International Distinguished Lecturer Program

Transcription

1 U International Distinguished Lecturer Program Ken-ya Hashimoto Chiba University Sponsored by The Institute of Electrical and Electronics Engineers (IEEE) Ultrasonics, Ferroelectrics and Frequency Control (UFFC) Society

2 -D Analysis Ken-ya Hashimoto Chiba University

3 Contents D Propagation and Guided Modes Scalar Potential Theory Observation by Laser Probe Suppression of Transverse Modes

4 Contents D Propagation and Guided Modes Scalar Potential Theory Observation by Laser Probe Suppression of Transverse Modes

5 Diffraction (a) Fresnel Region (Beam Propagation) W (a) (b) (b) Fraunhofer Region (Cylindrical Wave Propagation) x c Critical Length: x c =(1+γ)W /λ γ: Parameter Determined by Anisotropy(=0 for Isotropic)

6 Influence of Diffraction in Resonators Counter Measure

7 Transverse Mode Admittance G Inharmonic Resonance B Frequency ωr ω a

8 Wave Vector β β =π/λ:phase Delay per Unit Length y Wave Propagation Wave Propagation λ y λ β y =π/λ y β = β =π/λ λ x x β x =π/λ x V p (=fλ) does not Follow Vector Decomposition Rule! exp( jβ X) exp[ j( β x + β y + β z)] x y z

9 D Wave Equation D Wave Equation φ ω φ φ = + x x y V y V V x ω/v x0 ω/v y0 )] ( exp[ y x j y β x β φ + When 1 ) / ( ) ( ) / ( ) ( 0 0 = + ω β ω β y y x V x V β x β y

10 ω/v x0 ω/v y0 1 ) / ( ) ( ) / ( ) / ( ) ( 1 ) / ( ± = ± ω β ω ω β ω β x x y x x y y V V j V V β x β y ) )]exp( exp( ) exp( [ x j y j A y j A x y y β β β φ + + = +

11 Snell s Law Continuity of Wave Front at Boundary Medium 1 Medium Continuity of Lateral Wavelength Continuity of Lateral Wavevector Component

12 At Boundary Between Two media, S β x /ω θt β t /ω β x /ω S S 1 S 1 θ i θ i θ r θ r β i /ω β r /ω β i /ω β r /ω (a) Transmission (b) Total Reflection Slowness Surface (S=1/V p ) When S 1 >S β x =β 1 cosθ 1 =β cosθ n 1 cosθ 1 =n cosθ

13 Evanescent Field Since β x =β 1 cosθ 1 & β x + β y = β β y = β -(β 1 cosθ 1 ) Field Penetration at Total Reflection Exponential Decay (Energy Storage)

14 Tunneling Even for Total Reflection State, Wave Transmission Occurs when Medium is Thin No Phase Delay Through Transmission

15 Closed Waveguide Wavenumber of Mode λ g =π/β x θ t y x λ g For Phase Matching Between Incident and -Bounced Waves -βtcscθ+ Γ= βcotθ tcosθ+nπ Transverse Resonance Condition -β y t+ Γ=nπ Γ: Reflection Coef. at Boundary

16 Since β x +β y =β c β c : Velocity in Core Wavenumber of Guided Mode β x = β c -[( Γ nπ)/t] Normalized frequency 3.5 n= 1.5 n= n= Normalized wavenumber Relation Between β c and β x When Γ=0 or ±π

17 (a) Near Cutoff (b) Far from Cutoff Propagation of Waveguide Mode V p =ω/β x : Phase Velocity Frequency tan -1 (V g ) tan -1 (V p ) Wavenumber Propagation Speed of Phase Front V g = ω/ β x : Group Velocity Propagation Speed of Energy

18 Influence of Group and Phase Velocities on Signal Transfer t=l/v g t (a) Input Signal φ=ωl/v p t (b) Output Signal

19 At Cutoff (a) cut-off (b) (c) β R cut-off (d) β R β R Behavior as Evanescent Field

20 Even if not Cutoff β 1 R1 β R β1 R 1 β R Influence of Higher-Order Cutoff Modes

21 Open Waveguide θ t y x λ g Use of Total Reflection at Surfaces Energy Penetration to Outsides Transverse Resonance Condition -β y t+ Γ=nπ Γ is Frequency (or θ) dependent

22 Similarity with Closed Waveguide at Total Reflection Normalized frequency Critical Condition Normalized wavevector Relation between β c and β x If Total Reflection Condition is Not Satisfied?

23 Leaky Waveguide θ t When Reflection Coefficient at Surfaces is Large, Pseudo Mode Propagates with Energy Leakage to Outside If Reflection Coefficient at Surfaces is Small?

24 Propagation as Free Wave(Not Guided) Appearing When Velocities of Waveguide Mode and Free Wave are Close (Near Cutoff)

25 Contents D D Propagation and Guided Modes Scalar Potential Theory Observation by Laser Probe Suppression of Transverse Modes

26 w B w G w B Scalar Potential Analysis Region B Region G Region B y x -D Analysis Approximation as Uniform (Flat) IDT Field Expression(When w B = is Assumed for Simplicity) + φ B exp( α By y)exp( jβx) ( x + wg / ) + φ = { φg exp( jβgy y) + φg exp( + jβgy y)}exp( jβx) ( x wg / ) φ B exp( + α By y)exp( jβx) ( x wg / )

27 Due to Continuity of φ and φ/ y at y=±w g / Symmetric Mode (φ B+ =φ B-, φ G+ =φ G- ) φ + + B = φ G cos( βgywg / ) exp( α BywG α = β tan( β w / ) By Gy Gy G / ) Anti-Symmetric Mode (φ B+ =-φ B-, φ G+ =-φ G- ) φ + B + = jφg sin( βgywg / ) exp( α BywG / ) α = β By Gy cot( β Gy w G / )

28 ω/v x0 ω/v y ) / ( ) / ( x y x y x V V V ω β β = + β x β y ) / ( ) / ( ) / ( y x y x x x V V V V β ω ω β Parabolic Approximation

29 Parabolic Approximation for Slowness Surface S y S y S x S x (a) For Region G For Region B V G0-1 V G0-1 (b) ξ>0 ξ<0 β x x β ξ β β + ξ β β G0 G Gy / G0 α β B0 B By / B0 For Isotropic Case, ξ=0.5

30 Slowness Surface of SH-type SAW on 36-LT S y sec/km S x sec/km

31 Wavenumber of Grating Mode and Slowness Surface For Energy Trapping in Waveguide Real α By S y S y V -1 p S x V p -1 S x V B0-1 V G0-1 V G0-1 (a) For ξ>0 (b) For ξ<0 V G0 <V p <V B0 V B0 <V p <V G0 Higher-order Modes Appear in Higher Frequencies V B0-1 Higher-order Modes Appear in Lower Frequencies

32 Symmetric Mode + = + ˆ 1 ˆ tan ˆ 1 ˆ 1 V w V V G π + = + ˆ 1 ˆ cot ˆ 1 ˆ 1 V w V V G π Anti-Symmetric Mode When V B0 /V G0-1 «1, Where :Relative Phase Velocity :Relative Waveguide Width ˆ ˆ G G B p G G G B G B p V V V w w V V V V V V ξ λ = =

33 Relative SAW Velocity vs. Relative Aperture Relative phase velocity A 3 S 0.6 A 0.4 S 1 0. A S Relative aperture width Velocity in Region B Velocity in Region G

34 Equivalent Circuit for Multi-Mode Resonators (n) L m (3) L m () L m (1) L m C m (n) C m (3) C m () C m (1) C 0 R m (n) R m (3) R m () R m (1) 1 πv ω = p ( n) r = ( n) ( n) Cm Lm I ( n) p

35 Modes Propagate without Mutual Power Interaction + φk ( y) + φn( y) dy = φk ( y) dy + φn ( y) Mode Orthogonality + + φ * k ( y) φn ( y) dy = δ Field can be Expressed as Sum of Mode Fields Mode Completeness nk P k where P k + + = φ ( y) k dy dy φ( y) = k =1 A φ ( y) / k k P k

36 Fourier Transform ϕ n (x)=p -0.5 exp(nπjx/p) Orthogonality p p * φk x φn x dx = p 0.5 ( ) ( ) exp[πjx( n m) / p] dx = 0 0 Completeness φ( x) = p 0 k = 1 * n A φ ( x) k k p = 0 k = 1 p 0.5 k k = 1 k A k exp(kπjx φ( x ) φ ( x) dx A φ ( x) φ ( x) dx = = * n A δ nk / p) Multiplication of ϕ n* (x) & Integration give n p 0.5 A p n φ( x)exp( nπjx = 0 / p) dx

37 Difference of Waveguide Width w g with Finger Overlap Width w e w e w g Amplitude at Excitation Source φ( y) = φ 0 0 ( y ( y > w e w e /) /)

38 + = / / * 0 ) ( e e w w m m m dy y P A φ φ = + + = 1 * * ) ( ) ( / ) ( ) ( k m k k k m dy y y P A dy y y φ φ φ φ Multiplying φ m* (y) and Integrating Then 1D Analysis Gives A 0 =φ 0 w e 1 / / 0 (0) ) ( ) ( ) ( + + = = dy y w dy y A A C C n e w w n n m n m e e φ φ Since Motional Capacitance Power Excitation Efficiency,

39 Effective Electromechanical Coupling Factor vs. Relative Aperture Width (When w e =w g ) Relative coupling factor S Relative aperture width Zero Excitation Efficiency for Anti-Symmetric Modes S 1 S

40 Why Effective Coupling Factor Changes? (a) S 0 Mode (When w is small) Large Penetration (b) S 0 mode (When w is large) Small Penetration (c) S 1 mode Existence of Sign Inverted Region

41 Energy Trapping for FBAR Large Electrode Area for Small Impedance Generation of Transverse Mode Spurious Strongly Dependent on Lateral Coupling Strength Among Grains Fabrication Process Dependent = Good Films Offer Serious Transverse Modes

42 Scalar Potential Theory L Region B Region G Region B y x w G Field Expression (β x L=mπ) φ = + φ B exp( α By y)cos( β xx) + { φg exp( jβgy y) + φg exp( + φ B exp( + α By y)cos( β xx) jβ Gy ( x y)}cos( β x) x ( x + w ( x w G G / ) w G / ) / )

43 Exciter Distribution L Region B Region G Region B y x w G φ( y) = φ 0 0 ( y ( y > w w G G /) /)

44 Dispersion Relation 0.54 Normalized Frequency W/O Electrode With Electrode -0.j -0.1j Relative Wavenumber

45 Lamb Wave Dispersion on ZnO/Pyrex- Glass Structure (L S /L Z =0.56) ω/ω 0 1. TE mode 1 open short TE 1 mode Im(β y /β 0 ) Re(β y /β 0 )

46 Lamb Wave Dispersion on ZnO/Pyrex- Glass Structure (L S /L Z =1.3) ω/ω 0 1 TE mode 0.8 open short TE 1 mode Im(β y /β 0 ) Re(β y /β 0 )

47 Contents D D Propagation and Guided Modes Scalar Potential Theory Observation by Laser Probe Suppression of Transverse Modes

48 Data Logger IF Detector LO Source CCD camera Optics N Divider Stage Controller Precision Stage RF Source

49 SAW Resonator Impedance (Ω) 1k 100 (a) (b) (c) 1-port SAW Resonator Period: 7.18 µm Substrate: ST-Cut Quartz Electrode: Al (140 nm) Frequency (MHz) 0 min. for, pixels

50 Magnitude ( MHz) Magnitude ( MHz) Magnitude ( MHz)

51 Wavenumber Domain Analysis ( MHz) Spatial Modulation due to Periodic Variation of Optical Reflectivity Scattered Wave Spot (Guided Mode)

52 Processing in Wavenumber Domain Extraction, Move to Origin and IFFT Magnitude (Envelope)

53 Wavenumber Domain Analysis ( MHz) Processing Magnitude (Envelope)

54 Wavenumber Domain Analysis ( MHz) Processing Magnitude (Envelope)

55 Love-Wave Resonator 1-port SAW Resonator Periodicity: 10 µm Substrate: 15 YX-LiNbO 3 Electrode: Cu (560 nm) Frequency response S 11

56 Field Distribution (Magnitude) 0 min. for 900x30 pixels MHz MHz MHz MHz

57 Wavenumber Domain Analysis Y ( MHz) Field Distribution in IDT-Reflector Region +Y +X FFT X Vel m/s Angle 6 Field Distribution in Wavenumber Domain

58 Wavenumber Domain Analysis ( MHz) Y Field Distribution in IDT-Reflector Region FFT X +Y +X Field Distribution in Wavenumber Domain Vel m/s Angle 0 Vel m/s Angle 15

59 Identification of Propagation Mode Field Distribution in IDT-Reflector Region FFT Field Distribution in Wavenumber Domain IFFT IFFT Field Distribution for each peak Hybrid Mode Propagation Associating with Mode-Conversion at Bus-bar Boundary

60 Visualization by Laser Probe By Infineon

61 Contents D D Propagation and Guided Modes Scalar Potential Theory Observation by Laser Probe Suppression of Transverse Modes

62 One-Port Resonator on Cu/15 o YX-LiNbO 3 Input Admittance [S] RBW 1.7%! Transverse modes -0.1 Rayleigh mode Frequency [MHz]

63 Spectrum of Transverse Modes 850 Resonance Frequency, f r [MHz] S 0 S 4 S 3 S S S 5 Aperture, w/p Rayleigh mode

64 Use of Weighted IDT (Weighting for Excitation Efficiency) W G Reduction in Excitation Efficiency for Fundamental mode as well as Higher Modes

65 Influence of Dummy Electrodes (Reduction of Scattering at Discontinuities) W d W g W G W a Control in Field Distribution for Fundamental mode as well as Higher Modes

66 90 Influence of Dummy Electrodes Resonance Frequency [MHz] S 5A S 4B S 3B S A S 1B S 0A W d /W g Modes Due to Dummy Electrodes (W d Dependent)

67 Field Distribution of Spurious Modes 10 5 S A S 1B [free] [bus-bar] [dummy] x /λ 0 [grating] Energy Concentration at Dummy Electrode Region 0

68 Influence of Dummy Electrodes 90 Resonance Frequency [MHz] S 5A S 4B S 3B S A S 1B S 0A W d /W g Mode Coupling through Dummy Electrodes

69 Field Distribution at Coupling 10 5 [free] [bus-bar] [dummy] S A S 1B x /λ 0 [grating] Concentration at Dummy Electrodes Region 0

70 Metallization Ratio Resonance Frequency [MHz] Velocity at Dummy Electrode Region [m/s]

71 Weighted Dummy Electrodes for Suppression of Transverse Modes Scattering of Unnecessary Modes Through Coupling with Dummy Electrode Modes

72 Experiment Electrodes : Copper Substrates: 15 o YX-LiNbO 3

73 Dependence of dummy electrode length Reference (Without dummy electrodes) S 1 successfully suppressed! Too long.. (Q deteriorated)

74 Dependence of dummy electrode width With narrower dummy electrodes (S 1 trapped in IDT region) Reference (With uniform dummy electrodes) L/p =.5 With wider dummy electrodes (S 0 penetrates to dummy region)

75 0 Insertion loss [db] Frequency, f [GHz] W Weighted Dummy Electrodes W/O Weighted Dummy Electrodes

76 0 Insertion loss [db] Frequency, f [GHz] W Black Magic W/O Black Magic 1.3

77 Suppression of Transverse Modes (Infineon): Mass Placement on Upper Electrode resonator APLAC 7.80 User: Infineon Technologies AG Jun resonator APLAC 7.80 User: Infineon Technologies AG Jun spurious modes S11 W/O Mass Loading S11 W Mass Loading

78 Propagation of Transverse Mode z evanescent wave λ( f ) evanescent wave outside active resonator outside x 10 µm

79 Piston Mode BAW Resonator β y w~π/ Fundamental Resonance High f r Low f r β y w~π/ Low f r High f r Higher Order Resonance High f r Low f r Low f r High f r