Statistical Optimization of Wafer Yields - Lab Report Example

Statistical Optimization of Wafer Yields Name of Student Institution affiliation Date Abstract The incorporated circuits are significantly created on silicon chips which are slashed from silicon wafers. The manufacture of wafers experiences a number strides that decide the chips that will be fruitful. Through the Czochralski procedure where gems are developed, silicon wafers are delivered and inserted circuits are exchanged to the wafers by lithographic process. The accomplishment of exchange of the circuits relies on upon the control parameters set up at the manufacture plant. The quantity of fruitful chips are measured as yields and can be ascertained by use of likelihood techniques. The lab is a model of silicon wafers creation where the information acquired were utilized to ascertain the yields of the chips manufactured. Introduction The likelihood of acquiring the best Yield chips from the wafers is a component of the control techniques and measures set up. The creation procedure is thusly deliberately controlled to acquired even wafers of crystalline silicon. Crystalline silicon is most favored for two reasons: to start with, is that it is bottomless in the world's climate, being luxuriously found and as a component, coming next after oxygen. The second reason is that it is naturally agreeable. The silicon wafers contain a few number of chips that can be utilized for the manufacture of the Incorporated Circuits. The chips that are observed to be valuable in the creation of the ICs are the satisfactory yield. The quantity of effective yields rely on upon three parameters, to be specific: speed, weight and separation. These three parameters decide the thickness of the wafers and consequently the accomplishment of the yields. The control of the thickness in the plant requires that it be uniform the whole way across the wafer which is a troublesome undertaking for control specialists to accomplish. In this manner the control framework must be upgraded by enrolled factors. Taking the framework as (Y), the state factors then that decide the condition of the framework as: Speed (X1); Pressure (X2); distance (X3); By upgrading the control factors a more uniform thickness can be accomplished and henceforth create most extreme yields. In the test done underneath, measurable strategies are utilized to get the best and ideal conditions for the creation of most extreme yields. These is from trial information got. (Sullivan, 2005) Experimentation and Data The information was gathered and arranged in sheet 1 as appeared in the addendum with the thickness for the specimens researched taken at various thicknesses. For the primary test, both weight and separation were held consistent and the qualities were taken at rapid for the wafer testing. An aggregate of forty specimens were acquired and recorded in the sheet. The standard deviation of the thickness were made for the forty wafers. (W, 2013) Aims •To compose a minimized measurably nitty-gritty to give an account of thickness of the wafer affected by speed, distance and pressure •To decide the best condition for creating a uniform wafer. Results and Discussion Dataset Pressure and distance are unchanging at a low value in these 40 tests Sample number Low Speed Sample number High Speed 1 74 21 151.1 2 78.9 22 136.8 3 90.3 23 161.8 4 82.1 24 147 5 89.4 25 154.7 6 90.9 26 143.1 7 90.2 27 150.2 8 69.2 28 150.3 9 76.2 29 135.9 10 82.2 30 141.1 11 81.9 31 145.5 12 87.3 32 149.2 13 104.9 33 152.3 14 74.4 34 168.7 15 87.7 35 152.9 16 87.8 36 146.1 17 88.9 37 161.4 18 102.7 38 178.9 19 95.9 39 160.5 20 95.5 40 142.2 Question 1. Sources of variability in the data In the above data set, we have the independent data (speed) and the dependent data (thickness). This accounts for the variation of the speed. For the twenty samples that were taken in the laboratory the following chart can be used in comparison to the trends, where samples 21 – 40 represent high speed values while those of 1 to 20 represent low speed data. From the graph above it is deduced that the thickness are record-breaking high for qualities taken at fast when contrasted with those taken at low speed. An example is drawn for the sequencing of information at scopes of tests 8 to 13 for low speed and tests 29 to 34 for rapid. In both, a relentless ascent is noted with very nearly a comparative inclination then dives into a sorrow for both after the qualities were caught. It is additionally noticed that the pinnacle qualities were acquired at test 13 for low speed and test 34 for fast at qualities 104.9 and 168.7 separately. The above variety could be ascribed to various sources, which incorporates: Error due to instruments Calibration errors Reading errors Mode of measurement recording. i. Errors Due to instruments From the Czochralski wafer manufacture, the wafers are produced using crystalline silicon under controlled environment. Much of the time it is done in a vacuum to keep debasements from doping the silicon seed being developed. Nonetheless, uncleansed instruments can present remote material that would some way or another be supplanted into the silicon precious stones accordingly framing ununiformed surface of the wafer material in this way diminishing the yields. ii. Calibration errors Control instruments are inclined to shift their preset dimensions due to strain or by wear and tear. This acquaints balance blunder that need with be straightened out to such an extent that the gear record exact and exact esteem. The adjustment blunder additionally relies on upon the level of affectability of the hardware and control machines being used. iii. Reading errors These emerge mostly from parallax. It produces the greater part of transposition and interpretation blunders. Values read mistakenly are off base and see that they may reliable, they present variety and deviation of information from the standard. Therefore, the plotted qualities change from the normal. This sort of variety is not effortlessly revised but rather by astuteness and prescribed perusing practices, for example, opposite view. On account of computerized gear be that as it may, this sort of variety is missing particularly if the information is perused electronically and straightforwardly from the catch gadget. For this situation in any case with manual information passage, this blunder will undoubtedly be considered. iv. Mode of measurement recording a. Experience. This varies with the individual responsible for estimation. Hardness could bring about misread figures and values and consequently temperamental and conflicting information could be recorded. From the second specimen on rapid as appeared in the diagram above, it can be noticed that a sudden surge was noted in the recording. Such vacillations could be from flimsiness of gear or could likewise be as a consequence of hard recording and instrument set ups blunder will undoubtedly be considered. b. Profundity of the subject comprehension. This significantly influences the estimation of starting qualities to figure out if the conditions are legitimately set. Past test information demonstrate that this mistake however present is to the slightest. Consistency is a measure of value for this variety. (S., 1951) Question 2. Parametric and non-parametric test The null hypothesis is the claim that the consistency in covering is not the same for every speed. Taking the presumption that perception of the recorded information is autonomous and crude: which implies it has not been controlled, then the investigation can continue to test the appropriation bend. The appropriation of the qualities for the low speed and fast varieties can be portrayed from the radar graph plotted from the qualities which indicates the spreading over of the information. (Canover.) From the graph above, it can be noticed that for the low speed a noteworthy region traverses between the specimens 1 – 16 which favors of the way that the example surely takes after an ordinary appropriation bend. For the fast estimation the pinnacle esteem remains at test 18 which is near the found the middle value of check, and henceforth in like manner the example takes after suite. Question 3 Calculating the Significance value, β From the equation below: Method of least squares for equations with multiple independent variables are rewritten as following: Thus, in simpler term: Using the data provided from the experiment, the covariance matrix which is then used to solve for the values of β was calculated using the formula:   Where X1 = speed, X2 = pressure and X3 = distance Replacing the covariance values calculated above into the quadratic equation yields:   Solution for the simultaneous equations, yield: β = 0 Q4 Simpler derivation of Q3 A simpler method for deriving the values of 3 above is by drawing the regression line from the data provided. From the line the confidence line can be determined. However, the prerequisite is finding the covariance matrix. This is simpler since spreadsheet contains inbuilt statistical functions as described below: The elements of the covariance matrix are found from the formula: =COVARIANCE.S(Array1,Array2) Using the formular for elements 1 to 6 of the 3 x 3 matrix yields: The coefficients bo to b2 were obtained from the functions and were as shown: Variation chart shown in an xy scatter graph The chart indicates the span optimal value for the three parameters with limits on the sample between -1 and 1 References S., S. (1951). Mathematics, Measurement and psychophysics- Handbook of experimental psychology. New York: Jphn Wiley. Sullivan, L. (2005, December 07). Nonparametric Tests. Retrieved December 07, 2016, from sph.web.bumc.bu.edu. W, C. (2013). Practical Nonparametric Statistics (2nd ed.). New York: John Wiley and Sons. Read More

Results and Discussion Dataset Pressure and distance are unchanging at a low value in these 40 tests Sample number Low Speed Sample number High Speed 1 74 21 151.1 2 78.9 22 136.8 3 90.3 23 161.8 4 82.1 24 147 5 89.4 25 154.7 6 90.9 26 143.1 7 90.2 27 150.2 8 69.2 28 150.3 9 76.2 29 135.9 10 82.2 30 141.1 11 81.9 31 145.5 12 87.3 32 149.2 13 104.9 33 152.3 14 74.4 34 168.7 15 87.7 35 152.9 16 87.8 36 146.1 17 88.9 37 161.4 18 102.7 38 178.9 19 95.9 39 160.5 20 95.5 40 142.2 Question 1. Sources of variability in the data In the above data set, we have the independent data (speed) and the dependent data (thickness).

This accounts for the variation of the speed. For the twenty samples that were taken in the laboratory the following chart can be used in comparison to the trends, where samples 21 – 40 represent high speed values while those of 1 to 20 represent low speed data. From the graph above it is deduced that the thickness are record-breaking high for qualities taken at fast when contrasted with those taken at low speed. An example is drawn for the sequencing of information at scopes of tests 8 to 13 for low speed and tests 29 to 34 for rapid.

In both, a relentless ascent is noted with very nearly a comparative inclination then dives into a sorrow for both after the qualities were caught. It is additionally noticed that the pinnacle qualities were acquired at test 13 for low speed and test 34 for fast at qualities 104.9 and 168.7 separately. The above variety could be ascribed to various sources, which incorporates: Error due to instruments Calibration errors Reading errors Mode of measurement recording. i. Errors Due to instruments From the Czochralski wafer manufacture, the wafers are produced using crystalline silicon under controlled environment.

Much of the time it is done in a vacuum to keep debasements from doping the silicon seed being developed. Nonetheless, uncleansed instruments can present remote material that would some way or another be supplanted into the silicon precious stones accordingly framing ununiformed surface of the wafer material in this way diminishing the yields. ii. Calibration errors Control instruments are inclined to shift their preset dimensions due to strain or by wear and tear. This acquaints balance blunder that need with be straightened out to such an extent that the gear record exact and exact esteem.

The adjustment blunder additionally relies on upon the level of affectability of the hardware and control machines being used. iii. Reading errors These emerge mostly from parallax. It produces the greater part of transposition and interpretation blunders. Values read mistakenly are off base and see that they may reliable, they present variety and deviation of information from the standard. Therefore, the plotted qualities change from the normal. This sort of variety is not effortlessly revised but rather by astuteness and prescribed perusing practices, for example, opposite view.

On account of computerized gear be that as it may, this sort of variety is missing particularly if the information is perused electronically and straightforwardly from the catch gadget. For this situation in any case with manual information passage, this blunder will undoubtedly be considered. iv. Mode of measurement recording a. Experience. This varies with the individual responsible for estimation. Hardness could bring about misread figures and values and consequently temperamental and conflicting information could be recorded.

From the second specimen on rapid as appeared in the diagram above, it can be noticed that a sudden surge was noted in the recording. Such vacillations could be from flimsiness of gear or could likewise be as a consequence of hard recording and instrument set ups blunder will undoubtedly be considered. b. Profundity of the subject comprehension. This significantly influences the estimation of starting qualities to figure out if the conditions are legitimately set. Past test information demonstrate that this mistake however present is to the slightest.

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