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Residual stress analysis in 3csic thin films by substrate curvature method
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Thesis (M.S.M.E.)University of South Florida, 2010.
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ABSTRACT: Development of thin films has allowed for important improvements in optical, electronic and electromechanical devices within micrometer length scales. In order to grow thin films, there exist a wide variety of deposition techniques, as each technique offers a unique set of advantages. The main challenge of thin film deposition is to reach smallest possible dimensions, while achieving mechanical stability during operating conditions (including extreme temperatures and external forces, complex film structures and device configurations). Silicon carbide (SiC) is attractive for its resistance to harsh environments, and the potential it offers to improve performance in several microelectronic, microelectromechanical, and optoelectronic applications. The challenge is to overcome presence of high defect densities within structure of SiC while it is grown as a crystalline thin film. For this reason is important to monitor levels of residual stress, inherited from such grown defects, and which can risk the mechanical stability of SiC made thin film devices. Stoney's equation is the theoretical foundation of the curvature method for measuring thin film residual stress. It connects residual film stress with substrate curvature through thin plates bending mechanics. Important assumptions and vii simplifications are made about the filmsubstrate system material properties, dimensions and loading conditions; however, accuracy is reduced upon applying such simplifications. In recent studies of cubic SiC growth, certain Stoney's equation assumptions are violated in order to obtain approximate values of residual stress average. Furthermore, several studies have proposed to expand the scope of Stoney's equation utility; however, such expansions demand of more extensive substrate deflection measurements to be made, before and after film deposition. The goal of this work is to improve the analysis of substrate deflection data, obtained by mechanical profilometry, which is a simple and inexpensive technique. Scatter in deflection data complicates the use of simple processes such as direct differentiation or polynomial fitting. One proposed method is total variation regularization of differentiation process; and results are promising for the adaptation of mechanical profilometry for complete measurement of all components of nonuniform substrate curvature.
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Advisor: Alex A. Volinsky, Ph.D.
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Stress analysis
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Curvature analysis
Regularization
Noise reduction
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Residual Stress Analysis in 3C SiC Thin Films by Substrate Curvature Method by Jose M Carballo 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: Alex A. Volinsky, Ph.D. Jose L. F. Porteiro, Ph.D. Craig Lusk, Ph.D. Date of Approval: March 25, 2010 Keywords: stress analysis Stoney s equation curvature analysis, regularization, noise reduction Copyright 2010, Jose M Carballo
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Dedication I give thanks to my parents, brother, girlfriend, and friends for their constant support throughout my Masters program. Without their presence, I would have been greatly dispirited while completing this work.
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Acknowledgements I am thankful to Dr. Volinsky for his constant teaching attitude, support and motivation throughout this year. Also, special thanks to Dr. Saddow, and Chris Locke, for their contribution on samples for this work; and Richard Everly, N.N.R.C. staff, for his help on using the tools I needed. I also give thanks to my committee members, Dr. Porteiro and Dr. Lusk, and my group members, for their attention, and contribution to my work.
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i Table of Contents List of Tables iii List of Figures iv ABSTRACT vi Chapter 1. Introduction 1 1.1 Motivation for research 1 1.2 Residual Stress in thin films 5 1.3 Curvature method for measuring residual stress 8 1.4 Derivation of Stoneys equation 12 1.5 Limitations and modifications of Stoneys equation 21 Chapter 2. Analysis of Substrate Curvature 26 2.1 Substrate deflection measurements 26 2.2 Data analysis by polynomial curve fitting 29 2.3 Segmentation of substrate deflection data 34 2.4 Regularization method 37 2.5 Comparison of substrate curvature analysis methods 44 Chapter 3. Conclusions and Future Work 47 References 51 Appendices 61 Appendix A. Substrate curvature results for 3C SiC films on Si (100) substrates. 62 Appendix B. Substrate curvature results for W films on Si (100) substrates. 64 Appendix C. Total Variation Regularized Differentiation code using Matlab 67
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ii Appendix D. Qualitative selection of appropriate regularization 69 Appendix E. Local implementation of Stoneys equation with substrate curvature results for 3C SiC film on Si(100) samples. 70
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iii List of Tables Table 1. Selection of appropriate regularization parameter by visual inspection criteria. 69 Table 2. Magnitude and location of equibiaxial residual stress values. 70
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iv List of Figures Figure 1. Sequential analogy to thin film deposition on substrate. 6 Figure 2. Diagram of thin plate xz transverse section undergoing bending load ( M). 14 Figure 3. Transverse section in xz plane of a thin plate with a normal stress x profile caused by bending moment Mx. 17 Figure 4. State of stress and loads of a thin film substrate system. 19 Figure 5. Diagram of a Si (100) substrate with different scan orientation angles. 27 Figure 6. Thickness of CVD deposited 3C SiC films along two orthogonal orientations. 28 Figure 7. Substrate deflection data for 3C SiC films along two orthogonal orientations. 29 Figure 8. 3rd order polynomial fitting of substrate deflection for a) 0and b) 90 orie ntations, and c d) their second derivatives, respectively. 30 Figure 9. Average polynomial describing curvature change of 3C SiC on Si (100) systems. 32 Figure 10. Average polynomial describing curvature change of W on Si (100) systems. 33 Figure 11. Segmentation method applied to deflection data simulation. 36 Figure 12. Optimization of segmentation method by selecting number of segments that yields lower residual norm for all substrate deflection models used. 37 Figure 13. Visual inspection of regularization parameter ( ) effects, classified as a) too low, b) too high, and c) adequate. 42
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v Figure 14. Comparison of analysis methods for substrate curvature change of 3C SiC film on Si (100) along (a) 0 and (b) 90 scan orientations. 43 Figure 15. Comparison of analysis methods for substrate curvature change of W film on Si (100) along (a) 0 and (b) 90 scan orientations. 43 Figure 16. Residual film stress profiles after direct implementation of Stoneys equation. 49 Figure 17. 62 Figure 18. 62 Figure 19. of sample 040 from 0 scan. 62 Figure 20. of sample 040 from 90 scan. 62 Figure 21. of sample 043 from 0 scan. 63 Figure 22. of sample 043 from 90 scan. 63 Figure 23. of sample A1 from 0 scan. 64 Figure 24. of sample A1 from 15 scan. 64 Figure 25. of sample A1 from 30 scan 64 Figure 26. of sample A1 from 45 scan 64 Figure 27. of sample A1 from 60 scan. 65 Figure 28. of sample A1 from 75 scan. 65 Figure 29. of sample A1 from 90 scan. 65 Figure 30. of sample A1 from 105 scan. 65 Figure 31. of sample A1 from 120 scan 66 Figure 32. of sample A1, 135 scan. 66 Figure 33. of sample A1, 150 scan. 66 Figure 34. of sample A1, 165 scan. 66
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vi Residual Stress Analysis in 3C SiC Thin Films by Substrate Curvature Method Jose M Carballo ABSTRACT Development of thin films has allowed for important improvements in optical, electronic and electromechanical devices within micrometer length scale s. I n order to grow thin films, there exist a wide variety of deposition techniques, as each technique offers a unique set of advantages. The main challenge of thin film deposition is to reach smallest possible dimensions while achieving mechanical stability during operating conditions ( including extreme temperatures and external forces, complex film structures and device configurations). S ilicon carbide (SiC) is attractive for its resistance to harsh environments, and the potential it offers to improve perfor mance in several microelectronic, micro electromechanical, and optoelectronic applications. T he challenge is to overcome presence of high defect densities within structure of SiC while it is grown as a crystalline thin film For this reason is important to monitor levels of residual stress, inherited from such grown defects, and which can risk the mechanical stability of SiC made thin film devices. Stoneys equation is the theoretical foundation of the curvature method for measuring thin film residual stress. It connects residual film stress with substrate curvature through thin plates bending mechanics I mportant assumptions and
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vii simplifications are made about the film substrate system material properties, dimensions and loading conditions; however a ccuracy is reduced upon applying such simplifications In recent studies of cubic SiC growth, certain Stoneys equation assumptions are violated in order to obtain approximate values of residual stress average. Furthermore, several studies have proposed to expand the scope of Stoneys equation utility; however, such expansions demand of more extensive substrate deflection measurements to be made, before and after film deposition. The goal of this work is to improve the analysis of substrate deflection data, obtained by mechanical profilometry, which is a simple and inexpensive technique. S catter in deflection data complicates the use of simple processes such as direct differentiation or polynomial fitting. O ne proposed method is total variation regularization of di f ferentiation process; and results are promising for the adaptation of mechanical profilometry for complete measurement of all components of nonuniform substrate curvature.
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1 Chapter 1. Introduction 1.1 Motivation for research Because of its favorable mechanical and electrical properties, silicon carbide (SiC) raised plenty of inter est among fabricators of micro electronic, optoelectronic, and microelectromechanical thin film devices. SiC properties make this crystalline material preferable over currently used poly silicon for several electronic applications ; these properties are wide band gap, and high breakdown electric field strength, high thermal conductivity, saturated drift velocity, elastic modulus and hardness. Moreover, SiC is extremely tolerant to harsh environment, which is constituted by abrasive and corrosive subs tances, extremely high operating temperatures, and low levels of friction [1, 2]. Commercial use of SiC for electronic device s began with substrate fabrication for blue and green light emitting diodes (LED s); and actually, this has been one of very few successful commercial applications of SiC based thin film devices. T here is high interest however, in research of SiC based devices in a very wide variety of microscopic applications [2, 3]. In power applications with high voltage, SiC based field effec t transistors and power diodes have been developed with low onstate voltage drops and off state leakage, and fast switching characterist ics [4 6] Additionally SiC based chemical field effect
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2 transistors are being developed for gas sensing applications, such as exhaust monitoring in pistoncylinder, and turbine engines [7 9] Lastly, another SiC application example lies in micro electromechanical systems (MEMS) fabrication; and this presents an opportunity for SiC to outperform other materials ; because its hardness, highest next to diamond; and tolerance of extreme operating conditions [10, 11] SiC is a crystal that exists in more than 200 polyt ypes [12] Each SiC polytype corresponds to a unique stacking sequence formed by the SiC unit ; which can arrange itself in either a cubic, hexagonal, and rhombohedral form (such is the reason for polytype notation C, H or R, and prec e eding number corresponds to the number of layers involved in one sequence repetition). H exagonal polytypes 4H SiC and 6H SiC, have been widely used for bulk growth of substrates. Thin film growth of these two polytypes has also been achieved by the Chemical Vapor Deposition ( CVD) mainly for microelectronic applications; however, homoepitaxial growth has only been possible on substrates of the same material. Growth of SiC in both bulk and thin film forms is complicated and expens ive; and thus it is currently nonfeasible for device mass production. Because of the same properties that make SiC desirable, growth systems are required to meet very demanding thermodynamic conditions. Moreover, resultant defect density levels of SiC cry stal structure are too high to control device failure, and efficiently grow substrat es larger than 4 in [13, 14]
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3 On the other hand, only cubic 3C SiC can be epitaxially grown on a substrate of di fferent material by CVD [15] However, defect density is an ongoing issue of thin film quality; consequently, CVD process has been greatly improved over the last 2 decades, achieving significant reduction of defects, especially within layers above film substrate interfaces. What makes CVD an appropriate deposition technique is repeatability, and versatility of film composition results [1 1, 16, 17] Optimization of thin film quality is achieved by altering the CVD process ; hence, there are various types of CVD reactor s, which accomplish special conditions such as lower chamber pressure, higher temperature, flow orientation of reactants and plasmaenhanced reactions. Moreover, v ariable process parameters involve temperature and pressure in the reaction chamber, composition of reactants, substrate holder position, and substrate alignment. Consequently, each combination of variables gener ates a unique CVD process that achieves certain film qualities, e.g. thickness uniformity [18, 19] ; epitaxial or amorphous growth [20 24] ; film material purity [25 27] ; and composition homogeneity [28, 29] In the case of MEMS applications, 3C SiC offers significant advantage over materials currently used (i.e. silicon among others); especially for applications that require operation within harsh environments [30] Being a significant reason for developing 3C SiC growth, film quality for MEMS application is characterized in terms of the film mechanical properties; which are hardness strength, and elastic properties [31]
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4 The role of residual stress in thin films comes into play when studying mechanical integrity of thin films; it has a significant influence on film strength, and thus on device reliability. Additionally, high residual stresses can generate plastic deformation within material, and even promote inter diffusion of adjacent volumes of different compositions. For this reason characterization of 3C SiC film quality involves an accurate understanding of resultant levels of residual stres s. Measurement of thin film residual stress can be performed by several methods, which vary in terms of what measured quantity, and theoretical approach es are related to residual film stress. Each measuring method includes a unique set of advantages, and c hallenges; consequently, appropriate technique selection must consider how much accuracy is affected by the interpretation of measured quantity, and the involved theoretical assumptions. The present work explains how interpretation of measured quantity can be improved for a specific residual stress measurement technique, called the substrate curvature method, first proposed by Stoney for thin films deposited by electrolysis [32] This technique is perhaps the most practical in terms of implementation; as it is nondestructive, inexpensive, and involves simple tool usage and post measurement analysis. The following section will help the reader understand the mechanisms of formation of residual film stress, and how these affect its measurability. Next, and before presenting the work done with the curvature method, a brief review of residual stress measuring techniques will be given. Chapter 2 proposes a different approach for analyzing substrate
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5 deformation, and Cha pter 3 will provide conclusions and future recommendations. 1.2 Residual Stress in thin films W.D. Nix presents a visual analogy to the sequence of events that is very helpful to understand the resultant physical loading in a film substrate system; caused by the residual film stresses [33] In this analogy, the substrate and films are represented as two thin plates; each having different lateral dimensions, and t he film has a thickness much smaller than that of the substrate. Figure 1 a shows how both plates are initially unstrained. Then in Figure 1 b, film is uniformly strained among its volume by external forces located at the edges; causing the film lateral dimensions to match perfectly with those of the substrate ( e.g. if the film or iginally has smaller width and length, then the external forces need to be of tensile nature). Next, film and substrate adhere to each other, so bonds hold both plates together relieving the film from external forces As a result, there is a tendency for the film to recover its original geometry. The film substrate bonds are s hear ed at the edges of the plates; causing the substrate to deform into a new equilibrium st ate as shown in Figure 1 c. The mentioned te ndency is analogue to the causes of residual film stress. There are several mechanisms that independently cause residual stress. These stress formation mechanisms are classified as thermal, epitaxial, or
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6 intrinsic. First, thermal residual stress is generated due to the misfit in coefficients of thermal deformation, existing between film and substrate materials; moreover, film deformation is constrained by adhesion to the rigid substrate. Thermal stresses o ccur upon any temperature change; such as that experienced between film deposition event, and after deposition exposure. Figure 1 Sequential analogy to thin film deposition on substrate. a) Film and substrate are originally un strained. b) External forces strain film to match substrate. c) Substrate and film achieve equilibrium state after deformation. Epitaxial stresses are caused during epitaxial film growth, due to the misfit in crystal lattices existing between film and subs trate (or underlying film) materials. Similarly, adhesion to rigid substrates prevent s growing film from adapting to such lattice misfit, causing the mentioned epitaxial stress [33]
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7 Lastly, intrinsic stresses are ca used by volume changes that occur within the film material, and are induced by mechanisms of energy minimization. These mechanisms are related to epitaxial growth rate atom mobility, point defects, impurities, granular growth, and phase transformations A s a consequence, stresses arise intrinsically upon any volume change, and due to the constraint imposed by the same structure of the solid film, and by adhesion between film and substrate [34] In an actual film, epi taxial, thermal and intrinsic types of stresses are superimposed to a resultant residual stress, which can be large enough to generate significant deformations, or even failure of a thin film In other words, stresses within film may translate in the form of substrate cracking ; film delamination or buckling. Moreover, l ong term presence of stresses within a film can promote diffusionrelated processes within film substrate system such as densification of film material, and phase transformations. On the other hand lower levels of residual film stress are equilibrated by film substrate interactions [35 37] Deposition parameters can control residual stress for mation; however, because of the complexity and variety of mechanisms, studies are performed by focusing on specific combinations of material s and deposition technique. The CVD process parameters abovementioned are also directed towards the controlling of resultant residual stress in a CVD film; clear examples can be seen in [38 44]
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8 Evidently, it is important to understand what the mechanisms of residual film stress formation are; however, it is equally essential to understand at what length scale each stress source acts on. Intrinsic sources of stress keep the absolute residual film stress constant only over microscopic lengths; while effects of extrinsic and epitaxial stresses are macroscopically consistent, even throughout the entire film. In other words, the resultant residual stress made up by superimposed individual sources of stress, can be of a constant value over a certain volume of the film grown; or also can vary even microscopic ally, and thus averaging to a certain value, or zero, over macroscopic volumes. For this reason, residual stresses are also classified into macroscopic and micr oscopic stresses. This classification is based on the length scale over which the value of an individual stress source makes up one period of oscillation within the thin film structure [45] 1.3 Curvature m ethod fo r measuring residual stress The main purpose of this work is to present a technique to measure residual film stress induced on a film after deposition. It was previously explained that the determination of stress over a certain volume significantly depends on the length scale; accordingly, different techniques vary in their resolution range [45] Moreover, each currently available technique utilizes a certain theoretical approach which relates the residual stress in a film to a measureable property of
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9 thin film substrate system The following paragraphs will briefly reveal the wide variety of residual stress measuring techniques that are currently used. There are two measuring mechanism through which residual stress is frequently measured, one is by directly measuring strain, and the other by measuring bulk deflection of the film substrate body. Strainbased techniques have many useful capabilities besides just measuring residual stress, such as identifying material elements and compositions, and analyzing crysta lline structure s [46] The most significant theory of strainbased techniques is X ray diffraction (XRD), from which many different instruments and measurement procedures branch out. This technology is based on Braggs law and its use is mainly intended for analyzing internal structure and composition of crystalline materials ; nevertheless it has played a major role in the tas k of measuring residual film stresses [47] Bragg diffraction allows for measuring spacing between crystallographic planes which is unique for each specific crystal arrangement ; this then permits for valuable identification and analysis of crystal type, structure composition, and orientation Additionally, m aterial strain is quantified b y measuring the change of inter planar spacing that a film material undergoes after film deposition, with respect to its known unstrained spacing value. Subsequently, and under certain assumptions stress and strain are related through elasticity theory (Hookes law ) [48, 49] An e xtensive number of works has developed procedures based on XRD for me asuring residual stress effects; each work intending to overcome a certain
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10 challenge. For example, variation of strain with respect to film thickness has been determined through certain ways based on the sin2 technique [50 53] Glancing incidence XRD technique is used when small thicknesses only allow for small angles of diffraction [52, 53] High resolution XRD has better capabilities for determining complete strain tensor information, and thickness profile s [54, 55] In conclusion, XRD techniques can provide great detail of residual strain within materials at both microscopic inter granular and intragranular regions A limitation of Bragg diffraction techniques is that it only works for crystalline and polycrystalline materials. Ani sotropy of materials is analyzed with XRD by measuring strain at the same surface location from various independent perspectives (directions). On the other hand, more complicated diffraction approaches have been proposed for amorphous structures [51, 56] Raman spectroscopy is a different stress determination approach which analyzes light spectra emitted by specific materials. A light beam focused at a point location of the film substr ate system changes the internal energy of the compounds within the film material ; and a specific light spectrum is obtained, specific to the material composition The change of a certain peak can be related to the induced stress or strain. Implementation of Raman spectroscopy has been compared with other stress measuring techniques, while characterizing SiC deposited by CVS [57 60] The focus is switched now to techniques that measure deflection from a macroscopic perspective. These are developed from elasticity and mechanics of materials theories relating the measured change of curvature in the substrate
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11 plate due to bending loads caused by residual str esses. The most common, and also most simple theoretical relation between stress and substrate curvature change is Stoneys equation [32] Its derivation will be explained in the following section as well as how its idealistic assumptions deviate from actual film substrate conditions Popular stress measuring techniques that are based on this equation include optical interferometry [61 65] X ray double crystal diffraction topography [66 69] optical profiling [69] and mechanical profiling [38, 70] In summary, all techniques measure bending deflection of the substrate, caused by residual stresses. This work will make use of mechanical profilometry for the stress measurements. Before comparing the abovementioned techniques of stress measurement it is useful to remember the importance of the length scale in stress measurements ; explained in the previous section. Stress measurements taken at microscopic, or even higher scales, will determine the average stress value over that microscopic area, or v olume; this is the case for XRD and Raman Spectroscopy instruments. Microscopic st resses may not be accurate indicators of the average stress across the film surface, or even across its thickness ; however, they represent more precise ly values of absolute stress [71] On the other hand, macroscopi c deflectionbased techniques often make assumptions that imply an average stress value for the entire system, or at least for areas involving the entire thickness of the film; such measuring techniques do not bring microscopic stress variations into sight Stress results from the curvature
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12 (deflection) methods may present inaccuracies inside of an expected error range; but still are practical for average estimation of stress. Instead of being an inconvenience, the aforementioned difference between microsco pic strain based techniques, such as XRD and Raman spectroscopy, and macroscopic deflection based techniques can instead be of complementary advantage. In other words, average stresses that are evaluated macroscopically can be compared with stress values that correspond to microscopically scaled regions. Conversely different precision requirements can eliminate the use of a certain technique, or a certain instrument. Macroscopic deflection based methods have no restriction on the type of material sub ject of measurement, a s opposed to XRD, by which amorphous materials cannot be analyzed. Moreover, such techniques have the capability of measuring stress variations along lateral dimensions. Instrumentation for curvature measurement is practical for an industr ial environment, and also inexpensive, compared to XRD diffractometer 1.4 Derivation of Stoneys equation In order to set up the ground for Stoneys equation derivation; simplifications about t he states of stress and strain of the t hin film substrate sys tem are needed First thicknesses of such systems are small enough to be considered as thin plates For the applications involved in thi s work, thin films correspond to an approximate thickness no larger than 200 m; and su bstrates
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13 to slightly over 500 m thickness and 50 mm diameter. Notice that film is significantly thinner, and it entirely covers the surface of a substrate. It is also assumed that both film and substrate materials are isotropic and homogeneous. The film substrate system is mainly under bending loads due to residual stresses present in the film. E ffects from other loads types, including shear, can be neglected. Moreover, for the case of isotropic mechanical properties the substrate should bend into a spherical shape, showing a uniform curvature across its surface. Lastly, su bstrate deflections caused by bending moment will be considerably smaller than any dimension of the substrate, even its thickness Despite the fact that real internal loading of a film substrate system is slightly more complex, these assumptions are the foundation of Stoneys equation. A substrate is represented by a thin plate of uniform thickness. A plate is considered thin when its thickness is considerably smaller than its lateral dimensions. A Cartesian coordinate system shall be described such that x and y axes are horizontal, and parallel to the plates surface; z axis is oriented along the thickness dimension. The origin is placed on the volumetric center of the plate. The xy plane located at z=0 is called the midsurface of the plate, symmetrically dividing the plates crosssection in two parts Given the small thickness of the plate, bending loads will cause negl igible normal and shear strains parallel to the z axis. In other words, deformations f rom bending will not include any thicknes s change; and the plates cross section will remain perpendicular to the midsurface. These special conditions of deformation correspond to plane strain conditions, which define strain components as
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14 x ux y vy 0 z wz x v y uxy 0 x w z uxz 0 y w z yyz ( 1 ) where x, y, z are the components of normal strain oriented along x y and z coordinate axis respectively; xy, xz, yz are components of shear strain parallel to their corresponding coordinate plane; and u v and w are displacement s of material along the x y and z axes respectively. Figure 2 shows an originally flat thin plate under bending with the described coordinate system and plane strain conditions. The figure shows an exaggerated deflection w Figure 2 Diagram of thin plate xz transverse section undergoing bending load ( M ) Plate is bent concave upwards. Knowing that z equals zero implies that w is a function independent of z; thus it defines how the location of the midsurface varies from its original position, with respect to x and y coordinates. Pure bending conditions stipulate that the midsurface ( any point at z=0) does not undergo strain. For the case of a thin plate deformed concave upwards as shown above, material located on positive z axis will be under compression; while the opposite holds for negative z axis locations. Strain definitions described can be combined in order to derive that
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15 z x wx 2 2 z y wy 2 2 z y x wxy 22 ( 2 ) where the negative signs explain the negative (compressive) strain at points on the positive z axis and vice versa. Basic calculus helps explain that the curvature and radius of curvature r of a line w(x) is 2 / 3 2 2 21 1 x w x w rx x ( 3 ) where w(x) is the line describing midsurface deflection, only as a function of x coordinate. F or a specific location (x,y), certain radii of curvature rx, ry and rxy exist; and they are parallel to xz, yz and xy planes respectively. It was already noted that the applications of this work only involve small deflection values, such that the term 2) / ( x w is sufficiently small, and thus the equation above is simplified to 2 21 x w rx x and similarly, 2 21 y w ry y dy x w rxy xy 21 ( 4 )
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16 Strains x, y and xy within the thin plate can be expressed as functions of the corresponding location coordinates and cur vature k; this is done by relating equations ( 2 ) with ( 4 ) obtaining zx x zy y zxy xy 2 ( 5 ) It will be shown that this relation is very useful for the physical determination of intrinsic stress. By using Hookes law and equation ( 5 ) stress of a plate element will be related with its curvature. Based on the fact that substrate thickness and bending deflections are significantly small stress component z is neglected. Hence the stress state within any location of the plate will be described by y x y x xz E y w x w E E 2 2 2 2 2 2 21 1 1 x y x y xz E x w y w E E 2 2 2 2 2 2 21 1 1 xy xy xyz E y x w z E E 1 2 1 2 1 22 ( 6 ) where x and y are the functions of normal stress along x and y axis respectively; xy is the shear stress, parallel to the xy plane. E and are the Elastic Youngs modulus, and Poissons ratio, respectively. Like in any loading case of pure bending, each component of stress is distributed linearly, along the orientation perpendicular to the stress action. The
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17 neutral surface is that on which normal stress equals zero, as seen in equation ( 6 ) ; and because of symmetry properties, such surface coincides with the midsurface (i.e. any point at which z= 0). This is connected with the fact noted in equation ( 5 ) which results in zero strain at the mentioned surface z=0 As the location z varies towards the top and bottom plate surfaces, each stress component increases linearly, also depends on the c orresponding curvature =1/r, and on a constant E/(1+) which is called the biaxial elastic modulus of an isotropic material. Figure 3 shows a twodimensional diag ram of the stress profile alo ng a cross section of the plate parallel to the xz plane. Note that profiles are identical for a section in the yz plane. Figure 3 Transverse section in xz plane of a thin plate with a normal stress x profile caused by bending moment Mx. Figure 3 shows that stress distribution is symmetrical; hence it corresponds to zero net force, and to a certain bending moment component Mx.
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18 Accordingly, each stress distribution x, y and xy is re lated to bending moment components Mx, My, or Mxy respectively. These stress moment relations are 2 / 2 / t t x xdz z M 2 / 2 / t t y ydz z M 2 / 2 / t t xy xydz z M ( 7 ) where t is the plate thick n ess, and the integral is evaluated over the entire transverse section, along the z axis, and with integration boundaries from z= t/2 and z=t/2 If equations ( 6 ) are combined with ( 7 ) local moment components are related to the midsurface curvatur es, resulting in y x xD y w x w D M 2 2 2 2 x y yD x w y w D M 2 2 2 2 xy xyD y x w D M 1 12 ( 8 ) where ) 1 ( 122 3 t E D Due to the fact that the film is much thinner than the substrate, the film stress is interpreted as a point load on the top edge of the substrate. A force and a moment are reaction loads located at the center of the substrate section; and have the purpose of equilibrating the film stress. Figure 4 shows the loaded film substrate system with the corresponding internal stresses. Below, e quilibrium equations for the loads a nd moments are
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19 0 : s sf ft t F ( 9 ), 0 2 : x s f fM t t M ( 10), where the subscripts s and f indicate substrate and film, respectively; t is the thickness, as shown on Figure 4 Figure 4 State of stress and loads of a thin film substrate system. The moment Mx is described in equation ( 8 ) with the subscript x noting that it corresponds to the stress component x; however, this moment acts about the yaxis. Equations ( 8 ) and ( 10) are combined to describe film stresses i n terms of substrate curvatures:
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20 y x f s f xt t E 2 2 ,) 1 ( 6 x y f s f yt t E 2 2 ,) 1 ( 6 xy f s f xyt t E 2 2 ,) 1 ( 6 ( 11) where x,f, y,f and xy,f are the corresponding intrinsic film stresses in the respective directions.; in the other hand, material properties E and v correspond to those of substrate. Finally thicknesses of both film and substrate are assumed to be uniform in any direction of the whole system. Misfit strain existing in film is similarly uniform. For this reason, shear stress would not be present at the film substrate interface. Additionally, curvatures and stresses along any orthogonal set of directions are equal; or in other words, curvature and stresses are equibiaxial. These important simplifications have allowed Stoney in [32] to generate this famous relation, which summarizes the stress of thin film as a single value: f s ft t E 2 2) 1 ( 6 ( 12) where f = x,f = y,f, and = x = y. Next chapter will explain how is calculated, by first indicating that for initially deflected substrates, residual film stress f from equation ( 12) is actually dependent of the change of substrate curvature, which occurs after deposition process, and such change is noted as
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21 1.5 Limitations and modificati ons of Stoneys equation Stoneys equation is based on simplifying assumptions about the properties, and conditions of film substrate system; allowing for this relation to be extensively used for the estimation of residual film stress. There is no need for prior knowledge of substrate properties; and simple measurement techniques are required. These assumptions were previously described, and are now summarized: 1. Substrate and film are represented by a thin plate and a membrane, respectively; or, tf << ts. 2. Substrate bending deflections are small compared to any dimension. 3. Film and substrate material properties are homogeneous, isotropic or inplane symmetric and linear elastic. 4. Film stress is in plane isotropic, or equi biaxial. Shear stresses and out of pl ane stress components are negligible. 5. Substrate curvature change is uniform and equibiaxial. Twist curvature component is negligible. 6. All stress and curvature components remain constant across entire wafer A similar summary list can be read elsewhere [72] T hese assumptions imply s everal limitations to the applicability of Stoneys equation; and for this reason, actual film substrate system condition s often deviate from such idealizations However, Stoneys equation has d rawn enough interest, even for
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22 its misuse, on film substrate conditions that violate the implied limitations ; hence, errors in results are inherited. Often, the purpose of studies allows for less accura cy thus Stoneys equation results are then accompanied with a logical margin of error, and appropriate validation. A ssumption #1 may yield inaccuracies on cases when thickness of film is not constant over the entire surface. CVD thin films might become an example when gas reactants incidence is not uniform over the substrate surface ; in addition, patterned films are also clear examples of film thickness nonuniformity. Variations in film thickness generate non uniform substrate curvature, and film residual stress distributions across the entire film Solutions for these cases have been derived and recently proposed for several cases [72] (100) oriented crystals such as SiC and Si, are in plane isotropic ; for this reason, are not well described by the biaxial modulus E/(1 ) of equation ( 12 ) ; as a result, assumption #3 shall be relaxed by applying a modified version of this equation; which for a (100) oriented crystal has already been derived; namely, f s ft s s k t ) ( 612 11 2 ( 13) where 1/( s11 +s12) is the (100) crystal biaxial modulus; s11 and s12 are two of the three independent components of the material compliance matrix [33] Previous validation works have confirmed on the accuracy of such modifications of Stoneys equation; however, uniformity of stress and curvature remains valid for each in plane orientation of film and substrate [73 76]
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23 In the case of very high residual stresses, or when substrate is sufficiently flexible, large substrate deflections can become nonlinear. In this case, as sumption #2 is no longer valid; substrate deformation becomes nonaxis ymmetric, violating assumption #4. Solutions for these cases have been described several poss ible equilibrium states of a film substrate system [77 80] Past works have also concluded on a film stress threshold below which Stoneys equation remains valid. Above this cr itical stress level, nonlinear deformation s occur obtaining shapes other than spherical, such as cylindrical, or ellipsoidal, which achi e ve a lower energy state. These shapes would then be described by biaxial, or even non axisymmetric curvature, and stress components [81, 82] Another example of Stoneys equation extensions is proposed for nonaxisymmetric substrate deflection when radius of curvature is measured along two orthogonal orientations, R1 and R2, respectively [83] In this work, modified Stoneys equation would look like 1 1 1 6 12 1 1 2R R R t t Ef s s s f ( 14). Finally, film stress, and substrate curvature can be non uniform for many reasons; resulting in the nonvalidity of assumption #6. Reasons for this include nonuniform misfit (thermal or epitaxial) between material properties of films and substrate, structural defects, and nonuniform stress relaxation. This has been the most complicated case for numerical analysis to solve.
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24 Typically, such realistic issue s have been set aside, by simply measuring substrate curvature, either as average values through simple optics; or via pro filometry, as functions of horizontal position x, k(x) and tf(x) respectively. These curvature and thickness profiles (or functions of x) could be inserted into equation ( 12) for obtaining function (x); yielding a residual stress profile, which is assumed to be axis ymmetric, and vary across the substrate surface. The fact of substrate curvature change being nonu niform along a certain direction is enough evidence that nonuniform shear stress components existed in the film substrate interface; and consequently, equilibrium equation ( 10 ) would not be valid. Nevertheless, this localized approach has been consciously taken by validating results, e.g. by using finite element modeling; and propose an acceptable margin of error [38] Extensive work has been done in this matter by some of the authors already referenced in this section. Nonuniform stress and curvature components are derived from several driving conditions, such as nonuniform temperature distributions, arbit rary film thickness, and nonuniform misfit strain. Conclusions have been consistent, indicating that local residual stress values depend on local and even nonlocal curvature information, about the entire substrate. For this reason, the authors have suggested the need for measuring curvature components over the substrate, in order to obtain a full field profile of residual stress [72, 84 88] S everal techniques have been proposed to determine curvature across entire surface; however they require more expensive tools than mechanical
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25 profilometry [71] This work would serve as an aid to develop an appropriate procedure of full field deflection measurement via mechanical profilometry.
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26 Chapter 2. Analysis of Substrate Curvature 2.1 Substrate deflection measurement s Equation ( 4 ) requires measure of substrate deflection w i n order to use ; and determine substrate curvature induced by residual film stress Previous section explained that under thin film assumptions, w describes how the substrate d eflects in the z direction B oth w and are functions of the horizontal position x of a round substrate. E quation ( 4 ) was based on the assumption that deflection w is suff iciently small, such that the term 2/ 3 21 x w could be neglected. O therwise, the more complicated equation ( 3 ) would be required Blank substrates are almost flat when manufactured; nevertheless, small initial substrate curvature should be subtracted from the after deposition curvature. Accordingly, equation ( 4 ) would then look like 1 2 2 2 2 2w x w x ( 15) where w1 and w2 are the measured substrate deflections before and after deposition process respectively Curvature is now more appropriately called curvature change
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27 T he t hin plate approximation allows for the surface height of a substrate to be a direct measure of the midsurface deflection w from the original plane at z= 0 Surface profiling was performed with the a Tencor P 20 Profilometer, which uses a mechanical stylus that measures with a vertical (height) resolution of 10 Profiles were measured along lines collinear with the center point of circular subs trates. Figure 5 shows scan lines of different orientations in which profiles could be scanned. A coordinate system was defined with its origin located at the start p oint of every scan; in other words, w(0)= 0. In the case of films deposited by CVD, the opposite edge with respect to the silicon wafer flat is the x=0 point, with positive x axis oriented towards the flat edge. Reactants first arrive at this point; then follow a path along the 0 line, towards the opposite edge. All future plots that describe profiles of substrate deflections, and curvature, will utilize this coordinate system. Furthermore, scans will be centered about the wafer center, meaning that the middle point in the plotted x axis will correspond to the wafer center. Figure 5 Diagram of a Si (100) substrate with different scan orientation angl es
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28 3C SiC films were deposited by CVD on 2 inch diameter Si(100) substrates Deflection measurements were perf ormed at selected orientations in order to account for non uniformities that, as opposed Stoneys assumptions may arise from material anisotropy ; and non uniform film thickness, material composition and misfit strain. Furthermore, prior knowledge of the specific CVD reactor indicated that film thickness would follow variation in the form of linear, and parabolic profiles, along the 0, and 90 directions respectively [89] N onuniform film thickness could significantly limit the validity of Stoneys assumptions regarding uniformity and axis s ymmetry For this reason, thickness measurement s of deposit ed 3C SiC films were performed by FTIR spectrometry, along the 0 and 90 orientations [38] Example of film thickness data is shown and curvefitted in Figure 6 a) b) Figure 6 Thickness of CVD deposited 3C SiC films along two orthogonal orientations. Measurements are a) parallel and b) perpendicular with respect to fl ow of gases Following equation ( 15 ) before, and after deposition m easurement s of deflection were performed, and plotted as shown in Figure 7 Next, data from 2 2.2 2.4 2.6 2.8 3 3.2 3.4 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04Film thickness ( m)x (meters) Y = 2.0433e6 + 3.1403e5 *x Orientation: 0oSample: 035 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04x (meters) Y = 2.3688e6 + 4.0663e5 *x 0.001172 *x^2 Film thickness ( m)Orientation: 90oSample: 035
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29 previous experiments was added to the analysis of curvature for comparison purposes, as thickness nonuniformities in CVD grown SiC films present a challenge for analysis. Previous study [70] involved magnetronsputtered tungsten thin films on 100 mm diameter Si substrates. Besides W film thickness not being constant, it had axis symmetric profile variations. a) b) Figure 7 Substrate deflection data for 3C SiC films along two orthogonal orientations. Measurements are in a) parallel and b) perpendicular orientations with respect to gases flow. 2. 2 Data analysis by polynomial curve fitting Polynomial regression was used to fit deflection data of each blank substr ate; before an d after deposition. Thus functions describing substrate deflections along each measured orientation were obtained. Each function was then differentiated twice with respect to x for determining the terms on the right 3.5 1043 1042.5 1042 1041.5 1041 1045 1050 1000 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 Blank Substrate With thin film x (meters) 3CSiC flim, Si(100) substrate, wafer 035 Orientation: 0oSubstrate Deflection (meters) 3 1042.5 1042 1041.5 1041 1045 1050 1000 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 Blank Substrate With thin film x (meters) 3CSiC flim, Si(100) substrate, wafer 035 Orientation: 90oSubstrate Deflection (meters)
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30 side of equation ( 15 ) Finally, induced curvature change for e ach direction was obtained. C urvature change functions evaluated over the same domain of original data are plotted in Figure 8 This figure is also a clear indication that b lank Si substrates were indeed close to being flat prior to film deposition. The polynomials used for curve fitting were first selected to be of 3rd order. The same procedure was then repeated for implementing each polynomial degree between 3 and 10; hence, 8 different functions were obtained to represent the substrate curvature of each sampleorientation combination. Curvature change results for 3C S iC films on Si substrates are shown in Ap pendix A Moreover, variation among different fits of the same data set was quantified for analyzing consistency between fits, and for comparing with results from a another analysis method, which is based on regular ization of data (presented in the next section). a) b) Figure 8 3rd order polynomial fitting of substrate deflection for a) 0 and b) 90 orientations, and c d) their second derivatives, respectively. 3.5 1043 1042.5 1042 1041.5 1041 1045 1050 1000 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 Blank Substrate With thin film x (meters)Substrate Deflection (meters)Y = 7.0368e8 3.8823e5 *x .0182 *x^2 .0146 *x^3 Y = 1.8412e7 1.325e5 *x 0.3133 *x^2 + 2.4372 *x^33CSiC film, Si(100) substrate, wafer 035 Orientation: 0o 3 1042.5 1042 1041.5 1041 1045 1050 1000 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 Blank Substrate With thin film x (meters)Substrate Deflection (meters)3CSiC film, Si(100) substrate, wafer 035 Orientation: 90oY = 6.0465*108 2.3477*105 *x 0.0157 *x^2 0.0499 *x^3 Y = 4.0091*107 2.5596*105 *x 0.1980 *x^2 + 0.089 *x^3
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31 c) d) Figure 8. (Continued ) A verage with standard deviation of substrate cu rvature was extracted at every interval of the x axis, from all profiles of different degree polynomial s. Plots shown below would indicate how average curvature change results vary with respect to x, as well as how variation from this average is dependent of location along the substrate diameter Regardless of the not random nature of this variation, standard deviation was found useful for indicating inconsistencies between different fit s. For the case of 3C SiC films all Si substrates concaved downwards upon film deposition ; corresponding to negative values of curvature change, and compressive st ress (see equation ( 12 ) ). Figure 9 shows an example of average curvature change in a 3C SiC films on Si wafer along 0 and 90 orientations Along CVD gases flow (0 ) direction, amount of substrate curvature change varied in a decreasing manner ; starting between 0.3 m1 and 0.5 m1 at the x=0 edge, and ending between 0 m1 and 0.2 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 Blank Substrate With thin film Curvature change x (meters)Substrate Deflection (meters)Y = 0.0363 0.0878 *x Y = 0.6266 + 14.6235 *x Y = 0.5903 + 14.7113 *x 3CSiC film, Si(100) substrate, wafer 035 Orientation: 0o 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 Blank Substrate With thin film Curvature change x (meters)Substrate Deflection (meters)Y = 0.0314 0.2994 *x Y = 0.3959 + 0.5341 *x Y = 0.3645 + 0.8335 *x 3CSiC film, Si(100) substrate, wafer 035 Orientation: 90o
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32 m1 at the x 0.04 m Oppositely, along the 90 direction, there was a more constant changeof curvature trend, varying no more than 0.5 m1 over most of substrate scan lengths. a) b) Figure 9 Average polynomial describing curvature change of 3C SiC on Si (100) systems. Average profiles are derived from all polynomial fits along a) 0 and b) 90 orientations Plots from Appendix A and Figure 9 also show the corresponding variation f rom the average substrate curvature profile, represented by the shaded region. T here was significant variation of curvature change values along the 5 mm edge of the substrate. Without additional information, correct curvature values at the substrate edges are unknown. Analysis from the W films on Si substrates was also performed for comparison purposes; resultant curvature change plots are shown in Appendix B For this sample, there was data on 12 wafer orientations available ( all angles of measurement are shown back in Figure 5 ). Substrate curvature change resulted negative across the entire length of all scans (compressive residual stress) A maximum of 0.03 m1 curv ature change (approxi mately) in the substrate center; 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04x (meters) 3CSiC film, Si(100) substrate, wafer 035 Orientation: 0oAvg. substrate curvature change (meters1) 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04x (meters) 3CSiC film, Si(100) substrate, wafer 035 Orientation: 90oAvg. substrate curvature change (meters1)
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33 and minimum less than 0.01 m1, at the edge regions, were determined. F urthermore, consistency along 12 wafer orientations from 0 to 165 in steps of 15, indicated axis symmetry of wafer curvature change Significant variations are found at the edges of substrates and also, in the middle. Figure 10 shows 2, of the 12 orientations considered for curvature analysis The sole behavior of polynomials used for curve fitting can induce significant error upon calculation of second derivative. In general, fitting functions are constrained to describe all data points. However, polynomial behavior outside of the data domain is unique to their corresponding degrees, and the coefficient of its leading function term. When the polynomial degree is sufficiently high, degrees of freedom increase at the ends of data domain. By degrees of freedom, it is meant that the rate of change of the describing function is not entirely defined by the data, but by the sole nature of the specific polynomial. a) b) Figure 10 Average polynomial describing curvature change of W on Si (100) systems. Average profiles are derived from all polynomial fits along a) 0 and b) 90 orientations 0.035 0.03 0.025 0.02 0.015 0.01 0.005 0 0 0.01 0.02 0.03 0.04 0.05 0.06x (meters)Substrate curvature change (meters1)W film, Si(100) substrate, wafer A1 Orientation: 0o 0.03 0.025 0.02 0.015 0.01 0.005 0 0 0.01 0.02 0.03 0.04 0.05 0.06x (meters) W film, Si(100) substrate, wafer A1 Orientation: 90oSubstrate curvature change (meters1)
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34 Next when looki ng at the second derivatives these uncontrollable rates of change are represented by abrupt spikes or sudden peaks T he slope of a curvaturechange vs. x plot in these edge regions depends of the sign of the function leading term coefficient ; and also if the polynomial degree is an even, or an odd number. This explains the existing variation of cur vature change values at the edges of the substrate. Another source of er ror arises when low enough polynomial order limits the possible number of local maxima minima and inflection points that a fit can use to describe data. For this reason, a certain fit would not adapt well to a large presence of sharp curvature changes. S moothness is always forced upon curvature profile, regardless of the polynomial order ; hence, discontinuities created by surface scratches, may not be traced L ocal stress fields may have been disregarded because of such inaccuracies. A great inconvenience is that the abovementioned sources of error can all be present in a single curve fit Mo reover, an appropriate fit is impossible due to lack of knowledge of additional information about true substrate deflection. On the other hand, the locations of signi ficant variations are an indication that real substrate curvature is unrecognizable by polynomial fitting alone. 2.3 Segmentation of substrate deflection data Polynomials are not adequate for analyzing 2nd derivative of substrate deflection data; consequently, a different approach is taken for data curve fitting.
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35 It has been explained how a single polynomial fit can possibly over estimate substrate deflection, when excess degrees of freedom could lead to sharp curvature changes, when actual substrate curvature had been constant over corresponding length segment. Simultaneously, on different length segments of the same data set, the same polynomial fit might not trace abrupt changes of real substrate curvature, because of the flexibility constraint imposed by its order A different approach for data curve fitting is to generate different fits on every equally divided segment of a single data set. First, substrate deflection data was equally divided into a certain number of segments; next, each segment was fitted with a se cond order polynomial. As a result, the second derivative of each fit would then yield a constant curvature for each segment. Consecutive segments would be represented by margins that have coinciding boundaries; hence, it is assumed that real substrate cur vature at the edge of segments match with the margins. Continuity is implied as long as substrate does not have or discontinuities from cracking, film delaminating, buckling, or other forms of failure. A probable advantage of applying data segmentation is that different surface features, such as sharp curvature changes, and constant flat sections, could be analyzed independently. However, length of data segments should be kept large enough to avoid deceptive influence from data scatter, and small enough to offer flexibility to changes in substrate profile. Simulated data was generated in order to optimize the number of segments used in the abovedescribed method of substrate deflection analysis.
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36 Simulation of substrate deflection measurement was from a polyn omial fit of real substrate deflection data, and added Gaussian noise with certain signal to noise ratio (SNR), which simulates data scatter Several models were created with different polynomials, and SNR values; these are described below. Second derivati ves of simulated deflection profiles were then extracted to compare with results provided by segmentation method. a) b) Figure 11 Segmentation method applied to deflection data simulation. a) Segmented fits of deflection data b ) Segmented s ubstrate curvature change. Optimization was done by selecting the number of segments that generates least average residual between resultant curvature profile from segmentation method, and real curvature profile of each data model. Figure 11(a) shows segmented substrate deflection profile using 5 segments; and Figure 11(b) shows profiles of segmented substrate curvature, real (and continuous) substrate curvature of one data model. The squared norm of resultant residual between segmented curvature, and real curvature profiles was 2.5 1042 1041.5 1041 1045 1050 1000 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 Deflection data model Fit by segmentationSubstrate deflection (meters)x (meters) Model: 8th order poly. w/ white noise SNR = 1*106Segmentation: 5 segments 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 Curvature of model Curvature by segmentation x (meters) Model: 8th order poly. w/ white noise SNR = 1*106Segmentation: 5 segmentsSubstrate curvature change (m1)
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37 the main criteria utilized to judge how accurate results are when using a certain amount of segments. Figure 12 shows all resultant residuals generated by using each possible number of segments from 1 to 45, and each data model used. When using 10 segments, close to minimum residual is obtained, and resultant curvature segments follow the real curvature profiles well For this reason, division by 10 segments was applied to all substrate deflection data. R esults are also shown in Appendix A for 3C SiC films on Si (100) substrates, and on Appendix B for W films on Si (100) substrates. Figure 12 Optimization of segmentation method by selecting number of segments that yields lower residual norm for all substrate deflection models used 2.4 Regularization method The problem of determining the second derivative of a discrete data set which has certain degree of scatter constitutes a n illposed problem [90] 0 2 4 6 8 10 0 10 20 30 40 50 8th order SNR =1*106 8th order SNR =2*106 9th order SNR =3*106 9th order SNR =4*106Total Residual Norm (m1)# of Segments Optimization of Segmentation method
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38 Presence of small scatter amplitude is enough for magnifying error when differentiation of data is conventionally applied; especially double differentiation. FFT filtering can cause significant loss of information, whi ch would otherwise indicate local film stress fields Ad ditionally, FFT does not sufficiently eliminate noise magnification [91, 92] Studies in the field of image reconstruction and surface analysis have overcome this type of illposed problems through a process called Tikhonov Regularization [93] A regularized signal is that which has a reduced amount of scatter, variation, or other form of irregularities expected to signify error I nstead of directly applying this process to a measured signal, or data set; past works have implemented it to r egularize the process of differentiation itself and thus calculated a regularized derivative of a discrete signal, while concurrently avoiding propagation of error [94] The way that regularization of the differentiation process works is by minimizing the function R, f u I B u A u R ) ( ) ( ) ( ( 16) where A represents the size of noise, or scatter that is to be regularized from the desired solution u which is the derivative of the original signal f This first term is scaled by a preselected regularization parameter B quantifies the difference between f and the discrete integration of u determined by trapezoidal rule and evaluated over the entire domain of f at regular intervals x. A certain
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39 computation method is required to determine an optimum form of u that minimize s R R egularization parameter provides the correct balance between amount of regularization applied to the signal, and the accuracy that is sacrificed. Appropriate s election of t his value is not a straight forward procedure, and it depends on the relative amount of signal irregularity with respect to the expected true signal behavior Without an elaborate numerical approach, trial and error can be enough to find an appropriate par ameter value; however, some prior knowledge about what should the solution be, is required. The way that A and B from equation ( 16) are defined depends on the type of regularization used; and each provides a different effect on the solution. Tikhonov regularization utilizes the Euclidan, or L2 norm for defining functions A and B The effect of this type of regularization is that it forces smoothness upon u On the other hand, a slightly different method called Total Variation Regularization (TVR) is able to recognize noncontinuities in the solution [95] Total variation is the absolute amount of vertical distance that any function g(x) covers in a g vs. x plot, determined through the L1 norm of g namely, L x gdx g TV0 ) (' ( 17) Furthermore, Chartrand combined TVR with the objective of regularized differentiation into a regularization algorithm utilizing the gradient descent method for minimization purposes. This algorithm was proposed a s a tool for determining
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40 first derivative of noisy, nonsmooth one dimensional signals [96] Accordingly, R is modified as L Lf u I u u R0 2 0 ) (  2 1   ) ( ( 18) where the data domain is defined from x=0 to x=L ; and the terms A and B in equation ( 18) are respectively defined by the L1 norm of derivative solution u and L2 norm of the differential term described in equation ( 17 ) T he gradient descent method was utilized by the algorithm author, Chartrand, in order to minimize equation ( 18) In this work, s ubstrate deflection data was differentiated through total variation regularized differentiation (TVRD) explained above; however, the original algorithm (provided by the abovementioned author ) was modified for making u be the second, instead of the first derivative of f Modification involved the substitution of I to be defined as a double integral over the same domain. A s a result, curvature profile of deflection data was obtained. Integration c onstants were handled implicitly by prior translation, and rotation of data, so that w(0)=0 ; and that 00 xdx dw Appendix C of this work shows the minimization algorithm as written in Matlab syntax. A range of possible regularization parameters was selected by an evaluation process of TVDR results, using simulation of actual substrate deflection measurements. Simulation was based on several deflection data
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41 models, which were generated by polynomial fits of real substrate deflection data, and Gaussian noise was added with an appropriate signal to noise ratio (SNR) so that scatter from real measurements is well simulated Second derivatives of simulated deflection profiles were extracted for the purpose of validating TVDR results from actual deflection data. Selection of appropriate regularization parameter was done by iteration, and visual evaluation of TVDR results against real curvature profiles of deflection models. In simple words, too high values for yielded stiff and inaccurate curvature profiles; meaning that abrupt changes of slope were not recognized, and values of profile were significantly off the range of real curvature values. On the contrary, too low of curvature parameter produced exces sive scatter of results; thus a realistic profile cannot be observed Figure 13 will demonstrate, in plots, what the results from using too high, too low, and appropriate values of look like. Furthermore, Appendix D shows a list of regularization parameter values, evaluated through this qualitative criteria; and those considered appropriate. Appropriate regularization parameters resulted within 11012 and 11014 range of curvature values; therefore, these were all involved in the TVDR implementation to deflection data. Equation ( 15 ) was used again, along with TVDR result, to determine profiles of substrate curvature change. Similarly to the previous section, average and standard deviation were used to quantify consistency among differ ent regularization parameters; and also, to compare such method with the polynomial fitting method.
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42 a) b) c) Figure 13 Visual inspection of regularization parameter ( ) effects, classified as a) too low, b) too high, and c) ade quate. Figure 14 is an example of the average curvature profiles that the TVRD method generated at the 0 and 90 orientations for one of the 3C SiC on Si systems. Figure 15 shows the exact type of information; but instead, it corresponds to W films on Si (100) substrate combination. P olynomial fitting and segmentat ion method results are also shown in these figures Plots in Appendices B and C also show regularization results, facilitating this comparison through direct observation. These appendix sections will aid on the comparison of curvature analysis methods, whi ch is presented in the following section. 0.4 0.2 0 0.2 0.4 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 Real profile Regularized with alpha= 1e18 x (meters)Substrate curvature (meters1)"Too low" reg. parameter value 0.4 0.2 0 0.2 0.4 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 Real profile Regularized with alpha= 1e16 x (meters) "Too high" reg. parameter valueSubstrate curvature (meters1) 0.4 0.2 0 0.2 0.4 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 Real profile Regularized with alpha= 1e14 x (meters) Appropriate reg. parameter valueSubstrate curvature (meters1)
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43 a) b) Figure 14 Comparison of analysis methods for substrate curvature change of 3C SiC film on Si (100) along (a) 0 and (b) 90 scan orien tations TVRD generates curvature change profiles very similar to those derived pr eviously by polynomial fitting; with the same exact trends in both directions. However, amounts of variations in average substrate curvature change are smaller along every profile, indicating a higher consistency of results among the regularization parameter s used a) b) Figure 15 Comparison of analysis methods for substrate curvature change of W film on Si (100) along (a) 0 and (b) 90 scan orientations. 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 Polynomial fitting TVR Differentiation x (meters) Polynomial variation TVRD variation 3CSiC film, Si(100) substrate, wafer 035 Orientation: 0oSubstrate curvature change (meters 1) 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 Polynomial fitting TVR Differentiation x (meters) Polynomial variation TVRD variation 3CSiC film, Si(100) substrate, wafer 035 Orientation: 90oSubstrate curvature change (meters 1) 0.035 0.03 0.025 0.02 0.015 0.01 0.005 0 0 0.01 0.02 0.03 0.04 0.05 0.06 Polynomial fitting TVR Differentiation Polynomial variation TVRD variation x (meters)Substrate curvature change (meters 1)W film, Si(100) substrate, wafer A1 Orientation: 0o 0.035 0.03 0.025 0.02 0.015 0.01 0.005 0 0 0.01 0.02 0.03 0.04 0.05 0.06 Polynomial fitting TVR Differentiation Polynomial variation TVRD variation x (meters)Substrate curvature change (meters 1)W film, Si(100) substrate, wafer A1 Orientation: 90o
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44 A distinguishing trait observed on TVRD results was the ladder effect, which consists in small step changes that make any slope in the profile, and was predicted by author of this regularization algorithm [96] Figure 13 previously shown, reveals how real curvature profiles can be smoother that TVDR resultant profile; meaning that this ladder effect should be either removed, or disregarded. 2.5 Comparison of substrate curvature analysis methods The most noticeable difference between both analysis methods seemed to be in the presence of predefined behavior of polynomial functions This statement is evidenced over both edges of all curvature profi les substrate scan; and also along the middle segments of the W film on Si substrate data. TVDR results indicated that substrate curvature change remained constant over these segments O n the contrary, large profile abruptions were assumed by polynomial fi ts over these segments ; which were unclear due to significant variation that resulted among different polynomial degrees of fit In s ection 2.2, polynomial behavior was examined in terms of how it becomes more unpredictable as the ends of data domain are a pproached. Furthermore polynomials might have excessive degree of freedom to represent center portion of W on Si deflection data, as TVDR oppositely represent these segments as constant curvature sections.
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45 Results from segmentation method are very similar to those produced by polynomial fitting over entire data domain. This method did not show to be much more advantageous than polynomial fitting, mainly because curvature results are discrete, leaving uncertainty about curvature change on locations between segments. The appropriate way on how to determine the surrounding region of possible substrate curvature change is unknown. It was confirmed that if data was divided into more segments than the selected amount of 10, higher residual norm resulted when anal yzing data simulation; moreover, segmented curvature profiles would present significant scatter, which is illogical for continuous surfaces. Similar scatter was observed when utilizing a number of segments higher than 10. In TVDR implementation, there is no dependence on predefined function behavior; instead, it adjusts each single data point to a desired level of scatter reduction. Selection of a regularization parameter is what adjusts this reduction, better called regularization, so that it does not become destructive. The beauty of this comparison is that polynomial fitting is allowed for prior knowledge about the form of substrate curvature profile that can be expected. While t here is a perceptible relation between substrate curvature and deposited film thickness profiles, it is suspected that film thickness generates a residual stress gradient across the body of the film In the case of 3C SiC, a long 0 oriented substrate diameter, there is a linear increase in film thickness; while the re is a decrease in substrate curvature change. On the other hand, 90 orientated profiles do not show such correlation, although symmetrical film
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46 thickness profiles seem to go in hand with apparently symmetrical and almost constant substrate curvature cha nge profiles Consequently, t hickness profile along 0 direction appears to be significant for the resultant, nonuniform substrate curvature. A probable explanation for resultant nonuniform curvature change is that a film thickness gradient would somehow cause local residual film stress levels to vary across plane of the film. Normal stress gradient together with adhesion bonds at the film substrate interface would cause shear stresses to a rise ; hence, a varying internal bending moment would be generated on the substrate. TVDR results have the advantage of detecting abrupt changes in substrate curvature, which could be generated by either high nature of nonuniform residual film stress. Moreover regularization does not destroy information at the edges of substrate; for this reason, this analysis method shows favorable for developing a full field curvature measurement technique, which with appropriate numerical implementation, can be related to r esidual film stress.
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47 Chapter 3. Conclusions and Future Work A background about formation mechanisms and measurement of residual thin film stresses has been given with a focus on 3C SiC films on 100 Si substrates; and the present work has propose d a tool for the development of a more complete residual film stress measurement technique. Regardless of accuracy loss, t he substrate curvature method is attractive enough to use it beyond its limitations Proposed extensions of Stoneys equation require of substrate deflection measurements to determine all substrate curvature components existing along inplane directions Polynomial fitting is not an accurate indicator of substrate curvature change at the substrate edges Any degree of freedom might be enoug h to approximate substrate deflection data with negligible difference between measured and modeled results. H owever, great differences are obtained when the second derivatives of such fits are studied. Regularization, which is commonly used as an image reconstruction tool, has been proposed here for developing a more appropriate measurement procedure, via mechanical surface profilometry. It was confirmed that TVDR can well approximate real second derivatives from deflection data, which possesses misleadi ng scatter. S election of regularization parameter shall be based on
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48 accurate expectations about how much scatter does not correspond to actual substrate profile. N umerical analysis has been developed by several studies in order to expand scope of applicati ons for Stoneys equation. Conclusions from these techniques have settled on the need for measuring full field curvature change of substrate upon thin film deposition. While more complicated optical tools can be developed for such measurements current mechanical profilometry tools are inexpensive, and simple to use. B y using a different procedure, and more powerful method for analysis of deflection data, mechanical profilometry could probably be adjusted to meet the demands of Stoneys equation expansions TVDR is a potential complement to this development Non uniform, and nonaxis symmetric substrate curvatures that were observed on the samples of this work, can be attributed to film thickness nonuniformities. Nevertheless, Stoneys equation was enough to determine with adequate accuracy average residual film stress value at the point at which substrate curvature is equibiaxial. At this location, 0 and 90 oriented curvature change profiles coincide; hence orthogonal components are equal T he axis symmetric shape of a round substrate contributes for equibiaxial curvature location to be the circumferential center of the substrate. Nevertheless, thickness nonuniformity crystal structure defects and wafer flat cause equibiaxial curvature to deviate from such location. For the purpose of illustrating the equibiaxial stress within the substrate geometry, Stoneys equation was implemented locally, so that (x) results from
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49 utilizing equation ( 12 ) with functions k(x) and tf(x) ; which corresponded to the measured profiles presented above. It was clearly noted in Chapter 1 that stress nonuniformities are not accounted for by equation ( 12 ) ; hence, profile s of this nature are inaccurate, except for the single equibiaxial stress point, at which all orientations coincide. Figure 16 illustrates the equibiaxial film stress point for one of the 3C SiC film samples. While knowing that nonuniformities can be significant, this single value of equibiaxial stress has been utilized for estimation purposes [97] Because of having implemented different analysis methods, a certain stress profile was derived from each su bstrate curvature change result The rest of equibiaxial stress values for all samples considered are listed on Appendix E a) b) Figure 16 Residual film stress profiles after direct implementation of Stoneys equation. S u bstrate curvature change profiles are those obtained by a) Polynomial fitting, and b) TVDR method s Volinsky et al have estimated inaccuracies from using Stoneys equation in this manner by using a correction factor derived from a finite element model, 2.5 2 1.5 1 0.5 0 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0 orientation 90 orientationResidul film stress (GPa)x (meters) 3CSiC film on Si(100) wafer #035 Using k from polynomial fitting method 2.5 2 1.5 1 0.5 0 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0 orientation 90 orientationResidual film stress (GPa)x (meters) 3CSiC film on Si(100) wafer #035 Using k from TVDR method Equibiaxial stress Equibiaxial stress
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50 based on axis symmetric, nonuniform substrate curvature and film thickness. [38] This approach is an example for a temporary solution until proposed next steps are completed. T he next steps of this study shall f irst include validation of equibiaxial stress results presented above. Furthermore, it is necessary to develop a new measurement procedure, using mech anical profilometry instrument ( Tencor P 20 Profilometer ) with the goal of providing complete curvature change informat ion across the entire substrate. Because of nonsymmetric film thickness profiles are generated upon CVD, new procedure of measurement would require more than few scan lines, orthogonal to each other. Desired result is to obtain mo re than one component of curvature, and residual film stress could then be determined. F urther understanding of thin, bi layered plate mechanics is essential to provide appropriate derivation of curvaturestress relations that do not assume limiting condit ions, such as thickness uniformity equibiaxial, uniform curvature and stress components, and anisotropic materials. Numerical analysis shall be complemented by finite element modeling, which confirms obtained results. Moreover, numerical analysis that was referenced in section 2.5 could also be included for validation with finite element modeling
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51 References [1] Agarwal, A., and Saddow, S.E., 2004, "Advances in silicon carbide processing and applications," Artech House, Boston, p. 212. [2] Davis, R. F., Kelner, G., Shur, M., 1991, "Thin Film Deposition and Microelectronic and Optoelectronic Device Fabrication and Characterization in Monocrystalline Alpha and Beta Silicon Carbide," Proceedings of the IEEE, 79(5) pp. 677701. [3] Casady, J. B., and Johnson, R. W., 1996, "Status of Silicon Carbide (SiC) as a Wide Bandgap Semiconductor for HighTemperature Applications: A Review Solid State Electronics, 39(10) pp. 14091422. [4] Hefner, A. R., Jr., Singh, R., Jih Sheg Lai, 2001, "SiC Power Diodes Provide Breakthrough Performance for a Wide Range," IEEE Transactions on Power Electronics pp. 273280. [5] Lai, J. Huang, X., Yu, H., 2001, "High current SiC JBS diode characterization for hardand soft switching applications," Industry Appli cations Conference, 2001. Thirty Sixth IAS Annual Meeting. Conference Record of the 2001 IEEE 1 pp. 384390 [6] Elasser, A., and Chow, T. P., 2002, "Silicon Carbide Benefits and Advantages for Power Electronics Circuits and Systems," Proceedings of the IEEE, 90 (6) pp. 969986. [7] Spetz, A. L., Unus, L., Svenningstorp, H., 2001, "SiC Based Field Effect Gas Sen sors for Industrial Applications," Physica Status Solidi (a), 185 (1) pp. 15 25. [8] Fawcett, T. J., Wolan, J. T., Lloyd, A., 2006, "Thermal Detection Mechanism of SiC Based Hydrogen Resistive Gas Sensors Applied Physics Letters, 89 (18) p. 182. [9] Wing brant, H., 2003, "Using a MISiCFET Device as a Cold Start Sensor Sensors and Actuators.B, Chemical, 93(1) pp. 295. [10] Sundararajan, S., and Bhushan, B., 1998, "Micro/nanotribological Studies of Polysilicon and SiC Films for MEMS Applications," Wear, 2 17(2) pp. 251261.
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61 Appendices
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62 Appendix A. Substrate curvature results for 3C SiC films on Si (100) substrates. Figure 17 of sample 035 from 0 scan. Figure 18 of sample 035 from 90 scan. Figure 19 of sample 040 from 0 scan. Figure 20 of sample 040 from 90 scan. 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 Polynomial fitting Segmentation TVR DifferentiationSubstrate curvature change (meters1)x (meters) Orientation: 0o3CSiC on Si substrate# 035 Polynomial variation TVRD variation 0.45 0.4 0.35 0.3 0.25 0.2 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 Polynomial fitting Segmentation TVR Differentiationx (meters) Orientation: 90o3CSiC on Si substrate# 035Substrate curvature change (meters1) 0.5 0.4 0.3 0.2 0.1 0 0.1 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 Polynomial fitting Segmentation TVR DifferentiationSubstrate curvature change (meters1)x (meters) Orientation: 0o3CSiC on Si substrate# 040 0.22 0.2 0.18 0.16 0.14 0.12 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 Polynomial fitting Segmentation TVR DifferentiationSubstrate curvature change (meters1)x (meters) Orientation: 90o3CSiC on Si substrate# 040
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Appendix A. (Continued) 63 Figure 21 of sample 043 from 0 scan. Figure 22 of sample 043 from 90 scan. 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 Polynomial fitting Segmentation TVR DifferentiationSubstrate curvature change (meters1)x (meters) Orientation: 0o3CSiC on Si substrate# 043 0.36 0.34 0.32 0.3 0.28 0.26 0.24 0.22 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 Polynomial fitting Segmentation TVR DifferentiationSubstrate curvature change (meters1)x (meters) Orientation: 90o3CSiC on Si substrate# 043
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64 Appendix B. Substrate curvature results for W films on Si (100) substrates. Figure 23 of sample A1 from 0 scan. Figure 24 of sample A1 from 15 scan. Figure 25 of sample A1 from 30 scan Figure 26 of sample A1 from 45 scan 0.035 0.03 0.025 0.02 0.015 0.01 0.005 0 0 0.01 0.02 0.03 0.04 0.05 0.06 Polynomial fitting Segmentation TVR DifferentiationSubstrate curvature change (meters1)x (meters) Orientation: 0oW on Si substrate #A1 0.035 0.03 0.025 0.02 0.015 0.01 0.005 0 0 0.01 0.02 0.03 0.04 0.05 0.06 Polynomial fitting Segmentation TVR DifferentiationSubstrate curvature change (meters1)x (meters) Orientation: 15oW on Si substrate #A1 0.035 0.03 0.025 0.02 0.015 0.01 0.005 0 0 0.01 0.02 0.03 0.04 0.05 0.06 Polynomial fitting Segmentation TVR DifferentiationSubstrate curvature change (meters1)x (meters) Orientation: 30oW on Si substrate #A1 0.035 0.03 0.025 0.02 0.015 0.01 0.005 0 0 0.01 0.02 0.03 0.04 0.05 0.06 Polynomial fitting Segmentation TVR DifferentiationSubstrate curvature change (meters1)x (meters) Orientation: 45oW on Si substrate #A1
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Appendix B. (Continued) 65 Figure 27 of sample A1 from 60 scan. Figure 28 of sample A1 from 75 scan. Figure 29 of sample A1 from 90 scan. Figure 30 of sample A1 from 105 scan. 0.03 0.025 0.02 0.015 0.01 0.005 0 0 0.01 0.02 0.03 0.04 0.05 0.06 Polynomial fitting Segmentation TVR DifferentiationSubstrate curvature change (meters1)x (meters) Orientation: 60oW on Si substrate #A1 0.03 0.025 0.02 0.015 0.01 0.005 0 0 0.01 0.02 0.03 0.04 0.05 0.06 Polynomial fitting Segmentation TVR DifferentiationSubstrate curvature change (meters1)x (meters) Orientation: 75oW on Si substrate #A1 0.03 0.025 0.02 0.015 0.01 0.005 0 0 0.01 0.02 0.03 0.04 0.05 0.06 Polynomial fitting Segmentation TVR DifferentiationSubstrate curvature change (meters1)x (meters) Orientation: 90oW on Si substrate #A1 0.03 0.025 0.02 0.015 0.01 0.005 0 0 0.01 0.02 0.03 0.04 0.05 0.06 Polynomial fitting Segmentation TVR DifferentiationSubstrate curvature change (meters1)x (meters) Orientation: 105oW on Si substrate #A1
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Appendix B. (Continued) 66 Figure 31 of sample A1 from 120 scan Figure 32 of sample A1, 135 scan. Figure 33 of sample A1, 150 scan. Figure 34 of sample A1, 165 scan. 0.035 0.03 0.025 0.02 0.015 0.01 0.005 0 0 0.01 0.02 0.03 0.04 0.05 0.06 Polynomial fitting Segmentation TVR DifferentiationSubstrate curvature change (meters1)x (meters) Orientation: 120oW on Si substrate #A1 0.035 0.03 0.025 0.02 0.015 0.01 0.005 0 0 0.01 0.02 0.03 0.04 0.05 0.06 Polynomial fitting Segmentation TVR DifferentiationSubstrate curvature change (meters1)x (meters) Orientation: 135oW on Si substrate #A1 0.035 0.03 0.025 0.02 0.015 0.01 0.005 0 0 0.01 0.02 0.03 0.04 0.05 0.06 Polynomial fitting Segmentation TVR DifferentiationSubstrate curvature change (meters1)x (meters) Orientation: 150oW on Si substrate #A1 0.035 0.03 0.025 0.02 0.015 0.01 0.005 0 0 0.01 0.02 0.03 0.04 0.05 0.06 Polynomial fitting Segmentation TVR DifferentiationSubstrate curvature change (meters1)x (meters) Orientation: 165oW on Si substrate #A1
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67 Appendix C. Total Variation Regularized Differentiation code using Matlab Code is modified from original for the purpose of determining regularized second derivative of a data vector [95]. Matlab code is presented below. % tvdiffscp.m: totalvariation regularized differentiation % % Presumed input: vector e of noisy data to be differentiated. The % code assumes that e gives the values of a function at halfway between % the points of a uniform grid. % % Created output: column vector u, regularized derivative of e, given % at the grid points, so length(u)=length(e)+1. x=A1rotatedCHANGE(:,1); y=A1rotatedCHANGE(:,1); % parameters: dx=x(2)x(1); % grid spacing eps=0.000001; % constant used to avoid division by zero when u' = 0. % The code seems to tolerate very small values. Smaller values gets more accurate results, but slows convergence. Too small makes code unstable. alph=1e10; % regularization parameter. e=y; n=size(e,1); % construct operators of differentiation (D) and antidifferentiation(K) D=diag(ones(1,n+1))diag(ones(1,n),1); %ADJUSTED FROM ORIGINAL D=[zeros(n+1,1),D]; D(1,1)=1; D=D/dx; K=[zeros(n,1),zeros(n,1)+3/4,zeros(n,n)+0.25*eye(n)]; for i=1:n1 K=K+[zeros(n,2),[zeros(i,n);tril(ones(ni,ni)),zeros(ni,i)]]; K(i+1,2)=K(i+1,2)+i; K(i,1)=0.5*i; end clear i K=K*(dx)^2; % Second Integral % stopping criterion; when change in K*u is less than quit quit=1e6; k=1000; % initialize to naive derivative u=[0;0;0;diff(diff(e));0]./dx^2; %%ADJUSTED FROM ORIGINAL
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Appendix C. (Continued) 68 % since K*u(0)=0, we need to adjust ofst=e(1); %while k>quit for i=1:600 % solve lagged diffusivity equation. Alternate dg seems to give better results sometimes, for unknown reasons. dg=diag(1./sqrt(((u(2:(n+2))u(1:n+1))/dx).^2+eps)); % dg=diag(1./sqrt(((u(2:(n+1))u(1:n))).^2+eps)); L=dx*D'*dg*D; g=K'*(K*ue+ofst)+alph*L*u; H=K'*K+alph*L; s=H \ g; u=u+s; % check stopping condition k=norm((*s)); figure(9),plot(u,'ok'),drawnow; end
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69 Appendix D values The different models that were used for this selection process were described in terms of the polynomial degree used, and the SNR of the added Gaussian noise. The following table helps to visualize results that concluded on which regularization parameter values to select. Table 1 Selection of appropriate regularization parameter by visual inspection criteria. Model Model 1 (8th order / SNR= 1 10 6 ) Model 2 (8th order / SNR= 4 106) Model 3 (30th order / SNR= 1 106) Model 4 (30th order / SNR= 4 106) 1 10 10 Too low Too low Too low Too low 1 10 11 Too low Too low Too low Too low 1 10 12 Appropriate Appropriate Appropriate Appropriate 1 10 13 Appropriate Appropriate Appropriate Appropriate 1 10 14 Appropriate Appropriate Appropriate Appropriate 1 10 15 Too low Too low Too low Too low 1 10 16 Too low Too low Too low Too low
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70 Appendix E. Local implementation of Stoneys equation with substrate curvature results for 3C SiC film on Si(100) substrate samples. Table 2 Magnitude and location of equibiaxial residual stress values. Sample Equibiaxial stress using local Stoneys equation and k from polynomial fitting method Equibiaxial stress value using local Stoneys equation and k from TVDR method 3C SiC film on Si (100) wafer# 035 (0. 016550 m)= 1.091 GPa (0.016727 m)= 1.082 GPa 3C SiC film on Si (100) wafer# 040 (0. 013117 m)= 0.736 GPa (0.012947 m)= 0.719 GPa 3C SiC film on Si (100) wafer# 043 (0. 010803 m)= 1.215 GPa (0.012127 m)= 1.194 GPa
