IEOR 130 Methods of Manufacturing Improvement Fall, 2016, Prof. Leachman Homework Assignment #5, Due Thursday, Oct. 6, 2016

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1 IEOR 130 Methods of Manufacturing Improvement Fall, 2016, Prof. Leachman Homework Assignment #5, Due Thursday, Oct. 6, The management of a wafer fabrication facility is trying to sort out how much yield loss is coming from a stationary baseline distribution of defects vs. how much is coming from defect excursions and other systematic mechanisms of yield loss. A stacked wafer map of 450 wafers is analyzed including only wafers believed to not be involved in any defect excursions. Among 500 die sites on the wafer map, the best-yielding die site is one with a 65% yield, and this maximum yield is observed at only one site. The die size for the wafers in the stack is 0.5 sq cm. The average die yield over all wafers (including those involved in excursions) of this product is 35%. (a) Calculate the baseline defect-limited yield and the underlying baseline defect density. (b) Management is considering an upgrade of the air flow system costing $1.5 million. Engineering tests have been performed that indicate that this upgrade can be expected to cut baseline particle contamination on the wafers by 20%. However, particle excursions do not seem to be abated by the improved air flow. Estimate the improvement in baseline defect-limited yield and in the overall die yield if this upgrade is undertaken. (c) Management also is considering investment in a $1.5 million inspection system enabling increased process monitoring so that excursions can be detected earlier and thereby reduce yield losses. Engineering analysis and experiments indicate that total systematic and excursion yield losses could be cut 20% by this investment. Assuming the air flow system is NOT upgraded, what overall die yield would result from implementation of this inspection system? If only $1.5 million is available to spend, which is a better expenditure for improving yield the air flow system upgrade, or the new inspection system? 2. A product with 500 dice per wafer has an average die yield of 67% and a die area of 0.5 cm 2. The best die yield observed near the center of a wafer map made from 400 stacked wafers is 84%, and this yield was achieved at only one die site. (This wafer map was made from wafers in lots not subject to excursions.) (a) Estimate the stationary random yield (when systematic losses are not present), and estimate the systematic mechanisms limited yield. (b) Determine the Poisson defect density equivalent to the stationary random yield. (c) Clustering of the baseline (stationary) random defects has been studied. It has been found that the mean number of defects per die is 0.8 and the variance in the number of defects per die is 1.2. Revise the estimate of the stationary random defect density accordingly. (d) The following systematic mechanisms have been observed: 1

2 Mechanism Frequency Die Loss when occurs Edge losses all wafers 4.5% of all die on wafers Missing islands in all wafers 2.5% of all die on wafers gate photo exposure Particle excursions at 5% of lots, all wafers in 30% of the die on such wafers metal etch Particle excursions at 5% of lots, all wafers in 20% of the die on such wafers poly etch Poisoned vias 4% of lots, all wafers in 50% of the die on such wafers Clamp damage at 8% of lots, first wafer in 15% of the die on such wafers CMP Note: These failure mechanisms are not mutually exclusive, i.e., multiple failure mechanisms may be present in the same die. Assume lots have 25 wafers. What amount of systematic yield loss remains to be explained by other, unobserved mechanisms? 3. When the "O" ring in a low-pressure CVD machine fails, the machine down time averages 4.5 hours. If the "O" ring is replaced as part of weekly preventive maintenance procedures, the incremental downtime is only 0.5 hours. Data on the lifetimes of "O" rings in this machine type are as follows. (The notation pt denotes the fraction of the "O" ring population that fails during the t th week of operation.) t pt What policy for "O" ring replacement will maximize machine availability? 4. It is desired to schedule preventive maintenance (PM) procedures for a particular type of wafer processing machine. The expected cost to repair the machine after it goes down while in service is $200. The expected cost to perform a PM on a machine in working order is $50. Data on machine failures is as follows: 2

3 Number of Probability Days of Service of Failure (a) Determine the frequency at which the PM should be scheduled to minimize the expected costs. (b) During each PM or repair of the machine, the machine is thoroughly cleaned in order to requalify it for production. Immediately after such a clean, the die yield of the entire manufacturing process is 50%. With each additional day of service since the last clean, the machine becomes increasingly dirty. The increase in the number of fatal particles deposited on wafers is estimated to be 0.2 particles per sq cm per 24 hours. The fab makes only 10 wafers per day; each wafer has 100 die on it, and the good die sell for $2 apiece. The die area is 0.5 sq cm. In light of this new data, determine which is the best PM frequency: same as your answer to (a), or perform a PM every day? (Hint: Estimate the revenue loss resulting from the lower die yield and include this cost in the cost rate formula.) 5. A certain type of processing machine is subjected to three preventive maintenance procedures. The first procedure is performed once per week (the weekly PM). The second procedure is performed once every four weeks (the monthly PM). The third procedure is performed once every 12 weeks (the quarterly PM). Management needs to decide which wearable components of the machine should be replaced in each procedure. The machine includes many different kinds of components, each subject to random failure. When any component fails, 6.5 hours is required to take the machine out of service, replace the failed component and re-qualify the machine for production. Each kind of component has a 5 percent chance of failing during the t th week of service after replacement, t = 1, 2,..., 20. (No component survives longer than 20 weeks.) For each kind of component, there are only four possible maintenance strategies: (i) Planned replacement in weekly PM (ii) Planned replacement in monthly PM (iii) Planned replacement in quarterly PM (iv) No planned replacement, replace only when fails. 3

4 It is not allowed to schedule planned replacements at frequencies other than the above choices, e.g., one cannot schedule a planned replacement once every two weeks. Let c2 denote the incremental time required to replace a given component as part of any PM procedure. (The actual values of c2 vary over the many kinds of components, from seconds to hours.) Determine ranges on c2 for which each of the four possible maintenance strategies above are optimal. 6. A certain type of processing machine is subjected to three preventive maintenance procedures. The first procedure is performed once per week (the weekly PM). The second procedure is performed once every four weeks (the monthly PM). The third procedure is performed once every 12 weeks (the quarterly PM). Management needs to decide which wearable components of the machine should be replaced in each procedure. The machine includes many different kinds of components, each subject to random failure. When any component fails, 6.5 hours is required to take the machine out of service, replace the failed component and re-qualify the machine for production. The incremental time to replace each kind of component as part of any PM procedure is 0.5 hours. The maximum lifetime is different for each kind of component. Assume each component has an equal chance of failing in any week up to its maximum lifetime. For each kind of component, there are only four possible maintenance strategies: (i) Planned replacement in weekly PM (ii) Planned replacement in monthly PM (iii) Planned replacement in quarterly PM (iv) No planned replacement, replace only when fails. It is not allowed to schedule planned replacements at frequencies other than the above choices, e.g., one cannot schedule a planned replacement once every two weeks. Let τ denote the maximum lifetime of a given component. Determine ranges on τ for which each of the four possible maintenance strategies above are optimal. (Hint: To obtain closed-form expressions, you can make use of the following formula: t k = 1 t k = ( t + 1) To solve completely, you will need to apply the quadratic formula. You can receive almost full credit if for each maintenance strategy you express a functional inequality that τ must satisfy in order for that strategy to be optimal.) A diffusion furnace performs polysilicon depositions on four lots of wafers in one machine cycle. A machine cycle lasts 10 hours. At the start of the machine cycle, the load lock of the furnace is pumped down to vacuum. The load lock to the furnace incorporates an O ring that is 4

5 subject to failure. When the O ring fails, all 4 lots become contaminated and must be thrown out. It is not possible to determine if the O ring has failed until after the machine cycle is completed, at which point it is obvious if the O ring failed or not. When the O ring fails or when it is replaced before failure, it takes 10 hours to replace it and requalify the furnace for more production. Data on O ring lifetimes is as follows: # of furnace cycles, n fraction that fail in cycle n (a) Suppose the furnace is the bottleneck and it never has any idle time. In order to maximize the output rate, after how many furnace cycles should a planned replacement of the O ring occur? (b) Now suppose the furnace is not the bottleneck and it has considerable idle time. Will the best number of cycles until a planned O ring replacement become larger, or will it become smaller? Explain briefly. Do not solve numerically. 8. There are 200 electrical resistors wired into the circuitry of a burn-in board used in device testing. Because of small impacts experienced during handling of the boards, the resistors work loose and ultimately blow out. Replacing a blown resistor on a board costs $100; replacing the entire burn-in board costs $7,500. It is estimated that a random resistor has the following failure distribution: Month Prob. of Failure in that Month (a) If there is no replacement of burn-in boards, just replacement of blown out resistors, what is the expected cost per month? (b) Should there be planned replacements of burn-in boards? How often? Identify the cost savings compared to the policy in part (a). 5

6 9. A data-entry software application has been installed on 1,000 personal computers in a large semiconductor company. The application is run continuously on all 1,000 computers. Unfortunately, the application incorporates a random bug. This bug can lock up a computer and force it to be re-booted. Considering the time to re-enter lost data as well as the time to re-boot, log in again and re-open the application, there is an average of 40 minutes of down time on a computer every time the computer locks up. It is possible to re-install the application on all 1,000 computers at once by broadcasting it over the company s local area network, but all users would have to be logged off while this is done. Considering time to close the application, log off, log back on and re-open the application, the down time to re-install the application is 5 minutes on all computers. Data concerning the time until lock-up has been monitored at a number of computers, with the following results: Days of operation since last re-boot or re-install Probability of lock-up In order to minimize total computer down time per day, should the company periodically reinstall the application? How often? 6