Patent Publication Number: US-8534135-B2

Title: Local stress measurement

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application claims the benefit of Provisional Application No. 61/330,215, filed Apr. 30, 2010, which is incorporated by reference herein in its entirety. 
    
    
     BACKGROUND 
     Flat substrates, such as semiconductor substrates, are stressed during certain processing steps, e.g., depositing or etching thin films. Stress in deposited layers can warp the substrate, which can adversely affect subsequent process steps, device performance, reliability and line-width control. Thus, it is desirable to measure the radius of curvature of a substrate as well as measure the stress on a substrate that is associated with a processing step. 
     There are many measurement tools available for measurement of the radius of curvature and analysis of the stress associated with certain processing steps on substrates. Most of the available tools for the semiconductor industry use a laser displacement sensor to measure the radius of curvature and to monitor the change in radius of curvature of the substrate before and after the processing step. Generally, radius of curvature is used to describe the bow of the substrate over a larger scale, e.g., the diameter of the substrate. Typical metrology devices, however, do not measure the local topography, and thus, provide only a global Bow/Stress measurement. 
     SUMMARY 
     An optical metrology device determines the local stress in a film on a substrate in accordance with an embodiment of the present invention. The metrology device, which may be, e.g., a white light interferometer, maps the thickness of a substrate prior to processing. After processing, the metrology device determines the surface curvature of the substrate caused by the processing and maps the thickness of a film on the top surface of the substrate after processing. The surface curvature of the substrate may be parameterized over a set of basis functions. The local stress in the film is then determined using the mapped thickness of the substrate, the determined surface curvature, and the mapped thickness of the film. In one embodiment, the local stress may be determined using a version of Stoney&#39;s equation that is corrected for non-uniform substrate curvature, non-uniform film thickness, and non-uniform substrate thickness. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  shows a schematic view of a metrology device that may be used to determine localized stress in a substrate by characterizing the substrate warp and bow, substrate thickness variation, and film thickness variation. 
         FIG. 2  illustrates a white light interferometer, which may be used as the metrology device of  FIG. 1 . 
         FIGS. 3A and 3B  illustrate superposition of multiple wavelength interference patterns to produce white light interference. 
         FIG. 4A  illustrates measuring multiple locations on the substrate. 
         FIG. 4B  illustrates determining a height difference based on detected intensity signals for different pixels. 
         FIG. 5  is a perspective view of stress-free chuck with three retractable lift pins that support the substrate. 
         FIG. 6  is a side view of chuck and lift pins supporting the substrate. 
         FIG. 7  is a perspective view of a chuck that includes channels through which a vacuum may be applied to the back side of a substrate and shows the lift pins retracted. 
         FIG. 8  is a flow chart illustrating a process to determine localized stress in a thin film deposited on substrate by characterizing the substrate warp and bow, substrate thickness variation, and film thickness variation. 
         FIG. 9  is a flow chart illustrating the process of mapping the substrate thickness. 
         FIG. 10  is a flow chart illustrating determining the local differential curvature, for which the change in the substrate warp and bow due to the processing of the substrate. 
         FIG. 11  is a flow chart illustrating the process of mapping the film thickness. 
         FIG. 12  is an illustration of the process of determining localized stress in a substrate by characterizing the substrate warp and bow, substrate thickness variation, and film thickness variation. 
     
    
    
     DETAILED DESCRIPTION 
       FIG. 1  shows a schematic view of a metrology device  100  that may be used to determine localized stress in a patterned or unpatterned substrate  110  by characterizing the substrate warp and bow, substrate thickness variation, and film thickness variation. The metrology device  100  includes chuck  120  mounted on a stage  122 . The stage  122  is capable of horizontal motion in either Cartesian (i.e., X and Y) coordinates, as indicated by arrows  123  and  124 , or Polar (i.e., R and θ) coordinates or some combination of the two. The stage may also be capable of vertical motion. 
     Metrology device  100  includes an optical head  102  that is coupled to a computer  26 , such as a workstation, a personal computer, central processing unit or other adequate computer system, or multiple systems. If desired, multiple optical heads, i.e., different metrology devices, may be combined in the same metrology device  100 . The computer  26  may control the movement of the stage  122  and optical head  102 , as well as control the operation of the chuck  120 . In one embodiment, the chuck  120  may be held stationary while the optics move relative to the substrate  110  or both may move relative to the other. For example, the optical head  102  or a portion of the optical head  102 , e.g., an objective lens, may be movable in the vertical direction, as indicated by arrow  122   b.    
     In one embodiment, the optical head  102  may be white light interferometer  102  (shown in  FIG. 2 ), which produces two measurement beams  103 . Interferometer  102  includes a broadband light source  130  and a beam splitter  132 . Light from the beam splitter  132  is reflected towards an interference objective  134 , which includes a reference minor  136 . The interference objective  134  is coupled to an actuator  138 , which is controlled by computer  26 , to adjust the vertical position of the interference objective  134 . The interference objective produces a beam  103  that is incident on and reflects from the substrate  110 , passes back through the interference objective  134  and beam splitter  132  and focused by imaging lens  140  onto detector  142 , which is coupled to the computer  26 . 
     In operation, the white light interferometer  102  scans the interference objective  134 , as indicated by the arrow  135  collecting interference patterns in the image plane. White light interference is the superposition of multiple wavelength interference patterns, as illustrated in  FIGS. 3A and 3B .  FIG. 3B  illustrates the measured intensity of the light for a single pixel in detector  142 , where the vertical axis represents intensity and the horizontal axis represents the Z position (i.e., height) from the surface of the substrate  110 . When the peaks for the wavelengths are equal and all patterns have a common phase, the surface is detected (L=0). By measuring multiple locations in the illumination spot as illustrated by beamlets  103   a  and  103   b  in  FIG. 4A , i.e., by detecting intensity signals for different pixels in detector  142 , the height difference at the different locations can be determined, as illustrated in  FIG. 4B . By scanning the interference objective  134  parallel to the surface of the substrate  110 , the topography of the surface of the substrate  110  can be mapped as a three-dimensional image. White light interferometer  102  and its general operation are described in more detail in U.S. Pat. No. 5,398,113, which is incorporated herein by reference in its entirety. 
     While a white light interferometer  102  is described herein as providing the film thickness, substrate thickness, and surface curvature to determine localized stress, it should be understood that other types of metrology devices that alone or in combination that can characterize the substrate bow, substrate thickness variation, and film thickness variation may be used to determine localized stress as described herein. For example, metrology devices, such as confocal microscopes, reflectometers, ellipsometers, or other interferometers, including shear interferometers, may be used alone or in some combination within metrology device  100 . 
     To determine the localized stress in a thin film deposited on substrate  110 , the metrology device  100  characterizes the substrate warp and bow. In order to characterize substrate warp and bow, the substrate  110  is permitted to deform under gravity and internal stress using a stress-free chuck  120 .  FIG. 5  is a perspective view of stress-free chuck  120  with three retractable lift pins  126  that support the substrate  110  (illustrated with broken lines). Lift pins  126  lift a substrate off the top surface of the chuck  120  and support the substrate at a minimum of contact points to permit an accurate measurement of curvature of the substrate.  FIG. 6  is a side view of chuck  120  and lift pins  126  supporting the substrate  110 , which is illustrated with a film  112 . The bowing of substrate  110  is shown greatly exaggerated in  FIG. 6  for illustrative purposes. As illustrated in  FIG. 7 , the chuck  120  includes a plurality of channels  128  through which a vacuum may be applied to the back side of a substrate resting on the surface of the chuck. The vacuum ensures that the substrate lies flat during a thickness measurement. If desired, chuck  120  may use other means for holding the substrate  110  flat during thickness measurements, such as an electrostatic force, which is well known in the art.  FIG. 7  illustrates the lift pins  126  retracted. If desired, chuck  120  may include a slot  129  that allows access for a paddle (not shown) to place and retrieve a substrate on the top surface of the chuck  120 . Alternatively, the lift pins  126  may be used for loading and unloading the substrate. 
     Referring back to  FIG. 1 , the computer  26  controls the stage  122  and optical head  102 . The computer  26  also collects and analyzes the data from the optical head  102  to determine the substrate warp and bow, substrate thickness variation, and film thickness variation which is used to determine localized stress in a deposited thin film. A computer  26  is preferably included in, or is connected to or otherwise associated with optical head  102  for processing data detected by the optical head  102 . The computer  26 , which includes a processor  27  with memory  28 , as well as a user interface including e.g., a display  29  and input devices  30 . A non-transitory computer-usable storage medium  42  having computer-readable program code embodied may be used by the computer  26  for causing the processor to control the metrology device  100  and to perform the functions including the analysis described herein. The data structures and software code for automatically implementing one or more acts described in this detailed description can be implemented by one of ordinary skill in the art in light of the present disclosure and stored, e.g., on a computer readable storage medium  42 , which may be any device or medium that can store code and/or data for use by a computer system such as processor  27 . The computer-usable storage medium  42  may be, but is not limited to, magnetic and optical storage devices such as disk drives, magnetic tape, compact discs, and DVDs (digital versatile discs or digital video discs). A communication port  44  may also be used to receive instructions that are used to program the computer  26  to perform any one or more of the functions described herein and may represent any type of communication connection, such as to the internet or any other computer network. Additionally, the functions described herein may be embodied in whole or in part within the circuitry of an application specific integrated circuit (ASIC) or a programmable logic device (PLD), and the functions may be embodied in a computer understandable descriptor language which may be used to create an ASIC or PLD that operates as herein described. 
       FIG. 8  is a flow chart illustrating a process that may be performed by metrology device  100  to determine localized stress in a thin film deposited on substrate  110  by characterizing the substrate warp and bow, substrate thickness variation, and film thickness variation. As illustrated, the substrate thickness is mapped ( 210 ), e.g., using the white light interferometer  102 . It should be understood that the substrate may include multiple patterned and/or unpatterned films overlying a substrate. The local differential curvature is determined using local substrate warp and bow measurement both before and after processing ( 230 ). After processing, the film thickness is mapped ( 250 ), e.g., again using the white light interferometer  102 . The local stress can then be determined using the mapped substrate thickness, mapped film thickness, and the determined local differential curvature. 
       FIG. 9  illustrates the process of generating a thickness map of the substrate thickness ( 210 ). The surface of the chuck  120  is mapped by the metrology device  100  ( 212 ). The substrate  110  is loaded on to the chuck  120  and clamped against the chuck  120 , e.g., using vacuum or electrostic force ( 214 ). The surface of the substrate  110  is then mapped by the metrology device  100  ( 216 ). The thickness of the substrate which includes the substrate and any deposited films can then be determined based on the difference between the surface map of the substrate and the surface map of the chuck ( 218 ). The resulting map of substrate thickness variation, which consists of the total thickness variation of the substrate and films and any local topography changes, is stored in memory  28  and fed forward into the stress calculation. 
       FIG. 10  illustrates determining the local differential surface curvature ( 230 ), for which the change in the substrate warp and bow due to the processing of the substrate (e.g., deposition of a thin film or CMP (chemical mechanical polishing) process) is measured in a two-step (pre- and post-process) measurement. First, prior to deposition, the substrate  110  is loaded on lift pins  126  on the chuck  120  so that the substrate is allowed to deform under gravity and internal stress ( 232 ) and the substrate is mapped while on the lift pins  126  ( 234 ). By way of example, the substrate  110  may be loaded on lift pins  126  and mapped prior to or after mapping the thickness of the substrate ( 210 ) described above. The scanning white-light interferometer  102  may be used, for example, to map the height of the free standing substrate surface. After mapping the substrate  110  surface, the substrate  110  is loaded, processed, and reloaded on the lift pins  126  of the chuck  120  ( 236 ). The substrate  110  is again mapped ( 238 ), e.g., using the scanning white-light interferometer  102  to map the height of the free standing substrate surface. The same or different measurements sites measured in the pre-processing mapping may be measured. The difference between the two mapped surfaces is calculated to determine a surface difference ( 240 ). The change in the surface curvature is due to the change in the stress. The surface difference is then fit to a set of basis functions ( 242 ). By way of example, orthogonal basis functions, which may be polynomials or Fourier components, may be used. In one embodiment, Zernike polynomials of an order consistent with the length scale of the measurements may be used. Zernike polynomials are a set of orthogonal basis functions in cylindrical coordinates, which are well suited to characterizing disc shaped objects. The use of Zernike polynomials to describe the surface profile of substrate  110  is described in “Describing isotropic and anisotropic out-of-plane deformations in thin cubic materials by use of Zernike polynomials”, by Chang, Akilian, and Schattenburg, Appl Optics, 45, No. 3, (2006), pp. 432-37, which is incorporated herein by reference. The surface curvature is calculated by determining the second derivative of the basis functions analytically ( 244 ). The surface curvature is stored in memory  28  and fed into the stress calculation. 
       FIG. 11  illustrates the process generating a thickness map of the film ( 250 ). It should be understood that the film that is measured is the top film after further processing of the substrate  110 , i.e., the film may be a deposited film or the remaining film after a CMP process. The substrate  110  is clamped against the chuck  120 , e.g., using vacuum or electrostic force ( 252 ). This may be performed immediately before or after post-processing mapping of the free standing substrate surface ( 238 ) described above. The surface of the substrate  110  is then mapped by the metrology device  100  ( 254 ). The thickness of the film can then be determined ( 256 ). By way of example, for thick films, the film thickness may be measured directly using interferometry. Otherwise, the film thickness may be measured using Advanced Film Capability (AFC) analysis of Pupil Plane SWLI (PUPS) measurements as described by Peter J. de Groot and Xavier Colonna de Lega in “Transparent film profiling and analysis by interference microscopy” Interferometry XIV: Applications, Proc. Of SPIE, Vol. 706401, pp 1-6 (2008), and U.S. Pat. Nos. 6,545,763 and 7,061,623, both of which are incorporated herein by reference. The resulting map of film thickness variation is stored in memory  28  and fed into the stress calculation. 
       FIG. 12  is an illustration of the process of determining localized stress in a substrate  110  by characterizing the substrate warp and bow, substrate thickness variation, and film thickness variation. The substrate prior to additional processing  110   a  undergoes pre-measurement  202 , which includes mapping the substrate thickness  210 , as well as mapping the substrate surface  234 . The substrate  110   a  is removed from the metrology device for pre-measurement  202  and undergoes processing  204 , which may be thin film deposition  204  or any other desired processing, such as a CMP process, which may include depositing more than one layer, a lithography process, and etching process followed by polishing the top layer back via CMP. The substrate after processing  110   b  then undergoes post-measurement  206 , which includes mapping the film thickness  250 , as well as again mapping the substrate surface  238 . The results of the substrate surface mapping from the pre-measurement and post-measurement are combined to determine the surface difference  240 . The surface difference  240  is fit to orthogonal basis functions, such as Zernike polynomials  242 , and the second derivative of the polynomials is taken to determine the surface curvature  244 . The surface curvature  244 , along with the mapped substrate thickness  234  and the mapped film thickness  250  are fed into the stress calculation  260 , from which a local stress map of the substrate  262  is generated and which is stored in memory  28  and may be displayed or otherwise provided to the user. 
     In one embodiment, a single optical tool, such as white light interferometer  102  is used to perform the pre-measurement  202  and post-measurement  206 , which increases throughput as well as reduces cost of the device. 
     The stress calculation  260  is determined based on the surface curvature, the substrate thickness, and the film thickness. When the stress, substrate, and film are uniform and isotropic, the curvature is also uniform and isotropic and stress σ may be characterized by Stoney&#39;s Equation: 
                   σ   =         Eh   s   2       6   ⁢       h   f     ⁡     (     1   -     υ   s       )           ⁢   κ             eq   .           ⁢   1               
where E is the Young&#39;s modulus of the substrate, h f  is the thickness of the substrate, h f  is the thickness of the film, ν s  is the Poisson&#39;s ratio of the substrate, and κ is the curvature.
 
     When the stress, substrate, and film are non-uniform or not isotropic, however, the stress will not be related to the curvature through the simple Stoney&#39;s Equation, and, thus, Stoney&#39;s Equation cannot be applied to calculate local stress from measurements of local curvature. Accordingly, Stoney&#39;s Equation is modified with corrections for non-uniform substrate thickness, non-uniform film thickness, and non-uniform curvature. Information regarding Stoney&#39;s Equation is provided in D. Ngo, Y. Huang, A. J. Rosakis and X. Feng, “Spatially non-uniform, isotropic misfit strain in thin films bonded on plate substrates: the relation between non-uniform stresses and system curvatures”, Thin Solid Films 515 (2006), pp. 2220-2229; and D. Ngo, X. Feng, Y. Huang, A. J. Rosakis and M. A. Brown, “Thin film/substrate systems featuring arbitrary film thickness and misfit strain distributions: Part I. Analysis for obtaining film stress from nonlocal curvature information”, Int. J. Solids Struct. 44 (2007), pp. 1745-1754, both of which are incorporated herein by reference. 
     
       
         
           
             
               
                 
                   
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     Accordingly, the first correction term may be written as: 
     
       
         
           
             
               
                 
                   
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                                           n 
                                           m 
                                         
                                       
                                       + 
                                       
                                         
                                           1 
                                           
                                             r 
                                             2 
                                           
                                         
                                         ⁢ 
                                         
                                           
                                             ∂ 
                                             2 
                                           
                                           
                                             ∂ 
                                             
                                               θ 
                                               2 
                                             
                                           
                                         
                                         ⁢ 
                                         
                                           Z 
                                           n 
                                           m 
                                         
                                       
                                     
                                     ) 
                                   
                                   ⁢ 
                                   
                                     ⅆ 
                                     A 
                                   
                                 
                               
                             
                           
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   eq 
                   . 
                   
                       
                   
                   ⁢ 
                   10 
                 
               
             
           
         
       
     
     The additional correction terms may be written as: 
     
       
         
           
             
               
                 
                   
                     
                       s 
                       p 
                     
                     = 
                     
                       
                         
                           ( 
                           
                             1 
                             - 
                             υ 
                           
                           ) 
                         
                         
                           ( 
                           
                             1 
                             + 
                             υ 
                           
                           ) 
                         
                       
                       ⁢ 
                       
                         
                           ∑ 
                           k 
                         
                         ⁢ 
                         
                           
                             ( 
                             
                               k 
                               + 
                               1 
                             
                             ) 
                           
                           ⁢ 
                           
                             
                               r 
                               k 
                             
                             ⁡ 
                             
                               [ 
                               
                                 
                                   
                                     A 
                                     k 
                                   
                                   ⁢ 
                                   cos 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   k 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   θ 
                                 
                                 + 
                                 
                                   
                                     B 
                                     k 
                                   
                                   ⁢ 
                                   sin 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   k 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   θ 
                                 
                               
                               ] 
                             
                           
                         
                       
                     
                   
                   ; 
                 
               
               
                 
                   eq 
                   . 
                   
                       
                   
                   ⁢ 
                   11 
                 
               
             
             
               
                 
                   
                     
                       A 
                       k 
                     
                     = 
                     
                       
                         1 
                         π 
                       
                       ⁢ 
                       
                         ∫ 
                         
                           
                             ∫ 
                             A 
                           
                           ⁢ 
                           
                             
                               κ 
                               + 
                             
                             ⁢ 
                             
                               r 
                               k 
                             
                             ⁢ 
                             cos 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             k 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             θ 
                             ⁢ 
                             
                               ⅆ 
                               A 
                             
                           
                         
                       
                     
                   
                   ; 
                 
               
               
                 
                   eq 
                   . 
                   
                       
                   
                   ⁢ 
                   12 
                 
               
             
             
               
                 
                   
                     
                       A 
                       k 
                     
                     = 
                     
                       
                         1 
                         π 
                       
                       ⁢ 
                       
                         
                           ∑ 
                           
                             n 
                             , 
                             k 
                           
                         
                         ⁢ 
                         
                           
                             a 
                             
                               n 
                               , 
                               k 
                             
                           
                           ⁢ 
                           
                             ∫ 
                             
                               
                                 ∫ 
                                 A 
                               
                               ⁢ 
                               
                                 
                                   ( 
                                   
                                     
                                       
                                         
                                           ∂ 
                                           2 
                                         
                                         
                                           ∂ 
                                           
                                             r 
                                             2 
                                           
                                         
                                       
                                       ⁢ 
                                       
                                         Z 
                                         n 
                                         k 
                                       
                                     
                                     + 
                                     
                                       
                                         1 
                                         r 
                                       
                                       ⁢ 
                                       
                                         ∂ 
                                         
                                           ∂ 
                                           r 
                                         
                                       
                                       ⁢ 
                                       
                                         Z 
                                         n 
                                         k 
                                       
                                     
                                     + 
                                     
                                       
                                         1 
                                         
                                           r 
                                           2 
                                         
                                       
                                       ⁢ 
                                       
                                         
                                           ∂ 
                                           2 
                                         
                                         
                                           ∂ 
                                           
                                             θ 
                                             2 
                                           
                                         
                                       
                                       ⁢ 
                                       
                                         Z 
                                         n 
                                         k 
                                       
                                     
                                   
                                   ) 
                                 
                                 ⁢ 
                                 
                                   r 
                                   k 
                                 
                                 ⁢ 
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 k 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 θ 
                                 ⁢ 
                                 
                                   ⅆ 
                                   A 
                                 
                               
                             
                           
                         
                       
                     
                   
                   ; 
                 
               
               
                 
                   eq 
                   . 
                   
                       
                   
                   ⁢ 
                   13 
                 
               
             
             
               
                 
                   
                     B 
                     k 
                   
                   = 
                   
                     
                       1 
                       π 
                     
                     ⁢ 
                     
                       
                         ∑ 
                         
                           n 
                           , 
                           k 
                         
                       
                       ⁢ 
                       
                         
                           a 
                           
                             n 
                             , 
                             k 
                           
                         
                         ⁢ 
                         
                           ∫ 
                           
                             
                               ∫ 
                               A 
                             
                             ⁢ 
                             
                               
                                 ( 
                                 
                                   
                                     
                                       
                                         ∂ 
                                         2 
                                       
                                       
                                         ∂ 
                                         
                                           r 
                                           2 
                                         
                                       
                                     
                                     ⁢ 
                                     
                                       Z 
                                       n 
                                       k 
                                     
                                   
                                   + 
                                   
                                     
                                       1 
                                       r 
                                     
                                     ⁢ 
                                     
                                       ∂ 
                                       
                                         ∂ 
                                         r 
                                       
                                     
                                     ⁢ 
                                     
                                       Z 
                                       n 
                                       k 
                                     
                                   
                                   + 
                                   
                                     
                                       1 
                                       
                                         r 
                                         2 
                                       
                                     
                                     ⁢ 
                                     
                                       
                                         ∂ 
                                         2 
                                       
                                       
                                         ∂ 
                                         
                                           θ 
                                           2 
                                         
                                       
                                     
                                     ⁢ 
                                     
                                       Z 
                                       n 
                                       k 
                                     
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 r 
                                 k 
                               
                               ⁢ 
                               sin 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               k 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               θ 
                               ⁢ 
                               
                                 
                                   ⅆ 
                                   A 
                                 
                                 . 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   eq 
                   . 
                   
                       
                   
                   ⁢ 
                   14 
                 
               
             
           
         
       
     
     Advantageously, the fit coefficients α n,m  are outside the integrals in equations 10, 13, and 14. Thus, the Zernike polynomials Z n   k  can be integrated once prior to measurement and those values are saved and used as a multiple for the fit coefficients α n,m  once determined for an individual substrate. Accordingly, only the summations need be performed in the calculations, which increases speed compared to performing the integrations, as well as removes concerns about sparse data or boundaries in the integration. 
     To correct for varying film thickness, the actual film thickness h f (r,θ) is substituted in for each point of interest as follows: 
     
       
         
           
             
               
                 
                   
                     σ 
                     + 
                   
                   = 
                   
                     
                       
                         Eh 
                         s 
                         2 
                       
                       
                         6 
                         ⁢ 
                         
                           ( 
                           
                             1 
                             - 
                             
                               υ 
                               s 
                             
                           
                           ) 
                         
                         ⁢ 
                         
                           
                             h 
                             f 
                           
                           ⁡ 
                           
                             ( 
                             
                               r 
                               , 
                               θ 
                             
                             ) 
                           
                         
                       
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           κ 
                           + 
                         
                         + 
                         
                           s 
                           o 
                         
                         + 
                         
                           s 
                           p 
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   eq 
                   . 
                   
                       
                   
                   ⁢ 
                   15 
                 
               
             
           
         
       
     
     Although the present invention is illustrated in connection with specific embodiments for instructional purposes, the present invention is not limited thereto. Various adaptations and modifications may be made without departing from the scope of the invention. Therefore, the spirit and scope of the appended claims should not be limited to the foregoing description.