Neural network based hermite interpolator for scatterometry parameter estimation
View Patent ↗Generation of a meta-model for scatterometry analysis of a sample diffracting structure having unknown parameters. A training set comprising both a spectral signal evaluation and a derivative of the signal with respect to at least one parameter across a parameter space is rigorously computed. A neural network is trained with the training set to provide reference spectral information for a comparison to sample spectral information recorded from the sample diffracting structure. A neural network may be trained with derivative information using an algebraic method wherein a network bias vector is centered over both a primary sampling matrix and an auxiliary sampling matrix. The result of the algebraic method may be used for initializing neural network coefficients for training by optimization of the neural network weights, minimizing a difference between the actual signal and the modeled signal based on a objective function containing both function evaluations and derivatives.
1. A method of generating a meta-model for scatterometry analysis of a sample diffracting structure having unknown parameters, the method comprising:
generating a training set by rigorously computing a spectral signal evaluation and an evaluation of the spectral signal differentiated with respect to at least one parameter across a parameter space; and
training a neural network with the training set to provide reference spectral information for a comparison of the reference to spectral information recorded from the sample diffracting structure.
2. The method as in claim 1 , wherein the training further comprises:
initializing the neural network coefficients; and
regressing the neural network weights to minimize an objective function containing both function evaluations and derivatives penalizing the difference between the rigorously computed signal and the modeled signal.
3. The method as in claim 2 , wherein initializing the neural network coefficients further comprises:
determining a set of coefficients, the set including at least one coefficient corresponding to a coefficient resulting from a one hidden layer neural network trained by an algebraic method with a training set including derivative information.
4. The method as in claim 3 , wherein the neural network has two hidden layers and where the coefficients of one of the two hidden layers is initialized with values that make the one layer appear as a pass-through and the network functionally equivalent to a one hidden layer network.
5. The method as in claim 2 , wherein a derivative of the objective function with respect to the trainable weights is determined in part with a back propagation method and in part with a direct differentiation.
6. The method as in claim 2 , wherein the objective function is quadratic and a least squares difference between the rigorously computed signal and the modeled signal is minimized.
7. The method as in claim 1 , wherein the training is by an algebraic method further comprising:
evaluating the spectral signal and the spectral signal derivative for an input matrix P comprising M points selected from an N dimensional parameter space to generate an output matrix of primary sampling points;
selecting L points comprising fewer than or equal to MN points, from the N dimensional parameter space to generate a matrix R of auxiliary sampling points; and
setting the bias vectors d so as to center the basis functions of the neural network at both the primary and secondary sampling points.
8. The method of claim 7 , wherein setting the bias vectors d further comprises:
computing the bias vector d using values of a diagonal of the product of a forward weights matrix W and a matrix of the input matrix P and auxiliary matrix R.
9. The method as in claim 8 , wherein the training further comprises:
determining a neural network nodal output matrix S across both the primary and auxiliary sampling points;
determining a neural network nodal derivative matrix Δ; and
determining an output weight matrix v by solving a linear system including S, Δ a rigorously computed spectral signal matrix Z, and a rigorously computed spectral signal derivative matrix dZ.
10. The method as in claim 9 , further comprising:
employing v, d and W as initial conditions for training a second neural network with an optimization method.
11. A machine-accessible storage medium having instructions stored thereon which cause a data processing system to perform a method of generating a meta-model for scatterometry analysis of a sample diffracting structure having unknown parameters, the method comprising:
generating a training set by rigorously computing a spectral signal evaluation and an evaluation of the spectral signal differentiated with respect to at least one parameter across a parameter space; and
training a neural network with the training set to provide reference spectral information for a comparison to sample spectral information recorded from the sample diffracting structure.
12. The storage medium as in claim 11 , having instructions stored thereon which cause a data processing system to perform the method further comprising:
initializing the neural network coefficients; and
regressing the neural network weights to minimize an objective function which penalizes the difference between a rigorously computed signal and the modeled signal where the objective function contains both function evaluations and derivatives.
13. The storage medium as in claim 11 , having instructions stored thereon which cause a data processing system to perform the method further comprising:
determining a set of coefficients, the set including at least one coefficient corresponding to a coefficient resulting from a one hidden layer neural network trained by an algebraic method with a training set including derivative information.
14. The storage medium as in claim 13 , wherein the neural network has at least two hidden layers and where the coefficients of one of the two hidden layers is initialized with values that make the layer appear as a pass-through making it functionally equivalent to a one hidden layer network.
15. The storage medium as in claim 12 , wherein a derivative of the objective function with respect to the trainable weights is determined in part with a back propagation method and in part with a direct differentiation.
16. The storage medium as in claim 12 , wherein the objective function is quadratic and a least squares difference between the rigorously computed signal and the modeled signal is minimized.
17. The storage medium as in claim 11 , having instructions stored thereon which cause a data processing system to perform the method further comprising:
evaluating the spectral signal and the spectral signal derivative for an input matrix P comprising M points selected from an N dimensional parameter space to generate an output matrix of primary sampling points;
selecting L points comprising fewer than or equal to MN points, from the N dimensional parameter space to generate a matrix R of auxiliary sampling points; and
setting the bias vectors d so as to center the basis functions of the neural network at both the primary and secondary sampling points.
18. The storage medium as in claim 17 , having instructions stored thereon which cause a data processing system to perform the method further comprising:
computing the bias vector d using values of a diagonal of the product of a forward weights matrix W and a matrix of the input matrix P and auxiliary matrix R.
19. The storage medium as in claim 18 , having instructions stored thereon which cause a data processing system to perform the method further comprising:
determining a neural network nodal output matrix S across both the primary and auxiliary sampling points;
determining a neural network nodal derivative matrix Δ; and
determining an output weight matrix v by solving a linear system including S, Δ a rigorously computed spectral signal matrix Z, and a rigorously computed spectral signal derivative matrix dZ.
20. The storage medium as in claim 19 , having instructions stored thereon which cause a data processing system to perform the method further comprising:
employing v, d and W as initial conditions for training a second neural network with an optimization method.
21. An optical metrology system for analysis of a sample diffracting structure having unknown parameters, the system comprising:
a library generator employing a neural network meta-model configured to calculate spectral information for each of a plurality of parameterized diffracting structures, the calculated spectral information including a spectral signal and the spectral signal differentiated with respect to at least one parameter of the diffracting structure; and
a metrology processor to compare spectral information from the sample diffracting structure to the calculated spectral information.
22. The optical metrology system as in claim 21 , further comprising a reference spectra library to store the calculated spectral information.
23. The optical metrology system as in claim 21 , wherein the library generator calculates N spectral signal derivatives and N spectral signal evaluations for a parameterized diffracting structure having N degrees of freedom.
24. The optical metrology system as in claim 21 , wherein parameters of the diffracting structure are in at least one of the geometric domain, optical domain, or manufacturing process domain.