Abstract:
A method and apparatus for partially strapping two polysilicon lines, each having a first end and second end, uses a metal line having a plurality of spaced apart metal segments with each metal segment partially strapping a different portion of a polysilicon line. The metal segments are arranged from the first end to the second end with the signals propagating from the second end to the first end. Where two metal segments are used, the segments have lengths of 
     
       
         
           
             x 
             = 
             
               
                 2 
                  
                 L 
               
               
                 7 
               
             
           
         
       
     
     and L−X where L is the length between the first end and the second end. Where three segments are used, the segments have lengths of X=0.25 L, Y=0.48 L, and Z=0.27 L.

Description:
TECHNICAL FIELD 
       [0001]    The present invention relates to a metal line having a plurality of segments to partially strap a different portion of two polysilicon lines in a semiconductor integrated circuit device. 
       BACKGROUND OF THE INVENTION 
       [0002]    Semiconductor integrated circuit devices are well known in the art. In particular, in the field of memory devices, including both volatile (such as DRAM or SRAM) as well as nonvolatile (such as FLASH), a plurality of polysilicon lines are used as row lines to access various memory cells. Because these row lines are made out of polysilicon, they tend to have higher resistivity than conductors made out of metal. Thus, metal lines are used to “strap” various polysilicon lines. By strapping it is meant to connect a metal line in parallel with the polysilicon row line to reduce the resistivity of the polysilicon row line. 
         [0003]    As the scale of integration increases, i.e., the smallest dimension of a feature in a semiconductor integrated circuit device decreases, the width of the polysilicon lines will also decrease. However, the scale of integration for metal lines has not kept pace at the same rate as the scale of integration for other features such as the polysilicon lines. In other words, the width of the metal strapping lines has not been reduced at the same rate as the width of the polysilicon lines. 
         [0004]    Thus, a metal line having a width greater than a plurality of polysilicon lines, may be segmented into a plurality of metal segments with each metal segment used to strap a different portion of the polysilicon lines. 
         [0005]    The present invention deals with the optimal position for the strapping of such a partial strapped polysilicon line. U.S. Pat. No. 6,455,942 discloses one such technique, and its disclosure is incorporated herein by reference in its entirety. However, U.S. Pat. No. 6,455,942 deals with the problem of partial strapping by modeling the polysilicon lines as purely resistive loads without any capacitance. 
         [0006]    Using the delay of Elmore technique to solve the problem of delay in RC networks has been know in the prior art. See, for example, “Signal Delay in General RC Networks” by Tzu-Mu Lin and Carver Mead,  IEEE Transactions on Computer Aided Design,  vol. CAD-3, No. 4, October 1984 (pp. 331-349); “Splitting of RC Network for Accurate Model Reduction” by Patricia Renault and Pirouz Bazargan-Sabet,  IEEE,  2004 (pp. 734-737); and “Computing Signal Delay in General RC Networks by Tree/Link Partitioning” by Pak K. Chan and Kevin Karplus, 26 th ACM/IEEE Design Automation Conference,  1989 (pp. 485-490). 
       SUMMARY OF THE INVENTION 
       [0007]    In the present invention, a semiconductor integrated circuit device comprises a semiconductor substrate with a plurality of circuits in the semiconductor substrate. Two substantially parallel, laterally adjacent spaced part polysilicon lines, are insulated from the substrate and interconnect the plurality of circuits. Each of the plurality of polysilicon lines has a first end and a second end and has substantially the same length between the first end and the second end with electrical signals traversing from the second end to the first end. A metal line, spaced apart from the plurality of polysilicon lines, has two segments, with each segment having a length and wherein collectively the length of the two segments is substantially the length from the first end to the second end. Each segment of the metal line electrically connects to a different one of the two polysilicon lines. The metal line is arranged from the first end to the second end, 
         [0000]    
       
         
           
             x 
             = 
             
               
                 2 
                  
                 L 
               
               
                 7 
               
             
           
         
       
     
         [0000]    with one of the two segments having a length substantially of and with the other segment having the length of L−X where L is the length between the first end and the second end. 
         [0008]    The present invention also relates to a semiconductor integrated circuit device having two substantially parallel, laterally adjacent spaced part polysilicon lines, and strapped by a metal line having three segments. Finally, the present invention also relates to a method of forming such structures. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0009]      FIG. 1  is a top view of an integrated circuit device with a plurality of substantially parallel polysilicon lines to which the method and apparatus of the present invention can be applied. 
           [0010]      FIG. 2  is a top view of a first embodiment of the present invention in which a metal line, having two segments, straps two polysilicon lines. 
           [0011]      FIG. 3  is a schematic view of the first embodiment shown in  FIG. 2   
           [0012]      FIG. 4  is a schematic view of a second embodiment of the present invention in which a metal line, having three segments, straps two polysilicon lines. 
       
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
       [0013]    Referring to  FIG. 1 , there is shown a top level view of a semiconductor integrated circuit device  10  to which the method and apparatus of the present invention can be applied. The semiconductor integrated circuit device  10  comprises a semiconductor substrate in which a plurality of circuits  12  are formed. A plurality of substantially parallel laterally adjacent spaced apart polysilicon lines  14 ( a - n ) are formed above and insulated from the substrate. They interconnect the plurality of circuits  12 . Each of the plurality of polysilicon lines  14 ( a - n ) has a second end  20  and a first end  22  with substantially the same length between the first end  22  and the second end  20  with electrical signals traversing from the second end  20  to the first end  22 . 
         [0014]    In the preferred embodiment, the integrated circuit device  10  is a memory device in which the plurality of circuits  12  are row decoders. The plurality of polysilicon lines  14 ( a - n ) are word lines or row lines. The semiconductor integrated circuit device  10  can be a volatile memory device such as a DRAM or an SRAM or can be a non-volatile memory device such as a FLASH memory device or an EEPROM device. 
         [0015]    Referring to  FIG. 2 , there is shown a first embodiment of the present invention in which a single metal line  30  is used to strap two adjacent spaced-apart polysilicon lines  14   a  and  14   b.  The metal line  30  comprises two spaced apart segments  30 A and  30 B that are collinear and have a width of a dimension d. As previously stated, each of the polysilicon lines  14   a  and  14   b  has a first end  22  and a second end  20  and a length therebetween with each having a width of a dimension P, with P less than d. In fact, d overlaps the distance between the two polysilicon lines  14   a  and  14   b  as well as including their widths P. 
         [0016]    As previously stated, the metal line  30  has two segments  30   a  and  30   b  having a length of x and y respectively. Each of the lengths x and y is used to strap and is connected to a different one of the polysilicon lines  14   a  or  14   b.  Thus, the metal segment  30   a  is used to strap polysilicon line  14   a  through via holes  32   a  and  32   b.  The metal segment  30   b  is used to strap polysilicon line  14   b  through the via holes  34   a  and  34   b.    
         [0017]    Schematically this is shown in  FIG. 3 . Polysilicon line  1  extends from the first end  22  to the second end  20  through a line having resistance, R 1  and R 2 , as shown by the waviness of the line in  FIG. 3 . However, between the via holes  32   a  and  32   b,  the metal segment  30   a  having a length of L−p straps the polysilicon line  14   a.  There is essentially no resistance between the location of the via hole  32   a  and  32   b.  Each segment of the polysilicon line  1 , e.g. the segment between the via holes  32   a  and  32   b,  the polysilicon line has a resistance of R 1  and a capacitance of C 1 . 
         [0018]    Similarly, the polysilicon line  14   b  is shown as being “wavy” indicating that it has a resistance, R 3  and R 4 , between the first end  22  and the second end  20 . However, between the via hole  34   a  and  34   b  the metal segment  30   b  having a length of p is used to strap the polysilicon line  14   b.  Similar to the foregoing discussion for the polysilicon line  1 , each segment of the polysilicon line  2  also has an associated resistance R 3  and R 4  and an associated capacitance, C 3  and C 4 . Finally, one expects that the resistance of R 1  to be the same as the resistance of R 3  since the polysilicon segments would be virtually identical in length and width. Similarly, one expects R 2 =R 4 , and C 1 =C 3 , and C 2 =C 4 . 
         [0019]    Thus, each of the polysilicon lines  1  and  2  may be viewed as having a plurality of lumped resistances, as shown in  FIG. 3 , where L is the total length of the polysilicon line, p is the length of the un-strapped portion of the polysilicon line  1 , and nodes A, B, and C are the points of critical interest that will take the longest time to charge. Nodes B and C represent the midpoints of the strapped segments, while node A represents the endpoint of the array. 
         [0020]    The following identity can be used to apply the delay of Elmore technique to the delay for each of these nodes. 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       ∑ 
                       
                         i 
                         = 
                         1 
                       
                       n 
                     
                      
                     i 
                   
                   = 
                   
                     
                       
                         1 
                         2 
                       
                        
                       
                         n 
                         2 
                       
                     
                     + 
                     
                       
                         1 
                         2 
                       
                        
                       n 
                     
                   
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
     
         [0021]    The delays at nodes A and B may be computed as follows. 
         [0022]    Delay at Node A: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       t 
                       dA 
                     
                     / 
                     RC 
                   
                   = 
                   
                     
                       p 
                       + 
                       
                         ( 
                         
                           p 
                           - 
                           1 
                         
                         ) 
                       
                       + 
                       … 
                       + 
                       2 
                       + 
                       1 
                     
                     = 
                     
                       
                         
                           1 
                           2 
                         
                          
                         
                           p 
                           2 
                         
                       
                       + 
                       
                         
                           1 
                           2 
                         
                          
                         p 
                       
                     
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
         [0023]    Delay at Node B: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             t 
                             dB 
                           
                           / 
                           RC 
                         
                         = 
                           
                          
                         
                           L 
                           + 
                           
                             ( 
                             
                               L 
                               - 
                               1 
                             
                             ) 
                           
                           + 
                           … 
                           + 
                           
                             ( 
                             
                               p 
                               + 
                               1 
                             
                             ) 
                           
                           + 
                           
                             ( 
                             
                               p 
                               2 
                             
                             ) 
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           
                             ( 
                             
                               
                                 P 
                                 2 
                               
                               - 
                               1 
                             
                             ) 
                           
                           + 
                           … 
                           + 
                           2 
                           + 
                           1 
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               1 
                               2 
                             
                              
                             
                               L 
                               2 
                             
                           
                           + 
                           
                             
                               1 
                               2 
                             
                              
                             L 
                           
                           - 
                           
                             
                               3 
                               8 
                             
                              
                             
                               p 
                               2 
                             
                           
                           - 
                           
                             
                               1 
                               4 
                             
                              
                             
                               p 
                               . 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
         [0024]    To balance and minimize these delays we set t dA =t dB  to get 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         1 
                         2 
                       
                        
                       
                         L 
                         2 
                       
                     
                     + 
                     
                       
                         1 
                         2 
                       
                        
                       L 
                     
                     - 
                     
                       
                         3 
                         8 
                       
                        
                       
                         p 
                         2 
                       
                     
                     - 
                     
                       
                         1 
                         4 
                       
                        
                       p 
                     
                   
                   = 
                   
                     
                       
                         
                           
                             1 
                             2 
                           
                            
                           
                             p 
                             2 
                           
                         
                         + 
                         
                           
                             1 
                             2 
                           
                            
                           p 
                         
                       
                        
                       
                         
 
                       
                       ⇒ 
                       
                         
                           p 
                           2 
                         
                         + 
                         
                           
                             6 
                             7 
                           
                            
                           p 
                         
                         + 
                         
                           
                             4 
                             7 
                           
                            
                           
                             ( 
                             
                               
                                 - 
                                 
                                   L 
                                   2 
                                 
                               
                               - 
                               L 
                             
                             ) 
                           
                         
                       
                     
                     = 
                     
                       
                         0 
                          
                         
                           
 
                         
                         ⇒ 
                         p 
                       
                       = 
                       
                         
                           - 
                           
                             3 
                             7 
                           
                         
                         ± 
                         
                           
                             
                               9 
                               49 
                             
                             + 
                             
                               
                                 4 
                                 7 
                               
                                
                               
                                 ( 
                                 
                                   
                                     L 
                                     2 
                                   
                                   + 
                                   L 
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
           
         
       
     
         [0025]    or, for large L and choosing the positive root, 
         [0000]    
       
         
           
             p 
             ≅ 
             
               
                 2 
                 
                   7 
                 
               
                
               L 
             
             ≅ 
             
               0.76 
                
               
                 L 
                 . 
               
             
           
         
       
     
         [0026]    As can be seen, the foregoing result is independent of the value of the resistance R or the capacitance C, and based upon simulation, the result yields an approximate 24% improvement over the technique disclosed in U.S. Pat. No. 6,455,942, which assumes the polysilicon segments are only resistive segments. 
         [0027]    Referring to  FIG. 4  there is shown another embodiment of the present invention. In the embodiment shown in  FIG. 4 , two polysilicon lines,  1  and  2 , are strapped by a metal line  30 , having three segments:  30   a,    30   b,  and  30   c.  The metal segment  30   a  straps the portion of polysilicon line  1  over the length x, which is closest to the first end  22 , and the metal segment  30   c  straps the portion of polysilicon line  1  over the length z, which is closest to the second end  20 . The metal segment  30   b,  which is between the segments  30   a  and  30   c,  straps the portion of the polysilicon line  2  over the length y, which is between the lengths x and z. Each of the polysilicon lines ( 1  or  2 ) over the segments x, y, or z, has an associated resistance and an associated capacitance, as shown in  FIG. 4 . In reality, because the segment x is the same for polysilicon  1  and polysilicon  2 , R 1 =R 4  and C 1 =C 4 . Similarly R 2 =R 5 , and C 2 =C 5 . Finally, R 3 =R 6  and C 3 =C 6 . The embodiment shown in  FIG. 4  is advantageous to further reduce the delay in the worst case delay by using additional strapping segments. Nodes D, E, and F are the midpoints of the strapped segments with x, z, and y RC-stages respectively, whereas node G is a node at the second end  20 . 
         [0028]    Applying the delay of Elmore technique to the delay for each of these nodes, as before, one finds that the delay for the critical nodes D, E, F, and G are found to be: 
         [0029]    Delay at Node D: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             t 
                             dD 
                           
                           / 
                           RC 
                         
                         = 
                         
                           
                             
                               1 
                               2 
                             
                              
                             
                               
                                 ( 
                                 
                                   x 
                                   2 
                                 
                                 ) 
                               
                               2 
                             
                           
                           + 
                           
                             
                               1 
                               2 
                             
                              
                             
                               ( 
                               
                                 x 
                                 2 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             
                               1 
                               8 
                             
                              
                             
                               x 
                               2 
                             
                           
                           + 
                           
                             
                               1 
                               4 
                             
                              
                             x 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
         [0030]    Delay at Node E: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             t 
                             dE 
                           
                           / 
                           RC 
                         
                         = 
                           
                          
                         
                           
                             
                               1 
                               2 
                             
                              
                             
                               
                                 ( 
                                 
                                   L 
                                   - 
                                   x 
                                 
                                 ) 
                               
                               2 
                             
                           
                           + 
                           
                             
                               1 
                               2 
                             
                              
                             
                               ( 
                               
                                 L 
                                 - 
                                 x 
                               
                               ) 
                             
                           
                           - 
                           
                             
                               1 
                               2 
                             
                              
                             
                               z 
                               2 
                             
                           
                           - 
                           
                             
                               1 
                               2 
                             
                              
                             z 
                           
                           + 
                         
                       
                     
                     
                       
                           
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           
                             
                               1 
                               2 
                             
                              
                             
                               
                                 ( 
                                 
                                   z 
                                   2 
                                 
                                 ) 
                               
                               2 
                             
                           
                           + 
                           
                             
                               1 
                               2 
                             
                              
                             
                               ( 
                               
                                 z 
                                 2 
                               
                               ) 
                             
                              
                             
                                 
                             
                              
                             
                               ( 
                               
                                 x 
                                 = 
                                 
                                   L 
                                   - 
                                   y 
                                   - 
                                   z 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                     
                       
                           
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               1 
                               2 
                             
                              
                             
                               y 
                               2 
                             
                           
                           + 
                           
                             
                               1 
                               2 
                             
                              
                             y 
                           
                           + 
                           
                             
                               1 
                               8 
                             
                              
                             
                               z 
                               2 
                             
                           
                           + 
                           
                             
                               1 
                               4 
                             
                              
                             z 
                           
                           + 
                           yz 
                         
                       
                     
                     
                       
                           
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     11 
                      
                     a 
                   
                   ) 
                 
               
             
           
         
       
     
         [0031]    Delay at Node F: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             t 
                             dF 
                           
                           / 
                           RC 
                         
                         = 
                           
                          
                         
                           
                             
                               1 
                               2 
                             
                              
                             
                               L 
                               2 
                             
                           
                           + 
                           
                             
                               1 
                               2 
                             
                              
                             L 
                           
                           - 
                           
                             
                               1 
                               2 
                             
                              
                             
                               
                                 ( 
                                 
                                   y 
                                   + 
                                   z 
                                 
                                 ) 
                               
                               2 
                             
                           
                           - 
                           
                             
                               1 
                               2 
                             
                              
                             
                               ( 
                               
                                 y 
                                 + 
                                 z 
                               
                               ) 
                             
                           
                           + 
                         
                       
                     
                     
                       
                           
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           
                             
                               1 
                               2 
                             
                              
                             
                               
                                 ( 
                                 
                                   y 
                                   2 
                                 
                                 ) 
                               
                               2 
                             
                           
                           + 
                           
                             
                               1 
                               2 
                             
                              
                             
                               ( 
                               
                                 y 
                                 2 
                               
                               ) 
                             
                           
                         
                       
                     
                     
                       
                           
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               1 
                               2 
                             
                              
                             
                               L 
                               2 
                             
                           
                           + 
                           
                             
                               1 
                               2 
                             
                              
                             L 
                           
                           - 
                           
                             
                               3 
                               8 
                             
                              
                             
                               y 
                               2 
                             
                           
                           - 
                           
                             
                               1 
                               4 
                             
                              
                             y 
                           
                           - 
                           
                             
                               1 
                               2 
                             
                              
                             
                               z 
                               2 
                             
                           
                           - 
                           
                             
                               1 
                               2 
                             
                              
                             z 
                           
                           - 
                           yz 
                         
                       
                     
                     
                       
                           
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     11 
                      
                     b 
                   
                   ) 
                 
               
             
           
         
       
     
         [0032]    Delay at Node G: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             t 
                             dG 
                           
                           / 
                           RC 
                         
                         = 
                           
                          
                         
                           
                             
                               1 
                               2 
                             
                              
                             
                               L 
                               2 
                             
                           
                           + 
                           
                             
                               1 
                               2 
                             
                              
                             L 
                           
                           - 
                           
                             
                               1 
                               2 
                             
                              
                             
                               
                                 ( 
                                 
                                   y 
                                   + 
                                   z 
                                 
                                 ) 
                               
                               2 
                             
                           
                           - 
                           
                             
                               1 
                               2 
                             
                              
                             
                               ( 
                               
                                 y 
                                 + 
                                 z 
                               
                               ) 
                             
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           
                             
                               1 
                               2 
                             
                              
                             
                               z 
                               2 
                             
                           
                           + 
                           
                             
                               1 
                               2 
                             
                              
                             z 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               1 
                               2 
                             
                              
                             
                               L 
                               2 
                             
                           
                           + 
                           
                             
                               1 
                               2 
                             
                              
                             L 
                           
                           - 
                           
                             
                               1 
                               2 
                             
                              
                             
                               y 
                               2 
                             
                           
                           - 
                           
                             
                               1 
                               2 
                             
                              
                             y 
                           
                           - 
                           yz 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     11 
                      
                     c 
                   
                   ) 
                 
               
             
           
         
       
     
         [0033]    Comparing sums shows that t dD &lt;t dF  and t dD &lt;t dG , so that t dD  cannot be the node with worst case delay. To simplify analysis of the remaining nodes, replace x, y and z with x′, y′ and z′, such that x=x′L, y=y′L and z=z′L, and therefore 0&lt;x′, y′, z′&lt;1 and x′+y′+z′=1. Then equations (11(a-c)) become 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         t 
                         dE 
                       
                       / 
                       RC 
                     
                     = 
                     
                       
                         
                           1 
                           2 
                         
                          
                         
                           y 
                           ′2 
                         
                          
                         
                           L 
                           2 
                         
                       
                       + 
                       
                         
                           1 
                           2 
                         
                          
                         
                           y 
                           ′ 
                         
                          
                         L 
                       
                       + 
                       
                         
                           1 
                           8 
                         
                          
                         
                           z 
                           ′2 
                         
                          
                         
                           L 
                           2 
                         
                       
                       + 
                       
                         
                           1 
                           4 
                         
                          
                         
                           z 
                           ′ 
                         
                          
                         L 
                       
                       + 
                       
                         
                           y 
                           ′ 
                         
                          
                         
                           z 
                           ′ 
                         
                          
                         
                           L 
                           2 
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       
                         t 
                         dF 
                       
                       / 
                       RC 
                     
                     = 
                     
                       
                         
                           1 
                           2 
                         
                          
                         
                           L 
                           2 
                         
                       
                       + 
                       
                         
                           1 
                           2 
                         
                          
                         L 
                       
                       - 
                       
                         
                           3 
                           8 
                         
                          
                         
                           y 
                           ′2 
                         
                          
                         
                           L 
                           2 
                         
                       
                       - 
                       
                         
                           1 
                           4 
                         
                          
                         
                           y 
                           ′ 
                         
                          
                         L 
                       
                       - 
                       
                         
                           1 
                           2 
                         
                          
                         
                           z 
                           ′2 
                         
                          
                         
                           L 
                           2 
                         
                       
                       - 
                       
                         
                           1 
                           2 
                         
                          
                         
                           z 
                           ′ 
                         
                          
                         L 
                       
                       - 
                       
                         
                           y 
                           ′ 
                         
                          
                         
                           z 
                           ′ 
                         
                          
                         
                           L 
                           2 
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       
                         t 
                         dG 
                       
                       / 
                       RC 
                     
                     = 
                     
                       
                         
                           1 
                           2 
                         
                          
                         
                           L 
                           2 
                         
                       
                       + 
                       
                         
                           1 
                           2 
                         
                          
                         L 
                       
                       - 
                       
                         
                           1 
                           2 
                         
                          
                         
                           y 
                           ′2 
                         
                          
                         
                           L 
                           2 
                         
                       
                       - 
                       
                         
                           1 
                           2 
                         
                          
                         
                           y 
                           ′ 
                         
                          
                         L 
                       
                       - 
                       
                         
                           y 
                           ′ 
                         
                          
                         
                           z 
                           ′ 
                         
                          
                         
                           
                             L 
                             2 
                           
                           . 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
         [0034]    If L is large (the RC network is finely partitioned) then finding a segmentation that minimizes the loads across the three nodes of equations (12) is approximately equivalent to finding the y′ and z′ values which balance the following set of equations, where only terms proportional to L 2  have been retained, and where t′ dX =t dX L 2 . 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         t 
                         dE 
                         ′ 
                       
                       / 
                       RC 
                     
                     = 
                     
                       
                         
                           1 
                           2 
                         
                          
                         
                           y 
                           ′2 
                         
                       
                       + 
                       
                         
                           1 
                           8 
                         
                          
                         
                           z 
                           ′2 
                         
                       
                       + 
                       
                         
                           y 
                           ′ 
                         
                          
                         
                           z 
                           ′ 
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       
                         t 
                         dF 
                         ′ 
                       
                       / 
                       RC 
                     
                     = 
                     
                       
                         1 
                         2 
                       
                       - 
                       
                         
                           3 
                           8 
                         
                          
                         
                           y 
                           ′2 
                         
                       
                       - 
                       
                         
                           1 
                           2 
                         
                          
                         
                           z 
                           ′2 
                         
                       
                       - 
                       
                         
                           y 
                           ′ 
                         
                          
                         
                           z 
                           ′ 
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       
                         t 
                         dG 
                         ′ 
                       
                       / 
                       RC 
                     
                     = 
                     
                       
                         1 
                         2 
                       
                       - 
                       
                         
                           1 
                           2 
                         
                          
                         
                           y 
                           ′2 
                         
                       
                       - 
                       
                         
                           y 
                           ′ 
                         
                          
                         
                           z 
                           ′ 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
           
         
       
     
         [0035]    Numerical analysis of equations (13) shows that the ratio of three segments for minimizing delay at the three critical nodes is approximately given by 
         [0000]      x=0.25 L 
         [0000]      y=0.48 L 
         [0000]      z=0.27 L. 
         [0036]    From the foregoing and simulation, it can be seen that the ideal strapping ratios using three metal segments are also independent of the resistance and capacitance. Further, improvement to the embodiment shown in  FIGS. 2 and 3  is approximately 8%. Finally, by modeling the polysilicon lines as having both resistance and capacitance, improvement of the disclosure in U.S. Pat. No. 6,455,942 is also achieved.