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`IEEE TRANSACTIONS ON SEMICONDUCTOR MANUFACTURING, VOL. 15, NO. 4, NOVEMBER 2002
`
`Comparing the Economic Impact of Alternative
`Metrology Methods in Semiconductor Manufacturing
`
`Payman Jula, Costas J. Spanos, Fellow, IEEE, and Robert C. Leachman
`
`Abstract—Metrology is an essential part of advanced semi-
`conductor manufacturing. It accelerates yield improvement and
`sustains yield performance at every stage in both new and mature
`processes. Advances in metrology are needed to achieve chal-
`lenging industry goals, such as smaller feature sizes and reduced
`time for introduction of new materials and processes for future
`technology. To achieve difficult industry goals,
`it is expected
`that metrology practices will migrate from offline to inline, and
`ultimately, to in situ. Economic models are needed to study the
`costs and benefits of introducing new metrology technologies and
`to compare alternative metrology practices. Several qualitative
`and quantitative models are presented in this paper to study the
`elements of revenue and cost associated with different metrology
`tools and practices. Comparisons between in situ,
`inline and
`offline metrology systems are made. The cost components of the
`metrology methods are analyzed and discussed with respect to
`steady state process control as well as their effect on time to
`yield. Monte Carlo simulation models are used to study each
`system under different scenarios.
`
`Index Terms—Continuous-time Markov chain, economics,
`metrology, semiconductor manufacturing.
`
`I. INTRODUCTION
`
`H ISTORICALLY, semiconductor manufacturers rely on
`
`statistical process control (SPC) techniques for main-
`taining the processes within prescribed specification limits.
`While semiconductor manufacturing has continued to pursue
`ever-tightening specifications due to the well-known problems
`associated with the decreasing feature size, it has also become
`clear that
`there is a need for advanced-integrated process
`control. This approach requires a major shift in operational
`methods and requires the existence of complex, flexible archi-
`tectures to meet the above requirements. New metrology tools
`are introduced as an essential part of these architectures.
`Metrology accelerates yield improvement at every stage
`in both new and mature processes. Appropriate metrology
`practices can reduce the cost and cycle-time of manufacturing
`through better characterization of tools and processes. Ad-
`vances in metrology are needed to achieve difficult industry
`goals, such as smaller feature sizes and reduced time for intro-
`duction of new materials and processes for future technology.
`
`Manuscript received May 9, 2002; revised July 15, 2002.
`P.
`Jula and R. C. Leachman are with the Industrial Engineering
`and Operations Research Department, University
`of California
`at
`Berkeley, Berkeley, CA 94720 USA (e-mail: payman@ieor.berkeley.edu;
`leachman@ieor.berkeley.edu).
`C. J. Spanos is with the Electrical Engineering and Computer Sciences
`Department, University of California at Berkeley, Berkeley, CA 94720 USA
`(e-mail: spanos@eecs.berkeley.edu).
`Digital Object Identifier 10.1109/TSM.2002.804909
`
`To achieve these goals, it is expected that metrology practices
`will migrate from offline to inline, and ultimately be integrated
`in the tools (“in situ”) [1].
`Researchers have concentrated on the economic impact of
`particular aspects of metrology tools such as the sampling policy
`[2], [3] and the precision [4]. Dance et al. [5] tried to capture
`the economic behavior of metrology tools through a modified
`cost of ownership (COO) model. Still there is a need for more
`comprehensive models to identify elements of cost in complex
`metrology systems.
`Unless convinced otherwise, manufacturers are usually reluc-
`tant to adopt major equipment and technology changes because
`of the short-term uncertainties that arise during the introduc-
`tion of new technologies. Appropriate metrology models assist
`the semiconductor manufacturers to assess the costs that drive
`their businesses and help them in formulating the right opera-
`tional strategies. The ability to effectively identify cost drivers
`and manage cost reductions is a competitive advantage for any
`manufacturer. Therefore, accurate models are needed to study
`the costs and benefits of introducing new technologies and eval-
`uate different practices. Toward this goal, this paper introduces
`new analytical models to compare different metrology methods
`in a litho track system.
`Although this study tries to address the economics of
`metrology systems in a general form, the examples and illustra-
`tions are developed for litho track systems. Lithography steps
`are among the most crucial, and lithography tools are among
`the most expensive in semiconductor manufacturing. Most of
`the models offered in this document can easily be modified and
`extended to other equipment sets and metrology tools.
`Fig. 1 shows different metrology methods in a litho track
`system in terms of the position of the metrology tool in the
`system. Wafers first enter the track system, where they go
`through steps such as coating and baking in preparation for the
`main lithography process (stepper), in which small features
`are printed on the wafer. After lithography, wafers go through
`additional steps in the track system, such as post exposure bake
`(PEB) and development (DE).
`The qualities of the features defined during lithography
`(which in turn depends on the quality of the lithography
`process) have a direct effect on the quality of the final product.
`Therefore, we are interested in measuring and controlling the
`quality of the lithography step. The quality of the process
`(here the lithography step) is represented by measuring certain
`quantities on the wafer, such as the critical dimension (CD) of
`fine printed patterns.
`Offline systems, as depicted in Fig. 1(a), have traditionally
`been practiced by semiconductor manufacturers. In this method,
`
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`flows in our models, or attempt to evaluate the investments in
`terms of interest rates or discounted returns or tax benefits.
`
`II. ANALYTICAL MODELS OF METROLOGY METHODS
`
`In general, since metrology operations are in series with the
`processes, they reduce the throughput and increase the work
`in process (WIP) and the cycle time. WIP inventory between
`a process step and the subsequent inspection is at risk if the
`process drifts to an undesirable state. Manufacturers have been
`trying to reduce these risks using different methods such as
`changing the sampling policies and send-ahead samples.
`Simply reducing the number of samples may result in a better
`cycle time and WIP, but it negatively affects the throughput of
`good products. Product yields at subsequent steps depend on the
`quality of information extracted from the metrology data. The
`quality of information generated from the metrology measure-
`ments can be partly characterized by its accuracy, precision and
`sampling policy.
`It is desirable to identify bad products passing through the
`metrology tool and detect the out of control state of the process
`as soon as possible. This can be achieved by tightening the ac-
`ceptance criteria. If, however, these criteria are too tight, then
`good products may be rejected, or the system may be shut down
`unnecessarily, resulting in production loss.
`Another cause for production loss is the WIP between the
`process tool and the metrology tool. If the process drifts to
`an undesirable state, the process keeps manufacturing bad
`products until they are detected by the metrology tool. All the
`product in WIP processed since the process went out-of-control
`needs to be reworked or discarded. A send-ahead (also known
`as look-ahead) sample method eliminates the WIP risk but re-
`duces the process throughput and utilization. In the send-ahead
`sampling method, one or more wafers are processed and then
`submitted for measurement. The remaining wafers in the batch
`are processed after the measurements are complete, the results
`are released and the equipment is adjusted.
`Therefore, it is also desirable to minimize the WIP in the
`system. Assuming the same throughput for metrology tools, mi-
`grating from offline to inline and in situ usually reduces the WIP.
`In other words, integrated inline and in situ metrology operation
`minimizes the WIP lost with little impact on utilization. How-
`ever, the feasibility of these approaches and the quality of data
`collected by inline and in situ tools, along with the price tag
`of these types of equipment, should be considered in making a
`decision.
`
`A. Overall Equipment Efficiency (OEE)
`Overall equipment efficiency (OEE) is one of the most im-
`portant metrics for measuring equipment performance. OEE is
`defined as the ratio of the theoretical time needed to produce
`salable wafers in a given period, divided by the total time in
`that period [7]. Theoretical time refers to the time required by a
`machine in perfect working order performing the process spec-
`ification under ideal conditions.
`Since, in this study, we are mainly interested in understanding
`the differences among metrology practices, we classify the
`losses in equipment processing time into two main categories.
`
`Fig. 1. Different metrology methods applied to a Litho track system: (a)
`offline, (b) inline, and (c) in situ. “M ” indicates the position of the metrology
`tool.
`
`the metrology tool is located after the track system. Wafers are
`transported to the metrology tool by lots. Lots are then measured
`by the metrology tool with an appropriate sampling policy. Of-
`fline metrology tools are usually accurate and fast, but are also
`expensive and occupy significant clean room space.
`Newer inline systems occupy little footprint in the fab. Their
`accuracy and speed, however, is generally inferior to offline,
`though rapidly improving. In situ metrology systems are fully
`integrated and the measurements are done while the wafers are
`being processed or shortly after the process is completed. In situ
`lithography systems are under development and expected to be
`introduced with future generations of lithography tools.
`To study the elements of cost in the above system, several
`qualitative and quantitative models are introduced in this paper.
`In the next section, the major components of the costs and ben-
`efits for metrology practices are analyzed and two revenue and
`cost models are introduced. The effects of metrology methods
`on revenue during the steady state and the time to maturity
`are explained. Monte Carlo simulation studies are conducted
`to compare different scenarios in Section III. First, the results
`of analytical model are compared to those of simulation model
`for a simple system. Then, the effects of yield and price struc-
`ture, control policies, and the precision of metrology tools are
`examined in a series of scenarios. The results are presented and
`analyzed for each scenario. Recommendations are provided for
`each scenario and results are discussed. Conclusions and future
`avenues of study are explored at the end.
`Financing considerations should be addressed along with our
`models. In this paper, we do not account for the timing of cash
`
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`The first set of losses is associated with the metrology tool,
`its specifications, and the control policy chosen to detect and
`improve the bad process. The term “Bad process,” in this doc-
`ument, refers to the process that is out of control and produces
`out-of-spec products; the products that are not conforming to
`the required specifications set by the fab management. These
`specifications are those that are measured by the metrology
`tool. The crosshatched area between OEE and OEE in Fig. 2
`shows the first set of losses. These losses are the focus of this
`study and will be further explored.
`The second set of losses contains any loss that is not captured
`in the first set. These losses are those that occur regardless of
`the type of metrology tool and the control policy. Any loss of
`production due to unavailability of machine, bad utilization of
`equipment and slow process belongs to this category. The area
`between the OEE and 100% available time in Fig. 2 shows this
`set of losses.
`
`B. A Mathematical Model of Metrology Tools
`
`Assume the main process is up and in the “In Control” state
`for an exponential amount of time with the mean of mean time
`between failures (MTBF) of the process. The process goes to the
`“out-of-control” state and stays in this state until detected by the
`metrology tool. The quality of information extracted from the
`metrology measurements can be partly characterized by param-
`eters
`and . The type I error,
`, is the probability of rejecting a
`good product or process. The type II error,
`, on the other hand,
`shows the probability of accepting a bad product or process. The
`power of metrology,
`, is the probability of correctly re-
`jecting a process or product. Accuracy, precision, and sampling
`policy in metrology are among the factors that affect the quality
`of information extracted from the metrology tool.
`The time that is spent in the out-of-control state by the equip-
`ment is proportional to two factors; first, the time required for
`the results of the metrology tool to become ready, and second,
`the power of the metrology measurement. It is assumed that the
`equipment stays in the out-of-control state for an exponential
`amount of time with the mean of ACTM/
`, where
`is the power of the metrology tool and ACTM is the average
`cycle time to metrology. ACTM is the response time from the
`metrology tool, which depends on the amount of WIP between
`the process and the metrology tool. After the metrology tool
`gives the signal that the process is out of control, the process
`is shutdown and the repair starts.
`It is assumed that the tool stays in this state, which is called
`the “Failure Signal/Repair” state, for an exponential amount of
`time with the mean of the mean time to repair (MTTR). Be-
`cause of the metrology type I error
`, there is a probability
`that the metrology tool generates a failure signal even though
`the process is in the good (in control) state. During any time in-
`terval
`, in which the process is actually in the good state, the
`rate at which the equipment will be declared to be in the “Failure
`Signal/Repair” state is
`.
`The above system is a description of a continuous-time
`Markov chain consisting of three states: namely, “In Control,”
`“Out of Control” and “Failure Signal/Repair.” Fig. 3 shows this
`system.
`
`Fig. 2. The concept of OEE.
`
`Fig. 3. Continuous-time Markov chain model of a metrology system.
`
`Solving the limiting probability equations of this system [6]
`result in:
`
`(1)
`
`(2)
`
`are the long-term probabilities of the process
`and
`where
`being “in control” and “out of control,” respectively.
`The process under control produces acceptable products,
`while the out-of-control process produces bad products that
`must be reworked. The faster the out-of-control state is
`detected, the faster the process is calibrated; which limits
`the amount of required rework. Therefore, the cost of a bad
`metrology practice is twofold. First, there is the cost due to the
`lost time of equipment (metrology and litho track), including
`the expenses of investment in purchasing and installing the
`machines, maintenance, footprint, etc. The second cost element
`occurs because of WIP rework, resulting in material, energy
`and labor costs. These costs are further studied in this section.
`
`C. Revenue Models
`Let
`denote the number of machines of type
`stalled in the factory. Ignoring the requirement that
`an integer, Leachman et al. [7] have shown
`
`that are in-
`must be
`
`where 720 is the number of hours in a month. The left-hand
`side of this equation expresses the total machine-hours required
`
`(3)
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`is the designed
`wafers per month;
`to process
`output capacity and
`is the mature die yield.
`is the
`total theoretical process time per wafer (expressed in hours) on
`equipment type , considering all process steps performed by
`that equipment. The right-hand side is the total machine hours
`that can be devoted to processing (at theoretical rates) consid-
`ering the achieved equipment efficiency. Assuming a revenue of
`for each wafer for the current day, the total revenue per day
`in the near future can be calculated as
`
`(4)
`
`is the
`), where the
`with (
`Replacing the
`long run probability of the process being in the good (in-control)
`state, result in
`
`Fig. 4. Different phases of a process life cycle.
`
`Revenue/Day
`
`(5)
`
`As expected, the revenue increases with the decline of
`,
`ACTM and MTTR and decreases with the decline of MTBF.
`Over the long run, where the price is declining according to
`a continuous discount factor of
`, the total revenue realized up
`to time
`(expressed in days), assuming zero start-up and pro-
`duction delays, is expressed as
`
`,
`
`(6)
`
`D. The Effect of Metrology Tools on Ramp-Up
`Up to this point, the behavior of metrology tools was con-
`sidered for mature and stable process technology. However, as
`depicted in Fig. 4, each process goes through three different
`phases: development phase where the process is first introduced,
`the ramp phase where the volume of production is increased,
`and the mature phase where the process sustains high volume
`production.
`During the development phase, the equipment is installed and
`an appropriate recipe is applied. In this phase, the process usu-
`ally does not produce any marketable product. Therefore, this
`phase is not in our interest. The process starts producing sal-
`able products in the ramp phase. In the beginning of this phase,
`equipment fails more often. After some time, the process is cal-
`ibrated, the rate of failures declines, and the process becomes
`mature.
`Here, we are interested in studying the effect of the metrology
`tools on the ramp phase. For simplicity, we approximate the
`above curve with a step function, where the process has the
`average (MTBF
`) in the development and ramp phases and
`jumps to the mature phase (MTBF
`) at time
`(Fig. 5).
`There are many factors affecting the duration of the ramp
`phase
`. Studying the behavior of these factors is beyond the
`scope of this paper. However, it is known that the ramp-up du-
`ration, especially at lithography, depends on the knowledge and
`the experience of the engineers working with the process. Part
`
`Fig. 5. A simplified process life cycle.
`
`of the experience and knowledge comes from trial and error.
`Each equipment failure contributes to the knowledge about that
`equipment/recipe. Here, we assume the time to maturity is a
`function of the number of detected problems through time. The
`more problems are found, the more experienced the staff will
`become. Finally, after
`number of trial and errors, the equip-
`ment goes to the mature state and the failure rate decreases. We
`are interested in finding the effect of metrology tools and the
`control policies on the value of
`. Changes of
`can then be
`translated to cost.
`The number of required equipment is usually planned for the
`mature case; therefore, there is some lost revenue due to the
`unsatisfied demand in the development and ramp phases. Sim-
`ilar to (3), the satisfied demand in development and ramp phase
`, assuming the mature die yield, follows
`
`(7)
`
`Here, the
`is the long-term probability of the process
`being under control during the development and ramp phases
`and follows an equation similar to (1). All of the notation in
`this section concerns the equipment performance in the devel-
`opment and ramp-up phases and is similar to the notation for
`the mature phase. Using (3) and (7), the unsatisfied demand per
`month during the development and ramp phases can be calcu-
`lated as
`
`(8)
`
`The duration and the quantity of the lost demand during the
`ramp period will result in lost revenue during this period.
`
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`
`Considering the continuous-time Markov chain model for the
`development and ramp phases, therefore, the expected value of
`, the elapsed time for
`number of repairs, can be calculated
`as
`
`(9)
`
`The total possible revenue during the development and ramp
`phases, assuming all demands are satisfied, can be expressed as
`
`the amortized annual cost due to labor, material, and infrastruc-
`ture per wafer started.
`is the total amortized annual cost
`per wafer started. The last term captures the annual fixed cost
`and
`are the fixed labor cost and the
`of manufacturing.
`fixed space cost, respectively, that are independent of wafer start
`volume and the number of installed equipment.
`Using (1), (3) and (14), the total expenses of the machines per
`year can then be expressed as
`
`EPY(Machines)
`
`(10)
`
`Here,
`is the continuous discount factor for the exponentially
`declining sales price. The lost revenue can be calculated as
`
`The total lost revenue can be calculated as
`
`(11)
`
`(12)
`
`E. Comprehensive Revenue Model
`The comprehensive revenue model consists of the combined
`revenue obtained in the ramp phase and the mature phase. The
`total revenue obtained in the ramp phase can be expressed as
`
`(13)
`
`Given the duration of the mature phase, the total revenue
`obtained in the mature phase can be calculated by (6). The
`summation of (6) and (13) should be considered in selecting
`the metrology setup.
`The revenue models are more tailored toward the marketing
`department’s needs versus the manufacturing expenses. In other
`words, they only consider the incoming cash flow to the com-
`pany through sales. These models do not consider the outgoing
`cash flow and the expenses of the company. What if a metrology
`tool improves revenue, but the price of investment is high? How
`about the maintenance expenses and labor costs associated with
`each metrology system? These issues will be addressed by an-
`other model, called the cost model, in the following section.
`
`F. The Cost Model of Metrology Methods
`Leachman et al. [7] expressed the annual expense of a fab as
`
`(14)
`
`The first term captures the machine expenses.
`, and
`,
`are the amortized annual costs due to purchasing, labor, and
`foot-prints, respectively, per machine of equipment type .
`captures the total amortized annual cost per machine of equip-
`ment type . The second term captures the expenses related to
`the number of wafers started.
`,
`, and
`are respectively
`
`(15)
`
`The “litho” subscript represents the lithography system,
`which includes the exposure unit and the track line. The first
`term in (15) captures the effect of metrology in lithography
`costs through its effective processing time. The second term
`is the cost associated with the purchase, maintenance and the
`footprint of metrology devices. The third term captures all
`other equipment expenses in the fab.
`As discussed earlier, different metrology methods generate
`different amounts of WIP and rework. The rework consumes
`materials, energy and labor. Furthermore, the mask life, which
`is considered dependent on the number of exposures, causes the
`expenses to increase in proportion to the amount of rework. Ac-
`cording to our continuous-time Markov chain model, the total
`out-of-control machine-hours spent processing
`, the number
`of wafers in lithography to be reworked, will be:
`
`Considering (1)–(3), and (16), the total number of reworked
`wafers in lithography per month can be calculated based on the
`total monthly production rate as
`
`(16)
`
`(17)
`
`The fab total expense per year due to the number of wafers
`started includes two terms. The first term captures the expenses
`due to the reworked wafers in lithography steps. These expenses
`reflect material costs, energy, labor and masks. The second term
`includes all expenses that are functions of the number of wafers
`started. All the rework done on the other equipment sets (except
`lithography) are assumed to belong to this category. Therefore,
`the total expenses per year due to the number of wafer starts is
`
`EPY(Wafer started
`
`, are assumed to re-
`and
`The constant terms of (14),
`main unchanged after introducing different metrology methods.
`
`(18)
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`The difference between metrology methods can be calculated
`according to (15) and (18). This difference can be presented as
`
`EPY
`
`(19)
`
`To choose the best metrology method, manufacturers should
`consider the elements involved in (19). All the costs associated
`with acquiring, installing and maintaining the litho track tools
`should be considered. Special attention should be given to the
`quality of information extracted from the metrology tools. The
`failure rate, ease of repair and the position of metrology tool in
`the system should also be considered.
`
`III. MONTE CARLO SIMULATION MODELS OF METROLOGY
`METHODS
`
`In previous sections, several analytical models were pre-
`sented for litho track systems based on some simplifying
`assumptions. There is still a need to address the issues involved
`in more complex systems arising in industrial environments.
`Appropriate models can predict the behavior of these systems
`under different scenarios and help the decision makers in
`selecting the best practices in different environments. However,
`it is very difficult to capture the behavior of complex systems
`with closed-form mathematical models, similar to those pre-
`sented in the previous section. As an alternative, we use Monte
`Carlo (MC) simulation models to study the behavior of more
`complex systems.
`In these models, the results of five 24-hour days with five
`different initial random seeds are collected for each simulation
`run. The lithography throughput is considered to be 60 wafers
`per hour. To accommodate the behavior of a robot in an
`industrial system, a buffer (with the capacity of one wafer) is
`considered before and after each station. The revenue generated
`for each model is then plotted in sets of graphs. Each point
`in these graphs is based on the information that is statistically
`collected from 60
`24
`5
`5
`36 000 simulated wafers;
`each wafer includes 100 dice with individual characteristics.
`The data are collected after a warm-up period of 50 minutes.
`SIGMA [8] simulation software was modified and used as a
`platform for generating the data and collecting the information
`for these experiments.
`The values of the parameters used in these models are either
`the estimated values in the industry or what experts would
`expect to see in emerging technologies. The experiments are
`designed to assist the manufacturers with developing similar
`models. Decision-makers could develop similar experiments
`that address their specific needs and accommodate their partic-
`ular parameter values.
`For the center working point, MTBF is 240 minutes and
`MTTR is set at 20 minutes. For this working point, five samples
`are selected from each simulated wafer. The CDs of these
`
`samples are then measured and the 3 rule is used for the cutoff
`line. It is assumed that the results of the offline, inline, and in
`situ metrology are available after approximately 30, 15, and
`2 minutes, respectively. The performances of these systems
`are analyzed with respect to variation in MTBF, MTTR,
`,
`around the center working point for each of the inline, in situ
`and offline cases. Later in this document, the effect of control
`policies, yield/revenue structures, the precision of metrology
`tools and many other parameters are investigated.
`The reference of $1000 per chip for 250-nm technology
`along with the yield/revenue structure of products determines
`the revenue per chip in these cases. Total revenues on the
`order of millions of dollars are generated per day in these
`experiments. Different parameter values would certainly result
`in different values for revenue. However, readers should keep
`in mind that the absolute value of revenue is not our interest.
`We are interested in analyzing the changes in revenue based
`on the changes in the system. The relative differences will
`provide us with a better understanding of each system and help
`us predict the behavior of similar systems in similar working
`conditions. Therefore, revenues are presented in arbitrary units
`in this document.
`First, a simple model is developed to compare the results
`of analytical models with those of MC-simulation. The as-
`sumptions in this model are consistent with the assumptions
`under which the analytical models were developed. The second
`scenario enhances the first scenario by introducing a variance
`to the process and by considering more realistic structures
`for the yield and price. In the final scenario, more realistic
`conditions are introduced to the system. Different random er-
`rors are considered for each of the inline, offline and in situ
`tools to capture the different precision associated with each
`technology. Furthermore, wider and more continuous drifts
`are considered for the process.
`
`A. Analytical Approach Versus Monte Carlo Simulation
`A MC model is designed to verify the accuracy of the results
`generated from the analytical models presented in the previous
`section. The assumptions in this model are consistent with the
`assumptions of exponential failure times and repair times under
`which the analytical models were developed. The lithography
`targets a CD of 205 nm at in-control state. It produces bad prod-
`ucts with the CD of 225 at out-of-control state. For simplicity,
`the variance of the process is ignored at this stage; in the next
`section, the variance will be introduced to the system and its
`effect will be explored.
`Our study [9] shows the consistency between the analytical
`model and MC-simulation. For example, Fig. 6 shows the ef-
`fect on revenue from reducing the time between the process and
`the metrology tool. As shown, both the analytical model and
`MC-simulation predict a similar pattern. The figure shows an
`increase in revenue by migrating from offline to inline and in
`situ technology assuming that the same quality of information
`can be obtained from different metrology tools.
`
`B. The Effects of Process Variation on Revenue
`On many products in the semiconductor industry, it is well
`known that reduction of the critical dimension results in higher
`
`Applied Materials, Inc. Ex. 1014
`Applied v. Ocean, IPR Patent No. 6,836,691
`Page 6 of 10
`
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`460
`
`IEEE TRANSACTIONS ON SEMICONDUCTOR MANUFACTURING, VOL. 15, NO. 4, NOVEMBER 2002
`
`Fig. 6. Revenue per day (arbitrary unit) versus average cycle time to metrology
`tool (ACTM) for the analytical and simulation models.
`
`Fig. 8. Revenue (arbitrary unit) versus mean of the process for different
`standard deviation.
`
`205 nm. For the rest of this document, we assume a standard
`deviation of 10 nm associated with the process, and we try to
`keep the working point at 205 nm in order to gain the max-
`imum revenue. Assuming revenue of $1000 per chip for a CD
`of 250 nm will result in a revenue of $1315 per chip for a CD of
`205 (assuming the $7/nm decline rate). Another negative effect
`of variance on revenue is due to the risk involved in the quality
`of information extracted from the product measurements.
`Consider a process with a standard deviation of 10 nm
`that is targeted to work at 205 nm but it may go to the bad
`state of 225 nm after a random time with the distribution
`. (In this document
`notates a
`normal distribution with mean
`and standard deviation .) The
`process stays in the bad state until detected by the metrology
`tool. The shutdown/repair signal is generated when the average
`of the CDs measured from the sample points exceeds the cutoff
`line threshold. The process is then shut down and all the bad
`products in WIP are sent to rework. The process will be back
`in the good state after a random repair time with distribution
`. The $7/nm decline rate is observed
`in this case and there is no revenue for the products with
`CDs more than 220 nm, reflecting tight specifications set by
`management. Fig. 9 shows the in-control and out-of-control
`cases.
`Changing the number of sample points taken from each wafer
`and adjusting the cutoff line of the control policy affects the
`type I and type II errors. Suppose
`represents the point on
`the Standard Normal distribution
`with the probability
`of upper tail equal to . Then the following equations hold
`
`(20)
`
`is
`is the number of sample points in each