Patent Publication Number: US-6660637-B2

Title: Process for chemical mechanical polishing

Description:
TECHNICAL FIELD 
     This invention relates to manufacturing semiconductors, and more particularly, to reducing asymmetric polishing of a semi-conductive wafer during off-matched chemical mechanical polishing (“CMP”). 
     BACKGROUND 
     Manufacturing semiconductors involves a complex, multi-step process. One of the steps in this process includes planarizing a semi-conductive wafer in preparation for other steps, such as lithography. 
     Typically, manufacturing semiconductors employs chemical mechanical polishing to planarize a wafer. Chemical mechanical polishing is used because it provides a good overall planarizing performance when polishing the wafer. 
     CMP involves rotating a wafer and a polishing pad at select frequencies and touching the rotating wafer and pad to polish the wafer. A polishing chemical solution may also be applied to facilitate the polishing of the wafer. 
     One method of CMP, known as off-matched CMP, involves rotating a wafer at a first desired frequency and rotating a polishing pad at a second, unequal frequency. Despite better planarizing qualities (e.g., reducing scratches and other nonconformities on a wafer), off-matched CMP may result in an asymmetric polishing of the wafer. 
     SUMMARY 
     The invention relates to chemical mechanical polishing. In one aspect, the invention provides a process for reducing asymmetric polishing of a semi-conductive wafer in off-matched CMP. The process includes rotating a wafer having an alignment mark at a wafer rotation rate and a polishing surface at an off-matched rotation rate. For off-matched CMP, the wafer rotation rate and the off-matched rotation rate are not equal. 
     The wafer, rotating at the wafer rotation rate, and the polishing surface, rotating at the off-matched rotation rate, touch to polish points on the wafer. The wafer rotation rate and the off-matched rotation rate are then adjusted to achieve an approximately zero averaged rotation rate velocity for each point polished on the wafer with respect to the polishing surface, upon completion of a total polishing time. 
     In another aspect, the invention includes a wafer having an alignment mark rotating at a wafer rotation rate and a first polishing pad rotating at an off-matched rotation rate. A wafer carrier holds and rotates the wafer at the wafer rotation rate. Again, the wafer rotation rate and the off-matched rotation rate are not equal. 
     The wafer rotating at the wafer rotation rate and the first polishing pad rotating at the off-matched rotation rate touch to polish points on the wafer. The wafer and the first polishing pad touch for a portion of a total polishing time and then separate. Upon separation, the wafer rotation rate is adjusted to an adjusted wafer rotation rate and a second polishing pad is rotated at an adjusted off-matched rotation rate. 
     The wafer rotating at the adjusted wafer rotation rate and the second polishing pad rotating at the adjusted off-matched rotation rate touch to polish the plurality of points on the wafer. Together, the adjusted wafer rotation rate and adjusted off-match rotation rate cause an approximately zero averaged rotation rate velocity for each point on the wafer with respect to the rotation of a polishing surface. In this aspect the polishing surface is defined by the rotation of the first pad and the second pad polishing the wafer. 
     In yet another aspect of the invention, a CMP process for polishing a semi-conductive wafer includes rotating a wafer having an alignment mark at a wafer rotation rate and a polishing surface at an off-matched rotation rate Again, the wafer rotation rate and the off-matched rotation rate are not equal. 
     The wafer rotating at the wafer rotation rate and the polishing surface rotating at the off-matched rotation rate touch at an initial angle θ i  with respect to the polishing surface. The position of the wafer rotating at the wafer rotation rate is then adjusted with respect to the polishing surface in a manner to achieve an approximately zero averaged rotation rate velocity for each of the points on the wafer with respect to the polishing surface. 
     The details of one or more embodiments of the invention are set forth in the accompanying drawings and the description below. Other features, objects, and advantages of the invention will be apparent from the description and drawings, and from the claims. 
    
    
     DESCRIPTION OF DRAWINGS 
     FIG. 1 a  is a top view of a standard alignment mark. 
     FIG. 1 b  is a top view of a double edge alignment mark. 
     FIG. 2 defines relative rotational parameters for a wafer during CMP. 
     FIG. 3 a  graphs the instantaneous velocity components V x  and V y  for a point (θ equals 0.365Π, r c  equals 0.1 m and r cc  equals 0.3 m) on the wafer in FIG. 2 during matched CMP, where ω c  and ω p  both equal 100 revolutions per minute (“rpm”). 
     FIG. 3 b  graphs the instantaneous velocity components V x  and V y  for a point (θ equals 0.365Π, r c  equals 0.1 m and r cc  equals 0.3 m) on the wafer in FIG. 2 during off-matched CMP, where ω c  equals 120 rpm and ω p-off  equals 60 rpm. 
     FIG. 4 a  shows an alignment mark&#39;s topography after 100 seconds of symmetric polishing. 
     FIG. 4 b  shows the profile evolution of the alignment mark in FIG. 4 a  over periods of time. 
     FIG. 4 c  shows an alignment mark&#39;s topography after 100 seconds of asymmetric polishing. 
     FIG. 4 d  shows the profile evolution of the alignment mark in FIG. 4 c  over periods of time. 
     FIG. 5 is a process  50  according to one embodiment of the invention. 
     FIG. 6 a  graphs the instantaneous velocity components V x1  and V x2  for a point (θ equals 0.365Π, r c  equals 0.1 m and r cc  equals 0.3 m) on the wafer in FIG. 2, where the wafer and polishing surface rotate in the same direction at ω c  equals 120 rpm and ω p-off  equals 60 rpm. 
     FIG. 6 b  graphs the instantaneous velocity components V y1  and V y2  for the point in FIG. 6 a.    
     FIG. 6 c  graphs the average velocity vectors for points on the wafer in FIG. 2, where the wafer and polishing surface rotate in the same direction at ω c  equals 120 rpm and ω p-off  equals 60 rpm. 
     FIG. 6 d  graphs the average velocity vectors for points on the wafer in FIG. 2, at adjusted wafer and off-matched rotation rates to the rotations in FIG. 6 c.    
     FIG. 7 a  graphs the instantaneous velocity components V x1  and V x2  for a point (θ equals 0.365Π, r c  equals 0.1 m and r cc  equals 0.3 m) on the wafer in FIG. 2, where the wafer and polishing surface rotate in opposite directions at ω c  equals 120 rpm and ω p-off  equals 60 rpm. 
     FIG. 7 b  graphs the instantaneous velocity components V y1  and V y2  for the point in FIG. 7 a.    
     FIG. 7 c  graphs the average velocity vectors for points on the wafer in FIG. 2, where the wafer and polishing surface rotate in opposite directions at ω c  equals 120 rpm and ω p-off  equals 60 rpm. 
     FIG. 7 d  graphs the average velocity vectors for points on the wafer in FIG. 2, at adjusted wafer and off-matched rotation rates to the rotations in FIG. 7 c.    
     FIG. 8 is a process  80  according to one embodiment of the invention. 
     FIG. 9 is a process  90  according to one embodiment of the invention. 
     FIG. 10 is a top view of a wafer showing initial angle θ 1  and adjustment angles θ 2 , θ 3  and θ 4 . 
     FIG. 11 graphs the instantaneous velocity components V x1-4  and V y1-4  for a point (θ 1  equals 0.365Π, r c  equals 0.1 m and r cc  equals 0.3 m) on the wafer in FIG. 2 according the adjustment angles in FIG. 10, where ω c  equals 120 rpm and ω p-off  equal 60 rpm. 
     FIG. 12 graphs the sums V x  and V y  of the instantaneous velocity components V x1-4  and V y1-4  in FIG.  11 . 
    
    
     Like reference symbols in the various drawings indicate like elements. 
     DETAILED DESCRIPTION 
     Alignment marks,  12  or  14  (FIG.  1 ), are typically formed on a wafer  22  (FIG. 2) prior to CMP. Alignment marks  12  or  14  generally include depressed areas  13  and  15 . Depressed areas  13  and  15  typically operate to ensure the proper alignment of wafer  22  during the manufacturing process. 
     Asymmetric polishing of wafer  22  during off-matched CMP may distort alignment marks  12  or  14 . Any distortion in these marks may cause an inaccurate placement of wafer  22  during subsequent manufacturing steps. 
     The following embodiments reduce asymmetric polishing of wafer  22  during off-matched CMP. In particular, asymmetric polishing may be reduced by adjusting the relative rotation rates of wafer  22  and polishing surface  24  to achieve an approximately zero average rotation rate velocity for each point  23 ,  25  and  27  on wafer  22  with respect to polishing surface  24  over a total polishing time t p  (i.e., a zero averaged relative velocity vector for each point  23 ,  25  and  27  polished on wafer  22  by polishing surface  24  ). 
     CMP involves touching a wafer  22  (FIG. 2) rotating at a wafer rotation rate ω c  and a polishing surface  24  rotating at a pad rotation rate ω p  to polish a plurality of points  23 ,  25  and  27  on wafer  22 . A wafer carrier (not shown) rotates wafer  22  at wafer rotation rate ω c  and holds a surface, defined by points  23 ,  25  and  27  on wafer  22 , to polishing surface  24 . 
     CMP processes may be divided into two categories; matched CMP and off-matched CMP. In matched CMP, wafer  22  and polishing surface  24  rotate at the same rotation rate to polish points  23 ,  25  and  27 . In other words, wafer rotation rate ω c  and polishing pad rotation rate ω p  are equal in magnitude as wafer  22  and polishing surface  24  touch. As a result, the average rotation rate velocity for each point (e.g., point  25 ) polished on wafer  22  with respect to polishing surface  24  is approximately zero. 
     In off-matched CMP, wafer  22  and polishing surface  24  rotate at unequal or varying rotation rates to polish wafer  22 . In other words, wafer rotation rate ω c  and polishing pad rotation rate ω p , herein referred to as off-matched rotation rate ω p-off , are not equal in magnitude as wafer  22  and polishing surface  24  touch. As a result, the average rotation rate velocity for each point (e.g. point  25  ) polished on wafer  22  with respect to polishing surface  24  is non-zero. Heretofore, the non-zero average relative rotation rate velocity in off-matched CMP caused asymmetric polishing of wafer  22 . 
     The average rotation rate velocity (i.e., the relative velocity vector) for each point polished on wafer  22  may be determined by comparing the rotation of each point  23 ,  25  and  27  with the rotation of polishing surface  24  on pad  26 . For example, the average rotation rate velocity at point  25 , here (r c , θ), calculated with respect to the polishing surface  24  and the coordinate system shown in FIG. 2 is:          v   →     =             r   →     p     ×       ω   →     p       -         r   →     c     ×       ω   →     c         =               r   →     cc     ×       ω   →     p       +         r   →     c     ×     (         ω   →     p     -       ω   →     c       )              
     ⇒     v   →       =         r   cc          ω   p       |             cos        (       ω   c        t     )                 sin        (       ω   c        t     )             +       r   c        Δω       |           cos                   (   θ   )                 sin        (   θ   )                                   
     wherein 
     
       
         Δω=ω p −ω c   
       
     
     FIGS. 3 a  and  3   b  graph the instantaneous X and Y velocity components of point  25  at θ=0.365Π, r c =0.1 m and r cc =0.3 m on wafer  22  as polishing surface  24  and wafer  22  touch. The instantaneous X and Y velocity components V x  and V y  are graphed as functions of velocity, in meters per second, over time in seconds based on the movement of point  25  though Cartesian coordinate system  28  (FIG.  2 ). 
     FIG. 3 a  shows V x  and V y  for matched CMP, where wafer  22  and polishing surface  24  rotate at the same frequencies but in opposite directions to polish wafer  22  (e.g., ω c  and ω p  both equal at 100 rpm). Over each complete rotation of wafer  22  and polishing surface  24 , the average rotational rate velocity of point  25  in the X and Y direction is zero. This zero average rotation rate velocity causes point  25  to be symmetrically polished. 
     FIG. 3 b  shows V x  and V y  for off-matched CMP, where wafer  22  and polishing surface  24  rotate at unequal rotation rates to polish wafer  22  (e.g., ω c  equals 120 rpm and ω p-off  equals 60 rpm). Over each complete rotation of wafer  22 , the average rotational velocity of point  25  in the X and Y direction with respect to polishing surface  24  is non-zero. Hence, off-matched CMP causes point  25  to be polished asymmetrically. 
     FIGS. 4 a  and  4   b  show the result of symmetric polishing on alignment mark  44   a . In particular, FIG. 4 a  shows little or no distortion in the topography of mark  44   a  after 100 seconds of symmetric polishing. FIG. 4 b  shows the little or no distortion in the profile of mark  44   a  after 40 seconds, 120 seconds and 200 seconds of symmetric polishing. 
     FIGS. 4 c  and  4   d , on the other hand, show the result of asymmetric polishing on alignment mark  44   c . In particular, FIG. 4 c  shows a shift  45  in the topography of mark  44   c  in a dominant direction of the average velocity vector  46  after 100 seconds of asymmetric polishing. FIG. 4 d  shows an uneven, distorted profile of mark  44   c  after 40 seconds, 120 seconds and 200 seconds of asymmetric polishing. 
     Ensuring that the average rotation rate velocity of each point  23 ,  25  and  27  polished on wafer  22  is approximately zero with respect to polishing surface  24  reduces asymmetric polishing during off-matched CMP. The average rotation rate velocity for each point polished on wafer  22  may be determined as an X-component and Y-component with respect to polishing surface  24  based on the Cartesian coordinate system  28  (FIG.  2 ). The average rotational rate velocities in the X and Y-directions equal zero when:            v   _     x     =           ∑   t          (       v   x1     +     v   x2       )         t   p       =   0                 v   _     y     =           ∑   t          (       v   y1     +     v   y2       )         t   p       =   0                     
     Thus, asymmetric polishing of wafer  22  may be reduced during off-matched CMP by achieving an average rotation rate velocity for both X and Y directions (i.e., {overscore (ν)} x  and {overscore (ν)} y ) of approximately zero for every point polished on wafer  22 . 
     FIG. 5 shows a process  50  to reduce asymmetric polishing of wafer  22  during off-matched CMP. In particular, process  50  rotates ( 501 ) a wafer  22  having an alignment mark (e.g.,  12  or  14 ) at a wafer rotation rate ω c  and rotates ( 503 ) a polishing surface  24  at an off-matched rotation rate ω p-off . As explained above, wafer rate ω c  and off-matched rotation rate ω p-off  are not equal in magnitude for off-matched CMP. 
     Process  50  touches ( 505 ) wafer  22  rotating at wafer rotation rate ω c  and polishing surface  24  rotating at off-matched rotation rate ω p-off  to polish points  23 ,  25 , and  27 . Process  50  touches ( 505 ) wafer  22  rotating at ω c  and polishing surface  24  rotating at ω p-off  for a portion of a total polishing time t p . 
     Upon completion of the portion of total polishing time t p , process  50  adjusts ( 507 ) the wafer rotation rate ω c  of wafer  22  and the off-matched rotation rate ω p-off  of the polishing surface  24 . Process  50  adjusts ( 507 ) the rotation rates ω c  and ω p-off  to achieve an approximately zero averaged rotation rate velocity for each point  23 ,  25  and  27  polished on wafer  22  with respect to polishing surface  24  upon completion of the total polishing time t p . Here, the total polishing time t p  equals the amount of time needed to achieve a satisfactory polish of wafer  22 . Process  50  may adjust ( 507 ) the wafer rotation rate ω c  and off-matched rotation rate ω p-off  in any manner, including for example changing the frequency, direction and/or angle θ (FIG. 10) of wafer  22  with respect to polishing surface  24  to achieve an average relative rotation rate velocity for each polished point of approximately zero upon completion of a total polishing time t p . 
     For example, process  50  may simultaneously reverse the rotation of wafer  22  and polishing pad  24  in a continuous motion. 
     In another embodiment process  50  may separate wafer  22  and polishing surface  24  to adjust the wafer rotation rate ω c  and the off-matched rotation rate ω p-off  upon completion of the portion of the total polishing time and then touch wafer  22  and surface  24  rotating at adjusted rates to continue polishing of points  23 ,  25  and  27 . In both such embodiments, the average polishing time at the original ( 501  and  503 ) and adjusted ( 507 ) rotations may be approximately equal. In still other embodiments, several separations of wafer  22  and surface  24  may occur to adjust ( 507 ) the wafer and off-matched rotation rates to achieve the approximately zero averaged rotation rate velocity for each point  23 ,  25  and  27 . 
     FIGS. 6 a - 6   d  show relative instantaneous and average velocity information for one embodiment of the invention in which wafer  22  rotates in the same direction (e.g., clockwise) as polishing surface  24 , but at different rates, to polish points  23 ,  25  ands  27 . In particular, FIGS. 6 a-d  show that the average rotation rate velocity of wafer  22  with respect to polishing surface  24  will be zero for each point on wafer  22  touched by polishing surface  24  provided wafer  22  and polishing surface  24  rotate ( 501  and  503 ) in the same direction (e.g., clockwise) for approximately half of a total polishing time at wafer rotation rate ω c  and off-matched rotation rate ω p-off  and for the remainder of the total polishing time at an adjusted ( 507 ) wafer rotation ω c  and off-matched rate ω p-off , rotating at their previous frequency but in the reverse direction (i.e. counterclockwise). 
     FIGS. 6 a  and  6   b  graph the instantaneous X and Y velocity components of point  25  at θ=0.365Π, r c =0.1 m and r cc =0.3 m on wafer  22  as polishing surface  24  and wafer  22  touch. In particular, V x1 , V x2 , V y1  and V y2  are graphed as functions of velocity, in meters per second, over time in seconds based on the movements of point  25  though Cartesian coordinate system  28 , in FIG.  2 . 
     FIG. 6 a  shows V x1 , the relative instantaneous velocity of point  25  in the X-direction for an original ( 501  and  503 ) wafer rotation rate (e.g., ω c  equals 120 rpm) and an off-matched rotation rate (e.g., ω p-off  equal 60 rpm) rotating in the same direction (e.g. clockwise) will be negated by V x2 . Here, V x2  is the instantaneous relative velocity of point  25  in the X-direction at an adjusted ( 507 ) wafer rotation rate (i.e., ω c  equals 120 rpm) and an off-matched rotation rate (i.e., ω p-off  equals 60 rpm) rotating in the reverse direction (i.e. counterclockwise) over an equal time period, here t(s). Hence, by combining V x1  and V x2  process  50  achieves a zero average velocity in the X direction over total polishing time t p , here six seconds, or 2t(s). 
     FIG. 6 b  shows the same instantaneous velocity information as FIG. 6 a  for the Y-direction in the above example. Similarly, V y1  and V y2  also achieve a zero average velocity in the Y direction over the total polishing time t p . Thus, symmetric off-matched CMP of wafer  22  may be achieved for wafer  22  and polishing surface  24  rotating at different frequencies but in the same direction. 
     FIGS. 6 c  and  6   d  provide average velocity vector maps for wafer  22  with respect to polishing surface  24 . In particular, each arrow  62  in FIG. 6 c  represents the average velocity vector for that point on wafer  22  during the portion t(s) of the total polishing time at the original ( 501  and  503 ) wafer rotation rate and off-matched rotation rate (e.g., ω c  equals 120 rpm and ω p-off  equals 60 rpm), both rotating in the clockwise direction. Likewise, each arrow  64  in FIG. 6 d  represents the same information for that point on wafer  22  for the remainder t(s) of the total polishing time at the adjusted ( 507 ) wafer rotation rates and the adjusted off-matched rotation rate (i.e., ω c  equals 120 rpm and ω p-off  equals 60 rmp), both rotating in the counterclockwise direction. Adding each relative velocity vector  62  and  64  together on FIGS. 6 c  and  6   d  shows that a zero average relative velocity vector (i.e., a zero relative average velocity rate) may be achieved for each point on wafer  22  by reversing the direction of both wafer  22  and surface  24  after approximately half of a total polishing time, here t(s). 
     FIGS. 7 a - 7   d  show the same relative instantaneous and average velocity information as FIGS. 6 a-d  but for another embodiment of the invention. In particular, FIGS. 7 a-d  show that the average rotation rate velocity of wafer  22  with respect to polishing surface  24  will be zero for each point  23 ,  25  and  27  on wafer  22  touched by polishing surface  24 , provided wafer  22  and polishing surface  24  rotate ( 501  and  503 ) in opposite directions (e.g., wafer  22  rotates clockwise when polishing surface  24  rotates counterclockwise) for approximately half of a total polishing time at wafer rotation rate ω c  and off-matched rotation rate ω p-off  and for a remainder of the total polishing time at an adjusted ( 507 ) wafer rotation ω c  and off-matched rate ω p-off  rotating at their previous frequency but in reverse directions (i.e. wafer  22  rotates counterclockwise and polishing surface  24  rotates clockwise). 
     FIGS. 7 a  and  7   b  graph the instantaneous X and Y velocity components of point  25  at θ=0.365Π, r c =0.1 m and r cc =0.3 m on wafer  22  as polishing surface  24  and wafer  22  touch. In particular, V x1 , V x2 , V y1  and V y2  are graphed as functions of velocity, in meters per second, over time in seconds based on the movements of point  25  though Cartesian coordinate system  28 , in FIG.  2 . 
     FIG. 7 a  shows V x1 , the relative instantaneous velocity of point  25  in the X-direction for an original ( 501  and  503 ) wafer rotation rate (e.g., ω c  equals 120 rpm) and an off-matched rotation rate (e.g., ω p-off  equal 60 rpm) rotating in opposite directions (e.g. clockwise for wafer  22  and counterclockwise for polishing surface  24 ) will be negated by Vx 2 . Here, V x2  is the instantaneous relative velocity of point  25  in the X-direction at an adjusted ( 507 ) wafer rotation rate and an off-matched rotation rate rotating at their previous frequencies (i.e., ω c  equals 120 rpm and ω p-off  equals 60 rpm), but in reverse directions (i.e., counterclockwise for wafer  22  and clockwise for polishing surface  24 ) over an equal time period, here t(s). Thus, by combining V x1  and V x2  process  50  achieves a zero average velocity in the X-direction for point  25  with respect to polishing surface  24  over the total polishing time t p , here six seconds, or 2t (s). 
     FIG. 7 b  shows the same instantaneous velocity information as FIG. 7 a  for the Y-direction in the above example. Similarly, V y1  and V y2  also achieve a zero average velocity in the Y-direction for point  25  with respect to polishing surface  24  over the same total polishing time t p . Thus, symmetric off-matched CMP of wafer  22  may be achieved for wafer  22  and polishing surface  24  rotating at different frequencies and in opposite directions. 
     FIGS. 7 c  and  7   d  provide average velocity vector maps for wafer  22  with respect to polishing surface  24 . In particular, each arrow  72  in FIG. 7 c  represents the average velocity vector for that point on wafer  22  during the portion of the total polishing time at the original ( 501  and  503 ) wafer rotation rate and off-matched rotation rate (e.g., ω c  equals 120 rpm in the clockwise direction and ω p-off  equals 60 rpm in the counterclockwise direction), both rotating in opposite directions. Likewise, each arrow  74  in FIG. 7 d  represents the same information for that point on wafer  22  for the remainder of the total polishing time at the adjusted ( 507 ) wafer rotation rate and the adjusted ( 507 ) off-matched rotation rate (i.e., ω c  equals 120 rpm in the counterclockwise direction and ω p-off  equals 60 rmp in the clockwise direction), both rotating at their previous frequency but in opposite directions. Adding the relative velocity vectors  72  and  74  together FIGS. 7 c  and  7   d  shows that a zero average relative velocity vector (i.e., a zero average rotation rate velocity) may be achieved for each point on wafer  22  by reversing the direction of both wafer  22  and surface  24  after approximately half of a total polishing time, here t(s). 
     FIG. 8 shows an alternate embodiment process  80  to reduce asymmetric polishing of wafer  22  during off-matched CMP. In particular, process  80  rotates ( 801 ) a wafer  22  having an alignment mark (e.g.,  12  or  14 ) at a wafer rotation rate ω c  and rotates ( 803 ) a polishing surface  24  at an off-matched rotation rate ω p-off . 
     Process  80  touches ( 805 ) wafer  22  rotating at wafer rotation rate ω c  and polishing surface  24  rotating at off-matched rotation rate ω p-off  to polish points  23 ,  25 , and  27 . Process  80  touches ( 805 ) wafer  22  rotating at ω c  and polishing surface  24  rotating at ω p-off  for a portion of a total polishing time. 
     Upon completion of the portion of the total polishing time, process  80  separates ( 807 ) wafer  22  and polishing surface  24  and adjusts ( 809 ) the wafer rotation rate ω c . Process  80  also rotates ( 811 ) a second polishing pad at an adjusted off-matched rotation rate ω p-off . Process  80  touches ( 813 ) wafer  22  rotating at the adjusted ( 809 ) wafer rotation rate ω c  and the second polishing pad rotating at adjusted ( 811 ) off-matched rotation rate ω p-off  to polish the points  23 ,  25  and  27  on wafer  22  and achieve an approximately zero averaged rotation rate velocity for each point  23 ,  25  and  27  on wafer  22  with respect to polishing surface  24  upon completion of the total polishing time t p . 
     FIG. 9 shows an alternative embodiment process  90  to reduce asymmetric polishing of wafer  22  during off-matched CMP. In particular, process  90  rotates ( 901 ) a wafer  22  having an alignment mark (e.g.  12  or  14 ) at a wafer rotation rate ω c  and rotates ( 903 ) a polishing surface  24  at an off-matched rotation rate ω p-off . 
     Process  90  touches ( 905 ) wafer  22  rotating at wafer rotation rate ω c  and polishing surface  24  rotating at off-matched rotation rate ω p-off  at an initial angle θ i  (FIG. 10) with respect to polishing surface  24  to polish points  23 ,  25 , and  27 . Process  90  touches ( 905 ) wafer  22  rotating at ω c  and polishing surface  24  rotating at ω p-off  for a portion of a total polishing time. 
     Upon completion of the portion of the total polishing time, process  90  adjusts ( 907 ) the rotation of wafer  22  rotating at wafer rotation rates ω c  with respect to polishing surface  24  rotating at off-matched rotation rate ω p-off . Process  90  adjusts ( 907 ) to achieve an approximately zero average rotation rate velocity for each point  23 ,  25  and  27  polished on wafer  22  with respect to polishing surface  24  upon completion of a total polishing time t p . 
     Process  90  may adjust ( 907 ) the rotation of wafer  22  in any manner to achieve a zero average rotation rate velocity in both the X and Y-directions upon completion of total polishing time t p . For example, process  90  may divide the total polishing time t p  into a plurality of periods p and separate wafer  22  and polishing surface  24  upon completion of each of period p to adjust the position of wafer  22  with respect to polishing surface  24  by an adjustment angle θ a . Adjustment angle θ a  (i.e., the relative angular position of wafer  22  with respect to polishing surface  24  from initial angle θ i ) may be computed from a point  25  (r c , θ i ) on wafer  22  based on the number of periods p, provided all of the periods are apportioned equally, as follows: 
     
       
         θ a =θ i +2 Π/p   
       
     
     where p is the number of equally divided periods. 
     After adjusting ( 907 ) the position of wafer  22  by adjustment angle θ a , the adjusted wafer  22  and polishing surface  24  touch for the equally apportioned period of the total polishing time. Upon completions of all of the periods, and hence the total time period t p , the average rotation rate for each point polished on wafer  25  will be approximately zero with respect to polishing surface  24 . 
     FIG. 10 shows wafer  22  polished by process  90  in the manner as described above. Here, a total time period t p  is divided into four portions of equal time periods p 1 , p 2  p 3  and p 4 . Thus, the adjustment angle θ a  is θ i +Π/2 for period p 2 , θ 1 +Π for period p 3 , and θ i +3Π/4 for period p 4 . 
     FIGS. 11 a - 11   d  graph the instantaneous X and Y velocity components of point  25  at θ i =0.365Π, r c =0.1 m and r cc =0.3 m for the positions  1  (r c , θ i ),  2  (r c , θ 2  equals θ i+Π/2 )  3  (r c , θ 3  equals θ i+Π ) and  4  (r c , θ 4  equals θ i+3Π/4 ) on wafer  22  (FIG. 10) as polishing surface  24  and wafer  22  touch. The X and Y velocity components V x1-4  and V y1-4  are graphed as instantaneous functions of velocity, in meters per second, over time in seconds based on the movement of point  25  though Cartesian coordinate system  28  (FIG.  2 ). 
     FIG. 11 a  graphs instantaneous velocity V x1  and V y1  for point  25  in process  90 , where wafer  22  and polishing surface  24  rotates at wafer and off-matched rotation rates to polish wafer  22  (e.g., ω c  equals 120 rpm and ω p-off  equals 60 rpm). Here, the instantaneous velocity V x1  and V y1  graph point  25  on wafer  22  which first touches polishing surface  24  at an initial angle θ i  with respect Cartesian coordinate system  28 . 
     FIGS. 11 b-d  graph the instantaneous velocity V x1-4  and V y2-4  for point  25  adjusted by its proper adjustment angles (e.g., θ 2  equals θ i+Π/2 , θ 3  equals θ i+Π  and θ 4  equals θ i+3Π/4 ) to achieve a zero average rotation rate velocity for point  25  polished on wafer  22  with respect to polishing surface  24  upon completion of a total polishing time t p . 
     Finally, FIG. 12 graphs the sum of velocity components V x1-4  and V y1-4 , where V x  equals the sum of V x1-4  and V y  equals the sum of V y1-4 . Together, FIG.  11  and FIG. 12 show a zero average rotation rate velocity may be achieved by controlling the rotation of wafer  22  based on adjustment angles θ a  (e.g., θ 2 , θ 3  and θ 4 ) with respect to surface  24 . 
     A number of embodiments of the invention have been described. Nevertheless, it will be understood that various modifications may be made without departing from the spirit and scope of the invention. For example, the processes  50 ,  80  and  90  may be carried out in steps on different machines or on a single apparatus. The wafer rotation rate and/or off-matched rotation rate may be also adjusted in unequal intervals to produce a zero averaged rotation rate velocity during off-matched CMP. Furthermore, adjustment angles θ a  may be computed for any number of periods (p=2, 3, 4 . . .) desired by a manufacturer. Accordingly, other embodiments are within the scope of the following claims.