MSE 410/ECE 340
Electrical Properties of Materials
School of Materials Science & Engineering
Fall 2016/Bill Knowlton
Problem Set 1 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).
Note: Besides problem 1, the other problems are review problems for which answers can be found in your
ENGR 245 Introduction to Materials Science & Engineering, first and second semesters of physics courses,
and first semester chemistry course.
1. Ethics: [ABET question]
Program Outcome H. The broad education necessary to understand the impact of engineering solutions in a
global, economic, environmental, and societal context.
Go to my Courses Website at:
http://coen.boisestate.edu/bknowlton/course-information/82-2/
a. You will find a section called “Scientific and Professional Misconduct”. Read the articles about the
physicist Jan Hendrik Schön and one of the stem cell researchers, either Woo Suk Hwang or Haruko
Obokata, and the predicament they are in. Comment on what might have gone wrong and/or where they
went wrong in their careers that led them to where they are today.
An ethically responsible answer.
b. Read through the sections “Information on: Academic Honesty” and “Dismissal & Student Code of
Conduct” for Boise State. Comment on why Boise State takes this so seriously. Relate your comments
to your comments in part a.
An ethically responsible answer.
c. Explain how might your reflection and discussion with respect to part a and b affect your undergraduate
career and career beyond bachelorette degree?
An ethically responsible answer.
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MSE 410/ECE 340
Electrical Properties of Materials
School of Materials Science & Engineering
Fall 2016/Bill Knowlton
2. Waves and Wave Motion
a. Define the following for a wave in a sentence or two for each: Wavelength, Frequency, Velocity
Wavelength: For a periodic wave, the distance between two successive parts of the wave
that are in phase (that is, at identical points on the waveform)
Frequency: Number of cycles per second (f = 1/period or v = 1/period), that is, the
number of complete waveforms, or wavelengths that pass by a given point during each
second.
Velocity: Speed (i.e., magnitude of velocity) and direction of a wave. Velocity is a vector.
b. Look up and write down the equations that relates i) wavelength, frequency, velocity and ii) energy and
wavelength. Define all variables.
i)
velocity = λ f & E =
hc
λ
where λ is wavelength, f is frequency, h is Planck’s constant
and c is the speed of light and E is energy.
c. What is the difference between a transverse wave and a longitudinal wave? Use an illustrative
(drawing) example that you have created to aid your explanation. You may use phonons (wave packet
of crystalline lattice vibrations) in your illustration. That is, you may use examples of transverse
phonons and longitudinal phonons.
In a longitudinal wave, the displacement, u, (or motion) is parallel to the direction of the
wave velocity or wave vector, K. This can be seen in figure 1.[1] In a transverse wave, the
wave displacement, u, (or motion) is perpendicular to the direction of the wave velocity or
wave vector, K, as shown in figure 2.[1]
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MSE 410/ECE 340
Electrical Properties of Materials
School of Materials Science & Engineering
Fall 2016/Bill Knowlton
References:
[1] Charles Kittel, Introduction to Solid State Physics (8th Edition, 2005, John Wiley &
Sons, Inc.) Ch. 4, p. 90.
d. Using a graph that you have created with a mathematical program (not Excel) to help you, explain the
following:
i. Constructive interference
ii. Destructive interference
Perfect constructive interference is when two waves with the same magnitude are exactly
in phase. The crest of one wave coincides with the crest of the second wave. In the figure
directly below, the sinusoidal wave in red shows partial constructive interference (sin[x] +
cos[x]) where the initial two waves (cyan & green) have an amplitude of 1 and the final wave
has an amplitude of ~1.5 with the same frequency.
Perfect destructive interference is when two waves are completely out of phase (180
difference), that is a crest coincides with a trough. In the figure directly below, the
sinusoidal wave in blue shows partial destructive interference (sin[x]*cos[ x]) where the
initial two waves (cyan & green) have an amplitude of 1 and the final wave has half the
original amplitude. A decrease in frequency for the sinusoidal wave in blue is also observed.
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MSE 410/ECE 340
Electrical Properties of Materials
School of Materials Science & Engineering
Fall 2016/Bill Knowlton
The figure above plots Sin and Cos and examples of partial constructive and destructive
interference.
3. Electromagnetic spectrum: Calculate the range of frequencies or wavelengths (in meters) and energies
(in eV) for parts a-g using mathematical expressions for each (which you should show). Show and define
the mathematical expressions you use and the variables in the mathematical function you use in your
calculations and cite the references you used using a peer reviewed source. Hint: use problem 2b to help
you.
In this problem, we look up the wavelength or frequency range, list them, and calculate the
frequency or wavelength and energy (in eV) range for the following. Then, we can use the
relationships between energy, E, frequency, f, and wavelength, λ, given by:
=
E
hc hcf
=
= cf to calculate the ranges of each. Note that h is Planck’s constant, c is
λ 2π
the speed of light.
a. Radio and TV?
Frequency Range: 1x106 to 1x109 Hz
Wavelength: 1 to 600 m
Energy: 4.1x10-9 to 4.1x10-6 eV
b. Microwaves?
Frequency: 1x109 to 1x1011 Hz
Wavelength: 0.001 to 0.1 m
Energy: 4.1x10-6 to 4.1x10-4 eV
c. Infrared?
Frequency: 1x1011 to 4.3x1014 Hz
Wavelength: 1 x 10-3 to 7 x 10-7 m
Energy: 4.1x10-4 to 1.77 eV
d. Visible light?
Frequency: 7.5x1014 to 4.3x1014 Hz
Wavelength: 4 x 10-7 m to 7 x 10-7 m
Energy: 3.1 to 1.77 eV
e. Ultraviolet?
Frequency: 7x1014 to 1x1017 Hz
Wavelength: 1 x 10-7 to 1 x 10-10 m
Energy: 3 to 413 eV
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MSE 410/ECE 340
Electrical Properties of Materials
School of Materials Science & Engineering
Fall 2016/Bill Knowlton
f. X-Rays?
Frequency: 1x1017 to 1x1019 Hz
Wavelength: 1 x 10-10 to 1 x 10-12 m
Energy: 413 to 4.1x104 eV
g. Gamma Rays?
Frequency: above 1x1019 Hz
Wavelength: below 1 x 10-12 m
Energy: >4.1x104 eV
h. Using a mathematical program (not Excel), plot the electromagnetic (i.e. photon) energy (y-axis) from 0
eV to 10 eV as a function of wavelength (x-axis, units of nanometer) using the mathematical function you
used in the first part of this problem. On your plot, label thoroughly and show the energy and wavelength
ranges of the visible spectrum with lines that bound the region. Comment on whether or not you think
that the energy range of the visible spectrum is easier to remember in units of eV or units of Joules.
10
9
8
Energy (eV)
7
6
5
Visible regime within this
closed area
4
3
2
1
0
100
200
300
400
500
600
700
800
Wavelength (nm)
In the figure, the photon energy in electron-volts is plotted as a function of photon
wavelength in nm (solid blue line). The visible regime is the enclosed area by the intersection
of the lines on the energy axis at 1.77 eV and 3.1eV and the lines on the wavelength axis at
400nm and 700 nm. Since the visible range varies from about 1.77 to 3.1 eV, the visible regime
in electron-volts is much easier to recall for me. The plot was created in Mathematica. The
energy range in eV is much easier to remember (1.77 to 3.1 eV) than in Joules (2.84x10-19 to
4.97x10-19 Joules).
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MSE 410/ECE 340
Electrical Properties of Materials
School of Materials Science & Engineering
Fall 2016/Bill Knowlton
4. Bonding: The Lennard-Jones potential is one way to mathematically describe the bonding interaction
between two atoms. It provides the equilibrium bond length, ro, and the equilibrium bond energy, Eo.
a. Look up and write down the equation for the Lennard-Jones potential. Define all the variables. Cite the
reference where you obtained the Lennard-Jones potential.
b. Using a mathematical program (not Excel), plot the Leonard-Jones potential so that you obtain a graph
somewhat (but not exactly) similar to figure 1.10 in your book using the units of eV for energy and
nanometers for interatomic distance. Label your plot thoroughly including units, labeling ro, and labeling
Eo. Note: pay attention to the magnitude of energy and length in nanometers. If the values you obtain do
not make physically sense, then you probably made a mistake (usually with unit conversion).
c. For r > ro, why does your plot of energy never reach 0 or a positive value?
d. At a given E=f(r), describe the mathematical process to step by step find the bond energy and the bond
length of your plot. Mathematically, how could you ensure that ro was a minimum and not a maximum?
a.
A B
E (r ) =
− 6 + 12 (eV) where r is the interatomic distance and A and B are constants with
r
r
units that when divided by their respective denominators will provide the units of eV.
S.O. Kasap, “Principles of Electronic Materials & Devices, 3rd Edition, (McGraw-Hill, 2006)
page 23, example 1.4.
b. We take the approach that is shown in Example 1.4 in Kasap to use the Lennard-Jones
potential to find the van der Waals bond energy and length of solid Ar. We use the same
parameters as in the example problem. Unit conversion is key in this problem, otherwise we
would obtain physically unrealistic values for the bond length and energy. The problem is
solved using the following Mathematica code:
Clear En, r, M , B, m, , q , a, A
A
B
initializing variables
Assign Values
8 10 77 ;
A term in the L J potential; J m6
1.12 10 133
B term in the L J potential; J m12
Define Function
En r
:
r 10 9
Note that 10
A
nm
m
6
1.6022 10
9
r 10 9
19
B
nm
m
eV
J
12
;
;
Lennard Jones Potential or L J potential; p. 23 Kasap
is converting meters to nm & 1.6022 10
19
is to convert J to eV
Plotting E r in units of eV versus r in units of nm
Plot En r , r, .2, 1 , PlotRange
.1, .1 , Frame True,
RGBColor 1, 0, 0 , PlotLabel
GridLines Automatic, PlotStyle
"r nm ", "E r
eV "
FrameLabel
Zooming in on the plot
Plot En r , r, .3, .5 , PlotRange
0, .1 , Frame True,
RGBColor 1, 0, 0 , PlotLabel
GridLines Automatic, PlotStyle
"Solid Ar: Energy vs Interatomic Distance",
"Energy vs Atomic Radius, Enlarged", FrameLabel
"r
nm ", "E r
eV "
Solid Ar: Energy vs Interatomic Distance
0.10
eV
0.05
Er
0.00
0.05
0.10
Eo
0.2
ro
0.4
0.6
r nm
0.8
1.0
You were not required to do this, but to find the equilibrium bond length and bond energy
for solid Ar, we use the approach outlined in part d implementing the following Mathematica
6
MSE 410/ECE 340
Electrical Properties of Materials
School of Materials Science & Engineering
Fall 2016/Bill Knowlton
code and obtain the following solutions. Note that only one of the ro solutions is real while
the other solutions are imaginary. Using ro, we can plug that into En(ro) = Eo.
SolvingfortheBondLengthandBondEnergyUsingCalculustofindthelocalminimainrandEwhichisroandEo
Solving for the bond length by taking the derivative of the potential energy and setting
0, r
rosolns NSolve r En r
it equal to zero and solving for ro
ro r . rosolns 6 ; The only real positive solution is the 6th solution, so this command extracts the 6th solution
Printing out the solutions
", ro, " nm"
Print "Bond Length ro
", En ro , " eV"
Print "Bond energy Eo
r
0.375428 , r
Bond Length
ro
Bond energy
Eo
0.187714 0.32513
, r
0.187714 0.32513
, r
0.187714 0.32513
, r
0.187714 0.32513
, r
0.375428
0.375428 nm
0.0891631 eV
c. The reason that the plot of energy never reaches 0 or a positive value is because the
Lennard-Jones potential always assumes an interaction between Ar atoms. That is,
mathematically, the repulsion and attraction terms will never equal one another where r >
ro. One can prove this by setting Repulsive Term = Attractive Term and solving for r. One
will only obtain imaginary r values.
d. The mathematical process to step-by-step find the bond energy and the bond length of
the plot given the energy as a function of interatomic distance is given below:
Step one: We know that EN –versus- r has a minima at r = ro.
Step 2: Thus, we can take the derivative of E and set this to zero.
dEN
dr
= 0 ; That is, the slope is equal to zero.
r = ro
Step 3: Solve for r which is actually ro.
Step 4: Substitute ro into EN and this gives EN(ro). In essence, you are finding at what ro is
∑ Forces = 0 . That is, you are finding where molecular bond length is in
equilibrium.
To mathematically determine whether or not ro is a minimum and not a maximum,
one would take the second derivative of the energy equation, and substitute in ro.
If the number is negative, then it is concave down and is a maximum and if the
number is positive, then it is concave up and is a minimum.
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MSE 410/ECE 340
Electrical Properties of Materials
School of Materials Science & Engineering
Fall 2016/Bill Knowlton
5. Crystallography:: Showing all steps in your work including drawing a cubic unit cell, determine the Miller
indices for the following shown in the cubic unit cells. Show all work and label thoroughly (e.g., axes,
origin, unit lengths). Show all steps. You may use a table to show your steps.
a.
b.
Answers:
a.
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MSE 410/ECE 340
Electrical Properties of Materials
School of Materials Science & Engineering
Fall 2016/Bill Knowlton
b.
6. Crystallography: In your crystallography notes in table 1-3, one of the 14 space lattices is incorrect.
Which one is incorrect and why?
The last one (14), FCC, since not all faces have a lattice point.
7. Crystallography: List the two components necessary to describe a crystal structure.
crystal structure = bravais lattice + basis
or
crystal structure = space Lattice + motif
8. Defects: The formation energy of vacancies in Cu, Ag and Au is about 1 eV (p. 262 Kelly & Groves).
Calculate the equilibrium concentration of vacancies at 300 K in Cu. You will need to know the density and
AMU of the elements. What type of behavior does the plot of vacancy concentration versus 1/T exhibit?
Need bulk concentration, nbulk, for Cu which we can calculate using the density and atomic
mass of Cu with the help of dimensional analysis:
1 mole 6.0221023 atoms 8.96 g
nbulk =
8.49 ×1022 cm −3
×
×
=
63.546 g
1 mole
1 cm3
nv = nbulk
e
( )
− Ef
kT
− (1 )
= 8.49 ×10 cm
22
e
−3
(8.6174×10
−5
)
eV / K ( 300 K )
= 1.36 ×106 cm −3
The behavior exhibited by the plot of vacancy concentration versus 1/T is Arrhenius in
nature and follows the same trend as the plot diffusivity
9
MSE 410/ECE 340
Electrical Properties of Materials
School of Materials Science & Engineering
Fall 2016/Bill Knowlton
9. Defects: On the last page of the “Bulk Defects in Materials Notes”, cracks and pits are shown in GaN.
What is the rotational symmetry of the pits? Explain why you think the symmetry exists.
The pits are hexagonal in nature and thus have 6-fold or 3-fold rotational symmetry. The
reason is most likely because the crystal structure of GaN is based on the hexagonal
Bravais lattice and is Wurtzite.
10. Defects: Name and describe two of each type of defect listed in the following
a. Point defect
b. Line defect
c. Planar defect
Point defects: native point defects are defects that are intrinsic to the material such as
self interstitials and vacancies. Point defects are atoms and thus zero dimensional defects.
Line defects: dislocations are line defects described by the Burger’s (displacement) and line
(direction) vectors. An edge dislocation is a dislocation in which the Burger’s (displacement)
and line (direction) vectors are perpendicular while the Burger’s (displacement) and line
(direction) vectors are parallel for a screw dislocation.
Planar defects: defects that have an interface that is planar such as stacking faults and
grain boundaries are planar defects. Stacking faults are always bordered by partial
dislocations. Grain boundaries are the border between grains that form polycrystalline
materials and are highly defective containing point and line defects.
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