Local multivariable solver for optical proximity correction in lithographic processing method, and device manufactured thereby
View Patent ↗A multivariable solver for proximity correction uses a Jacobian matrix to approximate effects of perturbations of segment locations in successive iterations of a design loop. The problem is formulated as a constrained minimization problem with box, linear equality, and linear inequality constraints. To improve computational efficiency, non-local interactions are ignored, which results in a sparse Jacobian matrix.
1. A computer-implemented method comprising:
simulating a photolithography process using a design layout to produce a first simulated resist image;
perturbing each edge segment in the design layout by a selected amount to produce an initial perturbed layout;
simulating the photolithography process using the initial perturbed layout to produce a second simulated resist image;
determining a difference resist image value between the first simulated resist image and the second simulated resist image for each edge segment;
creating an n×n matrix J such that ΔRI=JΔC, where ΔRI is an n×1 vector of changes in resist image values and ΔC is an n×1 vector of changes in segment locations;
initializing the matrix J using the difference in resist image values divided by the perturbed amount;
determining a correction delta vector ΔC by minimizing |JΔC+RI| 2 +α|ΔC| 2 subject to constraints imposed on ΔC, wherein the correction delta vector includes a correction delta value for each edge segment and α is a non-negative scalar;
perturbing each edge segment in the perturbed layout by the corresponding correction delta value in the correction delta vector ΔC to create a further perturbed layout;
simulating the photolithography process using the further perturbed layout to produce a third simulated resist image;
using information from the third simulated resist image and the matrix J to produce an updated matrix J; and
updating the correction delta vector ΔC by minimizing |JΔC+RI| 2 +α|ΔC| 2 subject to constraints imposed on ΔC, where the updated matrix J is used in the minimization, wherein terms of the updated matrix J represent interactions between neighboring edge segments, and
wherein at least some of the steps of the method are performed using a computer.
2. The method as in claim 1 , further comprising:
for each of a plurality of iterations:
perturbing each edge segment in the perturbed layout by the corresponding correction delta value in the correction delta vector ΔC k to create a further perturbed layout at iteration k;
simulating the photolithography process using the further perturbed layout to produce a new simulated resist image RI k ;
using information from a most recently simulated resist image RI k and the matrix J k−1 to produce an updated matrix J k ; and
updating the correction delta vector ΔC k+1 by minimizing |J k ΔC k+1 +RI k | 2 +α|ΔC k+1 | 2 subject to constraints imposed on ΔC k+1 .
3. The method as in claim 1 , wherein ΔC is constrained by a box constraint ΔC min ≦ΔC≦ΔC max where ΔC min and ΔC max are n×1 vectors specifying minimum and maximum allowable values of ΔC.
4. The method as in claim 3 , wherein ΔC is further constrained by a linear equality constraint AΔC=b , where ΔC is a correction delta vector, A is an r×n matrix, and b is an r×1 vector specifying r equality constraints.
5. The method as in claim 4 , wherein ΔC is further constrained by a linear inequality constraint d≦EΔC≦f, where d and f are t×1 vectors and E is a t×n matrix specifying t linear inequalities.
6. The method as in claim 5 , wherein a total number of unknowns in the minimization is reduced by eliminating variables using the equality constraints.
7. The method as in claim 1 , wherein the matrix J is updated using Broyden's method.
8. The method as in claim 1 , wherein the matrix J is updated using a Kalman filter.
9. The method as in claim 1 , where the matrix J is a sparse matrix.
10. The method as in claim 2 , where the matrix J is a sparse matrix.
11. The method as in claim 3 , where the matrix J is a sparse matrix.
12. The method as in claim 4 , where the matrix J is a sparse matrix.
13. The method as in claim 5 , where the matrix J is a sparse matrix.
14. The method as in claim 6 , where the matrix J is a sparse matrix.
15. The method as in claim 7 , where the matrix J is a sparse matrix.
16. The method as in claim 8 , where the matrix J is a sparse matrix.
17. The method as in claim 13 , where a sparcity of the sparse matrix J is varied as a function of the iteration number.
18. The method as in claim 2 , wherein the plurality of iterations comprises a selected number of iterations.
19. The method as in claim 2 , wherein the plurality of iterations comprises continuing iterations until the simulated resist image achieves a selected figure of merit.
20. A non-transitory machine readable medium encoded with machine executable instructions for performing a method comprising:
simulating a photolithography process using a design layout to produce a first simulated resist image;
perturbing each edge segment in the design layout by a selected amount to produce an initial perturbed layout;
simulating the photolithography process using the initial perturbed layout to produce a second simulated resist image;
determining a difference resist image value between the first simulated resist image and the second simulated resist image for each edge segment;
creating an n×n matrix J such that ΔRI=JΔC, where ΔRI is an n×1 vector of changes in resist image values and ΔC is an n×1 vector of changes in segment locations;
initializing the matrix J using the difference in resist image values divided by the perturbed amount;
determining a correction delta vector ΔC by minimizing |JΔC+RI| 2 +α|ΔC| 2 subject to constraints imposed on ΔC, wherein the correction delta vector includes a correction delta value for each edge segment and α is a non-negative scalar;
perturbing each edge segment in the perturbed layout by the corresponding correction delta value in the correction delta vector ΔC to create a further perturbed layout;
simulating the photolithography process using the further perturbed layout to produce a third simulated resist image;
using information from the third simulated resist image and the matrix J to produce an updated matrix J, wherein terms of the updated matrix J represent interactions between neighboring edge segments; and
updating the correction delta vector ΔC by minimizing |JΔC+RI| 2 +α|ΔC| 2 subject to constraints imposed on ΔC, where the updated matrix J is used in the minimization,
wherein at least some of the steps of the method are performed using a computer.