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Stress diagnostics and crack detection in full-size silicon wafers using resonance ultrasonic vibrations
h [electronic resource] /
by Anton Byelyayev.
[Tampa, Fla.] :
b University of South Florida,
Thesis (Ph.D.)--University of South Florida, 2005.
Includes bibliographical references.
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ABSTRACT: Non-destructive monitoring of residual elastic stress in silicon wafers is a matter of strong concern for modern photovoltaic industry. The excess stress can generate cracks within the crystalline structure, which further may lead to wafer breakage. Cracks diagnostics and reduction in multicrystalline silicon, for example, are ones of the most important issues in photovoltaics now. The industry is intent to improve the yield of solar cells fabrication. There is a number of techniques to measure residual stress in semiconductor materials today. They include Raman spectroscopy, X-ray diffraction and infrared polariscopy. None of these methods are applicable for in-line diagnostics of residual elastic stress in silicon wafers for solar cells. Moreover, the method has to be fast enough to fit in solar cell sequential production line.
Adviser: Rudy Schlaf.
Co-adviser: Sergei Ostapenko
x Electrical Engineering
t USF Electronic Theses and Dissertations.
Stress Diagnostics and Crack Detection in Full-Size Silicon Wafe rs Using Resonance Ultrasonic Vibrations by Anton Byelyayev A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Electrical Engineering College of Engineering University of South Florida Date of Approval June 15, 2005 Co-Major Professor: Sergei Ostapenko, Ph.D. Co-Major Professor: Rudy Schlaf, Ph.D. Committee Member: John Wolan, Ph.D. Committee Member: Ashok Kumar, Ph.D. Committee Member: Julie Harmon, Ph.D. Keywords: Photovoltaics, sili con wafers, stress, cracks (C) Copyright 2005, Anton Byelyayev
2 Table of Contents List of Tables 4 List of Figures 5 Abstract 8 1. Introduction 1.1. Scope and motivation 1.2. Objectives 2. Literature Survey 2.1. Introduction 2.2. Silicon as solar cell material 2.2.1. Single-crystalline silicon 2.2.2. Multicrystalline silicon 2.2.3. Amorphous silicon 2.3. Elastic stress and cracks in commercial silicon solar cells 2.4. Experimental methods for measuring residual stress 2.4.1. Raman spectroscopy 2.4.2. X-Ray diffraction 2.4.3. Polariscopy technique 18.104.22.168. Linear scanning polar iscopy technique 22.214.171.124. Full-field near infrared polariscopy technique 2.5. Experimental methods for crack detection 2.5.1. Scanning acoustic microscopy 2.5.2. Ultrasound lock-in thermography 2.5.3. Millimeter waves 10 10 12 14 14 15 17 18 19 20 23 23 29 35 35 40 46 47 52 55 3. Experimental 3.1. Hardware 3.1.1. Setup for resonance ultrasonic vibrations (RUV) measurements 3.1.2. Infrared polariscopy setup fo r residual elastic stress measurements 3.1.3. Scanning acoustic microscope 3.2. Finite element modeling 3.3. Samples 59 59 59 62 64 67 68 4. Results and Discussion 4.1. Resonance ultrasonic vibrations a pproach for stress evaluation in full-sized silicon wafers 4.1.1. Single crystal Czochralski silicon 69 69 70
3 126.96.36.199.Analytical modeling 188.8.131.52.Resonance ultrasonic vibrations data 184.108.40.206.Chladni figures 4.1.2. Edge-defined film-fed growth silicon wafers 220.127.116.11.Scanning acoustic microscopy results 18.104.22.168.Residual elastic stress da ta by linear infrared polariscopy technique 22.214.171.124.Resonance ultrasonic vibrations data 126.96.36.199.Bending test 4.2. Crack detection and analyses using resonance ultrasonic vibrations 4.2.1. Initial measurements 4.2.2. Crack engineering 70 75 77 81 81 83 85 91 93 93 95 5. Conclusions and Recommendations 101 References 105 About the Author END page
4 List of Tables Table 1 The set of vital parameters of the methods for measuring residual elastic stresses 45 Table 2 The set of vital parameters of th e methods for crack inspection in wafers 58 Table 3 The set of roots of the equation (11) 73 Table 4 Resonant frequencies [Hz] of radial angular independent longitudinal vibrations 76 Table 5 Average thickness, average and peak stress, and resonance vibration frequencies of 100mm x 100mm EFG wafers 87 Table 6 Average experimental resonance frequencies [Hz] of the three selected longitudinal vibration modes 89
5 List of Figures Figure 1 A typical photovoltaic cell diagram 15 Figure 2 Energy distribution of scattered light 24 Figure 3 Energy level diagram for Raman scattering 26 Figure 4 Schematic diagrams of brittle fracture and ductile removal in lapping process when mediated by metallization of the surface 27 Figure 5 Typical Raman spectra taken from lapped silicon wafer surfaces 28 Figure 6 A Raman map from the edge of a wafer after edge-grinding 29 Figure 7 Different schematics of XRD techniques 30 Figure 8 X-ray images of a 7 m thickness epitaxial film on (100)oriented silicon wafer 33 Figure 9 Rocking curve of a heteroepitaxial Si0.80Ge0.20 film (150 nm) on (100) Si  34 Figure 10 Infrared birefri ngence experimental layout 36 Figure 11 Result of strain field mapping in EFG multicrystalline Si wafers. Mapping size 100mm x 100mm, step= 1.0mm 39 Figure 12 Experimental setup of the full field infrared residual stress polariscope 41 Figure 13 Fringe multiplier with two beam splitters 42 Figure 14 Mapping of residual stresses of the same EFG-Si wafer 44 Figure 15 SAM image of the initial crack on the wafer captured in the red box 49 Figure 16 SAM image of the same area of the wafer as Figure 15 after applied stress 50
6 Figure 17 SAM image of internal crack 51 Figure 18 Principle of ultrasound lock-in thermography 52 Figure 19 The ULT principal setup 53 Figure 20 ULT image of the inner crack in mc-Si wafer 54 Figure 21 Principal configuration of the millimeter wave measurement system 56 Figure 22 Millimeter wave imag e of the 125x125mm polycrystalline silicon wafer 57 Figure 23 Experimental set-up for re sonance ultrasonic vibration (RUV) measurements 60 Figure 24 Schematic of the setup for scanning linear infrared polariscopy measurements 62 Figure 25 Principal setup for SAM operation 65 Figure 26 Cross section of the transducer and the intensity of acoustic field 66 Figure 27 Normal frequencies of the long itudinal vibrations in circle silicon wafers of different diameters 74 Figure 28 Frequency scan measured on 300mm Cz-Si wafer with indicated by arrows first three radial modes 75 Figure 29 Chladni sand patterns on 300mm Cz-Si wafers for different mode shapes 78 Figure 30 Computed mode shape for 300mm Cz-Si wafer 79 Figure 31 ChladniÂ’s sand patter ns observed on 100x100mm EFG-Si wafers, m=2 81 Figure 32 SAM image of the 100mm x 100mm EFG wafer with periphery crack 82 Figure 33 Result of infrared polariscopy stress field mapping 84 Figure 34 Frequency scans at one of principal maximum on two EFG wafers 86
7 Figure 35 Experimentally measured frequencies of the resonance longitudinal vibration mode vers us average stress in a set of 100mm x 100mm EFG wafers 88 Figure 36 Finite element analysis calculations of the first two principal mode shapes 90 Figure 37 Schematics of the four-point bending test 91 Figure 38 Frequency curves of the resonan ce vibration mode at different load values using 5-point bending test 92 Figure 39 Full range ultrasonic frequenc y spectrum obtained on Cz-Si wafer #1 94 Figure 40 SAM image of the 125mm x 125mm Cz-Si wafer with introduced 28mm periphery crack 97 Figure 41 The A-mode spectra of non cracked wafer and wafers with different crack lengths 100
8 Stress Diagnostics and Crack Detection in Full-Size Silicon Wafe rs Using Resonance Ultrasonic Vibrations Anton Byelyayev ABSTRACT Non-destructive monitoring of residual elas tic stress in silicon wafers is a matter of strong concern for modern photovoltaic industr y. The excess stress can generate cracks within the crystalline structure, which fu rther may lead to wafer breakage. Cracks diagnostics and reduction in multicrystalline silicon, for example, are ones of the most important issues in photovoltaic s now. The industry is intent to improve the yield of solar cells fabrication. There is a number of techniques to measur e residual stress in semiconductor materials today. They include Raman spectroscopy, X-ray diffraction and infrared polariscopy. None of these methods are applicable for in-line di agnostics of residual elastic stre ss in silicon wafers for solar cells. Moreover, the method has to be fast enough to fit in solar cell seque ntial production line. In photovolta ics, fast in-line quality control has to be performed within two sec onds per wafer to match the throughput of the production lines. During this Ph.D. research we developed the resona nce ultrasonic vibration (RUV) approach to diagnose residual stress non-destructively in full-size multicrystalline
9 silicon wafers used in solar cell manufactur ing. This method is based on excitation of longitudinal resonance ultrasonic vibrations in the material us ing an external piezoelectric transducer combined with high sensitive ultrasonic probe and data acquisition of the frequency response to make the method suitabl e for in-line diagnostics during wafer and cell manufacturing. Theoretical and experimental analyses of the vibr ation mode in single crystal and multicrystalline silicon wafers were used to provide a benchmark reference analysis and validation of the approach. Importantly, we observe d a clear trend of increasing resonance frequency of the longitu dinal vibration mode with higher average in-plane stress obtained with scanning infrared polariscopy. Using the same experimental approach we assessed a fast crack detection and length determination in full-size solar-grade crystalline silicon wafers. We demonstrated on a set of identical non-processe d crystalline Si wafers with introduced periphery cracks that the crack shifts a selected RUV peak to a lower frequency and increases the resonance peakÂ’s half-width. Both characteri stics are gradually increased with the length of the crack. This was confirmed also theore tically by performing fin ite element analysis of longitudinal vibrations of wafers with cr acks. The frequency shif t and peak half-width were found to be reliable indicat ors of the crack appearance in silicon wafers suitable for mechanical quality control and fast waferÂ’s inspection. Resonance ultrasonic vibrations metrol ogy is a promising technique to provide quality control in full-size silicon wafers. Th is approach has the potential to be further developed into a diagnostic t ool to address the needs of silicon wafe r manufacturers, both in the microelectronic and the solar cell industries.
10 1. Introduction Â“Sooner or later we shall have to go directly to the Sun for our major supply of power. This probl em of the direct conversion from sunlight into power will occupy more and more of our attention as time goes on and eventually it must be solvedÂ…Â” Edison Pettit, Wilson Observatory, 1932 1.1. Scope and motivation The energy consumed by mankind now is mostly dependent on fossil fuels such as oil and coal. They are the cheapest and the most affo rdable sources of energy known to man today. Mankind is now facing the problem of shortage in fossil fuels and also the trend of environment pollution increase that has been observed over recent decades. Therefore, the possibility of using safe a nd virtually inexhaustible solar power draws a vast attention of scient ists and engineers. At the present moment, renewable ener gy technologies continue to grow and develop. Being one of its major players, the photovoltaic (PV) industr y, with crystalline silicon as a dominant segment, is expanding rapidly to meet growing energy demands all
11 over the world  However, for crystalline silicon (S i) cells to satisfy future energy requirements, there is still a significant need for improvements in manufacturing to increase throughput and production yield. One of the current technological problems is to identify and eliminate sources of mechanical defects such as thermo-elastic stress and cracks leading to the loss of wafer integrity and, ultimately, breakage of Si wafers and cells during the production process. Stress in solar-grade silicon wafers is one of the sources of wafer breakage which reduce the production yield of solar cells by up to 25% depending on wafer technology and processing steps. Moreover, stress can be a driving force for various types of defect reactions, such as precipitation of residual impurities at dislocations deteriorating the electronic quality of material . Ther e are a number of experimental techniques available to diagnose residual stress and detect cracks in wafers used in the semiconductor industry. However, none of thes e methods are suitable in their present form for rapid in-line stress diagnostics of as-grown a nd processed Si wafers. In the PV industry fast in-line quality control require s methods which achieve measurement times of around 2 seconds per wafer to match th e throughput of typical production lines. Ultrasonic testing is a well-establishe d approach in the family of methods classified as non-destructive analyses in ma terials and devices. One can use ultrasonic vibrations for quality assurance purposes detecting both micro and macro defects. An approach to measure stress and assess factors which affect the mechanical quality of fullsize silicon wafers was proposed by Ostapenko et al . It wa s based on the excitation of a specific flexural resonance vibration mode, assigned as Â“whistleÂ” mode in the wafer using an external piezoelectric transducer combined with a non-contact acoustic sensor
12 and high speed data acquisition to provide information on the frequency response over the entire wafer area. In the case of the circular 200 mm Czoc hralski silicon wafers, this approach allowed tracking of stresses in wafers with nanometer thick oxides, and also to assess wafers with as-grown bulk defects . A similar ultrasonic technique utilizing flexural vibrations was later extended to multicrystalline silicon wafers . The principal goal of this presented work wa s to establish the fundamental s of the resonance ultrasonic vibration (RUV) methodology, analyze its physi cal aspects in terms of wafer integrity and provide diagnostics on stress and detection of cracks in commercial grade full-size crystalline silicon wafers. 1.2. Objectives The objectives of the research project were: i) To develop a non-destructive methodology of the RUV technique applicable to full-size single-crystal and multi-crystalline silicon wafers primarily for photovoltaic application. ii) To develop a physical model of the RUV method based on excitation of the longitudinal ultrasonic vibrations in wafers of different shape and size. iii) To perform computer simulations us ing the physical model and compare theoretical results with experimental RUV data. iv) To apply the RUV method to non-destructiv e rapid diagnostics of residual elastic stress in multicrystalline silicon wa fers used in solar cell production.
13 v) To make the method suitable for crack de tection and its lengt h determination in full-size solar-grade crystalline silicon wafers. vi) To design the prototype hardware system for mechanical quality control and fast wafer inspection.
14 2. Literature Survey 2.1. Introduction The market for commercial photovoltaics today is fully dominated by siliconbased technology. It has been predicted that the mc-Si efficiency of commercial solar cells would reach 18 to 20% in 5 years, which is confirmed by record numbers of laboratory efficiencies of 17.8% (Georgia, Institute of T echnology, 2003) and crystalline silicon will be the dominant PV technology for the next 15-20 years . Besides, there is currently no promising technology that can co mpete with crystalline silicon. The ultimate challenges are still the same: to increase solar cell efficiency and to reduce cost, which is, obviously, difficult to achieve at the same ti me. There also will be continuous increases in wafer size, yield and power output, whil e the wafer thickness decreases. And one of the current technological problems is to id entify and eliminate s ources of mechanical defects such as thermo-elastic stress and cr acks leading to the loss of wafer integrity and ultimate breakage of as-grown and proces sed Si wafers and cells. The problem is enhanced as a result of the current stra tegy of reducing wafe r thickness down to 100 microns. For instance, cracks generated dur ing wafer sawing or laser cutting can propagate due to wafer handling and during solar cell processing such as, phosphorous diffusion, anti-reflecting coating, front back contact firing, and soldering of contact
15 Â“fingersÂ” (Figure 1). It is recognized that the developmen t of a methodology for fast inline crack detection and contro l is required to match the th roughput of typical production lines. Figure 1. A typical photovoltaic ce llÂ’s diagram (Property of DoE). 2.2. Silicon as solar cell material Solar cells can be made from a wide ra nge of semiconductor materials, which can be subdivided into three categories: crystallin e bulk silicon, polycrys talline thin films and single-crystalline thin films. Polycrystalline thin films include copper indium diselenide, cadmium telluride, and thin-film silicon. Single-crystalline thin films cover highefficiency materials such as gallium arseni de. Crystalline silicon segment of the PV market is the largest and still the most popular. Silicon is being used in three different
16 forms of crystallinity: crystalline Si, multicry stalline Si, and amorphous Si. Silicon is the second-most abundant element in the Earth's cr ust, after oxygen. However, to be useful as a semiconductor material in solar cells, silicon must be refined to a purity of 99.9999%. In single-crystalline silicon, the molecular structure, which is the arrangement of atoms in the material, is uniform, because th e entire structure is grown from the same crystal. This uniformity is ideal for transferri ng electrons efficiently through the material. To make an effective PV cell, however, silico n has to be doped with other elements to make it n-type or p-type. Polycrystalline silicon also known as mu lticrystalline Si (mc-Si), in contrast, consists of several smaller crystals or grains, which introduce boundaries. These boundaries impede the flow of electrons and en courage them to reco mbine with holes to reduce the power output of the solar cell. Ho wever, mc-Si is much less expensive to produce than single-crystalline silicon. Theref ore researchers are working on other ways to minimize the effects of grain boundaries and inter-g rain defects. Silicon is obtained from the reduction of quartzite or sand. The first technological step is to derive the metallurgical-grade si licon (MGS) which contai ns large quantities of impurities. Various refining processes are utilized to remove those impurities. These refining processes have to be effective but inexpensive Â– it is an ultimate benchmark for modern PV industry nowadays. So far, solar grad e Si is still a low qua lity Si, with a high concentration of impurities and defects (1012-1014 cm-3), whose effects may be crucial for the final photovoltaic devices .
17 2.2.1. Single-crystalline silicon In order to create silicon in a single-crystal state, one must to first melt highpurity silicon. One then causes it to solidify ve ry slowly in contact with a single crystal "seed." The silicon replicates th e crystal structure of the single-crystal seed as it cools and gradually solidifies. Several different proces ses can be used to grow a boule of singlecrystal silicon. The most well-known and re liable processes are the Czochralski (Cz) method and the float-zone (FZ) technique. Th e "ribbon-growth" tec hnique will also be discussed as it has been em ployed by the PV industry. In the Czochralski method, a seed crysta l is dipped into a crucible of molten silicon and withdrawn slowly, pulling a cy lindrical single crystal as the silicon crystallizes on the seed . The float-zone growth t echnique makes purer crysta ls than the Cz method, because they are not contaminated by the crucib le used in growing Cz crystals. In the FZ growth, a silicon rod is located atop a s eed crystal and then lowered through an electromagnetic coil. The coil's magnetic field induces an electric fi eld in the rod, heating and melting the interface between the rod an d the seed. Single-crystal silicon starts formation at the interface, growing upward as the electromagnetic coils are slowly raised . Once the single-crystal rods are produced, by either with the Cz or FZ method, they are usually sliced to form thin wafe rs a few hundred microns in thickness. The resulting thin wafers are doped, followed by a coating to reduce re flection, and coated
18 with electrical contacts to form functioning PV solar cells. However, such slicing is a source of wasting as much as 20% of the silic on ingot in the form of sawdust. Therefore, there is always a trend of search ing for cheaper ways of production. 2.2.2. Multicrystalline silicon Devices built of multicrystalline silicon (m c-Si) are generally less efficient than those of single-crystal silic on, but they are less expensiv e to make. There are several methods of producing multicrystal line silicon such as ingot ca sting, Tri-crystal growth or ribbon growth. The most popular methods use a casting process in which molten silicon is directly cast into a mold and allowed to solidify into an ingot. The starting material for this method is usually lower-grade silicon, unlike the higher-grade semiconductor grade required for single-crystal material. The coolin g rate is one factor that determines the final size of crystals in the ingot and the distribution of impurities . The "ribbon growth" technique edge-defined film-fed growth (EFG) starts with two crystal seeds that grow and capture a sh eet of material between them as they are pulled from a source of molten silicon. A fram e holds the thin sheet of material when pulled from the melt. This tec hnique does not waste much mate rial, but the quality of the material is not as high as Cz and FZ silicon . It should be noted that mc-Si wafers cont ain impurities and crystal defects, such as grain boundaries and disloc ations with a higher concen tration than found in singlecrystalline Si .
19 2.2.3. Amorphous silicon Amorphous silicon is produced in high frequency furnaces in a partial vacuum atmosphere. In the presence of a high freque ncy electrical field, gases like silane, B2H6 or PH3 are blown through the furnaces supplying silicon with boron or phosphorus . Amorphous silicon does not have the st ructural uniformity of singleor multicrystalline silicon. Small deviations in this material result in defects such as "dangling bonds", where atoms lack a nei ghbor to which they can bond. These defects provide places for electrons to recombine w ith holes, rather than contributing to the electrical circuit . Ge nerally speaking, amorphous silicon would be unacceptable for electronic devices, because def ects limit the flow of current. But it can be deposited so that it contains a small amount of hydrogen, in a process called "hydrogenation." The result is that the hydrogen atoms combine chemically with many of the dangling bonds, essentially removing them and permitting elec trons to move through the material . Today, hydrogenated amorphous silicon is used to produce low efficiency (~ 10%) solarpowered consumer devices that have low power requirements, such as wristwatches and calculators.
20 2.3. Elastic stress and cracks in commercial silicon solar cells One of the major current technological probl ems for the PV industry is to identify and eliminate potential sources of mechanical defects such as thermo elastic stress and cracks leading to the loss of wafer integrity and ultimate breakage of as-grown and processed silicon wafers and finished solar ce lls. The problem is of increase concern as a result of the current strategy of reducing wafer thickness down to 100 microns. Wafers having high level of residua l elastic stresses behave extremely unpredictably during processing and handling. The single-crystal Cz ingots for PV are pul led at growth rates that can be many times faster (~ cm/min) than that of the c onventional growth for microelectronics. These fast cooling rates are accompanied by excessi ve thermal stresses that lead to the generation of growth defects: di slocations, defect precipitates, thermo-elastic stresses. It should be noted, that residual elastic stresses caused by ingot sawing, the formation of the front and back contacts, diffusion of the emitter, and deposition of the antireflecting coating may even increase compared to as -grown material. Consequently, the singlecrystal material is expected to have high c oncentrations of nonequili brium point defects. In some cases, a portion of the ingot may acquire a high density of crystal defects (primarily dislocations) and even lose cr ystallinity and become multicrystalline . Speaking of which Â– it should be said that in a case of multicrys talline silicon, stress issues are much more severe. Stress levels in mc-Si wafers are orders of magnitude
21 higher than in single-crystalline silicon wafers. So, there is always a goal to keep stresses as low as possible after every proces sing step of solar cell production. For example, thin solar cells are difficu lt to interconnect with standard soldering techniques. High temperature du ring soldering, between 250-400 C, introduces stress on the joints and cells. This can cause warping and possible breakage of cells and decreases yield. The front side of cells suffers from extra stress caused by the tabs going from front to rear in series interconnection. Moreover, extra losses occur due to anti-reflecting coating deposition. Therefore, substituting conventional soldering technique for a low temperature joining method would avoid bu ilding up of mechanical stress, again, increasing process yield a nd reliability . Another issue is the cracks in silicon. Wa fer breakage during processing is a very high cost issue. This is particularly true wh en wafers fail during one of the print steps, generally resulting in several minutes of downtime while the operator cleans up the scattered parts and the wet paste. This is al so a source of potential contamination. It is believed that wafers frequently fail at the print steps because they come into the process already cracked and the crack then fails when it is stressed during the process step. Wafer cracks can also cause electrical failure at cell or module test . It was shown that cracks in readily processed solar cells lead to a weak recombination current. However, if a crack is already presented in the wafer before processing, or if it appears during proces sing before screen-printing the contact metallization, cracks may lead to serious ohmic shunts. If during screen-printing some metal paste penetrates a crac k, after firing this may produce especially strong shunts. The
22 shunts described above may also emerge if th ere are any holes presented in a cell, e.g., resulting from laser cutting . An example of such a crack in readily processed solar cell is shown in Figure 14.
23 2.4. Experimental methods for measuring residual stress 2.4.1. Raman spectroscopy Raman spectroscopy has been utilized to investigate the st resses and phase transformations in semiconductors [19, 20]. St ress maps of Si around indentations have been generated . Quantifica tion of residual stress in a s ilicon wafer in plane stress was also obtained . Intern al stress in semiconductors after machining has also been measured using Raman tensometry [23, 24] Perhaps, the greatest advantages of Raman spectroscopy are its non-destructive character, the simplicity of the equipment set-up and the short time required for obtaining data with essentially no sample preparation process required and no surface damage. It is also at tractive because it can detect both organic and inorganic species and measure the crystal linity of solids. In addition, it is free from charging effects that can influence el ectron and ion beam techniques. The light scattered from the surface of the wafer has basically three components (Fig. 2). There are Raleigh scattering (scatter ing with the incident frequency), Raman and also Brillouin scattering.
24 Figure 2. Energy distribu tion of scattered light. The interaction of the incident light with optical phonons is called Raman scattering while the interaction with acous tic phonons results in Brillouin scattering. Optical phonons have higher energies than acoustic phonons giving larger photon energy shifts. However, even for Raman scattering, th e energy shift is small. Since the intensity of Raman scattered light is very weak, Ra man spectroscopy is only practical when an intense monochromatic light s ource like laser is used  The Raman effect arises when a photon is incident on molecules comprising a solid state and interacts with the electric dipol e of the molecules. When light is scattered from a molecule most photons are elastically scattered. The scattered photons have the
25 same energy (frequency) and, therefore, wave length, as the incident photons. However, a small fraction of light (approximately 1 per 107 photons) is scattered at optical frequencies different from, and usually lowe r than, the frequency of incident photons (Fig. 3). The process leading to this inelastic scatter is the termed the Raman effect . Figure 3. Energy level diagram for Raman scattering (a) Stokes Raman scattering, (b) anti-Stokes Raman scattering. The energy difference between the incident and scattered photons is represented by the arrows of different lengths in Figure 3a. The energy is ultimately dissipated as heat. Because of the low intensity of Ra man scattering, the heat dissipation does not
26 cause a measurable temperature rise in a ma terial. A small fraction of the molecules are in vibrationally excited states. Raman scattering from excited molecules leaves the molecule in the ground state. The scattere d photon appears at a higher energy, as shown in Figure 3b. This anti-Stokes-shifted Raman spectrum is always weaker than the Stokesshifted spectrum, but at room temperature it is strong enough to be useful for frequencies less than about 1500 cm-1. The Stokes and anti-Stokes spec tra contain the same frequency information. The ratio of anti-Stokes to Stokes intensity at any vibra tional frequency is a measure of temperature. Anti-Stokes Raman scattering is used for contactless thermometry. The anti-Stokes spectrum is also used when the Stokes spectrum is not directly observable, for example because of poor detector response or spectrograph efficiency. Cz Si wafers were successfully st udied by Raman spectro scopy methods . In these studies wafers were investigated after being subjected by brittle fracture and ductile removal during the lapping process when medi ated by metallization of the surface (Fig. 4).
27 Figure 4. Schematic diagrams of brit tle fracture and du ctile removal in lapping process when mediated by metallization of the surface. (a) The abrasive grain comes in contact with the surface and a layer is removed by micro fractures, whic h leaves micro cracks in the underlying layer as deep as 5 to 10 m. (b) The abrasive grain comes in contact with the surface and pressure-induced metallization. Metallic Si is removed in a ductile manner, the surface is left with a transf ormed layer approximately 0.1 m thick.
28 Pristine Si peaks are shifted from 521 cm-1 to as much as 517 cm-1 due to residual tensile stress at many points of the lapped Cz-Si wafers (Fi g. 5). If a compressive stress exists before moving the tool, tensile stress is expected to occur on the surface behind the tool. A shift of 3.2 cm-1 corresponds to 1 GPa . Figure 5. Typical Raman spectra taken from lapped silic on wafer surfaces. (a) Pristine Si, (b) Si un der residual tensile stress. Hence, the evidence of residual stresse s comes in the form of shifted peak position. Up-shift of the Raman band is caused by compression wh ile a down-shift is caused by tension. Compression makes the latt ice spaces smaller which leads to an increase of the phonon frequencies. The reve rse effect, caused by te nsion, leads to the down-shift of the Raman band. Compressive stresses on the surface are non-uniform with significant variations across most areas.
29 Figure 6. A Raman map from the edge of a wafer after edge-grinding. (a) Optical image, (b) Raman map of intensity ratio of 350-515 cm-1 against 516-525 cm-1. Since the laser beam can be focused to a small diameter, one can measure stress in localized regions and do mapping of specific interested areas . 2.4.2. X-Ray diffraction X-Ray Diffraction (XRD) topography is a non destructive technique for measuring structural crystal defects . It is attract ive due to easy sample preparation and it can give structural information over entire se miconductor wafer. Unlike optical methods, the XRD image cannot be magnified because no lenses are use d. X-rays typically have special resolution of 100 microns and above.
30 If one will consider a defectless crysta l subjected to different monochromatic Xrays of wavelength from lattice planes spaced d There are many possibilities to see what happens to incident beam. One can detect either reflection or transmission of X-rays through the crystal (Fig. 7). Figure 7. Different schematics of XRD techniques: (a) Berg-Barrett reflection topography, (b) Lang transmission topography, (c) double crystal topography with a rocking curve.
31 First, consider the Berg-Barrett XRD tec hnique. The X-rays are incident on the sample at an angle as shown in Figure 7a. The diffracted beam emerges at twice the Bragg angle B defined by ) 2 / ( sin1dB (1) The diffracted X-rays are detected on a high-resolution, fine-grained photographic plate as close as possible to the sa mple without intercepting the in cident beam. If the lattice spacing or lattice plane orientation vary locally due to structural de fects, equation (1) no longer applies simultaneously to the perfect a nd the distorted regions Obviously there is a difference in X-ray intensity from the tw o regions. For example, the diffracted beam from dislocations is more intense than fr om an area without defects caused by Bragg defocusing. Dislocations produce a more heav ily exposed image on the film. The image is formed as a result of diffraction from an anomaly such as strain in the crystal but does not image the defect directly. One can define strain as the amount of elastic deformation unstrained strained unstrainedd d d S (2) Measuring d in unstrained and strained regions, one can determine the strain . It is the simplest X-ray topography method. There are ne ither lenses nor moving parts except for the sample alignment goniometer. This method is used to determine dislocation densities
32 up to about 106 cm-2. The resolution is about 10-4 cm, and areas as large as 200 mm diameter wafers can be examined with the Berg-Barrett method. The most popular XRD technique so far is illustrated in Fig. 7b and was introduced by Lang in 1959 . Monochrom atic X-rays pass through a narrow slit and hit the sample aligned in an appropriate Bragg angle. The tall and narrow primary beam is transmitted through the sample and strikes a lead screen. The diffracted beam falls on the photographic plate though a slit in the screen. X-rays are absorbed in a solid according to equation: x oe I x I ) / () ( (3) where ( / ) is the mass absorption coefficient, the detector material density, I(x) the Xray intensity in the detector, and Io the incident X-ray intensity. To Â“pick upÂ” defects, one usually uses a weakly diffracting plane. A uniform sample gives a featureless image. Structural defects cause stronger X-ray diffraction. For semiconductors, the Lang method is used primarily to study defects introduced during crystal growth or during wafer processing [ 32]. Transmission images provide information on defects through the entire sample; reflection images provide information of 10 to 30 m depth from the surface. X-ray images of silicon wafers are shown in Figure 8.
33 Figure 8. X-ray images of a 7 m thickness epitaxial film on (100)-oriented silicon wafer using the Lang and double crysta l topography methods (a), crystal defects by Lang transmission method (b)  Double-crystal diffraction provides highe r accuracy because the beam is more highly collimated than is possible with si ngle crystal topography . The technique consists of two successive Bragg reflections fr om reference and sample crystals (Fig. 7c). Reflection from the first Â“defectlessÂ” cr ystal produces a monoc hromatic and highly parallel beam to probe the sample. The doubl e crystal technique is used not only for topography, but also for rocking curve determina tion (Figure 9).
34 Figure 9. Rocking curve of a heteroepitaxial Si0.80Ge0.20 film (150 nm) on (100) Si . To record a rocking curve, the sample is slow ly rotated (Â“rockedÂ”) about an axis normal to the diffraction plane and the scattered intensit y is recorded as a function of the angle as shown in Fig. 7c. The rocking curve width at full width half maximum is a measure of crystal perfection. The narrower the curve, the more perfect is the material. For epitaxial layers it can give data on lattice mismatch, layer thickness, layer and substrate perfection, and wafer curvature.
35 2.4.3. Polariscopy technique 188.8.131.52. Linear scanning polariscopy technique If a crystalline material plastically deforms because of an external force, upon removal of that force, a Â“frozen-inÂ” (residual) stress will exist in the interior of the material. Stresses of this type have b een studied principally by XRD techniques. Unfortunately, these techniques are limited becau se the stresses vary over distances much greater than the width of th e X-ray beam. Another method fo r the investigation of stress utilizes the photoelastic effect in transparent crystals. Thes e photoelastic patterns give a direct picture of the magnitude, direction, and distribution of internal stress  When electromagnetic radiation is transmitte d through a material in all directions, equally, the material is assumed to be isotropic. This generally defines the optical character of crystals in the cubic system. Th e great majority of crystalline materials are anisotropic, however, and transmission of an el ectromagnetic wave will vary in velocity according to the direction of the ray in the crystal. Associated with anisotropy is a property of the birefringence, wh ich gives rise to a photoelastic pattern. Amorphous materials like gl ass behave as isotropic crystals for all vibration directions of light, but when in a state of stra in, however, they acquire the optical properties of anisotr opic crystals. Similarly, crysta lline silicon, which should be isotropic because of its cubic la ttice, has been observed to be birefringent when strained.
36 The presence of birefringence in silicon has be en interpreted from these studies to be the result of strain caused by severe thermal gr adients and uneven temper ature distribution in the crystal following solidification a nd cooling to room temperature. Silicon has a strong metallic reflection for incident radiation below 1.1 microns, but above that threshold silicon is tran sparent in the infrared range  The absorption coefficient is a function of wavelength in silicon. For wavelength about 1 micron, the absorption coefficient appears to be ~102 cm-1 [38, 39]. The typical infrared birefringence experimental layout also known as pol ariscopy is shown in Figure 10 . Figure 10. Infrared birefri ngence experimental layout.
37 The experimental details of scanni ng infrared polaris copy applied to multicrystalline Si wafers for solar cells are discussed in . For scanning measurements of the residual strains, a linear or circular polariscopy technique can be used. In linear polariscopy, collimated light from a 75 W halogen tungsten lamp is focused down to 60 m on the sample. The intensity of the optical transmission is measured in a spectral region of a transmi ssion window where Si is transparent, near = 1 3 m. Additionally, two infrared linear po larizers are used in the transmission experiment. They can be oriented with respect to a pre-selected crystal direction, such as a grain boundary in mc-Si. One of them (polar izer) is located in front of the sample and the second (analyzer) immediately behind th e sample. In general, the intensity of polarization transmission can be expressed as follows : ] 2 / sin ) ( 2 sin ) ( 2 sin )[cos 1 (2 2 2 R I Io (1) where I o is the intensity of incident light, R is the reflectivity, is the principal angle which determines the orientation of the stress axis at the plane, is the angle between polarizer and analyzer and is the azimuth angle of the polarizer (Figure 10b). The polarization intensity, I measured at selected orie ntations of the polarizer and analyzer, depends on the opt ical retardation parameter which in turn is directly related to the value of the residual elastic stra in. To derive this relationship, the following algorithm is applied . Two transmitted intensities of the polarized light are measured: one with the polarizer and analyz er parallel to each other, I = I ( = 0 ) and the second when the polarizer is or thogonal to the analyzer, I = I ( = / 2 ) By measuring the
38 angular -dependence of the ratio I/(I + I) one can determine quantitatively the optical retardation, and the direction of the principal stress angle, using the equation I/(I + I)= sin2()sin2/2. (2) As the last step of the analysis, two strain components, | ezzexx| and | exz|, can be calculated. The first term represents a difference in tensile strains along the crystallographic Z and X directions, while the second term is the shear strain component between Z and X The following can be used for calculating strains : ) 2 cos( )] /( )[ / (12 11 3 p p dn e eo xx zz (3a) 2 sin ) / )( / (44 3p dn eo xz (3b) where pij are photoelastic constants, d is sample thickness and no is the refractive index of the unstrained material. In the case of mc-Si wafers, the following values were used: = 1 3 m, no = 3 5, d = 300 m, p11 = 0 081, p12 = 0 001, p44 = 0 075 .
39 Figure 11. Result of strain field mapping in EFG multicrystalline Si wafers. Mapping size 100mm x 100mm, step= 1.0mm. A typical result of strain field mapping is presented in figure 11 for a full-size 100x100 mm2 wafers measured with a 1mm step. The strain field component, | ezz exx |, calculated using equation (3 a ), has a linear geometry and mirrors the orientation of the nearby grain
40 boundary. The maximum value of the stress is ~100 MPa in the high-stress wafer and 63 MPa in low-stress wafer. In mapping the entire 100 100 mm2 wafer, regions with a stress as high as 250 MPa were observed. Th is polariscopy technique will be applied in a current project to correlate th e data of the resonance acous tic measurements by measuring the elastic stress in EFG wafers. 184.108.40.206. Full-field near infrared polariscopy technique Similarly to linear polariscopy, the fullfield near infrared polariscopy technique measures changes in the polarization of transmitted light, which is caused by the residual stress-induced birefringence of principal shear stress . Near infrared optical irradiation, which is transparent to silicon, was used as a light source. The phase stepping technique was applied to determine photoelast ic parameters, which, then, could have been used to calculate the residual stress us ing the anisotropic piezo -optical law [45, 46]. The experimental setup of the full-field infrared polariscope is shown in Figure 12 . A fringe multiplied was used to increase the resolution of the system. The following was achieved by two beam-splitters placed before and behind the sample.
41 Figure 12. Experimental setup of the in frared residual stress polariscope. The light reflected by these two splitters passes through the sample several times as shown in Figure 13, and the photoelastic effect is amplified accordingly. The disadvantage of the fringe multiplier is the loss of light intensity and spatial resolution. However, the spatial resolution can be incr eased by minimizing distance between the two partial mirrors. To compensate the light loss, a high intensity light so urce and a sensitive camera were used .
42 Figure 13. Fringe multiplier with two beam splitters. Phase stepping provides the full-field ma pping of the relative retardation and the isoclinic angle. By selectin g six angular positions of th e quarter waveplate and the analyzer a sequence of simultaneous equations in relative retarda tion and the isoclinic angle. By selecting six angular positions of the quarter waveplate and the analyzer , a sequence of simultaneous equati ons in relative retardation and isoclinic angle were obtained. The simultaneous equations were solved for the isoclinic angle as: 4 6 3 52 tanI I I I (1) and the retardation
43 2 1 2 3 5 2 6 4) ( ) ( tanI I I I I I (2) where In() is the light intensity during phase ste pping as a function of different angles of waveplate, and analyzer, These parameters have been used togeth er with anisotropic piezo-optic law to extract the residual stresses [48, 49, 47]. The correlation experiments have been carried out . Six 100x100mm EFG-Si wafers were measured by the two polariscope systems. Typical results of the particular EFG wafer are shown in Figure 14. The maximu m shear stress in this sample is about 22MPa and the average is around 3MPa. As one can see, the results showed that linear and full-field infrared polariscopy methods were in good agreement.
44 (a) (b) Figure 14. Mapping of residual stresses of the same EFG-Si wafer measured by linear (a) and full-field (b) infrared polariscopy.
45 Table 1 represents a summary of the me thods for measuring residual stress in silicon wafers for photovoltaics. Table 1. The set of vital parameters of the methods for measuring residual stresses. Method Stress sensitivity level [MPa] Spatial resolution Additional comments Raman ~10 MPa >10-3mm Single crystalline wafers only. X-Ray n/a >10-3mm Qualitative analysis only. Linear IR ~1 MPa >1mm Time consuming if used in scanning mode. Full-field IR ~ 1 MPa >0.2mm Reference calibration needed.
462.5. Experimental methods for crack detection Cracks generated during wafer sawing or la ser cutting can propagate due to wafer handling and solar cell processing such as, phosphorous diffusion, anti -reflecting coating, front and back contact firing, and solderi ng of contact ribbons. It is recognized that development of a methodology for fast in-line crack detection and control is required to match the throughput of typical production lin es. At this time, there are several experimental methods, which address the pr oblem of crack detection. They include Scanning Acoustic Microscopy , ultrasonic lock-in thermography  and millimeter wave techniques . In this chapter, we discuss those three major methods, mentioned above, and compare ones to an alternative approach based on the recently developed RUV methodology, which also can be used to monitor elastic stress in multi-crystalline silicon solar-grade wafers . 2.5.1. Scanning acoustic microscopy The ultrasound methods for industry have been established in the early 1900s. The PV industry has recently entered the terr itory of ultrasonics with the introduction of high frequency immersion systems or, so cal led, C-Mode Scanning Acoustic Microscope (C-SAM). Initially, C-SAM tools have demons trated their usefulne ss for non-destructive testing of plastic and ceramic p ackaged integrated circuits at Texas Instruments in the late 1980s, early 1990s .
47 The first prototype of scanning acoustic microscope was designed and developed in Stanford University  But before that the C-scan was used by nondestructive testing industry (NDT) since the 1950s . Modern SAMs are the hybrids tools with characteristics of both the first StanfordÂ’s SAM and the C-scan. We will briefly review the characteristics of each of these methods. The term C-scan originated from early NDT specification. The C-scan image is an image of a planar region at a constant depth w ithin the sample. The C-scan is obtained by mechanically scanning a piezoelectric transduc er above the specimen and electronically gating the signal in time. The broad-band Cscan transducer has a small numerical aperture lens for sub-surface imaging. C-scan has played a major role in the microscopic imaging of sub-surface flaws in industrial components with resonance frequencies in the range of 1-10MHz. In the C-scan techniqu e the echo signal has traditionally been rectified for easier interpretation, so all phase information was not used . The scanning acoustic microscope was fi rst demonstrated by Lemmons and Quate at Stanford in 1973 [54, 57]. The SAM employs a large numerical aper ture lens in order to excite surface waves on the sample. Instead of the large water baths of C-scan, they use a water droplet to acousti cally couple the transducer a nd the sample. The lens are formed by grinding a hemispherical cavity in to the tip of a sapphire rod. The large difference in acoustic velocities between th e sapphire and the water droplet produces good focusing. Image contrast is formed by th e combined interference of longitudinal and flexural waves reflected from the surface a nd from beneath the surface. Narrow-band RF pulses with frequencies in the range of 100MHz-3GHz are used. Precision mechanical scanning is used for microscopic imaging. Both amplitude and phase of the reflected
48 pulses are measured to produce images of th e mechanical properties of the near-surface region. Sub-micron lateral resolu tion is achieved at the highest frequencies. In the C-scan method, lateral resolution is typically limited by absorption in the sample. In the case of SAM, lateral resolution is limited by high frequency attenuation in the coupling water droplet . SAM has started to be employed widely in the semiconductor field after it became commercially available in 1982. The applica tion of SAM in the semiconductor industry was initially the high frequenc y inspection of thin dielectri c layers and conductors on the device surface . Optional br oad-band transducers in the in termediate frequency range of 30-100MHz were available for sub-surface studi es such as die att ach inspections . SAM was used by the IC industry extens ively for a long time, although the PV manufacturers began to employ acoustic micr oscopy technique for crack detection in silicon wafers and solar modules relatively not long ago. For instance, one of the largest sola r cells manufacturers Â– BP Solar Â– in collaboration with the Nanotechnology and Na nomanufacturing Resear ch Center at the University of South Florida proposed to use SAM in order to characterize the formation of cracks under constant stre ss conditions and classify different types of cracks after various technological steps of solar cell manuf acturing . The init ial experiments were conducted with single crystal cells because a pr oduction yield problem with this cell type indicated that a significant fr action of these cells were susceptible to breakage. Figure 15 shows one of the cells as it was initially placed in the SAM system. The crack was initially ~15mm in length, but it eventually propagated across the wafer leading to wafer breakage, as shown in Figure 16. Figur e 16 looks exactly like the IR image of the
49 modules that incorporate these problem cells. The SAM system was able to see the small (micro-) crack (Figure 15) that late r lead to the break (Figure 16). Figure 15. SAM image of the initial cr ack on the wafer captured in the red box.
50 Figure 16. SAM image of the same area of the wafer as Figure 15 after applied stress. The initial crack had propagated across the waferÂ’s corner causing it to break. The SAM process was also able to identify internal cracks. An example is shown in Figure 17.
51 Figure 17. SAM image of internal crack. SAM has proven to be an accurate method for identifying cracks and microcracks in wafers and partially processed solar cells. In this case it provided BP Solar with information about where cracks and microcrack s were first occurring and therefore has provide valuable informati on to assist in reducing br eakage and increasing yield. However, each SAM wafer measurement took 20 minutes for sample set-up and data collection. This is clearly not an in-line production process. The data from this effort was very useful as the sample set evaluate d by SAM can now be used to calibrate and verify the accuracy of any new met hods developed for crack detection.
522.5.2. Ultrasound Lock-in Thermography Ultrasound lock-in thermography (ULT) wa s established a few years ago for remote non-destructive testing. It is based on propagation a nd reflection of thermal waves which are generated by ultrasound transducer from the surface into the inspected component by absorption of modulated radi ation . The schematics of the ULT principle are illustrated in Figure 18. Figure 18. Principle of ultr asound lock-in thermography. While thermography generates images wh ere the contrast is provided by local thermal emission, lo ck-in thermography means that one investigates a coded heat flow by analyzing the temper ature modulation that is i nduced by periodical heat deposition. Absorption of modulated optical ra diation results in a te mperature modulation that propagates as a thermal wave into the inspected component. As the thermal wave is
53 reflected at the boundaries of subsurface featur es, its superposition to the original thermal wave causes a signal change that depends on the depth of the hidde n boundary. [62, 63]. In ULT, the attached ul trasonic transducer drives m odulated acoustic waves into the sample where they propa gate until they disappear since they are converted into heat in the high-loss-angle areas of defects. That is why the defect reveals itself by the internally generated and emitted thermal wave that is phase sensitively monitored when it arrives at the sample surface. Therefore, only the defect is detected and not the intact areas of the inspected component. As the phase difference between the modulation and the thermal wave is proportional to the dept h where the defect is located, local depth information is obtained. The typical setup for ULT measurements is shown in Figure 19 . Figure 19. The ULT principal setup.
54 Again, the use of ULT technique for PV industry has initiated not long ago. Recently, a group of German researchers ha d employed ULT method for detection of cracks in silicon wafers and solar cells  The principle of crack detection by ULT is based on the heat created of the cracks fla nks because of their friction caused by the ultrasound driven into silicon wafer. The ULT image of one of the cracks in mc-Si wafer is illustrated in Figure 20. Figure 20. ULT image of the inne r crack in mc-Si wafer . The special resolution of the method de pends on the quality of IR camera incorporated into the ULT setup. Their lock -in thermography system allowed imaging of periodic surface temperature modulations at frequencies up to 54 Hz having an effective value as low as 10 mK using 1/2 hour measur e time. This is still far too long for using this method as an in-line module for solar cells manufacturing.
552.5.3. Millimeter waves Millimeter wave is an electromagnetic wave having a wavelength of 1 mm to 10 mm. Millimeter wave has an advantage that it can propagate well in air, unlike scanning acoustic microscopy and ultrasound lock-in thermography where a coupling medium is needed. The absence of coupling is extremely important in nondestructive testing. In the 1970s, some researchers have attempted usin g microwave to detect surface cracks in metallic components , where a horn ante nna was used. In recent years, some researchers have suggested the use of an open-ended rectangular waveguide in a nearfield fashion . Very recently, a millimeter wave system has been developed utilizing an open-ended coaxial line sensor for detect ion and evaluation of sm all fatigue cracks on metal surface . But the spatial resoluti on and scanning speed of millimeter wave setups used in papers referenced above were not sufficient to satisfy the PV assembly lines where 2-3 seconds per wafer are required. In 2004, Y. Ju et al. demonstrated the detection of small crack in the polycrystalline silicon substrate by using a slit aperture sensor with a high testin g speed of 35mm/s . In this paper, a millimeter wave signal of 110GHz was used and the amplitude of the reflection coefficient was measured. Th e configuration of the millimeter wave measurement system is shown in Figure 21. The design of the setup is attractive due to its simplicity. A network analyzer was used to ge nerate a continuous wave signal fed to the sensor and to measure the amplitude of reflection coefficient at the sensor aperture. A computer was used to synchronize the stag e movements for x-y scanning process. In
56 order to apply technique to in-line testing, a high speed testing of 35 mm/s was kept. Therefore, for testing an area of 50x35sq.mm only 1 second was needed. Figure 21. Principal configuration of the millimeter wave measurement system . Figure 22 shows the photogra ph of the sample. There is no indication of the crack size, but assuming that the silicon substrate had a dime nsion of 125x125mm then rough estimations gives the crack size to be about 23cm. It is still unclear whether or not the method is capable to detect cr acks not only in bare but processed, af ter metallization, wafers. A god example is when the crack is located beneath the metal contact.
57 Figure 22. Millimeter wave image of the 125x125mm polycrystalline silicon wafer . In Table 2 we presented a summary of th e methods for crack monitoring in silicon wafers for photovoltaics.
58 Table 2. The set of vital parameters of the methods for crack inspection in wafers. Method Time required to inspect 100x100mm wafer Minimum crack length Additional comments SAM ~ 10-15min (Depends on scanning resolution) >5-10 microns Acoustic coupling required. ULT ~ 5-10s >100 microns (Depends on cameraÂ’s resolution) Thermal coupling required. MW ~ 3-5s >400 microns Wafers before metallization only. As one can see there is no common and yet accepted method for mechanical quality control in solar-grade commercial fu ll-size silicon wafers. The in-line quality control requires methods which can achieve measurement times of around 2 seconds per wafer to match a throughput of typical produ ction lines. Therefore, the efforts of the current research in creating a reliable and ve rsatile tool for stress monitoring and crack detection were highly motivated by todayÂ’ s state of the photovoltaic industry.
59 3. Experimental All experimental work has been done using research facilities of the Spectroscopy Laboratory at USF. The setup for resonance ul trasonic vibrations for measuring flexural vibrations in silicon wafers was designed and developed previously and described in my masterÂ’s thesis  and with important upg rade, it was rebuilt for the longitudinal mode. The setup for scanning infrared polariscopy wa s built and described by Dr. I. Tarasov in . Scanning acoustic micros copy results were obtained on HS1000 HiSPEED produced by Sonix Inc. Finite element analysis and calculations were performed in collaboration with Mr. O. Polupan. 3.1. Hardware 3.1.1. Setup for resonance ultrasonic vibrations measurements In the resonant ultrasonic method, ultrason ic vibrations of a tunable frequency and adjustable amplitude are applied to the enti re silicon wafer. Ultr asonic vibrations are generated in the wafer using a resonati ng piezoelectric transducer. Two circular
60 transducers of different size (2 or 3 inches in diameter) were used. The transducers were manufactured from a piezoelectr ic ceramic material (PZT-5H). The transducer contains a central hole which provides a reliable vacuum coupling between the wafer and transducer by applying a small (~ 50 kPa) negative pressure to the backside of the wafer (Figure 23). Figure 23. Experimental set-up for re sonance ultrasonic vibration (RUV) measurements. Ultrasonic vibrations are pr opagated into the wafer from the transducer and form standing acoustic waves at sp ecific resonance frequencies. This vacuum method, to couple the wafer and transducer, allows fast wafer exchange and provides simple wafer alignment within 100 micron accuracy. The am plitude and spatial distribution of the standing waves are measured using a broad-ba nd ultrasonic probe. A function generator (WaveTek 10 MHz DDS 29) and a broadband power amplifier (Samson Servo-240)
61 provide an ac driving voltage to the transducer w ith a tunable frequency and adjustable amplitude. This geometry of acoustic loadi ng offers a quick sample exchange and does not damage the front surface or bulk region of the wafer. The ac voltage from the acoustic probe is recorded using a lock-in amplifier (Stanford Research Systems SR850), which is synchronized to the frequency, f, and phase of the driving generator. In the present design, the ultrasonic probe meas ures the longitudinal vibration mode characteristics by contacting the edge of the wafer with a controlled force. The resonance frequency of the longitudina l vibration mode is independent of the wafer thickness (h), in contrast to the flexural vi bration mode, which is proportional to h3/2 . This is especially beneficial in the mc-Si ribbon wafers which may have significant thickness variations of up to 20% ac ross the wafer, as well as from wafer to wafer. The RUV system is computer controlled to achieve fast data acquisition and analyses. Additionally, the transducer and wafer can be rotated using a rotary table. A 8Â” linear moving stage (New England Affiliated Technologies Inc.) was used for radial probe movement. Step motors were synchroni zed by a digital movement controller (Unidex-11 by Aerotech). This allowed us to acquire the spatial distribution data with a minimum step size of 10-microns.
623.1.2. Infrared polariscopy setup for res idual elastic stress measurements To characterize and quantify the level and spatial distribution of in-plane stresses in the material, we have used a scanning linear infrared (IR) polariscopy technique . In linear IR polariscopy, a collimated light from a 75 W halogen tungsten lamp can be focused from a 1 mm diameter down to 60 m. The intensity of the optical transmission is measured at a 1.3 m wavelength corresponding to a spectral region where Si is transparent (Fig. 24). Figure 24. Schematic of the setup for scanning linear infrared polariscopy measurements.
63 Additionally, two infrared linear polarizers ar e used and oriented with respect to a pre-selected crystal di rection, such as the growth dire ction of the mc-Si ribbon. One of them (the polarizer) is located in front of the sample and the second (the analyzer) immediately behind the sample. The polarizat ion intensity is measured at selected orientations of the polarizer and analy zer and depends on the optical retardation parameter, which in turn is directly related to the value of the residual elastic stress. In general, the intensity of polarization transm ission can be expressed as follows : ] 2 / sin ) ( 2 sin ) ( 2 sin [cos ) 1 (2 2 2 R I Io (1) where Io is the intensity of incident light, R is the reflectivity, is the principal angle which determines the orientation of the strain axis at the plane, is the angle between polarizer and analyzer and is the azimuth angle of the polarizer. The polarization intensity, I, measured at selected orientations of the polarizer and analyzer, depends on the optical retardation parameter To derive the relationship between and stress components, the following algor ithm is applied. Two transmitted intensities of the polarized light are measured : one with polarizer a nd analyzer parallel to each other, I = I ( = 0), and the second when the polarizer is orthogonal to the analyzer, I = I ( = /2). By measuring the angular -dependence of the ratio I/(I + I) one can determine quantitatively the optical retardation, and the direction of the principal stress angle, using the equation I/(I + I)=sin22()sin2/2. (2)
64 As the last step of the anal ysis, two strain components, |uzz-uxx| and |uxz|, can be calculated. The first term represents a difference in tensile strains along the crystallographic Z and X directions, while the second term is the shear strain component between Z and X. The following can be used for calculating strains: ) 2 cos( )] /( )[ / (12 11 3 p p dn u uo xx zz (3a) 2 sin ) / )( / (44 3p dn uo xz (3b) where pij are photoelastic constants, d is sample thickness and no is the refractive index of the unstrained material. In the case of EFGSi wafers, the following values were used: =1.3 m, no=3.5, p11=0.081, p12=0.001, p44=0.075. Strain component s can be converted to matching values of stress usi ng elastic constant s of silicon: c11 = 1.657e12 dyn/cm2, c12 = 0.639e12 dyn/cm2, c44 = 0.7956e12 dyn/cm2 . 3.1.3. Scanning acoustic microscope The HS1000 HiSPEEDÂ™ Scanning Acoustic Mi croscope (SAM) by Sonix Inc was used for surface morphology and structural (bulk) integrity evaluation. In SAM, a focused acoustic beam is scanned over the front and back surfaces of the wafer (Fig. 25).
65 Figure 25. Principal setup for SAM operation. The sound pulses are transmitted through the wafer and the reflection from the wafer interfaces is monitored. The ultrasonic pulses are generated by a high-frequency piezoelectric transducer. An electrical pulse fr om a high voltage transmitter is converted to mechanical energy. This activation cause s the transducer to vibrate at a specific frequency causing ultrasonic pulses to be transmitted from the transducer. These pulses travel through the material at the materials ve locity and are reflected at the interfaces of the material it strikes. The ultrasonic en ergy does not travel well through air, so the wafers have to be placed in a coupling medi um (deionized water ba th). The system uses the pulse echo technique and operates at frequency up to 250 MHz. The pulses are repeated at the repetition rate of 20 KHz, so echoes from one pulse do not overlap those
66 from the next. The returned echoes are receive d by the transducer and converted back to voltages. The voltage data is amplified and digiti zed, providing peak amplitude, peak phase and Â“time-of-flightÂ” (TOF) data The TOF represents the tim e required for a pulse to travel back and forth across th e wafer thickness. TOF data we re further used to obtain thickness maps on mc-Si wafers. We used a typical value for the longitudinal speed of sound in silicon of 8600m/s to calculate the thickness. E ach map was measured over a full wafer area, with a step size of 50 m. Steps as small as 5 m can also be used for high-resolution imaging, although this significa ntly decreases the data acquisition time. Figure 26. Cross section of the transdu cer (a) and the intensity of acoustic field (b).
67 Spatial resolution of the method is strongl y dependent on the ultrasonic beam spot size formed by the piezoelectric high-freque ncy transducer. Figure 26 shows the cross section of the transducer (a) and the intens ity of acoustic field (b ) generated by it. The beam spot size at 6dB is govern ed by following relationship: d f c F XdB 028 16, (1) where F is a focal length in water, c is the sound velocity of the sampleÂ’s material (silicon in our case), f is an operational frequency of the transducer, d is an active piezoelectric element diameter. Assuming, that F=5.9mm, c=8600m/s, f=250MHz and d=5mm, a beam spot size of about 4 microns was obtained. 3.2. Finite element modeling Modal analysis of the free edge silic on wafersÂ’ vibrations was performed using ANSYS 7.0-8.0 software packages based on the finite element analysis . The EFG-Si 100mm by 100 mm wafers were modeled using a 100x100 mesh, which makes the size of individual shell element to be equal to 1 mm 1mm. The square wafers are modeled as an isotropic material with a Young modulus of 1.671012 dyn/cm2, a PoissonÂ’s coefficient of 0.3 and a density of 2.329 g/cm3. In a case of Cz-Si 125mm by 125mm pseudo square wafers were modeled using the 125x125 mesh, with the size of individual elements being 1.25 mm x 1.25mm. The pseudo-square wafers are modeled with isot ropic eight-node planar elements with a
68 Young modulus of 1.671012 dyn/cm2, a PoissonÂ’s coefficient of 0.3 and a density of 2.329 g/cm3. 3.3. Samples Silicon wafers used during this research may be subdivided in two groups. The first group of wafers was formed by those whic h had been used for st ress evaluation. The second group was formed in order to perfor m experiments on crack detection in silicon wafers. Two types of silicon wafers have been included in the first group. The first type is circular single crystalline Cz-Si wafers of three different diam eters of 150, 200 and 300 mm, which are typically used in the microelectronic industry. The second type is square mc-Si ribbon wafers produced by the Edge-def ined Film-fed Growth (EFG) technique used in solar cell manufacturing. A set of twelve 100x100mm as-grown EFG wafers with thicknesses between 340 and 370 m was chosen. A set of identical 125mm x 125mm pseudosquare shaped (100) oriented Cz-Si wafers formed the second group. The pseudo-square shape represents one of the accepted photovoltaic industry standards. The wafers we re as-grown with a nominal thickness of 0.035 cm.
69 4. Results and Discussion 4.1. Resonance ultrasonic vibrations appr oach for stress evaluation in full-sized silicon wafers In this chapter a resonance vibra tion approach applied to measure nondestructively residual stress in full size multicry stalline silicon wafers used in solar cell manufacturing will be presented. This met hod is based on excitation of the longitudinal resonance ultrasonic vibrations in the material using an exte rnal piezoelectric transducer combined with a highly sensitive ultrasonic pr obe with data acquis ition of the frequency response to make the method suitable for in-line diagnostics during wafer and cell manufacturing. Theoretical and experimental analyses of the vibration mode in single crystal and multicrystalline silicon wafers are used to provide a benchmark reference analysis and validation of the approach.
70 4.1.1. Single crystal Czochralski (Cz) silicon The main purpose of this theoretical an alyses and experimental data on Cz-Si wafers was to validate our approach. Firs t the resonance ultrasonic methodology for circular single crystal silicon wafers was implemented. These wafers have typically internal stress values below 1MPa and can se rve here as model ob jects allowing solving analytically a general equati on of the free longitudinal vibrations and to calculate resonance vibration frequencies. 220.127.116.11. Analytical modeling Pure radial vibrations in a th in circular plate with radius R and thickness h, assuming that h<
71 where E and are Young modulus and Poisson coeffi cient in x-y plane respectively. For further consideration of the longitudinal vibratio ns it is convenient to rewrite (4) in polar coordinates r Let Ur and U be the respective component s of the strain vector U. Then, using the symmetry of the problem, these components may be written as U=0 Ur=U Thus the components of uij in polar coordinates are: 0 , r rru r U u r U u (5) Therefore the equations (4) in polar c oordinates can be expressed as follows: ) ( 1 ) ( 12 2 u u E u u Err rr rr (6) The equation for longitudinal vibrations in polar coordinates had been given in many sources (see for example ). In our case, for radial strain, we have: r r t Urr rr 2 2 (7) where is a density of Si. Substituting (6) in (7) and using the expressions for uij described by (5) one can apply the method of separation of the variables for equation (7). The above procedure transforms (7) to the following form:
72 0 ) 1 (2 2 2 2 2 U r q dr dU r d r U d r (8) where ) 1 (2E q (9) Equation (8) represents the Besse l equation of the first order and one of its solutions is ) ( ) (1qr AJ r U (10) where J1(qr) is the Bessel function of the first order. Applying the free edge boundary condition, 0 R r rr, it yields: 0 ) ( ) 1 ( ) (1 0 J J (11) where is a dimensionless variable equal to qR The first ten roots ( m=110 ) of equation (11) for =0.3 are presented in Table 3. The resp ective resonance frequencies of these modes, fm, can be calculated using the following relation:
73 ) 1 ( 22E R fm m (12) Table 3. The set of root s of the equation (11). m m 1 2.049 2 5.389 3 8.572 4 11.732 5 14.862 6 18.025 7 21.173 8 24.318 9 27.463 10 30.612 These values are plotted in Figure 27 for Cz-Si wafers of the three diameters 150, 200 and 300mm. In calculations of the reso nant frequency the following material constants E=1.671012 dyn/cm2, =0.3, =2.329 g/cm3 were used.
74 Figure 27. Normal frequencies of the longitudinal vibrati ons in circlular silicon wafers of different diamet ers: (circles) Â– 150mm, (triangles) 200mm, and (squares) 300mm.
75 18.104.22.168. Resonance ultrasonic vibrations data The calculated frequencies are in a good agreement with RUV data on the 300 mm Cz-Si wafer (Figure 28). Table 4 compares the first three theoretically predicted resonant frequencies for the 300 mm wafer to the experimentally measured resonant frequencies Figure 28. Frequency scan measured on 300mm Cz-Si wafer with indicated by arrows first three radial modes. The insert zooms in the mode for m=2. Finite element analysis (FEA) has also been employed to compare experimentally obtained data with theory presented in Table 4. As one can see, the measured, analytically calculated and FEA modeled reso nant frequencies are well correlated (with
76 deviations of less than 1%). This confirms the model that was applied to explain the origin and the type of ultrason ic excitations exhibited in the wafer. Some deviation of the experimental data is anticipated because of the ultrasonic transducer used in mechanical contact with wafer is a source of vibrations th at could cause perturba tion to the free wafer vibration spectrum. There are also a number of other resonant frequencies shown in Figure 28 which are generated by asym metric longitudinal vibrations. Table 4. Resonant frequencies [Hz] of radial angular independent longitudinal vibrations meas ured experimentally on 300mm diameter Cz-Si wafer compared to calculated using Eq. (12) and modeled with Finite Element Analyses. m Experiment [Hz] Theory [Hz] FEA [Hz] 1 20071 19425 19297 2 51809 51063 50760 3 80688 81224 80734
77 22.214.171.124. Chladni figures The theoretical model was further validat ed through classic Chladni type patterns presented in Figure 29. These were obtained by sprinkling fi ne sand on the wafer while dwelling at each mode frequency. The sand collects at the nodal lines of the mode shapes . Historically, this method is considered to be the oldest experimental way to study the nodes of vibration of circular and square plates. Ernst Chladni first demonstrated this at the French Academy of Sc ience in 1808, it caused such interest that the Emperor offered a kilogram of gold to the first person who could explain the patterns. Comparing the nodal lines from the calcula ted mode shapes in Figure 30a with the nodal sand lines in Figure 29 for mode with m=2 shows excellent agreement.
78 m=2, f=51809 Hz m=3, f=80688 Hz Figure 29. Chladni sand patterns on 300mm Cz-Si wafers for different mode shapes.
79 Figure 30a. Computed mode shap e for 300mm Cz-Si wafer, n=5
80 Figure 30b. Experimental radial distri bution of the normalized vibrationsÂ’ amplitude versus theoretical curve. In addition, notice that the same type of experiments were carried out on square shaped 100x100mm EFG-Si wafers, but unfort unately the transducer used disturbed significantly the sand patterns around the central area of the wafer. Ther efore, it would be problematic and inappropriate to compare ex perimental distribution with theoretically modeled. Some examples of Chladni patterns on 100x100mm EF G-Si wafers are presented in Figure 31.
81 Figure 31. Chladni sand pattern obse rved on 100x100mm EFG-Si wafers at f ~47kHz. 4.1.2. Edge-defined Film-fed Grow th (EFG) silicon wafers 126.96.36.199. Scanning acoustic microscopy results To study longitudinal vibrations in square shaped wafers using the RUV technique, twelve 100 mm x100 mm mc-Si EFG wafers with thicknesses ranging from 340 to 370 m and a set of 125mm x125mm cast wafe rs were screened using a high resolution SAM technique, as describe d above, to check that cracks over 10 m length
82 were not present at the wafer periphery. As an example, Figure 32 shows a full image of the EFG wafer with a mm-size peripheral cr ack. The wafer was rejected from the RUV experiments based on SAM screening. The cr ack initially had a length of a few mm and eventually propagated to cause the wafer breakage (Figure 32b). (a) (b) Figure 32. SAM image of the 100mm x 100mm EFG wafer with periphery crack: (a) image of the entire wafer, the crack is boxed out in white, (b) zooming area of the crack with 5 micron step.
8188.8.131.52. Residual elastic stress data by infrared polariscopy technique In the set of EFG wafers measured, the sp atial distribution of the in-plane stress using scanning linear IR polariscopy, and stre ss maps of all test wafers were obtained using the algorithm described in Section 3.1.2. Representative maps of Â“lowÂ”-stress and Â“highÂ”-stress wafers are presented in Figur e 33. Each stress map contains 100x100 data points obtained with a 1 mm spatial resoluti on. Figure 32a shows an example of a fairly uniform stress distribution over most of the low-stress EFG wafer #16 with stress average value of 2.8MPa. In contrast, significant vari ation in residual stress within EFG wafer #22 (average stress is 5.6MPa) is shown in Figure 33b. To quantify the stress mapping, we present in Figure 33c two line scans meas ured across the growth directions in both wafers. These show quite different stress di stributions. The values of measured wafer thickness and calculated average and peak stress are presented in Table 5. The stress values are corrected to account for the aver age wafer thickness, which was measured using the SAM TOF technique and also confirmed by data of the wafer weight and density (Si = 2.329 g/cm3).
84 (a) (b) (c) Figure 33. Result of infrared polariscopy stress field mapping: (a) low-stress wafer (#16, Avg. Stress = 2.8MPa) and (b) high-stress wafer ( #22, Avg. Stress = 5.6MPa). Contrast bar is [MPa] values. The dotted line on maps indicates the position of scan lines we have shown in (c).
8184.108.40.206. Resonance ultrasonic vibrations data Using the RUV technique described a bove, frequency scans from 10 to 100 kHz were performed on EFG wafers. A typical f-scan is presented in the insert in Figure 34. A dominant vibration mode at ~48 kHz was sel ected in order to correlate the resonant frequencies with the respective values of the elastic stress. A noticea ble variation of the resonance frequency on a set of identical test wafers (Table 5) was observed, which can be attributed to the stress variations. In Figure 35 the dependence of the resonance frequency on the average value of in-plane stress for these EFG wafers is presented. Data points show a trend of a resonance frequency sh ift to higher values with the increase of the average stress in the wafer. The resonanc e ultrasonic mode fre quency increases with increasing stress suggesting that residu al stress increases wafer stiffness. Modal analysis of the free edge square waferÂ’s vibrations is performed using ANSYS software package based on the finite element method . The wafer is modeled as a 100x100 mesh, which makes the size of indi vidual shell element to be equal to 1 mm 1mm or 1.25 mm x 1.25mm. The square wafers are modeled as an isotropic material with a Young modulus of 1.67 1012 dyn/cm2, a PoissonÂ’s coefficient of 0.3 and a density of 2.329 g/cm3. Table 6 demonstrates calculated resonant frequencies of 100mm x 100mm and 125mm x 125mm square wafers compared to the experimentally measured ones. Two computed vibration mode s are illustrated in Figure 36.
86 Figure 34. Frequency scans at one of principal maximum on two EFG wafers: high-stress wafer #22 with aver age stress of 5.65 MPa, and lowstress wafer #16 Â– with average stress of 2.81 MPa. The insert shows a typical frequency scan in a range from 10kHz to 100kHz.
87 Table 5. Average thickness, average and peak stress, and resonance vibration frequencies of 100mm x 100mm EFG wafers. Wafer I.D. [#] Thickness [microns] Average Stress [MPa] Peak Stress [MPa] Resonance frequency [KHz] 13 348 5.06 28.58 47.80 14 341 4.37 31.37 48.38 15 340 4.10 29.46 47.66 16 366 2.81 25.01 47.60 17 369 3.33 30.57 47.99 18 356 4.58 37.78 47.96 19 343 4.11 29.56 48.98 20 349 4.93 28.66 49.04 21 344 5.38 44.77 48.68 22 346 5.65 29.17 48.78 23 347 4.36 29.84 48.15 24 344 4.10 29.54 47.80
88 Figure 35. Experimentally measured fre quencies of the resonance longitudinal vibration mode versus average st ress in a set of 100mm x 100mm EFG wafers.
89 Table 6. Average experimental res onance frequencies [Hz] of the three selected longitudinal vibrati on modes for 100x100mm EFG-Si and 125x125mm cast-Si wafers compared with FEA calculated values. Experimental values show standard deviation of 1% (see Table 5). 100x100mm EFG-Si 125x125mm cast-Si FEA [Hz] Experiment [Hz] FEA [Hz] Experiment [Hz] 48772 48620510 39018 41210 60794 62540 48635 49520 85381 88350 59754 60450
90 (a) (b) Figure 36. Finite Element Analysis cal culations of the first two principal mode shapes at frequency of (a) 48772 Hz and (b) 60794 Hz. The X and Y are scaled in cm.
9220.127.116.11. Bending test Additional evidence of a stress dependent resonance frequency shift is achieved from the following 5-point wafer bending experi ment. In this test, a silicon wafer was loaded vertically upward using supporting pins applied to the four corners of the wafer while it was held in its center with a vac uum (Figure 37). Concurrently, the frequency scan was taken at each loading value and the resonance vibration frequency peak location determined (Figure 38). One can see the tendency of the wafer resonance frequency location to shift to higher values with the increasing load. This loading experiment provides additional experimental evidence linking the resonant peak frequency shift to stress, as found for EFG wafers. Figure 37. Schematics of the four-point bending test.
92 Figure 38. Frequency curves of the resona nce vibration mode at different load values using 5-point bending test. Insert shows the resonance frequency versus loading values.
934.2. Crack detection and analyses usin g resonance ultrasonic vibrations 4.2.1. Initial measurements A set of identical 125mm x 125mm pseudosquare shaped (100) oriented Cz-Si wafers were chosen for this study. This sh ape represents one of the accepted photovoltaic industry standards. The wafers were as-gro wn with a nominal thickness of 0.035 cm. All wafers were initially screened by SAM for st ructural integrity. The SAM maps have been measured with 100 m step and no periphery or bulk cr acks have been observed within this accuracy. Using the RUV vibrations t echnique frequency scans of the longitudinal vibrations from 10 to 100 kHz were measured on all wafers. A typical full range acoustic spectrum obtained on wafer #1 is shown in Figure 39. In the frequency spectrum, a set of distinctive resonance modes was observed, whic h are consistently reproduced from wafer to wafer in terms of the maximum amp litude, frequency position and bandwidth ( BW) One of these resonance modes, assigned he reafter as an A-mode, is observed at f=49,93020 Hz Tracking f-scan characteristics of the A-mode as a means of crack detection and control in the wafers was proposed. A pr imary criterion for the A-mode selection is a noticeable clearance from the ne ighbor peaks to avoid their overlapping and interference, which would redu ce the accuracy of the mode analyses. As seen from Figure 39 (insert), the A-mode is ind eed a stand-alone narrow peak with BW =90 10 Hz.
94 Figure 39. Full range ultrasonic freque ncy spectrum obtained on Cz-Si wafer #1. Insert shows the A-mode experi mental data (open circles) and fitting Lorentz approximation (solid line). BW parameter is shown by arrows. We emphasize that the A-mode frequency scan and shape analyses can be measured with sufficient signal-to-noise ra tio in a time range of a few seconds. This obviously makes the RUV approach feasible and attractive for potential implementation as an in-line wafer quality control module. The other important feature of the RUV method is that according to vibration th eory, the resonance frequencies of the longitudinal vibration modes are in dependent of the wafer thickness ( h ), in contrast to the flexural vibrations, which are proportional to h3/2 . This is especially beneficial in the multicrystalline ribbon silicon wafers, which ma y have significant (up to 20%) thickness variations across the wafer, as well as from wafer to wafer. This statement was tested by
95 varying the Cz-Si waferÂ’s thickness between 100 and 350 microns. Th e frequency of the principle vibration modes observed in the RUV scans show slight (1.6%) high-frequency shift with reduced wafer thickness, which can be attributed to stress enhancement as was previously observed in mc-Si ribbon wafers . It should be noted that every resonant peak shown in Figure 39 represents a cer tain longitudinal vi bration mode, both symmetrical and asymmetrical. But for now, only the A-mode was considered. The solid line in the insert in Figure 39 shows the numeric approximation of the experimental frequency sweep for the A-mode. This appr oximation is obtained with the Lorentz function and the following parameters have b een extracted: peak resonance frequency, f0, and bandwidth, BW 4.2.2. Crack engineering To introduce or engineer cracks with diffe rent sizes, few wafers were cracked by scribing the wafer edge with a diamond pin. As expected, the crack originates at the spot of the diamond application and are oriented along <110> crystallogr aphic directions. The length of each engineered crack has been measured using SAM in the precision mode with a 10 m step. The SAM image of one of thes e cracks is presented in Figure 40a. After careful SAM study, the RUV measuremen ts on the wafers with cracks have been carried out and the frequency spectra of the A-mode recorded. In Figure 40, the A-mode spectra in wafers with different crack si zes was demonstrated. Clearly, the A-mode frequency decreases with incr easing crack length (also see f line in insert in Figure 37).
96 It was noticed that the vibr ation spectrum of the wafer with the 28 mm crack is nonsymmetrical indicating sufficient asymmetry in wafer elasticity induced by the crack. In addition, the A-mode BW increases with crack length. One was able to clearly detect mm-size cracks by assessing A-mode line sh ift and broadening. Specifically, a 3mm crack is estimated from a 160Hz shift and an 8% increase in BW Thus the RUV approach offers sub-millimeter crack length sensitivity in Cz-Si wafers.
97 (a) (b) Figure 40. SAM image of the 125mm x 125mm Cz-Si wafer with introduced 28mm periphery crack (a). Vi bration mode at 51052Hz obtained with Finite Element Analysis calculations on the wafer with identical crack (b).
98 The observed decrease of the A-mode frequency and its dependence on crack length are consistent with fini te element analysis (FEA) . The wafer is modeled as a 125x125 mesh, with the size of individual elements being 1. 25 mm x 1.25mm. The pseudo-square wafers are modeled with isot ropic eight-node planar elements with a YoungÂ’s modulus of 1.67 1012 dyn/cm2, a PoissonÂ’s coefficient of 0.3 and a density of 2.329 g/cm3. The resonance frequencies up to 100KHz and respective mode shapes of the free edge longitudinal vibrations of the shell were calculat ed . We concentrated on the frequency shift closest to th e experiment vibration mode at f0=51,052Hz. Figure 40b shows a calculated profile of the vibration mo de on the wafer with identical geometry as the experimental Cz-Si wafers containing a 28mm periphery crack. The crack size and location are identical to the experimental va lues. The FEA shows a decrease in frequency shift with increased cr ack length which supports the expe rimental values in Figure 41. However, calculated values show a much larger frequency shift than observed experimentally. It was initially thought that one possibility for th is difference could be the increase in damping in the test wafers with increased crack length. However, even though the BW is proportional to damping, and the BW and therefore damping increases with crack length by about an order of magn itude in the tests pr esented, the actual damping levels associated with the longitudina l A-mode are very low for all test wafers. Therefore, the increased damping with crac k length does not notably decrease the modal frequency. To clarify this, the damping level can be quantified by the modal damping ratio nf BW 2 in terms of BW and modal undamped natural frequency fn. The modal damped natural frequency is related to undamped natural frequency and the damping
99 ratio as 21 n df f Since the damping ratio for the cracked wafers range from 0.001 to 0.01, the modal damped and undamped natural frequencies are essentially equal. A more likely possibility for the difference betw een experimental and numerical results is that the FEA models an ideal crack without contact elements and contact forces. Contact forces in a crack are inevitable when the test wafer is subjected to vibratory excitation. Such forces would tend to provide some stiffness and thereby result in the smaller frequency shift observed experimentally compared to the wafer model simulations without contact forces. Contact elements can be introduced in the FEA. Such efforts are currently in progress and are expected to improve the model and therefore estimation capabilities of the method.
100 Figure 41. The A-mode spectra of (a) no n cracked wafer, a nd wafers with different crack lengths: (b) 3.0 mm, (c) 18 mm and (d) 28 mm. The insert shows the dependence of resonant frequency shift and also BW of the longitudinal mode spectra versus crack length.
101 5. Conclusions and recommendations As the principal goal of this PhD research project, resonance ultrasonic vibration methodology was developed both experimental ly and theoretically and applied to the analysis of longitudinal vibrations in full-si ze single-crystal and multi-crystalline silicon wafers primarily for photovoltaic application. Theoretical analysis of the longitudi nal vibrations in single crystal and multicrystalline silicon wafers was created and used to provide a benchmark reference analysis and validation of the approach. Eigenfrequencies and mode shapes were calculated using analytical and finite element approximation approaches. The calculated frequencies were in a good agreement with RUV data obtained on 300 mm Cz-Si wafers. The finite element approximation data also correlated well with both experimental data and theoretical model. Specific vibration modes were identifie d in wafers with different geometric shapes and their resonant frequencies corr elated to in-plane residual stress. In multicrystalline silicon EFG wafers, a clear trend of increasing resonance frequency of the longitudinal vibration mode with higher average in-plane stress obtained with scanning IR polariscopy was observed. Additional evidence of a stress dependent
102 resonance frequency shift is achieved from the 5-point wafer bending experiment. One could see the tendency of the wafer resonance frequency location to sh ift to higher values with the increasing load. That loading experiment provides additional experimental evidence linking the resonant peak frequency shift to stress, as found in EFG wafers. The resonance ultrasonic vibrations method is suitable for fast crack detection and its length determination in full-size solar-grade Cz-Si wafers. A gradual downward shift of the resonant frequency and line broadening of the longitudinal vibration mode versus crack length has been shown. On a set of iden tical crystalline Si wafers with artificially introduced periphery cracks, it was demonstrat ed that the crack results in a frequency shift of a selected RUV peak to lower frequencies and increases the resonance peak band width. Both characteristics Â– peak position and bandwidth Â– are increased with the length of the crack. These dependences demonstrate a direct relation of the extracted parameters on crack length. The probable mechanism of the observed effect is attributed to a decrease in wafer stiffness thus affectin g the vibration mode frequency with the mmlength peripheral crack. Therefore, the freq uency shift and the bandwidth were found to serve as reliable indicators of the crack app earance in silicon wafers and are suitable for mechanical quality control and fast wafer inspection. It is suggested that the proposed mode l needs to be further investigated experimentally, for instance, using wafers with different mechanical parameters (materialÂ’s density, Young modulus) and also theoretically. The finite element modeling
103 of the longitudinal vibrations on wafers with periphery cracks is currently in progress by other students in the group. The RUV system, built on the idea of us ing frequency shift and bandwidth as indicators of wafer quality, allows fast da ta acquisition (as short as 3 seconds) and analyses matching the throughput of solar cell production lines. The results of this work were published or submitted to the following journals: i) A. Belyaev, O. Polupan, S. Ostapenko, D. Hess, and J. Kalejs, J. Appl. Phys. 2005 (in press). ii) A. Belyaev, O. Polupan, W. Dallas, S. Ostapenko, D. Hess, J. Wohlgemuth, Appl. Phys. Letters 2005 (in press). iii) A. Belyaev, S. Ostapenko, W. Dallas, J. Wohlgemuth, Proc. 31st Photovoltaic Specialists Conference, Orlando FL (U. S. A.), January 2005. iv) A. Belyaev, D. Hess, S. Ostapenko, O. Polupan, I. Tarasov, J. Kalejs, Proc. Photovoltaic Specialists Conference, 2004. v) A. Belyaev, S. Lulu, I. Tarasov, S. Ostapenko and J. P. Kalejs, Proc. MRS Spring Meeting, New Orleans, La, 2002.
104 vi) A. Belyaev, S. Lulu, S. Ostapenko, I. Tarasov, J. Kalejs, Proc. Photovoltaic Specialists Conference, 2002.
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112 About the Author Anton Byelyayev received his bachelor de gree in Physics with concentration in Solid State Physics at Kiev State University (Ukraine). His bachelor diploma work was dedicated to investigation of niobium nitride alloys as a Schottky contacts for metalsemiconductor diodes. In February 2000, Anton joined the Ma sterÂ’s program in Defect Engineering Laboratory supervised by Prof. Sergei Ostape nko at the University of South Florida. He successfully defended his thes is in July 2003 and in A ugust of the same year he continued to study the resonan ce ultrasound technique applied to full-sized silicon wafers as a Ph.D. candidate. His Ph.D. work wa s accomplished in May 2005 and in July 2005 Anton defended his dissertation. Now he works for Semiconductor Diagnostics, Inc. located in Tampa, FL.