Abstract:
An exemplary embodiment of the invention is a method and system for determining a radius of curvature of a two-dimensional curved feature. The system includes an image acquisition device for obtaining images of said curved feature. A processor is coupled to the image acquisition device for receiving the images and converting the images to n sets of coordinates corresponding to points on the perimeter of the curved feature. The processor chooses at least three sets of said coordinates to define at least one group and fits each set of said coordinates from each group to an equation for a circle and determines a radius of curvature by solving each equation simultaneously. A storage device is coupled to the processor for storing processor data. An output device is coupled to the processor for outputting processor data.

Description:
BACKGROUND OF THE INVENTION 
     The invention relates generally to semiconductor fabrication and, more specifically, to a method and system for performing radius of curvature measurements on a two-dimensional curved feature to accurately determine the feature&#39;s diameter and circularity. Critical dimension scanning electron microscopes “CD-SEM” are often used to perform measurements on semiconductor features during fabrication (e.g. contact hole diameter, line widths, etc.). Numerous scans, typically 32 scans or 64 scans, of the feature are performed in the x direction and/or in the y direction. For contact holes, the largest x measurement and the largest y measurement are recorded as the assumed diameter. Comparison of these two assumed diameter measurements gives a limited indication of circularity. 
     For example, some CD-SEMs can measure the diameter of contact holes in both the y and x directions. The measurements represent the maximum space length of the series of scans that are rastored over the contact hole. This technique is suitable for contact holes that are highly circular and exhibit little edge roughness. A more accurate measurement is obtained by performing scans in the vertical and horizontal directions and averaging the diameters. The ratio of the two numbers can also give information about the circularity of the feature. However, the results are often inaccurate given a feature that is highly non-circular. 
     Due to typical irregularities on the edge of a feature such as a contact hole in a semiconductor device, this numerous scan approach often results in erroneous measurements. Further, numerous scans cause charging and/or contamination buildup on a feature&#39;s edge. This may cause the apparent feature size to change. Also, the practice of using the largest measurement as the assumed diameter will give an erroneous result if a feature is a non-circular contact hole. A more accurate method is desired. 
     BRIEF SUMMARY OF THE INVENTION 
     An exemplary embodiment of the invention is a method and system for determining a radius of curvature of a two-dimensional curved feature. The system includes an image/scan acquisition device for obtaining images/scans of said curved feature. A processor is coupled to the image/scan acquisition device for receiving the images and converting the images to n sets of coordinates corresponding to points on the perimeter of the curved feature. The processor chooses at least three sets of said coordinates to define at least one group and fits each set of said coordinates from each group to an equation for a circle and determines a radius of curvature by solving each equation simultaneously. A storage device is coupled to the processor for storing processor data. An output device is coupled to the processor for outputting processor data. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     Referring now to the drawings, wherein like elements are numbered alike in several FIGURES: 
     FIG. 1 is a block diagram of a system for determining a radius of curvature, diameter and circularity of a two-dimensional curved feature in one embodiment of the invention. 
     FIG. 2 illustrates an exemplary procedure for determining the radius of curvature, diameter and circularity of a two-dimensional curved feature. 
     FIG. 3 depicts an exemplary schematic representation of the spatial reference of scans across a curved feature that is circular such as a contact hole in a semiconductor device. 
     FIG. 4 depicts an exemplary schematic image of waveforms taken across a curved feature that is circular such as a contact hole in a semiconductor device, derived from a CD-SEM. 
     FIG. 5 is an exemplary schematic representation showing sixteen scans across a curved feature, four selected scans with the corresponding eight sets of Cartesian coordinates and four groups of Cartesian coordinates selected from the eight sets. 
     FIG. 6 is an exemplary schematic representation showing two scans across a curved feature that is circular, and the corresponding sets of Cartesian coordinates. 
     FIG. 7 is an exemplary schematic representation showing four scans across a curved feature that is non-circular, and the corresponding sets of Cartesian coordinates. 
     FIG. 8 illustrates an exemplary procedure for determining the center of circularity, contact hole skew, and contact hole roughness. 
     FIG. 9 is an exemplary schematic representation of a curved feature with coordinates for calculating the center of circularity. 
     FIG. 10 is an exemplary schematic representation of a contact hole top and bottom. 
     FIG. 11 is an exemplary schematic representation of a curved feature with edge roughness. 
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     In the present invention, the initial calculation gives a radius of curvature rather than a diameter. Using the radius of curvature data allows for a more accurate determination of the diameter and circularity of a curved feature such as a contact hole in a semiconductor device. Further application to the roundness of vertical and horizontal line intersects could also be sought. 
     This method is also useful in determining the length of a foreshortened line-type feature. Currently, lines, e.g. poly gate, can be measured for line width only because the line length exceeds the CD-SEM range. Foreshortening is known to occur however, and line width measurements must be used to estimate the level of foreshortening. Since foreshortening occurs as feature end rounding, the present invention can be used to determine the feature end&#39;s curvature which can then be correlated to how much the feature foreshortened and hence, its length. 
     The present invention can be applied to any image with known spatial reference, thus, making it suited for determining the radius of curvature, diameter and circularity of any curvilinear object. Ideal situations where this method could be employed are scanning electron and atomic force microscopy where prolonged imaging can damage materials, however, the method may be applied to any digital image with spacial reference. 
     FIG. 1 is a block diagram of an exemplary system for determining the radius of curvature, diameter and circularity of a curved feature in one embodiment of the invention. A curved feature may be circular or non-circular. The system includes a target containing a two-dimensional curved feature, wherein the curved feature is scanned by an image/scan acquisition device  4 . The scanned images/scans are collected and transformed to sets of Cartesian coordinates by a processor  6 . The processor  6  may be implemented using a general-purpose computer executing a computer program for carrying out the processes described herein. The processor  6  is coupled to a storage device  8  and an output device  10 . The processor  6  executes software applications which may be implemented through computer programs. The computer programs may be stored on the processor  6  or may be stored on the storage device  8 . 
     FIG. 2 illustrates an exemplary procedure for determining the radius of curvature, diameter and circularity of two-dimensional curved features. In the preferred embodiment, a CD-SEM is used as the image acquisition device  4 . In steps  20 ,  22 , and  28 , the curved feature is determined to be either circular or non-circular. Also in the preferred embodiment, steps  24  and  30  perform sixteen incremental one-dimensional scans across the curved feature. The data  26  and  32  is then processed in step  34  to obtain thirty two sets of Cartesian coordinates for a curved feature that is circular or sixteen sets for a curved feature that is non-circular. 
     For example, FIG. 3 depicts an exemplary schematic of waveforms generated from CD-SEM scans across a curved feature, such as a contact hole in a semiconductor device. The edges of the curved feature are represented by peaks in the waveforms. In step  34  the peaks of the waveforms are converted to sets of Cartesian coordinates by the processor  6 . For example, the pixels of the scanned image correspond to (x,y) coordinates in plane space. The x-coordinate is obtained from the horizontal distance of a pixel of interest and the y-coordinate is determined from the line scan. The waveforms represent equidistant scans across the feature. Therefore, the vertical distance is a function of the scan height. The relationship between scan height, H and vertical distance, dvert is shown below: 
     
       
           d vert= H/n −1 
       
     
     where, n is the number of scans. One skilled in the art will appreciate that other methods may be substituted to determine the sets of Cartesian coordinates without departing from the spirit and scope of the present invention. Coordinate systems other than Cartesian may be used. 
     For a curved feature that is circular, such as a contact hole in a semiconductor device, the sixteen scans of the preferred embodiment are illustrated schematically in FIG.  4 . Thirty two sets of Cartesian coordinates are obtained from the sixteen scans. In step  36 , four of the sixteen scans are chosen, representing eight sets of Cartesian coordinates. The exact number of scans and subsequent groups of Cartesian coordinates is user defined. As part of step  36 , four groups containing three sets of Cartesian coordinates per group are determined. As discussed later, three sets of Cartesian coordinates are required to determine a radius of curvature. For a curved feature that is a closed object (such as a circle or ellipse), two groups contain sets of Cartesian coordinates from a first side of the curved feature, and the other two groups contain sets of Cartesian coordinates from a second side. The first and second sides of the curved feature are defined by an imaginary line perpendicular to the scan containing the sets of Cartesian coordinates with the maximum spacial distance there between. The line divides the feature in two halves of approximately equal size. For example, FIG. 5 is a schematic showing the scans and sets of Cartesian coordinates chosen to obtain four sample groups, each group containing three sets of Cartesian coordinates. 
     As part of step  38 , each group is fitted in the equation of a circle and solved simultaneously to obtain a radius of curvature. In other words, with three sets of Cartesian coordinates, three equations can be solved simultaneously, and the radius of curvature determined. The mathematical equation for a circle in Cartesian coordinates is given below: 
     
       
         ( x−h ) 2 +( y−k ) 2   =R   2   
       
     
     Where the curved feature will have radius R, and a center of curvature located at the Cartesian coordinate given by (h, k). Each group yields one radius of curvature value. In other words, in the above example, four radius of curvatures are determined. 
     In steps  40  and  42 , the radius of curvature data is used to determine the diameter and circularity of the curved feature. The diameter, D may be calculated as 
     
       
           D=r avg×2 
       
     
     Where ravg is simply the average of the radius of curvature values. 
     Also, in step  42 , the radius of curvature data is used to determine the circularity of the curved feature. First, the variance, V of each radius of curvature is determined. 
     
       
           V=|r−r avg|/ r avg 
       
     
     Then, circularity, C is the average variance in the radius of curvature. 
     
       
           C =( V   1 + V   2  . . .  Vn )/ n   
       
     
     A perfect circle is represented by a value of zero for C. In another embodiment, wherein scans are taken in orthogonal directions, the oblateness or prolateness of oval contact holes can be determined by the ratio of horizontal to vertical V values. Finally, in step  44 , the results are stored in the storage device  8  and/or exported to the output device  10 . 
     As shown in the following tables, a sample circularity calculation is made on a hypothetical contact hole. In this example, step  36  involves selecting eight sets of Cartesian coordinates from scan lines  5 ,  7 ,  11  and  13 . The sets of Cartesian coordinates are obtained in step  34  by the processor  6 , and measured in nanometers. Table 1 illustrates eight sets of Cartesian coordinates of the hypothetical contact hole, as measured in nanometers. For this example, four radius of curvatures are determined: two for the left edge of the contact hole and two for the right edge. Table 2 shows the resulting radius of curvatures, deviation (r-ravg) and variance. The ravg is determined to be 163.8, with the circularity determined to be 4.13% non-circular (C=0.0413). 
     
       
         
               
               
               
               
               
             
               
               
               
               
               
             
           
               
                   
                 TABLE 1 
               
               
                   
                   
               
               
                   
                 Line Scan 
                 X-Left 
                 X-Right 
                 Y 
               
               
                   
                   
               
             
             
               
                   
               
             
          
           
               
                   
                 5 
                 269.8 
                 465.9 
                 132.8 
               
               
                   
                 7 
                 210.8 
                 524.9 
                 199.2 
               
               
                   
                 11 
                 209.4 
                 529.3 
                 332 
               
               
                   
                 13 
                 255.1 
                 474.8 
                 398.4 
               
               
                   
                   
               
             
          
         
       
     
     
       
         
               
               
               
               
               
             
               
               
               
               
               
             
           
               
                 TABLE 2 
               
               
                   
               
               
                 Line Scan 
                 Edge 
                 Radius of Curvature 
                 Deviation (r-ravg) 
                 Variance 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                 7 
                 Left 
                 158.58 
                 5.22 
                 .0319 
               
               
                 13 
                 Left 
                 177.26 
                 13.46 
                 .0822 
               
               
                 11 
                 Right 
                 163.55 
                 0.25 
                 .0015 
               
               
                 5 
                 Right 
                 155.68 
                 8.12 
                 .0496 
               
               
                   
               
             
          
         
       
     
     In another embodiment, step  24  performs only two incremental one-dimensional scans across a curved feature that is a closed object. The data  26  is then processed in step  34  by the processor  6  to obtain four sets of Cartesian coordinates. FIG. 6 illustrates exemplary schematics of a curved feature that is a closed object with four points obtained from the respective scans. In step  36 , two groups, containing three sets of Cartesian coordinates each, are selected from the four sets of Cartesian coordinates. The two groups may contain the following sets of Cartesian coordinates: 
     Group 1: (x 1 ,y 1 ); (x 2 ,y 2 ); (x 3 ,y 1 ) 
     Group 2: (x 2 ,y 2 ); (x 3 ,y 1 ); (x 4 ,y 2 ). 
     As stated previously, in step  38 , the groups are fitted in the equation of a circle and solved simultaneously. In this embodiment, two radius of curvature results are obtained. In steps  40  and  42 , the radius of curvature data is used to determine the diameter and circularity of the curved feature, and in step  44 , the results are stored in the storage device  8  and/or exported to the output device  10 . 
     In another embodiment, step  24  performs a minimum of three incremental one-dimensional scans across a curved feature that is a open object. For example, FIG. 7 illustrates an open object with four scans. The data  26  is then processed in step  34  by the processor  6  to obtain four sets of Cartesian coordinates. In step  36 , two groups, containing three sets of Cartesian coordinates each, are selected from the four sets of Cartesian coordinates. The two groups may contain the following sets of Cartesian coordinates: 
     Group 1: (x 1 ,y 1 ); (x 2 ,y 2 ); (x 4 ,y 4 ) 
     Group 2: (x 2 ,y 2 ); (x 3 ,y 3 ); (x 4 ,y 4 ). 
     As stated previously, in step  38 , the groups are fitted in the equation of a circle and solved simultaneously. In this embodiment, two radius of curvature results are obtained. In steps  40  and  42 , the radius of curvature data is used to determine the diameter and circularity of the curved feature, and in step  44 , the results are stored in the storage device  8  and/or exported to the output device  10 . 
     Other embodiments use different combinations of scans and sets of Cartesian coordinates. For example, in a further embodiment, the scanning of steps  24  and  30  is performed both vertically and horizontally. In another embodiment involving a curved feature that is closed, the feature is schematically divided into four equal slices, or quadrants. The sets of Cartesian coordinates selected in step  36  are chosen from each of the quadrants, with three sets of Cartesian coordinates from each quadrant. 
     In order to calculate the center of circularity, skew and roughness for irregular shapes such as contact holes or vias in integrated circuits the exemplary procedure of FIG. 2 can be expanded. FIG.  9  and FIG.  10  and FIG. 11 illustrate how the concepts of FIG. 2 can be applied to such a structure and FIG. 8 is set of exemplary procedures for doing so. Referring to the coordinates referenced in FIG. 9, the center of circularity calculation procedure is charted in the upper half of FIG.  8 . First, at step  50  define g number of groups, from c number of coordinate pairs obtained completely form either top or bottom of contact hole, taken from n number of scans. Each group g i  will have at least one coordinate, c, from the opposite side of the contact. The maximum number of groups will be equal to the maximum number of possible independent combinations of three coordinate pairs from all coordinate pairs available. 
       g   max   =c C 3  where  c =2 n.   
     Next at step  52  for each group, g i , the center of circularity is determined as the coordinate (h i , k i ) as determined by the best fit equation for a circle comprised from the coordinates in g i . The center of circularity for the top and bottom of the contact are determined in steps  54  and  56 , respectively. 
     As shown in FIG. 10 the centers of circularity for the top and bottom holes are often skewed. The procedure described above can be applied to the determination of top and bottom contact via or hole skew. The procedure in FIG. 8 is applied to address this issue. Step  54  uses coordinate pairs determined from the “Top” of the contact hole determine the center of circularity for the “top”. Step  56  uses the coordinate pairs determined from the “Bottom” of the contact hole determine the center of circularity for the “Bottom.” Step  62  calculates the “contact skew.” Contact skew is defined by the vector difference between the top and bottom center of circularity. This vector can be defined relative to the “top” or “bottom” of the contact. For example, V the vector difference in “top” and “bottom” centers is represented as ([h top −h bottom ],[k top −k bottom ]). The vector represented in vector notation would be V=(h b −h t )i+(k b −k t )j. Alternately, the skew can be defined as the scalar, v with an angle of skew, θ. The orientation angle of skew, θ, is defined as 
     
       
         θ=tan −1 [( h   b   −h   t ) i /( k   b   −k   t )] 
       
     
     with a scalar distance of 
     
       
           v =[( h   b   −h   t ) 2 /( k   b   −k   t ) 2 ] ½   
       
     
     Therefore, the skew can either be represented as a vector, V, or as a scalar, v, with an angle θ. 
     The center of circularity procedure can also be used in the calculation of contact hole roughness. As illustrated in FIG.  11 . First, at steps  54  and  56  a center of circularity calculation is done for the top and bottom of the contact hole. Then at step  64  the hole roughness is calculated. The contact hole roughness is defined as the variance in radii as determined from the center of circularity. Where the radius, r, is the distance of a coordinate pair (x i ,y i ) from the center ((h avg k avg ). The average contact radius is r avg  and the roughness is the variance in deltas of all r i  from r avg . 
     All of the above calculations can be performed on any curved feature or one side of a curved feature open or closed. The description applying these procedures to contacts is merely illustrative. As described above, the present invention can include embodiments in the form of computer-implemented processes and apparatuses for practicing those processes. The present invention can also include embodiments in the form of computer program code containing instructions embodied in tangible media, such as floppy diskettes, CD-ROMs, hard drives, or any other computer-readable storage medium, wherein, when the computer program code is loaded into and executed by a computer, the computer becomes an apparatus for practicing the invention. The present invention can also include embodiments in the form of computer program code, for example, whether stored in a storage medium, loaded into and/or executed by a computer, or transmitted over some transmission medium, such as over electrical wiring or cabling, through fiber optics, or via electromagnetic radiation, wherein, when the computer program code is loaded into and executed by a computer, the computer becomes an apparatus for practicing the invention. When implemented on a general-purpose microprocessor, the computer program code segments configure the microprocessor to create specific logic circuits. 
     While the invention has been described with reference to exemplary embodiments, it will be understood by those skilled in the art that various changes may be made and equivalents may be substituted for elements thereof without departing from the scope of the invention. In addition, many modifications may be made to adapt a particular situation or material to the teachings of the invention without departing from the essential scope thereof. Therefore, it is intended that the invention not be limited to the particular embodiments disclosed for carrying out this invention, but that the invention will include all embodiments falling within the scope of the appended claims.