Method for determining contribution to a fingerprint

Abstract

A method, system and program for determining a fingerprint of a parameter. The method includes determining a contribution from a device out of a plurality of devices to a fingerprint of a parameter. The method includes obtaining parameter data and usage data, wherein the parameter data is based on measurements for multiple substrates having been processed by the plurality of devices, and the usage data indicates which of the devices out of the plurality of the devices were used in the processing of each substrate; and determining the contribution using the usage data and parameter data.

1 . A method for determining a contribution of a device out of a plurality of devices to a fingerprint of a parameter, the parameter being associated with processing of a substrate, the method comprising:
 obtaining parameter data and usage data, wherein the parameter data is based on measurements for multiple substrates having been processed by the plurality of devices, and the usage data indicates which of the devices out of the plurality of the devices were used in the processing of each substrate; and   determining the contribution using the usage data and parameter data.   
     
     
         2 . The method of  claim 1 , further comprising determining a matrix using the usage data and wherein determining the contribution to the fingerprint comprises solving an equation comprising the matrix using the parameter data. 
     
     
         3 . The method of  claim 2 , wherein the matrix equation is solved to determine the contribution of a device out of a plurality of devices by multiplying a transformed version of the matrix with the parameter data, optionally wherein the transformed version of the matrix is the inverse of the matrix. 
     
     
         4 . The method of  claim 2 , wherein a row of the matrix represents a substrate having been processed by at least one device in a first class of devices and at least one device in a second class of devices, and the column of the matrix represents one of the plurality of devices. 
     
     
         5 . The method of  claim 2 , wherein a row of the matrix has a non-zero entry corresponding to each of the devices used to process the substrate represented by that respective row, and a zero entry for the devices out of the plurality of devices not used to process the substrate represented by that respective row. 
     
     
         6 . The method of  claim 2 , wherein determining the matrix comprises:
 a. generating a first matrix, N, representing the possible combinations of the plurality of devices used to process a substrate;   b. determining a vector, n i , for each row, i, of the first matrix, N;   c. calculating a delta covariance matrix, ΔY i  for each row, wherein the delta covariance matrix is n i   T n i ;   d. selecting a row, i, from the first matrix, N, and storing the selected row as a second matrix, M;   e. determining if a stopping criteria has been met, wherein:
 if the stopping criteria is not met, continuing with step f using an updated first matrix, N, in which the selected row is removed; and 
 if the stopping criteria is met, using the second matrix, M, as the determined matrix; 
   f. calculating the pseudo-determinant corresponding to each remaining row of the updated first matrix, N;   g. determining a row with a preferred pseudo-determinant, updating the second matrix, M, to include the row with the preferred pseudo-determinant, and updating the first matrix, N, by removing the row with the preferred pseudo-determinant;   h. determining if a stopping criteria has been met, wherein:
 if the stopping criteria is not met, returning to step f using an updated first matrix, N, in which the row with the preferred pseudo-determinant is removed; and 
 if the stopping criteria is met, the updated second matrix, M, is used as the determined matrix. 
   
     
     
         7 . The method of  claim 2 , wherein the matrix comprises at least one sub design matrix. 
     
     
         8 . The method of  claim 7 , wherein the sub-design matrix may be a zero entry sub-design matrix or a non-zero entry sub-design matrix, wherein a zero entry sub-design matrix corresponds to the devices out of the plurality of devices not used to process the substrate and a non-zero sub-design matrix corresponds to each of the devices used to process the substrate, wherein a non-zero entry sub-design matrix is based on a modelled contribution to the fingerprint from the respective device and substrate. 
     
     
         9 . The method of  claim 2 , wherein the equation comprising the matrix is solved using:
 i. least squares fit, optionally using regularization, and further optionally using an L-curve method and/or leave-one-out cross validation; and/or   ii. Bayesian statistics.   
     
     
         10 . The method of  claim 1 , further comprising analyzing variation of the parameter data using the usage data, and wherein determining the contribution to the parameter for a device comprises grouping the data using the analyzed variation. 
     
     
         11 . The method of  claim 1 , wherein at least one of the devices is controlled based on the determined contribution to the fingerprint of the parameter of that device. 
     
     
         12 . The method of  claim 1 , wherein the parameter is one or more selected from:
 i. critical dimension, overlay, critical dimension uniformity, line edge placement, alignment, focus, pattern shift, line edge roughness, micro topology, and/or edge placement error; and/or   ii. a shape description of a feature, such as side-wall angle, resist height, and/or contact hole ellipticity; and/or   iii. a processing parameter such as a coating thickness, optionally bottom anti-reflective coating thickness and/or resist thickness, and/or an optical property of a coating which may optionally indicate a measure of absorption, such as refractive index and/or extinction coefficient; and/or   iv. a parameter determined from substrate measurements, such as a yield parameter, optionally defect and/or electrical performance.   
     
     
         13 . The method of  claim 1 , wherein the parameter data is based on processed measurements, and wherein the measurements are processed using principal component analysis or fitting of a model, desirably a polynomial model or a linear model. 
     
     
         14 . A system comprising a processor configured to determine a contribution of a device out of a plurality of devices to a fingerprint of a parameter, the parameter being associated with processing of a substrate, the processor configured to at least:
 obtain parameter data and usage data, wherein the parameter data is based on measurements for multiple substrates having been processed by the plurality of devices, and the usage data indicates which of the devices out of the plurality of devices were used in the processing of each substrate; and   determine the contribution using the usage data and the parameter data.   
     
     
         15 . A program to control determining a contribution of a device out of a plurality of devices to a fingerprint of a parameter, the parameter being associated with processing of a substrate, the program comprising instructions for carrying out at least:
 obtaining parameter data and usage data, wherein the parameter data is based on measurements for multiple substrates having been processed by the plurality of devices, and the usage data indicates which of the devices out of the plurality of devices were used in the processing of each substrate; and   determining the contribution using the usage data and the parameter data.