Method for determining stack configuration of substrate
A method for determining a stack configuration for a substrate subjected to a patterning process. The method includes obtaining (i) measurement data of a stack configuration with location information on a printed substrate, (ii) a substrate model configured to predict a stack characteristic based on a location of the substrate, and (iii) a stack map including a plurality of stack configurations based on the substrate model. The method iteratively determines values of model parameters of the substrate model based on a fitting between the measurement data and the plurality of stack configurations of the stack map, and predicts an optimum stack configuration at a particular location based on the substrate model using the values of the model parameters.
1. A non-transitory computer-readable medium comprising instructions stored therein that, when executed by one or more processors, cause the one or more processors to at least:
obtain (i) initial values including a starting point and a search region of model parameters of a computer model configured to predict a characteristic of a patterning process, (ii) measurement data corresponding to the characteristic of the patterning process, (iii) a predicted characteristic using the initial values of the model parameter and the measurement data, and (iv) an objective function, wherein the objective function comprises a first term related to a fit level, and a second term representing a penalty; and
determine values of the model parameters based on the starting point, the search region, and the objective function.
2. The medium of claim 1 , wherein the characteristic of the patterning process is a stack characteristic, wherein the model is a substrate model representing the stack characteristic, and wherein the fit level is indicative of a difference between a predicted characteristic and the measurement data.
3. The medium of claim 2 , wherein the stack characteristic is a substrate thickness, a thickness deviation, an overlay, and/or an alignment.
4. The medium of claim 1 , wherein the instructions are further configured to cause the one or more processors to predict an optimum stack configuration at a particular location based on the substrate model using the values of the model parameters.
5. The medium of claim 2 , wherein the substrate model includes one or more models corresponding to the stack characteristic of one or more layers of the substrate.
6. The medium of claim 4 , wherein the substrate model has a parabolic form.
7. The medium of claim 1 , wherein the search region is defined by a radius with a starting point as a center, wherein the radius is a distance from the center.
8. The medium of claim 1 , wherein the instructions configured to cause the one or more processors to determine the values of the model parameters are further configured to cause the one or more processors to:
determine a number of sample points to be selected from the search region based on a number of model parameters and a size of the search region; and
update the starting point and/or the search region based on the objective function.
9. The medium of claim 1 , wherein the determination of the values of the model parameters comprises an iterative process, wherein an iteration comprises:
determination of a number of sample points to be selected from the search region based on a number of model parameters and a size of the search region;
fitting of the model and the measurement data based on the selected sample points;
determination of a fit level based on the fitting;
evaluation of the objective function comprising the fit level;
evaluation of a fit quality based on the objective function; and
updating of the starting point and the search region based on the fit quality such that the objective function is reduced.
10. The medium of claim 9 , wherein the updating of the starting point and the search region comprises selection of a new starting point and adjustment of the search region, in response to the fit quality breaching one or more thresholds.
11. The medium of claim 1 , wherein the objective function comprises:
a first penalty term configured to maintain a positive value of coefficients of second order terms of the cost function; and/or
a second penalty term associated with a distance between a predicted characteristic and the measurement data.
12. The medium of claim 1 , wherein the determination of the values of the model parameters is an iterative process, wherein an iteration comprises:
generation of a stack map having a plurality of stack configurations based on simulation of the substrate model and a patterning process;
prediction of intermediate values of model parameters based on an optimization algorithm; and
fitting of the measurement data and the plurality of stack configurations of the stack map such that a cost function is reduced.
13. The medium of claim 1 , wherein the patterning process comprises a design for control process configured to automatically predict a stack configuration using the substrate model as perturbations.
14. The medium of claim 1 , wherein the measurement data comprises a metrology recipe used for measurement of one or more stack characteristics of a stack configuration at a particular location on the substrate.
15. The medium of claim 1 , wherein the instructions are further configured to cause the one or more processors to convert measurement data from a Cartesian coordinates to polar coordinates using a Zernike based conversion model.
16. A non-transitory computer-readable medium comprising instructions stored therein that, when executed by one or more processors, cause the one or more processors to at least:
obtain (i) measurement data of a stack configuration, with location information, for a substrate subjected to a patterning process, (ii) a substrate computer model configured to predict a stack characteristic based on a location associated with the substrate, and (iii) a stack map including a plurality of stack configurations based on the substrate model;
determine values of model parameters of the substrate model based on a fitting between the measurement data and the plurality of stack configurations of the stack map; and
predict a stack configuration at a particular location based on the substrate model using the values of the model parameters.
17. The medium of claim 16 , wherein the substrate model includes one or more models corresponding to the stack characteristic of one or more layers of the substrate.
18. The medium of claim 16 , wherein the substrate model is expressed in Cartesian coordinates having a first set of model parameters, and/or in polar coordinates having a second set of model parameters.
19. The medium of claim 16 , wherein the stack configuration comprises a plurality of layers of the substrate, wherein each layer is associated with a stack characteristic.
20. The medium of claim 16 , wherein the stack characteristic is a thickness of a layer of the substrate, a critical dimension of a feature of the substrate, a distance between adjacent features of the substrate, and/or a difference between a thickness of a layer and a selected thickness of the layer.