described respectively. as and are the electric displacement vector and electric field, respectively. S = ∇ u (3) The relation of strain and mechanical displacementJ caniJ i be described as where =∇∂ 0 0 0 ∂ ∂  (3) ∂x1 ∂x3 ∂x2 ∂ ∂ ∂ ∇iJ =  0 0 0  (4) where  ∂x2 ∂x3 ∂x1  0 0 ∂ ∂ ∂ 0 ∂ ∂x3∂ ∂x2 ∂∂x1 According to Maxwell’s000 equations and boundary conditions of SAW devices, the rela- ∂ ∂ ∂ tion among the electric displacement Di, electric field Ej, electric potential φj and charge density ρ are expressed as∂ ∂ ∂ s ∇ = 0 0 0 (4) ∂ ∂D = εE ∂ (5) ∂ ∂E = −∇∂ φ (6) 00 0 ∂ ∂∇· D∂= 0 (7)

According to Maxwell’s equations and boundary∇·D conditions= ρs of SAW devices, the re- (8) lation among the electric displacement , electric field , electric potential and charge density are expressed as =ε (5) =−∇ (6) ∇∙=0 (7)

∇∙= (8)

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h i where, the Nabla operator ∇ = ∂ ∂ ∂ , Equation (7) is applied in a piezoelectric ∂x1 ∂x2 ∂x3 medium, and Equation (8) is applied on the interface between the piezoelectric medium and electrode. In this work, we assume that the length of the aperture along the x2 direction is inﬁnite (∂/∂x2 = 0). Thus, the operator ∇iJ and Nabla operator ∇ can be expressed as the following equation. According to Equations (9) and (10), only the x1 and x2 coordinate axes are contained in the operator ∇iJ and the Nabla operator ∇. That is to say, the computation domain of the PDE-based 2D-FEM model is a 2D plane, as shown in Figure2. Obviously, the DOFs are signiﬁcantly less than that of the quasi-3D-FEM model reported by A. Shimko et al.  ∂ 0 0 0 ∂ 0  ∂x1 ∂x3 ∇ =  0 0 0 ∂ 0 ∂  (9) iJ  ∂x3 ∂x1  0 0 ∂ 0 ∂ 0 ∂x3 ∂x1 h i ∇ = ∂ 0 ∂ (10) ∂x1 ∂x3 Assuming that there is no external force applied, the equilibrium equation in the piezoelectric medium can be described with tensor form

.. ∇iJcJK∇Klul + ∇iJeJK∇φ = ρui (11)

− ∇iεim∇mφ + ∇eiK∇Kjuj = 0 (12) Correspondingly, the derived equilibrium Equations (11) and (12) can be converted with the matrix form as follows. Based on the 2D-FEM model, four solutions u1, u2, u3 and φ of the equilibrium equation can be obtained by solving Equation (13). In addition, Voigt’s notation is used in Equation (13).   c11 c15 c16 c14 c15 c13 e11 e31  c c c c c c e e   51 55 56 54 55 53 15 35   2u   c c c c c c e e  ∇u  ρω 1  61 65 66 64 66 64 16 36  1 2    ρω u2   c41 c45 c46 c44 c46 c44 e14 e34  ∇u2   2  − ∇·    = 0 (13)  ρω u3   c51 c55 c56 c54 c55 c53 e15 e35  ∇u3    0  c31 c35 c36 c34 c35 c33 e13 e33  ∇φ    e11 e15 e16 e14 e15 e13 −ε11 −ε13  e31 e35 e36 e34 e35 e33 −ε31 −ε33 where ω is angular frequency, and ρ is density. The main formulas and corresponding description of PDE-based 2D-FEM model are given in Supplementary Material. For clarity and space limitations, the detailed deduction process is not mentioned in this section. As FEM software COMSOL Multiphysics provides a mathematics module, it provides a general form PDE interface for solving piezoelectric Equation (13). In the case of four dependent variables u1, u2, u3, φ, a general form system of the equation takes the following form (14) [23].

∂2u ∂u elk k + dlk k − ∇·(c∇u + αu − γ) + β∇u + au = f in Ω (14) a ∂t2 a ∂t k k k k l

lk lk where the equation index l and k ranges from 1 to 4, ea is the mass coefficients and da is damping coefficients. c is the diffusion coefficient. α is the conservative flux convection coefficient. γl is the conservative flux source term, βl is the convection coefficient. a is the absorption coefficient. fl is the source term, Ω is the computational domain. Note that lk lk lk ea , da and ca are four-by-four matrices, γl, βl and fl are four-by-one matrices, α and a is scalar. According to Expression (13), the coefficient of Equation (14) is given by Micromachines 2021, 12, 5 6 of 13

, and are four-by-four matrices, , and are four-by-one matrices, and a is scalar. According to Expression (13), the coefficient of Equation (14) is given by Micromachines 2021, 12, 5 6 of 13 = 0 c=  ρ   ρ  elk =  a  ρ  0   c11 c15 c16 c14 c15 c13 e11 e31  c51 c55 c56 c54 c55 c53 e15 e35     c61 c65 c66 c64 c66 c64 −e16 e36−     c41 c45 c46 c44 c46 c44 e14 e34  c= − −   c c c c c c e e   51 =0,=0,=0,=0,a=0,55 56 54 55 53 15 =035 .   c c c c c c e e  Based on the above analysis, we assume31 35 that ∂36/∂ =0,34 the35 PDE-based33 13 2D-FEM33  model  e e e e e e −ε −ε  is introduced in detail and four solutions11 15 , ,16 and14 15 of Equation13 11 (13) 13can be ob- e31 e35 e36 e34 e35 e33 −ε31 −ε33 lk tained by solved according to Equation d(14)a =. 0,Therefore,α = 0, γ = 0,theβ =PDE-based0, a = 0, fl = 2D-FEM0. model is equivalent to the built-in quasi-3D FEM model of COMSOL Multiphysics reported by A. Based on the above analysis, we assume that ∂/∂x2=0, the PDE-based 2D-FEM model Shimko et al. [17]. Obviously,is introduced the in detailproposed and four method solutions for uSAW1, u2, u 3resonatorsand ]φ of Equation calculation (13) canhas be ob- natural advantages withtained fewer by solved DOFs. according In addition, to Equation to achieve (14). Therefore, quicker thecalculations PDE-based for 2D-FEM SAW model devices, the GPU-assistedis equivalent HCT to[20] the is built-in employed quasi-3D to FEMachieve model acceleration. of COMSOL Multiphysics reported by A. Shimko et al. [17]. Obviously, the proposed method for SAW resonators calculation has natural advantages with fewer DOFs. In addition, to achieve quicker calculations for SAW 2.3. GPU-Assisted HCTdevices, for Fu thell-Scale GPU-assisted Resonator HCT Simulation [20] is employed to achieve acceleration. To achieve quicker calculation for SAW devices, the GPU-assisted HCT is employed 2.3. GPU-Assisted HCT for Full-Scale Resonator Simulation to achieve acceleration. Figure 3a shows the multi-layered FEM mesh of the TC-SAW unit To achieve quicker calculation for SAW devices, the GPU-assisted HCT is employed block with a trapezoidto achieve electrode. acceleration. This FEM Figure 3modela shows is the achieved multi-layered easily FEM in mesh COMSOL of the TC-SAW Mul- unit tiphysics due to its user-friendlyblock with a trapezoid interface, electrode. flexible This way FEM ofmodel modeling is achieved and mesh easily generation, in COMSOL Multi- which can greatly reducephysics duethe tocomplicated its user-friendly process interface, of flexiblemodeling way and of modeling physical and properties mesh generation, definition. Figure 3bwhich shows can the greatly unit reduce block theis used complicated to cascade process into of modelinga finite-length and physical structure properties definition. Figure3b shows the unit block is used to cascade into a finite-length structure by by means of the continuity of the mechanical displacement and the electric potential lo- means of the continuity of the mechanical displacement and the electric potential located cated at the left- andat right-hand the left- and right-handedge. edge.

Figure 3. 2D-ﬁnite element method (FEM) mesh of TC-SAW devices. (a) Mesh of unit block with trapezoid shape electrode; (bFigure) the ﬁnite-length 3. 2D-finite TC-SAW element devices method consist (FEM) of a unit mesh block. of TC-SAW devices. (a) Mesh of unit block with trapezoid shape electrode; (b) the finite-length TC-SAW devices consist of a unit block. According to the literature [15], linear system equations of the unit block can be According to thewritten literature as the [15], following linear Equation system (15). equations Two ways of are the employed unit block to achieve can be acceleration:written as the following oneEquation is to transfer (15). A-matricesTwo ways from are RAMemployed to the GPU to achieve due to the acceleration: fact that GPUs one have is faster data processing capabilities than CPUs. The other is to eliminate the internal DOFs XI from to transfer A-matrices from RAM to the GPU due to the fact that GPUs have faster data processing capabilities than CPUs. The other is to eliminate the internal DOFs from the system in Equation (15) by HCT, and the system A-matrices can be greatly reduced in dimensionality.

Micromachines 2021, 12, 5 7 of 13

Micromachines 2021, 12, 5 7 of 13 0 0 0 = (15) 0 0 the system in Equation (15) by HCT, and the system A-matrices can be− greatly reduced in dimensionality. where A-matrices are the system matrix, , , are DOFs located on the left-hand,      interior domain,ALL andA LIright-hand0 unitALV block, respectively,XL 0which not only contains dis-  AIL AII AIR AIV  XI   0  placement ,, but also , the scalar  only contains=  the electric(15) potential on the  0 ARI ARR ARV  XR   0  surface of theA electrode,VL AVI isA theVR netA surfaceVV chargev on the −electrodeq surface. Subsequently, eliminating the internal DOFs from A-matrices, the B-matrix only where A-matrices are the system matrix, XL, XI, XR are DOFs located on the left-hand, includes , and . Therefore, the B-matrix can be written as interior domain, and right-hand unit block, respectively, which not only contains displacement u1, u2, u3 but also φ, the scalar v only contains the electric potential0 on the surface of the electrode, q is the net surface charge on the electrode surface. =0 (16) Subsequently, eliminating the internal DOFs X from A-matrices, the B-matrix only I − includes XL, XR and v. Therefore, the B-matrix can be written as The expression of the currents flowing  into the electrodes can be written as B11 B12 B13 XL 0 =  B21 B22 =−B23  XR − 0+ (16) (17) B31 B32 B33 v −q

2.4. The expressionElectromagnetic of the currents Model ﬂowingfor Filter into the electrodes can be written as

With the emergenceI =of− new-typeiω[(B31 − B SAW32)XL +devicesB33v] with higher frequencies(17) and smaller sizes, the packaging structure has an increasing influence on the electrical performance of 2.4. The Electromagnetic Model for Filter the SAW device by creating parasitic capacitance and inductance [12,21], mainly the par- With the emergence of new-type SAW devices with higher frequencies and smaller sizes,asitic the electromagnetics packaging structure of hasthe an socket increasing and influence bonding on wire the electrical of the SAW performance device. of This means the theelectromagnetic SAW device by creatingeffect of parasitic SAW capacitancedevices must and inductancebe considered [12,21], for mainly more the accuracy para- in simula- sitiction. electromagnetics Based on the ofcalculated the socket andacoustic-electric bonding wire of characteristic the SAW device. of This the means resonator the by the PDE- electromagneticbased 2D-FEM effect model, of SAW the devices unit block must be is considered obtainable for and more the accuracy practical in simulation. solid model of the RF Based on the calculated acoustic-electric characteristic of the resonator by the PDE-based filter package is constructed in the HFSS software, and then the electromagnetic effect of 2D-FEM model, the unit block is obtainable and the practical solid model of the RF filter packagethe entire is constructedpackage is in calculated the HFSS software, in HFSS, and as thenshown the electromagneticin Figure 4. Theref effectore, of the the multiphysics entirefield packageof the SAW is calculated filter, incontaining HFSS, as shown the acoust in Figureic4 .field, Therefore, electric the multiphysicsfield and electromagnetic fieldfield, of is the considered SAW filter, containingin the design the acoustic process field, and electric the behavior field and of electromagnetic SAW devices is analyzed field,with ishigh considered accuracy. in the design process and the behavior of SAW devices is analyzed with high accuracy.

Figure 4. 4.Electromagnetic Electromagnetic model model of the TC-SAWof the TC-SAW ladder ﬁlter. ladder filter.

3. Results and Discussion 3.1. Calculation of TC-SAW Resonator In this work, a 2D-FEM model of a unit block of TC-SAW devices was modeled; an SH-type SAW was excited on 5° YX-cut LiNbO3 substrate with copper electrode, with wavelength λ = 4 µm, aperture W = 104 µm, 5λ thick LiNbO3 substrate, 3λ thick PML, metallization ratio of IDT and reflector r = 0.44. The fundamental material constants of the substrate were taken from Kovacs constants [24]. Correspondingly, the mesh of the 2D-

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FEM model was made of free triangular elements and the discretization was set as quadratic element order, and triangular elements of about 456 and DOFs of about 4004 can be obtained. Micromachines 2021Figure, 12, 5 5 presents the admittance curve of the TC-SAW ladder resonator in the case8 of 13 of different SiO2 thickness and electrode thickness. The results show that the resonance frequency and anti-resonance frequency have a phenomenon of frequency shift, which 3. Results and Discussion depend not only on SiO2 thickness but also on electrode thickness. As shown in Figure 5a, 3.1. Calculation of TC-SAW Resonator the admittance curve of the TC-SAW ladder resonator with a 230 nm thick copper elec- In this work, a 2D-FEM model of a unit block of TC-SAW devices was modeled; trode is presented for comparison, with the increase◦ of SiO2 thickness, the resonance and an SH-type SAW was excited on 5 YX-cut LiNbO3 substrate with copper electrode, anti-resonance frequencywith wavelength decreasesλ =gradually 4 µm, aperture and theW =anti-resonance 104 µm, 5λ thick frequency LiNbO3 substrate, decreases 3λ thick more. Meanwhile, thePML, corresponding metallization ratio bandwidth of IDT and reﬂectordecreasesr = 0.44.as SiO The2 thickness fundamental increases. material constants In addition, spurious ofwaves the substrate occur when were taken SiO from2 thickness Kovacs constantsincreases [24 to]. Correspondingly,1200 nm, the reason the mesh be- of the 2D-FEM model was made of free triangular elements and the discretization was set as ing that the boundary conditions of the interface between the SiO2 coating and piezoelec- quadratic element order, and triangular elements of about 456 and DOFs of about 4004 can tric substrate were changedbe obtained. and the wave dispersion characteristic in the thin layer of SiO2 coating was also changed.Figure 5 presents the admittance curve of the TC-SAW ladder resonator in the case For comparison,of differentFigure SiO5b 2showsthickness the and admittance electrode thickness. of the TC-SAW The results ladder show that resonator the resonance frequency and anti-resonance frequency have a phenomenon of frequency shift, which de- with 800 nm thick SiO2. As can be seen, the admittance curve moves to the left with the pend not only on SiO thickness but also on electrode thickness. As shown in Figure5a, increase of electrode thickness. The 2spurious wave becomes considerably visible when the admittance curve of the TC-SAW ladder resonator with a 230 nm thick copper elec- electrode thicknesstrode increases is presented to 250 for comparison,nm. By comparing with the increase the ofdifferences SiO2 thickness, in admittance the resonance and curves affected by variationanti-resonance of SiO frequency2 thickness decreases and gradually electrode and thickness, the anti-resonance it is found frequency thatdecreases the frequency responsemore. characteristic Meanwhile, of the TC-SAW corresponding devices bandwidth is more decreases sensitive as SiOto the2 thickness change increases. of In addition, spurious waves occur when SiO2 thickness increases to 1200 nm, the reason electrode thickness. Therefore, the reasonable parameter selection of electrode and SiO2 being that the boundary conditions of the interface between the SiO2 coating and piezo- coating is very crucialelectric for substrate suppressing were changed the spurious and the wavewave dispersion when designing characteristic TC-SAW in the thin de- layer of vices. SiO2 coating was also changed.

20 h_SiO2=550 nm 20 h_Cu=200 nm h_Cu=230 nm 10 h_SiO2=800 nm Rayleigh wave h_Cu=250 nm 0 h_SiO2=1200 nm 0 SH wave SH wave -10

-20 -20

-30 Rayleigh wave

Admittance (dB) -40 -40 (dB) Admittance TC−SAW resonator TC−SAW resonator -50 5◦ YX−cut LiNbO 5◦ YX−cut LiNbO 3 -60 3 -60 h_Cu=230 nm h_SiO2=800 nm -70 n_IDT=240, n_GR=20 (each) n_IDT=240, n_GR=20 (each) -80 800 900 1000 1100 800 900 1000 1100 Frequency (MHz) Frequency (MHz) (a) (b)

◦ FigureFigure 5. The 5. The simulated simulated admittance admittance curve of thecurve TC− ofSAW the resonator TC−SAW on 5resonatorYX−cut LiNiOon 5°3 YXsubstrate.−cut LiNiO (a) The3 admittancesub- Y11strate.curve with(a) The SiO 2admittancethickness dependency. Y11 curve (b with) The admittanceSiO2 thickness Y11 curve dependency. with electrode (b) thickness The admittance dependency. Y11 curve with electrode thickness dependency. For comparison, Figure5b shows the admittance of the TC-SAW ladder resonator with 800 nm thick SiO2. As can be seen, the admittance curve moves to the left with the In this analysis,increase the finite-length of electrode thickness. TC-SAW The resonator, spurious waveconfigured becomes with considerably 280 electrodes visible when as IDT and 20 × 2 electrodeselectrode thickness as two increases side reflec to 250tors, nm. is By calculated. comparing theThe differences achieved in admittancesimulation curves speeds of PDE-basedaffected HCT by on variation the GPU of SiO 2werethickness about and 0.6 electrode s per thickness, frequency it is foundpoint, that which the frequency is about four times fasterresponse than characteristic that in the ofpaper TC-SAW by J. devices Koskela is more [15] sensitive(with 321 to theelectrodes change of need electrode thickness. Therefore, the reasonable parameter selection of electrode and SiO2 coating is 2.4 s per frequency verypoint). crucial For for example, suppressing the the frequency spurious wave ranged when from designing 800 to TC-SAW 1100 MHz, devices. the total calculational time Inof this finite-length analysis, the struct finite-lengthure is TC-SAW reduced resonator, to a quarter configured of the with previously 280 electrodes as reported method. ForIDT clarity, and 20 ×Table2 electrodes 1 shows as th twoe comparison side reflectors, in is calculation calculated. The speeds achieved between simulation the proposed and thespeeds previously of PDE-based reported HCT onmethods. the GPU were about 0.6 s per frequency point, which is about four times faster than that in the paper by J. Koskela [15] (with 321 electrodes

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Table 1. The comparison of other methods for SAW resonator simulation. T1 donate calculation needspeed 2.4 per s per frequency frequency point).point, ForT2 example,donate calculation the frequency speed ranged within from 800 the to range 1100 MHz, of f = 800 MHz–1100 theMHz. total calculational time of ﬁnite-length structure is reduced to a quarter of the previously reported method. For clarity, Table1 shows the comparison in calculation speeds between the proposed and the previously reported methods.Regular Resonator TC-SAW Resonator

Table 1. The comparison of other methods for SAW resonator(Reference simulation. [9]) T1 donate calculation (This Work) speed perNumber frequency point, of electrode T2 donate calculation speed within the 321range of f = 800 MHz–1100 MHz. 320

Calculation time T1 Regular Resonator 2.4 s TC-SAW Resonator 0.6 s Calculation time T2 (Reference [9]) 720 s (This Work) 180 s Number of electrode 321 320 Calculation time T1 2.4 s 0.6 s CalculationFigure 6 time shows T2 that the relative 720 s bandwidth (BW) 180of a s one-port resonator on 5° YX- cut LiNbO3 substrate with a copper electrode differs with SiO2 thickness and electrode ◦ thickness.Figure6 showsThe results that the relativeshow bandwidththat the relative (BW) of a bandwidth one-port resonator depends on 5 YX-not only on SiO2 thick- cut LiNbO3 substrate with a copper electrode differs with SiO2 thickness and electrode ness but also on electrode thickness. Furthermore, the BW decreases gradually as SiO2 thickness. The results show that the relative bandwidth depends not only on SiO2 thickness butthickness also on electrode increases thickness. and Furthermore,the BW decreases the BW decreases with the gradually increase as SiO of2 thickness electrode thickness, which increasesprovides and guidance the BW decreases for the with parameter the increase selection of electrode of thickness, electrode which and provides SiO2 coating to design the guidanceTC-SAW for devices the parameter with selection certain of BW. electrode and SiO2 coating to design the TC-SAW devices with certain BW.

11 h_Cu=200 nm 10 h_Cu=230 nm h_Cu=250 nm

8 BW (%) 7

TC−SAW resonator 6 ◦ 5 YX−cut LiNbO3 n_IDT=240, n_GR=20 (each) 5 600 800 1000 1200 1400 1600 1800 2000 h_SiO2 (nm) Figure 6. The simulated bandwidth (BW) of a TC-SAW ladder resonator on 5◦ YX-cut LiNbO Figure 6. The simulated bandwidth (BW) of a TC-SAW ladder resonator on3 5° YX-cut LiNbO3 sub- substrate with different copper electrode and SiO2 thickness. strate with different copper electrode and SiO2 thickness. 3.2. Calculation of TC-SAW Ladder Filters 3.2. TC-SAWCalculation ladder of filters TC-SAW are widely Ladder used Filters in wireless communication systems because of good temperature stability [25]. In this case, we consider one-port ladder resonators as buildingTC-SAW blocks of ladder filters. filters Figure are7 awidely shows the used configuration in wireless of aTC-SAW communication ladder systems because filterof good comprised temperature of a T-section stability branch formed[25]. In by this series case, (S) and we parallel consider (P) resonators one-port in ladder resonators as whichbuilding the frequency blocks characteristicsof ladder filters. determine Figure the response 7a shows of the ladderthe configuration filter. In order to of a TC-SAW ladder accurately simulate SAW devices, the corresponding layout of the TC-SAW ladder filter is designedfilter comprised in Figure7b withof a referenceT-section to Figure branch7a. The formed electromagnetic by series effect (S) of and the layout parallel (P) resonators in waswhich calculated the frequency in HFSS software. characteristics determine the response of the ladder filter. In order to accurately simulate SAW devices, the corresponding layout of the TC-SAW ladder filter is designed in Figure 7b with reference to Figure 7a. The electromagnetic effect of the layout was calculated in HFSS software.

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FigureFigure 7. 7. ConfigurationConfiguration of of the the TC-SAW TC-SAW ladder ladder filter. filter. ( (aa)) Electric Electric circuit circuit of of the the TC-SAW TC-SAW ladder ladder fil- filter; ter;(b) ( theb) the layout layout of theof the TC-SAW TC-SAW ladder ladder filter. filter.

BasedBased on on hybrid hybrid full-wave full-wave analysis, analysis, the the frequency frequency response response characteristic characteristic of of the the TC- TC- SAW ladder filter on 5◦ YX-cut LiNbO substrate with a copper electrode is researched SAW ladder filter on 5° YX-cut LiNbO33 substrate with a copper electrode is researched withinwithin the the frequency frequency range range of of 750 750 to to1050 1050 MHz. MHz. As shown As shown in Figure in Figure 8a, the8a, insertion the insertion loss curveloss curve of the of TC-SAW the TC-SAW ladder ladder filter with filter the with 230 the nm 230 thick nm electrode thick electrode is presented is presented for com- for comparison, and the results show that the laws of the frequency response characteristic parison, and the results show that the laws of the frequency response characteristic af- affected by SiO2 thickness is similar to that of the ladder resonator, and the resonance fected by SiO2 thickness is similar to that of the ladder resonator, and the resonance fre- frequency and anti-resonance frequency decreases gradually with the increase of SiO2 quency and anti-resonance frequency decreases gradually with the increase of SiO2 thick- thickness. The anti-frequency decreases more at the high end of the insertion loss curve. ness. The anti-frequency decreases more at the high end of the insertion loss curve. When When SiO2 thickness increases to 1200 nm, the insertion loss becomes large at the center SiO2 thickness increases to 1200 nm, the insertion loss becomes large at the center fre- frequency, and the reason is that the variation of SiO2 thickness yields a frequency shift of quency, and the reason is that the variation of SiO2 thickness yields a frequency shift of the one-port ladder resonator, which leads to an impedance mismatch of the ladder filter. the one-port ladder resonator, which leads to an impedance mismatch of the ladder filter. In Figure8b, the insertion loss curve of the TC-SAW ladder filter with the 800 nm thick In Figure 8b, the insertion loss curve of the TC-SAW ladder filter with the 800 nm thick SiO2 is illustrated for comparison. The results show that the regularity of the frequency SiO2 is illustrated for comparison. The results show that the regularity of the frequency response characteristic affected by electrode thickness coincided with that of the ladder responseresonator, characteristic and the overall affected insertion by electrod loss curvee thickness moves coincided to the right with with that the of decreasethe ladder of resonator,electrode thickness.and the overall When insertion electrode loss thickness curve moves decreases to the to right 200 nm,with the the insertion decrease loss of electrodebecomes thickness. large at center When frequency electrode duethickness to impedance decreases mismatch to 200 nm, caused the insertion by a frequency loss be- comesshift of large one-port at center ladder frequency resonator. due By to comparing impedance differences mismatch incaused frequency by a responsefrequency curves shift of one-port ladder resonator. By comparing differences in frequency response curves af- affected by variation of electrode thickness and SiO2 thickness, electrode thickness has fectedmore influenceby variation on of the electrode frequency thickness response and characteristic. SiO2 thickness, electrode thickness has more influence on the frequency response characteristic. 3.3. Experimental Verification To validate the simulation accuracy of the hybrid full-wave technique, a TC-SAW lad- ◦ der filter on 5 YX-cut LiNbO3 substrate, with 800 nm thick SiO2 and 230 nm thick copper electrode were fabricated. The center frequency of the ladder filter was designed to be 847 MHz. By means of comparison of the simulated insertion loss curve with the measured curve, the simulated result is in good agreement with the measured result, as shown in Figure9. The simulated curve is in good agreement with the measured result within the passband, and the small difference within the stopband is mostly due to two main reasons. First, considering the PDE-based 2D-FEM model perspective, this FEM model fully takes account of the propagation loss, dielectric loss, electrode resistance loss and electrode shape, making it consider practical factors as much as possible, but those parameters have a discrepancy with practical SAW devices. Second, from the manufacturing technology

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0 0 h_SiO =550 nm h_Cu=200 nm 2 -10 -10 h_Cu=230 nm h_SiO2=800 nm -20 h_Cu=250 nm -20 h_SiO2=1200 nm

-30 -30

-40 -40 (dB)

21 -50 (dB) -50 S 21

S -60 Micromachines 2021, 12, 5 -60 11 of 13 -70 -70 TC−SAW ladder filter TC−SAW ladder filter ◦ -80 ◦ 5 YX−cut LiNbO3 5 YX−cut LiNbO3 -80 h_Cu=230 nm -90 h_SiO2=800 nm -90 perspective,750 800 the 850angle 900 of the 950 trapezoid 1000 1050 shape electrode750 800 was 850set to 900 7◦, and 950 in practice, 1000 1050 the Micromachines 2021, 12, 5 Frequency (MHz) Frequency (MHz) 11 of 13 trapezoid shape electrode may be fabricated with inconsistent angles due to the instability (a) (b) of the process level. Figure 8. The simulated insertion loss of the TC-SAW ladder filter on 5° YX-cut LiNbO3 substrate. 0 (a) The insertion loss S21 curve with SiO2 thickness0 dependency. (b) The insertion loss S21 curve with electrode thicknessh_SiO =550 dependency.nm h_Cu=200 nm 2 -10 -10 h_Cu=230 nm h_SiO2=800 nm -20 h_Cu=250 nm -20 3.3. Experimental h_SiO Verification2=1200 nm -30 To validate the simulation accuracy-30 of the hybrid full-wave technique, a TC-SAW -40 -40 ladder filter on 5° YX-cut LiNbO3 substrate, with 800 nm thick SiO2 and 230 nm thick cop- (dB)

per electrode were fabricated. The 21 center-50 frequency of the ladder filter was designed to be (dB) -50 S 21

S 847 MHz. By means of comparison -60of the simulated insertion loss curve with the meas- -60 ured curve, the simulated result is in good agreement with the measured result, as shown -70 -70 in Figure TC−SAW9. The simulatedladder filter curve is in good agreement with theTC−SAW measured ladder filter result within ◦ -80 ◦ 5 YX−cut LiNbO3 5 YX−cut LiNbO3 -80 the passband, and the small difference within the stopband is mostly due to two main h_Cu=230 nm h_SiO =800 nm reasons. First, considering the PDE-based-90 2D-FEM model perspective,2 this FEM model -90 750 800 850fully 900takes account 950 1000 of the 1050 propagation750 loss, 800dielectric 850 loss, 900 electrode 950 resistance 1000 1050 loss and electrodeFrequency (MHz) shape, making it consider practical factorsFrequency as much (MHz) as possible, but those parameters(a) have a discrepancy with practical SAW devices.(b Second,) from the manufacturing technology perspective, the angle of the trapezoid shape electrode was set to 7°, and in Figure 8. The simulated insertion loss of the TC-SAW ladder ﬁlter on 5◦ YX-cut LiNbO substrate. (a) The insertion loss S Figure 8. The simulatedpractice, insertion the trapezoid loss of the shape TC-S electrodeAW ladder may filterbe fabricated on3 5° YX-cut with inconsistent LiNbO3 substrate. angles due21 to curve with SiO thickness dependency. (b) The insertion loss S curve with electrode thickness dependency. (a) The insertion2 lossthe S21 instability curve with of theSiO process2 thickness level.21 dependency. (b) The insertion loss S21 curve with electrode thickness dependency. 0 Simulation 3.3. Experimental Verification-10 Measurement To validate the simulation-20 accuracy of the hybrid full-wave technique, a TC-SAW ladder filter on 5° YX-cut-30 LiNbO3 substrate, with 800 nm thick SiO2 and 230 nm thick copper electrode were fabricated.-40 The center frequency of the ladder filter was designed to be

847 MHz. By means (dB) of-50 comparison of the simulated insertion loss curve with the meas- 21 S ured curve, the simulated-60 result is in good agreement with the measured result, as shown in Figure 9. The simulated curve is in good agreement with the measured result within -70 TC-SAW ladder filter the passband, and the small◦ difference within the stopband is mostly due to two main -80 5 YX-cut LiNbO3 reasons. First, consideringh-SiO the2 =800 PDE-based nm, h-Cu=230 nm 2D-FEM model perspective, this FEM model -90 fully takes account of 700the propagation 750 800 850 loss, 900 dielectric 950 1000 loss, 1050 electrode resistance loss and electrode shape, making it considerFrequency practica (MHz)l factors as much as possible, but those parameters have a discrepancyFigureFigure 9. 9. ComparisonComparison with practical of of the the measurement measurement SAW devices. with with simulated simulated Second, insertion insertion from loss loss the of of manufacturing the the TC-SAW TC-SAW filter ﬁlter on on 5° 5◦ technology perspective,YX-cutYX-cut LiNbO LiNbOthe angle33 substrate.substrate. of the trapezoid shape electrode was set to 7°, and in practice, the trapezoid4. Conclusions shape electrode may be fabricated with inconsistent angles due to

the instability of the processThis paper level. presents an effective technique for accurate and fast simulation of SAW devices based on hybrid full-wave analysis. First, the PDE-based 2D-FEM model was pro- 0 posed to calculate the acoustic-electric characteristic of SAW devices, and the practical solid model of the entire package Simulation was constructed and its parasitic electromagnetic effect was -10 calculated in HFSS software. Measurement With the use of the PDE-based 2D-FEM model to overcome the problem of a large number of DOFs, with the help of GPU-assisted HCT, the simulation -20 for a SAW filter with higher speed and accuracy was achieved in the frequency range. -30 The validity and accuracy of the proposed method were verified by comparing the simulated frequency response characteristics with that of the measurement of the TC-SAW -40 ◦ ladder filter on 5 YX-cut LiNbO3 substrate. Considering the generality of FEM, in future

(dB) -50 work, the proposed hybrid full-wave technique can be extended to accurately analyze the 21

S key factors affecting the performance (rectangularity, bandwidth, ﬂatness in passband) -60 of SAW devices with complex structures, and to ﬁnd out the corresponding relation be-

-70 TC-SAW ladder filter ◦ -80 5 YX-cut LiNbO3

h-SiO2 =800 nm, h-Cu=230 nm -90 700 750 800 850 900 950 1000 1050 Frequency (MHz) Figure 9. Comparison of the measurement with simulated insertion loss of the TC-SAW filter on 5° YX-cut LiNbO3 substrate.

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tween the performance and structure parameters, which is quite useful for the development of new-type SAW devices with high performance to meet the stringent requirements of RF devices.

Supplementary Materials: The following are available online at https://www.mdpi.com/2072-666 X/12/1/5/s1, title: Detailed derivation of the PDE model. Author Contributions: Z.C. wrote the manuscript and participated in design and calculation; Q.Z. supervised the writing and review of the manuscript and helped develop and refine the EM model; S.F. contributed to fabrication and experiment work; X.W. contributed to the analysis and the interpretation of the experimental results; H.W. and X.Q. proposed the concept of the PDE-based 2D-FEM model. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by the National Key Research and Development Program of China (Grant No. 2016YFC0104802), Beijing Science and Technology Project (Grant No. D171100004617001), Jiangsu Planned Projects for Postdoctoral Research Funds (Grant No.2019K053), and National Natural Science Youth Foundation of China (Grant No. 11904233). Institutional Review Board Statement: The authors follow International Committee of Medical Journal Editors (ICMJE) recommendation of authorship. All those designated as authors should meet all criteria for authorship, and all who meet the criteria should be identified as authors. Informed Consent Statement: The authors agree with the plan to submit/publish to micromachines; the contents of the manuscript; to being listed as an author; and to the conflicts of interest statement. Data Availability Statement: The authors have access to all the data in the study (for original research articles) and accept responsibility for its validity. Conflicts of Interest: The authors declare no conflict of interest.

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