Abstract:
A method for inspecting a sample, consisting of receiving a definition of image attributes that are characteristic of defects, and processing an image of the sample so as to identify candidate defects on the sample. The method further includes forming distributions of values of the respective attributes from the candidate defects, and selecting a set of the candidate defects that are characterized by respective candidate attribute values that fall in one or more tails of the distributions. The selected set is presented to a human operator, and respective classifications of the candidate defects in the selected set are received from the operator. A definition of the one or more tails of the distributions is refined responsively to the classifications. The method may be used as a filter to remove false alarms, or nuisances. The method may also be used to categorize the candidate defects into two or more classes.

Description:
CROSS-REFERENCE TO RELATED APPLICATION  
       [0001]     This application claims the benefit of U.S. Provisional Patent Application 60/631,912, filed Nov. 29, 2004, which is incorporated herein by reference. 
     
    
     FIELD OF THE INVENTION  
       [0002]     The present invention relates generally to semiconductor wafer fabrication, and specifically to detecting defects in the fabricated wafer.  
       BACKGROUND OF THE INVENTION  
       [0003]     An integral part of semiconductor wafer fabrication is detection of defects that lead to reduced performance of the die where the defect is located. A number of methods for performing such detection are known in the art. The methods usually include optical and/or charged particle scanning of the wafer, and analysis of the scanned image. One of the methods for detecting defects uses comparison of the image with other images, typically on a die-to-die basis and/or on a wafer-to-wafer basis, so that regions of the wafer which may have defects can be identified. Other methods are known in the art.  
         [0004]     One of the problems of defect identification is that identified defects may in fact not lead to reduced performance of the die. For example, the existence of metal grains in the die, of a rough edge on a conductor, or of anomalies under a scanned layer, typically do not reduce performance. It is thus useful to classify defects, and to use the classification to reduce the number of defects which are considered problematic.  
         [0005]     U.S. Pat. No. 5,966,459 to Chen, et al., whose disclosure is incorporated herein by reference, describes a method for determining classification codes for semiconductor wafer defects, and for storing the information used to determine the classification codes. A wafer is scanned after a first and subsequent manufacturing processes. After each scan images of selected defects of the wafer are examined, and are assigned a code. The code is modified according to the results of the subsequent scans.  
         [0006]     U.S. Pat. No. 5,978,501 to Badger, et al., whose disclosure is incorporated herein by reference, describes a system for detecting defects in the design of a photolithographic mask or of a semiconductor wafer. The system derives an adaptive inspection algorithm that is claimed to allow for a tighter inspection of a mask or a wafer to a data set which has repeatable differences. The inspection is also claimed to allow flexibility in removal of unimportant differences while maintaining a tight inspection capability.  
         [0007]     U.S. Pat. No. 6,483,938 to Hennessey, et al., whose disclosure is incorporated herein by reference, describes a system for generating a knowledge base for use in labeling anomalies on a manufactured object. A pixel-based representation of an image having an anomaly is decomposed into primitives. The anomaly is isolated, and is compared with primitives of known anomalies to locate the closest primitive set. A label of the set is presented to an operator using the system.  
         [0008]     U.S. Pat. No. 6,487,307 to Hennessey, et al., whose disclosure is incorporated herein by reference, describes a system for optically inspecting structures on an object on a moving platform. Structure edges within the object are delineated, and a sequence of images of the object are captured. The structure is detected in each image, and a histogram is produced for each image identifying the slope and length of each edge of the structure. The histograms are used to reduce differences between images and are claimed to be able to detect foreign objects and other defects in the object.  
         [0009]     U.S. Pat. No. 6,701,004 to Shykind, et al., whose disclosure is incorporated herein by reference, describes a method for detecting defects on a photomask by patterning alternating dice on a wafer with different process conditions. The different conditions, such as a length of exposure time and an optical focus condition, are configured to highlight and detect defect areas.  
         [0010]     U.S. Patent Application 2004/0028276 to Okuda, et al., whose disclosure is incorporated herein by reference, describes an automatic defect classifying system. A user defines a classifying class arrangement by combining classes supplied by the system itself or classes defined by the user. The user also provides the system with a priori knowledge on the defect class, the knowledge being used as a restriction so as to carry out restricted learning.  
       SUMMARY OF THE INVENTION  
       [0011]     In embodiments of the present invention, a wafer inspection system generates a set of defects of the wafer, and the system performs an initial filtration of the defects to generate candidate defects for further analysis. The inspection system generates values of image attributes for each of the candidate defects, the image attributes typically comprising expressions that are functions of measurements of the candidate defects made by the inspection system. The image attributes may be generated from one or more signals from the candidate defects.  
         [0012]     The inspection system forms distributions of each of the image attributes of the candidate defects, and selects candidate defects that are in tails of the distributions. The tails are defined by threshold values for the attribute distributions. The candidate defects that are in the tails are presented to a human operator in the form of a display on a monitor, and the operator classifies the presented defects as true defects or as false alarms, also herein termed nuisances.  
         [0013]     The inspection system uses the classifications of the operator to refine the values of the thresholds of the attribute distributions, typically in an iterative manner, until a required false alarm rate (FAR), also herein termed a nuisance rate, of classified defects is achieved by the threshold values.  
         [0014]     In an embodiment of the invention, a further modification of the threshold values is performed by analyzing correlations between the attribute values of defects lying in attribute tails.  
         [0015]     The present invention will be more fully understood from the following detailed description of the embodiments thereof, taken together with the drawings, a brief description of which follows.  
     
    
     BRIEF DESCRIPTION OF THE DRAWINGS  
       [0016]      FIG. 1  is a schematic illustration of a wafer inspection apparatus, according to an embodiment of the present invention;  
         [0017]      FIG. 2  is a flowchart showing an overall inspection process performed by the wafer inspection apparatus, according to an embodiment of the present invention;  
         [0018]      FIGS. 3A and 3B  are schematic histograms of a generic attribute of defects, according to an embodiment of the present invention;  
         [0019]      FIG. 4  is a schematic 2-dimensional graph of a first generic attribute and a second generic attribute of defects, according to an embodiment of the present invention; and  
         [0020]      FIG. 5  is a flowchart showing steps performed in a directed review of defects, according to an embodiment of the present invention.  
     
    
     DETAILED DESCRIPTION OF EMBODIMENTS  
       [0021]     Reference is now made to  FIG. 1 , which is a schematic illustration of a wafer inspection apparatus  10 , according to an embodiment of the present invention. Apparatus  10  is used to inspect a surface  25  of a wafer  26 , typically a semiconductor wafer, during one or more stages of fabrication of the wafer. Herein, by way of example, apparatus  10  is assumed to use an optical inspection scanning system  13 , but those having ordinary skill in the art will be able, mutatis mutandis, to adapt the following description to accommodate other inspection systems or combinations of inspection systems. Such systems include, but are not limited to, inspection of wafer  26  by substantially any type of electromagnetic radiation and/or charged particle scanning.  
         [0022]     Apparatus  10  comprises a source  12 , typically a laser, which radiates an optical beam  71  onto surface  25 . Reflected and/or scattered light that results from the irradiation is detected in a number of detectors  11 . Herein detectors  11  are assumed to comprise a bright field detector  36 , which receives specular reflection from surface  25  via a partially reflecting element  18 . Detectors  11  also comprise four generally similar gray field (GF) detectors  45  (for clarity, only two of the GF detectors are shown in  FIG. 1 ), which receive near field scattered radiation from surface  25  via a mirror  46  and a partially reflecting element  53 . Mirror  46  has a hole  47  in its center to allow beam  71  to have unimpeded access to surface  25 , as well as to allow specular reflection of the beam to be received by detector  36 . The four detectors  45  are arranged substantially symmetrically around beam  71 , so that each detector receives radiation scattered into different but approximately equal solid angles with respect to surface  25 . A fifth gray field detector  59 , herein termed the gray field union (GFU) detector, receives scattered radiation, via element  53 , corresponding to a sum of the individual radiations received by detectors  45 .  
         [0023]     Detectors  11  also comprise a number, typically four, of dark field detectors  49  which receive far field scattered radiation from surface  25 . For clarity, only one dark field detector is shown in  FIG. 1 .  
         [0024]     Apparatus  10  further comprises processing system  60  and a computing system  66  including a monitor  64 . Processing system  60  comprises two image processors (IPs)  61  and  63 , also referred to herein as IP 1  and IP 2  respectively, and a post-processor  62 . As shown in  FIG. 1 , processing system  60  receives respective signals from each of detectors  11 . The functions of systems  60  and  66  are described in more detail below with respect to  FIG. 2 .  
         [0025]     Apparatus  10  inspects surface  25  in order to locate defects on or close to the surface. By way of example, the inspection process is assumed to comprise a die-to-die comparison, although other inspection processes known in the art, such as a wafer-to-wafer comparison and/or comparison with results from a database, may also be used. The defects that are identified by the inspection process, prior to any filtering, typically comprise large numbers of defects that have little or no effect on the performance of the die to which the defect belongs, these type of defects being non-yield-limiting defects. Typical non-yield-limiting defects include metal grains and irregular edges of conductors in a die. Embodiments of the present invention enable the defects to be filtered so that only those defects with a high probability of limiting the yield are identified; these defects are herein termed true defects. Other defects, also herein termed false alarms, nuisance defects, or nuisances, are filtered out.  
         [0026]     Apparatus  10  is typically operated in an initial “learning” phase, and subsequently in a production phase. In the learning phase, surface  25  of wafer  26  is inspected, and a human operator  90  of apparatus  10  interacts with the apparatus, using results generated by the apparatus, to iteratively decide values of the results that have a high probability of being generated by yield limiting defects. In the production phase, surfaces of other wafers are inspected, these wafers being in substantially the same phase of fabrication as already-inspected wafer  26 . The production phase inspection uses the learning phase results derived from the inspection of wafer  26 . As a consequence, during the production phase the values determined in the learning phase enable apparatus  10  to automatically filter out nuisance defects.  
         [0027]      FIG. 2  is a flowchart showing an overall inspection process  200  performed by apparatus  10 , according to an embodiment of the present invention. Process  200  includes the learning phase referred to above. In a first step  202 , typically performed as beam  71  is scanned across surface  25 , data from detectors  11  is passed to image processors  61  and  63 . For each detector  11 , processors  61  and  63  compare the level from the detector with the level from the same detector for corresponding pixels of neighboring dies. From the comparisons generated processors  61  and  63  generate an initial set of suspicious pixels; suspicious pixels are also referred to herein as alarms.  
         [0028]     In a second step  204 , typically performed in post-processor  62  after surface  25  has been completely scanned, the alarms are grouped into clusters, and the clusters are classified as defects. Typically one cluster represents one defect. Post-processor  62  performs further filtering of each of the defects according to the properties of each defect, where the properties comprise the number of alarms, also herein termed the volume, in the cluster, and its grade, which is a function of individual alarm grades of each alarm. Defects clustered by the post-processor and not filtered are displayed as a candidate defect map on monitor  64 .  
         [0029]     In a third step  206 , operator  90  may use the displayed map of candidate defects produced in step  204  to adjust threshold values of parameters, herein termed attributes, associated with the candidate defects so that some of these defects are classified as nuisances. The initial threshold values are generated automatically by post-processor  62 . The attributes are derived from the volume and grade of the defect, as well as from additional parameters associated with the defects. The attributes, and the setting of the threshold values of the attributes, are described in more detail below with reference to  FIGS. 4 and 5 ; the setting of the threshold values of the attributes comprises the learning phase performed by apparatus  10 . When operator  90  has verified that the threshold values of the attributes display substantially only true defects on monitor  64 , process  200  concludes.  
         [0030]     It will be appreciated that the map of candidate defects generated after step  204 , and displayed on monitor  64 , may typically comprise an extremely large proportion of nuisance defects. The learning phase provided by step  206  reduces the numbers of nuisance defects displayed, using assumptions that are described below with reference to  FIGS. 3A, 3B , and  4 .  
         [0031]      FIG. 3A  is a schematic unimodal histogram  250  of an attribute of defects and  FIG. 3B  is a schematic multimodal histogram  240  of an attribute of defects, according to an embodiment of the present invention. A disclosed embodiment of the present invention uses twelve attributes, which are further described below with reference to Table I. The description herein, for  FIGS. 3A and 3B , refers to one generic attribute. Histograms  240  and  250  show the number of defects vs. the attribute value for the defects, where both the number and the attribute values are assumed for the purposes of this explanation to have arbitrary values. Typically, attribute distributions are similar to histogram  250 , which has a unimodal distribution.  
         [0032]     In some cases the attribute distributions may not be unimodal, for example, if two classes of defects are distinct in attribute space but yet arise from similar population sizes. In such cases, a histogram  240  of the attribute may have a bimodal shape, where a class A of attribute values cluster around value a low value  242  and a class B of attribute values cluster around a high value  244 .  
         [0033]     The inventor has found that, in attribute space, there are one or more sections of any attribute histogram in which there is a preponderance of the nuisances. For histogram  250 , for example, the one or more sections may comprise all or part of a central region  253 , and/or all or part of a region outside the central region. For histogram  240 , the one or more sections may comprise regions around values  242  and  244 . Outside these one or more sections, there is a preponderance of true defects. Thus, for histogram  240 , a preponderance of true defects are in a section  246  in a range between values  242  and  244 .  
         [0034]     Hereinbelow, by way of example, the one or more sections having a preponderance of nuisance values are assumed to comprise central region  253  of histogram  250 , which in turn defines histogram tail regions  256  and  258 . The regions are delineated by a low threshold  252  and a high threshold  254 . Consequently, below the low threshold and above the high threshold, i.e., in histogram tail regions  256  and  258 , there is a preponderance of true defects.  
         [0035]      FIG. 4  is a schematic 2-dimensional graph  260  of a first generic attribute and a second generic attribute of defects, according to an embodiment of the present invention. Each generic attribute is assumed to behave approximately as described above with reference to  FIG. 3A , so that each attribute has a respective low threshold  262 ,  264 , and a respective high threshold  266 ,  268 . In a region  270 , i.e., within a box  272  defined by the thresholds, defects will comprise a preponderance of nuisance defects, and defects outside the box will comprise a preponderance of true defects. The property exemplified by 2-dimensional graph  260  is amplified herein to apply to an N-dimensional graph of attributes, where N is any natural number.  
         [0036]     It will be understood that each threshold value referred to in reference to  FIGS. 3A and 4  may typically be chosen from a range of values. A threshold value that is set closer to a mean value of the attribute makes the tail larger; a value set farther from the mean value makes the tail smaller. Typically, as the size of the tail reduces, there is a corresponding increase in the ratio of true defects to nuisances in the tail; however, the reduction in tail size increases the number of true defects that are not in the tail. Embodiments of the present invention select the threshold values so as to optimize the numbers of true defects in the tails of the attribute distribution histograms.  
         [0037]     Referring back to step  206  of  FIG. 2 , operator  90  adjusts the threshold values of each of the attributes to optimize the number of true defects in regions equivalent the region outside box  272  ( FIG. 4 ), while maintaining the number of nuisances as low as possible.  
         [0038]     Table I below lists the twelve attributes Att n  calculated for each defect by the disclosed embodiment referred to above, where 1≦n≦12, nε1. The symbols used in Table I are defined and explained in Table II.  
                   TABLE I                       ATTRIBUTE           SYMBOL   DEFINITION                               Att 1     Log 10 V               Att 2               1     4   ⁢     N   DF         ⁢       ∑   iεDFalarms     ⁢       ∑     j   =   1     4     ⁢     ΔGL   i     (   j   )                               Att 3                          ∑       iεDF   +   ve     ⁢          ⁢   alarms       ⁢       ∑     j   =   1     4     ⁢       Q   j     ·     ΔGL   i     (   j   )                    1   /   2       /     1   4       ⁢       ∑       iεDF   +   ve     ⁢          ⁢   alarms       ⁢       ∑     j   =   1     4     ⁢     ΔGL   i     (   j   )                               Att 4               1     N   GFU       ⁢       ∑   iεGFUalarms     ⁢     ΔGL   i     (   5   )                             Att 5               1     4   ⁢     N   GF         ⁢       ∑   iεGFalarms     ⁢       ∑     j   =   1     4     ⁢     ΔGL   i     (     j   +   5     )                               Att 6                          ∑       iεGF   +   ve     ⁢          ⁢   alarms       ⁢       ∑     j   =   1     4     ⁢       Q   j     ·     ΔGL   i     (     j   +   5     )                    1   /   2       /     1   4       ⁢       ∑       iεGF   +   ve     ⁢          ⁢   alarms       ⁢       ∑     j   =   1     4     ⁢     ΔGL   i     (     j   +   5     )                               Att 7               1     4   ⁢     N   1         ⁢       ∑   iεalarms     ⁢     (       ∑     j   =   1     4     ⁢       GL   i     (   j   )       ⁡     (   ref   )         )                           Att 8               1     N   1       ⁢       ∑   iεalarms     ⁢       GL   i     (   5   )       ⁡     (   ref   )                             Att 9               1     4   ·     N   2         ⁢       ∑   iεalarms     ⁢       ∑     j   =   6     9     ⁢       GL   i     (   j   )       ⁡     (   ref   )                               Att 10               1     N   2       ⁢       ∑   iεalarms     ⁢       GL   i     (   10   )       ⁡     (   ref   )                             Att 11                 1     4   ⁢     N     G   ⁢   F           ⁢       ∑     iεGF   ⁢          ⁢   alarms       ⁢       ∑     j   =   1     4     ⁢     G   i     (     j   +   5     )             -       1     4   ⁢     N   DF         ⁢       ∑     iεDF   ⁢          ⁢   alarms       ⁢       ∑     j   =   1     4     ⁢     G   i     (   j   )                                 Att 12               1     N     G   ⁢   FU         ⁢       ∑     iεBF   ⁢          ⁢   alarms       ⁢     ΔGL   i     (   10   )                                
 
         [0039]     In Table I, vertical brackets represent the determinant of the corresponding matrix.  
                   TABLE II                       Notation   Definition                   V   Number of pixels in a defect       N 1     Number of alarms detected by image processor IP1 in a defect       N 2     Number of alarms detected by image processor IP2 in a defect       N DF     Number of dark field (DF) alarms in a defect       N GFU     Number of gray field union (GFU) alarms in a defect       N GF     Number of gray field (GF) alarms in a defect       N BF     Number of bright field (BF) alarms in a defect       G i   (j)     Grade of pixel i in detector j       GL i   (j)  (ref)   Reference gray level of detector j in pixel i       ΔGL i   (j)     Inspected gray level - Reference gray level of detector j in           pixel i.               Q j                               ⁢       [         0         2   ⁢     x   j     ⁢     y   j                 2   ⁢     x   j     ⁢     y   j           0         ]     ,     where   ⁢           ⁢     (       x   j     ,     y   j       )     ⁢           ⁢     is the relative spatial                         ⁢     coordinate of the j&#39;th gray level (GL) detector.     ⁢                                 ⁢       j   ⁢           ⁢   ɛ   ⁢           ⁢     {     1   ,   2   ,   3   ,   4     }       ,       x   j     ⁢           ⁢   ɛ   ⁢           ⁢     {     1   ,     -   1       }       ,       y   j     ⁢           ⁢   ɛ   ⁢           ⁢       {     1   ,     -   1       }     .                                            
 
         [0040]     In addition to the twelve attributes shown in Table I, an additional attribute, herein termed the die participation attribute (Att DP ), is calculated for each defect. A value assigned to Att DP  is a value representing the number of other dies for which a defect with corresponding or neighboring die coordinates has been found. The inventor has found that defects with large values of Att DP  broadly correspond to nuisance defects, and conversely, defects with small values of Att DP  broadly correspond to true defects.  
         [0041]      FIG. 5  is a flowchart  300  showing steps performed in a directed review of defects, according to an embodiment of the present invention. Flowchart  300  corresponds to the learning phase described above in reference to step  206  ( FIG. 2 ). In performing the steps of the flowchart, apparatus  10  automatically generates initial thresholds of the attributes listed in Table I having lowest mean values of Att DP . These thresholds are then adjusted to form final threshold values for the attributes. Flowchart  300  is divided into two stages, a first uni-dimensional stage  301  wherein each of the attribute distributions are analyzed substantially independently, and a second multi-dimensional stage  303  wherein correlations between the distributions are taken into account.  
         [0042]     In a first step  302  of stage  301 , operator  90  inputs a value of a prescribed false alarm rate (FAR) that apparatus  10  is to achieve. A typical value for FAR is 0.05, although any other suitable value may be used. Also in the first step, for each attribute post-processor  62  forms a distribution, also herein termed a histogram, of all candidate defects having the attribute. The candidate defects correspond to the defects, typically comprising a large proportion of nuisances, displayed at the conclusion of step  204  ( FIG. 2 ). In an embodiment of the present invention, each histogram is formed to have dynamic bin sizes, the number of bins n being given by expression (1):  
             n   =     ⌈         N   -     N   miss       N     ·     100   bin_percentile       ⌉             (   1   )             
 
         [0043]     where 
        N is the total number of defects having the attribute;     N miss  is the number of defects without the attribute; and     bin_percentile is a constant between 1 and 100, a typical value for the constant being of the order of 7.        
 
         [0047]     For example, assuming that bin_percentile=5, an attribute that is defined for all defects, so that N miss =0, is sorted into 20 bins; an attribute that is defined for 7% of the defects, so that  
             N   -     N   miss       N     =   0.07     ,       
 
 is sorted into two bins, and an attribute defined for less than 5% of the defects, so that  
             N   -     N   miss       N     &lt;   0.05     ,       
 
 is placed in one bin. 
 
         [0048]     The bin borders [x (1) , x (2) , x (3) , . . . , x (n) , x (n+1) ] are given by expression (2):  
             {               ⁢       x     (   1   )       =     x   min                       ⁢       x     (   i   )       =       x   ⁡     (       100   ⁢     (     i   -   1     )       n     )       ⁢     ∀     1   &lt;   i   ≤   n                           ⁢       x     (     n   +   1     )       =     x   max                       (   2   )             
 
         [0049]     where 
        x min  is the minimum value of the attribute;     x max  is the maximum value of the attribute; and     x(p) is the p&#39;th percentile of the attribute x.        
 
         [0053]     In a second step  304 , tails of each of the histograms are analyzed. For each tail, a mean value {overscore (Att DP )} of the die participation attribute of all the defects in the tail is calculated, and the K chosen  tails having the lowest values of {overscore (Att DP )} are reviewed further. K chosen  is any number greater than zero; a typical value for K chosen  is 5.  
         [0054]     For each histogram, each tail is assumed to comprise m extreme bins of the histogram, where m is defined according to expression (3):  
             m   =     ⌈       N   rev       N   class       ⌉             (   3   )             
 
         [0055]     where 
        N rev  is a number, greater than 1, of the total number of defects chosen for review for this attribute. A typical value for N rev  is 8; and     N class  is a number, greater than 1, of classified defects per bin. A typical value for N class  is 6. N class  represents a minimum number of defects of a bin to be reviewed.        
 
         [0058]     The steps following step  304  are typically iterated, as is explained in more detail below.  
         [0059]     In a third step  306 , in a first iteration, each tail selected according to step  304  has a subset of N rev  defects chosen from the bins of its tail. Thus, using the typical values given above with respect to equation (3), in the first iteration six defects from the most extreme bin and two defects from the next-most extreme bin are reviewed. In any subsequent iteration, rather than opening m bins from the extremities of the attributes, bins within the threshold values determined in step  312  below, and that do not have N class  classified defects, are opened so as to provide further subsets of defects for classification. Typically, for a given iteration sampling a low tail, the number of bins opened is bounded from above by the number of bins up to the current low threshold. A generally similar limitation applies to sampling the high tail.  
         [0060]     In a fourth step  308 , the defects to be reviewed are selected according to a ranking system that post-processor  62  applies to the defects. Defects that are as yet unclassified (according to step  310  below) are assigned a numerical rank equal to the number of tail bins, of the different attributes, they fall into. The tail bins are those bins selected in step  306  above. Defects with higher numerical ranks are selected first. It will be understood that the rank assigned to a particular defect may change as the process of flowchart  300  iterates, since the tail bins upon which the ranking depends may change according to step  306 .  
         [0061]     In a fifth step  310 , the defects selected for review in step  308  are displayed on monitor  64  for review by operator  90 . The operator classifies the displayed defects as true, nuisance, or unknown, and post-processor  62  uses the classification in subsequent steps of flowchart  300 . Alternatively or additionally, locations of the selected defects may be sent to another tool, for example an electron microscope, so that the defects may be reviewed using the tool. This method is advantageous for defects that may be too small to be manually classified using optical methods.  
         [0062]     In a sixth step  312 , after operator  90  has performed the classification of all the chosen tails, post-processor  62  calculates a lower threshold value T lo  and an upper threshold value T hi  for each of the attributes. The calculation is based on the prescribed false alarm rate (FAR) input in step  302 .  
         [0063]     T lo  is the maximum value of T for which the following criterion is correct:  
                   N   f     ⁡     (   T   )             N   f     ⁡     (   T   )       +       N   t     ⁡     (   T   )           &lt;     FAR     (     1   ⁢   D     )               (   4   )             
 
         [0064]     where 
        N f (T) is the number of classified nuisances with an attribute value &lt;T;     N t (T) is the number of classified true defects with an attribute value &lt;T; and     FAR (1D)  is a number less than 1, corresponding to a multiple of FAR. A typical value for FAR (1D)  is 3×FAR.        
 
         [0068]     T hi  is the minimum value of T for which the criterion of expression (4) is correct, with the following redefined variables: 
        N f (T) is the number of classified nuisances with an attribute value &gt;T; and     N t (T) is the number of classified true defects with an attribute value &gt;T.        
 
         [0071]     In a convergence step  314 , which is the last step in stage  301 , post-processor  62  checks to see that numbers of defects that have been reviewed are sufficient, in other words, that at least a minimum number N class  of defects have been reviewed in each tail bin. To perform this check, in a disclosed embodiment, in each of the K chosen  chosen tails, bins of each of the respective histograms are checked according to the following criteria: 
 
 For the low tail T lo :  
               N   rev     (   b   )       ≥       N   class     ⁢     ∀     b   ≤     b   0                   (   5   )             
 
 For the high tail T hi :  
               N   rev     (   b   )       ≥       N   class     ⁢     ∀     b   ≥     b   0                   (   6   )             
 
         [0072]     where 
        N rev   (b)  is the total number of reviewed and classified defects in bin b, and     b 0  is the bin which includes T.        
 
         [0075]     Post-processor  62  checks that expressions (5) and (6) are satisfied for all chosen tails. If the expressions are satisfied, the post-processor continues to multidimensional stage  303 . If the expressions are not all satisfied, post-processor  62  suggests to operator  90  that an additional iteration of directed review, comprising a repetition of steps  306 - 314 , be performed.  
         [0076]     The T lo  and T hi  levels determined in stage  301  may be visualized as forming an N-dimensional box, where in the example described herein N is 12. The N-dimensional box corresponds to 2-dimensional box  272  described in reference to  FIG. 4 . Within the N-dimensional box there are a preponderance of nuisance defects; outside the box there are a preponderance of true defects. The N-dimensional box is herein also referred to as the first-stage box.  
         [0077]     In multidimensional stage  303 , post-processor  62  adjusts the threshold values calculated in uni-dimensional stage  301 , by taking account of correlations between the attributes for specific defects. The adjustments correspond to expanding the N-dimensional first-stage box. The inputs for stage  303  comprise the set of thresholds {T lo , T hi } determined in stage  301 , and the manually classified defects that were so classified in that stage, but that lie outside the N-dimensional first-stage box. The adjustment is typically uni-directional, i.e., each T lo  is only lowered, and each T hi  is only raised, so that the size of a respective tail is reduced. The adjustment thus corresponds to expanding the first-stage box, and it is performed substantially automatically by post-processor  62 .  
         [0078]     In a first step  316  of stage  303 , post-processor  62  considers each of the false alarm defects individually lying outside the N-dimensional first-stage box formed in stage  301 . For each of these false alarms, post-processor  62  forms the smallest new box that encloses the first-stage box and the specific false alarm. The sides of each new box are given by expression (7):  
                 T   ^     lo     (   i   )       =       min   ⁢           ⁢     (       T   lo     (   i   )       ,     att   i     (   fa   )         )     ⁢           ⁢   and   ⁢           ⁢       T   ^     hi     (   i   )         =     max   ⁢           ⁢     (       T   hi     (   i   )       ,     att   i     (   fa   )         )     ⁢     ∀   i                 (   7   )             
 
         [0079]     where 
        {circumflex over (T)} lo   (i) ,{circumflex over (T)} hi   (i)  are the lower and upper threshold boundaries of the new box,     T lo   (i) ,T hi   (i)  are lower and upper threshold boundaries of the first-stage box, and att i   (fa)  is the value of the attribute of the specific false alarm, for the i th  attribute.        
 
         [0082]     In a step  318 , for each new box generated in step  316 , post-processor  62  calculates the number of true defects, ΔN T , which are in the new box but not in the corresponding first-stage box, using expression (8):  
                 T   ^     lo     (   i   )       ≤     att   i     (   true   )       &lt;       T   lo     (   i   )       ⁢           ⁢   or   ⁢           ⁢     T   hi     (   i   )         &lt;     att   i     (   true   )       ≤         T   ^     hi     (   i   )       ⁢           ⁢   for   ⁢           ⁢   any   ⁢           ⁢   i             (   8   )             
        where     att i   (true)  is the value of the attribute of the true defect for the i th  attribute.        
 
         [0085]     In a step  320 , post-processor  62  chooses the new box having the minimum value of ΔN T  as the box to be used in filtering nuisances, this box herein being termed the bounding box. The false alarm rate of the bounding box, FAR bb , is given by expression (9):  
               FAR   bb     =         N   FA     ⁡     (   T   )             N   FA     ⁡     (   T   )       +       N   T     ⁡     (   T   )                   (   9   )             
        where T represents the set of thresholds for the bounding box, N FA (T) is the number of false alarms determined by the box, i.e., outside the box, and N T (T) is the number of true defects determined by the box.        
 
         [0087]     In a decision step  322 , post-processor checks to see that the number of false alarms outside the bounding box is sufficiently small, using expression (10):  
                 N   FA     ⁡     (   T   )       &lt;     max   (       N   FA     (   max   )       ,       FAR     (   recipe   )       ·     (         N   T     ⁡     (   T   )       +       N   FA     ⁡     (   T   )         )                   (   10   )             
 
         [0088]     where 
        N FA   (max)  is a number, ≧0, which ensures that flowchart  300  gives satisfactory results when small numbers of defects are reviewed. A typical value of N FA   (max)  is 4.     FAR (recipe)  is the prescribed false alarm rate that is to be produced by flowchart  300 .        
 
         [0091]     If expression (10) is not satisfied, steps  316 ,  318 , and  320  of the multidimensional stage are reiterated.  
         [0092]     Satisfaction of expression (10) concludes the multidimensional stage, giving a final bounding box having a set of thresholds T={T fbb }.  
         [0093]     In optional steps  324 , post-processor  62  may display on monitor  64  a map of the true defects determined by the final bounding box, and {T fbb } may be approved or rejected by operator  90  on the basis of the display. If the display is approved, {T fbb } is used in analysis of production wafers in substantially the same stage of fabrication as wafer  22 ; if the display is rejected, process  300  may be repeated using a different value of FAR as the false alarm rate to be achieved by the process.  
         [0094]     While the process described above has assumed, by way of example, that only one wafer is used to determine {T fbb }, it will be appreciated that more than one wafer may be used, such wafers typically being in substantially the same stages of fabrication. Alternatively or additionally, the same wafer may be re-analyzed to give an additional set of distributions. Typically, the multiple sets of results derived by either or both these methods may be applied to find {T fbb } in a cumulative and/or an iterative fashion. It will also be appreciated that such a cumulative application may also be used to add, delete, or alter attributes from those used in an initial analysis of a wafer.  
         [0095]     In one embodiment of the present invention, modifications to {T fbb } are performed after two wafers in substantially the same stage of fabrication are analyzed, when there has been a process variation between the two wafers. In this case, typically the uni-dimensional and multidimensional stages of flowchart  300  are applied to the first wafer, but only the multidimensional stage of flowchart  300  is applied to the second wafer.  
         [0096]     It will be understood that the number of attributes used in embodiments of the present invention is not limited to a specific number such as the twelve attributes of the embodiment described above. Substantially any convenient number of attributes may be used, the number typically being chosen according to the detectors and/or the configuration of apparatus  10 . It will also be understood that the attributes described herein are exemplary, and that other attributes, generally similar to those described herein and enabling false alarms to be distinguished from true defects, may be used in embodiments of the present invention.  
         [0097]     It will further be understood that the scope of the present invention includes distinguishing between classes of defects, such as those exemplified in the Background of the Invention, as well as distinguishing false alarms from true defects. Distinguishing between classes of defects may be accomplished by selection of relevant attributes, and using these attributes in the processes described above with respect to  FIGS. 2 and 5 .  
         [0098]     It will also be understood that the detectors from which the attributes are generated may comprise substantially any type of detector or detecting system that generates a signal in response to scanning of wafer  26 . For example, a detecting system for an electromagnetic radiation scanning system may comprise detectors that are at least partly based on far field scattered radiation from wafer  26 , and/or on characteristics of returning radiation such as wavelength, amplitude, phase and/or polarization of the radiation. For scanning systems that comprise charged particle scanning, detectors based on substantially any measurable parameter of charged particle returning from wafer  26  may be used. Such parameters include, but are not limited to, the number, velocity and/or the energy of the charged particles.  
         [0099]     It will thus be appreciated that the embodiments described above are cited by way of example, and that the present invention is not limited to what has been particularly shown and described hereinabove. Rather, the scope of the present invention includes both combinations and subcombinations of the various features described hereinabove, as well as variations and modifications thereof which would occur to persons skilled in the art upon reading the foregoing description and which are not disclosed in the prior art.