Semiconductor yield management system and method

A system and method for yield management is disclosed wherein a data set containing one or more prediction variable values and one or more response values is input into the system. The system can pre-process the input data set to remove prediction variables with missing values and data sets with missing values. The pre-processed data can then be used to generate a model that may be a decision tree. The system can accept user input to modify the generated model. Once the model is complete, one or more statistical analysis tools can be used to analyze the data and generate a list of the key yield factors for the particular data set.

BACKGROUND OF THE INVENTION

This invention relates generally to a system and method for managing a semiconductor process and in particular to a system and method for managing yield in a semiconductor process.

The semiconductor industry is continually pushing toward smaller and smaller geometries of the semiconductor devices being produced since smaller devices generate less heat and operate at a higher speed than larger devices. Currently, a single chip may contain over one billion patterns. The semiconductor manufacturing process is extremely complicated since it involves hundreds of processing steps. A mistake or small error at any of the process steps or tool specifications may cause lower yield in the final semiconductor product, wherein yield may be defined as the number of functional devices produced by the process as compared to the theoretical number of devices that could be produced assuming no bad devices. Improving yield is a critical problem in the semiconductor industry and has a direct economic impact to the semiconductor industry. In particular, a higher yield translates into more devices that may be sold by the manufacturer.

Semiconductor manufacturing companies have been collecting data for a long time about various process parameters in an attempt to improve the yield of the semiconductor process. Today, an explosive growth of database technology has contributed to the yield analysis that each company follows. In particular, the database technology has far outpaced the yield management ability when using conventional statistical methods to interpret and relate yield to major yield factors. This has created a need for a new generation of tools and techniques for automated and intelligent database analysis for yield management.

Current conventional yield management systems have a number of limitations and disadvantages which make them less desirable to the semiconductor industry. For example, the conventional systems may require some manual processing which slows the analysis and makes it susceptible to human error. In addition, these conventional systems may not handle both continuous and categorical yield management variables. Some conventional systems cannot handle missing data elements and do not permit rapid searching through hundreds of yield parameters to identify key yield factors. Some conventional systems output data that is difficult to understand or interpret even by knowledgeable semiconductor yield management people. In addition, the conventional systems typically process each yield parameter separately, which is time consuming and cumbersome and cannot identify more than one parameter at a time.

Thus, it is desirable to provide a yield management system and method which solves the above limitations and disadvantages of the conventional systems and it is to this end that the present invention is directed.

SUMMARY OF THE INVENTION

The yield management system and method in accordance with the invention may provide many advantages over conventional methods and systems which make the yield management system and method more useful to semiconductor device manufacturers. In particular, the system may be fully automated and easy to use so that no extra training is necessary to make use of the yield management system. In addition, the system handles both continuous (e.g., temperature) and categorical (e.g., Lot1, Lot2, etc.) variables. The system also automatically handles missing data during a pre-processing step. The system can rapidly search through hundreds of yield parameters and generate an output indicating the one or more key yield factors/parameters. The system generates an output (a decision tree) that is easy to interpret and understand. The system is also very flexible in that it permits prior yield parameter knowledge (from users) to be easily incorporated into the building of the model in accordance with the invention. Unlike conventional systems, if there is more than one yield factor/parameter affecting the yield of the process, the system can identify all of the parameters/factors simultaneously so that the multiple factors are identified during a single pass through the yield data.

In accordance with a preferred embodiment of the invention, the yield management method may receive a yield data set. When a data set comes in, it first goes through a data preprocessing step in which the validity of the data in the data set is checked and cases or parameters with missing data are eliminated. Using the cleaned up data set, a Yield Mine model is built during a model building step. Once the model is generated automatically by the yield management system, the model may be modified by one or more users based on their experience or prior knowledge of the data set. Once the model has been modified, the data set may be processed using various statistical analysis tools to help the user better understand the relationship between the response and predict variables.

DETAILED DESCRIPTION OF A PREFERRED EMBODIMENT

The invention is particularly applicable to a computer-implemented software-based yield management system and it is in this context that the invention will be described. It will be appreciated, however, that the system and method in accordance with the invention has greater utility since it may be implemented in hardware or may incorporate other modules or functionality not described herein.

FIG. 1is a block diagram illustrating an example of a yield management system10in accordance with the invention implemented on a personal computer12. In particular, the personal computer may include a display unit14, that may be a cathode ray tube (CRT), a liquid crystal display or the like, a processing unit16and one or more input/output devices18that permit a user to interact with the software application being executed by the personal computer. In this example, the input/output devices may include a keyboard20and a mouse22, but may also include other peripheral devices such as printers, scanners and the like. The processing unit16may further include a central processing unit (CPU)24, a persistent storage device26, such as a hard disk, a tape drive, an optical disk system, a removable disk system or the like and a memory28. The CPU may control the persistent storage device and memory. Typically, a software application may be permanently stored in the persistent storage device and then may be loaded into the memory28when the software application is going to be executed by the CPU. In the example shown, the memory28may contain a yield manager30. The yield manager may be implemented as one or more software applications that are executed by the CPU.

In accordance with the invention, the yield management system may also be implemented using hardware and may be implemented on different types of computer systems, such as client/server systems, web servers, mainframe computers, workstations and the like. Now, more details of the implementation of the yield management system in software will be described.

FIG. 2is block diagram illustrating more details of the yield management system30in accordance with the invention. In particular, the yield management system may receive a data set containing various types of semiconductor process data including continuous/numerical data, such as temperature or pressure, and categorical data, such as the lot number of the particular device. The yield management system in accordance with the invention may process the data set, generate a model, apply one or more statistical tools to the model and data set and generate an output that may indicate, for example, the key factors/parameters that affected the yield of the devices that generated the current data set.

In more detail, the data may be input to a data preprocessor32that may validate the data and remove any missing data records. The output from the data preprocessor may be fed into a model builder34so that a model of the data set may be automatically generated by the system. Once the system has generated a model, the user may enter model modifications into the model builder to modify the model based on, for example, past experience with the particular data set. Once the user modifications have been incorporated into the model, a final model is output and made available to a statistical tool library36. The library may contain one or more different statistical tools that may be used to analyze the final model. The output of the system may be, for example, a listing of one or more factors/parameters that contributed to the yield of the devices that generated the data set being analyzed. As described above, the system is able to simultaneously identify multiple yield factors. Now, a yield management method in accordance with the invention will be described.

FIG. 3is a flowchart illustrating an example of a yield management method40in accordance with the invention. The method may include receiving an input data set in step41and preprocessing the input data set in step42to clean up the data set (e.g., validate the data and remove any data records containing missing, erroneous or invalid data elements). In step44, the cleaned up data set may be used to build a model and the user may enter model modifications in step46. Once the model is complete, it may be analyzed in step48using a variety of different statistical tools to generate yield management information such as key yield factors. Each of the above steps of the method will now be described in more detail to better understand the invention. In particular, the data preprocessing step in accordance with the invention will now be described.

The data preprocessing step42helps to clean up the incoming data set so that the later analysis may be more fruitful. The yield management system in accordance with the invention can handle data sets with complicated data structures. A yield data set typically has hundreds of different variables. These variables may include both a response variable, Y, and predictor variables, X1, X2, . . . , Xm, that may be of a numerical type or a categorical type. A variable is a numerical type variable if its values are real numbers, such as different temperatures at differenttimetimesduring the process. A variable is a categorical type variable if its values are of a set of finite elements not necessarily having any natural ordering. For example, a categorical variable couldtakestakevalues in a set of {MachineA, MachineB, MachineC} or values of (Lot1, Lot2or Lot3).

It is very common for a yield data set to have missing values. The data pre-processing step removes the cases or variables having missing values. In particular, the preprocessing first may remove all predictor variables that are “bad”. By “bad”, it is understood that either a variable has too much missing data, ≧MS, or, for a categorical variable, if the variable has too many distinct classes, ≧DC. In accordance with the invention, both MS and DC are user defined thresholds so that the user may set these values and control the preprocessing of the data. In a preferred embodiment, the defaultvaluevaluesare MS=0.05×N, DC=32, where N is the total number of cases in the data set.

Once the “bad” predictor variables are removed, then, for the remaining data set, data preprocessing may remove all cases with missing data. If one imagines that the original data set is a matrix with each column representing a single variable, then data preprocessing first removes all “bad” columns (variables) and then removes “bad rows” (missing data) in the remaining data set with the “good” columns.

FIG. 4is a diagram illustrating an example of the data preprocessing procedure in accordance with the invention. In particular, for this example, the MS variable is set to 2.FIG. 4shows an original data set50, a data set52once bad columns have been removed and a data set54once the bad rows have been removed. As shown, the original data set50may include three predictor variables (Pred1, Pred2and Pred3) and a numerical variable (Response) wherein three values for Pred3are unknown and one value for Pred2is unknown. Since the MS is set to 2 in this example, any predictor variables that have more than two unknown values are removed. Thus, as shown in the processed data set52, the column containing the Pred3variable is removed from the data set. Since the Pred2variablehavehasonly one missing value, it is not removed from the data set in this step. Next, any bad rows of data are removed from the data set. In the example shown inFIG. 4, the row with a Pred1value of 0.5 is removed since the row contained an unknown value for variable Pred2. Thus, once the preprocessing has been completed, the data set54contains no missing values so that the statistical analysis may produce better results. Now, the model building step in accordance with the invention will be described in more detail.

The yield management system uses a decision tree-based method. In particular, the method partitions the data set, D, into sub-regions. The tree structure may be a hierarchical way to describe a partition of D. It is constructed by successively splitting nodes (as described below), starting with the root node (D), until some stopping criteria are met and the node is declared a terminal node. For each terminal node, a value or a class is assigned to all the cases within the node. Now, the node splitting method and example of the decision tree will be described in more detail.

FIG. 5shows an example of a Yield Mine model decision tree100generated by the system. In this example, the data set contains 233 process step variables, 233 time variables corresponding to each process step, and 308 parametric test variables. All of the variables are not shown inFIG. 5for clarity. All of these variables are used in the Yield Mine model building as predictor variables. The response variable in this example is named “Good DielRing” and represents the number of good dies around the edge of a wafer produced during a particular process run.

In this example, out of all 774 predictor variables, the Yield Mine system using the decision tree prediction, identifies one or more variables as key yield factors. In this example, the key yield factor variables are PWELLASH, FINISFI, TI_TIN_RTP_, and VTPSP_. In this example, PWELLASH and FINISFI are time variables associated with the process variables PWELLASH_and FINISFI_andTI13TIN_RTP_TI—TIN—RTP_and VTPSP_are process variables. Note that, for each terminal node102in the decision tree, the value of the response variable at that terminal node is shown so that the user can view the tree and easily determine which terminal node (and thus which predictor variables) result in the best value of the response variable.

In the tree structure model in accordance with the invention, if a tree node is not terminal, it has a splitting criterion for the construction of its sub-nodes as will be described in more detail below with reference toFIG. 6. For example, the root node is split into two sub-nodes depending on the criterion of whether PWELLASH is before or after 3:41:00 am, 07/03/98. If PWELLASH is before 3:41:00 am, 07/03/98, the case is put in the left sub-node. Otherwise, it is put in the right sub-node. The left sub-node is further split into its sub-nodes using the criterion FINISFI <07/17/98 4:40:00 pm. The right sub-node is also further split into its sub-nodes using the criterion TI_TIN_RTP—=2RTP, whereTI13TIN_RTP_TI—TIN—RTP_is a process step parameter and 2RTP is one of its specification if the variable is continuous. For a terminal node, the average value of all cases under the node is shown. In this example, it is pretty clear to the user that when PWELLASH<07/03/98 3:41:00 am, the yield is higher, especially when the criterion FINISFI<07/17/98 4:40:00 pm is also satisfied. The worst case happens when PWELLASH>07/03/98 3:41:00 am, and TI_TIN_RTP—<>1RTP, and VTPSPE—ε{23STEP, 25STEP, 26STEP}.

To find the proper stopping criteria for tree construction is a difficult problem. To deal with the problem we first over-grow the tree and then apply cross validation techniques to prune the tree. Pruning the tree is described in detail in the following sections. To grow an over sized tree, the method may keep splitting nodes in the tree until all cases in the node having the same response value, or the number of cases in the node is less than a user defined threshold, n0. The default in our algorithm is n0=max{5, floor(0.02×N)} where N is the total number of cases in D, and the function floor(x) gives the biggest integer that is less than or equal to x. Now, the construction of the decision tree and the method for splitting tree nodes in accordance with the invention will be described.

FIG. 6is a flowchart illustrating a method110for splitting nodes of a decision tree in accordance with the invention. In step112, a particular node of the decision tree, T, is selected. The process is then repeated for each node of the tree. In step114, the method may determine if the number of data values in node T is less than a predetermined threshold, N. If the number of data values is less than N, then the splitting for the particular node is stopped in step116and the next node may be processed. If the number of data values for the node is not less than N, then in step118, the processing of the particular node is continued. In particular, for each predictor variable, i, where i=1 . . . n, the “goodness” of the split value, Φi, is calculated. Then, in step120, the predictor variables, j, are selected such that Φj=MAX {Φi|i=1, . . . n}. In step122, the method may determine if Φj>V wherein V is a user-defined threshold value as described below. If Φjis not greater than the threshold value, then in step124, the splitting process for the particular node is stopped and the processing begins with the next node.

If Φj>V, then in step126, the node, T, is split into one or more sub-nodes, T1, T2, . . . , Tm, based on the variable j. In step128, for each sub-node, Tkwhere k=1, . . . , m, the same node splitting method is applied. In this manner, each node is processed to determine if splitting is appropriate and then each sub-node created during a split is also checked for susceptibility to splitting as well. Thus, the nodes of the decision tree are split in accordance with the invention. Now, more details of the decision tree construction and node splitting method will be described.

A decision tree is built to find relations between the response variable and the predictor variables. Each split, S, of a node, T, partitions the node into m subnodes T1, T2, . . . , Tm, in hopes that the subnodes are less “noisy” than T as defined below. To quantify this idea, a real-value function that measures the noisiness of a node T, g(T), may be defined wherein NTdenotes the number of cases in T, and NTidenotes the number of cases in the ith sub-node Ti. The partition of T is exclusive, therefore, Σi=1mNTi=NT. Next, the method may define Φ(S) to be the goodness of split function for a split, S, wherein:

We say that the subnodes are less noisy than their ancestor if Φ(S)>0. In Yield Mine, a node split depends only on one predictor variable. The method may search through all predictor variables, X1, X2, . . . , Xn, one by one to find the best split based on each predictor variable. Then, the best split is the one to be used to split the node. Therefore, it is sufficient to explain the method by describing how to find the best split for a single predictor variable. Depending on the types of the response variable, Y, and the predictor variable, X, as being either categorical or numerical, there are four possible scenarios. Below, details for each scenario on how the split is constructed and how to assign a proper value or a class to a terminal node is described. Now, the case when Y and X are both categorical variables is described.

Y is Categorical and X isAtegoricalCategorical

Suppose that Y takes values in the set A={A1, A2, . . . , Ak}, and X takes values in the set B={B1,B2, . . . , Bl}. In this case, only binary splits are allowed. That is, if a node is split, it produces 2 sub-nodes, a left sub-node, TL, and a right sub-node, TR. A split rule has the form of a question: Is xΣBS, where BSis a subset of B. If the answer to the question is yes, then the case is put in the left sub-node TL. Otherwise it is put in the right sub-node TR. There are 2ldifferent subsets of B. Therefore, there are 2ldifferent splits.

Let NiTdenote the number of class i cases in node T. The function whichmeasuremeasuresthe noisiness of the node, g(T), is defined as:

Since there are only two sub-nodes, the goodness of split function, Φ(S), is:

The method thus searches through all positions 2lsplits to find the one that minimizes Φ(S). Now, the case where Y is categorical and X is numerical will be described.

Y is Categorical and X is Numerical

Suppose that Y takes values in the set A={A1,A2, . . . , Ak} and x1,x2, . . . , xNTdenotes the numerical predictor variable in each case. The split in this case is also binary as above. We will again use TLand TRto denote the two sub-nodes of T. A split rule takes one of the two forms.1) Is x≦α?2) Is α<x≦β?where α and β take values from the set {x1, x2, . . . , xNT}. For each case, if the answer to the split rule is true, it is put in TL. Otherwise it is put in TR.

Now, we define NTand g(T) in the same way as in the previous scenario. Since a split in this case has one more parameter than a splitinthe first case above, the method may define

Φ⁡(S)={g⁡(T)-NTLNT⁢g⁡(TL)-NTRNT⁢g⁡(TR)if⁢⁢S⁢⁢has⁢⁢form⁢⁢(1)c⁡(g⁡(T)-NTLNT⁢g⁡(TL)-NTRNT⁢g⁡(TR))if⁢⁢S⁢⁢has⁢⁢form⁢⁢(2)wherein c is between 0 and 1 and can be set by the user. The default is 0.9. There are only finite split rules of form (1) and (2). The method thus searches through all possible splits to find the one that minimizes Φ(S). Now, the case where Y is numerical and X is categorical will be described.

Y is Numerical and X is Categorical

In this case, the split rule is the same as the first case. The only difference is the way in which the noiseness function, g(T), is defined. In particular, since Y is numerical, let y1, y2, . . . , yNTdenote the response variable for all cases in T. g(T) is then defined as the LPnorm of the empirical distribution function of Y in T. In particular,

As in the first case, there are onlyafinite number of possible splits and the method searches through all possible splits to find the one that minimizes Φ(S). Now, a fourth case where Y and X are both numerical will be described.

Y is Numerical and X is Numerical

In this case, the split rule is defined the same way as in the second case above, and g(T) is defined the same way as in the third case. Thus, the method may search through all possible splits to come up with the split, S*, which minimizes Φ(S), where:

Then, a linear regression model, as set forth below, is fit
y=a0+a1x+ε,  (5)where ε is assumed to be i.i.d. Gaussian with mean 0 and variance σ2
Now, let ŷidenote the fitted value of the model for case i. Let r be the LPnorm of the residuals. That is,

h2={1if⁢⁢i<NT⁢⁢mod⁢⁢d0Otherwisewhere d is a user defined parameter. The default value of d is 4. Now, assigning a value or class to a terminal node will be described.

When a terminal node is reached, a value or a class, ƒ(T), is assigned to all cases in the node depending on the type of the response variable. If the type of the response variable is numerical, ƒ(T) is a real value number. Otherwise, ƒ(T) is set to be a class member of the set A={A1, A2, . . . , Ak}. Now, the cost function may be determined if Y is categorical or numerical.

Y is Categorical

Assume Y takes values in set A={A1, A2, . . . , Ak}. T is a terminal node with NTcases. Let NiTbe the number, Y, equal to Aiin T, iε{1,2, . . . , L}. If the node is pure (i.e., all the cases in the node has the same response Aj), then, ƒ(T)=Aj. Otherwise, the node is not pure. No matter which class, ƒ(T), is assigned to, there is at least one case misclassified in the node. Let u(i|j) be the cost of assigning a class j case to class i. Then the total cost of assigning ƒ(T) to node T is

If u(i|j) is constant for all i and j, then ƒ(T) is assigned to the biggest class in the node. When there is a tie for the best choice of ƒ(T) among several classes, ƒ(T) is picked arbitrarily among those classes. Now, the case where Y is numerical is described.

Y is Numerical

In this case, the cost function is the same function g(T) whichmeasuremeasuresthe “noisiness” of the node as described above. ƒ(T) is assigned to the value which minimizes the cost. It can be easily shown that, when g(T) is the L2norm of the node, ƒ(T) equals to the mean value of the node. Now, the pruning of the decision tree will be described.

By growing an oversized tree as described above, one encounters the problem of over fitting. To deal with this problem, cross validation is used to find the right size of the model. Then, the tree can be pruned to the proper size. Ideally, one would like to split the data into two sets. One for constructing the model and one for testing. But, unless the data set is sufficiently large, using only part of the data set to build the model reduces its accuracy. Therefore, cross validation is the preferred procedure.

An n-fold cross validation procedure starts with dividing the data set into n subsets, D1,D2, . . . , Dn. The division is random and each subset contains as nearly as possible, the same number of cases. Let Dicdenote the compliment set of Di. Then n tree structure models TR1, TR2, . . . , TRnare built using the D1c, D2c, . . . , Dnc. Now we can use the cases in Dicto test the validity of TRiand to find out what is the right size of the tree model.

A measure of the size of a tree structure model, g(TR), the complexity of TR, is defined as follows. Let TTdenote the set of terminal nodes of a tree node T. Let C(t) be the cost function of node t if all nodes under t are pruned. Thus,

where |TT| is the cardinality of TT, and

where P(t) is the probability function.

Next, one can define
g(TR)=max(g(T)|T is a node of TR)

Theorem: Let T0be the node, such that g(T0)=g(TR). Then, pruning off all sub-nodes of T0will not increase the complexity of the tree.

Proof

Let TRNbe the tree obtained by pruning off T0from TR. Every node TNin tree TRNcomes from the node T in TR. If we can show that, for every TN, g(TN)≦g(T), then, by definition, g(TRN)≦g(TR).

There are two scenarios. 1) For node TN, itscounter partcounterpartT contains T0as one of itssub-nodesub-nodes. 2) For node TN, itscounter partcounterpartT does not contain T0as a sub-node. In the second scenario, TNand Thashavethe same structure. Therefore, g(TN)=g(T). Now, let us consider the first scenario. If TNhas no sub-node, then, g(TN)=0≦g(T). Otherwise, by definition,

This theorem establishes a relationship between the size of a tree structure model and its complexity g(TR). In general, the bigger the complexity the more the number of nodes of the tree.

Cross validation can point out which complexity value v is likely to produce the most accurate tree structure. Using this v, we can prune the tree generated from the whole data set until its complexity is just below v. This pruned tree is used as the final tree structure model. Now, the model modification step will be described.

In some cases, the predictor variables can be correlated with each other. The splits of a node based on different parameters can produce similar results. In such cases, it is really up to the process engineer who uses the software to identify which parameter is the real cause of the Yield problem. To help the engineer to identify the possible candidates of parameters at any node split, all predictor variables are ranked according to their relative significance if the split were based on them. To be more precise, let Xibe the variable picked by the method which the split, S*, is based on.

For any j≠i let Sjdenote the best split based on Xj. Then, define

Since S* is the best split 0≦q(i)≦1 . Then, when double clicking on a node, a list of all predictor variables ranked by their q values is shown as illustrated inFIG. 5. If the user decides it is more appropriate to split on a predictor variable other than the one picked by the method, the user can then highlight it from the list, and a single click of the mouse will reproduce the tree and force node T to be split based on user picked parameters. For example,FIG. 7illustrates two trees130,140wherein the first tree130is built using the PWELLASH variable selected by the method and the second tree140is built using the user-selected parameter PWELLCB. Now, a method for tree prediction in accordance with the invention will be described in more detail.

FIG. 8is a flowchart illustrating a method150for tree prediction in accordance with the invention. In step152, the previously generated tree model is input. Next, the prediction value, X, of interest is input. In step156, the prediction using the tree starts at the root node, T. In step158, the method may determine if the node is terminal. If the current node is not terminal, then the method may assign X to one of the sub-nodes in step160according to the split rule described above. After the assignment in step160, the method loops back to step158to test if the next node is a terminal node. If the node is terminal, then the method outputs the prediction value in step162and the method is completed. Now, the analysis step in accordance with the invention will be described.

All the basic statistical analysis tools are available to help the user to validate the model and identify the yield problem. At each node, a right click of the mouse produces a list of tools available as shown inFIG. 9. Every analysis is done at the node level (i.e., it only uses the data from that particular node). An example of the analysis tools available at the right node after the first split is shown inFIG. 9. In this example, those analysis tools may include box-whisker chart, Cumsum control chart, Shewhet control chart, histogram, one-way ANOVA, two sample comparison and X-Y correlation analysis. The particular tools available to the user depend upon the nature of the X and Y parameters (e.g., continuous vs. categorical).

After each model is built, the tree can be saved for future predictions. If a new set of parameter values is available, it can be fed into the model and generate prediction of the response value for each case. This functionality can be very handy for the user.