1. 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

2. 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

3. Index Introduction Theory of SAW devices
Simulations (finite element method)
Fabrication
Characterization
Results

4. 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

5. 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

6. 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

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

8. 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

9. 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

10. Introduction Theory Simulation Fabrication Characterization Results
SAW devices
SAW Sensors
Chemical
Optical
Thermal
Pressure
Acceleration
Torque
Biological
SAW sensors – using SAW propagation speed

11. 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

12. 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

13. 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

14. 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

15. 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

16. 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)

17. 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.

18. 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

19. 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

20. 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

21. 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

22. 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

23. 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

24. 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

25. 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.

26. 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

27. 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

28. 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

29. 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

30. 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.

31. Introduction Theory Simulation Fabrication Characterization Results
Resonance at 5.49 GHz, for Single-electrode IDT with 200 nm finger width.

32. Introduction Theory Simulation Fabrication Characterization Results
Resonance at 2.79 GHz, of Split-electrode type IDT with 200 nm finger width.

33. 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.

34. 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

35. 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

36. 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

37. 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)

38. 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

39. 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

40. 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.

41. 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”

42. 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

43. 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

44. 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.

45. 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.

46. 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.

47. 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.

48. 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

49. 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

50. End of slide show, click to bananas.

53. 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

54. 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

55. 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.