Control of processing parameters for substrate polishing with angularly distributed zones using cost function
Generating a recipe for controlling a polishing system includes receiving a target removal profile that includes a target thickness to remove for a plurality of locations on a substrate that are angularly distributed around the substrate, and storing a first function defining a polishing rate for a zone from a plurality of pressurizable zones of a carrier head that are angularly distributed around a the carrier head. The first function defines polishing rates as a function of pressures. For each particular zone of the plurality of zones a recipe defining a pressure for the particular zone over time is calculated by calculating an expected thickness profile after polishing using the first function, and minimizing a cost function that incorporates a first term representing a difference between the expected thickness profile and a 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 removal profile that includes a target thickness to remove for a plurality of locations on a substrate that are angularly distributed around a center of the substrate;
store a first function defining a polishing rate for a zone from a plurality of pressurizable zones of a carrier head that are angularly distributed at different angular positions around a center of the carrier head, the first function defining the polishing rate for the zone as a function of one or more pressures of one or more zones from the plurality of pressurizable zones of the carrier head; and
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 using the first function, and to minimize a cost function that incorporates a first term representing a difference between the expected thickness profile and a target thickness profile, wherein an output of the minimization of the cost function is a scalar quantity representing a summed removal error.
2. The computer program product of claim 1 , comprising instructions to determine a polishing rate for each zone of the plurality of zones at a uniform frequency.
3. The computer program product of claim 2 , wherein the first term is represented at least in part by
∑
t
=
0
T
B
(
t
)
u
(
t
)
·
Δ
t
-
R
where B[t] is a selector matrix, u[t] is a vector representing polishing parameters as a function of time, R is the target thickness profile, and Δt is a time step in the summation.
4. The computer program product of claim 3 , wherein the selector matrix B[t] represents an orientation of the substrate relative the carrier head over time.
5. The computer program product of claim 1 , wherein the first function defining the polishing rate comprises a matrix based on a Preston equation.
6. The computer program product of claim 1 , wherein the cost function includes a second term representing variation in pressure over time.
7. The computer program product of claim 6 , wherein the second term is a sum of pressure differences between consecutive pressures applied by the particular zone.
8. The computer program product of claim 7 , wherein the second term is represented at least in part by:
λ
∑
t
=
Δ
t
T
u
(
t
)
-
u
(
t
-
Δ
t
)
where u[t] is a vector representing polishing pressures as a function of time, Δt is a time step in the summation, and a λ is a weight for the second term relative to the first term.
9. The computer program product of claim 8 , wherein the λ is about 0.01 to 0.1.
10. The computer program product of claim 1 , wherein a second term is a sum of addends with each addend being a polishing rate multiplied by time for which the polishing rate is applied.
11. A method for generating a recipe for controlling a polishing system, the method comprising:
receiving a target removal profile that includes a target thickness to remove for a plurality of locations on a substrate that are angularly distributed around a center of the substrate;
storing a first function defining a polishing rate for a zone from a plurality of pressurizable zones of a carrier head that are angularly distributed at different angular positions around a center of the carrier head, the first function defining the polishing rate for the zone as a function of one or more pressures of one or more zones from the plurality of pressurizable zones of the carrier head; and
for each particular zone of the plurality of zones, calculating a recipe defining a pressure for the particular zone over time by calculating an expected thickness profile after polishing using the first function and minimizing a cost function that incorporates a first term representing a difference between the expected thickness profile and a target thickness profile, wherein an output of the minimization of the cost function is a scalar quantity representing a summed removal error.
12. The method of claim 11 , comprising determining a polishing rate for each zone of the plurality of zones at a uniform frequency.
13. The method of claim 12 , wherein the first term is represented at least in part by
∑
t
=
0
T
B
(
t
)
u
(
t
)
·
Δ
t
-
R
where B[t] is a selector matrix, u[t] is a vector representing polishing pressures as a function of time, R is the target thickness profile, and Δt is a time step in the summation.
14. The method of claim 13 , wherein the selector matrix B[t] represents an orientation of the substrate relative the carrier head over time.
15. The method of claim 11 , further comprising polishing the substrate according to the recipe.
16. 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 on a substrate that are angularly distributed around a center of the substrate,
store a first function defining a polishing rate for each zone of a plurality of pressurizable zones of the carrier head that are angularly distributed at different angular positions around a center of the carrier head, the first function defining the polishing rate for the zone as a function of one or more pressures of one or more chambers from the plurality of pressurizable chambers of the carrier head, and
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 using the first function and minimizing a cost function that incorporates a first term representing a difference between the expected thickness profile and a target thickness profile, wherein an output of the minimization of the cost function is a scalar quantity representing a summed removal error.
17. The system of claim 16 , wherein the controller is configured to determine a polishing rate for each zone of the plurality of zones at a uniform frequency.
18. The system of claim 17 , wherein the first term is represented at least in part by
∑
t
=
0
T
B
(
t
)
u
(
t
)
·
Δ
t
-
R
where B[t] is a selector matrix, u[t] is a vector representing polishing pressures as a function of time, R is the target thickness profile, and Δt is a time step in the summation.
19. The system of claim 18 , wherein the selector matrix B[t] represents an orientation of the substrate relative the carrier head over time.