Shape roughness measurement in optical metrology

1 . A method of generating a simulated diffraction signal to be used in measuring shape roughness of a structure formed on a wafer using optical metrology, the method comprising:

(a) defining an initial model of the structure;

(b) defining a statistical function of shape roughness;

(c) deriving a statistical perturbation based on the statistical function;

(d) superimposing the statistical perturbation on the initial model of the structure to define a modified model of the structure; and

(e) generating a simulated diffraction signal based on the modified model of the structure.

2 . The method of claim 1 , wherein the initial model of the structure is defined by smooth lines, and has a rectangular shape when the structure is a line/space pattern or an elliptical shape when the structure is a contact hole.

3 . The method of claim 1 , wherein the initial model of the structure is defined by smooth lines and has a T-shaped island or an L-shaped island when the structure is a via.

4 . The method of claim 1 , wherein the initial model of the structure is defined by smooth lines and has a trapezoidal shape when the structure is a line/space pattern.

5 . The method of claim 1 , wherein the initial model of the structure is defined by smooth lines, and wherein the statistical function of shape roughness is defined in a lateral dimension, a vertical dimension, or lateral and vertical dimensions.

6 . The method of claim 1 , wherein the statistical function comprises root-mean-square roughness, autocorrelation function, or power spectrum density.

7 . The method of claim 1 , wherein generating a simulated diffraction signal comprises:

discretizing the modified model;

applying Maxwell's equations to the discretized model; and

solving Maxwell's equations using a numerical analysis technique to generate the simulated diffraction signal.

8 . The method of claim 7 , further comprising:

defining an elementary cell containing the modified model, wherein the modified model in the elementary cell is discretized.

9 . The method of claim 8 , wherein discretizing the model comprises:

dividing the elementary cell into a plurality of pixel elements; and

assigning an index of refraction and a coefficient of extinction (n & k) values to each pixel element.

10 . The method of claim 9 , wherein the numerical analysis technique is rigorous coupled-wave analysis.

11 . The method of claim 8 , wherein the elementary cell includes multiple periods of the structure.

12 . The method of claim 1 , further comprising:

deriving at least another statistical perturbation based on the statistical function of shape roughness defined in step (b);

superimposing the at least another statistical perturbation on the initial model defined in step (a) to define at least another modified model of the structure;

generating at least another simulated diffraction signal based on the at least another modified model of the structure; and

averaging the simulated diffraction signal generated in step (e) and the at least another simulated diffraction signal.

13 . The method of claim 1 , further comprising:

repeating steps (a) to (e) to generate a plurality of modified model and corresponding simulated diffraction signal pairs, wherein the statistical function in step (b) is varied to define varying modified models of the structure in step (d) and to generate varying simulated diffraction signals in step (e);

storing the plurality modified model and corresponding simulated diffraction signal pairs in a library;

obtaining a diffraction signal measured from directing an incident beam at a structure being examined (a measured diffraction signal); and

comparing the measured diffraction signal to one or more of the simulated diffraction signals stored in the library to determine the shape of the structure being examined.

14 . The method of claim 1 , further comprising:

obtaining a diffraction signal measured from directing an incident beam at a structure being examined (a measured diffraction signal);

comparing the measured diffraction signal to the simulated diffraction signal generated in step (e); and

when the measured diffraction signal and the simulated diffraction signal generated in step (e) do not match within a preset criteria:

repeating steps (a) to (e) to generate a different simulated diffraction signal, wherein the statistical function in step (b) is varied to define a different modified model of the structure in step (d) and to generate the different simulated diffraction signal in step (e); and

repeating the comparing step using the different simulated diffraction signal.

15 . The method of claim 1 , wherein the simulated diffraction signal is generated using a machine learning system.

16 . A method of generating a simulated diffraction signal to be used in measuring shape roughness of a structure formed on a wafer using optical metrology, the method comprising:

(a) defining an initial model of a deterministic basic feature of the structure;

(b) defining a statistical function of shape roughness;

(c) generating a statistical perturbation based on the statistical function;

(d) superimposing the statistical perturbation on the initial model to define a modified model of the structure; and

(e) generating a simulated diffraction signal based on the modified model of the structure.

17 . The method of claim 16 , wherein the initial model is defined by smooth lines, and has a rectangular shape when the structure is a line/space pattern or an elliptical shape when the structure is a contact hole.

18 . The method of claim 16 , wherein the initial model of the structure is defined by smooth lines and has a T-shaped island or an L-shaped island when the structure is a via.

19 . The method of claim 16 , wherein the initial model of the structure is defined by smooth lines and has a trapezoidal shape when the structure is a line/space pattern.

20 . The method of claim 16 , wherein the initial model of the structure is defined by smooth lines, and wherein the statistical function of shape roughness is defined in a lateral dimension, a vertical dimension, or lateral and vertical dimensions.

21 . The method of claim 16 , wherein the statistical function comprises root-mean-square roughness, autocorrelation function, or power spectrum density.

22 . The method of claim 16 , wherein generating a simulated diffraction signal comprises:

discretizing the modified model of the structure;

applying Maxwell's equations to the discretized model; and

solving Maxwell's equations using a numerical analysis technique to generate the simulated diffraction signal.

23 . The method of claim 22 , further comprising:

defining an elementary cell containing the modified model, wherein the modified model in the elementary cell is discretized.

24 . The method of claim 23 , wherein discretizing the model comprises:

dividing the elementary cell into a plurality of pixel elements; and

assigning an index of refraction and a coefficient of extinction (n & k) values to each pixel element.

25 . The method of claim 24 , wherein the numerical analysis technique is rigorous coupled-wave analysis.

26 . The method of claim 16 , wherein the simulated diffraction signal is generated using a machine learning system.

27 . A computer-readable storage medium containing computer executable instructions for causing a computer to generate a simulated diffraction signal to be used in measuring shape roughness of a structure formed on a wafer using optical metrology, comprising instructions for:

(a) defining an initial model of the structure;

(b) defining a statistical function of shape roughness;

(c) deriving a statistical perturbation based on the statistical function;

(d) superimposing the statistical perturbation on the initial model of the structure to define a modified model of the structure; and

(e) generating a simulated diffraction signal based on the modified model of the structure.

28 . A system to generate a simulated diffraction signal to be used in measuring shape roughness of a structure formed on a wafer using optical metrology, the system comprising:

an initial model of the structure;

a modified model of the structure defined by superimposing a statistical perturbation derived from a statistical function defined for the initial model of the structure; and

a simulated diffraction signal generated based on the modified model of the structure.