Endpoint detection using spectrum feature trajectories

1. A method of polishing, comprising:

polishing a substrate;

making a sequence of measurements of light reflected from the substrate while the substrate is being polished, at least some of the measurements of the sequence of measurements differing due to material being removed during polishing;

for each measurement in the sequence, determining a first value of a first characteristic and a second value of a different second characteristic of the light to generate a sequence of first values and second values;

storing a predetermined path in a coordinate space of the first characteristic and the second characteristic;

for each measurement in the sequence, determining a position on the predetermined path based on the first value and the second value; and

determining at least one of a polishing endpoint or an adjustment for a polishing rate based on the position on the path.

2. The method of claim 1 , wherein the first characteristic is an intensity at a first wavelength and the second characteristic is an intensity at a different second wavelength.

3. The method of claim 1 , wherein the first characteristic is a ratio of an intensity at a first wavelength to an intensity at a different second wavelength, and the second characteristic is a ratio of an intensity at a different third wavelength to an intensity at a different fourth wavelength.

4. The method of claim 3 , wherein the first wavelength and the second wavelength are constant during polishing of the substrate.

5. The method of claim 3 , wherein making the sequence of measurements comprises illuminating the substrate with a wide-band light beam that includes the first wavelength and the second wavelength.

6. The method of claim 5 , wherein making the sequence of measurements comprises measuring a sequence of spectra, and wherein determining the first value and the second value comprises extracting the first intensity at the first wavelength in the wide-band light beam and extracting the second intensity at the second wavelength in the wide-band light beam.

7. The method of claim 2 , wherein making the sequence of measurements comprises illuminating the substrate with a first narrow-band light beam that includes the first wavelength and illuminating the substrate with a second narrow-band light beam that includes the second wavelength.

8. The method of claim 1 , wherein the first characteristic is one of a location, width or intensity of a selected spectral feature and the second characteristic is another of the location, width or intensity of the selected spectral feature.

9. The method of claim 8 , wherein the selected spectral feature persists with an evolving location, width or intensity through the sequence of spectra.

10. The method of claim 9 , wherein the selected spectral feature comprises a peak or a valley.

11. The method of claim 8 , wherein making the sequence of measurements comprises measuring a sequence of spectra, and wherein determining the first value and the second value comprises extracting the one of the location, width or intensity of the spectral feature and the another of the location, width or intensity of the selected spectral feature.

12. The method of claim 1 , wherein determining the position on the path comprises determining a closest position on the path to a coordinate in the coordinate space comprising the first value and the second value.

13. The method of claim 12 , wherein determining the closest position comprises determining a Euclidean distance.

14. The method of claim 1 , wherein determining at least one of the polishing endpoint or the adjustment for the polishing rate comprises determining a distance travelled along the path.

15. The method of claim 14 , further comprising halting polishing when the distance crosses a threshold.

16. The method of claim 14 , wherein storing the predetermined path comprises storing a Bezier function, and determining the distance travelled along the path comprises determining a T-parameter of the Bezier function for the position.

17. The method of claim 1 , further comprising halting polishing when the position crosses a threshold.

18. The method of claim 1 , wherein storing the predetermined path comprises storing a Bezier function.

19. The method of claim 1 , wherein storing the predetermined path comprises storing a sequence of coordinates.

20. The method of claim 1 , wherein determining the position on the predetermined path generates a sequence of positions on the predetermined path, and wherein determining at least one of the polishing endpoint or the adjustment for the polishing rate comprises fitting a function to the sequence of positions over time.

21. The method of claim 1 , wherein the predetermined path contains no degeneracies.

22. The method of claim 21 , wherein the sequence of measurements is from a first portion of the substrate, and further comprising taking a second sequence of measurements of light reflected from a second portion of the substrate while the substrate is being polished, for each measurement in the second sequence, determining a third value of the first characteristic and a fourth value of the second characteristic and determining a second position on the path based on the third value and the fourth value.