Optical metrology system and metrology mark characterization device

Abstract

An optical metrology system is disclosed that has a measuring system configured to irradiate a metrology mark and record a portion of a reflected, a transmitted, or both, electromagnetic field and a characterization device configured to determine from the recorded field a mark shape parameter indicative of the structure of the metrology mark, the characterization device comprising: a field calculation unit configured to calculate an expected field for reflection, transmission, or both, from a theoretical reference mark based on an algebraic eigenvalue-eigenvector representation of the expected field, a field derivative calculation unit configured to calculate a first order derivative, a higher order derivative, or both, of the expected field with respect to the mark shape parameter by first deriving analytical forms for corresponding derivatives of eigenvalues and eigenvectors of the eigenvalue-eigenvector representation, and an optimization unit configured to use the outputs from the field and field derivative calculation units to determine an optimized mark shape parameter for which the expected field substantially matches the recorded field.

1 . An optical metrology system, comprising: 
 a measuring system configured to irradiate a metrology mark and record a portion of a reflected, a transmitted, or both, electromagnetic field; and    a characterization device configured to determine from the recorded field a mark shape parameter indicative of the structure of the metrology mark, the characterization device comprising:    a field calculation unit configured to calculate an expected field for reflection, transmission, or both, from a theoretical reference mark based on an algebraic eigenvalue-eigenvector representation of the expected field,    a field derivative calculation unit configured to calculate a first order derivative, a higher order derivative, or both, of the expected field with respect to the mark shape parameter by first deriving analytical forms for corresponding derivatives of eigenvalues and eigenvectors of the eigenvalue-eigenvector representation, and    an optimization unit configured to use the outputs from the field and field derivative calculation units to determine an optimized mark shape parameter for which the expected field substantially matches the recorded field.    
   
   
       2 . The system according to  claim 1 , wherein 
 the field calculation unit is configured to solve a second order differential equation for the field; and    the eigenvalues and eigenvectors of the eigenvalue-eigenvector representation are eigenvalues and eigenvectors of a coefficient matrix of the second order differential equation.    
   
   
       3 . The system according to  claim 1 , wherein the metrology mark of a substrate has a periodic structure along an axis within the plane of the substrate.  
   
   
       4 . The system according to  claim 1 , wherein the metrology mark comprises a one-dimensional grating, a two-dimensional grating, or both.  
   
   
       5 . The system according to  claim 1 , wherein the field calculation unit is arranged to calculate the expected field using Rigorous Coupled-Wave Analysis (RCWA), the C-method, or both.  
   
   
       6 . The system according to  claim 1 , wherein the optimization unit is arranged to start searching for an optimal mark shape parameter based on a starting parameter selected according to an expected distortion of the alignment mark.  
   
   
       7 . The system according to  claim 6 , wherein the expected distortion is derived based on a process history of a substrate.  
   
   
       8 . The system according to  claim 1 , wherein the algebraic eigenvalue-eigenvector representation of the field comprises a matrix A, characterized by eigenvector matrix W and eigenvalue matrix Λ according to the following eigenvalue system equation AW=WΛ.  
   
   
       9 . The system according to  claim 8 , wherein the field derivative calculation unit is configured to determine first derivatives of the eigenvalue matrix Λ, denoted by Λ′, using the following expression:  
     
       

       W 
       −1 
       A′W−Λ′=CΛ−ΛC,  

     
     where the eigenvector derivatives of W, denoted by W′, are projected onto the basis consisting of the eigenvectors W according to the relation W′=WC, with C as a coefficient matrix.  
   
   
       10 . The system according to  claim 8 , wherein the field derivative calculation unit is configured to determine second derivatives of the eigenvalue matrix Λ, denoted by Λ″, using the following expression:  
         W   −1   A″W−Λ″= 2 Λ′C+ 2 CΛ′+DΛ−ΛD+ 2Λ CC− 2 CΛC,    
     where the first eigenvector derivatives of W, denoted by W′, are projected onto the basis consisting of the eigenvectors W according to the relation W′=WC, with C as a coefficient matrix, and the second eigenvector derivatives of W, denoted by W″, are projected onto the basis consisting of the eigenvectors W according to the relation W″=WD, with D as a coefficient matrix.  
   
   
       11 . A lithographic apparatus, comprising: 
 an optical metrology system, including: 
 a measuring system configured to irradiate an alignment mark on a substrate and record a portion of a reflected, a transmitted, or both, electromagnetic field;  
 a characterization device configured to determine from the recorded field a mark shape parameter indicative of the structure of the alignment mark, the characterization device comprising: 
 a field calculation unit configured to calculate an expected field for reflection, transmission, or both, from a theoretical reference mark based on an algebraic eigenvalue-eigenvector representation of the expected field,  
 a field derivative calculation unit configured to calculate a first order derivative, a higher order derivative, or both, of the expected field with respect to the mark shape parameter by first deriving analytical forms for corresponding derivatives of eigenvalues and eigenvectors of the eigenvalue-eigenvector representation, and  
 an optimization unit configured to use the outputs from the field and field derivative calculation units to determine an optimized mark shape parameter for which the expected field substantially matches the recorded field; and  
 
   a substrate position determining device configured to derive a position of the substrate using the optimized mark shape parameter determined by the characterization device.    
   
   
       12 . The apparatus according to  claim 11 , wherein the substrate position determining device is configured to derive a position of the substrate by reference to a center of symmetry of the theoretical reference mark as defined by the optimized mark shape parameter.  
   
   
       13 . The apparatus according to  claim 11 , wherein the field calculation unit is arranged to calculate the expected field using Rigorous Coupled-Wave Analysis (RCWA), the C-method, or both.  
   
   
       14 . A method of characterizing a metrology mark, comprising: 
 irradiating a metrology mark and recording a portion of a reflected, a transmitted, or both, electromagnetic field;    determining from the recorded field a mark shape parameter indicative of the structure of the metrology mark;    calculating an expected field for reflection, transmission, or both, from a theoretical reference mark based on an algebraic eigenvalue-eigenvector representation of the expected field;    calculating a first order derivative, a higher order derivative, or both, of the expected field with respect to the mark shape parameter by first deriving analytical forms for corresponding derivatives of eigenvalues and eigenvectors of the eigenvalue-eigenvector representation; and    using the results of the calculating to determine an optimized mark shape parameter for which the expected field substantially matches the recorded field.    
   
   
       15 . The method according to  claim 14 , wherein the expected field is calculated using a Rigorous Coupled-Wave Analysis (RCWA), the C-method, or both.  
   
   
       16 . The method according to  claim 14 , wherein determining the optimized mark shape parameter starts with a starting parameter selected according to an expected distortion of the alignment mark.  
   
   
       17 . A device manufacturing method, comprising: 
 characterizing a metrology mark formed on a substrate by: 
 irradiating a metrology mark and recording a portion of a reflected, a transmitted, or both, electromagnetic field;  
 determining from the recorded field a mark shape parameter indicative of the structure of the metrology mark;  
 calculating an expected field for reflection, transmission, or both, from a theoretical reference mark based on an algebraic eigenvalue-eigenvector representation of the expected field;  
 calculating a first order derivative, a higher order derivative, or both, of the expected field with respect to the mark shape parameter by first deriving analytical forms for corresponding derivatives of eigenvalues and eigenvectors of the eigenvalue-eigenvector representation; and  
 using the results of the calculating to determine an optimized mark shape parameter for which the expected field substantially matches the recorded field; and  
   aligning the substrate using the result of the characterization.    
   
   
       18 . The method according to  claim 17 , wherein the expected field is calculated using a Rigorous Coupled-Wave Analysis (RCWA), the C-method, or both.  
   
   
       19 . The method according to  claim 17 , wherein determining the optimized mark shape parameter starts with a starting parameter selected according to an expected distortion of the alignment mark.