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X-ray diffraction applications in thin films and (100) silicon substrate stress analysis

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Title:
X-ray diffraction applications in thin films and (100) silicon substrate stress analysis
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Rachwal, James
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University of South Florida
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Thin films
Residual stress
X-Ray diffraction
Sin squared psi technique
Dissertations, Academic -- Mechanical Engineering -- Masters -- USF   ( lcsh )
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non-fiction   ( marcgt )

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Abstract:
ABSTRACT: Silicon is used as a substrate for X-ray mirrors for correct imaging. The substrate needs to be mechanically bent to produce a certain curvature in order to condition and focus the X-ray beam. The X-rays impinge a mirror at very shallow angles, in order to reduce the amount of intensity loss in the diffraction process. The X-ray mirrors need to be bent to an extremely precise profile, and even small distortions from this profile can reduce the effectiveness of the X-ray mirrors. The X-rays that impinge on the mirror also produce large amounts of heat that can change the temperature of the substrate, resulting in its thermal expansion and distortion. By measuring the distortions in-situ caused by these temperature changes it may be possible to correct for these errors. A four-point bending fixture was designed for in-situ X-ray bending experiments in order to measure the distortions to the (100) silicon sample caused by the bending setup. By being able to measure the distortion caused by the setup, in like manner it would be possible to measure distortion caused by thermal expansion. Several alignments were needed in order to obtain accurate results, including adding copper powder on top of the sample. The copper powder that was added is not under stress, and therefore will not shift its reflection peak when the sample is under bending stress, thus serving as a reference in order to make corrections. The strain results were then compared to values calculated from mechanical deflections from bending. Despite the efforts to control accuracy, a significant variation appeared in the values when the top surface was in compression. As an alternative an IONIC stress-gauge was used to measure the deflections of the sample rather than calculate them. Another alternative was to calculate the deflection of the substrate by first determining the stress in the layer deposited onto the mirror's substrate by using sin squared psi technique, then using Stoney's equation to determine the change in curvature of the substrate, with the stress in the layer being known. Several tests were performed to demonstrate the ability to measure these deflections.
Thesis:
Thesis (M.S.M.E.)--University of South Florida, 2010.
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Includes bibliographical references.
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by James Rachwal.
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ABSTRACT: Silicon is used as a substrate for X-ray mirrors for correct imaging. The substrate needs to be mechanically bent to produce a certain curvature in order to condition and focus the X-ray beam. The X-rays impinge a mirror at very shallow angles, in order to reduce the amount of intensity loss in the diffraction process. The X-ray mirrors need to be bent to an extremely precise profile, and even small distortions from this profile can reduce the effectiveness of the X-ray mirrors. The X-rays that impinge on the mirror also produce large amounts of heat that can change the temperature of the substrate, resulting in its thermal expansion and distortion. By measuring the distortions in-situ caused by these temperature changes it may be possible to correct for these errors. A four-point bending fixture was designed for in-situ X-ray bending experiments in order to measure the distortions to the (100) silicon sample caused by the bending setup. By being able to measure the distortion caused by the setup, in like manner it would be possible to measure distortion caused by thermal expansion. Several alignments were needed in order to obtain accurate results, including adding copper powder on top of the sample. The copper powder that was added is not under stress, and therefore will not shift its reflection peak when the sample is under bending stress, thus serving as a reference in order to make corrections. The strain results were then compared to values calculated from mechanical deflections from bending. Despite the efforts to control accuracy, a significant variation appeared in the values when the top surface was in compression. As an alternative an IONIC stress-gauge was used to measure the deflections of the sample rather than calculate them. Another alternative was to calculate the deflection of the substrate by first determining the stress in the layer deposited onto the mirror's substrate by using sin squared psi technique, then using Stoney's equation to determine the change in curvature of the substrate, with the stress in the layer being known. Several tests were performed to demonstrate the ability to measure these deflections.
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X-Ray Diffraction Applications in Thin Films and (100) Silicon Substrate Stress A nalysis by James D. Rachwal A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in Mechanical Engineering Department of Mechanical Engineering College of Engineering University of South Florida Major Professor: Dr. Alex A. Volinsky, Ph.D. Dr. Craig Lusk, Ph.D. Dr. Delcie Durham, Ph.D. Date of Approval: April 7, 2010 Keywords: Thin films, Residual stress, X-Ray diffra ction, Sin2 technique Copyright 2010, James D. Rachwal

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i Table of Contents List of Tables .................................... ................................................... ............................ v List of Figures ................................... ................................................... ........................... vi ABSTRACT .......................................... ................................................... ....................... ix Chapter 1. X-Ray Mirrors .......................... ................................................... ................... 1 1.1 Introduction to X-Ray Mirrors ..................... ............................................ 1 1.2 Distortions in X-Ray Mirrors ...................... ............................................. 2 1.3 Stress in the X-Ray Mirror Thin Film .............. ........................................ 3 Chapter 2. Introduction to Stress in Thin Films ... ................................................... ......... 4 2.1 Origin of Stress in Thin Films .................... ............................................. 4 2.2 Overview of Thin Film Deposition .................. ........................................ 4 2.3 Mechanics of (100) Silicon Substrate .............. ....................................... 5 2.3.1 Crystallographic Properties of (100) Silicon ...... ................................ 5 2.3.2 Elastic Properties of (100) Silicon ............... ...................................... 7 Chapter 3. Stress Characterization Techniques ..... ................................................... ...... 9 3.1 X-Ray Diffraction ................................. ................................................... 9 3.1.1 Lattice Parameter and Bragg’s Law ................. ................................. 9

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ii 3.1.2 Sin2 Technique ........................................ ......................................11 3.1.3 Strain Transformation ............................. .........................................13 3.1.4 Sin2 Equation.......................................... .......................................20 3.1.5 Calculation of Stress Using Sin2 .................................................. ..22 3.1.6 Sin2 Example .......................................... .......................................24 3.2 Curvature Methods ................................. ..............................................25 3.2.1 Stoney’s Equation.................................. ..........................................25 3.2.2 Curvature Measuring Techniques .................... ................................26 Chapter 4. In-Situ Bending Experiments in XRD ..... ................................................... ....29 4.1 Four-Point Bending Apparatus Design ............... ...................................29 4.2 System Alignments ................................. ..............................................31 4.2.1 Vertical Alignment................................. ...........................................31 4.2.2 Zeroing the Sample ................................ .........................................33 4.2.3 Rocking Scan ...................................... ............................................34 4.3 Sources of Error .................................. ..................................................3 6 4.3.1 Errors Due to Change in Height .................... ...................................37 4.4 Depth of X-Ray Penetration ........................ ..........................................38 4.4.1 Basis of X-Ray Absorption ......................... ......................................38 4.4.2 Depth of Attenuation .............................. ..........................................38 4.5 Scanning Procedure ................................ .............................................40

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iii 4.6 Peak Determination ................................ ..............................................40 4.6.1 Peak Height ....................................... ..............................................40 4.6.2 Peak Determination Functions ...................... ...................................41 4.6.3 Procedure for Peak Determination .................. .................................42 4.7 Error Corrections ................................. .................................................44 4.8 In-Situ XRD results ............................... ................................................45 4.9 Comparison of XRD Results with Calculated Bending S trains ..............47 Chapter 5. System Calibration and Experiment with I ONIC-System Stress-Gauge .......49 5.1 IONIC-Systems Profile Calibration ................. .......................................49 5.1.1 Factory Calibration ............................... ...........................................49 5.1.2 Taking Readings ................................... ..........................................49 5.1.3 Profile Calculation................................ ............................................53 5.2 IONIC-System Experiment ........................... .........................................54 5.2.1 IONIC-Systems Stress-Gauge Readings ............... ..........................54 5.2.2 Wafer Deflection Equation ......................... ......................................55 5.2.3 Variation Caused by Anisotropic Properties ........ .............................57 5.2.4 IONIC Stress-Gauge Error Analysis.................. ...............................58 5.2.5 Wafer Deflection Results .......................... .......................................59 Chapter 6. Conclusions and Future Work ............ ................................................... .......60 References ........................................ ................................................... .........................62 Apendix A ......................................... ................................................... ..........................68

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iv A-1 IONIC-Systems Stress-Gauge ........................ ......................................68 A-1.1 IONIC-Systems Stress-Gauge Calibration Setup ...... ......................74 A-1.2 Taking a Measurement on the IONIC-Systems Stress-Ga uge ........76

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v List of Tables Table 1. Sin2 Data for Tungsten Film on a (100) Silicon Wafer. .................................24 Table 2. Mechanical Properties of Tungsten. ...... ................................................... .......25 Table 3. Comparison of Calculated and XRD Strain R esults. .......................................48 Table 4. Results From Stress-Gauge Profile Calibra tion. ............................................. .54 Table 5. Readings From IONIC-Systems Stress-Gauge for Various Weights Added, and the Corresponding Displacements. ....... .......................................... 55 Table 6. Wafer Deflection Measurements and Calcula tions. .........................................59 Table 7. Errors From Deflection Calculations. .... ................................................... ........59

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vi List of Figures Figure 1. Illustration of a Gbel Mirror. ........ ................................................... ................. 1 Figure 2. Illustration of a Gbel mirror distorted by thermal expansion. ........................... 3 Figure 3. Crystal Structure of Silicon............ ................................................... ................ 5 Figure 4. (a) Identifying features of single crys tal silicon wafers. ............................... ..... 6 Figure 5. Determination of lattice spacing using B ragg’s law. ....................................... 11 Figure 6. Coordinate System Used for the sin2 Technique. ....................................... 12 Figure 7. Basic principles of sin2 stress determination. ............................ ................... 13 Figure 8. x Component of the 3D-Strain Transformation to the x’ Axis......................... 14 Figure 9. y Component of the 3D-Strain Transformation to the x’ Axis......................... 15 Figure 10. z Component of the 3D-Strain Transformation to the x’-Axis. ..................... 16 Figure 11. xz Component of the 3D-Strain Transformation to the x’-Axis. .................... 17 Figure 12. (a) xz Component of the 3D-Strain Transformation to the x’-Axis. (b) Close-Up of the Strain Transformation Components. ........................................ 18 Figure 13. (a) yz Component of the 3D-Strain Transformation to the x’-Axis. (b) Close-Up of the Strain Components. ................ .................................................. 19 Figure 14. d vs. Sin2 for (a) Linear Behavior, (b) Splitting Behavior, and (c) Oscillatory Behavior. Adapted from [24]. ......... .................................................. 21 Figure 15. Example of a d vs. Sin2 Plot for Tungsten Film. .......................... ............. 23 Figure 16. Forces Acting on a Cross-Sectional Area Act as the Basis of Stoney’s Equation. ......................................... ................................................... ............... 25 Figure 17. IONIC Stress-Gauge Measurement. ....... ................................................... .. 26

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vii Figure 18. (a) 4-Point Bending Fixture. (b) Descri ption of Parts. ................................... 29 Figure 19. Constant Radius of Curvature Between In ner Bending Points on a 4Point Bending Setup. .............................. ................................................... ........ 30 Figure 20. Alignment Test for Vertical Position. ................................................... ........ 31 Figure 21. Illustration of an Alignment Test, with the Wafer Too L4ow. ......................... 32 Figure 22. Illustration of an Alignment Test with the Wafer Too High. ........................... 32 Figure 23. The 2 -Scan with a Misaligned Wafer. .................... ..................................... 34 Figure 24. Relative Intensity of Diffracted X-Rays ................................................. ....... 35 Figure 25. Rocking-Scan Used to Determine the Alig nment of the Wafer. .................... 35 Figure 26. Using an Offset Angle to Correct for An gular Alignment. ............................. 36 Figure 27. Source of Error Due to Changing Height of Wafer Sample. ......................... 37 Figure 28. Schematic Representation of a Diffracti on Reflection, (a) Before, and (b) After Stripping Cu[k 2] Off of the Raw Data. .......................... ...................... 43 Figure 29. Four-Point Bending Setup for X-Ray Diff raction with Copper Powder Added. ............................................ ................................................... ................ 44 Figure 30. Illustration of Copper Powder Correctio n. ................................................ .... 45 Figure 31. Raw data from the X-ray diffraction sca ns, after adjusting for copper 220 XRD Reflections, Showing Cu[k 1], and Cu[k 2] Peaks.............................. 46 Figure 32. Setup of Digital Indicator on the IONI C-Systems Stress-Gauge for the Profile Calibration. .......................... ................................................... ........... 51 Figure 33. Data Used in Stress-Gauge Profile Calib ration. ........................................... 53 Figure 34. (a) Weight Producing a Small Circular Area Load on the Wafer. (b) Lip Produced by Machining Process. (c) Load from W eight Concentrated Along Outer Edge of Weight-Flat. .................. ................................................... 56 Figure 35. Back panel of IONIC-systems stress-gaug e................................................. 69

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viii Figure 36. Top of IONIC-Systems stress-gauge. .... ................................................... ... 70

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ix X-Ray Diffraction Applications in Thin Films and (100) Silicon Substrate Stress A nalysis James D. Rachwal ABSTRACT Silicon is used as a substrate for X-ray mirrors fo r correct imaging. The substrate needs to be mechanically bent to produce a certain curvature in order to condition and focus the X-ray beam. The X-rays impinge a mirror at very shallow angles, in order to reduce the amount of intensity loss in the diffract ion process. The X-ray mirrors need to be bent to an extremely precise profile, and even s mall distortions from this profile can reduce the effectiveness of the X-ray mirrors. The X-rays that impinge on the mirror also produce large amounts of heat that can change the t emperature of the substrate, resulting in its thermal expansion and distortion. By measuring the distortions in-situ caused by these temperature changes it may be possi ble to correct for these errors. A four-point bending fixture was designed for in-situ X-ray bending experiments in order to measure the distortions to the (100) silicon sample caused by the bending setup. By being able to measure the distortion caused by the setup, in like manner it would be possible to measure distortion caused by thermal ex pansion. Several alignments were needed in order to obtain accurate results, includi ng adding copper powder on top of the sample. The copper powder that was added is not un der stress, and therefore will not shift its reflection peak when the sample is under bending stress, thus serving as a reference in order to make corrections. The strain results were then compared to values calculated from mechanical deflections from bending Despite the efforts to control accuracy, a significant variation appeared in the v alues when the top surface was in

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x compression. As an alternative an IONIC stress-gau ge was used to measure the deflections of the sample rather than calculate the m. Another alternative was to calculate the deflection of the substrate by first determining the stress in the layer deposited onto the mirror’s substrate by using sin2 technique, then using Stoney’s equation to determine the change in curvature of th e substrate, with the stress in the layer being known. Several tests were performed to demonstrate the ability to measure these deflections.

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1 Chapter 1. X-Ray Mirrors 1.1 Introduction to X-Ray Mirrors Silicon is used as a substrate for X-ray mirrors co mmonly known as Gbel mirrors [1]. Gbel mirrors require extremely preci se profiles in order to be effective in Xray mirrors [2]. Commonly these mirrors are bent i nto either parabolic or elliptical shapes [3]. The shape of the mirror is used to foc us the X-ray beam, meaning that the X-rays will reflect off the mirror as parallel beam s. In this manner the X-rays are focused at an infinite distance [2,3], as shown in Figure 1 Figure 1. Illustration of a Gbel Mirror. In a synchrotron "white beam" X-rays are generated and diffract off a monochromator that leaves one frequency, and then r eflect off the Gbel mirror. The XX-ray source Mirror Parallel X-ray beams

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2 rays that are diffracted off the monochromator tend to get attenuated to some lesser degree [4]. Any amount of attenuation to the inten sity of the diffracted rays gets absorbed by the mirror in the form of heat energy [ 4], and this heat energy can have detrimental effects on the performance of the mirro r, including possibly damaging the mirror [5]. Additional heat energy causes thermal expansion of the mirror, and if the temperature of the mirror is allowed to exceed a ce rtain amount a chemical reaction will occur that will permanently damage the mirror [6]. This thesis focuses on the concerns of distortions to the mirror due to thermal expansi on. The X-rays are reflected off the mirror at extremely shallow angles [7,8], which cau ses a very small loss of intensity to the reflected X-rays, and therefore a minimal amoun t of heat energy will be absorbed by the mirror [7]. The reason for this is that most m aterials have a critical angle, below which a reflected beam experiences little loss of t he energy, and almost none of the energy will be absorbed by the material [7]. The a ngle at which X-rays are reflected off the mirror is very close to, but steeper than the c ritical angle; as a result some of the energy gets absorbed by the mirror. To compensate for the energy being absorbed by the mirror, Gbel mirrors are cooled with liquid ni trogen [9]. 1.2 Distortions in X-Ray Mirrors Despite the efforts to control the temperature of t he X-ray mirrors, thermal expansion still occurs. The distortions caused by thermal expansion can affect the performance of the X-ray mirrors [10], as shown in Figure 2. As a means to determine these strains, a four-point bending fixture was des igned for in-situ X-ray bending experiments.

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3 Figure 2. Illustration of a Gbel mirror distorted by thermal expansion. 1.3 Stress in the X-Ray Mirror Thin Film X-ray mirrors have multiple layers deposited on the m, most commonly these are alternating layers of In2O3/Ag, W/B4C, Mo/B4C, Re/Si, Re/C, W/Si, W/C, Ta/Si, W/Be, Mo/Be, Mo/Si, Mo/C, Ni/C, Au/C, AuPd/C, ReW/B, ReW/ C, Al/Be or V/C [11]. These layers may be amorphous, or crystalline [11]. A st ress in a film deposited onto a substrate will cause the substrate to bend [12]. S toney’s equation is used to determine the stress in the film, based on the change in curv ature due to the stress [32], however if the stresses were to be determined, then from the s tress in the film, the change in curvature of the substrate could be determined. Th e stress in the film could be determined by using sin2 method. Heat energy causing thermal expansion X-ray beams not focused Mirror X-ray source

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4 Chapter 2. Introduction to Stress in Thin Films 2.1 Origin of Stress in Thin Films Stress originates in thin films or substrates due t o substrate-film differences in thermal expansion, or due to epitaxial mismatch [13 -16]. For stresses due to thermal expansion, if substrate-film are in equilibrium at one temperature, and there is a change in temperature, the film, or the substrate will try to expand at different rates, resulting in a stress. For stresses due to epitaxial mismatch, th e atoms of the substrate have a particular spacing due to atomic arrangement and si ze. The film’s atoms that is deposited epitaxially will line-up with the atoms o f the substrate, however if the atomic spacing of the film and the substrate do not corres ponds, there will be a resulting stress. 2.2 Overview of Thin Film Deposition There are several methods of depositing thin films onto substrates, including physical vapor deposition (PVD), chemical vapor dep osition (CVD), and electroplating [17]. Chemical vapor deposition uses high temperat ures that can result in high residual stresses in thin films. Physical vapor deposition, however is done at lower temperature, and therefore is less prone to create residual stre sses due to mismatch in thermal coefficients [17].

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2.3 Mechanics of (100) 2.3.1 Crystallographic Properties of W afers are made from single crystal regular patterns on long rang silicon is face-centereddiamond Figure 3 This crystal structure can be represented by tak ing two FCC and adding them together in such a manner that one of th orientation as the first, but offset from the first on each axis by an amount of [18]. Since the s ilicon wafers are made from single crystal anisotropic; this means the properties wafers used are (100) wafers The orientation of the wafers is such that the (100 ) plane is parallel to the top surface of the wafer, and the major, or primary fla t of the wafer is parallel to the (1 plane. The orientation of s 5 (100) Silicon Substrate Crystallographic Properties of (100) Silicon afers are made from single crystal s ilicon, this means the atoms repeat in regular patterns on long rang e order throughout the material. The crystal struc ture of diamond -cubic [18]. The crystal structure for s ilicon is This crystal structure can be represented by tak ing two FCC crystal structures and adding them together in such a manner that one of th e cells is added in the same orientation as the first, but offset from the first on each axis by an amount of Figure 3. Crystal Structure of Silicon. ilicon wafers are made from single crystal s ilicon, the material is anisotropic; this means the properties are different in different directions. The wafers The orientation of the wafers is such that the (100 ) plane is parallel to the top surface of the wafer, and the major, or primary fla t of the wafer is parallel to the (1 plane. The orientation of s ilicon wafers are shown in Figure 4. ilicon, this means the atoms repeat in e order throughout the material. The crystal struc ture of ilicon is shown in crystal structures e cells is added in the same orientation as the first, but offset from the first on each axis by an amount of x = y = z = a0/4 ilicon, the material is are different in different directions. The silicon The orientation of the wafers is such that the (100 ) plane is parallel to the top surface of the wafer, and the major, or primary fla t of the wafer is parallel to the (1 10)

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6 Figure 4. (a) Identifying features of single crys tal silicon wafers. (b) Orientation of crystallographic planes relative to wafer features for (100) type wafer. Image courtesy of N. Maluf, An Introduction to Microelectromechanical Systems Engineering, 1st ed. Boston: Artech House, 2000. The minor or secondary flat of the wafer is used to identify the type of wafer, depending on where the secondary flat is relative t o the primary flat identifies which type of wafer it is. The secondary flat may be 45, 90 180 from the primary flat, or there

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7 may be no secondary flat on the wafer. If the wafe r has no secondary flat, it a {111} ptype wafer, if the secondary flat is 45 from the p rimary flat it is a {111} n-type, if the secondary flat is 90 from the primary flat it is a {100} p-type, and if the secondary flat is 180 from the primary flat it is a {100} n-type sil icon wafer. 2.3.2 Elastic Properties of (100) Silicon Silicon has a cubic crystal structure, and has cubi c symmetry; this means that the elastic properties can be described with only three independent properties [19]. These three independent properties can be either three el ements from the stiffness matrix (i.e. C11, C12, C44), or three elements from the compliance matrix (i. e. S11, S12, S44) [19]. If the properties are needed for a particular directionv the values can be calculated by using equation 2.1, and 2.2 [19]. () ()2 2 2 2 2 2 44 12 11 112 1 2 1 n l n m m l S S S S E + + =abg (2.1) In equation 2.2 , and are the angles between the x y and z axis respectively, and l m and n are the cosines of the angles , and respectively [19]. ( ) () 2 2 2 2 2 2 44 12 11 11 2 2 2 2 2 2 44 12 11 122 1 2 2 1b a b a b a b a b a b a abnn l n m m l S S S S n n m m l l S S S S + + n + + n + = (2.2) In equation 2.2, the subscripts and refer to the directions for the Poisson’s ratio, and l m and n are the cosines between the x y and z axis and either or respectively. Note that the directions and must be orthogonal to each other.

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8 In the case of a deflection of a (100) Silicon wafe r, the properties needed are for the directions [100], [110], and all the directions in-between. Neither the elastic modulus, nor the Poisson’s ratio are constants thro ughout this range of directions, however the biaxial modulus in the {100}-plane is a constant [21], where the biaxial modulus is defined in equation 2.3. () n= 1 E B (2.3) This biaxial modulus in the {100} plane has a value of B100 = 179.4 GPa [20].

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9 Chapter 3. Stress Characterization Techniques This chapter discusses different means of determini ng the stress in thin films by X-ray diffraction, curvature utilizing Stoney’s equ ation, and other methods. 3.1 X-Ray Diffraction This section discusses how X-ray diffraction, can b e used to determine the stress in a thin film by using the sin2 technique. 3.1.1 Lattice Parameter and Bragg’s Law Many materials are crystalline, that is their atoms are arranged in repeatable 3dimensional arrays. These crystals are formed of u nit cells, which contain the smallest number of atoms that repeat to form the 3-dimension al array. The unit cells can be cubic, hexagonal, or a few other types. The dimens ions of these unit cells are called the lattice parameter. In the case of a cubic cell (si licon being discussed in this thesis), requires only one lattice parameter to define its d imensions. One method of determining the lattice parameter of a crystalline solid is by using Bragg’s law with X-ray diffraction (XRD), to measure the interplanar spacing [22]. Th e interplanar spacing is the distance between two parallel planes of atoms in a crystalli ne material. The interplanar spacing can then be used to determine the lattice parameter By knowing the coordinates of the diffracting plane, the intercept of the diffraction plane with the x y and z -axis, the lattice parameter can be determined. The correlation betwe en the lattice parameter and the interplanar spacing is defined in equation 3.1,

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10 2 2 2 0l k h a d + + = (3.1) where h k and l are the intercepts of the diffracting plane with t he x y and z -axis respectively. In the case of the (100) single crys tal silicon sample that was used in the experiments, h = 1, k = 0, and l = 0, therefore the lattice parameter is equal to the interplanar spacing, that is d = a0. Measuring the lattice parameter using Bragg’s law i nvolves two concepts [23]. The first concept is interference of waves. When t wo waves come together, having the same wavelength, and frequency, the resultant wavef orm is the sum of the two waves. If the two waves of the same frequency are in phase i.e. their amplitude maxima occur at the same time, the resultant amplitude will be t he sum of the two amplitudes. If they are 180 out of phase, in other words, one amplitud e maxima occurs exactly midway between the amplitude maxima of the other wave, or rather the peak of one wave occurs when the other wave at its trough, when this occurs the resultant amplitude is the difference between the two amplitudes. If these tw o waves came from the same source, they would have the same initial amplitudes, and th erefore the difference between amplitudes would be zero, that is the two waves wou ld cancel each other out. The second concept of Bragg’s Law involves simple trigo nometry. When two waves hit atoms on two parallel lattice planes, one wave will travel an extra distance shown as l in Figure 5, and the same extra distance l after diffracting off of the atoms, therefore one wave will travel 2 l greater distance than the other. The distance 2 l that one wave travels farther than the other is a function of the distanc e between the two planes (shown as d in Figure 5), and the angle they make with the lattice plane. From Figure 5 it can be seen that d is the hypotenuse in a right triangle, and l is the side opposite angle therefore l =

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11 d sin( ). If the extra distance that one wave travels (2 l ) is exactly equal to one full wavelength ( ) or any integer multiples of (i.e. n ), then the waves will be back in phase again, and there will be a constructive interferenc e. Another way of stating this is that this is a phase shift of 360, which brings the two amplitude maxima back in phase. Since an amplitude maxima only occurs when the phas e shift is exactly equal to a multiple of wavelengths, the distance between two l attices ( d ) can be determined from the angle that the reflection occurs. Equation 3.2 is the Bragg’s law equation. ) sin( 2q l n d = (3.2) Figure 5. Determination of lattice spacing using B ragg’s law. 3.1.2 Sin2 Technique The sin2 technique is a method using X-ray diffraction, in conjunction with Bragg’s law in order to determine the full strain t ensor, or to determine the full stress tensor directly by applying Hook’s law [24-26]. Th e full strain, and also the full stress tensor contains six components, therefore a minimum of six measurements are needed, however more measurements are often taken in order to reduce the amount of uncertainty [26,27].

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12 The sin2 technique requires a coordinate system to be estab lished, that is the x y and z -directions need to be assigned, although these dir ections may be assigned using any orientation, a standard orientation to th e coordinate system is usually used. This standard orientation usually consists of the x and y -axis being in the plane of the sample surface, and therefore the z -axis is the direction normal to the surface. Only two angles need to be given in order to establish a uni que direction, the angles that are used to define a direction are and The -angle is the angle between the x -axis, and the projection of a vector-v onto the xy -plane. The –angle, also known as the tilt angle, is the angle between the z -axis, and the vector, as shown in Figure 6. The a nglementioned in the previous section is the angle at w hich the X-rays impinge on the plane of atoms, when Bragg’s law is satisfied in order to create a reflection. Figure 6. Coordinate System Used for the sin2 Technique. Before giving a detailed explanation of the sin2 technique, a simple basis of how the -angle, or rather the tilt angle plays a role in de termining the strain tensor, which is used to calculate the stress tensor [28]. This sim ple basis is as follows. x y z v ^

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13 First consider a lattice that is under tensile stre ss, and is parallel to the applied stress. This stress will result in a strain that e longates the lattice parameter. Second consider a lattice that is under tension, but this lattice is perpendicular to the applied stress. This lattice parameter will be reduced due to the Poisson’s ratio. Those lattice parameters that are at some angle other than parall el, or perpendicular to the applied stress, will experience an amount of elongation due to the component which is parallel, and some reduction due to the component that is per pendicular, as shown in Figure 7. Figure 7. Basic principles of sin2 stress determination. This explanation is used only to give a basic idea as to how the -angle plays a role in determining the full strain tensor, a much more complete explanation is needed to understand the sin2 technique. 3.1.3 Strain Transformation In order to calculate the stress using the sin2 method the first thing needed is to find the 3-dimensional strain projection, that is i t is needed to determine what the strain will be along a particular direction, given the str ains for a given coordinate system. The Lattice parameter Elongated due to stress Tensile Stress Tensile Stressa0Lattice parameter due to Poisson’s Ratio Reduced Oblique unit cell a combination of strains experiencing unstrained unit cell

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14 basis for this strain transformation is that each o f the strains from the original coordinates is projected onto the new axis [29,30]. The 2D-strain Transformation is a superposition of each of the individual strain comp onents; this means that each of the strain components can be added separately. The tra nsformation done here is a 3Dstrain transformation, as opposed to a 2D-strain tr ansformation. This requires two transformations for each component to project each component onto the new axis instead of just one transformation. In addition the 3D strain transformation has six components of strain, whereas a 2D strain transform ation has only three components. Figure 8. x Component of the 3D-Strain Transformation to the x’ Axis. As shown in Figure 8, x is multiplied by dx to give the displacement x dx This displacement is projected onto the ‘ -axis,’ which gives x dx cos This is projected onto the x’ -axis, which gives x dx cos sin The definition of the strain along the new axis is equation 3.3, where equation 3.4 is determined from trigonometry and solving for dx’ in x’[ x]

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15 terms of dx gives equation 3.4. Equation 3.5 can be plugged i nto equation 3.3 to give equation 3.6, which is the strain component along t he new axis. x ’ = x’/dx’ (3.3) dx = dx’ sin cos (3.4) dx’ = dx/ (sin cos ) (3.5) = = =y f y f e d e esin cos sin cos ' ] ['dx dx dx xx x xy f e2 2sin cos x (3.6) Figure 9. y Component of the 3D-Strain Transformation to the x’ Axis. As shown in Figure 9, y is multiplied by dy to give the displacement y dy This displacement is projected onto the ‘ -axis,’ this gives x dy sin Equation 3.7 is determined from trigonometry and solving for dx’ in terms of dy gives equation 3.8. x’[ y]

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16 Equation 3.8 can be plugged into equation 3.3 to gi ve the strain component along the new axis dy = dx’ sin sin (3.7) dx’ = dy/ (sin sin ) (3.8) = = =y f y f e d e esin sin sin sin ' ] ['dy dy dx xy y xy f e2 2sin sin y (3.9) Figure 10. z Component of the 3D-Strain Transformation to the x’-Axis. This component of the strain is independent of angl e, and therefore can be illustrated using a two-dimensional figure along th e ‘ -axis’ and the z -axis, as shown in Figure 10. z is multiplied by dz to give the displacement z dz Equation 3.10 is determined from trigonometry and solving for dx’ in terms of dz gives equation 3.11. Equation 3.11 can be plugged into equation 3.3 to g ive equation 3.12, which is the strain component along the new axis. dz = dx’ cos (3.10) dx’ = dz/ cos (3.11) x’[ z]

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17 = = =y y e d e ecos cos ' ] ['dz dz dx xz z xy e2cosz (3.12) Figure 11. xz Component of the 3D-Strain Transformation to the x’-Axis. As shown in Figure 11, xy is multiplied by dy to give the displacement xy dy This displacement is projected onto the ‘ -axis,’ this gives xy dy cos Next, this is projected onto the x’-axis, which gives xy dy cos sin The definition of the strain along the new axis is equation 3.3, where equation 3.7 is determined from trigonometry and solving for dx’ in terms of dy gives equation 3.8. Equation 3.8 can be plugged i nto equation 3.3 to give equation 3.13, which is the st rain component along the new axis. x’[ xy]

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18 = = =y f y f g d g esin sin sin cos ' ] ['dy dy dx xxy xy xy f g2sin ) 2 sin( xy (3.13) (a) (b) Figure 12. (a) xz Component of the 3D-Strain Transformation to the x’-Axis. (b) Close-Up of the Strain Transformation Component s. As shown in Figure 12, xz is multiplied by dz to give the displacement xy dz This displacement is projected onto the ‘ -axis,’ this gives xz dz cos Next, this is projected onto the x’ -axis, which gives xz dz cos sin The definition of the strain along the new axis is equation 3.3, where equation 3.10 is determ ined from trigonometry and solving for dx’ in terms of dz gives equation 3.11. Equation 3.11 can be plugged into equation 3.3 to give equation 3.14, which is the strain comp onent along the new axis. x’[ xz]

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19 = = =y y f g d g ecos sin cos ' ] ['dz dz dx xxz xz x) 2 sin( cosy f gxz (3.14) (a) (b) Figure 13. (a) yz Component of the 3D-Strain Transformation to the x’-Axis. (b) Close-Up of the Strain Components. As shown in Figure 13, yz is multiplied by dz to give the displacement yz dz This displacement is projected onto the ‘ -axis,’ this gives yz dz sin This is projected onto the x’ -axis, which gives yz dy sin sin The definition of the strain along the new axis is equation 3.3, where equation 3.10 is determined fro m trigonometry and solving for dx’ in terms of dz gives equation 3.11. Equation 3.15 is the yz strain component, along the new axis. x’[ yz]

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20 = = =y y f g d g ecos sin sin ' ] ['dz dz dx xyz yz xy y f gsin cos sin yz (3.15) By adding all the individual strain transformation components together, yields equation 3.16, which is the 3-dimensional strain tr ansformation. y f g y f g y f g y e y f e y f e e2 sin sin 2 sin cos sin 2 sin cos sin sin sin cos2 2 2 2 2 2 yz xz xy z y x x+ + + + + = (3.16) 3.1.4 Sin2 Equation Equation 3.16 can be changed to an equation of stre ss by utilizing the generalized form of Hook’s Law. Also by using the fact that sin2 = 1cos2 the terms may be grouped to form equation 3.17 [31]. y f t f t n s s s n s n y s f t f s f s n efy fy2 sin ) sin cos ( 1 ) ( 2 1 sin ) 2 sin sin cos ( 12 2 2 0 0 yz xz z y x z z xy y x E E E E d d d + + + + + + + + + + = = ( 3.17), where E is the elastic modulus of the film, n is Poisson's ratio of the film, and d0 is the unstrained lattice parameter. The equation for the stress along the -direction is defined in equation 3.18. f t f s f s sf2 sin sin cos2 2 xy y x+ + = (3.18) Since this stress is in a thin film, and therefore near the surface, the normal stress in the z -direction will not exist. By substituting equatio n 3.18 into equation 3.17, one derives:

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21 y f t f t n s s n y s n ef fy fy2 sin ) sin cos ( 1 ) ( 2 sin 12 0 0 yz xz y xE E E d d d + + + + + = =(3.19) If a plot is made with d vs. sin2 there will be three types of behavior, linear behavior, –splitting behavior, and oscillatory behavior [24], as shown in Figure 14. (a) (b) (c) Figure 14. d vs. Sin2 for (a) Linear Behavior, (b) Splitting Behavior, and (c) Oscillatory Behavior. Adapted from [24]. If shear stresses xz, and yz are zero, equation 2.13 will have a linear behavio r with sin2 If the shear stresses are non-zero, the plot of d vs. sin2 will have a – splitting behavior. If the plot of d vs. sin2 shows an oscillatory behavior it indicates that the material is textured and strain cannot be solved using equations mentioned above [24]. The methods of determining the stress in the material using the first two types of behavior are discussed in the next section

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22 3.1.5 Calculation of Stress Using Sin2 As mentioned in the previous section, a plot of d vs. sin2 will have three different types of behavior, linear, splitting, and oscillatory behavior, shown in Figur e 14. If the plot is linear, this means that the two shea r terms, xz, and yz are zero; therefore equation 3.19 is reduced to equation 3.20 ) ( 2 sin 12 0 0 y xE E d d ds s n y s n ef fy fy+ + = = (3.20) Differentiating equation 3.20 with respect to sin2 results in equation 3.21. f fys n yE d d + = 1 1 ) (sin0 2 (3.21) Using equation 3.21, and simply solving for one gets equation 3.22: n n n n + =y sfy f2 0sin 1 1 d d v Ef f (3.22) Equation 3.22 illustrates that the stress should be proportional to the slope of the d vs. sin2 plot. It would only be necessary to take measurem ents at two –tilts in order to determine the slope; however more points a re usually taken in order to reduce the amount of uncertainty [50]. As seen in Figure 15, the d vs. Sin2 plot, the behavior is not exactly linear, probably either due to a sma ll amount of shear stress, or due to some uncertainty in the measurements.

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23 Figure 15. Example of a d vs. Sin2 Plot for Tungsten Film. The slope is found by doing a linear regression of the data. In this case the material is Tungsten, the slope of the plot is -1.2 256x10-12 m, d0 is 2.2524x10-10 m, E is 300 GPa, and n is 0.3, therefore in this example the stress along the –axis is 1.26 GPa compressive, because the slope is negative. Howeve r, in order to calculate the full stress tensor, additional values are needed. To do this similar plots are taken at different -angles. Examining equation 3.18, if is set to 0, then sin2 = 0, sin(2 ) = 0, and cos2 = 1. As a result equation 3.18 reduces to x. If the angle is set to 90, then in equation 3.18, sin2 = 1, sin(2 ) = 0, and cos2 = 0, and equation 2.4 is reduced to y. Finally if is set to 45, then sin2 = 0.5, cos2 = 0.5, and sin(2 ) =1, and equation 3.18 is reduced to 0.5 x + 0.5 y + xy. Since x, and y are known, xy can be solved for. Any three different angles for can be used; however choosing the above values gre atly simplifies calculations. nnr nr nr nr nr nr nr nr nr rrrrrrd (m)sin 2 (unitless)

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24 If the plot shows –splitting behavior, the above method can be used, however in order to find the slope of d vs. sin2 an average between the slopes for positive, and negative –tilts needs to be used. In equation 3.22 the comp onent d/ sin2 is determined by doing a linear regression of the d, vs. sin2 data, as mentioned in the next section. 3.1.6 Sin2 Example Table-1 is an example of data of diffraction reflec tions that was taken from a tungsten film on a (100) silicon wafer, at various -tilts at the Technical University of Dresden. Table 1. Sin2 Data for Tungsten Film on a (100) Silicon Wafer. (deg) 2 (Pseudo Voigt) d sin 2 -40 40.1769 2.24182E-10 0.413175911 -24 40.1871 2.24128E-10 0.165434697 -10 40.1659 2.24241E-10 0.03015369 0 40.1707 2.24216E-10 0 5 40.1715 2.24211E-10 0.007596123 10 40.1670 2.24235E-10 0.03015369 12 40.1659 2.24241E-10 0.043227271 16 40.1623 2.2426E-10 0.075975952 24 40.1661 2.2424E-10 0.165434697 32 40.1716 2.24211E-10 0.280814427 40 40.1757 2.24189E-10 0.413175911 To calculate the slope of d vs. sin2 one would take the average slope of negative psi tilts and positive psi tilts. To calc ulate the slope for negative psi tilts a linear regression of all the negative psi tilts are taken, where d is the y -axis, and sin2 is the x -axis. Similarly a linear regression is taken for all the positive psi tilts using the same manner that was done for the negative psi tilts. A n average of the negative and positive slopes is then used in the calculations [50]. The values, including the elastic properties,

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25 and the slopes for the negative, and positive, as w ell as the average slope are given in table 2. In order to find the full stress tensor t he same procedure will be done for two other -angles. Table 2. Mechanical Properties of Tungsten. E= 3.00E+11 Pa n = 0.3 E/(1+ n )= 2.30769E+11 Pa slope[neg]= 8.51906E-13 slope[pos]= -5.63543E-13 slope[ave]= 1.44182E-13 d[110]=a[0]/sqrt(2) = 2.238134E-10 = 148.6625 MPa Tensile 3.2 Curvature Methods 3.2.1 Stoney’s Equation Stoney’s equation is used to determine the stress i n a thin film from the amount of curvature in the wafer. This formula is based o n simple beam bending formulas, where the tension and compressive forces act on the cross-section to create a moment at the cross-section resulting in the curvature, as shown in Figure 16. (a) (b) Figure 16. Forces Acting on a Cross-Sectional Area Act as the Basis of Stoney’s Equation. Substrate FilmFfMFs f x Ef E s

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26 Equation 3.23 is Stoney’s equation [32]. Stoney’s equation can also be written using variables for curvature r or r which is the reciprocal of the radius of curvatur e. () n n =1 2 21 1 1 6 R R h h v Ef s s s rs (3.23) The variables used in Stoney’s equation are: Es elastic modulus of the substrate, ns Poisson’s ratio of the substrate, h – film thickness, where subscripts s and f refer to substrate, and film respectively, R radius of curvature of the substrate, where subs cript1 refers to the radius of curvature before depositi on and subscript-2 refers to the radius of curvature after deposition [29]. 3.2.2 Curvature Measuring Techniques The key to using Stoney’s equation is to be able to determine substrate curvature with a sufficient enough accuracy. There are sever al methods to determining the curvature, including IONIC stress-gauge, profilomet er, and X-ray diffraction. The IONIC stress-gauge determines the curvature of the wafer at a single point, in the center by determining the height of the cent er, relative to a fixed height. Figure 17. IONIC Stress-Gauge Measurement. The IONIC stress-gauge uses the reflection of light from a fiber bundle in order to determine the curvature as shown in Figure 17. In this manner, the IONIC stress-gauge knife edge 100 mm [100] Si wafer light beam light bundle

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27 does not touch the surface of the wafer, thereby re ducing any damage to the wafer due to contact. Some of the key features of the IONIC stress-gauge setup are the knife edge, which has a very thin contact area. This knife edg e has a very small surface area in contact with the wafer, which helps ensure that the possibility of errors due to debris between the wafer, and the contact surface is very low. The knife edge is also very near the outer edge, this means that the radius is deter mined over the largest possible distance, to minimize the amount of uncertainty. O ne of the advantages in the IONIC stress-gauge is the simplicity of the setup. Some o f the disadvantages of the IONIC stress-gauge are that it only determines the deflec tion at a single point. In fact the curvature can vary over the entire wafer [33,34]. Another disadvantage is that the IONIC stress-gauge is very sensitive, and any deviation o f the reflectivity on the back surface of the wafer can cause very large inaccuracies. A profilometer measures the wafer height at several points, in this manner changes in curvature can be calculated over the sur face of the wafer. This device uses a stylus to scan from one side of the wafer to the other, recording the scan height. Profilometers can have a resolution approximately t o that of an atomic spacing, and therefore can accurately determine the radius of cu rvature. One disadvantage to profilometers is the data is scattered [35] that is it shows abrupt changes in height over a very small distance. Because of this data scatter, data smoothing techniques must be used [36]. There are different types of data smoot hing, one type of data smoothing is to fit the data to a polynomial [37]. If the polynomi al is a second order polynomial, the curvature will be constant, if small deflection for mulas are used. X-ray diffraction can also be used to determine cur vature profile [38]. The curvature determination is done by utilizing a rock ing curve scan on single crystal films.

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28 The rocking curve scan is where the emitter and col lector move in sync with each other, this is similar to having the wafer tilt as the emi tter and collector remain stationary. When a reflection occurs during a rocking scan, the normal to the wafer is the bisection of the angle between the emitter and collector. By taking readings at several points, the curvature can be determined [38].

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29 Chapter 4. In-Situ Bending Experiments in XRD An experiment was done at Technical University (TU) in Dresden Germany in order to demonstrate the ability to determine the d istortion in a (100) silicon wafer. The experiment consisted of using a piece of a (100) si licon wafer in a four-point bending setup that was deformed to a certain amount and the n the stress was determined [39]. The stress was determined by first measuring the la ttice parameter, and comparing it to a known unstressed lattice parameter. Then the str ess was calculated by using the elastic constants of the material. 4.1 Four-Point Bending Apparatus Design In order to perform the following measurements a fo ur point bending fixture needed to be designed and built, as shown schematic ally in Figure 18. (a) (b) Figure 18. (a) 4-Point Bending Fixture. (b) Descri ption of Parts. Carriage Winner bending points Adjustment -screw Lock-Nuts Outer Bending Points SupportPlate

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30 The four point bending fixture was designed by the author, and built by machinists at TU-Dresden. The reason for using a f our point bending fixture, as opposed to a three point bending fixture is that a four point bending fixture will have a constant radius of curvature between the inner bend ing points [40,41], as shown in Figure 19. Figure 19. Constant Radius of Curvature Between Inner Bending Points on a 4-Point Bending Setup. The stress between the inner points is constant. T his will help eliminate errors caused by stress being a function of the x-directio n, should the x-ray not hit the exact same spot on the (100) silicon wafer. The bending fixture consists of a threaded carriage providing the inner points for bending. A screw is fitted to the carriage to provide the motion; this same screw goes through an adjustment plate to provide the force on the carriage. The carriage is also fitted with support tabs. These tabs provide the inner bending points in order to put the bottom of the wafer into tension. The parts for the bending fixture are shown in Figure 18b. T he entire assembly is attached to the x-ray diffraction machine by means of elongated hol es to provide adjustment. The setup once installed was tested to be level by use of a c ommon carpenter’s level, and should it have needed leveling shims perpendicular to the mac hine would have been used. constant curvature

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31 4.2 System Alignments After the setup was properly installed several alig nment tests were performed. These adjustments included vertical alignment, basi c level alignment, and a rocking scan to more precisely level the setup. 4.2.1 Vertical Alignment The test for vertical alignment consisted of perfor ming a scan so that the emitter, and collector are directly facing each other initia lly. This setup is shown in Figure 20. Figure 20. Alignment Test for Vertical Position. It should be noted that in doing this test it is im portant to insert a copper shield in either the emitter, or collector aperture, in order to reduce the intensity of the X-rays that will be hitting the collector. This scan is done w ith the collector remaining stationary at the 180-mark during the scan. The emitter is init ially below the 0-mark, in other words the emitter is situated so that the X-rays are impi nging on the wafer, or the bending fixture. The emitter continues the scan moving upw ard, to a point that is both past the 0-mark, and past a point where the X-rays pass ove r the wafer, and are detected by the collector. After the scan is done, the angle at wh ich the collector first started to absorb the X-rays is observed. If the angle where the col lector first absorbed the X-rays is lower n r

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32 than the 0-mark, it indicates that the fixture is too low, and therefore needs to be raised, as shown in Figure 21. Figure 21. Illustration of an Alignment Test, with the Wafer Too Low. If on the other hand the angle where the collector first started to absorb X-rays is above the 0-mark, it indicates that the fixture is too high, and needs to be lowered, as shown in Figure 22. This type of scan is repeated until the angle where the collector first starts to absorb X-rays is within small tolerance f rom the 0-mark. At each point that the fixture was adjusted, it was checked for level. Figure 22. Illustration of an Alignment Test with the Wafer Too High. n n r

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33 4.2.2 Zeroing the Sample The first scan performed on the sample was done wit h no stress on the wafer in order to determine the unstrained lattice spacing o f the sample. It was necessary to locate the point where there would be no slack betw een the sample, and the setup, nor any initial stress in the sample. The sample was a djusted in order to produce a minimal amount of play between the sample, and the setup, o r a minimal amount of initial stress. This was done by initially adjusting the screw so t hat there is slack between the sample, and the setup, then the screw is adjusted s lightly, and then the sample is checked for play. In order to check for play, a pi ece of paper is used to attempt to move the sample in the fixture, if the sample moves, the re is still play in the setup, and the screw is turned an additional amount. This is repe ated until the sample is not able to be moved by the piece of paper. This process has a certain amount of resolution, th e resolution in this case means that there is an amount that the screw is tur ned between each time that the sample is checked for slack. The result of this is that the point where the slack in the sample is taken up is anywhere in between two point s where the sample is checked. In order to produce a finer resolution the process is done twice, the first time with the screw being turned a fairly large amount in-between each check, in order to get an approximate location of the zero point. The zeroin g process is repeated a second time, this time where the zero point is known to within a certain amount, and much finer turns of the screw being made between each time the sampl e is checked. In order to prevent a hysteresis error, the screw was only turned in on e direction between checking the sample.

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34 4.2.3 Rocking Scan To perform the rocking-scan on the setup, a regular 2 -scan was performed first. Leveling of the initial setup was performed to be a ble to determine a reflection during the 2 -scan. If a reflection cannot be determined at thi s point, it would indicate that the setup was not properly aligned. If a reflection oc curs during the 2 -scan, then a rocking scan can be done in order to make a much more accur ate alignment of the setup. The reflection that occurs during the initial 2 -scan will more than likely have a much lower intensity, than a reflection should be for a single crystal reflection, which is an indication that the wafer is not properly aligned to an accura te enough degree, as shown in Figure 23. Figure 23. The 2 -Scan with a Misaligned Wafer. The reason for having a lower intensity is because the diffracted X-rays have the highest intensity at the true diffraction angle, an d greatly diminish at angles other than the true diffracted angle as shown in Figure 24. Assumed orientation of wafer Actual orientation of wafer n

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35 Figure 24. Relative Intensity of Diffracted X-Rays First the 2 -angle at which the reflection occurred is recorded and the rocking scan is done so that there is a fixed angle between the emitter and collector as the scan is done, this is shown in Figure 25. Doing a rocki ng scan in this manner has the same effect as if the setup were to be rotated about a p oint where the centerline of the emitter and collector intersect. The rocking scan however can be done in a much more accurate manner, and it will rotate about a constan t point. Figure 25. Rocking-Scan Used to Determine the Alig nment of the Wafer. After the rocking scan is done the angle at which a reflection occurs is recorded, this will be used as the offset angle in the next s tep. The next step is to do a 2 -offsetscan. The angle where the reflection occurred durin g the rocking scan is used as the Angle “” is fixed during a Rocking-Scan Rocking-Scan n n Incident X-rays Diffracted X-rays

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36 offset angle for the 2 -offset-scan. If the misalignment in the initial s etup is large enough, it will produce a significant error when do ing the rocking-scan. If this happens it may require several iterations of rocking-scans, fo llowed by 2 -offset-scans in order to reduce the error caused by misalignment. The 2 -offset-scan is similar to the 2 -scan, with the exception that instead of the emitter and collector scanning at equal angles from the 90-mark, the emitter and collector are scanned at equal angles from a point that is offset from 90-mark, hence the term 2 -offset-scan. An illustration of the 2-offset-scan is shown in Figure 26. Figure 26. Using an Offset Angle to Correct for An gular Alignment. 4.3 Sources of Error There are several sources of error, or uncertainty in the measurements, these sources include alignment, peak determination, and the bending fixture. The alignment can cause errors due to the detector not picking up the true diffracted X-rays. Even if the setup is exactly aligned, there is an amount of uncertainty in locating the diffraction peaks. The bending fixture caused errors due to ch anging the height of the wafer. / 2 Offset Angle / 2 n

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37 4.3.1 Errors Due to Change in Height The bending fixture mentioned previously deflected the wafer by changing the height of the inner bending points; as a result thi s caused a change in the height of the wafer, at the point where measurements were taken, by an amount greater than the amount of adjustment of the carriage as shown in Fi gure 27. This was done partially due to a lack of foresight on the part of the autho r; however given the time frame of the experiment, and the amount of engineering needed, t he method used was the most practical means in order to carry out the experimen ts given the time frame, and so as to not place too heavily a burden on the machinists at TU-Dresden. The errors that were produced as a result of the changing height of the wafer sample needed to be addressed. Figure 27. Source of Error Due to Changing Height of Wafer Sample. Measured Angle Actual Angle n

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38 4.4 Depth of X-Ray Penetration 4.4.1 Basis of X-Ray Absorption Materials absorb X-rays as the X-rays pass through them, the amount of X-rays that are absorbed depends on several factors. The main factors that determine the intensity of the resulting beam as it passes throug h a material are the depth of the material that the X-rays have passed through, and t he material properties [42]. As a result the intensity of the X-rays after it has pas sed through a certain length of a material is according to equation 4.1 [42,43], xe I I -=m0 or xe I I -=r r m) / ( 0 (4.1) where I refers to the intensity of the beam at depth x I0 is the original intensity of the beam, is the density of the material, and is the mass attenuation coefficient. This equation follows the same principles for absorption of optical rays, known as BeerLambert’s law. 4.4.2 Depth of Attenuation According to equation 1.1, the depth that the X-ra y will penetrate is infinite, however the intensity will continuously decrease as the depth increases. This situation would prompt the question, what level of attenuatio n is considered to contribute to a reflection peak. Analyzing this one would realize that when the intensity of the X-rays have attenuated to a level at or below the intensit y of the noise data, then the diffracted rays would contribute to the same intensity of nois e data. In this manner equation 4.1 can be changed to equation 4.2, xe N S I I= =m/ /0 or xe N S I I = =r r m) / ( 0/ / (4.2)

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39 where S/N is the signal to noise ratio. Solving equation 4. 2 for x yields equation 4.3. m) / ln( N S x = (4.3) The X-rays however need to penetrate the material as incident rays, as well as diffracted rays, therefore the depth that the X-ray s will penetrate at this I / I0 ratio is x /2. The previous statement is based on the X-rays being incident perpendicular to the material, since the X-rays are incident at angles o ther than perpendicular, the depth that the X-rays penetrate is calculated according to equ ation 4.4 ) sin( 2 q = x d (4.4) where d is the penetration depth, x is the total penetration length, and is the incident angle. Therefore by substituting equation 4.3, int o 4.4 results in equation 4.5. ) sin( 2 ) / ln(q m = N S d (4.5) The values for the mass attenuation coefficients ha ve been determined for a number of materials, and records for them are kept by organizations such as the National Institute of Standards and Technology, kno wn simply as NIST. The value of for Cu[ k 1] is 14.46 0.07 mm-1 [43]. Plugging these values into equation 4.5 res ults in d = 0.147 mm, or 147 m. Since this value constitutes the total depth contri buting to diffracted rays making up the reflection peak the average depth contributi ng to the diffraction reflection would only be half of this value, or 0.075 mm, or 75 m.

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40 4.5 Scanning Procedure After making all the adjustments for errors several scans were taken. Each scan was allowed to repeat several times, and all the da ta was accumulated. Many of the scans were observed, in this manner if any gross an omalies were present they would be observed, and corrective action could be taken if n ecessary. Such corrective action might be to run the scan for a longer time [44-45]. Between each set of scans the adjustment screw was turned one-quarter turn, and t he scans repeated. The adjustment screw placed the wafer under a different amount of stress. The first setup put the top portion of the wafer under compressive stress. Wit h each turn of the screw it was expected that the reflection for the sample would s hift a little. Since turning the screw in the first setup placed the wafer under compressive stress the lattice parameter in the xdirection would decrease. Because of the Poisson's ratio the lattice parameter in the ydirection would increase. According to Bragg’s Law /2= d sin therefore d = /(2sin ), and therefore since d is increasing should decrease. 4.6 Peak Determination 4.6.1 Peak Height Peak height can be affected by two main factors, on e is shielding, and the other is due to alignment [46]. Shielding is necessary to reduce the intensity of the diffracted X-rays that are incident on the collector. Therefo re the height of the reflection is an important factor, as sufficient reflection intensit y is an indication of good alignment. In addition a higher reflection will reduce the effect of random errors in the noisy data [47,48]. As this section illustrates, having a suf ficient height of the reflection peak is necessary in reducing the uncertainty of the peak l ocation [49]. The amount of

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41 uncertainty of the peak location can be approximate ly calculated as a function of peak height as in equation 4.6 [50]. T CC W U / )3/2( (4.6) Where UC is the uncertainty of the peak location, CT is the total neutron or photon count in the peak and W is known as Full Width at Half Maximum, or FWHM, w hich is the width of the reflection, measured in radians at half of the reflection height. The uncertainty in the strain as a function of the peak height is illustrated in equation 4.7 [50]. TC W U / cot )3/1(qe(4.7) 4.6.2 Peak Determination Functions There are a few common methods for determining the peak of a reflection; these include Pseudo-Voigt method, parabola [51], and COG -center of gravity [52]. The Pseudo-Voigt method fits the diffracted data to a G aussian function. The parabola method fits the data to a parabola, and the Center of Gravity, or COG, does a weighted average to determine the center of the reflection [ 52]. An X-ray beam incident on a surface will form a ran dom distribution, which is somewhere between a Lorentzian and a Gaussian distr ibution, the diffracted X-rays will also form this same distribution. Therefore a refl ection from an X-ray diffraction scan will form this same type of distribution. The manner to determine the location of the peak of the reflection would be to fit the data to this sam e type of distribution. This method of fitting the data is referred to as a Pseudo-Voigt m ethod. It helps to greatly reduce the amount of error in locating the peak of a reflectio n. There are other methods of locating the peak of a reflection, one other such method is fitting the data to that of a parabola.

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42 Once the data is fitted to the parabola it is a sim ple matter of algebra to determine the peak. Yet another method of determining the peak o f a reflection is called COG, or center of gravity, this is simply dome by finding t he weighted average of the data, where the height of each data point is the weight of each data part. In practice these methods will be done by software, and several methods will be used in order for the user to get a feel of the con fidence of the peak locations. 4.6.3 Procedure for Peak Determination Determining the peak of a reflection is usually don e by software, as the process would be very time consuming, and subject to errors because of the numerous amounts of calculations needed to be done. With a diffraction scan, the user needs to select t he data that will be used in the peak determination. It is important to select enou gh data so that the entire reflection is included in the data, however selecting too much da ta will add errors to the result. A good rule to use is to select 3W of data, where W i s the width of the reflection at the point that is half of its height, known as full wid th at half maximum, or FWHM. The first thing that the software does to calculate the locat ion of the peak is to eliminate the noise from the data, which is data from random diffractio ns; this is seen as the data points that exist between reflections. After the noise data is removed Cu [K 2] needs to be removed from the data. The X-rays from Cu [K ] have two wavelengths Cu [K 1], and Cu [K 2] that are nearly identical, as a result the two p eaks from these wavelengths will occur within the same reflection. In order to corr ectly locate the peak of the reflection the Cu [K 2] data needs to be removed from the Cu [K 1] data. An example of this is shown in Figure 28. Figure 28a is the reflection d ata before stripping off Cu[k 2], and Figure 28b is after stripping Cu[k 2] off the data. At this point the software will c alculate

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43 the peak by using several methods, such as Pseudo-V oigt, parabola, and center of gravity COG. Figure 28. Schematic Representation of a Diffracti on Reflection, (a) Before, and (b) After Stripping Cu[k 2] Off of the Raw Data. The parabolic fit takes the data and fits it to a p arabolic function. Once the data is fit to a parabolic function (i.e. y = a x2 + b x + c ) finding the reflection is simply a matter of taking the derivative with respect to x (i.e. y’ =2 a x + b ) set y’ =0, and then simply solve for x (i.e. x = b /2 a ). The center of gravity takes the data, and does a we ighted average, as shown in equation 4.8 [53]. x = f ( xi) xi / f ( xi)(4.8) The Pseudo Voigt method takes the data and fits it to a normal distribution curve. This method assumes scatter of data is due to rando m errors [54].

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44 4.7 Error Corrections The vertical alignment done on the setup reduced th e amount of error in the vertical direction. However, an additional correct ion was necessary in order to reduce the errors to a more acceptable degree. Also when the wafer was adjusted in the bending setup, this changed the height of the wafer As a result it was necessary to make corrections for this adjustment. Copper powde r was placed on the wafer. Copper has a known diffraction reflection close to the ref lections of the (100) silicon wafer. Because the copper is in the form of a powder its d iffraction reflection will not be affected by the bending of the wafer. A picture of the setu p with the copper powder placed on the wafer sample is shown in Figure 29; the copper powd er is the dark substance in the middle of the sample. Figure 29. Four-Point Bending Setup for X-Ray Diff raction with Copper Powder Added. The copper powder may absorb some of the X-rays, bu t most of the X-rays will pass through the copper powder. Since copper has a known diffraction reflection fairly close to the (100) silicon reflection, the error wa s corrected by determining the difference between the reflection produced by the copper powde r, and where the reflection is Bending setup

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45 supposed to occur. This difference is then subtrac ted from the reflection produced by the (100) silicon wafer. Since the reflections occ ur fairly close to each other this should correct the errors to within an acceptable degree. An illustration of this is shown in figure 32. Note: Figure 30 is not a true scan, the (100) s ilicon, and (220) copper reflections were scanned separately, as copper needed a higher scan time to get accurate results, the scans were then put together using photo-editin g tools, in order to illustrate the method of correcting the reflections using copper p owder. Figure 30. Illustration of Copper Powder Correctio n. 4.8 In-Situ XRD results After all the measurements were taken the results w ere compared to calculations that were made based on the deflections made on the wafer pieces. The strain in the y direction was calculated from the basic definition of strain, shown in equation 4.9 d / d0 (4.9) Si [100] Cu [220]

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46 where d0 is the unstrained lattice. Equation 4.9 is expand ed in order to obtain equation 4.10, y [XRD] = ( dXRD – d0)/ d0, (4.10) where XRD refers to strains determined by X-ray dif fraction. The data that was collected from the X-ray diffract ion scans, after correcting for copper powder 220 reflections is shown in Figure 31 1011021031041051061076869707172737475 Si 100 reflections .5 offset .66 .25 offset .73 0 offset .77 -0 offset .725 -.25 offset .604 -.38 offset .615 -.5 offset .582 Intensity a.u.2Theta, degrees Cu 220 reflections Figure 31. Raw data from the X-ray diffraction sca ns, after adjusting for copper 220 XRD Reflections, Showing Cu[k 1], and Cu[k 2] Peaks. After collecting the data the peaks from the reflec tions were determined using the Pseudo Voigt method, by computer software. Once th is was done y[XRD] was determined using equation 4.10.

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47 4.9 Comparison of XRD Results with Calculated Bendi ng Strains After all the measurements were taken the results w ere compared with calculations that were made based on the deflection s made on the wafer pieces. This data was then compared to the strains that were cal culated using the deflections of the wafer piece. Since the wafer piece is bent using a four-point be nding setup, there is a constant radius of curvature between the two inner bending points. The line passing through the center of gravity of the cross-sectiona l area, perpendicular to the radial line to the center of bending, is the neutral axis. The neutral axis is where there is no longitudinal strain. From here the longitudinal st rain (i.e. x) at the surface can be calculated by simple geometry. The longitudinal str ain is the ratio of the change in length from the edge of the wafer to the center of the waf er, divided by the length at the center of the wafer. Since both segments create the same angle from the center of bending, the strain is the result of equation 4.11. x = ( SedgeScenter) / Scenter = ( Redge – Rcenter ) / Rcenter (4.11) In equation 4.11 the common factors and can be factored out resulting in equation 4.12 x = ( Redge Rcenter) / Rcenter (4.12) To determine y one would multiply x by the negative of the Poisson’s ratio (n .) Since Redge Rcenter is the thickness (i.e. t ), equation 4.13 can be derived from equation 4.12. y = n t /2 r (4.13)

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4 8 T h e c a l c u l a t e d v a l u e s o f y c a n b e c o m p a r e d t o t h e v a l u e s o f y [ X R D ] d e t e r m i n e d b y X r a y d i f f r a c t i o n A c o m p a r i s o n o f t h e t w o s e t s o f v a l u e s a r e s h o w n i n T a b l e 3 F r o m t h e r e s u l t s s h o w n i n T a b l e 3 t h e c a l c u l a t e d v a l u e s a r e m u c h c l o s e r t o t h e X R D v a l u e s t h a t w e r e a d j u s t e d f o r c o p p e r T h e e r r o r s f o r v a l u e s i n c o m p r e s s i o n a r e m u c h g r e a t e r t h a n t h o s e f o r t e n s i o n O n e o f t h e p o s s i b l e r e a s o n s f o r t h i s t y p e o f e r r o r i s t h a t t h e w a f e r p i e c e m a y h a v e b e e n u n d e r s o m e d e g r e e o f c o m p r e s s i o n p r i o r t o b e n d i n g T h i s c o u l d h a v e b e e n c a u s e d b y t h e l e a d s c r e w p u t t i n g p r e s s u r e o n t h e w a f e r a t t h i s p o i n t T h i s i s d e s p i t e t h e f a c t t h a t c a r e f u l a t t e m p t s w e r e m a d e u s i n g a p i e c e o f p a p e r t o m o v e t h e w a f e r t o e n s u r e t h a t t h e w a f e r w a s n o t u n d e r t e n s i o n t h e l e a d s c r e w w a s t h e n t i g h t e n e d i n s m a l l i n c r e m e n t s u n t i l t h e w a f e r c o u l d n o t b e m o v e d u s i n g a p i e c e o f p a p e r T a b l e 3 C o m p a r i s o n o f C a l c u l a t e d a n d X R D S t r a i n R e s u l t s S c r e w t u r n s 2 q d e g d [ 1 0 0 S i ] y [ X R D ] y [ c a l c ] V a r i a t i o n ( % ) 0 5 c o m p 6 8 9 9 3 1 3 5 8 E 1 0 4 1 6 8 E 0 4 1 3 9 0 E 0 3 2 3 4 9 % 0 2 5 c o m p 6 9 0 3 5 1 3 5 8 E 1 0 2 4 0 6 E 0 4 8 5 3 8 E 0 4 3 0 1 3 % 0 6 9 1 0 3 1 3 5 7 E 1 0 0 0 0 6 9 1 4 9 1 3 5 7 E 1 0 4 5 5 9 E 0 5 5 8 7 6 E 0 4 0 2 5 t e n s i o n 6 9 1 9 7 1 3 5 7 E 1 0 1 3 4 2 E 0 4 6 0 8 6 E 0 4 1 8 7 % 0 3 8 t e n s i o n 6 9 2 3 8 1 3 5 7 E 1 0 1 9 2 4 E 0 4 1 1 3 2 E 0 3 0 3 5 % 0 5 t e n s i o n 6 9 2 7 7 1 3 5 7 E 1 0 2 4 0 5 E 0 4 1 6 1 3 E 0 3 3 5 5 %

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49 Chapter 5. System Calibration and Experiment with IONIC-System Stress-Gauge 5.1 IONIC-Systems Profile Calibration 5.1.1 Factory Calibration It was determined that a calibration for the profil e on the stress-gauge should be done. The reason for this is that the stress-gauge and the software were purchased used, and the software may belong to a different st ress-gauge. Each stress-gauge is factory calibrated over its en tire measurement range. This calibration is done using a table of 50 data points These data points are used by the system software in order to convert the reading fro m the digital readout to distance measurement. Since the stress-gauge unit only need s to measure differences in distance, a digital indicator will be used to make the distance measurements. 5.1.2 Taking Readings Before doing the calibration of the profile for thi s stress-gauge, first the stressgauge needs to have the setup done as described in the section marked IONIC-Systems stress-gauge Setup. The system measures on both sides of the maximum in tensity point, however it does not measure readings at the maximum. The reas on for this is that at a relative maximum, the slope is always zero; this means that any change in distance at the maximum will not have any change in the digital rea ding. At points near the maximum the slope will be very shallow, this means that ver y small changes in the digital readout

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50 will correspond to large changes in distance. When this happens the uncertainty in the measurements are greatly increased. Therefore in or der to reduce the amount of uncertainty in the measurements, data points should be taken far from the maximum intensity point. The two areas where data points should be taken are referred to by the systems operator manual as the “A”-slope, and the “B”-slope where the “A”-slope refers to distances less than the distance at the maximum int ensity point, and the “B”-slope refers to distances that are greater than the distance at the maximum intensity point. Readings were taken on the “A” slope. These measur ements were taken using a Mitutoyo 543-253 digital indicator. The Mitutoyo d igital indicator has a resolution of 0.001 mm (1 m), and an accuracy of 0.003 mm (3 m). These instruments work by fixing one end onto a fixed surface, such as the ba se plate in this instance, the ball end on a movable arm is placed in contact with the surf ace whose motion is to be measured. The setup for taking these readings was done in a m anner to reduce the amount of uncertainty. The digital indicator uses a magnetic base, and a flexible setup in order to hold the digital indicator. The base plate not onl y serves as a place to attach the magnetic base, but also reduces the motion between the digital indicator, and the fiberoptic bundle. The fiber-optic bundle is mounted to a movable steel spindle. The spindle moves up and down with the probe-gap adjustment, an d the fiber-optic bundle moves along with the spindle. This spindle serves a good place to make the measurements with, using the digital indicator. The setup for m aking these measurements is shown in Figure 32

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51 Figure 32. Setup of Digital Indicator on the IONIC-Systems Stress-Gauge for the Profile Calibrat ion. Before taking any readings a calibration was done f or the digital indicator. This calibration consisted of turning the probe gap adju stment down until the spindle attached to the fiber-optic bundle was about as low as it could go. At this point a marker was placed on the probe-gap adjustment wheel, this would be used to accurately determine whole number of turns of the probe-gap ad justment wheel. The digital indicator was zeroed, and the probe-gap adjustment wheel was turned several turns. At this point a reading on the digital indicator was t aken, and the amount of distance that the fiber-optic bundle moved in one revolution of t he probe-gap adjustment wheel was determined. Then the setup for the stress-gauge wa s done as described in the section marked as IONIC-Systems stress-gauge setup, except that after finding the maximum

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52 point the probe gap was not adjusted to a reading o f 2500, but left at the reading of 5000, at which point readings were taken on the str ess-gauge. When the setup was done before doing this profile c alibration, the probe-gap was not adjusted to obtain an output reading of 2500 as described in the operator’s manual, but instead the probe-gap adjustment was left at th e maximum intensity point where it had a value of 5000 on the digital readout. This w as because the readings were taken over the entire range. This profile calibration me asures changes in distance so the initial distance measurement was given a value of zero. Ea ch measurement taken was compared to the values obtained from the software, in order to give a feel for the level of confidence in the readings. The initial reading fr om the digital output on the stressgauge was entered into the software as a “Before”-r eading, this “Before”-reading would remain for the entire profile calibration. Each su ccessive reading would be entered into the software as an “After”-reading, this would give distance values output from the software that would correspond to distance values t hat were taken manually. The values from the software should be within an accept able amount of tolerance from the values that were taken manually. At each measureme nt the probe-gap adjustment wheel was turned one-eighth of a turn, the output f rom the stress-gauge digital readout was entered into the software as an “After”-reading the distance value was calculated by the software, and all the values, including the digital output from the stress-gauge, the number of turns of the probe-gap adjustment wheel, and the result from the software were entered into a spreadsheet file. A total numb er of 25 data points were taken for each profile calibration. The profile calibration was repeated twice on the same wafer, and another two profile calibrations were done on a nother wafer, and the results were compared.

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53 5.1.3 Profile Calculation The data used in the profile calibration are shown in Figure 33. As can be seen in Table 4 the software has virtually no variance i n the data, as is expected, however there is some amount of variance from one profile t o another, as well as the average profile deviating from the software profile. Figure 33. Data Used in Stress-Gauge Profile Calib ration. For each profile that was taken a linear regression was done, using data points that appeared to be within the linear region. The selection of data points was simply done visually, taking notice of where the data appe ared to deviate from a linear function. The linear regression was done using the built-in f unctions of the spreadsheet software. The linear regression resulted in a total of five s lopes, including the slope for the software. The other four slopes were for each of t he profile calibrations. Since the software calculates the outputs based on internal f ormulas, no variation is expected with 900 1400 1900 2400 2900 3400 3900 4400 4900 -500-400-300-200-1000Signal (mV)Probe Travel ( m) V-software V-software2 trial-1 old disc trial2 old disc new disc trial1 new disc trial2

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54 the software, however the profiles for the samples that were taken manually a variance was expected, and an analysis of this variance was done. An average was taken for all the slopes, and a stan dard deviation was calculated. Also the difference between the averag e slope, and the slope obtained from the software was calculated. This is shown in Tabl e 4. Table 4. Results From Stress-Gauge Profile Calibra tion. Slope Source 25.49897 Software 26.92747 Disc1 Trial1 27.73408 Disc1 Trial2 28.62908 Disc2Trial1 25.20376 Disc2Trial2 Results 27.12359 Slope Average 1.456413 Standard-Deviation 1.624625 Slope Difference from Software 5.2 IONIC-System Experiment 5.2.1 IONIC-Systems Stress-Gauge Readings After doing the profile calibration measurements we re taken on a wafer that was placed under a load. First a setup was done for th e wafer that was being used, as described in the section marked IONIC-Systems stres s-gauge Setup. The load that was placed on the wafer was from common metal weights t hat were placed on the center of the wafer. The weights used were 1g, 25g, 100g, an d both the 1g and 25g weight. The deflection was then determined using the IONIC-Syst ems stress-gauge, and the

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55 deflection of the wafer was determined. The readin gs obtained from the IONIC-Systems stress-gauge are listed in Table 5. Table 5. Readings From IONIC-Systems Stress-Gauge for Various Weights Added, and the Corresponding Displacements. Digital output !" Before (no weight) r #" 1g r$$ #" 25g r%rr&% #" 26g rr%$%& #" 100g r$&$ #" 5.2.2 Wafer Deflection Equation The displacements for the various weights were also calculated for small deflections of a flat circular plate. In order to calculate the deflection of the wafer under the load, the type of loading needs to be determine d. The knife edge on the IONICSystems stress-gauge causes the wafer to be simply supported at the outer edge. The type of loading caused by the weights needs to be d etermined. The weights are made from machined aluminum, and ea ch weight has a flat area on the bottom with a diameter ofabout 0.625 in. Si nce the weights have a flat area on the bottom it seems that the loading should be that of a circular area in the center of the wafer, as shown in Figure 34a. In this case the de flection from the weight would be according to equation 5.1 [55], where the deflectio n is based on a uniform load over a very small central circular area, and edge simply s upported. However the machining process may have produced a lip as shown in Figure 34b. In this case the deflection would be according to equation 5.2 [55], where the deflection is from a uniform annular

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56 line load, as shown in Figure 34c. One test to see if a lip is present would be to feel the bottom area of the weight with the fingernail, howe ver a person is only able to sense approximately one-half of one thousandth of an inch or 0.0005-in [12.7 m], this would mean that it is possible that a small amount of lip could be present without being able to be detected by simple means. As a result the type of loading will be considered to be somewhere between the two types of loading. Both t ypes of loading will be calculated, which will result in an upper, and lower bounds for the calculations. Figure 34. (a) Weight Producing a Small Circular Area Load on the Wafer. (b) Lip Produced by Machining Process. (c) Load from Weight Concentrated Along Outer Edge of Weight-Flat. The equation for small circular area loading is equ ation 5.1, and the equation for a load concentrated along the weight-flat circumfer ence is equation 5.2, equations 5.35.5 are necessary in order to use equations 5.1, an d 5.2 [55, 56,]. n + + =n n p1 3 162D Wa yMax(5.1) case-16a, small circular area load n + =3 9 22 1 2 L L D Wa yMaxn (5.2) case-9a, annular line load

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57 r r r r n + n n + =2 0 0 0 91 4 1 ln 2 1 a r r a a r Ln n(5.3) r r r r n n + n n + n = 1 ln 1 42 0 0 2 0 0 3r a r a a r a r L (5.4) () 2 31 12n= Et D (5.5) where W = weight of load, a = radius of support, E = modulus of elasticity, n = Poisson’s ratio, and r0 = radius at outer edge of load. 5.2.3 Variation Caused by Anisotropic Properties As mentioned earlier, single crystal silicon is not an isotropic material, however the biaxial modulus in the {100} plane is a constan t. The value of the biaxial modulus for (100) silicon plane is B100 = 179.4 GPa [19], where the biaxial modulus in equati on 2.3 is repeated here. () n= 1 E B (2.3) repeated Since the biaxial modulus is a constant it is possi ble to have an equation for the deflection as a function of the elastic properties as yc = f(B,n), as opposed to yc = f(E,n). Note that since 1n2 = (1+ n )(1n ), the value for D when B is substituted for E /(1n ) is given in equation 5.5. () 2 31 12n= Et D (5.5) As stated in the previous section the biaxial modul us is a constant in the {100} plane, but the modulus of elasticity, and the Poiss on’s ratio are not. The modulus of

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58 elasticity ranges from 130 GPa, to 169 GPa, in the {100} plane, this is a ratio of 1.3 or a variation of 30%. The Poisson’s ratio on the other hand ranges from 0.064, to 0.28, in the {100} plane, this is a ratio of 4.375, which is a ratio greater than four to one. This range of an elastic constant for the Poisson’s rati o in the {100} plane would cause a significant amount of error in the calculations. T he biaxial modulus takes both of these elastic properties into account, and is constant th roughout the {100} plane, however the Poisson’s ratio is still part of the equation for d isplacement of the wafer. For the value of D which is used in the calculation of the displacem ent, this range would be (1+ nMAX)/(1+ nMIN), this is a ratio of 1.28/1.064, or 20% variation. This still leaves Poisson’s ratio terms in the deflection calculation s. When calculating the deflection using the maximum, and the minimum Poisson’s ratio the result varies by 6.5%, and 7%, for case-9a, and case-16a respectively. This resul t can change by as much as 7% due to the difference in Poisson’s ratio. 5.2.4 IONIC Stress-Gauge Error Analysis The output for the IONIC stress-gauge is in mV, the refore the deflection caused by a 1 mV change in output is approximately 0.037 m, therefore the displacement resolution of the stress-gauge is approximately 0.0 4 m. The IONIC stress-gauge has a precision to tolerance ratio of 0.28 [57], and from the profile calibration the standard deviation is 1.456413 mV/ m, multiplying this by the slope of the IONIC stres s-gauge gives a tolerance of 0.687 m.

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59 5.2.5 Wafer Deflection Results A (100) silicon wafer was placed in the IONIC stres s-gauge, and measurements were taken. These measurements that were taken wer e compared to the calculations mentioned previously, as shown in table 6. Table 6. Wafer Deflection Measurements and Calcula tions. Digital output !"' % none r #"r #"r #" 1g r$$ #"r$%r& #"r$$ #" 25g r%rr&% #"&% #"% #" 26g rr%$%& #"&&$%& #"r%& #" 100g r$&$ #"$%r$& #"$$$ #" The percent errors from the measured values were al so calculated as shown in table 7. Table 7. Errors From Deflection Calculations. ())*)% ())*) none +,+, 1g %( $( 25g %( $( 26g &r( $( 100g r$( $(

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60 Chapter 6. Conclusions and Future Work The methods of XRD to determine the strain in the m aterial show a potential in being able to determine distortions in X-ray mirror s. X-ray diffraction has been used successfully to determine the stress by measuring t he lattice spacing between crystallographic planes. These tests however were inconclusive because of the errors induced from the play in the bending mechanism. By using a more careful means of determining the curvature in the wafer sample it ma y be possible to more precisely define the results. The IONIC stress-gauge mention ed in this paper is one means to be able to determine the curvature in order to better compare the XRD results with other curvature methods. However there needs to be a mea ns to measure the deflection of the sample that causes a change in curvature of the sample in the bending fixture by either the IONIC stress-gauge, or some other mechan ism. Another reason for the cause of variation between t he calculated strain, and the XRD strain is in zeroing the sample. The variation in one direction has a very low variation, but there is significant variation in th e other direction, which seems to suggest that there was either play in the setup, or some in itial stress in the direction with greater variation. Another method that was mentioned in this paper was using the sin2 method. The sin2 method is used to determine the stress in the laye rs deposited on the mirror substrate, then by using the Stoney’s equation to d etermine the change in curvature after the stress was determined using the sin2 method. The sin2 method was used in

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61 this paper to determine the stress in a tungsten fi lm, not on X-ray mirrors, however the stress in the layers could be able to be measured u sing the same methods. These methods were not pursued during this research due t o lack of time, and other constraints. However, these methods show promise i n being able to determine changes in curvature of X-ray mirrors.

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62 References 1. C. Tang, M. Miller, & D. Laundy, CLRC Technical Report DL-TR-96-003. Daresbury Laboratory Warrington, England. (1996). 2. C. Michaelsen, P. Ricardo and D. Anders, Improved graded multilayer mirrors for xrd applications Institute of Materials Research, GKSS Research Ce nter, vol 42, 2000, pp 308-320 3. N. Gurker, R. Nell, G. Seiler, and J. Wallner, A tunable focusing beamline for desktop X-ray microtomography Institut fu¨r Angewandte und Technische Physik, Technische Universita¨t Wien, vol 70, 1999. 4. C. Rogers, D. Mills, and W. Lee, Performance of a liquid-nitrogen-&oled, thin silicon crystal monochromator on a high-power, focu sed wiggler synchrotron beam Scientific Instruments, v 66, pp 3494-9, 1995. 5. M. Rubel, P. Brunsell; R. Duwe, J. Linke, Molybdenum limiters for Extrap-T2 upgrade: Surface properties and high heat flux test ing, Fusion Engineering and Design v 49-50, p 323-329, 2000 6. D. Meyer,T. Leisegang, A.A. Levin, P. Paufler, A .A. Volinsky, Tensile crack patterns inMo/Si multilayers on Si substrates under high-temperature bending Institut fr Strukturphysik, Fachrichtung Physik de r Technischen Universitt, Materials Science & Processing, pp 303-305, 2003. 7. T. Burke, D. Huxley, R. Newport, R Cernik, An in situ X-ray diffraction method for the structure of amorphous thin films using shallow angles of incidence Review of Scientific Instruments, v 63, pp 1150-2, 1992. 8. J. Harvey, P. Thompson, C. Vernold, Understanding surface scatter effects in grazing incidence X-ray synchrotron applications Proceedings of the SPIE The International Society for Optical Engineering, v 34 47, pp 94-100, 1998 9. M. Rowen, J. Peck, T Rabedeau, Liquid nitrogen cooled X-ray monochromator for high total power loads Nuclear Instruments and Methods in Physics Research, Section A: Accelerators, Spectrometers, D etectors and Associated Equipment, v 467-468, p 400-403, July 21, 2001 10. C. Rogers, High heat flux X-ray monochromators: What are the l imits?, Proceedings of SPIE The International Society for Optical Engineering v 3151, pp 201-207, 1997, High Heat Flux and Synchrotron Ra diation Beamlines

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63 11. H. Goebel, inventor; 1994 Dec 13. X-ray diffractometer United States patent 5,373,544, 12. Y.C. Lin, J.Y. Li, W.T. Yen, Low temperature ITO thin film deposition on PES substrate using pulse magnetron sputtering Applied Surface Science, v 254, pp 3262-8, 2008 13. A A. Volinsky, The Role of Geometry and Plasticity in Thin, Ductil e Film Adhesion University of Minnesota, 2000 14. A. Zanattaa,F. Ferri, Crystallization, stress, and stress-relieve due to nickel in amorphous silicon thin films 2007, pp 15. H. Yu, J. Hutchinson, Delamination of thin film strips Thin Solid Films, 2003, pp 54-63 16. J. Lu, Handbook of Measurement of Residual Stresses. Lilbu rn GA: Fairmont Press, 1996. 17. P. Waters & A.A. Volinsky, Stress and Moisture Effects on Thin Film Buckling Delamination Experimental Mechanics, vol. 47, 2007, pp 163-170 18. W. O'Mara, R. Herring, L. Hunt, Handbook of semiconductor manufacturing technology William Andrew Inc., 1990 19. M. A. Hopcroft, Silicon Micromechanical Resonators for Frequency Re ferences Ph.D. dissertation, Stanford University, Stanford, CA USA, 2007. 20. M. A. Hopcroft, What is the Young’s Modulus of Silicon 2006 21. A. Jachim, Orthotropic Material Properties of Single Crystal S ilicon 1999 22. R.P. Vinci, E.M. Zielinski, J.C. Bravman, Thermal strain and stress in copper thin films Thin Solid Films, vol. 262, 1995, pp 142-153 23. W. Callister Jr., Materials Science and Engineering-7th Ed ., John Wiley & Sons, Inc, 2007 24. O. Anderoglu, Residual Stress Measurement Using X-ray Diffraction Mechanical Engineering, Texas A&M University, 2004 25. Noyan I. C. and Cohen J. B., Residual Stress: Measurements by Diffraction and Interpretation 1987, Springer-Verlag, New York. 26. Hauk V., Structural and Residual Stress Analysis by Nondestr uctive Methods 1997, Elsevier, Amsterdam. 27. Winholtz R. A. and Cohen J. B., Aust. J. Phys ., 41, 189-199 (1988).

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64 28. G. Beck, S. Denise, A. Simons, International Conference on Residual Stresses Elsevier Applied Science, 1989 29. Slaughter W, The Linearized Theory of Elasticity Boston: Birkhauser; 2002. 30. Nye J. F., Physical Properties of Crystals: Their Representati on by Tensors and Matrices 1985, Clarendon Press, Oxford. 31. Reddy JN, Mechanics of Laminated Composite Plates Boca Raton, FL: CRC Press; 1997. 32. G.G. Stoney, The Tension of Metallic Films Deposited by Electrol ysis Proc. Roy. Soc. London A 82, pp. 172-175 (1909) 33. M. A. Moram, M. E. Vickers, M. J. Kappers, and C. J. Humphreys, The effect of wafer curvature on X-ray rocking curves from galliu m nitride films 2008. 34. Barber, K. Samuel; Soldate, Paul; Anderson, H. Erik; Cambie, Rossana; McKinney, R. Wayne; Takacs, Z. Peter; Voronov, L. D mytro; Yashchuk, V. Valeriy, Development of pseudorandom binary arrays for calib ration of surface profile metrology tools, Journal of Vacuum Science and Technology B: Microelectronics and Nanometer Structures v 27, pp 3213-3219, 2009. 35. J.H. Selverian, Errors in curve fitting of profilometer data, Journ al of Vacuum Science & Technology A (Vacuum, Surfaces, and Films ) v 10, pp 3378-82, 1992 36. G. Chong, Smoothing noisy data via regularization: statistica l perspectives Inverse Problems, v 24, pp 034002 (20 pp.), June 20 08. 37. D. Koutsoyiannis, Broken line smoothing: a simple method for interpol ating and smoothing data series Environmental Modelling & Software, v 15, pp 13949, 2000. 38. A.J. Rosakisa, R.P. Singh, Y. Tsuji, E. Kolawa, N.R. Moore Jr., Full field measurements of curvature using coherent gradient s ensing: application to thin film characterization Thin Solid Films, vol. 325, 1998, pp 42–54 39. J Rachwal, A A. Volinsky, H Stcker, D. Meyer, X-ray Stress Analysis of Silicon Wafers Under Four-point Bending Technischen Universitt, Dresden, Germany, University of South Florida,2007 40. A.A. Benzerga, N.F. Shaver, Scale dependence of mechanical properties of single crystals under uniform deformation Scripta Materialia, vol. 54, 2006, pp 1937-1941 41. K. Bretzfield, F. Woeste, Joist Curvature versus Sheathing Curvature and the Probable Role of each on Ceramic Tile Performance 2008, TTMAC HARDSURFACE Magazine.

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65 42. V. Thomsen, D. Schatzlein, D. Mercuro, Tutorial : Attenuation of X-rays by matter, Spectroscopy v 20, pp 22-25, 2005. 43. L. Gerward, X-ray attenuation coefficients and atomic photoelec tric absorption cross sections of silicon Journal of Physics B (Atomic and Molecular Physic s), v 14, pp 3389-95, 1981. 44. J Taylor, Error Analysis The Study of Uncertanties University Science Books, 1982 45. R Figiola, D Beasley, Theory and Design for Mechanical Measurements 4th-E d ., John Wiley & Sons, Inc., 2006 46. Philip J. Withers, Mark R. Daymond and Michael W. Johnson, The precision of diffraction peak location 2001 47. Webster P. J. and King W. P., Optimization of neutron and synchrotron data collection and processing for efficient Gaussian pe ak fitting. Engineering Science group Technical Report ESG01/98 The Telford Institute of Structures and materials Engineering, University of Salford, March 1998. 48. H. Toraya, Estimation of errors in the measurement of unit-cel l parameters 2001 49. E.N. Dulov, D.M. Khripunov, Voigt lineshape function as a solution of the parabolic partial differential equation Radiative Transfer, vol. 107, 2007, pp 421428 50. M. E. Fitzpatric, A. Lodini, Analysis of Residual Stress by Diffraction using Neutron and Synchrotron Radiation Boca Raton, FL: Taylor & Francis, 2003 51. P.. Prevy, The Use of Pearson VII Distribution Functions in Xray Diffraction Residual Stress Measurement vol. 29. 1986, pp 103-111 52. E.N. Dulov, D.M. Khripunov, Voigt lineshape function as a solution of the parabolic partial differential equation, Radiative Transfer vol. 107, 2007, pp 421428 53. W. Parrish and J. I. Langford, International Tables for Crystallography Volume C: Mathematical, Physical and Chemical Tables, 2004 54. D Balzar, Voigt-function model in diffraction line-broadening analysis University of Colorado, 1999 55. Y. Warren, Roark's Formulas for Stress and Strain. 6th ed., New York: McgrawHill, 1989 56. P. Walter, Formulas for Stress, Strain, and Structural Matrice s Charlottesville, Virginia: John Wiley & Sons, Inc., 1994

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66 57. V.S. Dharmadhikari, Statistical approach to parameter study of stress i n multilayer films of phosphosilicate glass and silic on nitride Journal of Vacuum Science & Technology A (Vacuum, Surfaces, and Films ), v 9, pp 2497-502, 1991.

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67 Appendices

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68 Apendix A A-1 IONIC-Systems Stress-Gauge The IONIC-Systems stress-gauge determines the resid ual stress in the thin film on a wafer based on the change in curvature of the wafer; it does this by taking two measurements, before, and after processing in order to determine the deflection caused by processing the wafer. This system operates by m easuring the light intensity reflected off the back side of the wafer. The IONIC-System c an make very precise measurements using this method. This system determines the defl ection at the center of the wafer; as a result changes in curvature over the wafer diameter cannot be determined. The use of a light beam for measurement means that the measureme nts are nondestructive. Readings are taken before and after the wafer is pr ocessed, thus a change in curvature is determined. After the deflection is determined, the stress can be calculated using the deflection, and the properties of the wafer. The back panel of the IONIC-Systems stress-gauge co ntains the probe-gap adjustment, and lockdown, the lamp intensity adjust ment, and lockdown, and the computer interface shown in Figure 35. The probe-g ap adjustment consists of a large thumbwheel mounted on a threaded shaft that provide s the adjustment; the lockdown for the probe-gap consists of a threaded wheel that tig htens against the adjustment wheel. Next to the wheel are markings to indicate the dire ction for increasing, and decreasing the probe-gap, these markings are labeled as “INCR, ” and “DECR”, respectively. The back panel also contains a RS-232 computer interfac e, which allows the user to link the output reading to a computer.

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Appendix A: (Continued) 69 Figure 35. Back panel of IONIC-systems stress-gaug e. The top face of the IONIC-System contains the power switch, digital readout, and a base-plate. The power switch controls the power t o the stress-gauge. The digital readout is a four digit readout related to the refl ected light intensity. This readout is a measure of the voltage reading of the internal sens or, where a reading of 5000 corresponds to 5 Volts. The user does not need to relate this voltage to a light intensity, there is software for the user to input the readout on the system to determine the distance of the fiber-bundle to the back of the waf er. The top of the IONIC-Systems stress-gauge has a bas e-plate, with a fiber optic bundle, a series of knife edges, and positioning bl ocks as shown in Figure 36. The fiber optic bundle is at the center of the base plate; it can be adjusted up or down, and is used to make the measurements to determine the deflectio n of the wafer center. The knife edges are the series of concentric circles surround ing the fiber-optic bundle. These knife edges hold the wafer near the outer edge of t he wafer, so that the measurement is

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Appendix A: (Continued) 70 over most of the wafer’s diameter. The knife edges allow for different size wafers, and each series of knife edges is at a different height to allow clearance for larger wafers. The base plate also contains three different positi oning blocks. These positioning blocks can be placed accordingly to allow for different si ze wafers. These positioning blocks along with the knife edge constrain the wafer for a ccurate location. These positioning blocks are located at the 12-o’clock, 4-o’clock, an d 8-o’clock position. The positioning block at the 4-o’clock position is fixed, while the other two are spring loaded. The two spring-loaded blocks have different spring constant s, so that one will be seated, while the other is under compression. Figure 36. Top of IONIC-Systems stress-gauge. The wafer needs to be fixed in all degrees of freed om to help ensure accuracy in measurements.

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Appendix A: (Continued) 71 The knife edges constrain the wafer to a plane allo wing only three degrees of freedom, such as the x -direction, y -direction, and rotation. The positioning block at the four-o’clock position is fixed in place and constra ins the wafer in the y-direction. The other two positioning blocks are spring mounted, th e positioning block at the 8-o’clock position has a spring with a much stiffer spring co nstant than the one at the 12-o’clock position, and therefore the positioning block at th e 8-o’clock position will determine the position in the x-direction. This leaves only one degree of freedom remaining, the angular position. The major flat on the wafer is p ositioned against the block at the 8o’clock position, and the user needs to press the p ositioning block at the 12-o’clock position several times in order to properly seat th e flat of the wafer against the positioning block at the 8-o’clock position. After this is done the wafer is properly positioned. In this manner two of the positioning blocks are placing spring load on the wafer, in order to help ensure that the wafer will not drift out of position. The IONIC-System uses fiber-optic bundle to make me asurements. The fiber optic bundle is very sensitive, and any contact can cause damage, care to avoid contact with the fiber optic bundle is necessary. The fibe r optic bundle contains optical fibers half of these optical fibers transmit light, and ha lf of the optical fibers receive light. The measurement is taken by transmitting light from som e of the optical fibers onto the back of the wafer, the light is then reflected back to t he optical fiber bundle, and the receiving fibers transmit the light reflected back to a light sensor that reads the intensity. The light that is reflected back is compared to an intensity reading directly from the light source, in order to reduce certain types of errors. The light being reflected has a focal point within the range of measurement, the light intensity will maximize at this focal point, and therefore there is a relative maximum within the ra nge of measurement. Reading cannot

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Appendix A: (Continued) 72 be taken at this maximum point, because the derivat ive of this measurement with respect to light intensity will be zero. Readings may be taken on either side of the maximum reading, the manual refers to these two sid es of the maximum as slopes A, and B. In order to calculate the stress on the wafer the I ONIC-Systems stress-gauge comes with software in order to do the calculation using the digital readout. The software is a DOS based program that is included wi th the IONIC-Systems stressgauge. The software is on a 3.5-inch floppy disk, and contains all the system files necessary to run the software. The program file fo r the software is STRESS.EXE. Launching STRESS.EXE brings the user to the main me nu, where there are four options available, 1-Stress measurement, 2-Stress calculati on, 3-Others, and End. End simply exits the program as suggested. Stress measurement labeled as option-1 has five main options, Before, After, Wafer, View, and Esc, where Esc returns to the previous menu. Before any measurements are taken, the user needs t o enter a name for the wafer; this way information for several wafers may be stored at one time. Choosing the option marked “Wafer” allows the user to enter a name for the wafer. Each wafer must have some name to it before entering values for stress measurement into the software. Pressing “W”, while at the Stress-Measurement screen selects this option, and brings up a dialog box, wh ich prompts the user to enter a name for the wafer. The user can enter any name that th e user deems appropriate to identify the wafer, into the dialog box, and then press the enter key. The name that the user selects is limited to a certain number of character s. The user should choose option marked “Before”, prov ided that the wafer has been given a name to identify it. The method to en ter a name for a wafer is given in the

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Appendix A: (Continued) 73 previous paragraph. Pressing “B”, while at the Str ess-Measurement screen selects this option, and brings up a dialog box, which prompts t he user to enter a value taken from the digital readout on the IONIC-System stress-gaug e. After the value is entered the user should press the return key, to return to the previous menu. It is important that the user take a measurement, and enter this value into the software before any processing is done on the wafer. If no measurement is taken b efore any processing is done on the wafer, the software will not be able to properly de termine the stress in the wafer. The user should choose option marked “After”, when the wafer has been processed, and a measurement has been taken from di gital readout of the IONICSystem stress-gauge. Pressing “A”, while at the St ress-Measurement screen selects this option, and brings up a dialog box, which prom pts the user to enter a value taken from the digital readout on the IONIC-Systems stres s-gauge. After the value is entered the user should press the return key, to return to the previous menu. If the user has entered before, and after measurements into the sof tware at this point, the software will be ready to calculate a value for the stress on the wafer. By determining the difference between the two readings, a change in the wafer cur vature can be determined. The light intensity that is being measured may chan ge in two ways, one is due to a change in distance of the fiber bundle from the w afer, and the other is due to changes of the reflectivity in the back of the wafer. For this reason the back side of the wafer is used for measurements, and it is necessary to mask of the area on the back of the wafer that will reflect the light to make measurements. The stress-gauge constantly measures the light sour ce directly, and compares this reading to the light intensity that is being r eflected off the back side of the wafer. By

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Appendix A: (Continued) 74 making the measurements in this manner, errors due to fluctuations in the lamp intensity either because of age, or line voltage fluctuations are greatly reduced. A-1.1 IONIC-Systems Stress-Gauge Calibration Setup In order to setup the IONIC-Systems stress-gauge th e unit needs to warm up for a period of 72-hours before measurements. There ar e different sets of threaded holes to which the positioning blocks can be installed depen ding on the size of the wafer being used. The wafer should be placed so that the back side of the wafer, that is the side that will not be processed is down, that is in contact w ith the appropriate set of knife edges, and facing the optical-bundle. The major flat on t he wafer should be flush with the 8o’clock positioning block, and the user should use the procedure described in the previous section to ensure that this is properly do ne. Care needs to be used to avoid touching the optical-bundle, to avoid damage that m ay yield inaccurate results. After the wafer is positioned, a reading should be taken, and the wafer removed, then reinserted several times using the correct seating procedure. The user needs to check the reading each time the wafer is seated, and ensure that a re peatability of 10 counts on the digital readout is reached. With the wafer properly seated, the probe-gap adjus tment needs to be unlocked, and the user should adjust the probe gap so that th e reading on the indicator is increasing. If when adjusting the probe-gap the re adout decreases the user should make the adjustment in the other direction, also if the digital readout starts to approach the maximum reading, the user should unlock the lam p intensity adjustment, and adjust the lamp intensity to a value that is well within t he range of the digital reading, a value of 5000 is a good suggested value. The user should co ntinue the adjustment until the

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Appendix A: (Continued) 75 readout reaches a maximum. The user should obtain the maximum by continuing the adjustment past the maximum point until the readout starts to decrease, then reverse the adjustment until the returning to the previous maxi mum value. At this point the user should unlock the lamp intensity adjustment, and ad just the lamp intensity so that the digital readout is within 10 counts of 5000, and then lock the lamp intensity adjustment, ensuring that the readout does not drift out of spe cs in doing so. This is best done by first setting the value to 5000, and locking down t he adjustment first, and noting the amount of drift that occurs from doing so. Then th e user should unlock the lamp intensity again, and adjust the lamp intensity so a s to account for the amount of drift that occurs while locking the adjustment down. At this point the user should adjust the probe-gap in the decreasing direction, that is so t hat the gap between the fiber optic bundle, and the back side of the wafer will be decr easing. Note that turning the gap adjustment in either direction will cause a decreas ing value in the digital readout, so the user should refer to the back of the machine to ens ure they are moving the fiber-optic bundle in the “DECR”-direction, indicating a decrea sing gap between fiber-optical bundle, and wafer. The adjustment should be made t o obtain a digital readout within 100 counts of 2500 on the digital readout. The pro be-gap adjustment needs to be locked down at this point. The wafer is ready to m ake a reading. The user should record this reading on the digital readout, and ent er this value into the software for the IONIC-Systems stress-gauge, this should be entered as the before reading in the software. The previous section labeled IONIC-Syste ms software describes the manner of entering before, and after values.

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Appendix A: (Continued) 76 A-1.2 Taking a Measurement on the IONIC-Systems Str ess-Gauge In order to take a measurement on the IONIC-Systems stress-gauge the setup needs to be done for the system, as described in th e previous section. Once the setup is done a before-reading should have been taken, be fore the wafer is processed. The back of the wafer should be masked in order to avoi d changing the reflectivity of the back of the wafer, where the IONIC-Systems stress-g auge will take a measurement. Once the wafer is processed the wafer should be pla ced on the stress-gauge knife edge, and the wafer needs to be seated by repeatedly push ing, and releasing the block at the 12-o’clock position, as described in the previous s ection. The user should take the reading from the digital readout, and enter it into the software as an after reading.