Problem Set 3 Solutions 1. ABET PROBLEM: a. Ability to apply knowledge of mathematics, science, and engineering. (P) 1. You are a photoelectron spectroscopist in the new Boise State Center for Materials Characterization under the direction of Professor Rick Ubic. A company asks Professor Ubic to identify a material of unknown identity. The company representative, Dr. R.U. Qureus, states that the material is highly reactive when exposed to water and provides a sample of the material. Professor Ubic turns to you, the expert in photoelectron spectroscopy. He asks you to identify the material and submit a brief report on your findings. You perform photoelectron spectroscopy on the sample and obtain the following photoemission data (Excel.csv of frequency, v, in units of inverse seconds (s -1 ) versus KEmax in energy units of ev (ev) provided with the problem set). In your report, identify the material and show step by step how you identified the material. Also prove that the data is photoemission data. Include all plots (done by a mathematical program and/or Excel) and other aspects of your analysis. Include references used to identify the material. The report should have the following format: Title Name, affiliation, date Customer Name/contact Summary (2-3 sentences) Data Analysis Approach (include mathematical &/or graphical analysis software used, fitting used & fitting statistics, etc.; 1 paragraph) Data Analysis Results (1/2 to 1 page) Conclusion (2-3 sentences) References (minimum: 2 books or papers no websites) Writing style: use the style and approach used in journal articles (e.g., Science, Nature, Nature Materials). Do not use pronouns (I, you, they, we, etc.). Refrain from writing lists (e.g., I did this, then I did that, then I did this ). A succinct, clear, non-verbose writing approach is best. Note: it is important that you use your background in analyzing data as you have in the past problem sets relative to goodness of your analysis and the physical validity of your analysis. Be sure to provide definitions of each method you use. Leverage what you have used in the past in terms of methods and definitions. Solution: Brief Report: Title: Analysis of unknown material Photoelectron Spectroscopist: Bill Knowlton, Boise State Center for Materials Characterization Date: 9/9/16 Customer Name/Contact: Dr. R.U. Qureus Center Director: Dr. Rick Ubic 1
Summary: The work function of the material is 2.49 ev. Comparing the extracted work function to published data [1-2], the material is lithium (Li). Li is readily reactive with water. The data are indicative of photoemission data as the slope is very close to the value of Planck s constant, h. Data Analysis Approach: The data is plotted with KEmax as a function of frequency on a linear-linear plot. Linear regression is used to determine the degree of linearity of the data. The correlation coefficient/factor, R 2, is used to assess linearity. The closer R 2 is to 1, the better the fit and closer to linearity. The y-intercept of the data provides the negative value of the work function of the material. The work function can then be compared to published work functions.[1-2] In order to ensure that the data is photoemission data, the slope will be compared to Planck s constant, h. The % error (i.e., relative error; see [3] for example) is used to comparison the slope to h, and is given by: h slope % Error (or relative error) = 100 (1). h Data Analysis Results: Figure 1 shows a plot of both the photoemission data and the linear best fit. The linear regression data are shown in the Table 1. The correlation factor, R 2, is 0.99988 which is very close to 1 and is indicative of data that is linear. The negative value of the y- intercept is the work function, Φ, and is 2.49 ev. Examination of published data, for instance table 5.5 in Kasap [1] and Table 6.2 in Solymar and Walsh [2], reveals values of 2.5 ev and 2.48 ev, respectively, for Li. The value extracted from the data lies in between these published data which strongly indicates that the unknown material is Li. Furthermore, Li is highly reactive with water and oxidizes when in contact with water and air. Hence, storage in alcohol is necessary. When comparing the slope to Planck s constant using equation 1, the percent error is only -0.062%, which is very small. Therefore, the data are photoemission data. 2
Figure 1: KE max versus frequency of photoemission. Linear regression fit is shown in blue. The linear regression statistics are shown in the box in the lower right corner and in Table 1. The negative of regression curve s y-intercept is the material s work function, Φ. Table of fit parameters: Table 1: Linear Regression for Data. The fit equation was defined as the general photoelectron relation: KE = h*v workfunction. The R 2 =of the fit was 0.999949 which shows that the fit is very linear. Planck s constant for the fit, hfit, was an extracted value of 4.13828x10-15 ev/s and the work function of fit was 2.48561 ev. work function of fit: 2.48561 ev Planck s constant for the fit: 4.13828x10-15 ev/s 3
Conclusion: Data provided are photoemission data that are highly linear. Data analysis provided an extracted work function of 2.486 ev. Comparison to work function data in the literature identified the hydrophyllic material as Li. The physical validity of the fit can be assessed by comparing Planck s constant to the value of Planck s constant extracted from the fit. This is given by: %error for h: -0.062 % Which is a very small error. References: [1] S.O. Kasap, Principles of Electronic Materials and Devices, 3 rd Ed. (McGraw-Hill, 2006) p. 470 [2] L. Solymar and D. Walsh,, 7 th Ed. (Oxford, 2004) p.87 [3] Skoog, West and Holler, Fundamentals of Analytical Chemistry, 5 th Ed. (Saunders Publishing, 1988) p. 9 2. In class, we saw that the thermal conductivity and electrical conductivity of metals were directly proportional to one another. a. Look up and list the thermal conductivity data and the electrical resistivity or electrical conductivity data of four metals. Cite your references (e.g., Smithells Metals Reference Book, 8th Ed., Edited by Gale & Totemeier, 2010, 14-3 to 14-7, Wolfram Alpha, etc.). The Thermal Conductivities (κ) at 27 o C - from Smithell' s 14-3 to 14 7 [1]. KPb=35.3; KSb=16.7; KPt=71.6; KFe=80.2 all in units of W/m K Electrical resistivities with units of µω-cm at 27 o C - from Smithell's Ch. 14.[1] ρsb=59; ρpt=10.8; ρfe=9.98; ρpb=21.3 all in units of µω-cm. b. Plot the data as thermal conductivity (W/(m K)) versus electrical conductivity (1/Ω-cm) somewhat similarly to figure 2.21 in Kasap. Please include a figure caption. 4
A plot of the thermal conductivity (W/(K m) -1 ) versus electrical conductivity data in units of (ohm-cm) -1 is shown above. The data was acquired at 27 o C (300K). c. Fit the extracted data and plot the fit with the data. Please include a figure caption. Evaluate the goodness your fit by using at least two statistical parameters. Be sure to explain the statistical parameters that you use and cite your source. In order to calculate CWFL, the data of electrical conductivity data for in (ohmcm) -1 needs to be converted to (ohm-m) -1, plotted and fit. 5
A plot of the thermal conductivity (W/(K m) -1 ) versus electrical conductivity data in units of (ohm-m) -1 is shown above with data as points and a linear best fit as a green line. The data was acquired at 27 o C (300K). The fitting function that was obtained using Mathematica NonlinearModelFit command gave: Fitting Function: K =7.86832 x10-6 σ. From Mathematica, both a R 2 and Adjusted R 2 were determined. The R 2 = 0.998668 and the Adjusted R 2 = 0.998224. They are both very close to 1 signifying a very good fit. For the fit, note that I plotted K(W/K*m) versus σ (Ω*m) so that meters would cancel with regard to the units. The Coefficient of Determination, R 2, represents how well the model fits the data.[2] The value varies between 0 and 1, with 1 being a perfect fit, meaning that the model can accurately describe every data point. However, when adding additional variables to the fitting equation, R 2 can increase. Although the fit may be better (it usually is), adding additional adjustable parameters (i.e., degrees of freedom of the fit) is not a sound approach to use to obtain a better fit. That is, obtaining a good fit while minimizing the number of adjustable parameters is the best approach. The 2 Adjusted Coefficient of Determination, R provides a better assessment of the goodness of fit while taking into consideration the degrees of freedom of the fit.[2-3] 6
2 The Adjusted Coefficient of Determination, R, also represents how well the model fits the data, but also takes into account the degrees of freedom.[3] Hence, it can take on negative values.[3] d. One manner in which to substantiate the physical validity of your fit is to answer the following question - does the fit represent the WFL law? Explain using your results to support your explanation by determining the accuracy of your results. Be sure you define the physical validity approach you use. In order to determine whether or not the data follows the WFL law, equation 2.36 on page 150 of Kasap can be used. We see that equation 2.36 can be written as: C WFL κ 1 = σ T slope = T From the caption, we see that the temperature is 27 o C or 273 +27 = 300K. To determine the slope, we differentiate our linear fit. Then, we can divide by temperature: C = WFL slope = T [ linear fit] σ T From our linear fitting shown in part a and the above equations, we obtain the following: Fitting Function = 7.86832 x10-6 σ slope = 7.86832x10-6 ( Ω.W/K) CWFL = 2.62277 10-8 ( Ω.W/K 2 ) We see that from the text on page 150 of Kasap that: C WFL 2 2 π kb 8 WΩ = = 2.44 10 2 2 3e K The percent error (% error) [4] permits the accuracy of a calculated value obtained from the fit to be determined. The % error compares the calculated value from the fit to the actual value and is defined by: 7
calculated actual %Error = 100 actual 8 8 2.62277 10 2.44 10 = 100 8 2.44 10 = 7.49% The % error is not significantly large (below 10-15%), hence the fit DOES fairly accurately represent the WFL law relatively well as the figure caption states.[4] References: [1] Smithells Metals Reference Book, 8th Ed., Edited by Gale & Totemeier, 2010, Ch. 14-1 & 14-7. [2] H. Motulsky & A. Christopoulos, Fitting Models to Biological Data Using Linear & Nonlinear Regression A Practical Guide to Curve Fitting, Version 4 (GraphPad Prism Software, 2003). [3] Origin Labs section on Nonlinear Curve Fitting: http://www.originlab.com/www/helponline/origin/en/category/nonlinear_curve_fitting.html [4] Skoog, West, & Holler, Fundamentals of Analytical Chemistry, 5 Edition (Saunders College Publishing, 1988) p.8-11. 3. In no less than 90 words but no more than 150 words for each phenomenon, explain what are the consequences of each of the following phenomena (Hint: use your textbook): a. Skin effect. Explain. The skin effect makes it difficult to use ordinary, solid-metal core conductors in high frequency (AC frequencies >1GHz) because the magnetic field of an ac current in the conductor restricts the current flow to the surface region with a depth of δ (skin depth) < a (radius of a conductor). Since the current can only flow in the depth of δ, there is an increase in the resistance due to the decrease in the cross-sectional area for the current flow. The skin effect is important in electrical engineering as solid-core conductors can no longer be used in high-frequency applications. As the signal frequencies become greater than gigahertz (10 9 Hz) range, waveguides that have hollow cores must be used. (Taken from pg. 165 of Kasap 3rd Ed.) [1] b. Hall effect. Explain. The Hall effect is a phenomenon that helps explain the electron as a particle concept. If a magnetic field is applied, B z, in a perpendicular direction to an applied electric field, E x, that is driving the current in a material, another additional force, or Lorentz force, is experienced by the moving charges. The force is an electric field, E y = E H, or hall field. In 8
turn, a Hall voltage, V H, is induced and can be measured (see Kasap p. 157 fig. 2.27). The voltage is a result of the Lorentz force from the magnetic field pushing the electrons to the bottom of a sample causing a negative charge accumulation near the bottom of a sample and a positive charge near the top of a sample. This effect is called the Hall Field and is what causes V H.[1] c. Surface scattering of carriers in interconnects that are used in microelectronic circuits. Explain. Scattering of carriers by any defect will increase the resistivity of a material. The surface of a material, which is the interface between the material and air or another material, is considered a planar defect. As equation 2.59 in Kasap page 169 suggests, as a bulk material s thickness decreases to that of the order of the mean free path, λ, of a carrier (i.e., path an electron travels between scattering events), an increase of scattering of the carrier by the surfaces within the reduced dimension will occur. For metal interconnects, not only is the thickness of the interconnect on the order of λ, so is the width of the interconnect. So the resistivity of the interconnect is substantially increased by the reduced dimension of the interconnect which leads to slower electron transport and Joule heating due to the surface scattering.[1] References: [1] S.O. Kasap, Principles of Electronic Materials and Devices, 3 Edition (McGrawHill, 2006) p.8-11. 9
4. X-rays are photons with wavelengths in the range 0.01-10nm, with typical energies in the range of 100eV to 100 kev. When an electron transition occurs in an atom from the L to the K shell, the emitted radiation is generally in the X-ray spectrum. For all atoms with atomic number Z > 2, the K shell is full. Suppose that during scanning electron microscopy (SEM) of various materials, one of the electrons in the K shell of these materials has been knocked out by an energetic electron from the SEM impacting the atom (the electron from the SEM has been accelerated by a large voltage difference). The resulting vacancy in the K shell can then be filled by an electron in the L shell transiting down and emitting a photon. The emission resulting from the L to K shell transition is labeled the Kα line. The table shows the Kα line data obtained for various materials. Material Mg Al S Ca Cr Fe Cu Rb W Z 12 13 16 20 24 26 29 37 74 K α line (nm) 0.987 0.834 0.537 0.335 0.229 0.194 0.154 0.093 0.021 a. Let υ be the frequency of the emitted K α X-ray. Plot υ 1/2 against the atomic number Z of the element. Figure: Square root frequencies of the Ka line emissions of several elements are plotted against the atomic number, Z. A linear fit is shown with a R 2 = 0.999583 indicating the data is highly linear. Estimate StandardError t-statistic P-Value Out[617]= slope 5.20685 10 7 401920. 129.549 4.30688 10 13 b 9.60908 10 7 1.33349 10 7 7.20598 0.000176537 b. Use a linear fit of the υ 1/2 vs. Z data, to determine constants B and C in the equation below. υ 1/2 = B (Z C) 10
B and C are calculated from the coefficients of the linear fit in Part a using Mathematica. R 2 0.999865 Estimate StandardError t-statistic P-Value = B 5.20685 10 7 401920. 129.549 4.30688 10 13 c 1.84547 0.24425 7.55565 0.000131041 For the general Moseley relation: v 1 2 B Z C B 5.20685 10 7 Hz 1 2 C c of fit 1.84547 c. Evaluate the linearity of the data, and comment on whether or not it is appropriate to use a line to model the data. Use at least one reference to support your answer. The R 2 of the linear fit is 0.999865, which is very near 1. This shows the data is highly linear, and that line is an appropriate choice for a model.[1-2] d. Hyper Physics states that the emission frequency from a K α X-ray is related to the atomic number by Moseley s Relation: h v = (3/4) 13.6 ev (Z 1) 2 How well does the linear fit found in part b match this equation? That is, what is the physical validity of the parameters you have extracted from your fit? Explain. Additionally, extract h from the fit and compare it to the actual value of h. Comment on your findings. Rearrange Moseley s Relation to look like our fit: ( ) 1/2 3 / 4 13.6 ev ν = h ( Z 1) ( ) ev ( ) 3 / 4 13.6 3 / 4 13.6 ev = Z h h The equation in part b is given by: υ 1/2 = B (Z C) Comparing the two equations we see that: B = and BC = ( ) 3 / 4 13.6 ev h ( ) 3 / 4 13.6 ev h 11
This is because C << B which is supported in part b where C is 1.86 while B is 5.2x10 7 Hz 1/2. B GeneralFit B B GeneralFit MoseleyFit x100 = 2.17% Our parameter B has a relative error of 2.17% with respect to the Moseley s relation, which is quite good.[3] Relative to extracting h from the fit and compare it to the actual value of h, we find, using Mathematica, that: Out[1080]= Estimate StandardError t-statistic P-Value hfit 3.9313 10 15 5.23819 10 17 75.0508 1.10702 10 12 R 2 0.999645 h 3.9313 10 15 error for h extracted from the fit: 4.94224 The goodness of fit, evaluated by R 2 with a value of 0.9996, is very close to 1, hence, the fit is very good. The value of Planck s constant is 3.93 x 10-15 ev/s. which is within an order of magnitude of the actual Planck s constant of 4.1357 x 10-15. The percent error is about 5% which is quite good. The data and fit are shown below. What is nice about this fit relative to the first fit is that there is only one fit parameter, h. In the first fit, there were two fit parameters. The fewer the fits, the better, particularly if the fit parameter has physical meaning. In the first fit, the fit parameters did not have any physical meaning that one could readily see. Figure: Square root frequencies of the Ka line emissions of several elements are plotted against the atomic number, Z. A linear fit using Moseley s relation with h as a fit parameter is shown with a R 2 = 0.9996 indicating the data is highly linear. 12
References: [1] H. Motulsky & A. Christopoulos, Fitting Models to Biological Data Using Linear & Nonlinear Regression A Practical Guide to Curve Fitting, Version 4 (GraphPad Prism Software, 2003). [2] Origin Labs section on Nonlinear Curve Fitting: http://www.originlab.com/www/helponline/origin/en/category/nonlinear_curve_fitting.html [3] Skoog, West, & Holler, Fundamentals of Analytical Chemistry, 5 Edition (Saunders College Publishing, 1988) p.8-11. 13