Patent Publication Number: US-2022234163-A1

Title: Chemical mechanical polishing using time share control

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application is a divisional of U.S. application Ser. No. 16/688,604, filed Nov. 19, 2019, which claims priority to U.S. Provisional Application Ser. No. 62/816,015, filed Mar. 8, 2019, the disclosure of which is incorporated by reference. 
    
    
     TECHNICAL FIELD 
     The present disclosure relates to a chemical mechanical polishing of substrates. 
     BACKGROUND 
     An integrated circuit is typically formed on a substrate by the sequential deposition of conductive, semiconductive, or insulative layers on a silicon wafer. One fabrication step involves depositing a filler layer over a non-planar surface and planarizing the filler layer. For certain applications, the filler layer is planarized until the top surface of a patterned layer is exposed. A conductive filler layer, for example, can be deposited on a patterned insulative layer to fill the trenches or holes in the insulative layer. After planarization, the portions of the conductive layer remaining between the raised pattern of the insulative layer form vias, plugs, and lines that provide conductive paths between thin film circuits on the substrate. For other applications, such as oxide polishing, the filler layer is planarized until a predetermined thickness is left over the non planar surface. In addition, planarization of the substrate surface is usually required for photolithography. 
     Chemical mechanical polishing (CMP) is one accepted method of planarization. This planarization method typically requires that the substrate be mounted on a carrier or polishing head. The exposed surface of the substrate is typically placed against a rotating polishing pad. The carrier head provides a controllable load on the substrate to push it against the polishing pad. An abrasive polishing slurry is typically supplied to the surface of the polishing pad. 
     One issue in polishing is non-uniformity in the polishing rate across the substrate. For example, the edge portion of a substrate can polish at a higher relative to the central portion of the substrate. 
     SUMMARY 
     In one aspect, a method for chemical mechanical polishing includes rotating a polishing pad about an axis of rotation, positioning a substrate against the polishing pad, oscillating the substrate laterally across the polishing pad such that a central portion of the substrate and an edge portion of the substrate are positioned over a polishing surface of the polishing pad for a first duration such that the central portion of the substrate and the edge portion of the substrate are polished for the first duration, and holding the substrate substantially laterally fixed in a position such that the central portion of the substrate is positioned over the polishing surface of the polishing pad and the edge portion of the substrate is positioned over a polishing-control groove for a second duration such that the central portion of the substrate is polished the second duration. 
     In another aspect, a polishing system includes a rotatable platen to support a polishing pad, a carrier head radially movable across the polishing pad to hold a substrate against the polishing pad, an actuator to move the carrier head, and a controller coupled to the actuator. The controller is and configured to cause the carrier head to oscillate the substrate laterally across the polishing pad for a first duration, such that a central portion of the substrate and an edge portion of the substrate are positioned over a polishing surface of the polishing pad, and to hold the substrate substantially laterally fixed a position for a second duration such that the central portion of the substrate is positioned over the polishing surface of the polishing pad and the edge portion of the substrate is positioned over a polishing-control groove. 
     In another aspect, a polishing pad includes a polishing layer having a central region with a plurality of slurry-supply grooves formed therein and an outer region having a polishing-control groove formed therein, with the polishing control groove being wider than the slurry-supply grooves. 
     In another aspect, a method for determining dwell time includes determining a polishing rate of central region of a test substrate as polished by a polishing surface of a polishing pad, determining an edge polishing rate of an edge portion of the substrate as polished by the polishing surface of the polishing pad, determining a desired decrease in polishing rate for the edge portion of the substrate to reduce polishing non-uniformity between the central portion of the substrate and the edge portion of the substrate, and calculating a first duration and a second duration to provide the desired decrease in polishing rate. The first duration is for the device substrate to oscillate laterally across the polishing pad such that a central portion of the device substrate and an edge portion of the substrate edge are positioned on the polishing pad, and the second duration is for holding the device substrate substantially laterally fixed in a position such that the central portion of the substrate is positioned on the polishing surface of the polishing pad and the edge portion of the substrate is positioned over a groove. 
     Implementations may include one or more of the following features. 
     A ratio of the second duration to the second duration may be selected to provide a desired polishing rate for the edge portion. 
     The polishing pad may have slurry-supply grooves. The slurry supply grooves may be narrower than the polishing control groove. The slurry supply grooves may be concentric with the polishing control groove. The polishing control groove may surround the slurry supply grooves. The polishing control groove may be positioned near an edge of the polishing pad. The polishing control groove may be 5 to 30 mm wide. 
     Dwell time may be calculated using a radius of the groove, a radius of the substrate, and a distance from a center of the polishing pad to a center of the substrate. 
     Implementations may optionally include, but are not limited to, one or more of the following advantages. Polishing non-uniformity, e.g., caused by different polishing rates at different portions of the substrate, can be controlled and corrected. For example, controlling the position of the substrate relative to the non-polishing groove can provide edge-correction. Additionally, there is a minimal impact to throughput because the adjustment to polishing can be performed in the polishing station rather than as part of a separate module. Furthermore, a second station is not necessary to perform the edge-correction, reducing the footprint needed in the polishing station clean room. 
     The details of one or more implementations are set forth in the accompanying drawings and the description below. Other aspects, features, and advantages will be apparent from the description and drawings, and from the claims. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a schematic cross-sectional view of a chemical mechanical polishing system with a polishing pad having a groove. 
         FIG. 2  is a schematic cross-sectional view of a polishing pad having both slurry-supply grooves and a polishing-control groove. 
         FIG. 3A  is a schematic top view of a polishing pad having a groove that is concentric with the axis of rotation. 
         FIG. 3B  is a schematic top view of another implementation of a polishing pad having a groove that is concentric with the axis of rotation. 
         FIG. 3C  is a schematic top view of another implementation of a polishing pad having two grooves that are concentric with the axis of rotation. 
         FIG. 4A  is an exemplary graph of substrate position on platen versus time. 
         FIG. 4B  is an exemplary graph of substrate position on platen versus time for another implementation. 
         FIG. 4C  is an exemplary graph of substrate position on platen versus time for another implementation. 
         FIG. 5  is an exemplary graph showing a comparison of wafer edge uniformity from a base line removal and a time share control removal. 
     
    
    
     Like reference numbers and designations in the various drawings indicate like elements. 
     DETAILED DESCRIPTION 
     As noted above, when a substrate is polished by a polishing pad, the edge portion of the substrate can polish at a higher rate than a central portion of the substrate, resulting in a non-uniformly polished substrate. However, positioning, and holding, the substrate over a polishing control groove can reduce the non-uniformity of the polished substrate. The polishing control groove can be located nearer a perimeter of the polishing pad, or nearer a center of the polishing pad, or the polishing pad can include a first polishing control groove nearer the perimeter and a second polishing control groove nearer the center. 
       FIG. 1  illustrates an example of a polishing station of a chemical mechanical polishing system  20 . The polishing system  20  includes a rotatable disk-shaped platen  24  on which a polishing pad  30  is situated. The platen  24  is operable to rotate about an axis  25 . For example, a motor  26  can turn a drive shaft  28  to rotate the platen  24 . The polishing pad  30  can be a two-layer polishing pad with an outer polishing layer  32  and a softer backing layer  34 . The outer polishing layer  32  has a polishing surface  36 . 
     The polishing system  20  can include a supply port or a combined supply-rinse arm  92  to dispense a polishing liquid  94 , such as an abrasive slurry, onto the polishing pad  30 . The polishing system  20  can include a pad conditioner apparatus  40  with a conditioning disk  42  to maintain the surface roughness of the polishing surface  36  of the polishing pad  30 . The conditioning disk  42  can be positioned at the end of an arm  44  that can swing so as to sweep the disk  42  radially across the polishing pad  30 . 
     A carrier head  70  is operable to hold a substrate  10  against the polishing pad  30 . The carrier head  70  is suspended from a support structure  50 , e.g., a carousel or a track, and is connected by a drive shaft  58  to a carrier head rotation motor  56  so that the carrier head can rotate about an axis  55 . Optionally, the carrier head  70  can oscillate laterally, e.g., on sliders on the carousel, by movement along the track, or by rotational oscillation of the carousel itself. 
     The carrier head  70  includes a housing  72 , a substrate backing assembly  74  which includes a base  76  and a flexible membrane  78  that defines a plurality of pressurizable chambers  80 , a gimbal mechanism  82  (which may be considered part of the assembly  74 ), a loading chamber  84 , a retaining ring assembly  100 , and an actuator  122 . 
     The housing  72  can generally be circular in shape and can be connected to the drive shaft  58  to rotate therewith during polishing. There may be passages (not illustrated) extending through the housing  72  for pneumatic control of the carrier head  100 . The substrate backing assembly  74  is a vertically movable assembly located beneath the housing  72 . The gimbal mechanism  82  permits the base  76  to gimbal relative to the housing  72  while preventing lateral motion of the base  76  relative to the housing  72 . The loading chamber  84  is located between the housing  72  and the base  76  to apply a load, i.e., a downward pressure or weight, to the base  76  and thus to the substrate backing assembly. The vertical position of the substrate backing assembly  74  relative to a polishing pad is also controlled by the loading chamber  84 . The lower surface of the flexible membrane  78  provides a mounting surface for a substrate  10 . 
     In some implementation, the substrate backing assembly  74  is not a separate component that is movable relative to the housing  72 . In this case, the chamber  84  and gimbal  82  are unnecessary. 
     Referring to  FIG. 1 , the polishing pad  30  has at least one polishing control groove  102  formed in the polishing surface  36 . Each polishing control groove  102  is a recessed area of the polishing pad  30 . Each polishing control groove  102  can be an annular groove, e.g., circular, and can be concentric with the axis of rotation  25 . Each polishing control groove  102  provides an area of the polishing pad  30  that does not contribute to polishing. 
     The walls of the polishing control groove  102  are perpendicular to the polishing surface  36 . The bottom surface of the polishing control groove  102  is parallel with the polishing surface  36 , although in some implementations the bottom surface of the polishing control groove  102  can be angled relative to the polishing surface  36 . The bottom of the polishing control groove  102  can have a rectangular or a U-shaped cross-section. The polishing control groove  102  can be 10 to 80 mils, e.g., 10 to 60 mils, deep. 
     In some implementations, the pad  30  includes a polishing control groove  102   a  located near the outer edge of the polishing pad  30 , e.g., within 15%, e.g., with 10% (by radius) of the outer edge. For example, the groove  102   a  can be located at a radial distance of fourteen inches from the center of a platen having a thirty inch diameter. 
     In some implementations, the pad  30  includes a polishing control groove  102   b  located near the center of the polishing pad  30 , e.g., within 15%, e.g., with 10% (by radius) of the center or axis of rotation  25 . For example, the groove  102   b  can be located at a radial distance of one inch from the center of a platen having a thirty inch diameter. 
     In some implementations, the pad  30  includes only a polishing control groove  102   a  located near the outer edge of the polishing pad  30  (see  FIG. 3A ). In this case, there can be just a single control polishing groove  102  near the outer edge of the polishing pad  30 . In some implementations, the pad  30  includes only a polishing control groove  102   b  located near the center of the polishing pad  30  (see  FIG. 3B ). In this case, there can be just a single control polishing groove  102  near the center of the polishing pad  30 . In some implementations, the pad  30  includes a first polishing control groove  102   a  located near the edge of the polishing pad  30  and a first polishing control groove  102   b  located near the center of the polishing pad  30  (see  FIG. 3C ). In this case, there can be exactly two polishing grooves  102  on the polishing surface. 
     Returning to  FIG. 1 , the polishing control groove  102  is sufficiently wide that the by positioning a section of the substrate  10  over the groove, the polishing rate of that section will be materially reduced. In particular, for edge-correction, the groove  102  is sufficiently wide that an annular band at the edge of the substrate, e.g., a band at least 3 mm wide, e.g., a band 3-15 mm wide, e.g., a band 3-10 mm wide, will have a reduced polishing rate. The polishing control groove  102  can have a width of three to fifty, e.g., five to fifty, e.g., three to ten, e.g., ten to twenty, millimeters. 
     When the substrate  10  is positioned over the polishing surface  36  of the polishing pad  30 , the polishing surface  36  contacts and polishes the substrate  10 , and material removal takes place. On the other hand, when an edge of the substrate  10  is positioned above the polishing control groove  102 , there is no contact or polishing of the edge of substrate  10  to cause removal takes place. Optionally, the groove  102  can provide a conduit for polishing slurry to pass through without abrading the substrate  10 . 
     Referring now to  FIG. 2 , the polishing pad  30  can also include one or more slurry-supply grooves  112 . The slurry-supply grooves  112  can be annular grooves, e.g., circular grooves, and can be concentric with the polishing control groove  102 . Alternatively, the slurry supply grooves can have another pattern, e.g., rectangular cross-hatch, triangular cross-hatch, etc. The slurry supply grooves can have a width between about 0.015 and 0.04 inches (between 0.381 and 1.016 mm), such as 0.20 inches, and a pitch between about 0.09 and 0.24 inches, such as 0.12 inches. 
     The slurry-supply grooves  112  are narrower than the polishing control groove  102 . For example, the slurry-supply grooves  112  can be narrower by a factor of at least 3, e.g., at least 6, such as 6 to 100. The slurry supply grooves  112  can be uniformly spaced across the polishing pad  30 . The polishing control groove  102  can have a smaller, similar, or greater depth than the slurry-supply grooves  112 . In some implementations, the polishing control groove  102  is the only groove on the polishing pad wider than the slurry supply grooves  112 . In some implementations, the polishing control grooves  102   a  and  102   b  are the only grooves on the polishing pad wider than the slurry supply grooves  112 . 
     Referring now to  FIG. 3A , for a first duration, the substrate  10  can be positioned in a first position or first range of positions such that the central portion  12  of the substrate  10  and the edge portion  14  of the substrate  10  are both polished by the polishing surface  36  of the polishing pad  30 . As such, none of the substrate  10  overlaps the polishing control groove  102 . Although portions of the substrate  10  overlap the slurry-supply grooves  112 , the slurry-supply grooves  112  are relatively closely spaced and relative motion averages out any effects on the polishing rate. 
     For a second duration, the substrate  10  can be positioned such that the central portion  12  of the substrate  10  is polished by the polishing surface  36  and a region  14   a  of the edge portion  14  of the substrate  10  is above the polishing control groove  102 . During the second duration, the substrate  10  can be held laterally fixed in a second position. Thus the central portion  12  of the substrate  10  is polished during the second duration, whereas a region  14   a  of the edge portion  14  of the substrate  10 , positioned above the polishing control groove  102 , is not polished. Since some of the edge portion  14  (indicated by  14   b ) remains over the polishing surface  36 , the edge portion  14  will still be polished to some degree. Due to rotation of the substrate  10 , the edge portion  14  should still be polished in an angularly uniform manner, but at a lower rate than the central portion  12  due to lack of polishing in region  14   a . In addition, although portions of the substrate  10  overlap the slurry-supply grooves  112 , the slurry-supply grooves  112  are relatively closely spaced and relative motion averages out any effects on the polishing rate. A controller can cause the support to move the carrier head  70  to oscillate the substrate  10  laterally during the first duration, and to hold the substrate  10  at a fixed position laterally for a time at the second duration. 
     To reduce the removal of the edge portion  14  of the substrate  10 , and to obtain a more uniformly polished substrate  10 , a non-polishing area time share can be determined. For example, equation [1] can be used to determine the non-polishing area time share S 1 : 
     
       
         
           
             
               
                 
                   
                     S 
                     1 
                   
                   = 
                   
                     α 
                     180 
                   
                 
               
               
                 
                   [ 
                   1 
                   ] 
                 
               
             
           
         
       
     
     where α is the angle subtended across the substrate  10  by the polishing control groove  102   a  relative to the center of the substrate, and can be determined using equations [2]-[3]: 
     
       
         
           
             
               
                 
                   
                     
                       cos 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       α 
                     
                     = 
                     
                       
                         
                           R 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             1 
                             2 
                           
                         
                         - 
                         
                           r 
                           2 
                         
                         - 
                         
                           x 
                           2 
                         
                       
                       
                         2 
                         ⁢ 
                         xr 
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   and 
                 
               
               
                 
                   [ 
                   2 
                   ] 
                 
               
             
             
               
                 
                   α 
                   = 
                   
                     
                       cos 
                       
                         - 
                         1 
                       
                     
                     ⁢ 
                     
                       
                         
                           R 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             1 
                             2 
                           
                         
                         - 
                         
                           r 
                           2 
                         
                         - 
                         
                           x 
                           2 
                         
                       
                       
                         2 
                         ⁢ 
                         xr 
                       
                     
                   
                 
               
               
                 
                   [ 
                   3 
                   ] 
                 
               
             
           
         
       
     
     and where R1 is the radius of the inner edge of the groove  102   a, r  is the radius of the substrate  10 , and x is the distance from the center of the polishing pad  30  to the center of the substrate  10 . 
     Referring now to  FIGS. 3A and 4A , the central portion  12  of the substrate  10  and the edge portion  14  of the substrate  10  is positioned over the polishing area of polishing pad  30  for a first duration T 1  (t 0  to t 1 ). The substrate can be moved laterally in an oscillatory manner during this first duration. At the end of the first duration, the substrate is repositioned. The central portion  12  of the substrate  10  is positioned over the polishing area of the polishing pad  30  and the edge portion  14  of the substrate  10  is positioned and held over the non-polishing area of the polishing control groove  102  for a second duration T 2  (t 1  to t 2 ). 
     The process can be repeated so the substrate  10  oscillates between a first position—where the central portion and the edge portion of the substrate  10  is polished—for the first duration and a second position at the second duration, where the substrate is held at the second position—where the central portion of the substrate  10  is polished and the edge portion of the substrate  10  is not polished—for a duration calculated using the time share S. 
     The ratio of the first duration T 1  to the second duration T 2  can be selected so as to reduce the polishing rate of the edge portion  14  by a desired amount. For example the ratio T 1 /T 2 , can be selected to achieve a desired polishing rate at the edge, e.g., to achieve the same polishing rate as the center portion  12 . 
     The ratio T 2 /(T 1 +T 2 ) provides a percentage of time per cycle where the substrate, e.g., the edge portion  14   b , is positioned and held over the groove  102  (also known as the dwell time), where a cycle is determined by the amount of time it takes for a substrate to return to the same position during one oscillation. 
     In general, if P is the polishing rate of the edge portion  14  without using the polishing control groove, and P DES  is the desired polishing rate, then the ratio T 2 /(T 1 +T 2 ) can be set as follows: 
     
       
         
           
             
               
                 
                   
                     P 
                     DES 
                   
                   = 
                   
                     
                       P 
                       ⁡ 
                       
                         [ 
                         
                           1 
                           - 
                           
                             
                               S 
                               1 
                             
                             ⁡ 
                             
                               ( 
                               
                                 
                                   T 
                                   2 
                                 
                                 
                                   
                                     T 
                                     1 
                                   
                                   + 
                                   
                                     T 
                                     2 
                                   
                                 
                               
                               ) 
                             
                           
                         
                         ] 
                       
                     
                     . 
                   
                 
               
               
                 
                   [ 
                   4 
                   ] 
                 
               
             
           
         
       
     
       FIG. 5  is an exemplary graph showing a comparison of wafer edge uniformity from a base line polishing  120  of a substrate (e.g., substrate polishing without time share control) and a time share control polishing  130  of a substrate. The polishing of the central portions of the substrates is comparable between the base line polishing  120  and the time share control polishing  130 , but the polishing of the edge portions of the substrates is approximately 9% higher for the base line polishing  120  that it is for the time share polishing  130 . That is, following the base line removal  120 , the edge portion of the substrate would be approximately 9% thinner than the central portion of the substrate. By using time share control, the substrate edge area was positioned to spend approximately 91% of the time over the polishing area, and approximately 9% of the time over the non-polishing area (e.g., for the time share S). Using time share control polishing  130 , polishing non-uniformity was reduced. 
     Although  FIG. 3A  illustrates using a polishing pad with the polishing control groove  102   a  near the perimeter of the polishing pad  30 , a similar process can be carried out using a polishing pad with the polishing control groove  102   b  near the center of the polishing pad  30 . Referring to  FIGS. 3B and 4B , for the polishing pad with the polishing control groove near the center, P DES  can be calculated in a similar manner as described above, but using equations [5]-[8]. 
     In particular, equation [5] can be used to determine the non-polishing area time share S 2 : 
     
       
         
           
             
               
                 
                   
                     S 
                     2 
                   
                   = 
                   
                     β 
                     180 
                   
                 
               
               
                 
                   [ 
                   5 
                   ] 
                 
               
             
           
         
       
     
     where β is the angle subtended across the substrate  10  by the polishing control groove  102   b  relative to the center of the substrate, and can be determined using equations [6]-[7]: 
     
       
         
           
             
               
                 
                   
                     
                       cos 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       β 
                     
                     = 
                     
                       
                         
                           r 
                           2 
                         
                         + 
                         
                           x 
                           2 
                         
                         - 
                         
                           R 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             2 
                             2 
                           
                         
                       
                       
                         2 
                         ⁢ 
                         xr 
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   and 
                 
               
               
                 
                   [ 
                   6 
                   ] 
                 
               
             
             
               
                 
                   α 
                   = 
                   
                     
                       cos 
                       
                         - 
                         1 
                       
                     
                     ⁢ 
                     
                       
                         
                           r 
                           2 
                         
                         + 
                         
                           x 
                           2 
                         
                         - 
                         
                           R 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             2 
                             2 
                           
                         
                       
                       
                         2 
                         ⁢ 
                         xr 
                       
                     
                   
                 
               
               
                 
                   [ 
                   7 
                   ] 
                 
               
             
           
         
       
     
     and where R2 is the radius of the outer edge of the polishing control groove  102   b , r is the radius of the substrate  10 , and x is the distance from the center of the polishing pad  30  to the center of the substrate  10 . 
     Then if P is the polishing rate of the edge portion  14  without using the polishing control groove, and P DES  is the desired polishing rate, then the ratio T 2 /(T 1 +T 2 ) can be set as follows: 
     
       
         
           
             
               
                 
                   
                     P 
                     DES 
                   
                   = 
                   
                     
                       P 
                       ⁡ 
                       
                         [ 
                         
                           1 
                           - 
                           
                             
                               S 
                               2 
                             
                             ⁡ 
                             
                               ( 
                               
                                 
                                   T 
                                   2 
                                 
                                 
                                   
                                     T 
                                     1 
                                   
                                   + 
                                   
                                     T 
                                     2 
                                   
                                 
                               
                               ) 
                             
                           
                         
                         ] 
                       
                     
                     . 
                   
                 
               
               
                 
                   [ 
                   8 
                   ] 
                 
               
             
           
         
       
     
     Although  FIGS. 3A and 3B  illustrate using a polishing pad with a single polishing control groove  102 , a similar process can be carried out using a polishing pad with two polishing control groove  102   a ,  102   b . Referring now to  FIGS. 3C and 4C , the central portion  12  of the substrate  10  and the edge portion  14  of the substrate  10  is positioned over the polishing area of polishing pad  30  for a first duration T 1  (t 0  to t 1 ). The substrate can be moved laterally in an oscillatory manner during this first duration, or simply move linearly across the polishing pad. At the end of the first duration, the substrate is repositioned with the central portion  12  of the substrate  10  is positioned over the polishing area of the polishing pad  30  and the edge portion  14  of the substrate  10  is positioned over the non-polishing area of the polishing control groove  102   a  near the perimeter of the polishing pad. The substrate can be held at this position for a second duration T 2  (t 1  to t 2 ). Next, the central portion  12  of the substrate  10  and the edge portion  14  of the substrate  10  is positioned over the polishing area of polishing pad  30  for a third duration T 3  (t 2  to t 3 ). The substrate can be moved laterally in an oscillatory manner during this first duration, or simply move linearly across the polishing pad. At the end of the third duration, the substrate is repositioned with the central portion  12  of the substrate  10  is positioned over the polishing area of the polishing pad  30  and the edge portion  14  of the substrate  10  is positioned over the non-polishing area of the polishing control groove  102   b  near the center of the polishing pad. The substrate can be held at this position for a fourth duration T 2  (t 1  to t 2 ). 
     If P is the polishing rate of the edge portion  14  without using the polishing control groove, and P DES  is the desired polishing rate, then the ratios T 2 /(T 1 +T 2 +T 3 +T 4 ) and T 4 /(T 1 +T 2 +T 3 +T 4 ) can be set as follows: 
     
       
         
           
             
               
                 
                   
                     P 
                     DES 
                   
                   = 
                   
                     P 
                     ⁡ 
                     
                       [ 
                       
                         1 
                         - 
                         
                           
                             S 
                             1 
                           
                           ⁡ 
                           
                             ( 
                             
                               
                                 T 
                                 2 
                               
                               
                                 
                                   T 
                                   1 
                                 
                                 + 
                                 
                                   T 
                                   2 
                                 
                                 + 
                                 
                                   T 
                                   3 
                                 
                                 + 
                                 
                                   T 
                                   4 
                                 
                               
                             
                             ) 
                           
                         
                         - 
                         
                           
                             S 
                             2 
                           
                           ⁡ 
                           
                             ( 
                             
                               
                                 T 
                                 4 
                               
                               
                                 
                                   T 
                                   1 
                                 
                                 + 
                                 
                                   T 
                                   2 
                                 
                                 + 
                                 
                                   T 
                                   3 
                                 
                                 + 
                                 
                                   T 
                                   4 
                                 
                               
                             
                             ) 
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   [ 
                   9 
                   ] 
                 
               
             
           
         
       
     
     where S1 and S2 are calculated as described in equations [1]-[3] and [5]-[7]. 
     As used in the instant specification, the term substrate can include, for example, a product substrate (e.g., which includes multiple memory or processor dies), a test substrate, a bare substrate, and a gating substrate. The substrate can be at various stages of integrated circuit fabrication, e.g., the substrate can be a bare wafer, or it can include one or more deposited and/or patterned layers. The term substrate can include circular disks and rectangular sheets. 
     The above described polishing system and methods can be applied in a variety of polishing systems. Either the polishing pad, or the carrier head, or both can move to provide relative motion between the polishing surface and the substrate. The polishing pad can be a circular (or some other shape) pad secured to the platen. The polishing layer can be a standard (for example, polyurethane with or without fillers) polishing material, a soft material, or a fixed-abrasive material. Terms of relative positioning are used; it should be understood that the polishing surface and substrate can be held in a vertical orientation or some other orientation. 
     Particular embodiments of the invention have been described. Other embodiments are within the scope of the following claims. For example, the actions recited in the claims can be performed in a different order and still achieve desirable results.