Manufacturing process optimizations
This application is a continuation-in-part of application Ser. No. 07/440,057 filed Nov. 21, 1989 now abandoned. 
    
    
     BACKGROUND OF THE INVENTION 
     The present invention concerns a method for selecting manufacturing process variable values for manufacturing process optimization. Specifically, a non-iterative method for selecting process variable values based on a second order response surface mapping representation of the manufacturing process determines the optimal variable values in a single computation for optimizing manufacturing results. 
     Manufacture of product generally involves a series or sequence of interrelated activities collectively referred to as a manufacturing process. Each activity has one or more process variables which affect the efficiency or yield of the particular activity. In addition, the performance of an activity may, and usually does, affect the performance or requirements of another activity in the overall manufacturing process required to manufacture a product. It will therefore be evident that adjustment of the value or set point of any of the variables in any one activity can have a major impact on the manufacture of the final product. Therefore, it is necessary to calculate by computer and adjust all of the process variable values in order to optimize the manufacturing process. In the prior art, models of a manufacturing process have been developed in which the process variable values are adjusted iteratively by sequential perturbation of process parameters and observation of the effect of the adjustment upon the manufacturing process. In other prior methods, particularly methods associated with two controllers, the independent process variable values are simply adjusted to a predetermined reference value. 
     In manufacturing processes there are dependent variables or parameters which are the result of performing the process. Each process is associated with a design target value for each dependent variable. In addition, there are independent variables in a manufacturing process which are adjustable. The manufacturing process efficiency performs according to the adjusted values of the independent variables. In the present invention, a method is disclosed for determining the values for the independent variables which are required to optimize overall manufacturing system process performance. 
     The present invention employs a response surface mapping technique to represent the manufacturing process and to determine the optimal variable values in a single computation for achieving a specified product level target, such as yield. Additionally, in the design and manufacture of a product theoretical values for each of the dependent variables are calculated in order to optimize the manufacturing process. In calculating the values of the independent variables, the present invention seeks a solution most nearly approximating the theoretical values of the dependent variables, i.e., design parameters. 
     SUMMARY OF THE INVENTION 
     A principle object of the present invention is, therefore, the provision of a method for selecting manufacturing process variable values which achieve a specified process response. 
     Another object of the invention is the provision of a method for selecting manufacturing process variable values which yield a specified process response and also approximate design parameters. 
     A further object of the invention is the provision of a method for selecting manufacturing process variable values based on a second order response surface mapping used to represent the manufacturing process. 
     A still further object of the invention is the provision of a non-iterative method for selecting process variable values in a single computation based on a second order response surface mapping of the manufacturing process. 
     Further and still other objects of the present invention will become more apparent when the following specification is read in conjunction with the accompanying drawing. 
    
    
     BRIEF DESCRIPTION OF THE DRAWING 
     The sole FIGURE is a flow diagram of a preferred embodiment for practicing the present invention. 
    
    
     DETAILED DESCRIPTION 
     Typically, before a product is to be manufactured, product development and manufacturing engineering organizations design and test both a manufacturing process and a product manufactured according to the process. The manufacturing variable values determined, for instance, in a preproduction product development project are referred to as design parameters. In theory, when all the independent variables, e.g., temperature, energy level, time, etc., are set to achieve the design parameter values, the developed manufacturing process should be optimized for the product being manufactured. However, in practice, the designed manufacturing process variables require fine adjustment in order to optimize the manufacturing process itself. The present invention provides a method of adjusting the independent variable values of a manufacturing process while seeking a solution such that the adjusted value of the dependent variable approximates the design value. 
     The concept of the invention will best be understood by considering a manufacturing process comprising K independent, controllable variables, i.e., input variables, denoted by vector x of length K. The response variable, i.e., output variable, is denoted as dependent variable y having a specified target value of y T . The problem to be solved by use of a computer is the identification of a vector x r  which achieves a predetermined manufacturing process output target, i.e., set points vector. If it is not possible to achieve the desired target values, the values will be selected so that the closest values of y to the desired target values y T  result. If multiple solutions to the problem exists, the solution closest to the design parameters x s , is the selected solution. In order to demonstrate the present invention, an illustrative example will be described hereinafter. The invention is not limited to the following example which is provided for illustrative purposes only. 
     A simple planar CMOS capacitor manufacturing process comprises thermal oxidation of doped silicon wafer, followed by an implant step and a drive-in (anneal) step and final metallization. The process variables in the example are to be selected so that a predetermined capacitor threshold voltage is achieved. 
     The problem to be solved is the determination of various times, temperatures, energy levels, dosages and the like (i.e., values for all the independent variables) which will result in the achievement of the predetermined threshold voltage (or closest to the threshold voltage) while at the same time, the dependent variable values approximate the design parameters of the manufacturing process. Referring now to the figure, a manufacturing process with design or target values is defined in Step 10. 
     In Step 12, the manufacturing process is expressed as a second order response surface mapping of the process. The mapping is achieved in a known manner by a Design-of-Experiments technique or by simply fitting a second order multi-variant polynomial to manufacturing data obtained from the process. 
     The surface response is expressed as ##EQU1## where bi are the linear coefficients and B ij  are the quadratric coefficients of the second order surface. 
     In Step 14, the second order response surface obtained in Step 12 is transformed into its principle axes. The canonical transformation is accomplished by a translation followed by a rotation. The transformation is given by ##EQU2## where |X O  &gt;=-0.5B -1  |b&gt; and M is an orthogonal matrix whose columns are the (normalized) eigenvectors of matrix B and B -1  is the inverse of matrix B with elements B ij . The coordinates w i  are the new (transformed) coordinate axes, and the terms ##EQU3## represent design specifications in w-space. The response surface has become ##EQU4## where λ is a vector of the eigenvalues of matrix B, and y O  is a process response at coordinates x i  where i=1 . . . K. 
     In Step 16, a nonlinear transformation to new axes via W 2  →&#34;Z&#34; is performed. The second order surface in w-space becomes a first order surface (hyperplane) of dimension k in z-space. The process design specifications are given by Z si  =W si   2 . The response surface is now y=y o  +Σλ i  Z i . 
     In Step 18, the response surface dependent variable y is set to its target value y T . The response surface becomes a hyperplane in K-1 dimension where every point on the hyperplane satisfies the target value requirement. Only positive solutions correspond to the &#34;real world&#34; solutions. The response surface is now C=y T  -y=Σλ i  Z i , where C is a scalar. 
     Next, it is necessary in Step 20 to find the vector Z T  on the hyperplane (in the positive z-space) closest to the process specifications. The result is a desired process values vector in z-space. The vector is ##EQU5## where scalar ##EQU6## If Z T  contains negative components, the negative components must be set to zero and removed. The formula is recalculated with the corresponding components of Z s  and λ removed. The resulting vector represents direction and distance from a point to a hyperplane. 
     The z-space coordinates are converted back into the original x-space parameters in Step 22. The converted x-space parameters are the desired result. 
     The vector components ##EQU7## represent process variable values which achieve the target value of the dependent variable or approximates the value of the target value while concurrently being closest to the process design value specifications. The signs of the square root terms are chosen to match the signs belonging to the vector W s . 
     In Step 24, the results of Step 22 are used to adjust the independent process variable values x in the manufacturing process. 
     The above-described process was tested on a simulation fabrication of a planar CMOS capacitor. The manufacturing (fabrication) process comprised an oxidation step (having the two variables of time and temperature), of an n doped silicon wafer (having a doping concentration variable), followed by an implant step (having dosage and energy variables) followed by an anneal step (having two additional variables of time and temperature) and a metallization step. The response parameter was threshold voltage of the capacitor. Implementation of the above process resulted in the correct determination of all seven listed process variable values in every test case, representing all possible outcomes. 
     While there has been described and illustrated a method of set point optimization for manufacturing processes with specified target values, variations and modifications will be apparent to those skilled in the art without deviating from the broad scope of the invention which shall be limited solely by the scope of the claims appended hereto.