Control of processing parameters for substrate polishing with substrate precession
Generating a recipe for a polishing process includes receiving a target removal profile that includes a target thickness to remove for a plurality of locations spaced angularly around a center of a substrate, storing a first function providing substrate orientation relative to the carrier head over time, storing a second function defining a polishing rate below a zone of the zone as a function of one or more pressures of one or more zones from a plurality of pressurizable zones of the carrier head that are spaced angularly around the center of the substrate, and for each particular zone of the plurality of zones, calculating a recipe defining a pressure for the particular zone over time. Calculating the recipe includes calculating an expected thickness profile after polishing from the second function defining the polishing rate and the first function providing substrate orientation relative to the zone over time, and applying a minimizing algorithm to reduce a difference between the expected thickness profile and the target thickness profile.
1. A computer program product for generating a recipe for controlling a polishing system, the computer program product encoded on a non-transitory computer readable medium and comprising instructions for causing one or more computers to:
receive a target thickness profile that includes a target thickness for a plurality of locations spaced angularly around a center of a substrate;
store a first function providing substrate orientation relative to a carrier head over time;
store a second function defining a polishing rate below a pressurizable zone from a plurality of pressurizable zones of the carrier head as a function of one or more pressures of one or more zones from the plurality of pressurizable zones of the carrier head, the plurality of pressurizable zones spaced angularly around an axis of rotation of the carrier head;
for each particular zone of the plurality of zones, calculate a recipe defining a pressure for the particular zone over time, wherein the instructions to calculate the recipe include instructions to calculate an expected thickness profile after polishing from the second function defining the polishing rate and the first function providing substrate orientation relative to the zone over time, and instructions to apply a minimizing algorithm to reduce a difference between the expected thickness profile and the target thickness profile; and
cause the carrier head to apply, for each pressurizable zone of the plurality of zones, the pressure over time as defined by the recipe.
2. The computer program product of claim 1 , wherein the first function maps physical zones of the head to locations on the substrate.
3. The computer program product of claim 1 , wherein the first function assumes a constant precession rate of the substrate relative to the carrier head.
4. The computer program product of claim 3 , wherein the first function includes calculating angular offset based on an initial angular offset and the precession rate.
5. The computer program product of claim 4 , wherein the first function includes calculating the precession rate based on a carrier head rotation rate.
6. The computer program product of claim 1 , wherein the second function comprises a matrix based on the Preston equation.
7. The computer program product of claim 1 , wherein calculating the recipe includes minimizing a cost function that incorporates the first function and the second function.
8. The computer program product of claim 7 , wherein the cost function is consistent with x[t]=B[t]*u[t], wherein x[t] is a vector representing pressures on the substrate as a function of time, u[t] is a vector representing polishing parameters as a function of time, and B[t] is a selector matrix that varies over time consistent with the first function.
9. The computer program product of claim 8 , wherein B[t] depends on an angular position of a region on the substrate with respect to a zone on the carrier head as a function of time (t).
10. The computer program product of claim 7 , wherein the cost function is subject to evolution constraint of x[t+1]=A*x[t]+B[t]*u[t].
11. The computer program product of claim 10 , wherein A is a matrix that transforms coordinate positions based on rotation of substrate orientation relative to the carrier head over time.
12. The computer program product of claim 7 , wherein the cost function is subject to the constraint x[t]=C[t]*y[t], where y[t] represent measurements over time from an in-situ monitoring system.
13. The computer program product of claim 12 , wherein C[t] represents a matrix that provides mapping of individual measurements from a zone on the carrier head to a location on the substrate.
14. A polishing system, comprising:
a platen to support a polishing pad;
a carrier head to hold a substrate against the polishing pad, the carrier head including a plurality of pressurizable chambers spaced angularly around an axis of rotation of the carrier head; and
a controller configured to
receive a target removal profile that includes a target thickness to remove for a plurality of locations spaced angularly around a center of a substrate,
store a first function providing substrate orientation relative to the carrier head over time,
store a second function defining a polishing rate as a function of one or more pressures of one or more zones from a plurality of pressurizable zones of the carrier head, the plurality of pressurizable zones spaced angularly around an axis of rotation,
for each particular zone of the plurality of zones, calculate a recipe defining a pressure for the particular zone over time by calculating an expected thickness profile after polishing from the second function defining the polishing rate and the first function providing substrate orientation relative to the zone over time, and applying a minimizing algorithm to reduce a difference between the expected thickness profile and the target thickness, and
cause each respective chamber of the plurality of pressurizable chambers to apply the pressure in accordance with the recipe.
15. The system of claim 14 , wherein the first function assumes a constant precession rate of the substrate relative to the carrier head.
16. The system of claim 15 , wherein the first function includes calculating angular offset based on an initial angular offset and a precession rate.
17. The system of claim 16 , wherein the first function includes calculating the precession rate based on a carrier head rotation rate.