System and method for removing noise from roughness measurements
Systems and methods are disclosed that remove noise from roughness measurements to determine roughness of a feature in a pattern structure. In one embodiment, a method for determining roughness of a feature in a pattern structure includes generating, using an imaging device, a set of one or more images, each including measured linescan information that includes noise. The method also includes detecting edges of the features within the pattern structure of each image without filtering the images, generating a biased power spectral density (PSD) dataset representing feature geometry information corresponding to the edge detection measurements, evaluating a high-frequency portion of the biased PSD dataset to determine a noise model for predicting noise over all frequencies of the biased PSD dataset, and subtracting the noise predicted by the determined noise model from a biased roughness measure to obtain an unbiased roughness measure.
1. A method for determining roughness of a feature in a pattern structure, said method comprising:
generating, using an imaging device, a set of one or more images, each image of the set including one or more instances of a feature within a respective pattern structure, each image including measured linescan information corresponding to the pattern structure that includes noise;
detecting edges of the features within the pattern structure of each image of the set without filtering the images;
generating a biased power spectral density (PSD) dataset representing feature geometry information corresponding to the edge detection measurements of the set of images;
evaluating a high-frequency portion of the biased PSD dataset to determine a noise model for predicting noise over all frequencies of the biased PSD dataset; and
subtracting the noise predicted by the determined noise model from a biased roughness measure to obtain an unbiased roughness measure.
2. The method of claim 1 , wherein said evaluating a high-frequency portion of the biased PSD dataset to determine a noise model comprises:
measuring a noise floor value in a high-frequency portion of the biased PSD dataset to determine a uniform value of predicted noise over all frequencies of the biased PSD dataset.
3. The method of claim 1 , wherein said evaluating a high-frequency portion of the biased PSD dataset to determine a noise model comprises:
analyzing the biased PSD dataset to determine a set of noise model parameters defining a pink noise model having a non-uniform value of predicted noise over all frequencies of the biased PSD dataset.
4. The method of claim 3 , wherein said analyzing the biased PSD dataset comprises:
measuring a noise floor value in a high-frequency portion of the biased PSD dataset;
subtracting the noise floor value from the biased PSD dataset to obtain an intermediate PSD dataset;
fitting a PSD roughness model to the intermediate PSD dataset;
determining a transition frequency at which the PSD roughness model decreases to less than a predetermined percentage of the measured noise floor value;
fitting the pink noise model to the biased PSD dataset over a range of frequencies greater than the transition frequency, to determine the set of noise model parameters for predicting noise over all frequencies of the biased PSD dataset.
5. The method of claim 3 , wherein said analyzing the biased PSD dataset comprises:
fitting a combined model that accounts for both roughness and noise to the biased PSD dataset to obtain roughness parameters that represent an unbiased PSD dataset, and to determine the set of noise model parameters for predicting noise over all frequencies of the biased PSD dataset.
6. The method of claim 5 , further comprising:
determining, before said fitting step, an initial estimate for each of the parameters in the combined roughness and noise model to use as starting values for the fitting step;
wherein said initial estimates are determined by:
measuring a noise floor value in a high-frequency portion of the biased PSD dataset;
subtracting the noise floor value from the biased PSD dataset to obtain an intermediate PSD dataset;
fitting a PSD roughness model to the intermediate PSD dataset;
determining a transition frequency at which the PSD roughness model decreases to less than a predetermined percentage of the measured noise floor value;
fitting a pink noise model to the biased PSD dataset over a range of frequencies greater than the transition frequency, to determine the set of noise model parameters for predicting noise over all frequencies of the biased PSD dataset;
subtracting the noise predicted by the pink noise model from the biased PSD dataset to obtain an unbiased PSD dataset; and
fitting a roughness model to the unbiased PSD dataset to obtain unbiased roughness parameters corresponding to the feature geometry information.
7. The method of claim 3 , wherein the pink noise model is of the form:
PSD
noise
(
f
)
=
PSD
low
-
PSD
high
1
+
2
π
f
ξ
noise
B
+
PSD
high
where PSD low represents the noise level in the low-frequency region, PSD high represents the noise level in the high-frequency region, ξ noise represents a correlation length of the noise, and B represents an exponential power that defines how fast the noise transitions from the low-frequency region to the high-frequency region.
8. The method of claim 1 , wherein said detecting edges of the features within the pattern structure of each image of the set without filtering the images comprises:
measuring the pattern structure of each image of the set using an inverse linescan model (ILM) to detect the feature edges.
9. The method of claim 8 , wherein said measuring the pattern structure comprises:
applying the measured linescan information to an inverse linescan model that relates measured linescan information to feature geometry information; and
determining, from the inverse linescan model, feature geometry information that describes feature edge positions of the predetermined feature of the pattern structure that corresponds to the measured linescan information.
10. The method of claim 1 , wherein:
said biased roughness measure comprises the biased PSD dataset; and
said unbiased roughness measure comprises an unbiased PSD dataset.
11. The method of claim 10 , further comprising:
analyzing the unbiased PSD dataset to obtain unbiased roughness parameters corresponding to the feature geometry information.
12. The method of claim 11 , wherein analyzing the unbiased PSD dataset comprises:
fitting a roughness model to the unbiased PSD dataset to obtain the unbiased roughness parameters.
13. The method of claim 1 , wherein said generating a biased PSD dataset comprises:
averaging individual PSD datasets from each measured feature of each image of the image set.
14. The method of claim 1 , wherein the feature geometry information comprises at least one of feature width, or feature placement position.
15. A system for determining roughness of a feature in a pattern structure, said system comprising:
an imaging device for generating a set of one or more images, each image of the set including one or more instances of a feature within a respective pattern structure, each image including measured linescan information corresponding to the pattern structure that includes noise; and
a processor coupled to receive the measured linescan information from the imaging device, wherein said processor is configured to:
detect edges of the features within the pattern structure of each image of the set without filtering the images;
generate a biased power spectral density (PSD) dataset representing feature geometry information corresponding to the edge detection measurements of the set of images;
evaluate a high-frequency portion of the biased PSD dataset to determine a noise model for predicting noise over all frequencies of the biased PSD dataset; and
subtract the noise predicted by the determined noise model from a biased roughness measure to obtain an unbiased roughness measure.
16. The system of claim 15 , wherein, to evaluate a high-frequency portion of the biased PSD dataset to determine a noise model, the processor is further configured to:
measure a noise floor value in a high-frequency portion of the biased PSD dataset to determine a uniform value of predicted noise over all frequencies of the biased PSD dataset.
17. The system of claim 15 , wherein, to evaluate a high-frequency portion of the biased PSD dataset to determine a noise model, the processor is further configured to:
analyze the biased PSD dataset to determine a set of noise model parameters defining a pink noise model having a non-uniform value of predicted noise over all frequencies of the biased PSD dataset.
18. The system of claim 17 , wherein, to analyze the biased PSD dataset, the processor is further configured to:
measure a noise floor value in a high-frequency portion of the biased PSD dataset;
subtract the noise floor value from the biased PSD dataset to obtain an intermediate PSD dataset;
fit a PSD roughness model to the intermediate PSD dataset;
determine a transition frequency at which the PSD roughness model decreases to less than a predetermined percentage of the measured noise floor value;
fit the pink noise model to the biased PSD dataset over a range of frequencies greater than the transition frequency, to determine the set of noise model parameters for predicting noise over all frequencies of the biased PSD dataset.
19. The system of claim 17 , wherein, to analyze the biased PSD dataset, the processor is further configured to:
fit a combined model that accounts for both roughness and noise to the biased PSD dataset to obtain roughness parameters that represent an unbiased PSD dataset, and to determine the set of noise model parameters for predicting noise over all frequencies of the biased PSD dataset.
20. The system of claim 19 , wherein the processor is further configured to determine an initial estimate for each of the parameters in the combined roughness and noise model to use as starting values for fitting the combined model to the biased PSD dataset, wherein said initial estimates are determined by:
measuring a noise floor value in a high-frequency portion of the biased PSD dataset;
subtracting the noise floor value from the biased PSD dataset to obtain an intermediate PSD dataset;
fitting a PSD roughness model to the intermediate PSD dataset;
determining a transition frequency at which the PSD roughness model decreases to less than a predetermined percentage of the measured noise floor value;
fitting a pink noise model to the biased PSD dataset over a range of frequencies greater than the transition frequency, to determine the set of noise model parameters for predicting noise over all frequencies of the biased PSD dataset;
subtracting the noise predicted by the pink noise model from the biased PSD dataset to obtain an unbiased PSD dataset; and
fitting a roughness model to the unbiased PSD dataset to obtain unbiased roughness parameters corresponding to the feature geometry information.
21. The system of claim 17 , wherein the pink noise model is of the form:
PSD
noise
(
f
)
=
PSD
low
-
PSD
high
1
+
2
π
f
ξ
noise
B
+
PSD
high
where PSD low represents the noise level in the low-frequency region, PSD high represents the noise level in the high-frequency region, ξ noise represents a correlation length of the noise, and B represents an exponential power that defines how fast the noise transitions from the low-frequency region to the high-frequency region.
22. The system of claim 15 , wherein, to detect edges of the features within the pattern structure of each image of the set without filtering the images, the processor is further configured to:
measure the pattern structure of each image of the set using an inverse linescan model (ILM) to detect the feature edges.
23. The system of claim 22 , wherein, to measure the pattern structure, the processor is further configured to:
apply the measured linescan information to an inverse linescan model that relates measured linescan information to feature geometry information; and
determine, from the inverse linescan model, feature geometry information that describes feature edge positions of the predetermined feature of the pattern structure that corresponds to the measured linescan information.
24. The system of claim 15 , wherein:
said biased roughness measure comprises the biased PSD dataset; and
said unbiased roughness measure comprises an unbiased PSD dataset.
25. The system of claim 24 , wherein the processor is further configured to analyze the unbiased PSD dataset to obtain unbiased roughness parameters corresponding to the feature geometry information.
26. The system of claim 25 , wherein, to analyze the unbiased PSD dataset, the processor is further configured to fit a roughness model to the unbiased PSD dataset to obtain the unbiased roughness parameters.
27. The system of claim 15 , wherein the processor is further configured to generate the biased PSD dataset by averaging individual PSD datasets from each measured feature of each image of the image set.
28. The system of claim 15 , wherein the feature geometry information comprises at least one of feature width, or feature placement position.