Problem Set 2 Solutions

Transcription

1 Problem Set 2 Solutions Reminder From Syllabus: Problem Set Format Checklist: At the top of the page, include your name, problem set number, course name & number, and date. Submitted problem sets must be legible, neat and decipherable. Show all work. Complete work means step-by-step. Multiple page problem sets must be stabled with problems kept in sequential order. Multiple page problem sets must be stabled except for ABET questions. Answers must be circled/boxed. If the answer has units, include the units as they are required. All graphs must be thoroughly labeled, have axis titles and units, figure captions, and detailed explanations. Include comments on trends you observe and what you want your audience to take away from the graph. Using legends, arrows, and text will facilitate trends you want to point out. Using color does not help if you print your problem set in black and white. Extra credit many times is given for exceptional graphs with thoughtful and thorough explanations. All software program code must be thoroughly documented/commented and follow the above criteria. Include units. If unit conversions are being performed, the manner in which they are performed (e.g., dimensional analysis) need to be complete. "ABET Problems" - These problems are to be handed in separately and follow the criteria above. ABET problems with multiple pages should not be stapled. ABET problems will be assessed more thoroughly than other problems relative to completeness, correctness, legibility, neatness and decipherability, so extra care should be taken when answering these questions. Grading The TA/grader will provide a cursory review of your solution and provide a grade based on being able to follow and understand your solution. The grade will be based on the final solution, the completeness and validity of the path to the solution, and the Problem Set Format Checklist. If the grader cannot decipher your solution, you will be graded accordingly. Ultimately, you are responsible for understanding the solution to each problem. Citing - Remember to cite your references using references that have been peer-reviewed (yes, need to do for this assignment as well). 1. Diffusion: You are working in a company that grows boules of the semiconductor silicon (Si) for the semiconductor industry using a growth process known as Czochralski growth which uses a quartz crucible in which to melt Si. Oxygen from the quartz crucible diffuses into the Si boule. A customer asks you about oxygen interstitial (O i ) diffusion in the semiconductor silicon (Si) and what the diffusivity is of O i in Si during thermal annealing of dopants in Si at 750 o C. In order to determine O i diffusion in Si, you must first determine the activation energy (E A ) for O i diffusion in Si and the pre-exponential (D o ). You know Prof. Hartmut Bracht (University of Munster, Germany) wrote a very nice review article about point defect diffusion mechanisms in Si (H. Bracht, Diffusion Mechanisms and Intrinsic Point Defect Properties in Silicon, MRS Bulletin (June 2000) p. 22) which contains a comprehensive plot of impurity or foreign atom diffusion in Si. Using this plot: Solutions: a. Extract the O i diffusion in Si data from the plot (Note: see digitizing plots approaches on course website next to where you downloaded this Problem Set). The data was extracted using WebPlotDigitizer ( Plot the data on two different Arrhenius plots (be sure to provide a figure captions for both): 1

2 i. using a log 10 scale Figure: Extracted data of Log 10 of O i diffusivity in Si versus the inverse of temperature. The linear trend of the data is indicative of Arrhenius behavior. ii. using a natural log scale (log e scale) Figure: Extracted data of Log e of O i diffusivity in Si versus the inverse of temperature. The linear trend of the data is indicative of Arrhenius behavior. 2

3 b. Fit the data using a mathematical program and plot both the data and the fit on the same graph. Extract the activation energy in units of ev and pre-exponential of diffusivity in units of cm 2 /s from the data. Note that the activation energy should be in the range of 0.5eV to 5eV. Use at least two goodness of fit approaches to show that your fit is adequate. (Note: see data analysis approaches on course website next to where you downloaded this Problem Set). Define each goodness of fit parameter you use and please reference. Figure: A linear fit of extracted data of Log e of O i diffusivity in Si versus the inverse of temperature. The linear fit indicates that the data is indeed linear and thus the data shows Arrhenius behavior. Mathematica was used to plot and fit the data. To find the activation energy, we use the following approach. E k slope A B ln D2 ln D1 kb 1 1 T2 T For log[d] versus plot, EA is given by: T EA kb slope log D2 log D1 ( ) kb T2 T1 log D log D ( ) kb T Where: T ' 2 ' 1 2 T1 2 1 ' ' 2 T & T T Mathematica was used to extract the slope using a linear fit model and then used calculate E A and these values are giving below. The equation of line provided by the fit is given below as well as the extracted slope and the y- intercept with the Standard Error. y x Estimate Standard Error t-statistic P-Value Out[350]= slope b

4 Slope: E A ev R 2 1. Adjusted R 2 1. Diffusion activation energies range from 100 s of mev s to ~3eV, so this activation energy appears to be physically valid. The goodness of fit parameters I ve used here are the coefficient of determination (R 2 ) and adjusted coefficient of determination (adjusted R 2 ). Shown above, they were determined to be: R 1 Adjusted R 1 A value of 1 signifies a perfect fit, so both goodness of fit parameters show an excellent fit. The definitions of both are below: o Coefficient of Determination, R 2 : The coefficient of determination, R 2, is a measure of the goodness of a fit. It is determined by taking the difference between the value 1 and the ratio of two sum-of-squares values: SS reg and SS tot. SS reg is the sum of the square of the difference between the mean y-value and the actual y-value. SS tot is the sum of the square of the difference between the actual y-value and the best fit curve. Hence, the closer R 2 is to 1, the better the fit.[1] o Adjusted Coefficient of Determination, Adjusted R 2 When adding additional fitting parameters to the model, R 2 will increase, however, this does not suggest a better fit. The adjusted R 2 negates this effect. To avoid this effect, we can examine the adjusted R 2 : As with R 2, the closer AdjR 2 is to 1, the better the fit. To find pre-exponential of diffusivity, D o, we use the Arrhenius equation for diffusivity given by: E A D Do Exp kt B And we solve for D o : E A Do DExp 1 kt B 1 where ' log[ D1 ] 1000 D1 10 & T ' T 1 4

5 Note that D 1 and T 1 is specific to one data point extracted from the plot as shown above. Thus, D o was calculated using Mathematica: D o = cm 2 /s c. Using the information you extracted in part b, calculate the diffusivity in units of cm 2 /s of O i diffusion in Si during the thermal annealing of dopants in Si. Substituting the E A, D o and the temperature for dopant processing in Kelvin, T = K, into the Arrhenius equation for diffusivity given by: E A D Do Exp kt B And using Mathematica, we obtain: D(750 o C) = 4.55x10-14 cm 2 /s d. Find another journal article that provides the activation energy and pre-exponential for diffusivity of oxygen interstitial diffusion in silicon (e.g., use Google Scholar to perform the search for a journal article). Using this information, examine (i.e., determine and comment on) the physical validity of the activation energy and preexponential that you extracted from the data. Mikkelsen [3] found that the activation energy for oxygen interstitial diffusion in silicon is 2.44eV and the pre-exponential, D o, is cm 2 s 1. One can examine the physical validity of the activation energy by determining the accuracy of the activation energy via the percent error. E E 2.55eV 2.44eV eV 4.70% A, calculated A, actual % Error for E A 100 EA, actual A 5% error is small, hence the accuracy and thus physical validity of the extracted E A is supported. Do, calculated D o, actual % Error for Do 100 Doactual, 2 2 cm 2 cm s s cm 710 s 99.97% The % Error for D o is nearly 100% which is poor, hence the validity of D o is not supported. However, D o is not a materials constant as is E A nor is D o a fundamental constant, hence, it is not surprising that the error is so high. References: [1] Interpreting Regression Results - from OriginLabs. 5

6 [2] H. Motulsky & A. Christopoulos, Fitting Models to Biological Data using Linear and Nonlinear Regression -A practical Guide to Curve Fitting, Version 4 (GraphPad Prism; 2003) p [3] J. C. Mikkelsen Jr., Diffusivity of oxygen in silicon during steam oxidation, Appl. Phys. Lett. 40, 336 (1982). 6

7 2. Create a problem analogous to the one below for another monovalent metal using your own reasonable values. Make sure you follow all the steps in the problem below. Question: The electrons in a certain material have the following properties: drift velocity is 10 3 m/s, the mobility 2 cm is 40 and the mean thermal energy is 10 mev. Determine the following additional properties of the Vs electrons. Assume the Drude model. a. Determine the electric field experienced by the electrons in units of V/cm. b. Determine the temperature of electrons in units of Kelvin. c. Determine the mean time between electron collisions in units of seconds. d. Determine the diffusivity of the electrons in units of cm 2 /s. e. Assume the certain material in this problem is Cu. Determine the conductivity for Cu in units of (ohm cm) -1. f. Assume the certain material in this problem is Al, which is trivalent. Determine the conductivity for Al in units of (ohm cm) -1. 7

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9 c. 3. The resistivity of pure nickel (assume no Table 1 Resistivity of Nickel at increasing defects are present) as a function of temperatures. Data collected from Smithell s Metals temperature will be explored in this problem. Reference Book Section 14-5 The data in Table 1 lists the resistivity of nickel at various temperatures. You will fit the data Temperature ( C) Resistivity (μω*cm) using a mathematical program (e.g., both Mathematica and Matlab both have nonlinear regression functions and very good statistical analysis of the fit), examine the goodness of the fit, and assess the physical validity of the fit. Before you do this, you will thoroughly define both goodness of fit and validity of fit. Hint: Use the two handouts provided with problem set 2 on course website: [1] H. Motulsky & A. Christopoulos, Fitting Models to Biological Data using Linear and Nonlinear Regression -A practical Guide to Curve Fitting, Version 4 (GraphPad Prism; 2003) p [2] Interpreting Regression Results - from OriginLabs. a. Mattheissen s rule gives a linear relationship in temperature for the resistivity of pure metals. In part b, you will prove that this is the case for the data given in Error! Reference source not found. by fitting the data to Mattheissen s equation. For part a, consider that you are teaching an algebra class and you are giving a lecture using this problem as an application problem for the subject you are lecturing on called Lines and Their Uses in Engineering. For the algebra students to which you will lecture, provide a bulleted list of steps (~5-6 steps) necessary to demonstrate this application starting with Mattheissen s rule. Include mathematical relationships showing the relation of Mattheissen s rule to that of a line (i.e., equation of a line), defining each of its components, in the steps and connect Mattheissen s rule to the line concept so that the students understand the application to Lines and Their Uses in Engineering. In your steps, include the concepts of examining the goodness of the fit, and assessing the physical validity of the fit. Hint: remember that this is a pure metal for which we assume no defects are present, so think about what happens at the origin of the plot when you fit the data. 9

10 Mathiessen s rule: ρ = ρ T + ρ I = AT + ρ I Equation of a line: y = m x +b where m = slope and b is the y-intercept Comparing the two equations, one sees that the slope m = A and the y- intercept b = ρ I Mathiessen s rule for a pure metal: ρ I = 0, so ρ = ρ T = AT. If ρ I is used, its physical validity would need to be assessed. Now one can use the equation of a line to fit the data and the y-intercept should be zero and thus the line should go through the origin. The fitting parameter is A, which is the slope, and can be extracted from the fit. A should be compared to a physical aspect of the resistivity of the data. In this case, mobility and A are related. Mobility can be b. Approaches used for fitting data are linear and non-linear regression. A typical regression method is called least squares method of fitting. Define the method of least squares method of fitting including the assumptions. Use a graph schematic to help you illustrate the least squares method of fitting (i.e., draw a graph with data points and a fit line). List the assumptions of the least squares method of fitting. Cite your sources. 10

11 [1] In the figure above, note the following: yi : the y-value of the data point yˆ i : the y-value of the fit y : the average y-value i Assumption of the nonlinear least-squares method: (taken verbatim from [2] page 30). (1) x is known precisely, all the error is in y. (2) The variation in y follows a known distribution. Almost always, this is assumed to be a Gaussian distribution. (3) Standard nonlinear regression assumes that the amount of scatter (the standard deviation of the residuals) is the same all the way along the curve. This assumption of uniform variance is called homoscedasticity. Weighted nonlinear regression assumes that the scatter is predictably related to the Y value. (4) Observations are independent. When you collect a y value at a particular value of x, it might be higher or lower than the average of all y values at that x value. 11

12 Regression assumes that this is entirely random. If one point happens to have a value a bit too high, the next value is equally likely to be too high or too low. [1] Interpreting Regression Results - from OriginLabs. [2] H. Motulsky & A. Christopoulos, Fitting Models to Biological Data using Linear and Nonlinear Regression -A practical Guide to Curve Fitting, Version 4 (GraphPad Prism; 2003) p [3] D. Skoog, D. West, & F. Holler, Fundamentals of Analytical Chemistry, 5 th Ed. (Sanders College Publishing, 1988) Ch. 2. c. When fitting data, one often examines the goodness of the fit and physical validity of the fit. With significant detail, define goodness of the fit and physical validity of the fit and describe the difference between goodness of the fit and physical validity of the fit. List (i.e., give examples) of several methods used for goodness of the fit and for physical validity of the fit and define two of the examples each for goodness of the fit and for physical validity of the fit. Use schematics to help you with your definitions. Be sure to cite your sources. Goodness of Fit: When data is fit or described by a mathematical relation, a comparison of (x,y) data to the (x,y) of the fit can be performed (i.e., regression method) to see how close the fit data point is to the actual data point. This is typically done using non-linear regression. The closer the difference, the better (i.e., gooder ) the fit. From the figure above, we see that: yi : the y-value of the data point yˆ i : the y-value of the fit y : the average y-value i Using these definitions, we can then define the following: o Total sum of squares (TSS) about the mean (variation between data n points & Mean): 2 TSS y y i1 i Several methods used to check the Goodness of Fit include: o Residuals or Residual sum of squares or reduced chi-square: n Regression sum of squares: 2 SS yˆ y i is the portion of variation reg i i i1 that is explained by the regression model n Residual sum of squares: 2 RSS y yˆ i i is the portion of variation i1 that is not explained by the regression model. Residuals should be random. If they are not random, the linear fit should be replaced with a nonlinear regression fit. 12

13 o Coefficient of Determination, R 2 : R SS RSS 1 1 TSS TSS 2 reg i1 n n i1 y yˆ y i i i y i The coefficient of determination, R 2, is a measure of the goodness of a fit. It is determined by taking the difference between the value 1 and the ratio of two sum-of-squares values: SS reg and SS tot. SS reg is the sum of the square of the difference between the mean y-value and the actual y-value. SS tot is the sum of the square of the difference between the actual y-value and the best fit curve. Hence, the closer R 2 is to 1, the better the fit.[1] o Adjusted Coefficient of Determination, Adjusted R 2 When adding fitting parameters to the model, R 2 will increase, however, this does not suggest a better fit. To avoid this effect, we can examine the adjusted R 2 : SS 2 reg RSS dferror R 1 TSS TSS dftotal Like R 2, the closer AdjR 2 is to 1, the better the fit. o Confidence levels: we need to define the following: Population standard deviation: 2 2 N i1 x 2 N where N is the sample number and μ is the mean of the population. σ is used when N is infinitely large. Sample standard deviation: s N i1 x x 2 i N 1 where N is the sample number and x is the mean of the sample population. S is used when N is finite. When s is a good approximation to σ, then we can define the confidence limits (in %) by the percent area under a Gaussian curve (figure 2-6). The x-axis is defined by the difference between the two means and normalized by the population standard deviation. Hence, each unit on the x-axis is a standard deviation σ, from the difference between the two means. So if the deviation is 0.67σ, this equates to a 50% confidence level that the experimentally determined mean, x, is within that limit. i [3] 13

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15 Figure: 95% confidence and prediction limits and intervals of a data set in which x p is the point of interest.[2] Physical validity of fit: When data is fit or described by a mathematical relation, the independent variable (x) is varied and the dependent variable (y) is determined using a regression method. Other parameters in the mathematical relation can be used to help fit the data. These relations can have physical meaning. The physical validity of these variables should be checked to see if they make physical sense. o If you have many fits, than one can examine the accuracy of the fit parameters using the mean, standard deviation, variance, or the coefficient of variation. o Precision: used for single values of a parameter and when a known value for the parameter exists Relative error = (fit parameter value known value)/(known value) [3] Absolute error = (fit parameter value known value) [3] d. Using the previous parts of this problem, prove that Matthiessen s rule gives a linear relationship in temperature for the resistivity of pure metals using the data given in Error! Reference source not found. by fitting the data to Matthiessen s equation. Provide the equation of the fit, define each variable (i.e., dependent variable, independent variable, fit parameters and names of each) and provide units of cm V -1 s -1. Note: Convert the temperature to Kelvin before fitting the data. Show both the data set and the fit on the same plot. To help prove Mattheiessen s rule gives a linear relationship for the data in Table 1, examine and discuss the goodness of the fit using two goodness of fit approaches. Note: Ensure that the scales of the axes are expansive enough such that they allow for any divergence of the fit from the data to be observed. Matthiessen's rule for conduction in metals ρ(t) = AT + B (Kasap p. 128, equation [2.17]), ρ(t) is the resistivity and the dependent variable T is the temperature and the independent variable A is the slope and a fit parameter B is the y-intercept and a fit parameter and is zero for pure metals (Kasap p. 128). Although it is not required, the data was fit with two equations. First approach: ρ(t) = AT + B where B = 0 (pure metals) 15

16 First approach: ρ(t) = AT + B where B not equal to 0 assuming that the data does not correlate to the origin and thus not constraining that the fit goes through the origin. The plot, fit and extracted parameters are below. The plot shows the resistivity for nickel (points are data) as a function of temperature. The fits (lines) were generated using Matheissen s rule for either a pure metal (B=0, blue line) and by allowing B to vary (green line). The latter produces a better fit. For ρ(t) = AT: ρ(293k) = μω-cm A = μω -cm-k -1 ρ293k 3.17μΩ cm and A μΩ cm K^1 For ρ(t) = AT + B: ρ(293k) = 6.02 μω-cm A = μω-cm-k -1 B = μω-cm A negative B does not make sense, but not constraining the fit to go through the origin provides a superior R 2. See part e. 16

17 Although we are to examine two goodness of fit parameters, I examine three goodness of fit parameters. R 2 = for ρ(t) = AT Where ρ(293k) = μω-cm and A = μω-cm-k -1 R 2 = for ρ(t) = AT + B R Adjusted R A cm K 1 B cm 293K cm Where ρ(293k) = 6.02 μω-cm and A = μω-cm-k -1 and B = μω-cm A negative B does not make sense, but not constraining the fit to go through the origin provides a superior R 2. That is, from both the plot and the R 2 s, it can be seen that ρ(t) = AT + B gives the superior fit although it does not assume B = 0. Both R 2 s show that the goodness of fits are fine. R 2 for the fit performed above is for B=0 or for B not equal to zero, which indicates that the fit models the data with an accuracy of 97.4% or 99.4%, respectively. The above plot shows both the data and the fits, further reinforcing the validity of the fit. Notice the fits do model the behavior of the data well in that the fits do not deviate away from any data point(s). Plot of the Residuals of the fits are below. A residual is the difference between each original y-data point and its fitted value. The smaller the difference, the better the fit.[1-2] 17

18 Figure: Residuals with a difference of about 8 over a range of 10 data points. The first 6 data points are negative while the last 4 data points are negative indicating a possible pattern. In assessing residuals, one looks for patterns. In the first residuals plot, the first 6 residuals show all negative and the next four show all positive indicating a linear fit may be ill-advised and a NonLinearFitModel[ ] fit is more appropriate. Figure: Residuals with a difference of about 6 over a range of 10 data points. A random pattern seems to be apparent. In the second residuals plot, the trend seems to show a random behavior and the residuals, the difference between each original y-data point and its fitted value, are smaller in the second plot than the first plot indicating that the goodness of fit is sound. 18

19 e. Discuss the physical validity of your fit using the fit model you found earlier in the problem by calculating the electron drift mobility at 293K and compare the result to the value of 6.40 cm V -1 s -1 for the electron drift mobility as determined by Dr. G. Bradley Armen (Department of Physics, University of Tennessee). Note: Your answer should have units of cm V -1 s -1. There are two ways to check the physical validity of your fit: I. the approach outlined in part f. II. relating A and B to the TCR, α o, as well as to ρ o and T o. A o o B 1 T o o o So if one knows ρ o and T o, then one can find α o for Ni and calculate A. That A could be compared to the extracted A. Here, we shall use approach I. From equation 2.17 on page 128 in Kasap, it is known that =AT+ o. Where: is the resistivity o is the resistivity at 0 C A is the constant of proportionality (the slope of the fit found in the Data Analysis section) T is the temperature It is assumed that the nickel is pure (no impurities). Therefore, Matthiessen's rule may be used and is given by: total (T)= T + I (Kasap, Page 127, equation 2.15 and 2.16) which can be written as: total (T)= +(Kasap, Page 128), equation 2.17) In pure metals, we know that I =0 (thus, B=0) (Kasap, Page 127 and 128), equation 2.15), so we can write: total (T) = T ( I =0 or B=0 in pure metals). We know that =1/, and that = en d. d =1/(e n total (T)) = 1/(e n T). n is the number of free carriers per unit volume (page 119 Kasap). From example 2.2 on page 119, it is known that the number of carriers per volume is: dn A n v M atomic where: d is the density (g/cm 3 ) N A is Avogadro's Number (mol -1 ) M at is the atomic mass (g/mol) v is the number of valence electrons ( electrons/cm 3 ). 19

20 One can also use dimensional analysis to determine n if given d, N A, M at and v. Using Mathematica, n = 1.42 x electrons/cm 3. The drift mobility, µmob, is given by: T Note the 6 10 is to convert μω to Ω. drift 1 in units of cm 6 e n T 10 Vs Using Mathematica: T drift cm Vs 2 So the electron drift mobility in nickel as calculated using the fit is 7.33 cm 2 /(V s). According to Hall Effect data of data from G. Bradley Armen from the University of Tennessee, the drift mobility in nickel at 293K is ~ 6.4 cm 2 /(V s). The % error, calculated using Mathematica, is given by: actual fit % error = 100 = 14.6% actual A -14.6% percent error as determined using the value provided by Armen, is less than 20%, which indicates that the mobility extracted by the fit is physically valid; in other words, it is within an order of magnitude of the measured value and not an unrealistic value. The ~15% error is somewhat high but the extracted mobility is certainly fine to use when assessing whether or not the material should be used for electronic applications. 20