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MSE310/ECE 340: Electrical Properties of Materials Fall 2014

Department of Materials Science and Engineering Boise State University

Practice Midterm Exam

October 2014

Read the questions carefully Label all figures thoroughly Please circle your answers Make sure to include units with your answers Show all steps in your derivations to obtain full credit. Be thorough. Allowed: Pen/Pencil, Eraser, Equation sheet and Integral tables provided by Bill &

Donald (Calculator not allowed)

Problem Total Points Points Obtained Grand Total:

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Key to doing well on the exam:

Do all the practice exam problems correctly

o Show all steps – do not take short cuts

o Some students do the problems, but assume their solutions are correct but they are incorrect – check with your peers and with me

Redo all your quizzes (low average quizzes usually make it on the exam)

Review your problem sets and the solutions Attend Office hours and Discussion Sections for help If you do all of these things, you will do well on the exam

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REVIEW

CRYSTALLOGRAPHY AND BONDING:

1. Draw a conventional cubic unit cell and label all the features thoroughly. Within the unit

cell, draw 430 . Show each step of how you came to your answer. (10 pts)

2. Draw a conventional cubic unit cell and label all the features thoroughly. Within the unit

cell, draw 430 . Show each step of how you came to your answer. (10 pts)

3. Palladium (Pd) has the FCC crystal structure, has a density of 12.0 g cm-3 and an atomic mass of 106. 4 g mol-1. What is the atomic concentration, lattice parameter a, and atomic radius of palladium?

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THERMODYNAMICS:

4. Name the three distribution functions used in statistical thermodynamics and their corresponding particles that follow their distribution. Write down each distribution.

5. Use both explanations and mathematical equations to describe the three conditions required for equilibrium. Use the Fundamental Equation of Thermodynamics (FEOT) to help you.

DEFECTS, DIFFUSION and THERMALLY ACTIVATED PROCESSES:

6. What is the driving force for diffusion? Explain.

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7. You are studying diffusion of a new dopant, boreium (Bo), in Si for potential application in integrated circuits. You perform experiments at various temperatures and determine the diffusivity at these temperatures. The data you have acquired are plotted below.

5 6 7 8 9 1010-15

10-14

10-13

10-12

10-11

10-10

10-9

10-8

Diff

usiv

ity (

cm2/s

)

104/T

a. What type of behavior does the data in the plot exhibit? (2 points)

Using the plot above, determine the following parameters for Si diffusion in the novel material:

b. Activation energy in units of eV. (10 points)

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c. Pre-exponential in units of cm2/s. (8 points)

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8. The data below tabulates the vacancy concentration in Cu for several temperatures. Use the data to answer the questions below while showing all steps of your work. Hint: what is the minimum number of points to determine a line and thus a slope?

Table 1: Vacancy concentration versus temperature. Vacancy Concentration

(cm-3) Temperature (K)

1/T (K-1)

1x105 421.6 0.00237

1x1012 691.7 0.00145

1x1016 1091 0.000917

From the data, determine the following while showing all work:

a. Activation energy in units of eV. Comment on your answer. (10 points)

b. Pre-exponential in units of cm-3. (8 points)

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CONDUCTION IN SOLIDS: FORCE-ENERGY RELATIONSHIPS

9. What is the mathematical relationship between force and energy? Define all variables (5 pts)

10. What is the mathematical relationship between energy and voltage (assume that voltage and electric field are static, i.e., non-varying)? Define all variables (5 pts)

11. What is the mathematical relationship between force and electric field (assume that voltage and electric field are static, i.e., non-varying)? Define all variables (5 pts)

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12. The resistivity of a metal is 1 at a temperature T1 and 2 at a temperature T2. The TCR was determined at a temperature To and is constant over the temperature range of the resistivities given. Derive a formula for the TCR as a function of 1, T1, 2, and T2 but not o. Show all work. (10 points)

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13. The electrons in a 1 m Ga wire with a diameter of 1 mm have the following properties: a

trivalent metal, drift velocity is 6x103 m/s, the mobility is 35 2cm

V s and the mean thermal

energy is 10 meV. Determine the following additional properties of the electrons. Assume the Drude model. Note: AMUGa = 69.72 g/mol; DensityGa = 5.91 g/cm3

a. Determine the temperature of the electrons in units of K.

b. Determine the electric field experienced by the electrons in units of MV/cm.

c. Determine the mean time between electron collisions in units of seconds.

d. Determine the diffusivity of electrons in units of cm2/s.

e. Show how to determine n using two different approaches.

f. Determine the conductivity in units of (ohm cm)^-1.

g. Calculate the thermal conductivity of Ga at 0 C.

h. Calculate the thermal resistance of Ga at 0 C.

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Extra work space:

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14. Determine Nordheim’s coefficient from the plot below. The points are data while the line is a best linear fit. Show all work.

0.00 0.05 0.10 0.15 0.20 0.250

20

40

60

80

100

120

140

160

Tr in Zp: Data Linear RegressionR

esid

ual R

esis

tivi

ty (

n m

)

X(1-X)

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15. A plot of thermal conductivity versus electrical conductivity data is given below in which the data is fit and an R2 of 0.992 is obtain. Extract CWFL from the data. Comment on the goodness of fit. Discuss the physical validity of the fit. You must determine some quantitative assessment of the physical validity of the fit.

Sb

Pb

Pt

Fe

T = 300 K

0 2 μ 106 4 μ 106 6 μ 106 8 μ 106 1 μ 107

0

20

40

60

80

Electrical conductivity, 1

.m

Ther

mal

cond

uctiv

ity,

W K.m

Line: fit; Scatter plot: data

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16. Is the Reynolds and Hough rule (equation [2.28]) used for a single phase material system or a two phase materials system?

17. State the two special conditions or cases for which the equation for the Reynolds and Hough rule can be reduced.

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MODERN PHYSICS and QUANTUM MECHANICS: 18. Sketch the spectral irradiance versus energy for three different temperatures of a material

emitting blackbody radiation. Label your plot thoroughly and show the trend in temperature. What was the significance of Planck’s ability to fit his blackbody radiation data?

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19. Table 1 shows the photoemission data from a photo electron effect experiment. Determine Planck’s constant and the work function from the data. Show all work.

Table 1: Kinetic energy as a function of frequency. Kinetic Energy (eV) Frequency (s-1)

0.19 8.20x1014

1.35 1.10x1015

2.18 1.30x1015

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20. Evidently, a new wave function, called the hyper spectral wave function, has recently been discovered for the hydrogen atom. The wave function is given by:

2( , , ) o

r

ar r Sin Cos e

.

Assume the wave function is not normalized. The integration table is given below.

2

2

12

10

2

2

2

2

3

110

2

Integration Table:

1 for = even

!

!

2

( )

Cos2

2 2

4 8 42

2 43

( )d

xnan

n axn

n axn

nx e dx a n

nn

x e dxa

nx e dx

a

Cos bxSin bx bx

bCos bx xSin bxx

b bSin b

Sin bx x

xx

b

Sin

Sin bx dx

b

Si

x

n b

x

x dx

3

4 12

Cos bx Cos bx

b b

a. Normalize the wave function. (10 points)

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b. Find the eigenvalue of the momentum. (5 points)

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c. Find the expectation value of the momentum. (15 points)

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Extra work space:

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21. For the particle in a box problem where the box length is a and V = 0 inside the box and is infinite everywhere else and the wave function is given by:

( ) ikx ikxx Ae Be

For this problem, you are given:

2

22

2

2 222 2

2 3

2

2 42 2

4 8 4

2 ( 1 2 ) 2

6 4 8

2

Sin bxxSin bx dx

bCos bx xSin bxx

xSin bx dxb b

xCos bx b x Sin bxxx Sin bx dx

b bCos bx

Sin bx Cos bx dxb

a. Using boundary conditions and solving for coefficients, show that ( )x can be written

as in terms of the Sin function.

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b. Determine the energy, En, of the particle in the box. Is the energy an eigenvalue?

c. Determine En in terms of n and a.

d. Describe what happens to E and E as a increases and decreases.

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e. Normalize the wave function to obtain: 2( )

nx i Sin x

a a

f. Find <H> in the box.

g. Find <p2>

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Extra work space:

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22. For the following two barrier problems:

Barrier 1: Infinite potential barrier

Barrier 2: Finite potential barrier

Do the following a. Draw the barrier with all labels on the x-axis and y-axis

b. Write down the wave functions in each region for both cases:

i. E > Vo

ii. E < Vo

c. Write down the Energies in each region for both cases:

i. E > Vo

ii. E < Vo

d. Write down the k’s in each region for both cases:

i. E > Vo

ii. E < Vo

e. Write down the boundary conditions.

f. Write down the equations and show how one would solve for T(E) using the probability current densities, J's and show what ratio of coefficients should be obtained.

g. Describe in detail and map out a path by using the boundary conditions of how one would obtain the ratio of coefficients to solve for T(E).

h. Why is T(E) important to know in the field of electronic devices? Use mathematical relationships in your arguments.

i. Solve for T & R for the case of E > Vo using the appropriate J’s (i.e., wave functions). You do not need to solve the coefficients in terms of k’s.

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Extra work space:

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23. For the finite barrier problem where Ee-<Vo, solve for T(E) using the probability current densities, J's, to show that T(E) is a ratio of wave function coefficients. Shall all steps. (4 points)

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24. The following questions are based on the Kronig-Penney Model. :

a. Sketch the plot that results from the Kronig-Penney model.

b. Describe the implications of the Kronig-Penney Model

c. State at least 3 names for the type of plot the Kronig-Penney Model represents.

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25. The Kronig-Penney Model is based on a 1 dimensional (1D) crystal structure (lattice + basis) in which each lattice point is decorated with one atom (basis). The spacing between the atoms (or lattice points) we defined as ao = a + b. For a 3D lattice, we typically define the lattice spacing as the d-spacing. For a cubic Bravais lattice, the d-spacing is defined by

2 2 2

ad

h k l

where (h k l) are the Miller indices for a given plane (Note: the k here is not the same as the k-space). Hence, ao is the d-spacing between atoms. We also defined the first Brillouin Zone where the wave vector (i.e., k, or reciprocal lattice vector, or k-space vector) is given

by 2

o

ka

. Derive an equation for the wave vector, k, as a function of the d-spacing and

the lattice constant a to determine the k value for the Brillouin zone at the X-point (i.e., (100)), L-point (i.e., 111), and K-point (i.e., (110) ) of a Brillouin zone for a FCC lattice.

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26. A tight binding model for a particular semiconductor describes the conduction band as E(k)cb=2k2+4. Determine equations for the group velocity and the effective mass for an electron in the conduction band of the semiconductor. Do not forget to include units.

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27. Why are the group velocity and effective mass important parameters to know in the field of electronic devices? Use mathematical relationships including but not limited to device current density in your arguments.

28. Draw and thoroughly label a E versus k diagram for the conditions below. Include (but not limited to) the X, L and gamma points, heavy and light hole valence bands, conduction band, etc.

a. A direct band gap semiconductor and an indirect band gap semiconductor.

b. A direct band gap semiconductor with a large effective mass

c. A direct band gap semiconductor with a small effective mass

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29. Draw and thoroughly label the energy band diagram in real space of a conductor, semiconductor and nonconductor.

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30. What does the chemical potential have to do with the energy band diagrams in the previous question? Explain.