Method, system and medium for controlling manufacturing process using adaptive models based on empirical data

US 20080177408A1
Filed: 12/20/2007
Published: 07/24/2008
Est. Priority Date: 07/19/2002
Status: Active Grant

First Claim

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1. A computer method for generating parameters for a manufacturing apparatus to produce wafers, the method comprising:

identifying one or more input parameters that cause a change in output characteristics of the wafers produced by the manufacturing apparatus;

estimating the output characteristics of the wafers based on changes to the input parameters;

minimizing a first mathematical function to yield optimized output values; and

generating a recipe for controlling the manufacturing apparatus for producing the wafers based on the optimized output values.

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Abstract

A method, system, and medium of modeling and/or for controlling a manufacturing process is disclosed. The method includes the steps of identifying one or more input parameters that cause a change in output characteristics, defining global nodes using estimated maximum and minimum values of the input parameters, and defining a mathematical equation that calculates a predicted output characteristic for each node. The method also includes the steps of receiving at least one empirical data point having one or more input parameter values and at least one empirical output value and adjusting the predicted output values at the nodes based on a difference between the at least one empirical output value and the predicted output characteristic calculated using the mathematical equation based on the one or more input parameter values.

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25 Claims

1. A computer method for generating parameters for a manufacturing apparatus to produce wafers, the method comprising:
- identifying one or more input parameters that cause a change in output characteristics of the wafers produced by the manufacturing apparatus;
  
  estimating the output characteristics of the wafers based on changes to the input parameters;
  
  minimizing a first mathematical function to yield optimized output values; and
  
  generating a recipe for controlling the manufacturing apparatus for producing the wafers based on the optimized output values.
- View Dependent Claims (2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12)
- - 2. The method of claim 1, wherein one of the output characteristics is a thickness of a region of a layer on a wafer polished by a CMP process.
  - 3. The method of claim 1, wherein estimating the output characteristics comprises:
    - predicting an output value for a set of values of the input parameters;
      
      receiving at least one empirical data point having empirical values for the input parameters and an empirical output value; and
      
      defining the first mathematical function based on a difference between the received empirical output value and a new predicted out value.
  - 4. The method of claim 3, wherein predicting the output value for the set of values of the input parameters comprises:
    - estimating a minimum value and a maximum value for the input parameters, wherein the minimum value and the maximum value are characterized by first nodes defining an input domain;
      
      dividing the input domain into a plurality of cells, each at least one cell having one or more extreme values characterized by second nodes;
      
      estimating output values at the second nodes, wherein an output value numerically represents the output characteristics;
      
      assigning at least one second mathematical function to each at least one cell, the at least one second mathematical function defined by the estimated output values at the second nodes; and
      
      generating the predicted output value using the assigned at least one second mathematical function for the cell based on the set of values of input parameters.
  - 5. The method of claim 3, wherein defining the first mathematical function based on the difference between the received empirical output value and the new predicted output value comprises:
    - changing the estimated output value at the second nodes using the received empirical output value;
      
      changing the second mathematical function using the changed estimated output value; and
      
      using the received empirical values for the input parameters in the changed second mathematical function to predict the new output value.
  - 6. The method of claim 4, wherein the at least one second mathematical function is a ruled interpolation equation.
  - 7. The method of claim 1, wherein the first mathematical function is:
    - $S = \sum_{i, \forall i} {W_{i} (y_{i}^{*} - y_{i}^{predicted})}^{2}$ $with$ $\min_{z_{k} \forall k} s, where$ $y_{i}^{predicted} = \sum_{(by . all . nodes)} α_{ik} z_{k},$ W_i—
      
      a weight coefficient, y_i*—
      
      an empirical output value at an empirical data point i, y_i^predicted—
      
      an output value calculated by the mathematical equation, z_k—
      
      an output value at node k, α
      
      _ik—
      
      a coefficient that depend solely on the location of point i and node k;
      
      i—
      
      an index for an empirical data point, and k—
      
      an index for a node.
  - 8. The method of claim 7, further comprising:
    - adjusting a value assigned to the coefficient, W_i, based on a precision requirement or when a particular empirical data point is received compared with other empirical data points.
  - 9. The method of claim 1, wherein the first mathematical function is:
    - $\min_{z_{k} \forall k} [\sum_{i = 1}^{N_{data}} {W_{i} (y_{i} - y_{i}^{predicted})}^{2} + \sum_{k = 0}^{N_{nodes}} ({W_{k} (z_{k} - z_{k}^{*})}^{2})]$ $s . t . z_{k, \max} \geq z_{k} \geq z_{k, \min}, \forall k Δ y_{j, \max}^{predicted} \geq \frac{\partial y}{\partial x_{j}} \geq Δ y_{j, \min}^{predicted}, \forall j$ W_i—
      
      coefficients which weight the importance of one data point versus another, W_k—
      
      coefficients which weight allowable deviations from a previous model to a new model, y_i—
      
      the i^thobserved output value, y_i^predicted—
      
      the prediction of the i^thobserved output value based on the ruled interpolation, formula using all model inputs corresponding to the i^thobserved output value, y_i^predicted=f(z₁, z₂, . . . , z_m, x₁, x₂, . . . x_n)+z_o, x_j—
      
      a model input from a set which consists of measurements, recipe parameters, or any other value which may be used to facilitate a prediction, z_k—
      
      an output value at node k, z_o—
      
      a bias term for z_k, z_k*—
      
      an output value at node k of a previous model, z_k,max,z_k,min—
      
      maximum and minimum, respectively, for the value of nodes which are updated in the model, Δ
      
      y_j,max,Δ
      
      y_j,min—
      
      maximum and minimum, respectively, for the change in predicted output values with respect to changes in x_j. i—
      
      an index for an empirical data point, and k—
      
      an index for a node.
  - 10. The method of claim 1, wherein minimizing the first mathematical function comprises:
    - solving the first mathematical function as potential energy of a virtual mechanical system, wherein the solving further comprises;
      
      representing the first and second nodes as free moving bodies each of which is configured to move along a guide, wherein the guide is a fixed unmovable body disposed orthogonally to an input parameter space in which input parameter values are located;
      
      providing a representation of at least one member capable of being attached to springs, the member fixed between two of the free moving bodies;
      
      representing the empirical data points as fixed points in the virtual mechanical system; and
      
      connecting a representation of a spring between at least one of the fixed points and the at least one member to force the member to reach an equilibrium position.
  - 11. The method of claim 10, further comprising:
    - representing the free moving bodies as ring-shaped bodies;
      
      representing the fixed unmovable bodies as rods on which the free moving bodies are guided; and
      
      representing the at least one member as at least one of a bar tied to the free moving bodies and configured to move in association with movements of the ring-shaped bodies, or a surface, corners of which are tied to the free moving bodies and configured to move in association with movements of the ring-shaped bodies.
  - 12. The method of claim 1, wherein generating the recipe is performed using a third mathematical equation:
    - $\min_{u_{r} \forall r} [\sum_{l = 1}^{N_{CO}} {W_{l} (y_{l}^{desired} - y_{l}^{predicted})}^{2} + \sum_{r = 0}^{N_{MV}} ({W_{r} (u_{r} - u_{r}^{*})}^{2})]$ $s . t . y_{l, \max} \geq y_{l}^{predicted} \geq y_{l, \min}, \forall l$ $Δ u_{r, \max} \geq Δ u_{r} \geq Δ u_{r, \min}, \forall r$ $u_{r, \max} \geq u_{r} \geq u_{r, \min}, \forall r$ W_l—
      
      coefficients which weight the importance of one controlled output versus another, W_r—
      
      coefficients which weight allowable deviations from the previous value of a manipulated variable to a new value of a manipulated variable, y_i^desired—
      
      the desired value of the l^thcontrolled output, y_i^predicted—
      
      the prediction of the l^thcontrolled output value based on the ruled interpolation formula using all previously determined node values and model inputs which are either calculated or collected, y_i^predicted=f(z₁, z₂, . . . , z_m, u₁, u₂, . . . u_n)+z_o, u_r—
      
      a model input from a set which consists of measurements, manipulated variables, or any other value which may be used to facilitate a prediction, z_k—
      
      an output value at node k, where z_ocorresponds to a bias term, y_l,max,y_l,min—
      
      maximum and minimum, respectively, for the value of predictions which are updated in the model, Δ
      
      u_r,max,Δ
      
      u_r,min—
      
      maximum and minimum, respectively, for the change of manipulated variable values with respect to previous values, and u_r,max,u_r,minmaximum and minimum, respectively, for manipulated variable values.

13. A computer-assisted system configured to generate parameters for a manufacturing apparatus to produce wafers, comprising:
- means for identifying one or more input parameters that cause a change in output characteristics of the wafers produced by the manufacturing apparatus;
  
  means for estimating the output characteristics of the wafers based on changes to the input parameters;
  
  means for minimizing a first mathematical function to yield optimized output values; and
  
  means for generating a recipe for controlling the manufacturing apparatus for producing the wafers based on the optimized output values.
- View Dependent Claims (14, 15, 16, 17, 18, 19)
- - 14. The system of claim 13, wherein one of the output characteristics is a thickness of a region of a layer on a wafer polished by a CMP process.
  - 15. The system of claim 13, wherein the means for estimating the output characteristics comprises:
    - means for predicting an output value for a set of values of the input parameters;
      
      means for receiving at least one empirical data point having empirical values for the input parameters and an empirical output value; and
      
      means for defining the first mathematical function based on a difference between the received empirical output value and a new predicted output value.
  - 16. The system of claim 13, wherein the first mathematical function is:
    - $S = \sum_{i, \forall i} {W_{i} (y_{i}^{*} - y_{i}^{predicted})}^{2}$ $with$ $\min_{z_{k} \forall k} s, where$ $y_{i}^{predicted} = \sum_{(by . all . nodes)} α_{ik} z_{k},$ W_i—
      
      a weight coefficient, y_i*—
      
      an empirical output value at an empirical data point i, y_i^predicted—
      
      an output value calculated by the mathematical equation, z_k—
      
      an output value at node k, α
      
      _ik—
      
      a coefficient that depend solely on the location of point i and node k, i—
      
      an index for an empirical data point, and k—
      
      an index for a node.
  - 17. The system of claim 13, wherein means for minimizing the first mathematical function comprises:
    - means for solving the first mathematical function as potential energy of a virtual mechanical system, wherein the means for solving further comprises;
      
      means for representing the first and second nodes as free moving bodies each of which is configured to move along a guide, wherein the guide is a fixed unmovable body disposed orthogonally to an input parameter space in which input parameter values are located;
      
      means for providing a representation of at least one member capable of being attached to springs, the member fixed between two of the free moving bodies;
      
      means for representing the empirical data points as fixed points in the virtual mechanical system; and
      
      means for connecting a representation of a spring between at least one of the fixed points and the at least one member, to force the member to reach an equilibrium position.
  - 18. The system of claim 17 further comprising:
    - means for representing the free moving bodies as ring-shaped bodies;
      
      means for representing the fixed unmovable bodies as rods on which the free moving bodies are guided; and
      
      means for representing the at least one member as at least one of a bar tied to the free moving bodies and configured to move in association with movements of the ring-shaped bodies, or a surface, corners of which are tied to the free moving bodies and configured to move in association with movements of the ring-shaped bodies.
  - 19. The system of claim 13, wherein the means for generating a recipe is performed using a third mathematical equation:
    - $\min_{u_{r} \forall r} [\sum_{l = 1}^{N_{CO}} {W_{l} (y_{l}^{desired} - y_{l}^{predicted})}^{2} + \sum_{r = 0}^{N_{MV}} ({W_{r} (u_{r} - u_{r}^{*})}^{2})]$ $s . t . y_{l, \max} \geq y_{l}^{predicted} \geq y_{l, \min}, \forall l$ $Δ u_{r, \max} \geq Δ u_{r} \geq Δ u_{r, \min}, \forall r$ $u_{r, \max} \geq u_{r} \geq u_{r, \min}, \forall r$ W_l—
      
      coefficients which weight the importance of one controlled output versus another, W_r—
      
      coefficients which weight allowable deviations from the previous value of a manipulated variable to a new value of a manipulated variable, y_i^desired—
      
      the desired value of the l^thcontrolled output, y_i^predicted—
      
      the prediction of the l^thcontrolled output value based on the ruled interpolation formula using all previously determined node values and model inputs which are either calculated or collected, y_i^predicted=f(z₁, z₂, . . . , z_m, u₁, u₂, . . . u_n)+z_o, u_r—
      
      a model input from a set which consists of measurements, manipulated variables, or any other value which may be used to facilitate a prediction, z_k—
      
      an output value at node k, where z₀corresponds to a bias term, y_l,max,y_l,min—
      
      maximum and minimum, respectively, for the value of predictions which are updated in the model, Δ
      
      u_r,max,Δ
      
      u_r,min—
      
      maximum and minimum, respectively, for the change of manipulated variable values with respect to previous values, and u_r,max,u_r,min—
      
      maximum and minimum, respectively, for manipulated variable values.

20. A computer readable medium comprising executable instructions which when executed on a processing system cause said processing system to generate parameters for a manufacturing apparatus to produce wafers, the method comprising:
- identifying one or more input parameters that cause a change in output characteristics of the wafers produced by the manufacturing apparatus;
  
  estimating the output characteristics of the wafers based on changes to the input parameters;
  
  minimizing a first mathematical function to yield optimized output values; and
  
  generating a recipe for controlling the manufacturing apparatus for producing the wafers based on the optimized output values.
- View Dependent Claims (21, 22, 23, 24)
- - 21. The computer readable medium of claim 20, wherein estimating the output characteristics comprises:
    - predicting an output value for a set of values of the input parameters;
      
      receiving at least one empirical data point having empirical values for the input parameters and an empirical output value; and
      
      defining the first mathematical function based on a difference between the received empirical output value and a new predicted output value.
  - 22. The computer readable medium of claim 20, wherein the first mathematical function is:
    - $S = \sum_{i, \forall i} {W_{i} (y_{i}^{*} - y_{i}^{predicted})}^{2}$ $with$ $\min_{z_{k} \forall k} s, where$ $y_{i}^{predicted} = \sum_{(by . all . nodes)} α_{ik} z_{k},$ W_i—
      
      a weight coefficient, y_i*—
      
      an empirical output value at an empirical data point i, y_i^predicted—
      
      an output value calculated by the mathematical equation, z_k—
      
      an output value at node k, α
      
      _ik—
      
      a coefficient that depend solely on the location of point i and node k, i—
      
      an index for an empirical data point, and k—
      
      an index for a node.
  - 23. The computer readable medium of claim 22, wherein minimizing the first mathematical function comprises:
    - solving the first mathematical function as potential energy of a virtual mechanical system, wherein the solving further comprises;
      
      representing the first and second nodes as free moving bodies each of which is configured to move along a guide, wherein the guide is a fixed unmovable body disposed orthogonally to an input parameter space in which input parameter values are located;
      
      providing a representation of at least one member capable of being attached to springs, the member fixed between two of the free moving bodies;
      
      representing the empirical data points as fixed points in the virtual mechanical system; and
      
      connecting a representation of a spring between at least one of the fixed points and the at least one member, to force the member to reach an equilibrium position.
  - 24. The computer readable medium of claim 20, wherein generating the recipe is performed using a third mathematical equation:
    - $\min_{u_{r} \forall r} [\sum_{l = 1}^{N_{CO}} {W_{l} (y_{l}^{desired} - y_{l}^{predicted})}^{2} + \sum_{r = 0}^{N_{MV}} ({W_{r} (u_{r} - u_{r}^{*})}^{2})]$ $s . t . y_{l, \max} \geq y_{l}^{predicted} \geq y_{l, \min}, \forall l$ $Δ u_{r, \max} \geq Δ u_{r} \geq Δ u_{r, \min}, \forall r$ $u_{r, \max} \geq u_{r} \geq u_{r, \min}, \forall r$ W_l—
      
      coefficients which weight the importance of one controlled output versus another, W_r—
      
      coefficients which weight allowable deviations from the previous value of a manipulated variable to a new value of a manipulated variable, y_i^desired—
      
      the desired value of the l^thcontrolled output, y_i^predicted—
      
      the prediction of the l^thcontrolled output value based on the ruled interpolation formula using all previously determined node values and model inputs which are either calculated or collected, y_i^predicted=f(z₁, z₂, . . . , z_m, u₁, u₂, . . . u_n)+z_o, u_r—
      
      a model input from a set which consists of measurements, manipulated variables, or any other value which may be used to facilitate a prediction, z_k—
      
      an output value at node k, where z₀corresponds to a bias term, y_l,max,y_l,min—
      
      maximum and minimum, respectively, for the value of predictions which are updated in the model, Δ
      
      u_r,max,Δ
      
      u_r,min—
      
      maximum and minimum, respectively, for the change of manipulated variable values with respect to previous values, and u_r,max,u_r,min—
      
      maximum and minimum, respectively, for manipulated variable values.

25. A computer implemented method for generating parameters for a manufacturing apparatus to produce wafers, the method comprising:
- identifying one or more input parameters that cause a change in output characteristics of the wafers produced by the manufacturing apparatus;
  
  predicting an output value for a set of values of the input parameters;
  
  receiving at least one empirical data point having empirical values for the input parameters and an empirical output value;
  
  defining a first mathematical function based on a difference between the received empirical output value and a new predicted output value;
  
  minimizing the first mathematical function to yield optimized output values; and
  
  generating a recipe for controlling the manufacturing apparatus for producing the wafers based on the optimized output values.

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Granted Patent

US 7,970,588 B2
Time in Patent Office

Days
Field of Search
US Class Current

700/97
CPC Class Codes

G05B 13/048 using a predictor

G05B 17/02 electric

Method, system and medium for controlling manufacturing process using adaptive models based on empirical data

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Method, system and medium for controlling manufacturing process using adaptive models based on empirical data

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