Methods and systems for x-ray scatterometry measurements employing a machine learning based electromagnetic response model
Methods and systems for estimating values of parameters of interest from X-ray scatterometry measurements with reduced computational effort are described herein. Values of parameters of interest are estimated by regression using a trained, machine learning (ML) based electromagnetic (EM) response model. A training data set includes sets of Design Of Experiments (DOE) values of parameters of interest and corresponding DOE values of a plurality of electromagnetic response metrics. In some examples, values of parameters of interest are determined from measured images based on regression using a sequence of trained ML based electromagnetic response models. In some examples, input values employed to train the ML based EM response model are scaled based on model output variation.
1 . A metrology system comprising:
an x-ray illumination source configured to generate an amount of x-ray illumination radiation directed to one or more semiconductor structures under measurement formed on a wafer surface;
an x-ray detector configured to detect one or more images of a plurality of diffraction orders of an amount of radiation scattered from the one or more structures in response to the x-ray illumination radiation incident on the one or more structures; and
a computing system configured to:
estimate values of parameters of interest characterizing the one or more semiconductor structures by regression on the one or more detected images using a trained, machine learning based electromagnetic response model, wherein the trained machine learning based electromagnetic response model maps values of the parameters of interest to values of a plurality of electromagnetic response metrics; and
train the machine learning based electromagnetic response model based on a training data set including sets of Design Of Experiments (DOE) values of the parameters of interest and corresponding DOE values of a plurality of electromagnetic response metrics, wherein the training involves a regression on the training data set to minimize an objective function characterizing errors between the DOE values of the plurality of electromagnetic response metrics and values of the plurality of electromagnetic response metrics determined using the machine learning based electromagnetic response model.
2 . The metrology system of claim 1 , wherein each of the plurality of electromagnetic response metrics is any of a pixel intensity at the x-ray detector, a scattering coefficient, a scattering order efficiency, and a scattering order value in Q-space.
3 . The metrology system of claim 1 , wherein the regression on the one or more detected images includes a metrology system model, wherein the metrology system model maps the values of the plurality of electromagnetic response metrics to one or more images at the x-ray detector.
4 . The metrology system of claim 1 , the computing system further configured to:
determine the DOE values of the plurality of electromagnetic response metrics corresponding to each set of values of the parameters of interest by simulation using a physics based electromagnetic solver.
5 . The metrology system of claim 1 , wherein the training data set is derived from measurements performed by a trusted, reference metrology system.
6 . The metrology system of claim 1 , wherein the objective function is based on a residual value associated with each of the plurality of electromagnetic response metrics.
7 . The metrology system of claim 6 , the computing system further configured to:
scale the DOE values of a parameter of interest based on a range of values of the residual associated with each of the plurality of electromagnetic response metrics determined based on the range of DOE values of the parameter of interest.
8 . The metrology system of claim 1 , wherein the training involves reducing a dimension of the plurality of electromagnetic response metrics.
9 . The metrology system of claim 1 , wherein the regression involves any of a multi-seed regression, a multi-pass regression, and a regularized regression.
10 . The metrology system of claim 1 , the computing system further configured to:
select additional sets of DOE values of the parameters of interest based on the estimated values of the parameters of interest;
determine DOE values of the plurality of electromagnetic response metrics corresponding to each additional set of values of the parameters of interest by simulation using a physics based electromagnetic solver, the additional sets of DOE values of the parameters of interest and the corresponding DOE values of the plurality of electromagnetic response metrics comprising a second training data set;
train a second machine learning based electromagnetic response model based on the second training data set; and
estimate refined values of the parameters of interest characterizing the one or more semiconductor structures by regression on the one or more detected images using the second trained, machine learning based electromagnetic response model.
11 . The metrology system of claim 1 , the computing system further configured to:
estimate a goodness of fit between the DOE values of each of the plurality of electromagnetic response metrics and the values of each of the plurality of electromagnetic response metrics determined using the machine learning based electromagnetic response model, wherein the regression on the one or more detected images employs the trained, machine learning based electromagnetic response model to determine a value of a electromagnetic response metric of the plurality of electromagnetic response metrics if the goodness of fit associated with the electromagnetic response metric is greater than a predetermined threshold value, and wherein the regression on the one or more detected images employs a physics based electromagnetic solver to determine the value of the electromagnetic response metric of the plurality of electromagnetic response metrics if the goodness of fit associated with the electromagnetic response metric is less than a predetermined threshold value.
12 . A method comprising:
illuminating one or more semiconductor structures formed on a wafer surface by an amount of x-ray illumination radiation generated by an x-ray illumination source;
detecting one or more images of a plurality of diffraction orders of an amount of radiation scattered from the one or more structures onto an x-ray detector, the amount of radiation scattered from the one or more structures in response to the x-ray illumination radiation incident on the one or more structures; and
estimating values of parameters of interest characterizing the one or more semiconductor structures by regression on the one or more detected images using a trained, machine learning based electromagnetic response model, wherein the trained machine learning based electromagnetic response model maps values of the parameters of interest to values of a plurality of electromagnetic response metrics; and
training the machine learning based electromagnetic response model based on a training data set including sets of Design Of Experiments (DOE) values of the parameters of interest and corresponding DOE values of a plurality of electromagnetic response metrics, wherein the training involves a regression on the training data set to minimize an objective function characterizing errors between the DOE values of the plurality of electromagnetic response metrics and values of the plurality of electromagnetic response metrics determined using the machine learning based electromagnetic response model.
13 . The method of claim 12 , wherein each of the plurality of electromagnetic response metrics is any of a pixel intensity at the x-ray detector, a scattering coefficient, a scattering order efficiency, and a scattering order value in Q-space.
14 . The method of claim 12 , wherein the regression on the one or more detected images includes a metrology system model, wherein the metrology system model maps the values of the plurality of electromagnetic response metrics to one or more images at the x-ray detector.
15 . The method of claim 12 , wherein the objective function is based on a residual value associated with each of the plurality of electromagnetic response metrics.
16 . The method of claim 12 , further comprising:
reducing a dimension of the plurality of electromagnetic response metrics.
17 . The method of claim 12 , further comprising:
scaling the DOE values of a parameter of interest based on a range of values of the residual associated with each of the plurality of electromagnetic response metrics determined based on the range of DOE values of the parameter of interest.
18 . The method of claim 12 , further comprising:
selecting additional sets of DOE values of the parameters of interest based on the estimated values of the parameters of interest;
determining DOE values of the plurality of electromagnetic response metrics corresponding to each additional set of values of the parameters of interest by simulation using a physics based electromagnetic solver, the additional sets of DOE values of the parameters of interest and the corresponding DOE values of the plurality of electromagnetic response metrics comprising a second training data set;
training a second machine learning based electromagnetic response model based on the second training data set; and
estimating refined values of the parameters of interest characterizing the one or more semiconductor structures by regression on the one or more detected images using the second trained, machine learning based electromagnetic response model.
19 . The method of claim 12 , further comprising:
estimating a goodness of fit between the DOE values of each of the plurality of electromagnetic response metrics and the values of each of the plurality of electromagnetic response metrics determined using the machine learning based electromagnetic response model, wherein the regression on the one or more detected images employs the trained, machine learning based electromagnetic response model to determine a value of an electromagnetic response metric of the plurality of electromagnetic response metrics if the goodness of fit associated with the electromagnetic response metric is greater than a predetermined threshold value, and wherein the regression on the one or more detected images employs a physics based electromagnetic solver to determine the value of the electromagnetic response metric of the plurality of electromagnetic response metrics if the goodness of fit associated with the electromagnetic response metric is less than a predetermined threshold value.
20 . A metrology system comprising:
an x-ray illumination source configured to generate an amount of x-ray illumination radiation directed to one or more semiconductor structures under measurement formed on a wafer surface;
an x-ray detector configured to detect one or more images of a plurality of diffraction orders of an amount of radiation scattered from the one or more structures in response to the x-ray illumination radiation incident on the one or more structures; and
a non-transitory, computer-readable medium including instructions that when executed by one or more processors of a computing system cause the computing system to:
estimate values of parameters of interest characterizing the one or more semiconductor structures by regression on the one or more detected images using a trained, machine learning based electromagnetic response model, wherein the trained machine learning based electromagnetic response model maps values of the parameters of interest to values of a plurality of electromagnetic response metrics; and
train the machine learning based electromagnetic response model based on a training data set including sets of Design Of Experiments (DOE) values of the parameters of interest and corresponding DOE values of a plurality of electromagnetic response metrics, wherein the training involves a regression on the training data set to minimize an objective function characterizing errors between the DOE values of the plurality of electromagnetic response metrics and values of the plurality of electromagnetic response metrics determined using the machine learning based electromagnetic response model.