Methods and systems for measurement of semiconductor structures with multi-pass statistical optimization
Methods and systems for measuring physical properties of a specimen by iterative solution of an optimization function including both current and prior measurement information are described herein. In one aspect, a Maximum A Posteriori (MAP) estimation approach is employed to integrate prior measurement information with current measurement information in the optimization function. In some examples, prior measurement information associated with the measurement of a semiconductor structure is available from a number of exogenous sources. In a further aspect, prior measurement information includes current measurements of a semiconductor structure under measurement as part of a multiple pass measurement. Prior statistics are determined based on an initial set of values of parameters of interest. The prior statistics are included in a subsequent MAP analysis of the same measurement data set employed in a first measurement pass. In another aspect, prior statistics are tracked to determine their impact on current measurements.
1. A system comprising:
an illumination source configured to provide an amount of illumination radiation directed to a measurement site on a surface of a semiconductor wafer;
a detector configured to detect an amount of radiation collected from one or more structures disposed on the semiconductor wafer at the measurement site in response to the amount of illumination radiation; and
a computing system configured to:
generate an amount of measurement data characterizing the detected amount of radiation from the measurement site; and
estimate values of one or more parameters of interest characterizing the one or more structures based on the amount of measurement data and an amount of prior measurement information, wherein the estimating of the values of the one or more parameters of interest involves a first iterative solution of an optimization function including a first term associated with the amount of measurement data and a term associated with the amount of prior measurement information.
2. The system of claim 1 , the computing system further configured to:
estimate prior values of at least one of the one or more parameters of interest characterizing the one or more structures based on the amount of measurement data, wherein the estimating of the prior values of at least one of the one or more parameters of interest involves second iterative solution of a prior optimization function including a second term associated with the amount of measurement data, and wherein the amount of prior measurement information includes the prior values of at the least one of the one or more parameters of interest.
3. The system of claim 2 , wherein the iterative solution of the optimization function is based on a Maximum A Posteriori (MAP) analysis, and the iterative solution of the prior optimization function is based on a Maximum Likelihood Estimation (MLE) analysis.
4. The system of claim 3 , wherein the optimization function is a negated log likelihood function, wherein the first term associated with the amount of measurement data is a likelihood function, and wherein the term associated the amount of prior measurement information includes a positive function characterized by a statistical model of the amount of prior measurement information.
5. The system of claim 4 , wherein the statistical model of the amount of prior measurement information is parameterized by wafer location.
6. The system of claim 4 , wherein the statistical model of the amount of prior measurement information characterizes a statistical spread in a measurement ensemble of the amount of prior measurement information and a measurement error of an exogenous measurement employed to measure the amount of prior measurement information.
7. The system of claim 1 , wherein the first term associated with the amount of measurement data includes a statistical model of measurement noise, and wherein the statistical model assumes a Gaussian distribution of the measurement noise.
8. The system of claim 1 , wherein the term associated with the amount of prior measurement information includes a positive function characterized by a statistical model of the amount of prior measurement information, wherein the statistical model assumes a Gaussian distribution of prior values of at the least one of the one or more parameters of interest.
9. The system of claim 1 , wherein the first term associated with the amount of measurement data includes a statistical model of measurement noise, and wherein the statistical model assumes a Poisson distribution of the measurement noise.
10. The system of claim 1 , wherein the term associated with the amount of prior measurement information includes a positive function characterized by a statistical model of the amount of prior measurement information, wherein the statistical model assumes a generalized bullseye distribution of normed values, wherein the normed values are determined from prior values of at least two parameters of interest.
11. The system of claim 10 , the computing system further configured to:
determine a value of a dimensionality of the generalized bullseye distribution such that the statistical model best matches a mean and standard deviation of an ensemble of normed values.
12. The system of claim 1 , wherein the amount of prior measurement information incorporated in the optimization function is derived from one or more of exogenous sources.
13. The system of claim 1 , the computing system further configured to:
track values of one or more statistical measures characterizing prior wafer statistics characterizing the amount of prior measurement information; and
update the prior measurement information based on the values of the one or more statistical measures.
14. The system of claim 1 , wherein the illumination source and the detector comprise an optical metrology system or an X-Ray based metrology system.
15. The system of claim 1 , wherein the amount of prior measurement information includes assumed values of the one or more parameters of interest, wherein the assumed values are within an expected range of values of the one or more parameters of interest, and wherein the assumed values are characterized by an assumed statistical distribution.
16. A method comprising:
providing an amount of illumination radiation directed to a measurement site on a surface of a semiconductor wafer;
detecting an amount of radiation collected from one or more structures disposed on the semiconductor wafer at the measurement site in response to the amount of illumination radiation;
generating an amount of measurement data characterizing the detected amount of radiation from the measurement site; and
estimating values of one or more parameters of interest characterizing the one or more structures based on the amount of measurement data and an amount of prior measurement information, wherein the estimating of the values of the one or more parameters of interest involves a first iterative solution of an optimization function including a first term associated with the amount of measurement data and a term associated with the amount of prior measurement information.
17. The method of claim 16 , further comprising:
estimating prior values of at least one of the one or more parameters of interest characterizing the one or more structures based on the amount of measurement data, wherein the estimating of the prior values of at least one of the one or more parameters of interest involves an second iterative solution of a prior optimization function including a second term associated with the amount of measurement data, and wherein the amount of prior measurement information includes the prior values of at the least one of the one or more parameters of interest.
18. The method of claim 16 , wherein the optimization function is a negated log likelihood function, wherein the first term associated with the amount of measurement data is a likelihood function, and wherein the term associated the amount of prior measurement information includes a positive function characterized by a statistical model of the amount of prior measurement information.
19. The method of claim 18 , wherein the statistical model is parameterized by wafer location.
20. The method of claim 16 , wherein the first term associated with the amount of measurement data includes a statistical model of measurement noise, and wherein the statistical model assumes a Poisson distribution of the measurement noise.
21. The method of claim 16 , wherein the term associated with the amount of prior measurement information includes a positive function characterized by a statistical model of the amount of prior measurement information, wherein the statistical model assumes a generalized bullseye distribution of normed values, wherein the normed values are determined from prior values of at least two parameters of interest.
22. The method of claim 16 , further comprising:
tracking values of one or more statistical measures characterizing prior wafer statistics characterizing the amount of prior measurement information; and
updating the prior measurement information based on the values of the one or more statistical measures.
23. A system comprising:
an illumination source configured to provide an amount of illumination radiation directed to a measurement site on a surface of a semiconductor wafer;
a detector configured to detect an amount of radiation collected from one or more structures disposed on the semiconductor wafer at the measurement site in response to the amount of illumination radiation; and
a non-transient, computer-readable medium storing instructions that, when executed by one or more processors, causes the one or more processors to:
generate an amount of measurement data characterizing the detected amount of radiation from the measurement site; and
estimate values of one or more parameters of interest characterizing the one or more structures based on the amount of measurement data and an amount of prior measurement information, wherein the estimating of the values of the one or more parameters of interest involves a first iterative solution of an optimization function including a first term associated with the amount of measurement data and a term associated with the amount of prior measurement information.
24. The system of claim 23 , the non-transient, computer-readable medium further storing instructions that, when executed by the one or more processors, causes the one or more processors to:
estimate prior values of at least one of the one or more parameters of interest characterizing the one or more structures based on the amount of measurement data, wherein the estimating of the prior values of at least one of the one or more parameters of interest involves an second iterative solution of a prior optimization function including a second term associated with the amount of measurement data, and wherein the amount of prior measurement information includes the prior values of at the least one of the one or more parameters of interest.