Patent Publication Number: US-7912679-B2

Title: Determining profile parameters of a structure formed on a semiconductor wafer using a dispersion function relating process parameter to dispersion

Description:
BACKGROUND 
     1. Field 
     The present application generally relates to optical metrology of a structure formed on a semiconductor wafer, and, more particularly, to generating a simulated diffraction signal using a dispersion function relating process parameter to dispersion. 
     2. Related Art 
     Optical metrology involves directing an incident beam at a structure, measuring the resulting diffracted beam, and analyzing the diffracted beam to determine one or more profile parameters of the structure. In semiconductor manufacturing, optical metrology is typically used for quality assurance. For example, after the fabrication of a structure on a semiconductor wafer, an optical metrology tool is used to determine the profile of the structure. By determination of the profile of the structure, the quality of the fabrication process utilized to form the structure can be evaluated. 
     In one conventional optical metrology process, a measured diffraction signal is compared to simulated diffraction signals. The simulated diffraction signals are generated using an optical metrology model. The optical metrology model includes a number of parameters that are varied in generating the simulated diffraction signals. Increasing the number of parameters of the optical metrology model can increase the accuracy of the optical metrology process when those parameters are not correlated to each other. However, parameters are typically correlated to each other to some degree. Thus, increasing the number of parameters of the optical metrology model that are varied can also increase the instability of the model. 
    
    
     
       BRIEF DESCRIPTION OF DRAWINGS 
         FIG. 1  is an architectural diagram illustrating an exemplary fabrication tool and an exemplary optical metrology tool; 
         FIG. 2  depicts the exemplary optical metrology tool in more detail; 
         FIG. 3  is a flowchart of an exemplary process of examining a structure formed on a semiconductor wafer; 
         FIG. 4  depicts an exemplary structure characterized using a set of profile parameters; 
         FIG. 5A  is a flowchart of an exemplary process of defining a dispersion function relating a process parameter to a dispersion; 
         FIG. 5B  is a flowchart of an exemplary process of defining a dispersion function relating a process parameter to a dispersion using a process simulator; 
         FIG. 6  is a flowchart of an exemplary process of controlling the fabrication tool; 
         FIG. 7  is an exemplary flowchart for automated process control using a library; 
         FIG. 8  is an exemplary flowchart for automated process control using a trained machine learning system; and 
         FIG. 9  is an exemplary block diagram of a system for determining and utilizing profile parameters and process parameters for automated process and equipment control. 
     
    
    
     DETAILED DESCRIPTION OF EXEMPLARY EMBODIMENT(S) 
     In order to provide a more thorough understanding of the present invention, the following description sets forth numerous specific details, such as specific configurations, parameters, examples, and the like. It should be recognized, however, that such description is not intended as a limitation on the scope of the present invention, but is intended to provide a better understanding of the exemplary embodiments. 
     In order to facilitate description of the present invention, a semiconductor wafer may be utilized to illustrate an application of the concept. The methods and processes equally apply to other work pieces that have a structure. 
       FIG. 1  depicts one or more wafers  102  processed in an exemplary fabrication tool  104 . One or more semiconductor fabrication processes can be performed on one or more wafers  102  in fabrication tool  104 . Typically a number of wafers  102  are processed as a batch, commonly referred to as a wafer lot, in fabrication tool  104 . For example, a wafer lot of 25 wafers  102  can be processed as a batch in fabrication tool  104 . It should be recognized, however, that the number of wafers  102  in a wafer lot can vary. 
     One or more process parameters are used in performing the one or more semiconductor fabrication processes. For example, the one or more process parameters can include deposition conditions, annealing conditions, etching conditions, temperature, gas pressure, vaporization speed, and the like. The etching conditions can include surface property changes, etching residual components, and the like. 
     Typically, the one or more process parameters are set to define a recipe. Also, the same recipe (i.e., a setting of the one or more process parameters) is used to process the wafers in one wafer lot. One or more individual process parameters of a particular recipe can be adjusted while processing the wafers in one wafer lot. The one or more process parameters can also be set to different values to define different recipes. Different recipes can be used to process different wafer lots. Thus, one recipe can be used to process one wafer lot, and another recipe can be used to process another wafer lot. 
     Fabrication tool  104  can be various types of fabrication tools, such as a coater/developer tool, plasma etch tool, cleaning tool, chemical vapor deposition (CVD) tool, and the like, that perform various fabrication processes. For example, when fabrication tool  104  is a coater/developer tool, the fabrication process includes depositing/developing a photoresist layer on one or more wafers  102 . The one or more process parameters can include temperature. Thus, in this example, variations in temperatures used to perform the deposition/development process can result in a variation in the photoresist layer, such as the thickness of the photoresist layer, deposited/developed using the coater/developer tool. 
     As depicted in  FIG. 1 , after one or more semiconductor fabrication processes are performed on one or more wafers  102  in fabrication tool  104 , one or more wafers  102  can be examined using optical metrology tool  106 . As will be described in more detail below, optical metrology tool  106  can be used to determine one or more profile parameters of a structure formed on one or more wafers  102 . 
     As depicted in  FIG. 2 , optical metrology tool  106  can include a photometric device with a source  204  and a detector  206 . The photometric device can be a reflectometer, ellipsometer, hybrid reflectometer/ellipsometer, and the like. 
     A structure  202  formed on wafer  102  is illuminated by an incident beam from source  204 . Diffracted beams are received by detector  206 . Detector  206  converts the diffracted beam into a measured diffraction signal, which can include reflectance, tan (Ψ), cos (Δ), Fourier coefficients, and the like. Although a zero-order diffraction signal is depicted in  FIG. 2 , it should be recognized that non-zero orders can also be used. 
     Optical metrology tool  106  also includes a processing module  208  configured to receive the measured diffraction signal and analyze the measured diffraction signal. Processing module  208  can include a processor  210  and a computer-readable medium  212 . It should be recognized, however, that processing module  208  can include any number of components in various configurations. 
     In one exemplary embodiment, processing module  208  is configured to determine one or more profile parameters of structure  202  using any number of methods which provide a best matching diffraction signal to the measured diffraction signal. These methods can include a library-based process, or a regression based process using simulated diffraction signals obtained by rigorous coupled wave analysis and machine learning systems. See, U.S. Pat. No. 6,943,900, titled GENERATION OF A LIBRARY OF PERIODIC GRATING DIFFRACTION SIGNALS, filed on Jul. 16, 2001, issued Sep. 13, 2005, which is incorporated herein by reference in its entirety; U.S. Pat. No. 6,785,638, titled METHOD AND SYSTEM OF DYNAMIC LEARNING THROUGH A REGRESSION-BASED LIBRARY GENERATION PROCESS, filed on Aug. 6, 2001, issued Aug. 31, 2004, which is incorporated herein by reference in its entirety; U.S. Pat. No. 6,891,626, titled CACHING OF INTRA-LAYER CALCULATIONS FOR RAPID RIGOROUS COUPLED-WAVE ANALYSES, filed on Jan. 25, 2001, issued May 10, 2005, which is incorporated herein by reference in its entirety; and abandoned U.S. patent application Ser. No. 10/608,300, titled OPTICAL METROLOGY OF STRUCTURES FORMED ON SEMICONDUCTOR WAFERS USING MACHINE LEARNING SYSTEMS, filed on Jun. 27, 2003, which is incorporated herein by reference in its entirety. 
     With reference to  FIG. 3 , an exemplary process  300  is depicted of examining a structure formed on a semiconductor wafer using an optical metrology model. It should be recognized that exemplary process  300  can be performed prior or subsequent to the structure actually being formed on the semiconductor wafer. 
     In step  302 , an optical metrology model is obtained. The optical metrology model has one or more profile parameters, one or more process parameters, and a dispersion. A dispersion function is defined to relate the process parameter to the dispersion. The dispersion function can be expressed in various forms, such as wavelength by wavelength tablet form, the optical property of each wavelength can be a function of process parameters, or using dispersion models with model parameters varied with the process parameters. It should be recognized, however, that the optical metrology model can also include any number of additional parameters. 
     The profile parameters of an optical metrology model characterize the geometric characteristics of a structure. For example,  FIG. 4  depicts an exemplary structure characterized using a set of profile parameters. In particular, the bottom width of the first layer of the structure is characterized using profile parameter w 1 . The top width of the first layer and the bottom width of the second width of the structure are characterized using profile parameter w 2 . The top width of the second layer of the structure is characterized using profile parameter w 3 . The height of the first layer of the structure is characterized using profile parameter h 1 . The height of the second layer of the structure is characterized using profile parameter h 2 . It should be recognized that the structure can have various shapes and can be characterized using any number of profile parameters. 
     As described above, the process parameters of an optical metrology model characterize one or more process conditions of a process for fabricating a structure. For example, with reference to  FIG. 1 , the process parameter can characterize a process condition in fabrication tool  104  used to fabricate a structure on wafer  102 . Examples of process parameters include deposition conditions (such as temperature, gas pressure, vaporization speed, etc.), annealing conditions, etching conditions (surface property change, etching residual components, etc.), and the like. 
     The dispersion characterizes optical properties of a material of the structure, the structure being formed by the process. For example, the dispersion can include the refractive indices (n) and the extinction coefficients (k) of a material. The optical metrology model can include separate dispersions for each material of the structure. For example, with reference again to  FIG. 4 , a first dispersion (n s  &amp; k s ) can correspond to the material of the substrate on which the structure is formed, such as silicon. A second dispersion (n 1  &amp; k 1 ) can correspond to the material of the first layer of the structure, such as oxide. A third dispersion (n 2  &amp; k 2 ) can correspond to the material of the second layer of the structure, such as poly-silicon. 
     With reference again to  FIG. 3 , in step  304 , a dispersion function is obtained that relates the dispersion to at least one of the process parameters. For example, with reference to  FIG. 4 , the dispersion function can relate the dispersion corresponding to the material of the second layer of the structure (n 2  &amp; k 2 ) to the temperature used in fabricating the structure in fabrication tool  104  ( FIG. 1 ). An exemplary process of developing the dispersion function will be described in more detail below. 
     With reference again to  FIG. 3 , in step  306 , a simulated diffraction signal is generated using the optical metrology model and a value for the at least one process parameter that is related to the dispersion by the dispersion function obtained in step  304  and a value for the dispersion. The value of the dispersion is calculated using the value for the at least one process parameter and the dispersion function. By relating the process parameter to the dispersion, the number of varying parameters in the optical metrology model is reduced with the desired accuracy, which increases the stability of the model. 
     For example, with reference again to  FIG. 4 , assume that the dispersion function obtained in step  304  ( FIG. 3 ) related the dispersion corresponding to the material of the second layer of the structure (n 2  &amp; k 2 ) to the temperature used in fabricating the structure in fabrication tool  104  ( FIG. 1 ). Assume that in generating a simulated diffraction signal, a value for the temperature is specified (T 1 ). Thus, the value of n 2  &amp; k 2  is calculated using the dispersion function obtained in step  304  ( FIG. 3 ) and T 1 . 
     As described in more detail below, the values of the remaining dispersions (n s  &amp; k s , n 1  &amp; k 1 ) can be fixed to set values or floated, meaning that the values can be varied when additional simulated diffraction signals are generated. A set of values is specified for the profile parameters. The simulated diffraction signal is then generated using the values for the profile parameters, the process parameters, and the dispersions. In particular, the simulated diffraction signal can be generated by applying Maxwell&#39;s equations and using a numerical analysis technique to solve Maxwell&#39;s equations, such as Rigorous Coupled Wave Analysis (RCWA). See, U.S. patent application Ser. No. 09/770,997, titled CACHING OF INTRA-LAYER CALCULATIONS FOR RAPID RIGOROUS COUPLED-WAVE ANALYSES, filed on Jan. 25, 2001, now U.S. Pat. No. 6,891,626, which is incorporated herein by reference in its entirety. The simulated diffraction signal can be generated using a machine learning system (MLS) employing a machine learning algorithm, such as back-propagation, radial basis function, support vector, kernel regression, and the like. See abandoned U.S. patent application Ser. No. 10/608,300, titled OPTICAL METROLOGY OF STRUCTURES FORMED ON SEMICONDUCTOR WAFERS USING MACHINE LEARNING SYSTEMS, filed on Jun. 27, 2003, which is incorporated herein by reference in its entirety. 
     With reference again to  FIG. 3 , in step  308 , a measured diffraction signal measured off the structure is obtained. For example, as described above, with reference to  FIG. 1 , after the structure is formed using fabrication tool  104 , optical metrology tool  106  can be used to measure a diffraction signal off the structure formed on the wafer. 
     With reference again to  FIG. 3 , in step  310 , the measured diffraction signal is compared to the simulated diffraction signal generated in step  306 . In step  312 , one or more profile parameters of the structure are determined based on the comparison of the measured diffraction signal to the simulated diffraction signal. 
     In particular, if the measured diffraction signal and the simulated diffraction signal match within one or more matching criteria such as goodness of fit and/or cost function, then the profile parameters used to generate the matching simulated diffraction signal are assumed to characterize the geometric shape of the structure. The dispersion used to generate the matching simulated diffraction signal is assumed to characterize the optical properties of the one or more materials of the structure to which the dispersion corresponds. The process parameters used to generate optical properties of the one or more materials and then generate the matching simulated diffraction signal are assumed to characterize the process conditions used to fabricate the structure. 
     If the measured diffraction signal and the simulated diffraction signal do not match within one or more matching criteria such as goodness of fit and/or cost function, then the measured diffraction signal is compared to one or more additional simulated diffraction signals until a match is found. The additional simulated diffraction signals are generated using a value of at least one profile parameter, process parameter, or dispersion that is different than the simulated diffraction signal that did not match the measured diffraction signal. 
     Values of one or more of the profile parameters (e.g., w 1 , w 2 , w 3 , h 1 , h 2  of  FIG. 4 ) can be changed in generating the additional simulated diffraction signals. Values of one or more of the process parameters can be changed in generating the additional simulated diffraction signals. If the values of the process parameters that are related to the dispersions in the dispersion function obtained in step  304  are changed, then the values of the dispersions are recalculated using the values of the process parameter and the dispersion function obtained in step  304 . The values of the dispersions that are not related to process parameters in the dispersion function obtained in step  304  are fixed to set values or are allowed to float, meaning they are changed in generating the additional simulated diffraction signals. 
     For example, assume that the dispersion function obtained in step  304  related temperature to the dispersion corresponding to the material of the second layer of the structure (n 2  &amp; k 2 ) depicted in  FIG. 4 . Assume that a first value of the temperature (T 1 ) was used in generating the first simulated diffraction signal. As described above, a first value of n 2  &amp; k 2  was calculated using the dispersion function obtained in step  304  and the first value of T 1 . Assume now that a second value of the temperature (T 2 ) is used in generating an additional simulated diffraction signal (i.e., a second simulated diffraction signal). Thus, in the present example, a second value of n 2  &amp; k 2  is calculated using the dispersion function obtained in step  304  and T 2 . 
     As described above, the remaining dispersions, which are not related to process parameters, can be fixed to set values or floated. For example, values of n s  &amp; k s , which correspond to the material of the substrate on which the structure in  FIG. 4  is formed, can be fixed to set values in generating the additional simulated diffraction signals. Values of n 1  &amp; k 1 , which corresponds to the material of the first layer of the structure in  FIG. 4 , can be floated (varied) in generating the additional simulated diffraction signals independent of the changes to the values for the process parameters. 
     In a library-based process, the additional simulated diffractions signals are generated in advance and stored in a library. In a regression-based process, the additional simulated diffraction signal is not generated until after the measured diffraction signal is found not to match the simulated diffraction signal to which it was compared. 
     With reference to  FIG. 5A , an exemplary process  500  is depicted of developing the dispersion function relating a process parameter to a dispersion. It should be recognized that exemplary process  500  can be performed in advance of exemplary process  300  ( FIG. 3 ). It should also be recognized that exemplary process  500  can be performed in advance or subsequent to forming the structure to be examined. 
     In step  502 , a first wafer is fabricated using a first value of the process parameter to be correlated to a dispersion. For example, assume the process parameter is temperature. Also assume that the dispersion is n 2  &amp; k 2 , which corresponds to the second layer of the structure depicted in  FIG. 4 . Thus, with reference to  FIG. 1 , the first wafer is fabricated using a first value of temperature in fabrication tool  104 . In particular, the second layer of the structure depicted in  FIG. 4  is formed using the first value of temperature in fabrication tool  104 . 
     With reference again to  FIG. 5A , in step  504 , a first value of the dispersion is measured from the fabricated first wafer. Returning to the example above, the value of the n &amp; k of the second layer of the structure depicted in  FIG. 4  is measured. 
     In step  506 , a second wafer is fabricated using a second value of the process parameter. In step  508 , a second value of the dispersion is measured from the fabricated second wafer. 
     In step  510 , a third wafer is fabricated using a third value of the process parameter. In step  512 , a third value of the dispersion is measured from the fabricated third wafer. 
     The first, second, and third values for the process parameter used in steps  502 ,  506 , and  510  are different from each other. In one exemplary embodiment, all other process parameters are kept constant in fabricating the first, second, and third wafers. 
     For example, assume again that the process parameter is temperature. Thus, in this example, with reference to  FIG. 1 , the first, second, and third wafers are fabricated in fabrication tool  104  using a first, second, and third temperature settings (T 1 , T 2 , and T 3 ) that are different. All other process parameters, such as pressure, vaporization speed, and the like, are kept constant in fabricating the first, second, and third wafers in fabrication tool  104 . 
     In one exemplary embodiment, the first, second, and third values of the dispersion can be measured using optical metrology tool  106 . In particular, assume test structures with the shape shown in  FIG. 4  are fabricated on the first, second, and third wafers using fabrication tool  104  ( FIG. 1 ). Measured diffraction signals are measured off the test structures using optical metrology tool  106  ( FIG. 1 ). The measured diffraction signals are compared to simulated diffraction signals to determine the first, second, and third values of the dispersions (e.g., n &amp; k) of the second layers of the test structures on the first, second, and third wafers. Note, the simulated diffraction signals used in determining the first, second, and third values of n &amp; k of the second layers of the test structures are generated by floating the values of n &amp; k. It is understood that more than three sets of wafers can be fabricated with different values of the one or more process parameters in order to provide a bigger data sample for fitting the dispersion function. 
     In step  514 , the dispersion function is defined using the first, second, and third values of the dispersion and the first, second, and third values of the process parameter. For example, a second-order polynomial can be fitted to the first, second, and third values of the dispersion and the first, second, and third values of the process parameter. In one exemplary embodiment, the dispersion function is defined for various wavelengths λ. Stated more generally, the second-order polynomial can be fitted to any number of dispersions (n &amp; k) and process parameters (p) with coefficients: an 0 (λ), an 1 (λ), an 2 (λ), ak 0 (λ), ak 1 (λ), and ak 2 (λ) by minimizing the following error functions: 
                     Enr   ⁡     (   λ   )       =       ∑     j   =   1     k     ⁢       (         an   0     ⁡     (   λ   )       +         an   1     ⁡     (   λ   )       ·     p   j       +         an   2     ⁡     (   λ   )       ·     p   j   2       -       n   j     ⁡     (   λ   )         )     2               (   1.1   )                 Ekr   ⁡     (   λ   )       =       ∑     j   =   1     k     ⁢       (         ak   0     ⁡     (   λ   )       +         ak   1     ⁡     (   λ   )       ·     p   j       +         ak   2     ⁡     (   λ   )       ·     p   j   2       -       k   j     ⁡     (   λ   )         )     2               (   1.2   )               
This fitting can be performed wavelength by wavelength. The number of samples (test structures) is larger than the number of the polynomial order plus one. It should be recognized that various functions can be fitted, such as a Taylor series in multi-dimensions.
 
     Although a linear relationship of the variables is used in the examples above, it should be recognized that non-linear functional relationships between the variables can be used, such as arbitrary functions, composite functions, and the like. Least squares fit solutions to a polynomial may include the determinant, matrix, independent parameter solutions, and the like. Least squares fit to an arbitrary function may include nonlinear fitting methods, searching parameter space methods, grid search methods, gradient search methods, expansion methods, the Marquardt method, and the like. For a more detailed discussion of these techniques, see Bevington, et al., “Data Reduction and Error Analysis for the Physical Sciences,” Third Edition, pages 116-177, which is incorporated herein by reference. 
     As described above, in step  306 , a simulated diffraction signal using the optical metrology model is generated with the value of the dispersion being calculated using the value for the process parameter and the dispersion function. Continuing with the foregoing example, the value of n &amp; k can be calculated for process parameter (p) at a wavelength as follows:
 
 n (λ)= an   0 (λ)+ an   1 (λ)· p+an   2 (λ)· p   2   (2.1)
 
 k (λ)= ak   0 (λ)+ ak   1 (λ)· p+ak   2 (λ)· p   2   (2.2)
 
     The simulated diffraction signal is best matched with a measured diffraction signal. The profile parameters and process parameters that are used to generate the simulated diffraction signal that best matches the measured diffraction signal are assumed to characterize one or more profile parameters of the structure from which the measured diffraction signal was measured and the process parameters used to fabricate the structure. 
     The profile parameters and process parameters can be used to evaluate the fabrication tool and/or fabrication process used to fabricate the structure. For example, the profile parameters can be used to evaluate the fabrication tool used in the fabrication process. The process parameters can be used to monitor the process condition of the fabrication process performed using the fabrication tool. Thus, fabrication tool and/or fabrication process can be controlled to achieve a more stable performance and increase yield. 
     With reference to  FIG. 5B , an exemplary process  550  is depicted of developing the dispersion function relating one or more process parameters to a dispersion using process simulation. As mentioned above, it should be recognized that exemplary process  550  can be performed in advance of exemplary process  300  ( FIG. 3 ). In step  552 , one or more process parameters of the fabrication process are selected. As mentioned above, the one or more process parameters can include deposition conditions, annealing conditions, etching conditions, temperature, gas pressure, vaporization speed, and the like. The etching conditions can include surface property changes, etching residual components, and the like. If the fabrication process is a deposition, the temperature of the chamber and/or the gas pressure may be selected as the process parameters. In step  554 , a set of values of the one or more process parameters is determined based on empirical data or experience with the application or recipe. The one or more process parameters selected are the process parameter that will be correlated to the dispersion. 
     In step  556 , fabrication of the wafer structure is simulated using a process simulator for each set of values of the selected one or more process parameters determined in step  554 . Simulation of a fabrication process is typically done with process simulators such as Athena™ from Silvaco International, Prolith™ from KLA-Tencor, Solid-C from Sigma-C Gmbh, TCAD™ from Synopsis, and the like. Specifics of the recipe and the wafer structure and process parameters are provided to the process simulator. One of the typical output of the process simulation is the structure profile after the simulated process is completed. For example, assume the process parameter of interest is temperature. Also assume that the dispersion is n 2  &amp; k 2 , which corresponds to the second layer of the structure depicted in  FIG. 4 . Thus, simulation of fabrication of the wafer structure is simulated using a value of temperature. In particular, simulation of formation of the second layer of the structure depicted in  FIG. 4  is done using the first value of temperature utilized by the process simulator. 
     With reference again to  FIG. 5B , in step  558 , a value of the dispersion is determined for the simulated wafer structure for each value of the one or more process parameters selected in step  552 . Returning to the example above, the value of the n &amp; k of the second layer of the structure depicted in  FIG. 4  may be obtained by setting up n &amp; k as an output of the process simulator or may be calculated using n &amp; k simulation software. Examples of n &amp; k simulation software include n &amp; k Analyzer™ from n&amp;k Technology, Prolith™ from KLA-Tencor, and the like. In one exemplary embodiment, all other process parameters are kept constant in the process simulation runs needed for step  558 . For example, assume again that the process parameter is temperature. Thus, in this example, all other process parameters used in the process simulator, such as pressure, vaporization speed, and the like, are kept constant. 
     In step  560 , the dispersion function is defined using the values of the dispersion and the values of the selected one or more process parameters. For example, if only one process parameter is selected, a second-order polynomial can be fitted to the values of the dispersion and the values of the process parameter. In one exemplary embodiment, the dispersion function is defined for various wavelengths λ. As described above, the second-order polynomial can be fitted to any number of dispersions (n &amp; k) and process parameters (p) with coefficients: an 0 (λ), an 1 (λ), an 2 (λ), ak 0 (λ), ak 1 (λ), and ak 2 (λ) by minimizing the following error functions expressed in equation (1.1) and (1.2) above. This fitting can be performed wavelength by wavelength. The number of samples (test structures) is larger than the number of the polynomial order plus one. It should be recognized that various functions can be fitted, such as a Taylor series in multi-dimensions. 
     As mentioned above, a linear relationship of the variables is used in the examples above, it should be recognized that non-linear functional relationships between the variables can be used, such as arbitrary functions, composite functions, and the like. Least squares fit solutions to a polynomial may include the determinant, matrix, independent parameter solutions, and the like. Least squares fit to an arbitrary function may include nonlinear fitting methods, searching parameter space methods, grid search methods, gradient search methods, expansion methods, the Marquardt method, and the like. For a more detailed discussion of these techniques, see Bevington, et al., “Data Reduction and Error Analysis for the Physical Sciences,” Third Edition, pages 116-177, which is incorporated herein by reference. 
     As described above, in step  306  of  FIG. 3 , a simulated diffraction signal using the optical metrology model is generated with the value of the dispersion being calculated using the value for the one or more process parameters and the dispersion function. Continuing with the foregoing example, the value of n &amp; k can be calculated for process parameter (p) at a wavelength as depicted in equations (2.1) and (2.2). The simulated diffraction signal is best matched with a measured diffraction signal. The profile parameters and the one or more process parameters that are used to generate the simulated diffraction signal that best matches the measured diffraction signal are assumed to characterize one or more profile parameters of the structure from which the measured diffraction signal was measured and the one or more process parameters used to fabricate the structure. As also mentioned above, the profile parameters and one or more process parameters can be used to evaluate the fabrication tool and/or fabrication process used to fabricate the structure. 
     With reference to  FIG. 6 , an exemplary process  600  is depicted for determining one or more profile parameters and/or one or more process parameters for automated process control. In step  602 , an optical metrology model is created for a structure formed on a semiconductor wafer. The optical metrology model comprises one or more profile parameters, which characterize one or more geometric characteristics of the structure, one or more process parameters, which characterize one or more process conditions for fabricating the structure, and a dispersion, which characterizes optical properties of a material of the structure. 
     In step  604 , a dispersion function is obtained that relates the dispersion to at least one of the one or more process parameters. In step  606 , a simulated diffraction signal is generated using the optical metrology model and a value for the at least one of the process parameters and a value for the dispersion. The value for the dispersion is calculated using the value for the at least one of the process parameter and the dispersion function. 
     In step  608 , a measured diffraction signal of the structure is obtained using an optical metrology tool. In step  610 , the measured diffraction signal is compared to the simulated diffraction signal. In step  612 , one or more profile parameters of the structure and one or more process parameters are determined based on the comparison in step  610 . In step  614 , the fabrication tool is controlled based on the determined one or more profile parameters of the structure. 
     With reference to  FIG. 7 , an exemplary process  700  is depicted for determining one or more profile parameters and one or more process parameters for automated process control using a library. In step  702 , an optical metrology model is created for a structure formed on a semiconductor wafer. The optical metrology model comprises one or more profile parameters, which characterize one or more geometric characteristics of the structure, one or more process parameters, which characterize one or more process conditions for fabricating the structure, and a dispersion, which characterizes optical properties of a material of the structure. 
     In step  704 , a dispersion function is obtained that relates the dispersion to at least one of the one or more process parameters. In step  706 , a simulated diffraction signal library is generated using the optical metrology model and a value for the at least one of the process parameters and a value for the dispersion. The value for the dispersion is calculated using the value for at least one process parameter and the dispersion function. 
     In step  708 , a measured diffraction signal of the structure is obtained using an optical metrology tool. In step  710 , a best match to the measured diffraction signal is obtained from the library. In step  712 , one or more profile parameters of the structure and one or more process parameters are determined based on the comparison in step  710 . In step  714 , the fabrication tool is controlled based on the determined one or more profile parameters of the structure or the determined one or more process parameters. 
     With reference to  FIG. 8 , an exemplary process  800  is depicted for determining one or more profile parameters and one or more process parameters for automated process control using a trained machine learning system. In step  802 , an optical metrology model is created for a structure formed on a semiconductor wafer. The optical metrology model comprises one or more profile parameters, which characterize one or more geometric characteristics of the structure, one or more process parameters, which characterize one or more process conditions for fabricating the structure, and a dispersion, which characterizes optical properties of a material of the structure. 
     In step  804 , a dispersion function is obtained that relates the dispersion to at least one of the one or more process parameters. In step  806 , a set of simulated diffraction signals is generated using the optical metrology model and a set of values for the at least one of the process parameters and a set of values for the dispersion. The value for the dispersion is calculated using the value for the at least one of the process parameter and the dispersion function. 
     In step  808 , a machine learning system is trained to process the set of simulated diffraction signals as input and generate the value of the one or more profile parameters and one or more process parameters as output. In step  810 , a measured diffraction signal of the structure is obtained using an optical metrology tool. In step  812 , the measured diffraction signal is input into the trained machine learning system. In step  814 , one or more profile parameters of the structure and one or more process parameters are determined based on the output of the machine learning system. In step  816 , the fabrication tool is controlled based on the determined one or more profile parameters of the structure or the determined one or more process parameters. 
     The determined one or more process parameters can include deposition conditions, annealing conditions, or etching conditions, and the one or more fabrication processes can include a deposition process, annealing process, or etching process, respectively. The deposition conditions can include temperature, gas pressure, and vaporization speed. The etching conditions can include surface property changes, and etching residual components. 
       FIG. 9  is an exemplary block diagram of a system for determining and utilizing profile parameters and process parameters for automated process and equipment control. System  900  includes a first fabrication cluster  902  and optical metrology system  904 . System  900  also includes a second fabrication cluster  906 . Although the second fabrication cluster  906  is depicted in  FIG. 9  as being subsequent to first fabrication cluster  902 , it should be recognized that second fabrication cluster  906  can be located prior to first fabrication cluster  902  in system  900  (e.g. and in the manufacturing process flow). 
     For example, a photolithographic process, such as exposing and/or developing a photoresist layer applied to a wafer, can be performed using the first fabrication cluster  902 . In one exemplary embodiment, optical metrology system  904  includes an optical metrology tool  908  and processor  910 . Optical metrology tool  908  is configured to measure a diffraction signal off the structure. Processor  910  is configured to compare the measured diffraction signal against the simulated diffraction signal. If the measured diffraction signal and the simulated diffraction signal match within a matching criterion, such as goodness of fit and/or cost function, one or more values of the profile parameters associated with the simulated diffraction signal are determined to be the one or more values of the profile parameters and the one or more process parameters associated with the measured diffraction signal are determined to be the one more values of the process parameters used to fabricate the structure. 
     In one exemplary embodiment, optical metrology system  904  can also include a library  912  with a plurality of simulated diffraction signals and corresponding one or more profile parameters and one or more process parameters. As described above, the library can be generated in advance; metrology processor  910  can compare a measured diffraction signal off the structure to the plurality of simulated diffraction signals in the library. When a matching simulated diffraction signal is found, the one or more values of the profile parameters and process parameters associated with the matching simulated diffraction signal in the library is assumed to be the one or more values of the profile parameters and process parameters used in the wafer application to fabricate the structure. 
     System  900  also includes a metrology processor  916 . In one exemplary embodiment, processor  910  can transmit the one or more values of the one or more profile parameters and/or process parameters to metrology processor  916 . Metrology processor  916  can then adjust one or more process parameters or equipment settings of first fabrication cluster  902  based on the one or more values of the one or more profile parameters and/or process parameters determined using optical metrology system  904 . Metrology processor  916  can also adjust one or more process parameters or equipment settings of the second fabrication cluster  906  based on the one or more values of the one or more profile parameters and/or process parameters determined using optical metrology system  904 . As noted above, fabrication cluster  906  can process the wafer before or after fabrication cluster  902 . In another exemplary embodiment, processor  910  is configured to train machine learning system  914  using the set of measured diffraction signals as inputs to machine learning system  914  and profile parameters and process parameters as the expected outputs of machine learning system  914 . 
     Furthermore, a computer readable medium (not shown) such as computer memory, disk, and/or storage may be used to store the instructions and computer programs to generate a simulated diffraction signal using a dispersion function relating process parameter to dispersion or to determine one or more profile parameters of a structure and one or more process parameters using an optical metrology model and generated simulated diffraction signals using the aforementioned method. In another embodiment, computer-executable instructions may be stored in a computer readable medium to determine one or more profile parameters and process parameters using a library comprising simulated diffraction signals and corresponding profile parameters and process parameters, and a measured diffraction signal. In another embodiment, computer-executable instructions may be stored in a computer readable medium to train a machine learning system to use a measured diffraction signal as input and generate one or more profile parameters and process parameters as output. In yet another embodiment, similar computer-executable instructions may be stored in a computer readable medium to control a photolithography cluster or other fabrication cluster using determined one or more profile parameters and process parameters to control a fabrication cluster. 
     Although exemplary embodiments have been described, various modifications can be made without departing from the spirit and/or scope of the present invention. Therefore, the present invention should not be construed as being limited to the specific forms shown in the drawings and described above.