Abstract:
A system and method for sizing semiconductor wafer defects combines contiguous light intensity values over a defect area of the wafer to provide a defect sizing metric. The light intensity values resulting from a light source applied to the wafer are summed and the summation is compared to known metric values related to known defect sizes. The result of the comparison provides an estimate for the defect size under consideration. The combination of the light intensity values produces defect sizing estimates in saturated and unsaturated ranges of operation for the wafer inspection equipment. Calculations applied to the combined light intensity values vary depending upon whether the light intensity component is from a saturated or an unsaturated measurement. The defect sizing metric can be applied continuously in saturated and unsaturated ranges of operation to avoid additional processing steps, such as recalibration of the inspection equipment.

Description:
CROSS REFERENCE TO RELATED APPLICATIONS 
   This application is a continuation-in-part application of U.S. patent application Ser. No. 11/321,689 filed Dec. 29, 2005 now U.S. Pat. No. 7,302,360 entitled DEFECT SIZE PROJECTION, which is a continuation-in-part application, of U.S. patent application Ser. No. 10/971,694 filed Oct. 22, 2004, now U.S. Pat. No. 7,184,928 entitled EXTENDED DEFECT SIZING, which claims benefit of U.S. Provisional Patent Application No. 60/514,289 filed Oct. 24, 2003 entitled EXTENDED DEFECT SIZING. 

   STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT 
   Not applicable 
   BACKGROUND OF THE INVENTION 
   1. Field of Invention 
   The present invention relates generally to systems and methods of inspecting semiconductor wafers, and relates more specifically to a semiconductor wafer inspection system, and method capable of detecting and measuring wafer defect&#39;s in which the scattering power of the defect exceeds the dynamic range of the system. 
   2. Description of Related Art 
   Systems and methods of inspecting workpieces and especially semiconductor wafers are known for detecting and measuring defects occurring on a surface of a semiconductor wafer. For example, a conventional laser-based surface scanning inspection system is typically configured to detect localized light scatters on a semiconductor wafer surface. Such localized light scatters may be indicative of one or more defects on the wafer surface that may render an integrated circuit(s) (IC) fabricated on the wafer to be inoperative. In a typical mode of operation, the conventional surface scanning inspection system sweeps a laser light beam in a predetermined direction, while the wafer being inspected rotates under the swept beam at an angle of about 90° to the predetermined sweep direction. Next, the conventional surface scanning inspection system detects a light beam reflected from the wafer surface, and samples the detected signal in both the predetermined direction of the swept beam and in the direction of rotation to obtain a two-dimensional array of data. When the light beam sweeps over a defect on the wafer surface, the data obtained by the wafer inspection system generally corresponds to the beam shape of the laser spot power at the wafer surface. This is because such wafer surface defects are generally much smaller than the spot size of the laser beam. After the conventional surface scanning inspection system has detected a defect, the system may attempt to measure the size of the defect by determining the value of the maximum scattering power of the defect, and may also determine the location of the defect on the surface of the wafer. 
   One drawback of the above-described conventional laser-based surface scanning inspection system is that the maximum scattering power of a detected defect may exceed the dynamic range of the system. As a result, the electronics within the wafer inspection system may saturate, thereby causing at least some of the defect size measurements performed by the system to be at a power level at which the measurements become nonlinear due to the saturation effects. 
   One way of addressing the effects of saturation on defect size measurements made by the conventional laser-based surface scanning inspection system is to employ a data extrapolation technique. However, such data extrapolation techniques are often difficult to perform in conventional wafer inspection systems. Alternatively, the conventional surface scanning inspection system may perform a nonlinear least squares fit of the measurements to a given Gaussian shape, which may be characterized by a number of parameters including an estimated amplitude, an estimated inverse correlation matrix, and an estimated pulse center location. However, conventional algorithms for performing such nonlinear least squares fit techniques often require a significant amount of processing time. Further, relatively small changes in the data resulting from, e.g., noise or a non-ideal signal, may lead to significantly large changes in the estimated parameters. 
   One methodology that can measure very large defects takes advantage of correlations between a scatter light response for a defect of unknown size, and a scatter light response for a defect of known size. For these large defects, a set-point threshold for scatter light intensity, or the equivalent voltage representation is used to contribute to defining a scatter light response area where the set-point threshold is exceeded. The defined response area is compared to response areas for known defect sizes, determined through calibration processes, and an estimated size of the measured unknown defect can be obtained. The methodology uses an empirical calibration between area and defect or particle size, and has proven to be quite robust. There is a drawback to the methodology in that it adds some difficulty because it uses a completely separate calibration process, which can be time consuming and cumbersome. 
   Another methodology calculates the area of a cross section of a pulse shape representation of the scatter light response at multiple non-saturated signal levels. The methodology calculates a fit line for area versus logarithm of the signal level. The fit line and non-saturated measurements are used to extrapolate to a pulse cross-section of zero area to estimate peak voltage corresponding to a peak of the pulse shape representation. The methodology also incorporates a verification of the slope of the fit line to match a slope from an expected Gaussian pulse response. This methodology has the drawback of requiring an unclipped or unsaturated signal level. If the signal is clipped, correction of areas for filter effects is imperfect and performance may be degraded. This methodology is also limited in range over which extrapolation is useful, as well as being computationally intensive. 
   Another technique uses an area of a scatter light response cross section at a single unsaturated signal level and then extrapolates to determine a response peak using a slope expected from a Gaussian pulse response. However, this technique also uses an unclipped signal and can be extremely sensitive to variations from an ideal Gaussian response. 
   It would therefore be desirable to have an improved system and method of inspecting semiconductor wafers that can measure the size and determine the location of a defect on a surface of a semiconductor wafer while avoiding the drawbacks of conventional wafer inspection systems and methods. 
   BRIEF SUMMARY OF THE INVENTION 
   Briefly described, there is provided in accordance with the present disclosure a system and method for sizing semiconductor wafer defects that combines contiguous light intensity values over a defect area of the wafer to provide a defect sizing metric. The light intensity values resulting from a light source applied to the wafer are summed and the summation is compared to known metric values related to known defect sizes. The result of the comparison provides an estimate for the defect size under consideration. The combination of the light intensity values produces defect sizing estimates in saturated and unsaturated ranges of operation for the wafer inspection equipment. Calculations applied to the combined light intensity values vary depending upon whether the light intensity component is from a saturated or an unsaturated measurement. The defect sizing metric can be applied continuously in saturated and unsaturated ranges of operation to avoid additional processing steps, such as recalibration of the inspection equipment. 
   According to an exemplary embodiment of the present disclosure, a size of a sample defect or particle on a surface of a wafer is estimated by deriving an intensity response shape volume value for the sample particle based on a three-dimensional intensity response shape. The three-dimensional intensity response shape is representative of the intensity of scatter light generated as a laser light traverses the sample particle. The intensity response shape may be defined as a three-dimensional shape comprising the outline and volume of a region of space having as two of its coordinates a two-dimensional location on a wafer at which the scatter light intensity is measured, the third coordinate being the intensity response measured at the location. 
   A threshold detection intensity may be used to qualify the intensity response shape volume, such as by forming the response volume mapping with scatter light intensity values that are above a given threshold. For example, the threshold may be used to help eliminate low level “haze” or noise that might cause spurious scatter light intensity readings. 
   The estimated defect or particle size may be obtained according to the following process. The scatter light response is mapped to a three-dimensional representation that tends to be formed as a pulse shape. The sample particle size estimate is based on the intensity response shape volume value derived from the pulse shaped mapping representation for the sample particle. The system and method identifies a relationship between defect or particle size and intensity response shape volume, using particles of known sizes and intensity response shape volume values associated with the known particle sizes. The system and method uses the relationship between the particle size and intensity response shape volume to identify a particle size that can be associated with the intensity response shape volume value obtained for the sample particle. 
   According to another exemplary embodiment, a method for determining the defect size includes combining a contiguous group of light intensity responses resulting from application of a light source to a surface of the wafer to obtain a representative value. The method compares the representative value to known values related to known defect sizes to obtain a measured defect size. The representative value may be a summation of the light intensity responses for a scanned defect area that results in a volume value. Each light intensity measure is taken in a discrete area that is defined by the in-scan spacing and cross-scan pitch. The light intensity responses may be filtered prior to being combined. 
   The combined light intensity responses are compensated if there are saturated measurements included in the response. The compensation is based on the recognition that the combination is linear for estimating defect sizes when not saturated, and semi-logarithmic when saturation components are present. 
   The comparison of the measured combination with known combinations related to known defect sizes may be based on one or more response curves. Alternately, or in addition, the comparison may be based on one or more tables for comparing the representative value to the known values. 
   According to an aspect of the disclosed system and method, a defect sizing metric contributes to providing a sample particle size estimate. The defect sizing metric is developed based on the volume of the portion of the intensity response shape that is above a threshold detection intensity. The detected light is filtered to produce the volume based sizing metric using an intensity response shape volume value for defect particles. The shape volume values are mapped to a defect size. The mapping may then be used to estimate sample defect sizes. 
   The intensity response shape volume values are formed by obtaining filtered signals representative of the intensity of light scattered on a surface of the wafer at various surface locations. The intensity response shape volume value is the volume of the portion of the intensity response shape that is above a threshold detection intensity. For example, a set of filtered signals associated with contiguous surface locations with intensity values exceeding a background haze level are identified. Locations of potential defects are determined based on the representations of the filtered signals. Each filtered signal is associated with an area of the surface location, which has a particular light intensity response value. The value is multiplied by an amount associated with a signal intensity level at the surface location to obtain a signal volume value. The volume values from all the contiguous locations in the set of filtered signals are summed to create an intensity response shape volume value. The summed volume values form a map that can be used to estimate sample defect sizes. 
   Estimation of sample defect sizes is performed by identifying a relationship between defect or particle size and intensity response shape volume. The known intensity response shape volume value for defects or particles of known sizes is used together with a relationship between particle size and intensity response shape volume. The known values and the relationship contribute to identifying a particle size associated with the intensity response shape volume value obtained for the sample particle. 
   According to one aspect of the present invention, a threshold value may be selected at which intensity responses are subject to being clipped. The relationship between particle size and intensity response shape volume permits estimates to be made for filtered responses above a threshold level. Various constants that are defined by system parameters contribute to identifying a clipping threshold value used in calculating the volume value. 
   The extended dynamic range obtained through estimates using the relationship between particle size and intensity response shape volume permit the equipment to have an extended factor of operation. The extended factor is a function of the peak of an intensity response to a particle of known size without clipping, and the clipping intensity level. For responses below the clipping threshold, as defined by system characteristics or parameters, a linear relationship exists between the intensity response shape volume values and the extended factor with which they are associated. Above the clipping threshold level, a linear relationship exists between intensity response shape volume values and the logarithm of the extended factor with which the values are associated. Volume values are calculated based on either the linear or semi-logarithm relationship, depending on whether the measured intensity is clipped. 
   Once the volume sizing metric is calculated, a size of a sample defect can be assigned based on a value of the volume sizing metric. The relationship between particle size and intensity response shape volume can be mapped with a response curve that relates intensity response shape volume values to the extended factor for particle size. The response curve may be represented in linear form or semi-log form to obtain intensity response shape volume values below and above a system clipping threshold, respectively. 
   Alternately, or in addition, a table having representative intensity response shape volume values and extended factor values for particle sizes may be used to obtain the defect sizing estimate. 

   
     BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWINGS 
     The invention will be more fully understood with reference to the following Detailed Description of the Invention in conjunction with the drawings of which: 
       FIG. 1  is a block diagram of a laser-based wafer surface scanning inspection system according to the present invention, in which the system performs a scan of a laser beam on a surface of a semiconductor wafer to detect defects on the wafer surface; 
       FIG. 2  is a functional illustration of components included in the surface scanning inspection system of  FIG. 1 ; 
       FIG. 3  is a diagram of a first geometric Gaussian shape in three-dimensional space, the first Gaussian shape representing non-saturated data collected by the surface scanning inspection system of  FIG. 1 ; 
       FIG. 4  is a diagram of an elliptical cross-sectional area of the first Gaussian shape of  FIG. 3 , the cross-sectional area being obtained by conceptually cutting the first Gaussian shape in an x-y plane corresponding to a predetermined height of the Gaussian shape; 
       FIG. 5  is a diagram of a second geometric Gaussian shape in three-dimensional space, the second Gaussian shape representing saturated data collected by the surface scanning inspection system of  FIG. 1 ; 
       FIG. 6  is a diagram illustrating a linear least squares fit of the data represented by the second Gaussian shape of  FIG. 5 ; 
       FIGS. 7   a - 7   h  are three-dimensional graphs showing various Gaussian response pulses with a clipping threshold height; 
       FIGS. 8   a - 8   h  are three-dimensional graphs illustrating filtered Gaussian response shape volumes for various clipping thresholds; 
       FIG. 9  is a semi-logarithmic graph illustrating volume sizing metric versus extension factor; 
       FIG. 10  is a semi-logarithmic graph illustrating volume sizing metric versus extension factor for various haze values; and 
       FIG. 11  is a graph illustrating a relationship between volume sizing metric and extension factor for various background haze values. 
   

   DETAILED DESCRIPTION OF THE INVENTION 
   A system and method of inspecting a semiconductor wafer is disclosed that is capable of measuring the size and determining the location of a defect on a surface of a semiconductor wafer. The presently disclosed wafer inspection system can perform such sizing and locating of wafer surface defects whether or not the scattering power associated with the defect exceeds the dynamic range of the system. 
     FIG. 1  depicts an illustrative embodiment of a laser-based wafer surface scanning inspection system  100 , in accordance with the present invention. In the illustrated embodiment, the surface scanning inspection system  100  comprises an optical module including a surface scanning mechanism  102 , and optics  104 . Optics  104  include a light detector with a channel for detecting reflected light, often referred to as a “light channel”. A light detector with a channel for detecting scattered light, often referred to as a “dark channel” is also included in optics  104 . For example, the surface scanning mechanism  102  may be an acousto-optic deflector (AOD) or any other suitable surface scanning mechanism, and the optics  104  may comprise a quadcell photodetector or any other suitable light detector. As shown in  FIG. 1 , the AOD  102  is configured to emit at least one collimated beam of laser light  108  toward a surface  107  of a semiconductor wafer  106  at an oblique angle of incidence θ i . Further, the optics  104  are configured to detect a light beam  110  specularly reflected from the wafer surface  107  at an angle of reflection θ I . Specifically, the optics  104  are configured to detect specular distortions in the reflected light beam  110 . It is noted that the wafer  106  may also be inspected from the backside by inverting the wafer in the surface scanning inspection system  100 . 
   For example, the AOD  102  may include a solid state laser such as a 532 nm wavelength diode-pumped solid state laser, or any other suitable type of laser. In the illustrative but not necessarily preferred embodiment, the AOD  102  emits the laser light beam  108  to produce a focused laser spot having a diameter of about 30 microns for scanning the wafer surface  107 , in which the incident angle θ i  of the emitted light beam  108  is about 65 degrees. It should be understood that the laser light beam  108  may alternatively be emitted by the AOD  102  at any suitable angle of incidence to produce any suitable spot size on the wafer surface. The surface scanning inspection system  100  further includes a theta stage  103  upon which the wafer  106  is held during inspection. The theta stage  103  is configured to rotate and to translate the wafer  106  through a scan line  112  produced by the AOD  102 , thereby generating a spiral pattern of light used to inspect the wafer surface  107 . The theta stage  103  includes an encoder such as an optical encoder that provides counts indicative of the rotational position of the stage  103  relative to a predetermined reference point. It is noted that the structure and operation of the theta stage  103  are known to those skilled in this art and therefore need not be described in detail herein. 
     FIG. 2  depicts a plurality of functional components included in the above-described surface scanning inspection system  100  (see  FIG. 1 ). As shown in  FIG. 2 , the surface scanning inspection system, shown as Wafer Inspection System  100 , comprises a turning mirror  206 , the AOD  102  including a beam expander  204 , a cylinder lens  202 , an objective lens  208 , the optics  104 , and a processor  208  and associated memory  210 . In the illustrated embodiment, the AOD  102  is configured to generate the narrow angle light beam  108  by exciting a crystal with a high frequency sound wave. The beam expander  204  is configured to expand the light beam  108  before the beam enters an aperture of the AOD  102  to obtain a desired angle of deflection. The cylinder lens  202  is disposed at the output of the AOD  102 , and is configured to compensate for parasitic cylinder lens loss that may be induced by the deflector. The scan is relayed through the objective lens  208  to the surface  107  of the wafer  106  (see also  FIG. 1 ). The optics  104  are configured to receive the reflected light beam  110 , and to detect any losses in light intensity resulting from specular distortion or deflection of the light beam  110 . 
   In the illustrative but not necessarily preferred mode of operation, the optics  104  (see  FIG. 1 ) are also configured to detect localized light scatters from the surface  107  of the wafer  106 . For example, such localized light scatters may be indicative of one or more defects on the wafer surface  107  that may render an integrated circuit(s) (IC) fabricated on the wafer  106  to be inoperative. Specifically, the AOD  102  emits the laser light beam  108  toward the wafer surface  107  at the angle of incidence θ i  and sweeps the light beam  108  in a predetermined radial direction, while the theta stage  103  rotates under the swept beam  108  at an angle of about 90° to the predetermined radial direction. Next, the optics  104  detect the laser light beam  110  reflected from the wafer surface  107  at the angle of reflection θ I , and detects the light scattered from the wafer surface and samples the detected signal in both the radial and rotational directions to obtain a two-dimensional array of data. The scattered light from the wafer surface is detected in the optics  104  in a dark channel. It is noted that the sampling of the data is generally non-orthogonal. The processor  208  included in the surface scanning inspection system  100  is operative to process the sampled data by executing one or more programs out of its associated memory  210  (see  FIG. 2 ). 
   In the presently disclosed embodiment, the corresponding location of each data sample on the wafer surface  107  is expressed as
 
x in,xs , y in,xs ,  (1)
 
in which the index “in” designates samples in the radial or “in scan” direction, and the index “xs” designates samples in the tangential or “cross scan” direction.
 
   When the light beam  108  sweeps over a defect on the wafer surface  107 , the data samples obtained by the surface scanning inspection system  100  generally correspond to the beam shape of the laser spot on the surface  107 . This is because wafer surface defects are normally much smaller than the spot size of the laser beam  108 . For example, the data samples may be represented by a geometric Gaussian shape that is non-isotropic due to the angle of incidence θ i  and the non-orthogonal sampling of the data. 
   The locations (x in,xs , y in,xs ) of the data samples on the wafer surface  107  may be expressed as a column vector, i.e., 
                   z   →     =       [           x       i   ⁢           ⁢   n     ,   xs                 y       i   ⁢           ⁢   n     ,   xs             ]     .             (   2   )               
Accordingly, the optical laser spot power at the wafer surface  107  may be expressed as
 power( {right arrow over (z)} )= P   0 exp(−( {right arrow over (z)}−{right arrow over (z)}   0 )′ R   −1 ( {right arrow over (z)}−{right arrow over (z)}   0 )),  (3) 
in which “P 0 ” is the maximum scattering power of the defect, “{right arrow over (z)} 0 ” denotes the location of the defect, and “R” is a positive definite symmetric matrix describing the beam shape.
 
   For example, if a laser spot is Gaussian, meaning that the intensity profile of the laser spot is a two-dimensional Gaussian function, and has a density of density(x)=e −x     2     /2σ     2   , then the 1/e 2  full-width may be expressed as 4σ. For an illustrative 50 micron 1/e 2  full-width beam, which strikes a wafer at a 65-degree incident angle, then the density at the wafer surface may be expressed as
 
density( x,y )= e   −x     2     /2(12.5μ)     2      e   −y     2     /2(12.5μ/cos(65°))     2   .  (4)
 
Equation (4) above may be rewritten as
 
                   density   ⁡     (     x   ,   y     )       =       exp   ⁡     (             [         x           y         ]     t     ⁡     [             (     12.5   ⁢           ⁢   μ     )     2         0           0           (     12.5   ⁢           ⁢     μ   /     cos   ⁡     (     65   ⁢   °     )           )     2           ]         -   1       ⁡     [         x           y         ]       )       .             (   5   )               
Accordingly, for this illustrative example,
 
   
     
       
         
           
             
               
                 R 
                 = 
                 
                   
                     [ 
                     
                       
                         
                           
                             
                               ( 
                               
                                 12.5 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 μ 
                               
                               ) 
                             
                             2 
                           
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           
                             
                               ( 
                               
                                 12.5 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   μ 
                                   / 
                                   
                                     cos 
                                     ⁡ 
                                     
                                       ( 
                                       
                                         65 
                                         ⁢ 
                                         ° 
                                       
                                       ) 
                                     
                                   
                                 
                               
                               ) 
                             
                             2 
                           
                         
                       
                     
                     ] 
                   
                   . 
                 
               
             
             
               
                 ( 
                 6 
                 ) 
               
             
           
         
       
     
   
   In the event the sampled data comprises non-saturated data (i.e., the data sampling is linear), the surface scanning inspection system  100  may determine the value of P 0  in equation (3) above by identifying the largest value in the collection of measured data points, which may be expressed as
 
power(x in,xs , y in,xs ).  (7)
 
   This technique for determining the value of P 0  generally does not yield useful results when the maximum scattering power of a detected defect exceeds the dynamic range of the surface scanning inspection system  100 , i.e., when the sampled data comprises saturated data. In practice, the gain of the surface scanning inspection system  100  is set high to permit detection of small defects. Thus, the power level of surface scanning inspection system  100  may be relatively high to detect and measure small defects. As a result, at least some of the defect size measurements performed by the wafer inspection system may be at a power level at which the measurements become nonlinear due to the saturation effects. 
   One prior technique measures the size and determining the location of a defect on a surface of a semiconductor wafer when the maximum scattering power of a detected defect exceeds the dynamic range of the surface scanning inspection system  100 , i.e., the sampled data collected by the wafer inspection system comprises saturated data.  FIG. 3  depicts a geometric Gaussian shape  302  in a space defined by x, y, and z axes, in which the Gaussian shape  302  represents non-saturated data collected by the surface scanning inspection system  100  (see  FIG. 1 ). If the Gaussian shape  302  is conceptually cut by an x-y plane  304  at a predetermined amplitude (“cut height”) along the z-axis, then the resulting cross-sectional area of the Gaussian shape  302  in the x-y plane  304  has the shape of an ellipse  402  (see  FIG. 4 ). The area of the ellipse  402  may be determined by solving for the area of a region defined by
 
power( {right arrow over (z)} )&gt;height,  (8)
 
in which “power({right arrow over (z)})” is expressed as indicated in equation (3) above. Substituting this expression for power({right arrow over (z)}) in equation (8) yields
 
                       (       z   →     -       z   →     0       )     ′     ⁢       R     -   1       ⁡     (       z   →     -       z   →     0       )         &lt;       ln   ⁡     (     P   0     )       -       ln   ⁡     (   height   )       .     
     ⁢   Let               (   9   )                 Area   =     ∫       ∫           (       z   →     -       z   →     0       )     ′     ⁢       R     -   1       ⁡     (       z   →     -       z   →     0       )         &lt;       ln   ⁡     (     P   0     )       -     ln   ⁡     (   height   )             ⁢     ⅆ   z           ,     
     ⁢   and           (   10   )                 y   =       R     -     1   2         ⁡     (       z   →     -       z   →     0       )         ,     
     ⁢     dy   =            R          -     1   2         ⁢   dz       ,     
     ⁢     dz   =            R          1   2       ⁢     dy   .                 (   11   )               
Accordingly,
 
   
     
       
         
           
             
               
                 
                   Area 
                   = 
                   
                     ∫ 
                     
                       
                         ∫ 
                         
                           
                              
                             y 
                              
                           
                           &lt; 
                           
                             
                               
                                 ln 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     P 
                                     0 
                                   
                                   ) 
                                 
                               
                               - 
                               
                                 ln 
                                 ⁡ 
                                 
                                   ( 
                                   height 
                                   ) 
                                 
                               
                             
                           
                         
                       
                       ⁢ 
                       
                         
                           
                              
                             R 
                              
                           
                           
                             1 
                             2 
                           
                         
                         ⁢ 
                         
                           ⅆ 
                           y 
                         
                       
                     
                   
                 
                 , 
               
             
             
               
                 ( 
                 12 
                 ) 
               
             
           
           
             
               
                 
                   Area 
                   = 
                   
                     
                       ∫ 
                       0 
                       
                         2 
                         ⁢ 
                         π 
                       
                     
                     ⁢ 
                     
                       
                         ∫ 
                         
                           
                             
                               ln 
                               ⁡ 
                               
                                 ( 
                                 
                                   P 
                                   0 
                                 
                                 ) 
                               
                             
                             - 
                             
                               ln 
                               ⁡ 
                               
                                 ( 
                                 height 
                                 ) 
                               
                             
                           
                         
                       
                       ⁢ 
                       
                         
                           
                              
                             R 
                              
                           
                           
                             1 
                             2 
                           
                         
                         ⁢ 
                         r 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ⅆ 
                           r 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ⅆ 
                           θ 
                         
                       
                     
                   
                 
                 , 
                 and 
               
             
             
               
                 ( 
                 13 
                 ) 
               
             
           
           
             
               
                 Area 
                 = 
                 
                   π 
                   ⁢ 
                   
                     
                        
                       R 
                        
                     
                     
                       1 
                       2 
                     
                   
                   ⁢ 
                   
                     
                       ( 
                       
                         
                           ln 
                           ⁡ 
                           
                             ( 
                             
                               P 
                               0 
                             
                             ) 
                           
                         
                         - 
                         
                           ln 
                           ⁡ 
                           
                             ( 
                             height 
                             ) 
                           
                         
                       
                       ) 
                     
                     . 
                   
                 
               
             
             
               
                 ( 
                 14 
                 ) 
               
             
           
         
       
     
   
   Equation (14) above shows that the area of a geometric Gaussian shape conceptually cut at a predetermined height (e.g., the area of the ellipse  402 ; see  FIG. 4 ) is a linear function of the natural logarithm (ln) of the predetermined cut height. As indicated by equation (14), the cross-sectional area is equal to zero when the cut height equals the scattering power P 0  of the defect. Further, the slope of the line defined by equation (14) is equal to
 
π|R| 1/2 ,  (15)
 
in which “|R| 1/2 ” is the square root of the determinant of the positive definite symmetric matrix describing the beam shape. It is noted that “π|R| 1/2 ” is equal to the “1/e” area of the Gaussian shape. Accordingly, after plotting the area values as a function of the natural logarithm (ln) of the predetermined cut heights, and applying a least squares fit to the plot to form a linear plot, the intercept at which the area is zero is equal to the natural logarithm of the scattering power P 0 , and the slope of the linear plot is equal to the 1/e area of the Gaussian shape.
 
   One technique for measuring the size and determining the location of a defect on a semiconductor wafer surface is illustrated by the following example.  FIG. 5  depicts a geometric Gaussian shape  502  in x, y, z coordinate space, in which the Gaussian shape  502  comprises saturated data collected by the surface scanning inspection system  100  (see  FIG. 1 ). In this example, the Gaussian shape  502  is conceptually cut in the x-y plane at a plurality of predetermined cut heights along the z-axis, namely, at cut heights of 0.2, 0.4, 0.6, 0.8, and 1.0 units. Next, the respective cross-sectional areas of the Gaussian shape  502  conceptually cut at these predetermined heights are determined. The values of the cross-sectional areas are then plotted versus the natural logarithm (ln) of the respective cut heights, and a least squares fit is applied to the plot to produce a linear plot  602  of the collected data, as depicted in  FIG. 6 . As shown in  FIG. 6 , the linear plot  602  includes the data points  604 ,  606 ,  608 ,  610 , and  612  corresponding to the predetermined cut heights 0.2, 0.4, 0.6, 0.8, and 1.0, respectively. In this illustrative example, the linear plot  602  may be expressed as
 
 y =−624 x +440,  (16)
 
in which the variable “y” represents the cross-sectional area of the Gaussian shape  502  and the variable “x” represents the natural logarithm of the predetermined cut height.
 
   Accordingly, equation (16) above indicates that the cross-sectional area (y) is equal to zero when the natural logarithm of the cut height (x) equals about 0.705. The cut height at which the cross-sectional area equals zero may therefore be obtained by taking the inverse natural logarithm of 0.705, which is about 2.02. Because the cross-sectional area is equal to zero when the cut height equals the scattering power P 0  of a wafer surface defect, as indicated in equation (14) above, P 0  is equal to about 2.02. In this example, the actual height of the illustrative Gaussian shape  502  (i.e., the height that would be observed in the absence of saturation effects) is 2.0. Further, the slope of the linear plot  602 , as expressed by equation (16) above, is equal to −624, which is the 1/e area of the Gaussian shape. In this example, the actual 1/e area of the Gaussian shape  502  (i.e., the 1/e area that would be observed in the absence of saturation effects) is 200π, or about 628. Based on these results, a correlation coefficient may be calculated as 0.9999. In general, if the correlation coefficient is much less than unity, then the linear least squares fit is considered to be poor. Because the correlation coefficient is equal to 0.9999 in this illustrative example, the linear least square fit is considered to provide an accurate measure of the actual height of the Gaussian shape  502 . 
   The above-discussed methodology extends the linear dynamic range of the wafer scanning system by factor of approximately ten in the optical power domain. By using techniques that involve curve fitting and knowledge of the beam shape, the methodology permits determination of a voltage equivalent magnitude of a defect. With the calculated voltage magnitude, optical models may be used to convert the voltage equivalent magnitude to a scattered power, and then to defect size. 
   However, the above-described methodology begins to degrade in performance when applied to extend the linear dynamic range of the system beyond the factor of ten. For example,  FIGS. 7   a - 7   h  illustrate Gaussian pulse responses determined from a light detector where the responses extend beyond a system threshold when measuring larger defects. As can be seen, the larger defects prevent the beam from accurately retaining a Gaussian shape. That is, as the observed defect becomes larger in size, the outlying lower-power sections of the beam influence the curve fit estimations and averages. Because the optics are not perfect, the beam is not perfectly Gaussian, and the imperfections can degrade the accuracy of the above methodology in the case of very large defects. 
   Other techniques have been used to detect and measure very large defects that produce pulses outside the dynamic range of the equipment. To measure the very large defects, a cross-sectional area of a Gaussian pulse generated by the defect is determined, based on a given set point threshold, or height at which the cross-sectional area is taken. The thus determined cross-sectional area is compared to areas generated under similar conditions for known defects or particle sizes. An estimate for the defect or particle size can be made based on the comparison and interpolation techniques. 
   While the results of the above-described technique are fairly robust, an empirical calibration is made between the area of the cross-section and the particle size. That is, a completely separate calibration process is used to set up the measurement, since the nature of the very large defect dramatically increases the range of extrapolation for determining pulse height and thus defect sizing. When the correlation coefficient of the previously described methodology is much less than unity, meaning that a very large defect is detected that introduces some nonlinearity into the methodology, the linear least squares fit is considered to be poor. The impact on the methodology can be observed by visualizing the truncated pulse of  FIG. 5  as approaching the shape of a column, as seen in  FIGS. 7   b - 7   h , for example, indicating a very high pulse, as well as an ill-conditioned fit for the linear relationship used to determine pulse height. 
   To improve detection of small defects when the gain of surface scanning inspection system  100  is set high, filtering is typically used to reduce shot noise caused by the arrival of discrete photons at the detector. Reliable detection of small defects results from the application of two-dimensional matched filtering to enhance the signal-to-noise ratio. Matched filtering is commonly used for obtaining measurements from systems having additive Gaussian noise. Matched filtering is described in detail in the U.S. Pat. No. 6,529,270, which is entitled APPARATUS AND METHOD FOR DETECTING DEFECTS IN THE SURFACE OF A WORKPIECE and which is hereby incorporated by reference, for background. 
   The matched filtering is typically performed in two steps related to operation of the surface scanning inspection system  100 . First, in-scan matched filtering is applied to the outputs of the optical detectors, followed by cross-scan matched filtering. The output of the optical detectors is a voltage that represents the instantaneous optical power of the received light. The application of the two-dimensional matched filtering tends to decrease the peak values of the input voltage signals representing the instantaneous optical power. Accordingly, even when a saturated response is observed, the two-dimensional matched filtering tends to smooth the aggregate defect light intensity response. A representation of such a smoothed response is illustrated in  FIGS. 8   a - 8   h.    
   The present invention provides a system and method for determining defect size when system measurements exceed the dynamic range of the equipment. The system and method derive an intensity response shape volume value based on a three-dimensional intensity response shape that is representative of the intensity of scatter light generated as a laser light traverses the sample defect. The intensity response shape volume value is used to estimate the defect size based on a relationship of a known intensity response shape volume value related to a known sample particle. The relationship may be used to extend the range of the measuring equipment without extraordinary calibration of the equipment, and is more robust than the previously used estimation techniques. 
   The intensity response shape volume value may be calculated as the volume of the portion of the intensity response shape that is above a given threshold detection intensity. The disclosed system and method estimates the defect or particle size based on the measured response shape volume value in comparison with a known intensity response shape volume value that corresponds to a known particle size. A set of known intensity response shape volume values and associated known particle sizes develop the relationship that may be used to extend the system measurement capability. By using the measured intensity response shape volume value and the known relationship, large defect sizes can be accurately estimated over a wide range of sizes. 
     FIGS. 8   a - 8   h  illustrate three-dimensional intensity response shapes representative of the intensity of scatter light generated with respect to a defect or particle under examination.  FIG. 8   a  similarly shows the filtered three-dimensional intensity response shape related to an unclipped signal shown in  FIG. 7   a .  FIGS. 8   b - 8   h  relate to three-dimensional filtered intensity response shapes corresponding to  FIGS. 7   b - 7   h  for clipped signals that would have been 2, 5, 10, 20, 50, 100 and 200 times the maximum unclipped signal, respectively. 
   A volume based sizing metric is formed by creating intensity response shape volume values for known defect particles and mapping the values to a defect size. The mapping is then used to estimate measured defect sizes. The intensity response shape volume values are formed by filtering signals that represent the intensity of light scattered on the surface of the wafer at specific locations.  FIGS. 8   a - 8   h  are filtered representations of the light intensity response shapes shown in  FIGS. 7   a - 7   h , respectively.  FIG. 7   a  shows an unclipped light intensity response, at a maximum unclipped height for the unfiltered intensity response. The filtered signals are grouped together as a set to represent contiguous surface locations where intensity response values exceed a background haze level. The set of signals represent a potential defect location. With a set of filtered signals representing light intensity responses at discrete wafer locations, the disclosed system and method obtains a volume value by combining the set of filtered signals. The volume value is related to the defect or particle size at the surface location from which the set of filtered signals is obtained. 
   According to one exemplary embodiment, the set of volume values are summed from all the contiguous locations in the set that relate to a given defect. An intensity response shape volume value results. The volume value represents a mapping from light intensity response to defect or particle size to assist with estimating the defect or particle size. 
   Once the intensity response shape volume value is known, the value is compared to known values associated with defects or particles of known size. The result of the comparison provides an estimate for the size of the defect or particle under examination. A form of interpolation may be used to determine the estimated defect or particle size based on known defect or particle sizes. 
   According to one exemplary embodiment, the intensity response shape volume value is derived from the portion of the light intensity response that is above a threshold detection intensity. The threshold detection intensity is set to avoid spurious inputs or noise in the light detection optics or devices. Measurements taken above the threshold detection intensity represent defects or particles to be analyzed. 
   The disclosed system and method works well for both clipped and unclipped signals. When light intensity response signals are unclipped, there is a linear relationship between the calculated volume value and the estimated size of the defect or particle under examination. When light intensity response signals are clipped, there is a semi-log relationship between the calculated volume value on the estimated size of the defect or particle. By knowing a priori the point at which the light detection device or detection system saturates, the appropriate linear or semi-log calculations can be applied to determine the appropriate relationship to use the estimate defect or particle size. In this way, the same metric may be used to estimate defect or particle size in a continuous range that encompasses a saturation point. The continuity of the metric permits defect detection over a wide range of saturated and unsaturated response without having to change detection methods. One advantage of this continuity is the avoidance of time consuming and costly recalibration. 
   In the range of clipped response, the disclosed system and method extends the range of the detection equipment by an extension factor. When the intensity responses are subject to being clipped at a selected intensity level, the relationship maps intensity response shape volume values to values identified with an extension factor. The extension factor is a function of the peak of an intensity response to a particle x of known size that would have been reached if the response had not been clipped. The extension factor is also a function of the clipping intensity level, which is the intensity level at which the response is clipped. The relationship of the extension factor is 
   
     
       
         
           
             
               
                 
                   EF 
                   ⁡ 
                   
                     ( 
                     x 
                     ) 
                   
                 
                 = 
                 
                   
                     p 
                     ⁡ 
                     
                       ( 
                       x 
                       ) 
                     
                   
                   C 
                 
               
             
             
               
                 ( 
                 17 
                 ) 
               
             
           
         
       
     
   
   In equation (17), p(x) equals the intensity response to a particle x of known size that would have been reached if it had not been clipped. C is a constant equal to or related to the clipping intensity level for the surface inspection system. In practice, a continuous relationship for inspection of defects with a pulse response below the threshold and with a pulse response above the threshold may be defined. The intensity response shape volume for particle x is thus equal to 
   
     
       
         
           
             
               
                 
                   
                     
                       
                         V 
                         ⁡ 
                         
                           ( 
                           x 
                           ) 
                         
                       
                       = 
                       
                         
                           
                             
                               B 
                               * 
                               
                                 EF 
                                 ⁡ 
                                 
                                   ( 
                                   x 
                                   ) 
                                 
                               
                             
                           
                           
                             
                               
                                 for 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   EF 
                                   ⁡ 
                                   
                                     ( 
                                     x 
                                     ) 
                                   
                                 
                               
                               &lt; 
                               1 
                             
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       = 
                       
                         
                           
                             
                               
                                 A 
                                 * 
                                 
                                   
                                     log 
                                     10 
                                   
                                   ⁡ 
                                   
                                     [ 
                                     
                                       EF 
                                       ⁡ 
                                       
                                         ( 
                                         x 
                                         ) 
                                       
                                     
                                     ] 
                                   
                                 
                               
                               + 
                               B 
                             
                           
                           
                             
                               
                                 for 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   EF 
                                   ⁡ 
                                   
                                     ( 
                                     x 
                                     ) 
                                   
                                 
                               
                               ≥ 
                               1 
                             
                           
                         
                       
                     
                   
                 
               
             
             
               
                 
                   
                     
                       ( 
                       
                         18 
                         ⁢ 
                         a 
                       
                       ) 
                     
                   
                 
                 
                   
                     
                       ( 
                       
                         18 
                         ⁢ 
                         b 
                       
                       ) 
                     
                   
                 
               
             
           
         
       
     
   
   In equations (18a) and (18b), A and B are constants defined by characteristics of the surface inspection system. For EF(x)&lt;1, there is linear relationship between intensity response shape volume values and the extension factor with which the volume values are associated. When EF(x) is greater than or equal to 1, there is a linear relationship between intensity response shape volume values and the log of the extension factor with which they are associated. 
   As an example, if a light intensity response is just clipped, so that p(x)=C, then V(x)=B. Accordingly, the constant B provides a system dependent value that represents the point at which unfiltered light intensity responses begin to saturate. Using the known value for B, the point at which linear or semi-log calculations are applied can be determined. 
   Referring now to  FIG. 9 , graph  90  illustrates the behavior of the volume sizing metric for multiple scanning configurations over a wide range of clipped inputs. As can be seen in graph  90 , the relationship between the volume sizing metric and the extension factor is linear with respect to the log of the extension factor. The extension factor is a multiple of the clipping level that the peak of an unclipped response would have reached if it had not been clipped due to system limitations. The extension factor ranges from 1, which indicates that the response is near the edge of being clipped, to a value of 200, indicating severe clipping. The scanning configurations represent cross-scan pitches of 12 μm, 6 μm, and 3 μm, all of them normal in-scan pitch. There is also the representation of a high-resolution configuration with a cross-scan pitch of 3 μm and an in-scan pitch that is ⅕ the normal in-scan pitch. The resulting plot, exemplified in graph  90  approximates a straight line in the semi-log graph, indicating very small variance for the volume sizing metric with respect to a specific scanning configuration. Indeed, the plots for the different scanning configurations fall nearly on top of each other. Accordingly, calibration of the equipment is simplified due to the fact that calibration need not be changed to obtain an accurate result over a wide range of measurements using the disclosed system and method. 
   Referring to  FIG. 10 , an illustration of background haze on the volume sizing metric is illustrated in a graph  100 . The background haze has an impact that reduces the difference between the haze level and the saturation level. The semi-log graph  100  shows plots for haze levels of 0 mV, 50 mV, and 100 mV. The relationship between the volume sizing metric remains linear in semi-logarithmic plot  100  for each haze level, with the slope and offset varying slightly with the haze. For intermediate haze values, interpolation may be used to establish the relationship between volume sizing metric and extension factor. 
   In addition to haze, spot size of the laser used in the measurement system can have a strong impact on the volume sizing metric. Table 1 below illustrates coefficients that may be used for a linear semi-log relationship between the extension factor and the volume sizing metric for spot sizes that vary from 40 μm to 55 μm, and haze levels that vary from 0 mV to 100 mV. 
   
     
       
             
           
             
             
             
           
             
           
             
             
             
           
             
           
             
             
             
           
             
           
             
             
             
           
             
           
             
             
             
           
             
           
             
             
             
           
             
           
             
             
             
           
             
           
             
             
             
           
             
           
             
             
             
           
             
           
             
             
             
           
             
           
             
             
             
           
             
           
             
             
             
           
             
           
             
             
             
           
             
           
             
             
             
           
             
           
             
             
             
           
             
           
             
             
             
           
             
           
             
             
             
           
         
             
               TABLE 1 
             
           
           
             
                 
             
             
               Volume = A * log10 (Extension Factor) + B 
             
           
        
         
             
                 
               A 
               B 
             
             
                 
                 
             
           
        
         
             
               spotSize = 40 um 
             
           
        
         
             
                0 mv haze: 
               24.3247 
               10.5638 
             
             
               25 mv haze: 
               24.2082 
               10.4830 
             
             
               50 mv haze: 
               24.0921 
               10.3995 
             
             
               75 mv haze: 
               23.9737 
               10.3158 
             
             
               100 mv haze:  
               23.8583 
               10.2254 
             
           
        
         
             
               spotSize = 41 um 
             
           
        
         
             
                0 mv haze: 
               25.5562 
               11.0983 
             
             
               25 mv haze: 
               25.4335 
               11.0144 
             
             
               50 mv haze: 
               25.3123 
               10.9271 
             
             
               75 mv haze: 
               25.1891 
               10.8399 
             
             
               100 mv haze:  
               25.0644 
               10.7513 
             
           
        
         
             
               spotSize = 42 um 
             
           
        
         
             
                0 mv haze: 
               26.8177 
               11.6473 
             
             
               25 mv haze: 
               26.6894 
               11.5599 
             
             
               50 mv haze: 
               26.5606 
               11.4713 
             
             
               75 mv haze: 
               26.4339 
               11.3782 
             
             
               100 mv haze:  
               26.3010 
               11.2882 
             
           
        
         
             
               spotSize = 43 um 
             
           
        
         
             
                0 mv haze: 
               28.1103 
               12.2074 
             
             
               25 mv haze: 
               27.9753 
               12.1169 
             
             
               50 mv haze: 
               27.8412 
               12.0240 
             
             
               75 mv haze: 
               27.7033 
               11.9341 
             
             
               100 mv haze:  
               27.5718 
               11.8323 
             
           
        
         
             
               spotSize = 44 um 
             
           
        
         
             
                0 mv haze: 
               29.4332 
               12.7819 
             
             
               25 mv haze: 
               29.2914 
               12.6894 
             
             
               50 mv haze: 
               29.1505 
               12.5937 
             
             
               75 mv haze: 
               29.0097 
               12.4958 
             
             
               100 mv haze:  
               28.8632 
               12.4009 
             
           
        
         
             
               spotSize = 45 um 
             
           
        
         
             
                0 mv haze: 
               30.7855 
               13.3706 
             
             
               25 mv haze: 
               30.6374 
               13.2739 
             
             
               50 mv haze: 
               30.4897 
               13.1747 
             
             
               75 mv haze: 
               30.3426 
               13.0736 
             
             
               100 mv haze:  
               30.1940 
               12.9720 
             
           
        
         
             
               spotSize = 46 um 
             
           
        
         
             
                0 mv haze: 
               32.1695 
               13.9703 
             
             
               25 mv haze: 
               32.0149 
               13.8701 
             
             
               50 mv haze: 
               31.8609 
               13.7670 
             
             
               75 mv haze: 
               31.7068 
               13.6616 
             
             
               100 mv haze:  
               31.5482 
               13.5619 
             
           
        
         
             
               spotSize = 47 um 
             
           
        
         
             
                0 mv haze: 
               33.5832 
               14.5852 
             
             
               25 mv haze: 
               33.4214 
               14.4826 
             
             
               50 mv haze: 
               33.2592 
               14.3775 
             
             
               75 mv haze: 
               33.0961 
               14.2715 
             
             
               100 mv haze:  
               32.9347 
               14.1641 
             
           
        
         
             
               spotSize = 48 um 
             
           
        
         
             
                0 mv haze: 
               35.0273 
               15.2121 
             
             
               25 mv haze: 
               34.8582 
               15.1059 
             
             
               50 mv haze: 
               34.6890 
               14.9972 
             
             
               75 mv haze: 
               34.5210 
               14.8870 
             
             
               100 mv haze:  
               34.3531 
               14.7734 
             
           
        
         
             
               spotSize = 49 um 
             
           
        
         
             
                0 mv haze: 
               36.5022 
               15.8521 
             
             
               25 mv haze: 
               36.3260 
               15.7426 
             
             
               50 mv haze: 
               36.1504 
               15.6298 
             
             
               75 mv haze: 
               35.9753 
               15.5161 
             
             
               100 mv haze:  
               35.7947 
               15.4050 
             
           
        
         
             
               spotSize = 50 um 
             
           
        
         
             
                0 mv haze: 
               38.0074 
               16.5064 
             
             
               25 mv haze: 
               37.8238 
               16.3937 
             
             
               50 mv haze: 
               37.6411 
               16.2775 
             
             
               75 mv haze: 
               37.4577 
               16.1603 
             
             
               100 mv haze:  
               37.2752 
               16.0397 
             
           
        
         
             
               spotSize = 51 um 
             
           
        
         
             
                0 mv haze: 
               39.5428 
               17.1725 
             
             
               25 mv haze: 
               39.3514 
               17.0559 
             
             
               50 mv haze: 
               39.1618 
               16.9355 
             
             
               75 mv haze: 
               38.9702 
               16.8158 
             
             
               100 mv haze:  
               38.7807 
               16.6925 
             
           
        
         
             
               spotSize = 52 um 
             
           
        
         
             
                0 mv haze: 
               41.1084 
               17.8535 
             
             
               25 mv haze: 
               40.9102 
               17.7325 
             
             
               50 mv haze: 
               40.7102 
               17.6115 
             
             
               75 mv haze: 
               40.5134 
               17.4857 
             
             
               100 mv haze:  
               40.3118 
               17.3659 
             
           
        
         
             
               spotSize = 53 um 
             
           
        
         
             
                0 mv haze: 
               42.7053 
               18.5456 
             
             
               25 mv haze: 
               42.4985 
               18.4223 
             
             
               50 mv haze: 
               42.2927 
               18.2957 
             
             
               75 mv haze: 
               42.0865 
               18.1676 
             
             
               100 mv haze:  
               41.8781 
               18.0413 
             
           
        
         
             
               spotSize = 54 um 
             
           
        
         
             
                0 mv haze: 
               44.3316 
               19.2529 
             
             
               25 mv haze: 
               44.1161 
               19.1266 
             
             
               50 mv haze: 
               43.9032 
               18.9950 
             
             
               75 mv haze: 
               43.6899 
               18.8631 
             
             
               100 mv haze:  
               43.4741 
               18.7312 
             
           
        
         
             
               spotSize = 55 um 
             
           
        
         
             
                0 mv haze: 
               45.9888 
               19.9726 
             
             
               25 mv haze: 
               45.7658 
               19.8417 
             
             
               50 mv haze: 
               45.5424 
               19.7081 
             
             
               75 mv haze: 
               45.3210 
               19.5724 
             
             
               100 mv haze:  
               45.0972 
               19.4377 
             
             
                 
             
           
        
       
     
   
   The volume sizing metric can be extended to extension factors in the range of 0-1 indicated in equations (18a) and (18b) above. In the range of 0-1, there is no non-linearity due to saturation. Accordingly, the relationship in the range on an extension factor between 0 and 1 is linear instead of semi-log. Referring to  FIG. 11 , the linearity of the relationship is illustrated between extension factor values of 0 and 1, while the semi-logarithmic relationship is illustrated in extension factors between 1 and 2. The overlap of the two ranges in graph  110  shows how the disclosed system and method can provide continuity of measurement across saturation or clipping thresholds. Accordingly, the same technique may be used to measure sample particles affected by saturation and sample particles that are not affected by saturation. The continuity provides an advantage for avoiding changing measurement techniques or methods over the course of a scan. Another advantage permitted by the measurement continuity is the simplification of calibration for the equipment. 
   The calculation of the volume sizing metric is used to ultimately size the sample defect or particle. The known relationship between volume and defect or particle size is applied to compare the calculated volume value to volume values obtained from known particle sizes. The relationship may be determined using at least one response curve showing intensity response shape volume values as a function of the extension factor for the particle size. One response curve may be represented in semi-log form to show the linear relationship between intensity response shape volume values and the log of the extension factor. This response curve may be used for extension factors greater than or equal to 1 in accordance with equations (18a) and (18b). 
   Alternately, the relationship may be determined using a table that contains representative intensity response shape volume values and extension factors for known particle sizes. The use of a table containing representative values may be helpful in situations in which it is desirable to obtain measurements that are less computationally intensive. 
   The volume sizing metric provides a simplified and straightforward means for locating and sizing defects on a wafer surface, even when the measuring equipment becomes saturated. The metric permits simple calibration of the equipment while allowing the equipment to operate in both saturated and unsaturated ranges of operation without recalibration. The disclosed system and method obtain very good correlation with actual defect sizes and consistency among defect measurements. 
   It will be appreciated by those of ordinary skill in the art that further modifications to and variations of the above-described extended defect sizing technique may be made without departing from the inventive concepts disclosed herein. Accordingly, the invention should not be viewed as limited except as by the scope and spirit of the appended claims.