USF Libraries
USF Digital Collections

Detection of cracks in single-crystalline silicon wafers using impact testing

MISSING IMAGE

Material Information

Title:
Detection of cracks in single-crystalline silicon wafers using impact testing
Physical Description:
Book
Language:
English
Creator:
Hilmersson, Christina
Publisher:
University of South Florida
Place of Publication:
Tampa, Fla
Publication Date:

Subjects

Subjects / Keywords:
Audible
Modes
Frequency response
Solar cells
Vibration
Dissertations, Academic -- Mechanical Engineering -- Masters -- USF
Genre:
bibliography   ( marcgt )
theses   ( marcgt )
non-fiction   ( marcgt )

Notes

Abstract:
ABSTRACT: This thesis is about detection of cracks in single-crystalline silicon wafers by using a vibration method in the form of an impact test. The goal to detect cracks from vibration measurements introduced by striking the silicon wafer with an impact hammer. Such a method would reduce costs in the production of solar cells. It is an inexpensive, relatively simple method which if commercialized could be used as an efficient in-line production quality test. A hammer is used as the actuator and a microphone as the response sensor. A signal analyzer is used to collect the data and to compute frequency response. Parameters of interest are audible natural frequencies, peak magnitudes, damping ratio and coherence.The data reveals that there are differences in frequency between the cracked silicon wafers and the non-cracked silicon wafers. The resonant peaks in the defective wafers were not as sharp (i.e., lightly damped) and occurred at lower frequencies (i.e., lower stiffness) with a lower magnitude and a higher damping ratio. These differences could be used to detect damaged product in a solar cell production line.
Thesis:
Thesis (M.A.)--University of South Florida, 2006.
Bibliography:
Includes bibliographical references.
System Details:
System requirements: World Wide Web browser and PDF reader.
System Details:
Mode of access: World Wide Web.
Statement of Responsibility:
by Christina Hilmersson.
General Note:
Title from PDF of title page.
General Note:
Document formatted into pages; contains 104 pages.

Record Information

Source Institution:
University of South Florida Library
Holding Location:
University of South Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 001795399
oclc - 153895659
usfldc doi - E14-SFE0001563
usfldc handle - e14.1563
System ID:
SFS0025881:00001


This item is only available as the following downloads:


Full Text
xml version 1.0 encoding UTF-8 standalone no
record xmlns http:www.loc.govMARC21slim xmlns:xsi http:www.w3.org2001XMLSchema-instance xsi:schemaLocation http:www.loc.govstandardsmarcxmlschemaMARC21slim.xsd
leader nam Ka
controlfield tag 001 001795399
003 fts
005 20070709094000.0
006 m||||e|||d||||||||
007 cr mnu|||uuuuu
008 070709s2006 flu sbm 000 0 eng d
datafield ind1 8 ind2 024
subfield code a E14-SFE0001563
040
FHM
c FHM
035
(OCoLC)153895659
049
FHMM
090
TJ145 (ONLINE)
1 100
Hilmersson, Christina.
0 245
Detection of cracks in single-crystalline silicon wafers using impact testing
h [electronic resource] /
by Christina Hilmersson.
260
[Tampa, Fla] :
b University of South Florida,
2006.
3 520
ABSTRACT: This thesis is about detection of cracks in single-crystalline silicon wafers by using a vibration method in the form of an impact test. The goal to detect cracks from vibration measurements introduced by striking the silicon wafer with an impact hammer. Such a method would reduce costs in the production of solar cells. It is an inexpensive, relatively simple method which if commercialized could be used as an efficient in-line production quality test. A hammer is used as the actuator and a microphone as the response sensor. A signal analyzer is used to collect the data and to compute frequency response. Parameters of interest are audible natural frequencies, peak magnitudes, damping ratio and coherence.The data reveals that there are differences in frequency between the cracked silicon wafers and the non-cracked silicon wafers. The resonant peaks in the defective wafers were not as sharp (i.e., lightly damped) and occurred at lower frequencies (i.e., lower stiffness) with a lower magnitude and a higher damping ratio. These differences could be used to detect damaged product in a solar cell production line.
502
Thesis (M.A.)--University of South Florida, 2006.
504
Includes bibliographical references.
516
Text (Electronic thesis) in PDF format.
538
System requirements: World Wide Web browser and PDF reader.
Mode of access: World Wide Web.
500
Title from PDF of title page.
Document formatted into pages; contains 104 pages.
590
Adviser: Daniel P. Hess, Ph.D.
653
Audible.
Modes.
Frequency response.
Solar cells.
Vibration.
690
Dissertations, Academic
z USF
x Mechanical Engineering
Masters.
773
t USF Electronic Theses and Dissertations.
4 856
u http://digital.lib.usf.edu/?e14.1563



PAGE 1

Detection of Cracks in Single-Crystallin e Silicon Wafers Using Impact Testing by Christina Hilmersson A thesis submitted in partial fulfillment of the requirement s for the degree of Master of Science in Mechanical Engineering Department of Mechanical Engineering College of Engineering University of South Florida Major Professor: Daniel P. Hess, Ph.D. Autar Kaw, Ph.D. Craig Lusk, Ph.D. Sergei Ostapenko, Ph.D. Date of Approval: March 29, 2006 Keywords: audible, modes, frequency response, solar cells, vibration Copyright 2006, Christina Hilmersson

PAGE 2

Acknowledgement I would like to thank Dr. Daniel Hess for letting me work on this project, for being an outstanding adviser and helping me to complete this thesis. I also want to thank Dr. Sergei Ostapenko for his advice, support and for being in the committee. I like to thank William Dallas for making the cracks in the silicon wafers and scanning them in SAM. I w ould like to thank Dr. Autar Kaw and Dr. Craig Lusk for being on my committee and taki ng their time to read my thesis and give me excellent feedback. I would like to thank all the people in the Mechanical Engineering Department for t heir support, I especially like to thank Sue Britten for always being there for me. I would like to thank the Un iversity of South Florida for providing me with a first-rate education. I would like to thank all my family especially my parents, my si ster and her family for all their support. I would also like to thank all my friends for always listeni ng to me. I would really like to thank my roommate for putting up with me during the completion of this work. The work was supported by the National Renewable Energy Lab (NREL) Subcontract Nos. AAT-2 -31605-06 and ZDO-2-30628-03.

PAGE 3

i Table of Contents List of T ables .......................................................................................................iv List of Fi gures...................................................................................................... vi Abstract ...............................................................................................................xii Chapter 1: Introduc tion and Back ground..............................................................1 1.1 Over view..............................................................................................1 1.2 Backg round..........................................................................................1 1.3 Out line.................................................................................................3 Chapter 2: Experim ental Se tup.............................................................................4 2.2 Introdu ction..........................................................................................4 2.2 Sens ors................................................................................................4 2.3 Anal yzer...............................................................................................5 2.4 Frequency respons e............................................................................5 2.5 Test specim ens....................................................................................7 2.5.1 Large cra ck wafer set.............................................................8 2.5.2 Miscellaneous wafer set.......................................................12 2.5.3 Small cra ck wafer se t...........................................................13 2.6 Setu p.................................................................................................15 2.6.1 Position of hammer and mi crophone.................................... 16 Chapter 3: Test Matrix ........................................................................................19 3.1 Introdu ction........................................................................................19

PAGE 4

ii 3.2 Randomi zation ...................................................................................19 3.3 Test with la rge cra cks........................................................................21 3.4 Test with sm all cra cks........................................................................22 3.5 Tests of misce llaneous cr acks...........................................................22 Chapter 4: Test Result s......................................................................................24 4.1 Introdu ction........................................................................................24 4.2 Sample frequency response da ta......................................................24 4.3 Extracted parameter s.........................................................................26 4.4 Natural frequencies ............................................................................27 4.4.1 Large cra ck wafer se t...........................................................27 4.4.2 Miscellaneous wafer set.......................................................31 4.4.3 Small cra ck wafer se t...........................................................32 4.5 Normalized frequencies .....................................................................32 4.5.1 Large cra ck wafer se t...........................................................33 4.5.2 Miscellaneous wafer set.......................................................34 4.5.3 Small cra ck wafer se t...........................................................37 4.6 Peak m agnitudes ...............................................................................37 4.6.1 Large cra ck wafer se t...........................................................37 4.6.2 Miscellaneous wafer set.......................................................39 4.6.3 Small cra ck wafer se t...........................................................40 4.7 Damping ratio....................................................................................41 4.7.1 Large cra ck wafer se t...........................................................41 4.7.2 Miscellaneous wafer set.......................................................43

PAGE 5

iii 4.7.3 Small cra ck wafer se t...........................................................44 4.8 Discuss ion.........................................................................................45 Chapter 5: Conclusion and Recommendat ions..................................................46 5.1 Conclu sions.......................................................................................46 5.2 Recomm endations .............................................................................47 Referenc es.........................................................................................................48 Appendice s.........................................................................................................49 Appendix A : R un order ............................................................................50 Appendix B : Frequency response dat a...................................................53 Appendix C : Data from the large crack wa fer set....................................63 Appendix D : Data from t he miscellaneous wa fer set..............................79 Appendix E : Data from the small crack wa fer set...................................95 Appendix F : Set up fr equency response data.......................................103

PAGE 6

iv List of Tables Table 3.1 Run order for t he large crack wafer set..............................................20 Table 3.2 Test number for t he large crack wafer set.........................................21 Table 3.3 Large crack wa fers............................................................................21 Table 3.4 Small crack wa fers.............................................................................22 Table 3.5 Miscella neous wa fers........................................................................23 Table 4.1 Second mode frequen cy for large specim ens....................................28 Table 4.2 Second mode peak magnitude for the large cra ck wafer set.............38 Table 4.3 Second mode peak damping rati o for the large crack wafer set........42 Table A.1 Test number for t he small crack wafer se t.........................................50 Table A.2 Run order for t he small crack wafer set.............................................50 Table C.1 First mode natural frequen cy for large cra ck wafer set.....................63 Table C.2 First mode normalized fr equency for large cra ck wafer set...............64 Table C.3 Second mode natural frequen cy for large cra ck wafer set................65 Table C.4 Second mode normalized fr equency for large crack wafer set..........66 Table C.5 Third mode natural fre quency for large cra ck wafer set....................67 Table C.6 Third mode normalized fr equency for large cra ck wafer set..............68 Table C.7 Fourth mode natural fr equency for large crac k wafer set..................69 Table C.8 Fourth mode normalized fr equency for large crack wafer set...........70 Table C.9 First mode peak magnit ude for large cra ck wafer set........................71 Table C.10 Second mode peak magnit ude for large cra ck wafer set................72

PAGE 7

v Table C.11 Third mode peak magni tude for large cra ck wafer set....................73 Table C.12 Fourth mode peak magni tude for large crac k wafer set..................74 Table C.13 First mode damping ra tio for large cra ck wafer set.........................75 Table C.14 Second mode damping rati o for large cra ck wafer set....................76 Table C.15 Third mode damping ra tio for large cra ck wafer set........................77 Table C.16 Fourth mode damping ra tio for large cra ck wafer set......................78 Table D.1 First mode natural frequen cy for miscellaneous wafer set................79 Table D.2 First mode normalized fr equency for miscellaneous wafer set..........80 Table D.3 Second mode natural fr equency for miscellaneous wafer set...........81 Table D.4 Second mode normalized fr equency for miscellaneous wafer set....82 Table D.5 Third mode natural fre quency for miscellaneous wafer set...............83 Table D.6 Third mode normalized fr equency for miscellaneous wafer set........84 Table D.7 Fourth mode natural fr equency for miscellaneous wafer set.............85 Table D.8 Fourth mode normalized fr equency for miscellaneous wafer set......86 Table D.9 First mode peak magnit ude for miscellaneous wafer set..................87 Table D.10 Second mode peak magni tude for miscellaneous wafer set...........88 Table D.11 Third mode peak magni tude for miscellaneous wafer set...............89 Table D.12 Fourth mode peak magni tude for miscellaneous wafer set.............90 Table D.13 First mode damping ra tio for miscellaneou s wafer set....................91 Table D.14 Second mode damping rati o for miscellaneou s wafer set...............92 Table D.15 Third mode damping ra tio for miscellaneou s wafer set...................93 Table D.16 Fourth mode damping ra tio for miscellaneous wafer set.................94

PAGE 8

vi List of Figures Figure 2.1 Large center crack wafer number 39..................................................9 Figure 2.2 Large center crack wafer number 31..................................................9 Figure 2.3 Large center crack wafer number 35..................................................9 Figure 2.4 Large center crack wafer number 48..................................................9 Figure 2.5 Large offset crack wafer number 32.................................................10 Figure 2.6 Large offset crack wafer number 40.................................................10 Figure 2.7 Large offset crack wafer number 36.................................................10 Figure 2.8 Large offset crack wafer number 27.................................................10 Figure 2.9 Large crack other direction wafe r number 8.....................................11 Figure 2.10 Large crack other direction wafe r number 6...................................11 Figure 2.11 Large crack other direction wafe r number 33.................................11 Figure 2.12 Large crack other direction wafe r number 41.................................11 Figure 2.13 Miscellan eous wafer num ber 11.....................................................12 Figure 2.14 Miscellan eous wafer num ber 23.....................................................12 Figure 2.15 Miscellan eous wafer num ber 25.....................................................12 Figure 2.16 Miscellan eous wafer num ber 45.....................................................12 Figure 2.17 Miscellan eous wafer num ber 46.....................................................13 Figure 2.18 Miscellan eous wafer num ber 7.......................................................13 Figure 2.19 Miscellan eous wafer num ber 47.....................................................13 Figure 2.20 Miscellan eous wafer num ber 8.......................................................13

PAGE 9

vii Figure 2.21 Small cr ack wafer num ber 21.........................................................14 Figure 2.22 Small cr ack wafer num ber 22.........................................................14 Figure 2.23 Small cr ack wafer num ber 23.........................................................14 Figure 2.24 Small cr ack wafer num ber 25.........................................................14 Figure 2.25 Small cr ack wafer num ber 26.........................................................14 Figure 2.26 Small cr ack wafer num ber 27.........................................................14 Figure 2.27 Small cr ack wafer num ber 29.........................................................14 Figure 2.28 Small cr ack wafer num ber 30.........................................................14 Figure 2.29 Small cr ack wafer num ber 31.........................................................14 Figure 2.30 Small cr ack wafer num ber 32.........................................................14 Figure 2.31 Pictur e of the set up. .......................................................................15 Figure 2.32 Position of hammer a nd microphone relative to wafer....................16 Figure 2.33 Hammer (x=83 mm, y=48 mm).......................................................17 Figure 2.34 Hammer (x=39 mm, y=60 mm).......................................................17 Figure 2.35 Hammer (x=53 mm, y=48 mm).......................................................18 Figure 4.1 Frequency response of crack-free wafe r number 29........................25 Figure 4.2 Frequency response of large crack wafe r number 35.......................25 Figure 4.3 Second mode frequen cy for large specim ens..................................28 Figure 4.4 Second mode natural frequenc ies for miscellaneous wafer set........36 Figure 4.5 Second mode normalized fr equencies for miscellaneous wafer set.36 Figure 4.6 Second mode peak magnitude for the large cra ck wafer set............39 Figure 4.7 Second mode peak damping rati o for the large crack wafer set.......43 Figure B.1 Frequency response data of crack-free wafe r number 29................53

PAGE 10

viii Figure B.2 Frequency response data of crack-free wafe r number 34................53 Figure B.3 Frequency response data of crack-free wafe r number 38................54 Figure B.4 Frequency response data of crack-free wafe r number 47................54 Figure B.5 Frequency response data of cracked wafe r number 39...................55 Figure B.6 Frequency response data of cracked wafer number 31 (Test 1)......55 Figure B.7 Frequency response data of cracked wafer number 31 (Test 2)......56 Figure B.8 Frequency response data of cracked wafer number 31 (Test 3)......56 Figure B.9 Frequency response data of cracked wafe r number 35...................57 Figure B.10 Frequency response data of cracked wafe r number 48.................57 Figure B.11 Frequency response data of cracked wafe r number 32.................58 Figure B.12 Frequency response data of cracked wafe r number 40.................58 Figure B.13 Frequency response data of cracked wafe r number 36.................59 Figure B.14 Frequency response data of cracked wafer number 27 (Test 1)....59 Figure B.15 Frequency response data of cracked wafer number 27 (Test 2)....60 Figure B.16 Frequency response data of cracked wafer number 27 (Test 3)....60 Figure B.17 Frequency response data of cracked wafe r number 8...................61 Figure B.18 Frequency response data of cracked wafe r number 6...................61 Figure B.19 Frequency response data of cracked wafe r number 33.................62 Figure B.20 Frequency response data of cracked wafe r number 41.................62 Figure C.1 First mode natural fre quency for large cra ck wafer set....................63 Figure C.2 First mode normalized fr equency for large cra ck wafer set..............64 Figure C.3 Second mode natural frequen cy for large cra ck wafer set...............65 Figure C.4 Second mode normalized frequency for large crack wafer set........66

PAGE 11

ix Figure C.5 Third mode natural frequen cy for large crac k wafer set...................67 Figure C.6 Third mode normalized frequen cy for large crack wafer set............68 Figure C.7 Fourth mode natural fr equency for large cra ck wafer set.................69 Figure C.8 Fourth mode normalized fr equency for large crack wafer set..........70 Figure C.9 First mode peak magnit ude for large cra ck wafer set......................71 Figure C.10 Second mode peak magnit ude for large cra ck wafer set...............72 Figure C.11 Third mode magnitude for large crack wafer set............................73 Figure C.12 Fourth mode peak magni tude for large cra ck wafer set.................74 Figure C.13 First mode damping ra tio for large cra ck wafer set........................75 Figure C.14 Second mode damping rati o for large cra ck wafer set...................76 Figure C.15 Third mode damping ra tio for large cra ck wafer set.......................77 Figure C.16 Fourth mode damping ra tio for large cra ck wafer set.....................78 Figure D.1 First mode natural fre quency for miscellaneous wafer set...............79 Figure D.2 First mode normalized fr equency for miscellaneous wafer set........80 Figure D.3 Second mode natural fr equency for miscellaneous wafer set..........81 Figure D.4 Second mode normalized fr equency for miscellaneous wafer set...82 Figure D.5 Third mode natural frequen cy for miscellaneous wafer set..............83 Figure D.6 Third mode normalized fr equency for miscellaneous wafer set.......84 Figure D.7 Fourth mode natural fr equency for miscellaneous wafer set............85 Figure D.8 Fourth mode normalized frequency for miscellaneous wafer set.....86 Figure D.9 First mode peak magnit ude for miscellaneous wafer set.................87 Figure D.10 Second mode peak magnit ude for miscellaneous wafer set..........88 Figure D.11 Third mode peak magnit ude for miscellaneous wafer set..............89

PAGE 12

x Figure D.12 Fourth mode peak magni tude for miscellaneous wafer set............90 Figure D.13 First mode damping ra tio for miscellaneous wafer set...................91 Figure D.14 Second mode damping ra tio for miscellaneous wafer set..............92 Figure D.15 Third mode damping ra tio for miscellaneous wafer set..................93 Figure D.16 Fourth mode damping ra tio for miscellaneous wafer set................94 Figure E.1 First mode natural fre quency for small cra ck wafer set....................95 Figure E.2 First mode normalized fr equency for small cra ck wafer set.............95 Figure E.3 Second mode natural frequency for small wafer set........................96 Figure E.4 Second mode normalized frequency for small crack wafer set........96 Figure E.5 Third mode natural frequen cy for small crac k wafer set...................97 Figure E.6 Third mode normalized frequen cy for small crack wafer set............97 Figure E.7 Fourth mode natural fr equency for small cra ck wafer set.................98 Figure E.8 Fourth mode normalized fr equency for small crack wafer set..........98 Figure E.9 First mode peak magnit ude for small cra ck wafer set......................99 Figure E.10 Second mode peak magnit ude for small cra ck wafer set...............99 Figure E.11 Third mode peak magnit ude for small crack wafer set.................100 Figure E.12 Fourth mode peak magnit ude for small crack wafer set...............100 Figure E.13 First mode damping rati o for small crack wafer set......................101 Figure E.14 Second mode damping rati o for small crack wafer set.................101 Figure E.15 Third mode damping rati o for small crack wafer set.....................102 Figure E.16 Fourth mode damping ra tio for small cra ck wafer set...................102 Figure F.1 Hammer (x =33 mm, y= 45 mm)......................................................103 Figure F.2 Hammer (x =43 mm, y= 45 mm)......................................................103

PAGE 13

xi Figure F.3 Hammer (x =38 mm, y= 40 mm)......................................................104 Figure F.4 Hammer (x =38 mm, y= 50 mm)......................................................104

PAGE 14

xii Detection of Cracks in Single-Crystalline Silicon Wafers Using Impact Testing Christina Hilmersson ABSTRACT This thesis is about detection of cra cks in single-crystalline silicon wafers by using a vibration method in the form of an impact test. The goal to detect cracks from vibration measurements intr oduced by striking the silicon wafer with an impact hammer. Such a method would reduce costs in the production of solar cells. It is an inexpensive, relatively simple method which if commercialized could be used as an efficient in-line production quality test. A hammer is used as the actuator and a microphone as the response sensor. A signal analyzer is used to co llect the data and to compute frequency response. Parameters of interest are audible natural frequencies, peak magnitudes, damping ratio and coherence. The data reveals that there are di fferences in frequency between the cracked silicon wafers and the non-cra cked silicon wafers. The resonant peaks in the defective wafers were not as sharp (i.e., lightly damped) and occurred at lower frequencies (i.e., lower stiffness) with a lower magnitude and a higher damping ratio. These differences coul d be used to detect damaged product in a solar cell production line.

PAGE 15

1 Chapter 1: Introduction and Background 1.1 Overview The renewable energy market is gro wing and so are the photovoltaic industries. The thought of using the sun’ s power for generation of electricity is not new. The concept dates back to the industrial revolution [1]. Crystalline Silicon is the most common material used in the photovoltaic market with over 95% market share [2]. The reason that the photovoltaic cell is not more widespread is cost, particularly co st of cell production. During crystal growth and processing of silicon wafers, imperfections (such as cracks, residual stresses and sub-surface damage) are intr oduced. Breakage during production due to defects is currently 6-15%, but the industry wants to get this down to 1% [2], the method used in this thesis to det ect the cracks could help facilitate this goal. There is a need for fast in-line mech anical quality control methods to detect these imperfections during the production of silicon solar cells. This could reduce the further processing of defective products and reduce overall costs. This thesis focuses on vibration impact testing of wafers for crack detection. 1.2 Background During the production of the silicon cells it is not uncommon that cracks and residual stresses are introduced. T here currently exist some methods to detect residual stresses. These are X -ray diffraction, transmission electron

PAGE 16

2 microscopy (TEM), micro-Raman spectrosc opy [3], scanning infrared polariscopy [4], resonance ultrasonic vibrations (RUV) [5] and audible vibration [6]. The cracks introduced are on a cleav age plane are almost closed and are not visible with the human eye. The critical lengths of such cracks is about 1 cm. At 1 cm or more, they propagate in processing from handling etc. The methods that have been used to detect cracks are scanning acoustic microscopy (SAM), ultrasound lock-in t hermography, and millimeter wave [7]. More recently, resonance ultrasonic vi bration (RUV) has been investigated to assess the presence or lack of cracks. Ult rasonic vibrations are applied to the wafers and the frequency spectra analyzed [8]. The SAM method of determining the pr esence of cracks is not feasible during mass production of photovoltaic cells since the time required to scan a 100 mm by 100 mm wafer is between 10 to 15 minutes. Addition ally, the wafer has to be submerged in a water bath or covered with a water droplet. The SAM method, however, does allow for cracks as small as 5 to 10 microns to be detected. Ultrasound Lock-in Thermography can det ect cracks with lengths as small as 100 microns. It takes 5-10 seconds to inspect a 100 mm by 100 mm wafer. The major disadvantage is still the time required to test the wafer. The Millimeter Wave method can be used to inspect a 100 mm by 100 mm wafer in as little as 3 to 5 seconds, but can only be employed for wafers before the metallization process. The crack l ength that can be detected is 400 microns and larger [7].

PAGE 17

3 The foundation behind the approach taken in th is thesis is that impact on a cracked surface sounds different than im pact on a non-cracked surface. An easy physical experiment is to tap a gla ss with a spoon and compare it to a glass which has a crack. One can hear the difference with a human ear. The same demonstration can be performed on silic on wafers. One can hear with the human ear which wafers are significantly cracked. These cracks are obviously large enough to be seen with your eye as well. It would be desirable to detect cracks that are not visible wit h the eye. In this thesis, an impact hammer is used to tap numerous silicon wafer specimens and the vibratory response is recorded by a microphone. The silicon wafers of interest are singl e crystalline silicon wafers grown by the Czochralski method [9]. This thes is assesses the use of an impact test method to detect cracks. This could be a relatively fast, inexpensive and nondestructive method that could be us ed for in-line quality testing. 1.3 Outline This thesis presents two different ty pes of data: data from crack-free wafers and data from cracked wafers. In Chapter 2, a description of the setup as well as a description of each instrum ent used is given. The makeup of the test matrix is covered in Chapter 3. This includes wafer number, type of crack, crack length, wafer thickness and pictur e number. In Chapter 4, the results obtained are given. Finally, Chapter 5 presents the conclusion and future recommendations.

PAGE 18

4 Chapter 2: Experimental Setup 2.1 Introduction This chapter presents the experiment al setup and describes the sensors and the analyzer used. The specimens us ed are single-crystalline Czochralski (Cz) silicon wafers. Since the purpose is to detect cracks in wafers there are different types of specimens tested. In this research, the cracked specimens have been deliberately damaged wit h a diamond pin. In all, thirty different cracked specimens were made and tested. 2.2 Sensors An impact hammer and a sound level meter are the two sensors used in this experiment. The im pact hammer, model PCB 084A17, is made by PCB Piezotronics Inc. The sensitivity of the impact hammer is 22.5 mV/N. The hammer’s weight is 2.9 grams and t he aluminum handle is 101.6 mm long, the hammer has a stainless steel head with a di ameter of 6.3 mm and a red vinyl tip with a 2.5 mm diameter. The hammer is connected by a 0.18G10 coaxial cable, which is 3 m long, with a 5-44 connector te rminating in a 10-32 connector that is connected with a BNC to the SigLab dynamic analyzer. The sound level meter used is a m odel 2900 manufactu red from Quest Technologies. The meter is set to m easure sound pressure in the range of 60120 dB. The sensitivity of the sound level meter is 5V/120 dB. The sound level

PAGE 19

5 meter and the SigLab dynamic analyzer are connected from the ac output of the meter with a 6 ft shielded cable 1/8” plug to an RCA plug. The RCA plug is connected with a gold plated RCA to BNC adapter and connected with the female BNC connection of the analyzer. 2.3 Analyzer The analyzer is SigLab model 20 -42 and is manufactured by DSP Technology Division. The SigLab has 4 input channels and 2 output channels. The impact hammer is connected to input channel 1 and the sound level meter is connected to input channel 3. The anal yzer calculates the frequency response with the impact force as t he input and the sound pressure as the output. A laptop is connected to the SigLab with a Slim SCSI PC card, the PC runs the SigLab software which is written in MatLab R12. In the SigLab so ftware, the bandwidth is set to 1.0 kHz and the record length is 8192, which gives a delta frequency of 0.313 Hz and a record time of 0.3 seconds Also, the sensitivity of the hammer and the sensitivity of the sound level mete r are included in the analyzer setup. The hammer sensitivity is set to 44.4 N/V for channel 1 and the sound pressure level sensitivity is set to 24 dB/V for channel 3. 2.4 Frequency response The frequency response is computed with the impact force, F (in units of Newtons) applied from the hammer as the input and the sound pressure level, S (in units of dB) from sound meter as the output. Time trace measurements of the

PAGE 20

6 input and output are obtained. The measurements are wi ndowed (i.e., box window for the input and ex ponential window for the response) and the Fast Fourier Transforms of the windowed time traces are computed. The measurements are repeated eight times, n, and then averaged. Power spectra (PFF(f),PSS(f)) and cross spectra (PSF(f)) are computed as [10,11] n f F f F f PFF) ( ) ( ) ( (1a) n f S f S f PSS) ( ) ( ) ( (1b) n f S f F f PSF) ( ) ( ) ( (1c) where F(f) is the Fourier transform of F S(f) is the Fourier transform of S and the is the complex conjugate. The frequency response is then computed as ) ( ) ( ) ( f P f P f FRFF SF (2) An m-file was written in Matlab to graph the magnitude (in units of dB/N), phase (in units of degrees) and coher ence (non-dimensional) versus frequency (Hertz). The coherence, ) (2f, is a function of frequency and is computed as ) ( ) ( ) ( ) ( ) (2f P f P f P f P fSS FF SF SF (3) Coherence is a number between 0 and 1, w here 1 means that all the output is caused by the input whereas a number of 0 means that none of the output is caused by the input.

PAGE 21

7 2.5 Test specimens The test specimens are single crysta lline (100) Czochralski (Cz) silicon wafers. They are pseudo square (see s pecimen corners in Figure 2.1) with dimensions 127 x 127 mm. The thickness of the wafers were find by weighing each wafers and knowing that the dens ity of the wafer is 2.329 g/cm3. Some of the specimens are crack-free and some of the specimens have cracks introduced in different orientations. The cra ck-free wafers are scanned in a Scanning Acoustic Microscopy (SAM) before and after testing to ensure that the wafers had not been damaged during the impact testing. Cracks are introduced in the wafers using a diamond pin and pressing carefully on the edge of the wafers with a lig ht force (equivalent to the force used in writing). By doing this, the human ear can hear the wafer crack. To quantify the cracks, the wafers ar e scanned in the SAM befor e and after the impact testing. Since the cracks are not visible usi ng optical methods the cracks images were created in the SAM. The wafers were placed under a water bath. A transducer (work as a transmitter and receiver) moves above the wafer and produced sound waves. These sound waves are at high frequencies and that is the reason why the wafer is in the wa ter bath since the high frequencies do not propagate through air. Three different types of sets are pres ented in this thesis: the large crack wafer set, the miscellaneous crack wafer set and the small crack wafer set. First the small crack wafer set was investigated. Since the small crack wafer did not

PAGE 22

8 show significant differences, larger cra cks were made. These exploratory larger cracks are presented in t he miscellaneous wafer set. After analyzing the miscellaneous wafers, the larger cra ck set was made and are presented as the large crack wafer set. 2.5.1 Large crack wafer set The large cracks have crack lengths varying from 38 mm to 55 mm. Some of the cracks begin at the center of an edge of the s pecimen, others are offset from the center of the edge. Some have segmented cracks (meaning the cracks are not continuous; instead they have small cracks in sequence). If zooming in on the crack and use high re solution of the SAM image one can see that wafer numbers 39, 32, 36, 40, 6, 8 and 41 has segmented crack. Wafer numbers 48 and 33 are also segmented but t he initial crack from the edge is longer than the others. Wafer number 27 initially had segmented crack but during the tests it became cont inuous and wafer numbers 31 and 35 have continuous cracks. The large center cracks are crack ed in the center of an edge of the specimen and are numbered 39, 31, 35, 48. These cra ck lengths vary between 38.6 to 52.7 mm and are shown in Figures 2. 1-2.4. The large offset cracks are cracked offset from the center of an edge of the wa fer are numbered 32, 40, 36, 27 (see Figures 2.5-2.12). The length of these cracks are 41.8-54.8 mm.

PAGE 23

9 Figure 2.1 Large center crack wafer number 39 Figure 2.2 Large center crack wafer number 31 Figure 2.3 Large center crack wafer number 35 Figure 2.4 Large center crack wafer number 48

PAGE 24

10 Figure 2.5 Large offset crack wafer number 32 Figure 2.6 Large offset crack wafer number 40 Figure 2.7 Large offset crack wafer number 36 Figure 2.8 Large offset crack wafer number 27

PAGE 25

11 Figure 2.9 Large crack other direction wafer number 8 Figure 2.10 Large crack other direction wafer number 6 Figure 2.11 Large crack other direction wafer number 33 Figure 2.12 Large crack other direction wafer number 41

PAGE 26

12 2.5.2 Miscellaneous wafer set The miscellaneous cracks were made to explore larger cracks so the cracks starts at different location of t he edge of the wafers. The cracks are made in different directions and have various crack lengths. The crack lengths vary from 18.5 mm to 52.7 mm. Wafer num bers 45, 46, and 7 have segmented cracks (meaning the cracks are not conti nuous instead they have small cracks in sequence). Wafer number 11 is smaller t han the other, wafer number 23, 25, 47, 8 have continued cracks. The SAM im ages of the miscellaneous wafers are shown in Figures 2.13-2.20. Figure 2.13 Miscellaneous wafer number 11 Figure 2.14 Miscellaneous wafer number 23 Figure 2.15 Miscellaneous wafer number 25 Figure 2.16 Miscellaneous wafer number 45

PAGE 27

13 Figure 2.17 Miscellaneous wafer number 46 Figure 2.18 Miscellaneous wafer number 7 Figure 2.19 Miscellaneous wafer number 47 Figure 2.20 Miscellaneous wafer number 8 2.5.3 Small crack wafer set The small cracks have crack lengths from 2.3 to 7.6 mm. Some of the cracks are a single crack and ot hers have a V-shape. Note that all cracks in this

PAGE 28

14 work initiate at an edge of the wafer. For the V-shape cracks, the point of the V is at the wafer edge. The location of the small cracks are arbitr ary. All the small crack specimens are shown in Figures 2.21 – Figure 2.30. Figure 2.21 Small crack wafer number 21 Figure 2.22 Small crack wafer number 22 Figure 2.23 Small crack wafer number 23 Figure 2.24 Small crack wafer number 25 Figure 2.25 Small crack wafer number 26 Figure 2.26 Small crack wafer number 27 Figure 2.27 Small crack wafer number 29 Figure 2.28 Small crack wafer number 30 Figure 2.29 Small crack wafer number 31 Figure 2.30 Small crack wafer number 32

PAGE 29

15 2.6 Setup The test setup is shown in Figure 2.31 The specimen is set on a piece of convoluted foam of dimensions 7 x 33 x 26.5 cm. The sound level meter is attached to a rigid fixture and the mi crophone is set at 1.2 cm above the specimen. The microphone is set perpendi cular to the wafer. The impact hammer is connected to channel 1 of the SigLab analyzer and the sound level meter is connected to channel 3 of the SigLab analyzer. Figure 2.31 Pict ure of the set up

PAGE 30

16 2.6.1 Position of ha mmer and microphone The horizontal position of the ha mmer and the sound level meter with respect to the specimen is shown in Figure 2.32. Figure 2.32 Position of hammer and microphone relative to wafer all the units are in mm The decision on were to locate the hammer and microphone with respect to the wafer was made by keeping t he hammer in the same place and moving the microphone, and then moving the hamme r while keeping the microphone in the same location. Figures 2.3335 show the frequency response with the microphone located in the same position as Figure 2.32 and the hammer is changing position. Figure 2.33 show s one dominate audible mode at 600 Hz,

PAGE 31

17 Figure 2.34 and Figure 2.34 show 4 dom inated modes with different peak magnitudes. The hammer and microphone locations used and shown in Figure 2.32 gave the most response (magnitudes ) for the four audible modes. 0 200 400 600 800 1000 0 0.5 1 Coherence 0 200 400 600 800 1000 0 500 1000 Magnitude (dB/N) 0 200 400 600 800 1000 -200 0 200 Phase (deg)Frequency (Hz) Figure 2.33 Hammer (x=83 mm, y=48 mm) 0 200 400 600 800 1000 0 0.5 1 Coherence 0 200 400 600 800 1000 0 500 1000 Magnitude (dB/N) 0 200 400 600 800 1000 -200 0 200 Phase (deg)Frequency (Hz) Figure 2.34 Hammer (x=39 mm, y=60 mm)

PAGE 32

18 0 200 400 600 800 1000 0 0.5 1 Coherence 0 200 400 600 800 1000 0 500 Magnitude (dB/N) 0 200 400 600 800 1000 -200 0 200 Phase (deg)Frequency (Hz) Figure 2.35 Hammer (x=53 mm, y=48 mm) After deciding the location of the hammer and microphone as shown in Figure 2.32 both the hammer and the microphone were moved 5 mm. These results show small variations in magnitude (see Appendix F, Figure F.1-F.4).

PAGE 33

19 Chapter 3: Test Matrix 3.1 Introduction This chapter lists all the wafers that were tested in this study and how the run order was determined. Three different groups of cracked wafers are listed: large crack wafer group, small crack wa fer group and miscellaneous wafer group. This miscellaneous wafer group was not tested as a set, but is included because it was used to explore larger cracks. 3.2 Randomization The order of the tests wa s randomized to eliminate bias error. All the wafers were tested three times each (8 impacts per test) and the test order is shown in Table 3.1 (for small crack test s see Table A.1). A table with the wafer number and test number (three tests per wafer) is also shown in Table 3.2 (for small crack tests see Table A.2). This table was used to define the random run order. The test order was determined by assigning each test a random number and sorting the random numbers.

PAGE 34

20 Table 3.1 Run order for the large crack wafer set Run order Wafer number Test number Random Number in ascending order 1 8 38 48 2 32 26 49 3 38 7 75 4 40 30 111 5 8 37 137 6 40 28 148 7 31 16 149 8 47 11 176 9 39 15 207 10 33 45 218 11 34 6 253 12 39 13 275 13 32 25 286 14 34 5 298 15 27 35 302 16 36 33 347 17 36 32 350 18 40 29 352 19 6 42 382 20 38 9 492 21 33 43 497 22 6 40 498 23 32 27 503 24 29 1 512 25 36 31 524 26 47 10 525 27 27 36 562 28 41 48 565 29 47 12 597 30 29 2 623 31 39 14 641 32 35 21 643 33 29 3 650 34 6 41 690 35 27 34 691 36 48 24 741 37 35 20 750 38 8 39 752 39 33 44 753 40 31 17 757 41 41 46 784 42 38 8 786 43 34 4 799 44 31 18 805 45 48 22 854 46 41 47 860 47 48 23 879 48 35 19 928

PAGE 35

21 Table 3.2 Test number for the large crack wafer set Wafer number Test Number 29 1 2 3 34 4 5 6 38 7 8 9 47 10 11 12 39 13 14 15 31 16 17 18 35 19 20 21 48 22 23 24 32 25 26 27 40 28 29 30 36 31 32 33 27 34 35 36 8 37 38 39 6 40 41 42 33 43 44 45 41 46 47 48 3.3 Test with large cracks The large crack test set contains 16 specimens from which 12 specimens are cracked and the other 4 are crack-free. Table 3. 3 shows the wafer number, type of crack, if segmented or continuous, the length of the crack, the thickness of the wafer and the figur e number of the image. Table 3.3 Large crack wafers Wafer number Type of crack Segmented Crack length [mm] Wafer thickness [ m] Photo Figure number 29 Crack Free Crack Free 0 305 N/A 34 Crack Free Crack Free 0 305 N/A 38 Crack Free Crack Free 0 306 N/A 47 Crack Free Crack Free 0 306 N/A 39 Center Crack Yes 38.6 306 2.1 31 Center Crack No 51.4 305 2.2 35 Center Crack No 52.7 305 2.3 48 Center Crack Yes 51.9 306 2.4 32 Offset Crack Yes 41.8 305 2.5 40 Offset Crack Yes 42.5 307 2.6 36 Offset Crack Yes 47.5 306 2.7 27 Offset Crack No 48.5 305 2.8 8 Offset Crack Yes 43.3 305 2.9 6 Offset Crack Yes 47.2 305 2.10 33 Offset Crack Yes 52.9 306 2.11 41 Offset Crack Yes 54.8 306 2.12

PAGE 36

22 3.4 Test with small cracks Thirty small crack wafers were tested. Twenty wafers were crack-free and ten wafers had cracks introduced to the wafers. Table 3.4 shows the wafer number, type of crack, the l ength of the crack, the thi ckness of the wafer and the figure number of the image. Table 3.4 Small crack wafers Wafer number Type of crack Crack length [mm] Wafer thickness [ m] Photo Figure number 11 Crack Free 0 294 N/A 12 Crack Free 0 294 N/A 13 Crack Free 0 294 N/A 14 Crack Free 0 292 N/A 15 Crack Free 0 291 N/A 18 Crack Free 0 291 N/A 19 Crack Free 0 291 N/A 20 Crack Free 0 292 N/A 33 Crack Free 0 294 N/A 34 Crack Free 0 294 N/A 35 Crack Free 0 294 N/A 36 Crack Free 0 294 N/A 37 Crack Free 0 294 N/A 38 Crack Free 0 293 N/A 39 Crack Free 0 294 N/A 40 Crack Free 0 294 N/A 41 Crack Free 0 294 N/A 42 Crack Free 0 293 N/A 43 Crack Free 0 294 N/A 44 Crack Free 0 293 N/A 21 V-shape 4.3 4.5 291 2.21 22 Single 7.4 291 2.22 23 V-shape 4.1 7.7 291 2.23 25 V-shape 4.6 – 6.3 291 2.24 26 V-shape 4.1 4.4 293 2.25 27 V-shape 4.2 4.9 292 2.26 29 Single 6.3 295 2.27 30 V-shape 2.3 7.0 293 2.28 31 V-shape 4.0 4.6 294 2.29 32 Single 7.6 294 2.30 3.5 Tests of miscellaneous cracks The tests from the miscellaneous cracks set we re not performed in a totally randomized manner. The wafers were tested at different periods of time.

PAGE 37

23 However, these tests are grouped put t ogether for presentation. This set includes twelve wafers total: four are crack-free and eight wafers are cracked. Table 3.5 shows the wafer number, type of crack, if segment ed or continuous, the length of the crack, t he thickness of the wafer and the figure number of the image. Table 3.5 Miscellaneous wafers Wafer number Type of crack Segmented Crack length [mm] Wafer thickness [ m] Photo Figure number 20 Crack free Crack Free 0 305.6 N/A 42 Crack free Crack Free 0 293.0 N/A 49 Crack free Crack Free 0 293.8 N/A 2 Crack free Crack Free 0 295.2 N/A 11 Cracked No 9 305.1 2.13 23 Cracked No 41.5 305.8 2.14 25 Cracked No 25.7 305.5 2.15 45 Cracked Yes 18.5 306.0 2.16 46 Cracked Yes 25.8 305.6 2.17 7 Cracked Yes 43.8 294.0 2.18 47 Cracked No 43.3 293.3 2.19 8 Cracked No 52.7 294.3 2.20

PAGE 38

24 Chapter 4: Test Results 4.1 Introduction This chapter presents the test data and results. This includes frequency response data with four audible modes. The following parameters are extracted from these 4 modes: natural frequencies, peak magnitudes, damping ratios and coherence. 4.2 Sample frequency response data Typical frequency response data from a crack-free wafer is shown in Figure 4.1. The graph shows a range of frequencies from 0-1000 Hz for coherence, magnitude and phase. Four dominant modes are found at the following frequencies: 420 Hz, 590 Hz, 840 Hz and 960 Hz. By comparing the frequency response data of a non-cracked wafer with a wafer with a large crack, as shown in Fi gure 4.2, one can still see four different modes but the frequencies are lo wer, the damping is lar ger (peaks not as sharp) and the peak magnitudes are lower. In the next section, t hese parameters are extracted from the frequency response data.

PAGE 39

25 0 200 400 600 800 1000 0 0.5 1 Coherence 0 200 400 600 800 1000 0 500 1000 Magnitude (dB/N) 0 200 400 600 800 1000 -200 0 200 Phase (deg)Frequency (Hz) Figure 4.1 Frequency response of crack-free wafer number 29 0 200 400 600 800 1000 0 0.5 1 Coherence 0 200 400 600 800 1000 0 500 Magnitude (dB/N) 0 200 400 600 800 1000 -200 0 200 Phase (deg)Frequency (Hz) Figure 4.2 Frequency response of large crack wafer number 35

PAGE 40

26 The coherence ranges between 0 and 1, and it measures the amount of output that is caused by t he input. A coherence value of 1 means that 100% of the output is caused by the input. In Figure 4.1 and Figure 4.2 the coherence versus frequency is plotted. From these figures, one can see that the coherence is close to one around the four domi nant modes. All the frequency response data from the large crack wafer set ar e presented in Figures B.1-B.20. The lowest coherence at some peaks is 0.9. The small crack wafer set is not in cluded, since they did not show any change compared to the crack-free wafers when the cracks are less than 8 mm. However, the data was collected and the ex tracted parameters are presented in the next section and may be compared wit h the large crack wafer set and the miscellaneous wafer set. 4.3 Extracted parameters The following parameters are extracted from the four dominant modes in the frequency response data: natur al frequencies, peak magnitudes and damping ratio. The data from the large-crack wafer set are shown in Tables C.1C.16 and Figures C.1-C.16, t he miscellaneous wafer set in Tables D.1-D.16 and Figures D.1-D.16, and the sm all crack wafer set in Figures E.1-E.16.

PAGE 41

27 4.4 Natural frequencies 4.4.1 Large crack wafer set The data from the second mode frequencie s of the large crack wafer set are most representative and ar e shown in Table 4.1 (as well as in Appendix C). The first 4 specimens in the table are the crack-free wafers and the following 12 specimens have cracks as defined in Table 3.3 and shown in Figures 2.1-2.12. In the crack-free wafers, the sec ond mode frequency ranges from 591.6 to 594.1 Hz. For an individual crack-free wafer, the frequen cy deviation is less than 0.3 Hz, which is the frequency resolution of the measurements. The frequency deviation across all four crack-free wafers is 2.5 Hz. For the 12 large crack wafers, th e second mode frequency ranges from 565.9 to 592.8 Hz and all the wafers are within 26.9 Hz. Si x of the cracked wafers have frequencies that fall within t he crack-free frequency range, two of the cracked wafers have slightly lower fr equencies (590.3-590.9 Hz) and four of the cracked wafers have significantly lo wer frequencies from 565.9 to 587.8 Hz. These 4 large crack specimens ar e numbered 31, 35, 48 and 27 and are italicized in Table 4.1. Comparing the “italicized” cracked wafers in Table 4.1 with Table 3.3, one sees that the continuous cracked wafers all show significant changes in frequency.

PAGE 42

28 Table 4.1 Second mode frequency for large specimens Specimen number Test 1 [Hz] Test 2 [Hz] Test 3 [Hz] Mean [Hz] 29 591.6 591.6 591.6 591.6 34 592.2 592.2 592.2 592.2 38 592.8 593.1 593.1 593.0 47 594.1 594.1 594.1 594.1 39 592.8 592.8 592.8 592.8 31 571.9 571.9 569.7 571.1 35 565.9 565.9 566.3 566.0 48 584.4 583.4 582.8 583.5 32 590.6 590.9 590.6 590.7 40 592.5 592.8 592.8 592.7 36 591.9 591.9 591.9 591.9 27 587.8 587.8 570.9 582.2 8 590.3 590.6 590.6 590.5 6 592.2 592.2 592.2 592.2 33 591.9 591.6 591.9 591.8 41 592.8 592.8 592.8 592.8 550 555 560 565 570 575 580 585 590 595 600293438473931354832403627863341 Wafer NumberFrequency [Hz] Test 1 Test 2 Test 3 Figure 4.3 Second mode frequency for large specimens The data in Table 4.1 is graphed in Figur e 4.3. As seen in Figure 4.3 all the specimens have a low frequency deviati on (less than 0.3 Hz), except large

PAGE 43

29 crack wafer numbers 31, 48, and 27. Cr acked wafer 31 has a deviation of 2.2 Hz, wafer number 48 has a deviation of 1.6 Hz, and wafer number 27 has a deviation of 16.9 Hz. By looking at the before and after images for wafer number 27(segmented) and wafer number 31(cont inuous), one can see that the cracks did elongate with testing (3 tests with 8 impacts each) thus reducing stiffness and frequency; however wafer number 48 (segment) did not show a notable change. Similar data for the first frequency m ode are presented in Table C.1 and Figure C.1 (Appendix C). The crack-free wafers first mode frequency range is 418.1-420 Hz. For an individual crack-fr ee wafer the frequency deviation is less than 0.6 Hz. The frequency deviation across a ll four crack-free wafers is 1.9 Hz. For the 12 large crack wafers the first mode frequency range is 389.4420.9 Hz and all the wafers are within 31.6 Hz. Five of the cracked wafers have frequencies that fall within t he crack-free frequency range, four of the cracked wafers have slightly lower frequencies ( 412.8-416.9 Hz) and thr ee of the cracked wafers have significant lower frequencies from 389.4 to 415.9 Hz. These 3 large crack specimens are numbered 31, 35 and 27. All the specimens have a frequency deviation of less than 0.6 Hz except large crack wafers numbered 31, 35, 48, and 27. Cracked wafer 31 has a deviation of 4.4 Hz, wafer number 35 has a deviation of 0.9 Hz, wafer number 48 has a deviation of 1.3 Hz, and wafer number 27 has a deviation of 10.6 Hz.

PAGE 44

30 Table C.5 and Figure C.5 shows the th ird frequency mode. The crack-free wafers third mode frequency range is 839. 1-842.8 Hz. For an individual crackfree wafer the frequency deviation is less than 0.3 Hz. The frequency deviation across all four crack-free wafers is 3.8 Hz. For the 12 large crack wafers the third mode frequency range is 812.5843.4 Hz and all the wafers are within 30.9 Hz. Eight of the cracked wafers have frequencies that fall within the crack-free frequency ra nge, one of the cracked wafers has slightly lower frequencies (834.1-835 Hz) and three of the cracked wafers have significant lower frequencie s from 812.5 to 835 Hz. These three cracked specimens are numbered 31, 35, and 27. All the specimens have a frequency deviation less than 0.3 Hz except large crack wafers numbered 31, 35, 48, and 27. Cracked wafer 31 has a deviation of 13.4 Hz, wafer number 35 has a deviation of 0.6 Hz, wafer number 48 has a deviation of 0.9 Hz, and wafer num ber 27 has a deviation of 18.4 Hz. Table C.7 and Figure C.7 s hows mode four. The cra ck-free wafers fourth mode frequency range is 960.6-964.7 Hz. Fo r an individual crack-free wafer the frequency deviation is less than 0.3 Hz. The frequency deviation across all four crack-free wafers is 4.1 Hz. For the 12 large crack wafers the fourth mode frequency range is 930.6962.2 Hz all the wafers are within 31.6 Hz Five of the cracked wafers have frequencies that fall within t he crack-free frequency range, three of the cracked wafers have slightly lower frequencies ( 958.4-960.3 Hz) and four of the cracked

PAGE 45

31 wafers have significant lower frequenc ies from 930.6-954.1 Hz. These four cracked specimens are num bered 31, 35, 48 and 27. All the specimens have a frequency deviation less than 0.6 Hz except large crack wafer numbers 48, and 27. Cracked wafer number 48 has a deviation of 2.5 Hz and wafer number 27 has a deviation of 23.4 Hz. 4.4.2 Miscellaneous wafer set The data from the second mode fr equencies of the miscellaneous crack wafer set are shown in Table D.3 and Figure D.3 (Appendix D). The first 4 specimens in the table are the crack-free wafers and the following 8 wafers have cracks as defined in Table 3.5 and shown in Figures 2.13-2.20. The second mode frequency range for t he crack-free wafers is from 569.1 to 592.8 Hz. For an individual crackfree wafer the frequency deviation is less than 0.3 Hz. The frequency deviation across all the crack-free wafers is 23.8 Hz. For the 8 cracked wafers the se cond mode frequency ranges from 554.7 to 593.1 Hz, all cracked wafers are within 38.4 Hz and the deviation within a wafer is less than 2.2 Hz. The deviation across the crack-free wa fers are higher than found for the large crack specimens because these miscellaneous wafers have various thickness 293-306m. This problem will be solved by normalizing in section 4.5. Data for the first mode is also shown in Table D.1 and Figure D.1, for the third mode Table D.5 and Figure D.5, and for the fourth mode in Table D.7 and Figure D.7.

PAGE 46

32 4.4.3 Small crack wafer set The second mode frequencies for the sm all crack wafer set are shown in Figure E.3. The first 20 specimens in the Table 3.4 are the crack-free wafers and the following 10 wafers have cracks as defined in Table 3.4 and shown in Figures 2.21-2.30. The crack-free wafers second m ode frequencies range from 564.1 to 571.9 Hz. For an individua l crack-free wafer the fr equency deviation is less than 0.6 Hz, the frequency deviation across all twenty crack-free wafers is 7.8 Hz. For the 10 small crack wafers the second frequency mode ranges from 565 to 572.5 Hz and all are within 7.5 Hz. The deviation in an individual wafer is less than 0.3 Hz. Note that the ranges ar e very similar to the crack-free wafers and that the cracks are small less than 8 mm. Data for the first mode is presented in Figure E.1, for the third mode in Figure E. 5, and for the fourth mode in Figure E.7. 4.5 Normalized frequencies Some of the wafers were found to have slightly different thickness. Since natural frequency is directly proportional to the thickness to the three-halves power, the frequencies are norma lized with respect to the thickness of the wafers by 2 / 3h f fnorm (4)

PAGE 47

33 where fnorm is the normalized frequency, f is the measured natural frequency (in Hz) and h is the thickness of the wafer (in m) [12]. The normalization is necessary for the miscellaneous wafer set since the thickness varied and also to compare the three different type of sets. 4.5.1 Large crack wafer set The normalized frequencies for the se cond mode (presented in Table C.4 and Figure C.4) of the large crack wafer set did not show a large difference from section 4.4.1 since the thickness range of these wafers are similar 305-307 m. The crack-free wafers second mode normalized frequency ranges from 0.0008 to 0.0010 Hz/m2 / 3. For an individual crack-free wafer, the frequency deviation is less than 0.0001 Hz/m2 / 3. The frequency deviation across all four crack-free wafers is 0.0002 Hz/m2 / 3. For the 12 large crack wafers, th e second mode normalized frequency ranges from 0.1062 to 0.1111 Hz/m2 / 3 and all the wafers are within 0.0049 Hz/m2 / 3. Four of the cracked wafers have normalized frequencies that fall within the crack-free normalized frequency r ange, four of the cracked wafers have slightly lower normalized frequencies (0.1103-0.1108 Hz/m2 / 3) and four of the cracked wafers have significantly lower normalized frequencies from 0.10620.1102 Hz/m2 / 3. These 4 large crack specimens are numbered 31, 35, 48 and 27. All the specimens have a frequen cy deviation of less than 0.0001 Hz/m2 / 3 except large crack wafer numbers 31, 48, and 27.

PAGE 48

34 Cracked wafer 31 has a deviation of 0.0004 Hz/m2 / 3, wafer number 48 has a deviation of 0.0003 Hz/m2 / 3, and wafer number 27 has a deviation of 0.0032 Hz/m2 / 3. Since the thickness for the large crack wafer set does not vary significantly, the normalization of the fi rst, third and fourth frequency modes are not discussed in this section but are pr esented in Appendix C for completeness. Data for the first mode in Table C.2 and Fi gure C.2, for the second mode in Table C.4 and Figure C.4, for the third mode in Table C.6 and Figure C.6, and for the fourth mode in Tabl e C.8 and Figure C.8. 4.5.2 Miscellaneous wafer set The miscellaneous wafers second mode frequencies are graphed in Figure 4.4 and the normalized second mode frequencies are graphed in Figure 4.5. In Figure 4.4 wafer numbers 42, 49, 2, 23, 7 ,47, 8 had significantly lower frequencies than the other 5 wafers. T he reason that the frequencies are lower is not because of any cracks instead it is because the thickness of the wafers varied from 293-306 m. The crack-free wafers second mode normalized frequency ranges from 0.1110 to 0.1135 Hz/m2 / 3. For an individual crack-free wafer, the normalized frequency deviation is less than 0.0001 Hz/m2 / 3. The normalized frequency deviation across all four crack-free wafers is 0.0025 Hz/m2 / 3.

PAGE 49

35 For the 12 large crack wafers, th e second mode normalized frequency ranges from 0.1043 to 0.1129 Hz/m2 / 3 and all the wafers are within 0.0086 Hz/m2 / 3. Three of the cracked wafers have normalized frequencies that fall within the crack-free normalized frequency r ange, four of the cracked wafers have slightly lower normalized frequencies (0.1099-0.1109 Hz/m2 / 3) and one (number 23) of the crack ed wafers has significantly lower normalized frequencies from 0.1043-0.1047 Hz/m2 / 3. All the specimens have a normalized frequency deviation of less than 0.0002 Hz/m2 / 3 except large crack wafer numbers 23 and 47 they have a deviation of 0.0004 Hz/m2 / 3. Data for mode 1 are provided in Table D.2 and Figure D.2, for mode 3 in Table D.6 and Fi gure D.6, and for mode 4 in Table D.8 and Figure D.8.

PAGE 50

36 550 555 560 565 570 575 580 585 590 595 600 204249211232545467478 Wafer NumberFrequency [Hz] Test 1 Test 2 Figure 4.4 Second mode natural fr equencies for miscellaneous wafer set 0.104 0.106 0.108 0.110 0.112 0.114 204249211232545467478 Wafer numberNormalized frequency [Hz/ m3/2] Test 1 Test 2 Figure 4.5 Second mode normalized frequencies for miscellaneous wafer set

PAGE 51

37 4.5.3 Small crack wafer set The second mode normalized frequencies for the small crack wafer set are shown in Figure E.4. Those wa fers thickness varies between 291 and 295m. The twenty crack-free wafers and the ten cracked wafers have the same normalized second mode frequency r ange from 0.1132 to 0.1138 Hz/m2 / 3. The deviation for an individual wafer is less than 0.0001 Hz/m2 / 3 and the deviation across all the wafers are 0.0006 Hz/m2 / 3. The normalization of the small crack wafer set did improve the second m ode frequency data (compare Figure E.3 with E.4). Data for the first mode is provided in Figure E. 2, for the third mode in Figure E.6, and for the fourth mode in Figure E.8. 4.6 Peak magnitudes 4.6.1 Large crack wafer set The data from the second mode peak m agnitude of the large crack wafer set are shown in Table 4.2. The first 4 specimens in the table are the crack-free wafers and the following 12 specimens hav e cracks as defined in Table 3.3 and shown in Figures 2.1-2.12. The crack-free wafers second mode peak magnitude ranges from 895 to 1025 dB/N. For an individual crack-free wafer, the magnitude deviation is less than 91 dB/N. The magnitude deviation ac ross all four crack-free wafers is 130 dB/N.

PAGE 52

38 For the 12 large crack wafers, th e second mode peak magnitude ranges from 397 to 1006 dB/N. All the wafers are within 609 dB/N. Four of the cracked wafers have magnitudes that fall within t he crack-free magnitude deviation range, one of the cracked wafers (number 27) wa s in the crack-free range but during the test, the wafer crack grew and fell outsi de the crack-free r ange (591-968 dB/N). Seven of the cracked wafers have significantly lower magnitudes (397-848 dB/N). These 7 large crack specimens are numbered 39, 31, 35, 48, 32, 40, 36, 27 and are italicized in Table 4.2. For all the cracked wafers, the individual deviation is less than 80 dB/N. With the exception of wafer number 39 (a deviation of 126 dB/N), wafer number 48 (a deviation of 134 dB/N), wafer number 27 (a deviation of 377 dB/N) and wafer num ber 8 (a deviation of 145 dB/N). Based on this second mode data, t he peak magnitude par ameter is a better indicator of large crack faults then natural frequency par ameter itself. Notice that all the wafers with conti nuous cracks show a significant change in peak magnitude (see Tables 3.3 and 4.2). Table 4.2 Second mode peak magnit ude for the large crack wafer set Specimen number Test 1 [dB/N] Test 2 [dB/N] Test 3 [dB/N] Mean [dB/N] 29 908 992 978 959 34 976 895 960 944 38 1025 944 936 968 47 900 923 991 938 39 848 723 754 775 31 487 468 487 481 35 398 397 403 399 48 801 731 667 733 32 805 736 736 759 40 675 710 685 690 36 712 718 692 707 27 949 968 591 836 8 1006 861 919 929 6 923 979 947 950

PAGE 53

39 Table 4.2 (Continued) 33 888 836 871 865 41 903 824 826 851 0 200 400 600 800 1000 1200 293438473931354832403627863341 Wafer numberMagnitude [dB/N] Test 1 Test 2 Test 3 Figure 4.6 Second mode peak magnit ude for the large crack wafer set Note that wafer number 27 did not lo wer its peak magnitude until the last test since the segmented cracks elongated to a conti nuous crack during test. Data for mode 1,2,3 and 4 are referenc ed in Tables C.9-C.12 and Figures C.9C.12. 4.6.2 Miscellaneous wafer set The data from the second mode peak magnitude of the miscellaneous crack wafer set are shown in Table D.10. The first 4 specimens in the table are the crack-free wafers and the follo wing 8 specimens have cracks.

PAGE 54

40 The crack-free wafers second mode peak magnitude range from 927 to 1010 dB/N. For an individual crack-free wafer, the magnitude deviation is less than 50 dB/N. The magnitude deviation ac ross all four crack-free wafers is 83 dB/N. For the 12 large crack wafers t he second mode peak magnitudes range from 371-1028 dB/N. All the wafers ar e within 657 dB/N. Two of the cracked wafers have magnitudes that fall within t he crack-free magnitude deviation range. Six of the cracked wafers have signifi cantly lower magnitudes (371-836N). These 6 cracked specimens are numbered 23, 25, 45, 46, 47 and 8. Again, the magnitude parameter is a better indicator of large crack faul ts than natural frequency parameter itself. Data for modes 1,2,3 and 4 are refe renced in Tables D.9-D.12 and Figures D.9-D.12. 4.6.3 Small crack wafer set The data from the second mode peak m agnitude of the small crack wafer set are shown in Figure E.10. The 20 crack-free wafers second mode peak magnitudes range from 760 to 1073 dB/N. For an individual cra ck-free wafer, the magnitude deviation is less than 153 dB/N The magnitude deviation across all twenty crack-free wafers is 303 dB/N. For the 10 small crack wafers, t he second mode peak magnitudes range from 811 to 1087 dB/N. All the wafers ar e within 276 dB/N. Again the crack-free wafers do not deviate from the cracked wa fers when the crack size is small. Data for mode 1,2,3 and 4 are referenced in Figures E.9-E.12. Therefore, the

PAGE 55

41 vibration impact test used in this thesis does not appear to be suitable for detecting small edge cracks. 4.7 Damping ratio The damping ratio is found by zooming in on each peak and subtracting 3 dB on each side of the peak and recording the corresponding frequencies. The damping ratio is then calculat ed by the following equation, n 2 ) (1 2 (5) where is the damping ratio, 2 is the frequency 3 dB down on the right side of the peak, 1 is the frequency 3 dB down on the left side of the peak and n is the frequency at the peak. 4.7.1 Large crack wafer set The damping ratio for the second m ode frequencies of the large crack wafer set are shown in Table 4.3. The first 4 specimens in the table are the crack-free wafers and the followi ng 12 specimens have cracks. The crack-free wafers second mode dam ping ratio range from 0.0014 to 0.0016. For an individual crack-free wafe r, the damping ratio deviates less than 0.0002. The damping ratio deviation across a ll four crack-free wafers is 0.0002. For the 12 large crack wafers, th e second mode damping ratios range from 0.0015 to 0.0054. All the wafers ar e within 0.0017. Five of the cracked wafers have damping ratios that fall with in the crack-free damping ratio range,

PAGE 56

42 three of the cracked wafers have a sli ghtly higher damping ra tio (0.0017-0.0021), one of the cracked wafers, number 27 (0 .0019-0.0036). Three of the cracked wafers have significantly higher damping ratios (from 0.0032 to 0.0054). This shows that the damping ratio parameter can be used to detect large continuous cracks. The other large crack specimens that do not show significant deviation are not continuous. These 3 large cr ack specimens are numbered 31, 35, 48 and are italicized in Table 4.3. Note that all of the wafers with continuous cracks show significant changes in damping ratio (see Tables 3.3 and 4.3) Table 4.3 Second mode peak damping ra tio for the large crack wafer set Specimen number Test 1 [non-dim] Test 2 [non-dim] Test 3 [non-dim] Mean [non-dim] 29 0.0015 0.0014 0.0016 0.0015 34 0.0015 0.0016 0.0015 0.0016 38 0.0015 0.0015 0.0016 0.0015 47 0.0015 0.0016 0.0014 0.0015 39 0.0016 0.0018 0.0016 0.0017 31 0.0049 0.0054 0.0052 0.0052 35 0.0044 0.0047 0.0046 0.0046 48 0.0032 0.0033 0.0036 0.0034 32 0.0015 0.0018 0.0020 0.0018 40 0.0020 0.0019 0.0019 0.0019 36 0.0019 0.0021 0.0021 0.0020 27 0.0020 0.0019 0.0036 0.0025 8 0.0015 0.0017 0.0016 0.0016 6 0.0016 0.0015 0.0016 0.0016 33 0.0017 0.0018 0.0018 0.0017 41 0.0015 0.0016 0.0015 0.0015

PAGE 57

43 0.000 0.001 0.002 0.003 0.004 0.005 0.006 293438473931354832403627863341 Wafer NumberDamping Ratio [non-dim ] Test 1 Test 2 Test 3 Figure 4.7 Second mode peak damping ra tio for the large crack wafer set Data for mode 1,2,3 and 4 are referenced in Tables C.13-C.16 and Figures C.13-C.16. 4.7.2 Miscellaneous wafer set The damping ratio for the second m ode frequencies of the miscellaneous crack wafer set are shown in Table D.14 and Figure D.14. The first 4 specimens in the table are the crack-free wafers and the following 8 wafers are cracked. The second mode damping ratio range for the crack-free wafers is from 0.0014-0.0016. For an individual crack-fr ee wafer the damping ratio deviation is less than 0.0001. The damping ratio deviation across all the crack-free wafers is 0.0002.

PAGE 58

44 For the 8 cracked wafers, the se cond mode damping ratio ranges from 0.0015-0.0041. All cracked wafe rs are within 0.0027. Four of the cracked wafers have damping ratios that fall within the crack-free damping ratio deviation range, four of the cracked wafers have signi ficantly higher damping ratio from 0.00230.0041, those 4 wafers are numbered 23, 25, 47 and 8. All the specimens have a damping ratio deviation of less than 0.0001 except large crack wafer numbers 23 (a deviation of 0.0014), 25 (a deviation of 0.0007), 47 (a deviation of 0.0007) and wafe r number 8 (a deviation of 0.0004). Data for the first mode is shown in Tabl e D.1 and Figure D.1, for the third mode in Table D.5 and Figure D.5, and for t he fourth mode in Table D.7 and Figure D.7. 4.7.3 Small crack wafer set The second mode damping ratio for the small crack wafer set is shown in Figure E.14. For the 20 crack-free wa fers, the second mode damping ratios range from 0.0013 to 0.0018 and are shown first, followed by 10 cracked wafers whose second mode damping rati os range 0.0014-0.0017. For the crack-free wafers and the wafe rs with a small crack, the deviation of an individual wafer is less than 0.0004, and the range across all the crack-free wafers is 0.0005 and for the crack ed wafers the range is 0.0003. Data for mode 1,2,3 and 4 are referenced in Tables E.9-E.12 and Figures E.13-E.16.

PAGE 59

45 4.8 Discussion For the large crack wafer set, four wafers (numbered 31,35,48,27) show significant deviation in the natural frequencies for the four modes. For the magnitude peaks, eight wafers (numbered 39, 31, 35, 48, 32, 40, 36, 27) show a significant difference. Fo r the damping ratio, four of the wafers (numbered 31, 35, 48, 27) show a significant ly difference. Only four from the twelve large crack wafers set showed significant deviati on in frequency, magnitude and damping ratio. These four large crack spec imens have continuous cracks as opposed to segmented cracks as in the other 8 large crack specimens. From the miscellaneous wafer set, wafer numbers 23 and 8 show a difference in the normalized frequency. Looking at the magnitude, six of the cracked wafers show a difference com pared to the crack-free wafers. The damping ratio was higher for number 23, 25, 47 and 8. In other words, 50% of the cracked wafers were different from the crack-free data set considering the damping ratio and the magnitude. The small crack wafer set did not s how any notable change in frequency, magnitude nor damping ratio. The crack l ength of the wafers were to small to detect the cracks using the impact me thod described in this thesis.

PAGE 60

46 Chapter 5: Conclusions and Recommendations 5.1 Conclusions The results showed some deviations in the four dominate audible modes that were measured for cracked versus crack-free wafers. A difference in the natural frequencies and in the magnitudes were found by the test. Also, the cracked wafers had higher damping ratios than the crack-free wa fers. This is expected due to frictional damping introduced within the crack. For the large crack wafers consider ing the second audible mode, 33% of the cracked wafers showed a significant difference in frequency, 67% had a significant difference in peak magnitude, and 33% had a significant difference in damping. Note that only 33% of the large cracked wafe rs had continuous cracks. Therefore, 100% of the wafers with c ontinuous large cracks showed significant differences in all 3 param eters. For the miscella neous wafers, 25% of the cracked wafers had a notable difference in frequency, 75% had a notable difference in peak magnitudes and 50% had a notable difference in damping. The small crack wafers did not show any notable difference between the crackfree wafers and the wafers with cracks for the frequency, magnitude, or damping. Overall, the data showed that the peak magnitude wa s the most sensitive to cracked wafers, followed next in s ensitivity by the damping ratio and the natural frequency.

PAGE 61

47 5.2 Recommendations For future work, it is reco mmended to make the impact hammer automated instead of manually hitting the hammer. This would facilitate a quicker repeatable test process possibly su itable for in-line production use. A non-automated impact test ta kes about 15-30 seconds. If automated this could be reduced to a few seconds. Different crack lengths should be inve stigated to establish a quantitative sensitivity limit for the millim eter size cracks. In this thesis, the crack lengths investigated were less than 8 mm or larger than 38-55 mm. Al so, explore further tests to different crack lo cations could be studied. It is also recommended to develop an endurance test to investigate how many impacts can be applied on the cracked wafer with a critical length of 1 cm before it break. This would represent an endurance “go” or “nogo” test.

PAGE 62

48 References 1. History of solar power, http://www.solarexpert.com/pvbasics2.htlm 2006. 2. S. Ostapenko, Priv ate corresponding, 2006. 3. Y. Gogotsi, C. Baek and F. Kirscht, Raman micro-spectroscopy study of processing– included phase transformations and residual stress in silicon. Semiconductor Science and Technology 14, 936. 1999. 4. M. Yamanda, Quantitative photoelastic measurem ent of residual strain in undoped semi-insulating GaAs. 1985 Appl. Phys. Lett. 47, 365-367 5. A. Belyaev, O. Polupan, S. Os tapenko, D. Hess and J. P. Kalejs, Resonance ultrasonic vibration diagnosti cs of elastic stress in full-size silicon wafers. Semicond. Sci. Technology. 21, 254 (2006). 6. S. R. Best, D.P. Hess, A. Belyaev, S. Ostapenko, J.P.Kalejs, Audible vibration diagnostics of thermo-elastic residual stress in multi-crystalline silicon wafers Appl. Acoustics, Volume 67, issue 6, 2006, p 541-549. 7. A. Byelyayev, Stress Diagnostics and Crack Detection in Full-Size Silicon Wafer Using Resonance Ph.D dissertation, USF, June 2005. 8. A. Belyaev, O. Polupan, W. Dallas, S. Ostapenko and D.Hess, Crack detection and analyses using resonance ultrasonic vibrations in full-size crystalline silicon wafers Appl. Phys. Lett. 88, 111907, 2006. 9. Czochralski process, http://en.wihipedia.org/ wiki/Czochralski_process 2006. 10. Bendat, J. S., and A. G. Piersol: Engineering Application of Correlation and spectral Analysis Wiley, New York. 1980. 11. Bendat, J. S., and A. G. Piersol: Random Data: Analysis and Measurement Procedures, 2nd ed. Wiley, New York. 1986. 12. R. D. Belvins, 2001 Formulas fo r natural frequencies and mode shapes. Florida: Krieger Publishing.

PAGE 63

49 Appendices

PAGE 64

50 Appendix A: Run order Table A.1 Test number for the small crack wafer set Wafer number Test Number 11 1 2 3 12 4 5 6 13 7 8 9 14 10 11 12 15 13 14 15 18 16 17 18 19 19 20 21 20 22 23 24 33 25 26 27 34 28 29 30 35 31 32 33 36 34 35 36 37 37 38 39 38 40 41 42 39 43 44 45 40 46 47 48 41 49 50 51 42 52 53 54 43 55 56 57 44 58 59 60 21 61 62 63 22 64 65 66 23 67 68 69 25 70 71 72 26 73 74 75 27 76 77 78 29 79 80 81 30 82 83 84 31 85 86 87 32 88 89 90 Table A.2 Run order for the small crack wafer set Run order Wafer number Test number Random numbers in ascending order 1 20 24 1 2 11 2 2 3 44 60 7 4 27 76 28 5 29 79 43 6 32 89 47 7 33 25 53 8 21 61 66 9 44 59 69 10 33 26 88 11 14 11 116 12 38 40 125 13 35 31 137

PAGE 65

51 Table A.2 (Continued) 14 27 77 148 15 25 71 156 16 31 86 163 17 41 49 169 18 22 64 185 19 43 56 195 20 36 35 229 21 39 45 231 22 40 46 240 23 35 32 262 24 32 90 264 25 40 48 272 26 43 55 289 27 12 4 302 28 42 53 303 29 15 13 309 30 41 50 313 31 29 81 340 32 18 18 352 33 18 17 375 34 11 3 379 35 34 29 397 36 31 85 411 37 13 9 412 38 21 63 417 39 22 65 423 40 37 38 426 41 26 73 433 42 39 44 492 43 18 16 502 44 42 54 503 45 21 62 505 46 11 1 507 47 20 23 519 48 37 39 531 49 20 22 535 50 36 36 542 51 27 78 558 52 33 27 574 53 26 74 592 54 19 21 597 55 42 52 603 56 34 28 608 57 25 72 614 58 37 37 618 59 38 41 629 60 30 84 662 61 31 87 664 62 23 69 687 63 12 5 710 64 43 57 722 65 22 66 732 66 13 8 737

PAGE 66

52 Table A.2 (Continued) 67 19 19 742 68 12 6 746 69 19 20 750 70 39 43 752 71 44 58 756 72 34 30 784 73 25 70 796 74 40 47 803 75 36 34 813 76 30 83 835 77 23 67 875 78 29 80 883 79 13 7 907 80 15 14 922 81 30 82 930 82 38 42 935 83 41 51 947 84 15 15 958 85 26 75 959 86 35 33 961 87 32 88 966 88 14 12 987 89 23 68 991 90 14 10 998

PAGE 67

53 Appendix B : Frequency response data 0 200 400 600 800 1000 0 0.5 1 Coherence 0 200 400 600 800 1000 0 500 1000 Magnitude (dB/N) 0 200 400 600 800 1000 -200 0 200 Phase (deg)Frequency (Hz) Figure B.1 Frequency response data of crack-free wafer number 29 0 200 400 600 800 1000 0 0.5 1 Coherence 0 200 400 600 800 1000 0 500 1000 Magnitude (dB/N) 0 200 400 600 800 1000 -200 0 200 Phase (deg)Frequency (Hz) Figure B.2 Frequency response data of crack-free wafer number 34

PAGE 68

54 Appendix B (Continued) 0 200 400 600 800 1000 0 0.5 1 Coherence 0 200 400 600 800 1000 0 500 1000 Magnitude (dB/N) 0 200 400 600 800 1000 -200 0 200 Phase (deg)Frequency (Hz) Figure B.3 Frequency response data of crack-free wafer number 38 0 200 400 600 800 1000 0 0.5 1 Coherence 0 200 400 600 800 1000 0 500 1000 Magnitude (dB/N) 0 200 400 600 800 1000 -200 0 200 Phase (deg)Frequency (Hz) Figure B.4 Frequency response data of crack-free wafer number 47

PAGE 69

55 Appendix B (Continued) 0 200 400 600 800 1000 0 0.5 1 Coherence 0 200 400 600 800 1000 0 500 1000 Magnitude (dB/N) 0 200 400 600 800 1000 -200 0 200 Phase (deg)Frequency (Hz) Figure B.5 Frequency response dat a of cracked wafer number 39 0 200 400 600 800 1000 0 0.5 1 Coherence 0 200 400 600 800 1000 0 500 Magnitude (dB/N) 0 200 400 600 800 1000 -200 0 200 Phase (deg)Frequency (Hz) Figure B.6 Frequency response data of cracked wafer number 31 (Test 1)

PAGE 70

56 Appendix B (Continued) 0 200 400 600 800 1000 0 0.5 1 Coherence 0 200 400 600 800 1000 0 500 Magnitude (dB/N) 0 200 400 600 800 1000 -200 0 200 Phase (deg)Frequency (Hz) Figure B.7 Frequency response data of cracked wafer number 31 (Test 2) 0 200 400 600 800 1000 0 0.5 1 Coherence 0 200 400 600 800 1000 0 500 Magnitude (dB/N) 0 200 400 600 800 1000 -200 0 200 Phase (deg)Frequency (Hz) Figure B.8 Frequency response data of cracked wafer number 31 (Test 3)

PAGE 71

57 Appendix B (Continued) 0 200 400 600 800 1000 0 0.5 1 Coherence 0 200 400 600 800 1000 0 500 Magnitude (dB/N) 0 200 400 600 800 1000 -200 0 200 Phase (deg)Frequency (Hz) Figure B.9 Frequency response dat a of cracked wafer number 35 0 200 400 600 800 1000 0 0.5 1 Coherence 0 200 400 600 800 1000 0 500 1000 Magnitude (dB/N) 0 200 400 600 800 1000 -200 0 200 Phase (deg)Frequency (Hz) Figure B.10 Frequency response data of cracked wafer number 48

PAGE 72

58 Appendix B (Continued) 0 200 400 600 800 1000 0 0.5 1 Coherence 0 200 400 600 800 1000 0 500 1000 Magnitude (dB/N) 0 200 400 600 800 1000 -200 0 200 Phase (deg)Frequency (Hz) Figure B.11 Frequency response dat a of cracked wafer number 32 0 200 400 600 800 1000 0 0.5 1 Coherence 0 200 400 600 800 1000 0 500 1000 Magnitude (dB/N) 0 200 400 600 800 1000 -200 0 200 Phase (deg)Frequency (Hz) Figure B.12 Frequency response data of cracked wafer number 40

PAGE 73

59 Appendix B (Continued) 0 200 400 600 800 1000 0 0.5 1 Coherence 0 200 400 600 800 1000 0 500 1000 Magnitude (dB/N) 0 200 400 600 800 1000 -200 0 200 Phase (deg)Frequency (Hz) Figure B.13 Frequency response data of cracked wafer number 36 0 200 400 600 800 1000 0 0.5 1 Coherence 0 200 400 600 800 1000 0 500 1000 Magnitude (dB/N) 0 200 400 600 800 1000 -200 0 200 Phase (deg)Frequency (Hz) Figure B.14 Frequency response data of cracked wafer number 27 (Test 1)

PAGE 74

60 Appendix B (Continued) 0 200 400 600 800 1000 0 0.5 1 Coherence 0 200 400 600 800 1000 0 500 1000 Magnitude (dB/N) 0 200 400 600 800 1000 -200 0 200 Phase (deg)Frequency (Hz) Figure B.15 Frequency response data of cracked wafer number 27 (Test 2) 0 200 400 600 800 1000 0 0.5 1 Coherence 0 200 400 600 800 1000 0 500 1000 Magnitude (dB/N) 0 200 400 600 800 1000 -200 0 200 Phase (deg)Frequency (Hz) Figure B.16 Frequency response data of cracked wafer number 27 (Test 3)

PAGE 75

61 Appendix B (Continued) 0 200 400 600 800 1000 0 0.5 1 Coherence 0 200 400 600 800 1000 0 1000 2000 Magnitude (dB/N) 0 200 400 600 800 1000 -200 0 200 Phase (deg)Frequency (Hz) Figure B.17 Frequency response data of cracked wafer number 8 0 200 400 600 800 1000 0 0.5 1 Coherence 0 200 400 600 800 1000 0 500 1000 Magnitude (dB/N) 0 200 400 600 800 1000 -200 0 200 Phase (deg)Frequency (Hz) Figure B.18 Frequency response data of cracked wafer number 6

PAGE 76

62 Appendix B (Continued) 0 200 400 600 800 1000 0 0.5 1 Coherence 0 200 400 600 800 1000 0 500 1000 Magnitude (dB/N) 0 200 400 600 800 1000 -200 0 200 Phase (deg)Frequency (Hz) Figure B.19 Frequency response data of cracked wafer number 33 0 200 400 600 800 1000 0 0.5 1 Coherence 0 200 400 600 800 1000 0 500 1000 Magnitude (dB/N) 0 200 400 600 800 1000 -200 0 200 Phase (deg)Frequency (Hz) Figure B.20 Frequency response dat a of cracked wafer number 41

PAGE 77

63 Appendix C: Data from the large crack wafer set Table C.1 First mode natural fr equency for large crack wafer set Specimen number Test 1 [Hz] Test 2 [Hz] Test 3 [Hz] Mean [Hz] 29 418.1 418.1 418.1 418.1 34 418.4 418.8 419.1 418.8 38 419.1 419.1 419.4 419.2 47 419.7 420.0 419.7 419.8 39 419.4 419.4 419.7 419.5 31 402.5 402.8 398.4 401.3 35 390.3 390.3 389.4 390.0 48 415.0 414.4 413.8 414.4 32 418.1 418.4 418.4 418.3 40 420.9 420.9 420.9 420.9 36 419.1 419.1 419.1 419.1 27 415.9 415.9 405.3 412.4 8 416.6 416.9 416.9 416.8 6 416.6 416.3 416.6 416.5 33 413.1 412.8 413.4 413.1 41 419.1 419.4 419.4 419.3 370 380 390 400 410 420 430 293438473931354832403627863341 Wafer NumberFrequency [Hz] Test 1 Test 2 Test 3 Figure C.1 First mode natural fr equency for large crack wafer set

PAGE 78

64 Appendix C (Continued) Table C.2 First mode normalized frequency for large crack wafer set Specimen number Test 1 [Hz] Test 2 [Hz] Test 3 [Hz] Mean [Hz] 29 0.0785 0.0785 0.0785 0.0785 34 0.0784 0.0785 0.0785 0.0785 38 0.0784 0.0784 0.0785 0.0784 47 0.0783 0.0783 0.0783 0.0783 39 0.0784 0.0784 0.0784 0.0784 31 0.0754 0.0754 0.0746 0.0752 35 0.0733 0.0733 0.0731 0.0732 48 0.0776 0.0775 0.0773 0.0775 32 0.0784 0.0784 0.0784 0.0784 40 0.0783 0.0783 0.0783 0.0783 36 0.0782 0.0782 0.0782 0.0782 27 0.0780 0.0780 0.0760 0.0773 8 0.0784 0.0784 0.0784 0.0784 6 0.0781 0.0780 0.0781 0.0781 33 0.0773 0.0772 0.0774 0.0773 41 0.0785 0.0785 0.0785 0.0785 0.069 0.071 0.073 0.075 0.077 0.079 0.081 293438473931354832403627863341 Wafer numberNormalized frequency [Hz/ m3/2] Test 1 Test 2 Test 3 Figure C.2 First mode normalized frequency for large crack wafer set

PAGE 79

65 Appendix C (Continued) Table C.3 Second mode natural fr equency for large crack wafer set Specimen number Test 1 [Hz] Test 2 [Hz] Test 3 [Hz] Mean [Hz] 29 591.6 591.6 591.6 591.6 34 592.2 592.2 592.2 592.2 38 592.8 593.1 593.1 593.0 47 594.1 594.1 594.1 594.1 39 592.8 592.8 592.8 592.8 31 571.9 571.9 569.7 571.1 35 565.9 565.9 566.3 566.0 48 584.4 583.4 582.8 583.5 32 590.6 590.9 590.6 590.7 40 592.5 592.8 592.8 592.7 36 591.9 591.9 591.9 591.9 27 587.8 587.8 570.9 582.2 8 590.3 590.6 590.6 590.5 6 592.2 592.2 592.2 592.2 33 591.9 591.6 591.9 591.8 41 592.8 592.8 592.8 592.8 550 560 570 580 590 600 293438473931354832403627863341 Wafer Number Frequency [Hz] Test 1 Test 2 Test 3 Figure C.3 Second mode natural fr equency for large crack wafer set

PAGE 80

66 Appendix C (Continued) Table C.4 Second mode normalized frequency for large crack wafer set Specimen number Test 1 [Hz] Test 2 [Hz] Test 3 [Hz] Mean [Hz] 29 0.1110 0.1110 0.1110 0.1110 34 0.1110 0.1110 0.1110 0.1110 38 0.1109 0.1110 0.1110 0.1109 47 0.1108 0.1108 0.1108 0.1108 39 0.1108 0.1108 0.1108 0.1108 31 0.1071 0.1071 0.1067 0.1070 35 0.1062 0.1062 0.1063 0.1063 48 0.1092 0.1091 0.1089 0.1091 32 0.1107 0.1108 0.1107 0.1107 40 0.1103 0.1103 0.1103 0.1103 36 0.1104 0.1104 0.1104 0.1104 27 0.1102 0.1102 0.1071 0.1092 8 0.1111 0.1111 0.1111 0.1111 6 0.1110 0.1110 0.1110 0.1110 33 0.1107 0.1107 0.1107 0.1107 41 0.1110 0.1110 0.1110 0.1110 0.104 0.106 0.108 0.110 0.112 0.114 293438473931354832403627863341 Wafer numberNormalized frequency [Hz/ m3/2] Test 1 Test 2 Test 3 Figure C.4 Second mode normalized frequency for large crack wafer set

PAGE 81

67 Appendix C (Continued) Table C.5 Third mode natural fr equency for large crack wafer set Specimen number Test 1 [Hz] Test 2 [Hz] Test 3 [Hz] Mean [Hz] 29 839.1 839.1 839.4 839.2 34 840.3 840.3 840.3 840.3 38 840.9 841.3 841.3 841.1 47 842.8 842.8 842.5 842.7 39 842.2 842.2 842.2 842.2 31 828.8 828.8 815.3 824.3 35 813.1 812.8 812.5 812.8 48 835.0 834.7 834.1 834.6 32 838.8 838.8 838.8 838.8 40 843.1 843.4 843.4 843.3 36 841.9 841.9 841.9 841.9 27 834.7 835.0 816.6 828.8 8 838.4 838.8 838.8 838.6 6 841.6 841.3 841.6 841.5 33 841.9 841.9 841.9 841.9 41 841.3 841.6 841.6 841.5 790 800 810 820 830 840 850 293438473931354832403627863341 Wafer NumberFrequency [Hz] Test 1 Test 2 Test 3 Figure C.5 Third mode natural fr equency for large crack wafer set

PAGE 82

68 Appendix C (Continued) Table C.6 Third mode normalized frequency for large crack wafer set Specimen number Test 1 [Hz] Test 2 [Hz] Test 3 [Hz] Mean [Hz] 29 0.1575 0.1575 0.1576 0.1575 34 0.1575 0.1575 0.1575 0.1575 38 0.1573 0.1574 0.1574 0.1574 47 0.1572 0.1572 0.1572 0.1572 39 0.1574 0.1574 0.1574 0.1574 31 0.1552 0.1552 0.1527 0.1544 35 0.1526 0.1526 0.1525 0.1526 48 0.1561 0.1560 0.1559 0.1560 32 0.1572 0.1572 0.1572 0.1572 40 0.1569 0.1570 0.1570 0.1569 36 0.1570 0.1570 0.1570 0.1570 27 0.1565 0.1566 0.1531 0.1554 8 0.1578 0.1578 0.1578 0.1578 6 0.1578 0.1577 0.1578 0.1577 33 0.1575 0.1575 0.1575 0.1575 41 0.1575 0.1576 0.1576 0.1576 0.152 0.154 0.156 0.158 0.160 0.162 293438473931354832403627863341 Wafer number Normalized frequency [Hz/ m3/2] Test 1 Test 2 Test 3 Figure C.6 Third mode normalized frequency for large crack wafer set

PAGE 83

69 Appendix C (Continued) Table C.7 Fourth mode natural frequency for large crack wafer set Specimen number Test 1 [Hz] Test 2 [Hz] Test 3 [Hz] Mean [Hz] 29 960.6 960.6 960.6 960.6 34 961.9 961.9 961.9 961.9 38 962.8 962.8 962.8 962.8 47 964.4 964.7 964.7 964.6 39 961.6 961.6 961.9 961.7 31 935.0 935.0 935.0 935.0 35 935.0 934.4 934.7 934.7 48 950.6 949.1 948.1 949.3 32 958.4 958.8 959.1 958.8 40 961.9 961.9 961.9 961.9 36 960.9 960.9 960.9 960.9 27 954.1 954.1 930.6 946.3 8 958.8 959.1 959.1 959.0 6 961.3 961.6 961.3 961.4 33 960.3 960.3 960.3 960.3 41 962.2 962.2 962.2 962.2 910 920 930 940 950 960 970 293438473931354832403627863341 Wafer NumberFrequency [Hz] Test 1 Test 2 Test 3 Figure C.7 Fourth mode natural frequency for large crack wafer set

PAGE 84

70 Appendix C (Continued) Table C.8 Fourth mode normalized frequency for large crack wafer set Specimen number Test 1 [Hz] Test 2 [Hz] Test 3 [Hz] Mean [Hz] 29 0.1803 0.1803 0.1803 0.1803 34 0.1803 0.1803 0.1803 0.1803 38 0.1801 0.1801 0.1801 0.1801 47 0.1799 0.1799 0.1799 0.1799 39 0.1797 0.1797 0.1797 0.1797 31 0.1751 0.1751 0.1751 0.1751 35 0.1755 0.1754 0.1755 0.1755 48 0.1777 0.1774 0.1772 0.1775 32 0.1797 0.1797 0.1798 0.1797 40 0.1790 0.1790 0.1790 0.1790 36 0.1792 0.1792 0.1792 0.1792 27 0.1789 0.1789 0.1745 0.1775 8 0.1804 0.1805 0.1805 0.1805 6 0.1802 0.1803 0.1802 0.1802 33 0.1797 0.1797 0.1797 0.1797 41 0.1802 0.1802 0.1802 0.1802 0.170 0.172 0.174 0.176 0.178 0.180 0.182 0.184 0.186 293438473931354832403627863341 Wafer numberNormalized frequency [Hz/ m3/2] Test 1 Test 2 Test 3 Figure C.8 Fourth mode normalized frequency for large crack wafer set

PAGE 85

71 Appendix C (Continued) Table C.9 First mode peak magn itude for large crack wafer set Specimen number Test 1 [dB/N] Test 2 [dB/N] Test 3 [dB/N] Mean [dB/N] 29 720.3 734.2 743.9 732.8 34 729.4 642.6 721.8 697.9 38 741.0 735.1 744.2 740.1 47 663.7 718.7 741.2 707.9 39 703.4 666.8 655.6 675.3 31 265.0 266.9 252.4 261.4 35 175.9 180.8 180.8 179.2 48 476.7 445.8 426.5 449.7 32 536.6 629.7 649.4 605.2 40 600.4 594.7 595.9 597.0 36 646.4 627.8 614.7 629.6 27 575.9 617.0 314.5 502.5 8 725.8 620.7 687.4 678.0 6 499.1 491.0 532.0 507.4 33 504.8 484.6 501.5 496.9 41 725.3 725.8 742.3 731.1 0 100 200 300 400 500 600 700 800 293438473931354832403627863341 Wafer number [non-dim]Magnitude [dB/N] Figure C.9 First mode peak magni tude for large crack wafer set

PAGE 86

72 Appendix C (Continued) Table C.10 Second mode peak magni tude for large crack wafer set Specimen number Test 1 [dB/N] Test 2 [dB/N] Test 3 [dB/N] Mean [dB/N] 29 907.7 992.4 978.4 959.5 34 976.4 894.9 960.0 943.8 38 1025.3 944.5 935.7 968.5 47 899.9 922.8 990.8 937.8 39 848.5 722.9 754.3 775.2 31 487.0 468.5 487.1 480.8 35 398.1 397.3 402.7 399.4 48 801.1 730.8 666.7 732.9 32 805.4 736.4 735.7 759.2 40 674.5 709.6 684.8 689.7 36 711.5 718.0 691.6 707.0 27 949.1 967.9 591.4 836.1 8 1005.9 860.8 919.0 928.6 6 923.2 979.0 946.8 949.7 33 887.6 835.7 870.5 864.6 41 903.0 824.1 826.0 851.0 300 400 500 600 700 800 900 1000 1100 293438473931354832403627863341 Wafer numberMagnitude [dB/N] Test 1 Test 2 Test 3 Figure C.10 Second mode peak magnitude for large crack wafer set

PAGE 87

73 Appendix C (Continued) Table C.11 Third mode peak magnitude for large crack wafer set Specimen number Test 1 [dB/N] Test 2 [dB/N] Test 3 [dB/N] Mean [dB/N] 29 818.8 859.1 799.1 825.6 34 845.6 796.8 821.6 821.3 38 968.4 879.3 865.2 904.3 47 766.3 771.5 827.7 788.5 39 769.4 715.2 687.6 724.1 31 828.8 271.4 304.9 468.4 35 403.3 405.1 394.0 400.8 48 685.3 581.5 517.1 594.6 32 789.0 757.5 774.9 773.8 40 924.9 845.2 816.5 862.2 36 783.7 774.6 755.6 771.3 27 972.8 969.4 379.6 773.9 8 873.5 688.6 734.6 765.6 6 684.1 700.0 685.0 689.7 33 658.1 637.4 638.9 644.8 41 813.0 780.2 816.5 803.2 0 200 400 600 800 1000 1200 293438473931354832403627863341 Wafer number [non-dim]Magnitude [dB/N] Test 1 Test 2 Test 3 Figure C.11 Third mode magnitu de for large crack wafer set

PAGE 88

74 Appendix C (Continued) Table C.12 Fourth mode peak m agnitude for large crack wafer set Specimen number Test 1 [dB/N] Test 2 [dB/N] Test 3 [dB/N] Mean [dB/N] 29 289.8 319.2 299.7 302.9 34 292.2 318.4 297.6 302.8 38 368.9 291.3 308.8 323.0 47 288.0 289.8 275.5 284.5 39 245.7 232.1 219.4 232.4 31 174.9 165.3 202.8 181.0 35 181.9 187.2 193.3 187.5 48 224.1 219.6 236.1 226.6 32 339.9 294.6 276.5 303.7 40 298.2 261.9 279.8 280.0 36 278.7 260.0 282.9 273.9 27 195.2 202.1 268.7 222.0 8 320.2 291.5 252.8 288.1 6 308.7 305.3 308.0 307.3 33 369.1 341.8 345.6 352.2 41 257.9 248.4 248.0 251.4 100 150 200 250 300 350 400 293438473931354832403627863341 Wafer numberMagnitude [dB/N] Test 1 Test 2 Test 3 Figure C.12 Fourth mode peak m agnitude for large crack wafer set

PAGE 89

75 Appendix C (Continued) Table C.13 First mode damping ratio for large crack wafer set Specimen number Test 1 [non-dim] Test 2 [non-dim] Test 3 [non-dim] Mean [non-dim] 29 0.0025 0.0024 0.0024 0.0024 34 0.0026 0.0029 0.0025 0.0027 38 0.0026 0.0024 0.0025 0.0025 47 0.0026 0.0024 0.0023 0.0024 39 0.0024 0.0027 0.0025 0.0025 31 0.0086 0.0086 0.0087 0.0086 35 0.0091 0.0094 0.0091 0.0092 48 0.0034 0.0040 0.0040 0.0038 32 0.0033 0.0028 0.0026 0.0029 40 0.0025 0.0027 0.0025 0.0026 36 0.0026 0.0027 0.0027 0.0027 27 0.0026 0.0025 0.0054 0.0035 8 0.0024 0.0027 0.0026 0.0026 6 0.0023 0.0024 0.0024 0.0024 33 0.0031 0.0033 0.0033 0.0032 41 0.0024 0.0025 0.0023 0.0024 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.010 293438473931354832403627863341 Wafer NumberDamping Ratio [non-dim ] Test 1 Test 2 Test 3 Figure C.13 First mode damping ratio for large crack wafer set

PAGE 90

76 Appendix C (Continued) Table C.14 Second mode damping ra tio for large crack wafer set Specimen number Test 1 [non-dim] Test 2 [non-dim] Test 3 [non-dim] Mean [non-dim] 29 0.0015 0.0014 0.0016 0.0015 34 0.0015 0.0016 0.0015 0.0016 38 0.0015 0.0015 0.0016 0.0015 47 0.0015 0.0016 0.0014 0.0015 39 0.0016 0.0018 0.0016 0.0017 31 0.0049 0.0054 0.0052 0.0052 35 0.0044 0.0047 0.0046 0.0046 48 0.0032 0.0033 0.0036 0.0034 32 0.0015 0.0018 0.0020 0.0018 40 0.0020 0.0019 0.0019 0.0019 36 0.0019 0.0021 0.0021 0.0020 27 0.0020 0.0019 0.0036 0.0025 8 0.0015 0.0017 0.0016 0.0016 6 0.0016 0.0015 0.0016 0.0016 33 0.0017 0.0018 0.0018 0.0017 41 0.0015 0.0016 0.0015 0.0015 0.001 0.002 0.003 0.004 0.005 0.006 293438473931354832403627863341 Wafer Number Damping Ratio [non-dim ] Test 1 Test 2 Test 3 Figure C.14 Second mode damping ra tio for large crack wafer set

PAGE 91

77 Appendix C (Continued) Table C.15 Third mode damping ratio for large crack wafer set Specimen number Test 1 [non-dim] Test 2 [non-dim] Test 3 [non-dim] Mean [non-dim] 29 0.0013 0.0013 0.0013 0.0013 34 0.0013 0.0014 0.0013 0.0013 38 0.0013 0.0013 0.0013 0.0013 47 0.0014 0.0013 0.0012 0.0013 39 0.0013 0.0014 0.0013 0.0013 31 0.0030 0.0037 0.0021 0.0029 35 0.0034 0.0034 0.0034 0.0034 48 0.0018 0.0019 0.0022 0.0020 32 0.0014 0.0014 0.0014 0.0014 40 0.0013 0.0012 0.0013 0.0013 36 0.0013 0.0013 0.0014 0.0013 27 0.0014 0.0014 0.0016 0.0014 8 0.0013 0.0015 0.0013 0.0014 6 0.0014 0.0015 0.0013 0.0014 33 0.0012 0.0013 0.0012 0.0013 41 0.0013 0.0013 0.0012 0.0013 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035 0.0040 293438473931354832403627863341 Wafer Number [non-dim]Damping Ratio [non-dim ] Test 1 Test 2 Test 3 Figure C.15 Third mode damping ratio for large crack wafer set

PAGE 92

78 Appendix C (Continued) Table C.16 Fourth mode damping ratio for large crack wafer set Specimen number Test 1 [non-dim] Test 2 [non-dim] Test 3 [non-dim] Mean [non-dim] 29 0.0014 0.0014 0.0013 0.0014 34 0.0014 0.0013 0.0014 0.0014 38 0.0014 0.0013 0.0015 0.0014 47 0.0013 0.0013 0.0014 0.0013 39 0.0015 0.0016 0.0015 0.0015 31 0.0037 0.0038 0.0034 0.0036 35 0.0032 0.0033 0.0033 0.0033 48 0.0029 0.0029 0.0028 0.0029 32 0.0016 0.0017 0.0017 0.0016 40 0.0020 0.0020 0.0020 0.0020 36 0.0019 0.0019 0.0019 0.0019 27 0.0019 0.0020 0.0030 0.0023 8 0.0013 0.0013 0.0013 0.0013 6 0.0017 0.0018 0.0018 0.0018 33 0.0016 0.0018 0.0018 0.0017 41 0.0014 0.0013 0.0014 0.0014 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035 0.0040 0.0045 293438473931354832403627863341 Wafer NumberDamping Ratio [non-dim ] Test 1 Test 2 Test 3 Figure C.16 Fourth mode damping ratio for large crack wafer set

PAGE 93

79 Appendix D: Data from the miscellaneous wafer set Table D.1 First mode natural fr equency for miscellaneous wafer set Wafer number Test 1 [Hz] Test 2 [Hz] Average [Hz] 20 418.8 418.8 418.8 42 402.2 401.3 401.7 49 402.2 402.2 402.2 2 405.0 405.0 405.0 11 417.2 417.5 417.3 23 381.9 373.1 377.5 25 420.0 420.0 420.0 45 417.5 417.8 417.7 46 416.3 417.2 416.7 7 401.6 401.9 401.7 47 402.2 401.9 402.0 8 391.9 390.3 391.1 370 380 390 400 410 420 430 204249211232545467478 Wafer NumberFrequency [Hz] Test 1 Test 2 Figure D.1 First mode natural fr equency for miscellaneous wafer set

PAGE 94

80 Appendix D (Continued) Table D.2 First mode normalized frequency for miscellaneous wafer set Wafer number Test 1 [Hz/ m^(3/2)] Test 2 [Hz/ m^(3/2)] Average [Hz/ m^(3/2)] 20 0.0784 0.0784 0.0784 42 0.0802 0.0800 0.0801 49 0.0799 0.0799 0.0799 2 0.0799 0.0799 0.0799 11 0.0783 0.0783 0.0783 23 0.0714 0.0698 0.0706 25 0.0787 0.0787 0.0787 45 0.0780 0.0781 0.0780 46 0.0779 0.0781 0.0780 7 0.0797 0.0797 0.0797 47 0.0801 0.0800 0.0800 8 0.0776 0.0773 0.0775 0.069 0.071 0.073 0.075 0.077 0.079 0.081 204249211232545467478 Wafer numberNormalized frequency [Hz/ m3/2] Test 1 Test 2 Figure D.2 First mode normalized frequency for miscellaneous wafer set

PAGE 95

81 Appendix D (Continued) Table D.3 Second mode natural frequency for miscellaneous wafer set Wafer number Test 1 [Hz] Test 2 [Hz] Average [Hz] 20 592.8 592.8 592.8 42 569.1 569.1 569.1 49 570.3 570.3 570.3 2 573.1 573.4 573.3 11 591.9 592.2 592.0 23 559.7 557.5 558.6 25 591.3 592.2 591.7 45 593.1 593.1 593.1 46 592.2 592.5 592.3 7 569.1 569.1 569.1 47 566.9 565.0 565.9 8 555.6 554.7 555.2 550 560 570 580 590 600 204249211232545467478 Wafer Number Frequency [Hz] Test 1 Test 2 Figure D.3 Second mode natural frequency for miscellaneous wafer set

PAGE 96

82 Appendix D (Continued) Table D.4 Second mode normalized frequency for miscellaneous wafer set Wafer number Test 1 [Hz/ m^(3/2)] Test 2 [Hz/ m^(3/2)] Average [Hz/ m^(3/2)] 20 0.1110 0.1110 0.1110 42 0.1135 0.1135 0.1135 49 0.1133 0.1133 0.1133 2 0.1130 0.1131 0.1130 11 0.1111 0.1111 0.1111 23 0.1047 0.1043 0.1045 25 0.1107 0.1109 0.1108 45 0.1108 0.1108 0.1108 46 0.1108 0.1109 0.1109 7 0.1129 0.1129 0.1129 47 0.1129 0.1125 0.1127 8 0.1100 0.1099 0.1100 0.104 0.106 0.108 0.110 0.112 0.114 204249211232545467478 Wafer numberNormalized frequency [Hz/ m3/2] Test 1 Test 2 Figure D.4 Second mode normalized frequency for miscellaneous wafer set

PAGE 97

83 Appendix D (Continued) Table D.5 Third mode natural fr equency for miscellaneous wafer set Wafer number Test 1 [Hz] Test 2 [Hz] Average [Hz] 20 841.3 840.9 841.1 42 805.9 806.3 806.1 49 807.8 807.8 807.8 2 811.9 811.9 811.9 11 839.7 839.7 839.7 23 820.9 820.6 820.8 25 842.5 842.2 842.3 45 843.1 843.1 843.1 46 841.9 842.2 842.0 7 805.9 805.9 805.9 47 805.6 805.3 805.5 8 801.3 800.0 800.6 790 800 810 820 830 840 850 204249211232545467478 Wafer NumberFrequency [Hz] Test 1 Test 2 Figure D.5 Third mode natural frequency for miscellaneous wafer set

PAGE 98

84 Appendix D (Continued) Table D.6 Third mode normalized frequency for miscellaneous wafer set Wafer number Test 1 [Hz/ m^(3/2)] Test 2 [Hz/ m^(3/2)] Average [Hz/ m^(3/2)] 20 0.1575 0.1574 0.1574 42 0.1607 0.1608 0.1608 49 0.1604 0.1604 0.1604 2 0.1601 0.1601 0.1601 11 0.1576 0.1576 0.1576 23 0.1535 0.1535 0.1535 25 0.1578 0.1578 0.1578 45 0.1575 0.1575 0.1575 46 0.1576 0.1576 0.1576 7 0.1599 0.1599 0.1599 47 0.1604 0.1603 0.1604 8 0.1587 0.1584 0.1586 0.152 0.154 0.156 0.158 0.160 0.162 204249211232545467478 Wafer numberNormalized frequency [Hz/ m3/2] Test 1 Test 2 Figure D.6 Third mode normalized frequency for miscellaneous wafer set

PAGE 99

85 Appendix D (Continued) Table D.7 Fourth mode natural frequency for miscellaneous wafer set Wafer number Test 1 [Hz] Test 2 [Hz] Average [Hz] 20 962.8 962.8 962.8 42 925.0 924.1 924.5 49 926.3 925.9 926.1 2 931.6 931.6 931.6 11 960.9 961.3 961.1 23 920.0 918.1 919.1 25 959.7 959.7 959.7 45 959.7 959.7 959.7 46 958.8 959.4 959.1 7 923.4 923.1 923.3 47 919.7 916.6 918.1 8 923.1 922.2 922.7 910 920 930 940 950 960 970 204249211232545467478 Wafer NumberFrequency [Hz] Test 1 Test 2 Figure D.7 Fourth mode natural frequency for miscellaneous wafer set

PAGE 100

86 Appendix D (Continued) Table D.8 Fourth mode normalized frequency for miscellaneous wafer set Wafer number Test 1 [Hz/ m^(3/2)] Test 2 [Hz/ m^(3/2)] Average [Hz/ m^(3/2)] 20 0.1802 0.1802 0.1802 42 0.1845 0.1843 0.1844 49 0.1840 0.1839 0.1839 2 0.1837 0.1837 0.1837 11 0.1803 0.1804 0.1803 23 0.1721 0.1717 0.1719 25 0.1798 0.1798 0.1798 45 0.1793 0.1793 0.1793 46 0.1794 0.1796 0.1795 7 0.1832 0.1831 0.1831 47 0.1831 0.1825 0.1828 8 0.1828 0.1826 0.1827 0.170 0.172 0.174 0.176 0.178 0.180 0.182 0.184 0.186 204249211232545467478 Wafer numberNormalized frequency [Hz/ m3/2] Test 1 Test 2 Figure D.8 Fourth mode normaliz ed frequency for miscellaneous wafer set

PAGE 101

87 Appendix D (Continued) Table D.9 First mode peak magn itude for miscellaneous wafer set Wafer number Test 1 [dB/N] Test 2 [dB/N] Average [dB/N] 20 606.7 636.5 621.6 42 676.2 489.9 583.0 49 464.0 425.1 444.6 2 693.8 691.0 692.4 11 671.3 601.0 636.2 23 180.8 189.0 184.9 25 388.2 585.4 486.8 45 779.8 704.3 742.1 46 600.4 620.0 610.2 7 605.3 652.4 628.9 47 480.2 394.2 437.2 8 316.4 279.3 297.8 0 100 200 300 400 500 600 700 800 204249211232545467478 Wafer numberMagnitude [dB/N] Test 1 Test 2 Figure D.9 First mode peak magni tude for miscellaneous wafer set

PAGE 102

88 Appendix D (Continued) Table D.10 Second mode peak m agnitude for miscellaneous wafer set Wafer number Test 1 [dB/N] Test 2 [dB/N] Average [dB/N] 20 930.1 964.9 947.5 42 949.3 927.2 938.2 49 976.3 944.2 960.3 2 959.9 1009.6 984.7 11 940.0 863.3 901.7 23 522.8 638.1 580.5 25 386.5 370.7 378.6 45 836.3 785.2 810.7 46 692.1 624.5 658.3 7 1027.7 995.9 1011.8 47 701.4 543.3 622.3 8 555.6 590.2 572.9 300 400 500 600 700 800 900 1000 1100 204249211232545467478 Wafer numberMagnitude [dB/N] Test 1 Test 2 Figure D.10 Second mode peak magni tude for miscellaneous wafer set

PAGE 103

89 Appendix D (Continued) Table D.11 Third mode peak m agnitude for miscellaneous wafer set Wafer number Test 1 [dB/N] Test 2 [dB/N] Average [dB/N] 20 784.9 901.4 843.1 42 528.9 700.6 614.7 49 391.3 386.8 389.1 2 512.2 509.6 510.9 11 827.1 756.8 792.0 23 180.6 171.6 176.1 25 506.2 651.2 578.7 45 783.1 756.3 769.7 46 796.9 872.0 834.4 7 566.3 644.4 605.4 47 675.0 619.1 647.1 8 490.0 421.2 455.6 0 200 400 600 800 1000 1200 204249211232545467478 Wafer number [non-dim]Magnitude [dB/N] Test 1 Test 2 Figure D.11 Third mode peak magni tude for miscellaneous wafer set

PAGE 104

90 Appendix D (Continued) Table D.12 Fourth mode peak m agnitude for miscellaneous wafer set Wafer number Test 1 [dB/N] Test 2 [dB/N] Average [dB/N] 20 314.8 346.6 330.7 42 219.3 294.0 256.6 49 200.3 253.7 227.0 2 151.9 164.9 158.4 11 280.1 278.8 279.4 23 368.2 390.1 379.2 25 218.5 224.8 221.7 45 209.6 228.4 219.0 46 253.5 242.7 248.1 7 130.6 155.5 143.0 47 198.0 135.9 167.0 8 159.0 167.8 163.4 100 150 200 250 300 350 400 204249211232545467478 Wafer number [non-dim]Magnitude [dB/N] Test 1 Test 2 Figure D.12 Fourth mode peak m agnitude for miscellaneous wafer set

PAGE 105

91 Appendix D (Continued) Table D.13 First mode damping ratio for miscellaneous wafer set Wafer number Test 1 [non-dim] Test 2 [non-dim] Average [non-dim] 20 0.0029 0.0028 0.0028 42 0.0028 0.0041 0.0034 49 0.0041 0.0044 0.0043 2 0.0024 0.0025 0.0024 11 0.0026 0.0023 0.0025 23 0.0073 0.0083 0.0078 25 0.0037 0.0027 0.0032 45 0.0024 0.0025 0.0025 46 0.0017 0.0024 0.0021 7 0.0025 0.0023 0.0024 47 0.0029 0.0032 0.0031 8 0.0050 0.0059 0.0054 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.010 204249211232545467478 Wafer NumberDamping Ratio [non-dim ] Test 1 Test 2 Figure D.13 First mode damping ratio for miscellaneous wafer set

PAGE 106

92 Appendix D (Continued) Table D.14 Second mode damping ra tio for miscellaneous wafer set Wafer number Test 1 [non-dim] Test 2 [non-dim] Average [non-dim] 20 0.0015 0.0015 0.0015 42 0.0016 0.0016 0.0016 49 0.0016 0.0015 0.0015 2 0.0015 0.0014 0.0014 11 0.0015 0.0015 0.0015 23 0.0039 0.0025 0.0032 25 0.0034 0.0041 0.0037 45 0.0017 0.0015 0.0016 46 0.0017 0.0017 0.0017 7 0.0016 0.0015 0.0015 47 0.0023 0.0030 0.0027 8 0.0034 0.0038 0.0036 0.001 0.002 0.003 0.004 0.005 0.006 204249211232545467478 Wafer NumberDamping Ratio [non-dim ] Test 1 Test 2 Figure D.14 Second mode damping ratio for miscellaneous wafer set

PAGE 107

93 Appendix D (Continued) Table D.15 Third mode damping ratio for miscellaneous wafer set Wafer number Test 1 [non-dim] Test 2 [non-dim] Average [non-dim] 20 0.0013 0.0014 0.0013 42 0.0014 0.0016 0.0015 49 0.0016 0.0017 0.0017 2 0.0014 0.0013 0.0013 11 0.0013 0.0012 0.0013 23 0.0024 0.0020 0.0022 25 0.0018 0.0015 0.0016 45 0.0013 0.0013 0.0013 46 0.0014 0.0013 0.0013 7 0.0014 0.0014 0.0014 47 0.0015 0.0015 0.0015 8 0.0017 0.0019 0.0018 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035 0.0040 204249211232545467478 Wafer NumberDamping Ratio [non-dim ] Test 1 Test 2 Figure D.15 Third mode damping ratio for miscellaneous wafer set

PAGE 108

94 Appendix D (Continued) Table D.16 Fourth mode damping ratio for miscellaneous wafer set Wafer number Test 1 [non-dim] Test 2 [non-dim] Average [non-dim] 20 0.0012 0.0013 0.0013 42 0.0018 0.0017 0.0018 49 0.0019 0.0017 0.0018 2 0.0019 0.0018 0.0019 11 0.0013 0.0013 0.0013 23 0.0024 0.0018 0.0021 25 0.0021 0.0021 0.0021 45 0.0017 0.0016 0.0017 46 0.0015 0.0015 0.0015 7 0.0031 0.0035 0.0033 47 0.0026 0.0040 0.0033 8 0.0018 0.0018 0.0018 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035 0.0040 0.0045204249211232545467478Wafer NumberDamping Ratio [non-dim ] Test 1 Test 2 Figure D.16 Fourth mode damping ratio for miscellaneous wafer set

PAGE 109

95 Appendix E: Data from the small crack wafer set 370 380 390 400 410 420 430111213141518192033343536373839404142434421222325262729303132Wafer Number [non-dim]Frequency [Hz] Test 1 Test 2 Test 3 Figure E.1 First mode natural fr equency for small crack wafer set 0.069 0.071 0.073 0.075 0.077 0.079 0.081111213141518192033343536373839404142434421222325262729303132Wafer NumberNormalized Frequency [Hz/ m^(3/2)] Test 1 Test 2 Test 3 Figure E.2 First mode normalized frequency for small crack wafer set

PAGE 110

96 Appendix E (Continued) 550 560 570 580 590 600111213141518192033343536373839404142434421222325262729303132Wafer NumberFrequency [Hz] Test 1 Test 2 Test 3 Figure E.3 Second mode natural frequency for small wafer set 0.104 0.106 0.108 0.110 0.112 0.114111213141518192033343536373839404142434421222325262729303132Wafer NumberNormalized Frequency [Hz/ m3/2] Test 1 Test 2 Test 3 Figure E.4 Second mode normalized frequency for small crack wafer set

PAGE 111

97 Appendix E (Continued) 790 800 810 820 830 840 850111213141518192033343536373839404142434421222325262729303132Wafer NumberFrequency [Hz] Test 1 Test 2 Test 3 Figure E.5 Third mode natural fr equency for small crack wafer set 0.152 0.154 0.156 0.158 0.160 0.162111213141518192033343536373839404142434421222325262729303132Wafer NumberNormalized Frequency [Hz/ m3/2] Test 1 Test 2 Test 3 Figure E.6 Third mode normalized frequency for small crack wafer set

PAGE 112

98 Appendix E (Continued) 910 920 930 940 950 960 970111213141518192033343536373839404142434421222325262729303132Wafer NumberFrequency [Hz] Test 1 Test 2 Test 3 Figure E.7 Fourth mode natural frequency for small crack wafer set 0.170 0.172 0.174 0.176 0.178 0.180 0.182 0.184 0.186111213141518192033343536373839404142434421222325262729303132Wafer Number Normalized Frequency [Hz/ mm^(3/2) ] Test 1 Test 2 Test 3 Figure E.8 Fourth mode normalized frequency for small crack wafer set

PAGE 113

99 Appendix E (Continued) 0 100 200 300 400 500 600 700 800111213141518192033343536373839404142434421222325262729303132Wafer NumberMagnitude [dB/N] Test 1 Test 2 Test 3 Figure E.9 First mode peak magni tude for small crack wafer set 300 400 500 600 700 800 900 1000 1100111213141518192033343536373839404142434421222325262729303132Wafer NumberMagnitude [dB/N] Test 1 Test 2 Test 3 Figure E.10 Second mode peak magni tude for small crack wafer set

PAGE 114

100 Appendix E (Continued) 0 200 400 600 800 1000 1200111213141518192033343536373839404142434421222325262729303132Wafer NumberMagnitude [dB/N] Test 1 Test 2 Test 3 Figure E.11 Third mode peak magni tude for small crack wafer set 100 150 200 250 300 350 400111213141518192033343536373839404142434421222325262729303132Wafer NumberMagnitude [dB/N] Test 1 Test 2 Test 3 Figure E.12 Fourth mode peak m agnitude for small crack wafer set

PAGE 115

101 Appendix E (Continued) 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.010111213141518192033343536373839404142434421222325262729303132Wafer NumberDamping Ratio [non-dim] Test 1 Test 2 Test 3 Figure E.13 First mode damping ratio for small crack wafer set 0.001 0.002 0.003 0.004 0.005 0.006111213141518192033343536373839404142434421222325262729303132Wafer NumberDamping Ratio [non-dim] Test 1 Test 2 Test 3 Figure E.14 Second mode damping ratio for small crack wafer set

PAGE 116

102 Appendix E (Continued) 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035 0.0040111213141518192033343536373839404142434421222325262729303132Wafer NumberDamping Ratio [non-dim] Test 1 Test 2 Test 3 Figure E.15 Third mode damping ratio for small crack wafer set 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035 0.0040 0.0045111213141518192021222325262729303132333435363738394041424344Wafer NumberDamping Ratio [non-dim ] Test 1 Test 2 Test 3 Figure E.16 Fourth mode damping ratio for small crack wafer set

PAGE 117

103 Appendix F: Set up frequency response data 0 200 400 600 800 1000 0 0.5 1 Coherence 0 200 400 600 800 1000 0 500 1000 Magnitude (dB/N) 0 200 400 600 800 1000 -200 0 200 Phase (deg)Frequency (Hz) Figure F.1 Hammer (x=33 mm, y=45 mm) 0 200 400 600 800 1000 0 0.5 1 Coherence 0 200 400 600 800 1000 0 500 1000 Magnitude (dB/N) 0 200 400 600 800 1000 -200 0 200 Phase (deg)Frequency (Hz) Figure F.2 Hammer (x=43 mm, y=45 mm)

PAGE 118

104 Appendix F (Continued) 0 200 400 600 800 1000 0 0.5 1 Coherence 0 200 400 600 800 1000 0 500 1000 Magnitude (dB/N) 0 200 400 600 800 1000 -200 0 200 Phase (deg)Frequency (Hz) Figure F.3 Hammer (x=38 mm, y=40 mm) 0 200 400 600 800 1000 0 0.5 1 Coherence 0 200 400 600 800 1000 0 500 1000 Magnitude (dB/N) 0 200 400 600 800 1000 -200 0 200 Phase (deg)Frequency (Hz) Figure F.4 Hammer (x=38 mm, y=50 mm)