Control of processing parameters during substrate polishing using cost function
Controlling a polishing system includes receiving from an in-situ monitoring system, for each region of a plurality of regions on a substrate being processed by the polishing system, a sequence of characterizing values for the region. For each region, a polishing rate is determined for the region, and an adjustment is calculated for at least one processing parameter. Calculation of the adjustment includes minimizing a cost function that includes, for each region, i) a difference between a current characterizing value or an expected characterizing value at an expected endpoint time and a target characterizing value for the region, and ii) a plurality of a projected future pressure changes over time for the region and/or a plurality of differences between projected future pressures over time and a baseline pressure for the region.
1. A computer program product for controlling a polishing system comprising a non-transitory computer readable medium comprising instructions for causing one or more computers to:
receive from an in-situ monitoring system, for each region of a plurality of regions on a substrate being processed by the polishing system, a sequence of characterizing values for the region;
for each region, determine a polishing rate for the region;
for each particular parameter update time of a plurality of parameter update times, calculate an adjustment for at least one polishing control parameter of the polishing system, wherein calculating the adjustment includes minimizing a cost function that includes, for each region
i) a difference between a current characterizing value or an expected characterizing value at an expected endpoint time and a target characterizing value for the region, and
ii) a plurality of projected future pressure changes to occur at parameter update times subsequent to the particular parameter update time for the region and/or a plurality of differences between projected future pressures to occur at parameter update times subsequent to the particular parameter update time and a baseline pressure for the region; and
for each particular parameter update time, adjust to the at least one polishing control parameter of the polishing system based on the calculated adjustment.
2. The computer program product of claim 1 , wherein the cost function includes, for each region, the plurality of projected future pressure changes to occur at parameter update times subsequent to the particular parameter update time for the region.
3. The computer program product of claim 1 , wherein the cost function includes, for each region, the plurality of differences between projected future pressures to occur at parameter update times subsequent to the particular parameter update time and the baseline pressure for the region.
4. The computer program product of claim 1 , wherein the cost function includes, for each region, the plurality of projected future pressure changes to occur at parameter update times subsequent to the particular parameter update time for the region and the plurality of differences between projected future pressures to occur at parameter update times subsequent to the particular parameter update time and the baseline pressure for the region.
5. The computer program product of claim 4 , wherein the cost function is a function of a square of each difference between the expected characterizing value at the end of polish and the target characterizing value for the region, a function of a square of each projected future pressure change, a square of each difference between the projected future pressure and the baseline pressure, and a weighted vector norm of expected future polish rates.
6. The computer program product of claim 5 , wherein the cost function includes a first constant for each region and the cost function is a function of the first constant multiplied by the square of the difference between the expected characterizing value and the target characterizing value for the region.
7. The computer program product of claim 6 , wherein the cost function includes a second constant for each region and the cost function is a function of the second constant multiplied by the square of each projected future pressure change.
8. The computer program product of claim 4 , wherein the cost function includes terms corresponding to
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where τ represents an ordinal number in a sequence projected future pressure changes, x(τ) represents a vector including differences between target values for the characterizing values at τ and measured characterizing values at τ, polishing rates at τ, and differences between estimated pressures at τ baseline pressures at τ, Q(τ) represents a weighting matrix for x(τ), u(τ) represents estimated pressure changes at τ, and R represents a weighting matrix for u(τ), x(T) represents a vector including differences between target values for the characterizing values at a final time T and estimated characterizing values for the final time T, estimated rates at a final time T, and the difference between estimated pressures and baseline pressures at a final time T, and Q f represents a weighting matrix for x(T).
9. The computer program product of claim 8 , comprising instructions to calculate x(τ+1) as Ax(τ)+Bu(τ), where A and B are pre-defined values.
10. The computer program product of claim 1 , wherein the cost function is subject to a state evolution constraint.
11. The computer program product of claim 1 , wherein the cost function is optimized by linear quadratic regulation.
12. The computer program product of claim 1 , wherein the characterizing value is thickness.
13. The computer program product of claim 1 , wherein the polishing control parameter is pressure of a chamber in a carrier head of the polishing system.
14. A polishing system, comprising:
a platen to support a polishing pad;
a carrier head to hold a substrate in contact with the polishing pad;
a motor to generate relative motion between the carrier head and the polishing pad;
an in-situ monitoring system to, for each region of a plurality of regions on the substrate being polished, generate a sequence of characterizing values for the region;
a controller configured to
for each region, receive the sequence of characterizing values,
for each region, determine a polishing rate for the region,
for each particular parameter update time of a plurality of parameter update times, calculate an adjustment for at least one polishing control parameter of the polishing system, wherein calculating the adjustment includes minimizing a cost function that includes, for each region
i) a difference between a current characterizing value or an expected characterizing value at an expected endpoint time and a target characterizing value for the region, and
ii) a plurality of projected future pressure changes to occur at parameter update times subsequent to the particular parameter update time for the region and/or a plurality of differences between projected future pressures to occur at parameter update times subsequent to the particular parameter update time and a baseline pressure for the region, and
for each particular parameter update time, adjust the at least one polishing control parameter of the polishing system based on the calculated adjustment.
15. The system of claim 14 , wherein the cost function includes, for each region, the plurality of projected future pressure changes to occur at parameter update times subsequent to the particular parameter update time for the region and the plurality of differences between projected future pressures to occur at parameter update times subsequent to the particular parameter update time and the baseline pressure for the region.
16. The system of claim 15 , wherein the cost function is a function of a square of each difference between the expected characterizing value at the end of polish and the target characterizing value for the region, a function of a square of each projected future pressure change, a square of each difference between the projected future pressure and the baseline pressure, and a weighted vector norm of the expected future polish rates.
17. The system of claim 16 , wherein the cost function includes a first constant for each region and the cost function is a function of the first constant multiplied by the square of the difference between the expected characterizing value and the target characterizing value for the region.
18. The system of claim 17 , wherein the cost function includes a second constant for each region and the cost function is a function of the second constant multiplied by the square of each projected future pressure change.
19. A method for controlling a polishing system, the method comprising:
receiving from an in-situ monitoring system, for each region of a plurality of regions on a substrate being processed, a sequence of characterizing values for the region;
for each region, determining a polishing rate for the region;
for each particular parameter update time of a plurality of parameter update times, calculating an adjustment for at least one polishing-control parameter of the polishing system, wherein calculating the adjustment includes minimizing a cost function that includes, for each region
i) a difference between a current characterizing value or an expected characterizing value at an expected endpoint time and a target characterizing value for the region, and
ii) a plurality of projected future pressure changes to occur at parameter update times subsequent to the particular parameter update time for the region and/or a plurality of differences between projected future pressures to occur at parameter update times subsequent to the particular parameter update time and a baseline pressure for the region; and
for each particular parameter update time, adjust to the at least one polishing control parameter of the polishing system on the calculated adjustment.
20. A computer program product for controlling a semiconductor processing system comprising a non-transitory computer readable medium comprising instructions for causing one or more computers to:
receive from an in-situ monitoring system, for each region of a plurality of regions on a substrate being processed, a sequence of characterizing values for the region;
for each region, determine a rate of change of the characterizing values for the region; and
for each particular parameter update time of a plurality of parameter update times, calculate an adjustment for at least one processing control parameter of the processing system, wherein calculating of the adjustment includes minimizing a cost function that includes, for each region
i) a difference between a current characterizing value or an expected characterizing value at an expected endpoint time and a target characterizing value for the region, and
ii) a plurality of projected future changes to the at least one processing parameter to occur at parameter update times subsequent to the particular parameter update time for the region and/or a plurality of differences between projected future parameter values to occur at parameter update times subsequent to the particular parameter update time and a baseline parameter value for the region, and
for each particular parameter update time, adjust to the at least one processing control parameter of the processing based on the calculated adjustment.