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**Theory Athanasios Margiolakis University of Crete 2012**

FEM analysis of Surface Acoustic Wave resonators of Piezoelectric Gallium Nitride on Silicon substrate for frequencies above 5 GHz Athanasios Margiolakis University of Crete 2012

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Purpose The purpose of this work is to develop an accurate numerical methodology that simulates SAW resonators for use with new technologies on wireless connections and telecommunications above 5 GHz

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**Index Introduction Theory of SAW devices**

Simulations (finite element method) Fabrication Characterization Results

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**Surface Acoustic Wave (SAW)**

Introduction Theory Simulation Fabrication Characterization Results Surface Acoustic Wave (SAW) Acoustic wave on surface of a material exhibiting elasticity Amplitude decays exponentially with depth into the substrate Generated by piezoelectric material stimulated by electrodes

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**Piezoelectric materials**

Introduction Theory Simulation Fabrication Characterization Results Introduction Theory Simulation Fabrication Characterization Results Piezoelectric materials Charge accumulates when force is applied Deforms when electrical field is applied

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**Interdigital Transducers (electrodes)**

Introduction Theory Simulation Fabrication Characterization Results Introduction Theory Simulation Fabrication Characterization Results Interdigital Transducers (electrodes) Alternating electric potential on IDTs is applied Electric field on piezoelectric material under IDTs Piezoelectric deforms creating waves Waves with wavelength similar to IDT dimensions survive

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**SAW propagation animation**

Introduction Theory Simulation Fabrication Characterization Results Introduction Theory Simulation Fabrication Characterization Results SAW propagation animation

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**Frequency control f = υ/λ**

Introduction Theory Simulation Fabrication Characterization Results Introduction Theory Simulation Fabrication Characterization Results Using SAW Frequency control f = υ/λ Dimensions of IDTs Wavelength λ Material properties Wave propagation speed υ

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**Introduction Theory Simulation Fabrication Characterization Results**

SAW Devices SAW filters - Frequency filter Used in telecommunications and wireless applications SAW Resonators Radio Transmitters Remote controls Radio links No channelization devices SAW Filters Mobile telephones Radio receivers

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**Introduction Theory Simulation Fabrication Characterization Results**

SAW devices SAW Sensors Chemical Optical Thermal Pressure Acceleration Torque Biological SAW sensors – using SAW propagation speed

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**Introduction Theory Simulation Fabrication Characterization Results**

Advantages SAW have better Performance Cost Size Compared to other technologies such as quartz crystals LC filters waveguide filters

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**Introduction Theory Simulation Fabrication Characterization Results**

Why above 5 GHz Current commercial telecommunication filter technology limited to 5 GHz Need for new technologies above 5 GHz Faster data transfers at higher frequencies Better performance filters (higher Q factor) Commercial products like WiMAX (2-11 GHz) and 4G mobile networks require higher frequencies

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**Introduction Theory Simulation Fabrication Characterization Results**

Materials Substrate material Silicon (Si) with diamond cubic structure, structural and electric properties highly anisotropic. Piezoelectric layer Gallium Nitride (GaN) is a binary III/V direct bandgap semiconductor with Wurtzite crystal structure

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**Introduction Theory Simulation Fabrication Characterization Results**

Material benefits Gallium Nitride (for piezoelectric material) Can be monolithically fabricated on microelectronic devices GaN SAW material is superior to other existing materials Silicon (for substrate material) Most common and low cost material for such applications

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**Why Numerical modeling**

Introduction Theory Simulation Fabrication Characterization Results Why Numerical modeling Need to accurately determine resonant frequencies Complex and non-linear problem Analytical methods not accurate Need for higher frequencies lead to dimension reduction Stronger effects of IDT dimensions and geometry on resonant frequencies Modeling using numerical methods Finite Element Method (FEM) analysis

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**Mechanical motion of SAW**

Introduction Theory Simulation Fabrication Characterization Results Mechanical motion of SAW Physical motion is time dependent elliptical displacement of the surface. The unbounded direction (y-axis) vibrates more than the bounded (x-axis). Amplitudes become negligible for penetration depths greater than few acoustic wavelengths λ(=ν/f)

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**Introduction Theory Simulation Fabrication Characterization Results**

Stress and Strain T=F/A Stress T is the force F applied per unit area A of the solid. S=ΔL/L Strain S is fractional deformation ΔL of solid length L T=cS c = elastic stiffness, Young’s modulus (N/m²) 𝑇 𝑗𝑘 = 𝑐 𝑗𝑘𝑙𝑚 𝑆 𝑙𝑚 tensor equation, 𝑇 𝑗𝑘 = 𝐹 𝑗 / 𝐴 𝑘 , 𝑆 𝑙𝑚 = 𝛥𝐿 𝑙 / 𝐿 𝑚 j denotes direction of force F, k direction of vector of area A Likewise deformation directions defined similarly.

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**Piezoelectric interactions**

Introduction Theory Simulation Fabrication Characterization Results Introduction Theory Simulation Fabrication Characterization Results Piezoelectric interactions The E established by V will distort the molecular charge distribution and result in accumulation of surface charge. 𝐷=[𝑒][𝑆]+[𝜀]𝐸 Electrical displacement density matrix equation 𝑒 piezoelectric constant, 𝐸 electric field intensity, [𝜀] dielectric permittivity [𝑒]= 𝑒 31 𝑒 31 𝑒 𝑒 𝑒

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**General equations of surface waves**

Introduction Theory Simulation Fabrication Characterization Results Introduction Theory Simulation Fabrication Characterization Results General equations of surface waves S = Strain T = Stress ρ = mass density u = mechanical displacement D = displacement density E = electric field intensity Φ = electric potential Equations of motion 𝜕 𝑇 𝑖𝑗 𝜕 𝑥 𝑖 =𝜌 𝜕 2 𝑢 j 𝜕 t 2 From Maxwell’s equations 𝜕 𝐷 𝑖 𝜕 𝑥 𝑖 =0 Electric field intensity 𝐸 𝑖 =− 𝜕𝛷 𝜕 𝑥 𝑖 Piezoelectric mechanical stress 𝑇 𝑖𝑗 = 𝑐 𝑖𝑗𝑘𝑙 𝑆 𝑘𝑙 − 𝑒 𝑛𝑖𝑗 𝐸 𝑛 Piezoelectric displacement density 𝐷 𝑚 = 𝑒 𝑚𝑘𝑙 𝑆 𝑘𝑙 + 𝜀 𝑚𝑛 𝐸 𝑛 Linear strain displacement 𝑆 𝑘𝑙 = 1 2 𝜕 𝑢 𝑘 𝜕 𝑥 𝑙 + 𝜕 𝑢 𝑙 𝜕 𝑥 𝑘

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**Introduction Theory Simulation Fabrication Characterization Results**

IDT types Single-electrode Basic IDT type Split-electrode reduce undesirable finger reflection effects reflections from each split-electrode pair cancel out at center frequency requires increased lithographic resolution DART (Distributed Acoustic Reflection Transducer) Variable reflectivity can be achieved to give greater design capability Reduce pass band ripples

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**Modeling SAW device Analytic method**

Introduction Theory Simulation Fabrication Characterization Results Modeling SAW device Analytic method Know absolutely how the model will behave Works only for simple models Linearized approximation Early SAW filter models (analytical solution) Delta function Crossed-Field Impulse-response Other analytical methods for SAW design Equivalent circuit model S-matrix model P-matrix model Coupling-of-modes

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**Introduction Theory Simulation Fabrication Characterization Results**

Finite Element Method Nanosized SAW devices require numerical simulations Initial values of the variables equations new values over small Δt FEM is dividing up a problem into small elements that can be solved in relation to each other Benefits of FEM for SAW modeling domain changes, precision varies over the entire domain, solving complex elasticity problems Comsol Multiphysics platform Solves a set of differential equations on grid using FEM

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**Geometry design Non-linear model with high computational requirements**

Introduction Theory Simulation Fabrication Characterization Results Introduction Theory Simulation Fabrication Characterization Results Geometry design Non-linear model with high computational requirements For typical 2-port SAW resonator, delay line 1000λ, aperture 500λ, depth 10λ, minimum 20 first order elements per wavelength = 108 elements Four degrees of freedom, displacement (Ux,Uy,Uz) and potential (φ) Total number of unknowns = 4×108

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**Approach of the model Reducing the size of the models dimensions**

Introduction Theory Simulation Fabrication Characterization Results Introduction Theory Simulation Fabrication Characterization Results Approach of the model Reducing the size of the models dimensions Dimension in the direction that the wave propagates is one wavelength Dimension of perpendicular to motion is discarded due to the shape of IDTs The depth of the substrate is set to 10 wavelengths of resonance frequency p Si substrate height 10 μm GaN piezoelectric layer 1.6 μm IDT electrode height 80 nm IDT width, p=130 nm , 200 nm

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**Complete/Unit cell model**

Introduction Theory Simulation Fabrication Characterization Results Introduction Theory Simulation Fabrication Characterization Results Complete/Unit cell model Comparing the two models For 1 hour of simulation of the compact model 50 years of simulations are needed for the complete model Compact model used 4 GB of RAM For the complete model 2 PB of RAM would be needed.

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**Model Parameters Boundary conditions (b.c.)**

Introduction Theory Simulation Fabrication Characterization Results Introduction Theory Simulation Fabrication Characterization Results Model Parameters Boundary conditions (b.c.) Periodic boundary conditions are defined on both sides of the substrate Stress-free boundary conditions for the surface and the electrodes Piezoelectric and substrate have “zero charge/symmetry” electric b.c. Electrodes are grounded or have 1 Volt electric potential Subdomain (material properties) Young’s Modulus, Poisson’s ratio, density, thickness, elasticity matrix, coupling matrix, relative permittivity and electric conductivity of substrate material and piezoelectric layer

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**Introduction Theory Simulation Fabrication Characterization Results**

Model Parameters In finite element model problem domain is discretized in smaller regions, called elements, connected at nodes Mesh Detail of mesh has major effect in simulation time Finer elements are used closer to the surface for better accuracy Bulkier elements to the bottom for faster calculations

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**Model Parameters Substrate thickness**

Introduction Theory Simulation Fabrication Characterization Results Introduction Theory Simulation Fabrication Characterization Results Model Parameters Substrate thickness Small values decrease the accuracy of the simulation Large values increase the computational time Values of few wavelengths provide fast and accurate results Height of 7 wavelengths were used

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**Model Parameters IDT width and in between spacing (p) SAW topologies**

Introduction Theory Simulation Fabrication Characterization Results Introduction Theory Simulation Fabrication Characterization Results Model Parameters IDT width and in between spacing (p) Define the wavelength of the resonant frequency SAW topologies Single electrode Split-electrode DART p Single-electrode model Split-electrode model

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**Introduction Theory Simulation Fabrication Characterization Results**

Resonance at 7.96 GHz with electric potential and total displacement maps plotted respectively, for Single-electrode IDT with 130 nm finger width.

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**Introduction Theory Simulation Fabrication Characterization Results**

Resonance at 5.49 GHz, for Single-electrode IDT with 200 nm finger width.

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**Introduction Theory Simulation Fabrication Characterization Results**

Resonance at 2.79 GHz, of Split-electrode type IDT with 200 nm finger width.

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**Resonance at 2.08 GHz, of DART type IDT with 200 nm finger width.**

Introduction Theory Simulation Fabrication Characterization Results Introduction Theory Simulation Fabrication Characterization Results Resonance at 2.08 GHz, of DART type IDT with 200 nm finger width.

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**Resonance at 5.49 GHz, of DART type IDT with 200 nm finger width.**

Introduction Theory Simulation Fabrication Characterization Results Introduction Theory Simulation Fabrication Characterization Results Resonance at 5.49 GHz, of DART type IDT with 200 nm finger width. DART array with multiple resonant wavelengths for the same IDT dimensions

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**IDT fabrication process**

Introduction Theory Simulation Fabrication Characterization Results IDT fabrication process Employing PMMA resist of 200 nm thickness on a Gallium Nitride wafer Direct electron beam lithography writing A metal layer of Ti/Au (5 nm/75 nm) was subsequently deposited on wafer using an e-beam evaporator Followed by a lift-off procedure to form the interdigitated electrodes(IDTs) Materials used for fabrication

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**Photolithography mask**

Introduction Theory Simulation Fabrication Characterization Results Introduction Theory Simulation Fabrication Characterization Results Photolithography mask Photolithography mask that was used, with different IDT characteristics Variable delay lines IDT finger lengths Crystal orientation Reflectors exist or not

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**Introduction Theory Simulation Fabrication Characterization Results**

Mix and Match fabrication IDTs were fabricated in “IMT” using e-beam lithography (gold) Rest of the device elements were fabricated in “FORTH” using conventional optical lithography (green)

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**Post fabrication images of the devices**

Introduction Theory Simulation Fabrication Characterization Results Introduction Theory Simulation Fabrication Characterization Results Post fabrication images of the devices Complete IDT structure with connecting pads at magnification x130 IDT fingers with reflectors at magnification x3000 Single electrode IDT type magnification x50000

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**Post fabrication images of the devices**

Introduction Theory Simulation Fabrication Characterization Results Introduction Theory Simulation Fabrication Characterization Results Post fabrication images of the devices Single electrode type IDT array, finger width 130nm and metallization ratio η=0.5 Split electrode type IDT array, finger width 200nm and metallization ratio η=0.5 DART type IDT array, finger width 200nm and metallization ratio η=0.5

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**Characterization Signal Ground**

Introduction Theory Simulation Fabrication Characterization Results Characterization Signal Ground A couple of three finger probes were placed on the pads of the IDTs The middle electrode was grounded and the upper/bottom were put to 1 Volt electric potential A frequency scan performed from 2 to 9 GHz depending the IDT type.

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**Introduction Theory Simulation Fabrication Characterization Results**

S-parameters Scattering (S) parameters describe the response of an N-port network to voltage signals at each port The first number refers to responding port, while second to incident port. S21 means response at port 2 due to signal at port 1 For 2-port network, incident voltage denoted by “a”, leaving by “b”

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**Introduction Theory Simulation Fabrication Characterization Results**

Y-parameters Admittance (Y) is a measure of how easily a circuit or device will allow a current to flow. It is the inverse of the impedance. Y-parameters describe the small-signal response of non-linear networks. For all ports the currents may be defined in terms of the Y-parameter matrix and the voltages I = YV Y is an N × N matrix 2-port network transformation from S-parameters to Y-parameters Y0 characteristic admittance

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**Simulated with measured results comparison**

Introduction Theory Simulation Fabrication Characterization Results Introduction Theory Simulation Fabrication Characterization Results Simulated with measured results comparison The simulations for the Single-electrodes with 200nm width had a frequency drift of 187 MHz to lower frequencies that is 3.29% difference from the characterized device. Ripples occur from interference of the reflections

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**Simulated with measured results comparison**

Introduction Theory Simulation Fabrication Characterization Results Introduction Theory Simulation Fabrication Characterization Results Simulated with measured results comparison The simulations for the Single-electrodes with 130nm width had a frequency drift of 184 MHz to lower frequencies that is 2.25% difference from the characterized device.

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**Simulated with measured results comparison**

Introduction Theory Simulation Fabrication Characterization Results Introduction Theory Simulation Fabrication Characterization Results Simulated with measured results comparison The simulations for the Split-electrodes with 200nm finger width had a frequency drift of 343 MHz to higher frequencies that is 4.14% difference from the characterized device.

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**Simulated with measured results comparison**

Introduction Theory Simulation Fabrication Characterization Results Introduction Theory Simulation Fabrication Characterization Results Simulated with measured results comparison The simulations of the Split-electrodes with 200nm width for a wider frequency spectrum had a harmonic resonance drift of 73 MHz to lower frequencies that is 3.42% difference from the characterized device. The main resonant frequency was compared in the previous figure.

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**Simulated with measured results comparison**

Introduction Theory Simulation Fabrication Characterization Results Introduction Theory Simulation Fabrication Characterization Results Simulated with measured results comparison The simulations for the DART electrodes with 200nm width had a frequency drift of 185 MHz to lower frequencies that is 3.25% difference from the characterized device.

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**Conclusions Efficient numerical methodology for determination of fs**

Introduction Theory Simulation Fabrication Characterization Results Conclusions Efficient numerical methodology for determination of fs SAW devices (GaN/Si) designed and implemented with fs up to 9GHz Measured fs to predicted fs deviation in the order of 3% Further improvement of the predictability of resonant frequency with reasonable cost in computational time

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**Acknowledgements I would like to thank Alexandra Stefanescu**

Dan Neculoiu Mircea Dragoman Adrian Dinescu George Konstantinidis Eleftherios Iliopoulos Alexandros Pantazis Panos Tzanetakis Alexandros Georgakilas ΠΑΡ for the financial support

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**End of slide show, click to bananas.**

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**Fabrication process GaN/Si wafers grown by MOCVD**

Introduction Theory Simulation Fabrication Characterization Results Fabrication process GaN/Si wafers grown by MOCVD Epitaxial relationship of GaN(0001) c-axis on Si(111) with ~17% mismatch

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**Model Parameters IDT height**

Introduction Theory Simulation Fabrication Characterization Results Model Parameters IDT height Greater height has wider cross-section, thus lower resistivity IDTs A shift in resonance frequency is observed for different heights Results did not provide any outcome that can be used to optimize the efficiency of the simulation

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**Introduction Theory Simulation Fabrication Characterization Results**

Conclusions determine the parameters which affect the bandwidth and center frequencies a series of simulations were done for various IDT structure types like Single-electrode, Split-electrode and DART For two different finger widths the simulated results compared with the experimental presented differences of less than 5% in center resonance frequency of all the tested devices.

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