Download EE 3311 Semiconductor Devices

EE 3311 Semiconductor Devices
Agendas and Homework Assignments
list for TA's for 8 lab session and number of student in each lab.
----------------------------------------------------------------------------------------------------------------------------Group 1: Tuesday 8:00am to 10:00am
TA's: Christina & Nikunj
No. of Students: 4
Group 3: Tuesday 12:30pm to 2:30pm
TA's: Nilay & Freddie
No. of Students: 4
Group 4: Tuesday 3:30pm to 5:30pm
TA's: Nilay & Freddie
No. of Students: 4
Group 5: Wednesday 8:00am to 10:00am
TA's: Christina & Sudip
No. of Students: 4
Group 6: Wednesday 10:00am to 12:00pm TA's: Nilay & Sudip
No. of Students: 4
Group 7: Wednesday 3:30pm to 5:30pm
TA's: Freddie & Nikunj
No. of Students: 4
Group 8: Thursday 8:00am to 10:00am
TA's: Nikunj & Sudip
No. of Students: 3
Group 11: Thursday 3:30pm to 5:30pm
TA's: Nilay & Freddie
No. of Students: 4
------------------------------------------------------------------------------------------------------------------------------Here is the list of all students for 8 lab session.
------------------------------------------------------------------------------------------------------------------------------Group 1: Tuesday 8:00am to 10:00am
Group 3: Tuesday 12:30pm to 2:30pm
Group 4: Tuesday 3:30pm to 5:30pm
Group 5: Wednesday 8:00am to 10:00am
Group 6: Wednesday 10:00am to 12:00pm
Group 7: Wednesday 3:30pm to 5:30pm
Group 8: Thursday 8:00am to 10:00am
Group 11: Thursday 3:30pm to 5:30pm
John Wensowitch, Charles Viviano, Parker Holloway, William Templeton
Danton Zhao, Camille Aucoin, Rami Abdelqader, Charles Stevens
Clayton Matthie, Parker Meinecke, Arianna Santiago, Serdar Halac
Allison Garcia, Sam Coday, Matthew Smallwood, Olivia Schmid
Omar Dominguez, Vivian Trinh, Graciela Garcia, Jose De Labra
Andrew Campbell, Amaan Charania, Tara Kezic, John Gilchrist
Student's Name: Odai Abdelqader, Seok Hee Lee, Roderick Brogan
Student's Name:
Osaid Jaffery, Laura Hatcher, Jenna Kleiman, Krystle Vela
---------------------------------------------------------------------------------------------------------------------------------
Monday, August 24, 2015
Course Philosophy
x1. Review Syllabus (this course may not be for you…)
I have two major objectives for students taking this course:
A. Make working p-channel MOSFET devices. Understand the fabrication process.
Understand the concepts of photolithography, oxidation, diffusion, etching and metallization.
B. Understand how basic semiconductor devices work—in particular how a p-n junction and
a MOSFET transistor works.
x2. Lab and Lab Safety (J. Kirk), Safety Quiz (see Blackboard) and Material Safety Data Sheets
(MSDS)
Our plan is to have a maximum of 4 students in each group. For safety reasons, you cannot
enter the clean room without appropriate dress: long pants, covered torso and shoes that
completely cover your feet (no open toes, no sandals).
SAFETY QUIZ MUST BE UPLOADED TO BLACKBOARD BY FRIDAY, MIDNIGHT,
AUGUST 28th, 2015.
x3. Submit your first, second and third choices for lab times by email to Nilay Shah
([email protected]) before noon on Wednesday August 26th from the list below.
Lab Time Possibilities (from which we will choose 8 groups)
Group 1: Tuesday, 8:00 am - 10:00 am
Group 2: Tuesday 10:00 am - 12:00 pm
Group 3: Tuesday, 12:30 pm - 2:30 pm
Group 4: Tuesday, 3:30 pm - 5:30 pm
Group 5: Wednesday, 8 am - 10 am
Group 6: Wednesday, 10 am to 12: pm
Group 7: Wednesday, 3:30 pm - 5:30 pm
Group 8: Thursday, 8:00 am - 10:00 am
Group 9: Thursday, 10:00 am - 12:00 pm
Group 10: Thursday, 12:00 pm – 2:00 pm
Group 11: Thursday, 3:30 - 5:30 pm
x4. Plagiarism: Not just an ethical question. The internet makes plagiarism easy—and the
internet makes it even easier to detect. Government funding agencies routinely check proposals
and reports for plagiarism. Penalties for plagiarism include banning individuals, institutions and
corporations from applying for grants and contracts.
x5. Teaching Baseball or Engineering? “this is the course I would have liked to have had as a
student”
--Old 3311 (all theory) and New 3311 (lab)
--discrete college courses and Boeing 727s…
--New 3311 brings together most everything you have had in all your classes: chemistry,
physics, advanced math, circuits, materials, (thermodynamics, heat transfer, quantum
mechanics, …)
Historical Background (Dallas and SMU related)
x6.
Jack St. Clair Kilby (November 8, 1923 - June 20, 2005) won the Nobel Prize (with
Herbert Kroemer and Zhores l. Alferov for heterojunctions) in physics in 2000 for his
invention of the integrated circuit during the summer of 1958 at Texas Instruments. He also
is credited with the invention of the handheld calculator and thermal printer. TI Archives and
the Jack St. Clair Kilby Archives are at the DeGolyer Library at Southern Methodist
University.
x7.
DRAM inventor Robert Denard received an honorary doctorate from SMU in 1997. He
received his BSEE and MSEE from SMU in 1954 and 1956.
x8. LED inventors Bob Biard and Gary Pittman--Bob Biard received an honorary doctorate from
SMU at the May 2013 graduation ceremony. Gary Pittman could not attend because of poor
health. Pittman received his BS in Chemistry from SMU in 1954. Gary Pittman died
October, 2013.
x9. Looking Backwards (mechanical logic and computers to microprocessors) and Forward
(silicon photonics): History and Future of Computing
(HistoryAndFutureComputingF2014.ppt on Blackboard)
10. JaegerCh01overview2013 powerpoint slides (JaegerCh01overview2013.ppt on Blackboard)
(stopped at discussion of 1 T+ transistors/chip on 8/24)
11. How are we going to make a MOSFET?
--Overview of Fabrication Process (MOSFETFab_Fall2014.ppt or most recent posted)
12. Basics of Materials: MaterialsBasicsAr12015.ppt (stopped on slide 27 on 8/26)
13. What do we need to know to understand how to make a MOSFET?
--thermal oxidation of silicon
--lithography and photoresist
--etching of oxide, metals (and semiconductors)
--diffusion (and ion implantation) of dopants
--testing, packaging
Wednesday, August 26, 2015
Agenda
1. Lab sessions (tentatively) start Tuesday, September 1 at 8:30 AM (maybe?)
2. SAFETY QUIZ MUST BE UPLOADED TO BLACKBOARD BY FRIDAY, MIDNIGHT,
AUGUST 28th, 2015. Safety related articles are on Blackboard in the Safety Information folder
under Course Documents.
3. Jay Kirk, Lab Instructor—Safety and Laboratory Protocol emphasis on Monday.
4. Lab Tours (maybe during first lab)
--Safety, Safety Quiz and Material Safety Data Sheets (MSDS)
x5. JaegerCh01overview2015 powerpoint slides (JaegerCh01overview2015.ppt on Blackboard)
(completed)
x6. Overview of Fabrication Process (MOSFETFab_Fall2014.ppt or most recent posted)
review 3rd week of class in detail.
7. Basics of Materials: MaterialsBasicsAr12015.ppt (stopped on slide 27 on 8/26)
8. What do we need to know to understand how a MOSFET works?
--electrons and holes in semiconductors, doping of semiconductors
--p-n junctions
--resistivity
9. What do we need to know and understand to make a MOSFET?
--thermal oxidation of silicon
--lithography and photoresist
--etching of oxide (dielectrics), metals, and semiconductors
--diffusion (and ion implantation) of dopants
--testing, packaging
Monday, August 31, 2015
Agenda
1. Lab sessions (tentatively) start Tuesday, September 1 at 8:30 AM
list for TA's for 8 lab session and number of student in each lab.
-----------------------------------------------------------------------------------------------------------------------------
Group 1: Tuesday 8:00am to 10:00am
TA's: Christina & Nikunj
Group 3: Tuesday 12:30pm to 2:30pm
TA's: Nilay & Freddie
Group 4: Tuesday 3:30pm to 5:30pm
TA's: Nilay & Freddie
Group 5: Wednesday 8:00am to 10:00am
TA's: Christina & Sudip
Group 6: Wednesday 10:00am to 12:00pm TA's: Nilay & Sudip
Group 7: Wednesday 3:30pm to 5:30pm
TA's: Freddie & Nikunj
Group 8: Thursday 8:00am to 10:00am
TA's: Nikunj & Sudip
Group 11: Thursday 3:30pm to 5:30pm
TA's: Nilay & Freddie
No. of Students: 4
No. of Students: 4
No. of Students: 4
No. of Students: 4
No. of Students: 4
No. of Students: 4
No. of Students: 3
No. of Students: 4
------------------------------------------------------------------------------------------------------------------------------Here is the list of all students for 8 lab session.
------------------------------------------------------------------------------------------------------------------------------Group 1: Tuesday 8:00am to 10:00am
Group 3: Tuesday 12:30pm to 2:30pm
Group 4: Tuesday 3:30pm to 5:30pm
Group 5: Wednesday 8:00am to 10:00am
Group 6: Wednesday 10:00am to 12:00pm
Group 7: Wednesday 3:30pm to 5:30pm
Group 8: Thursday 8:00am to 10:00am
Group 11: Thursday 3:30pm to 5:30pm
John Wensowitch, Charles Viviano, Parker Holloway, William Templeton
Danton Zhao, Camille Aucoin, Rami Abdelqader, Charles Stevens
Clayton Matthie, Parker Meinecke, Arianna Santiago, Serdar Halac
Allison Garcia, Sam Coday, Matthew Smallwood, Olivia Schmid
Omar Dominguez, Vivian Trinh, Graciela Garcia, Jose De Labra
Andrew Campbell, Amaan Charania, Tara Kezic, John Gilchrist
Student's Name: Odai Abdelqader, Seok Hee Lee, Roderick Brogan
Student's Name:
Osaid Jaffery, Laura Hatcher, Jenna Kleiman, Krystle Vela
---------------------------------------------------------------------------------------------------------------------------------
2. HW#1 due Tuesday, September 1st before 11:59 PM. Review of HW #1: Odai Abdelqader
3. SAFETY QUIZ SHOULD HAVE BEEN UPLOADED TO BLACKBOARD BY FRIDAY,
MIDNIGHT, AUGUST 28th, 2015. Safety related articles are on Blackboard in the Safety
Information folder under Course Documents.
4. Jay Kirk, Lab Instructor—Safety and Laboratory Protocol emphasis on Monday.
5. Lab Tours (maybe during first lab)
--Safety, Safety Quiz and Material Safety Data Sheets (MSDS)
x6. JaegerCh01overview2015 powerpoint slides (JaegerCh01overview2015.ppt on Blackboard)
(stopped at discussion of 1 T+ transistors/chip on 8/24; continued to completion; completed
MOSFET fab traveler; began Basic Materials)
x7. Overview of MOSFET Fabrication Process (MOSFETFab_Fall2014.ppt or most recent
posted)
8. Basics of Materials: MaterialsBasicsAr12015.ppt (stopped on slide 27 on 8/26)
9. What do we need to know to understand how a MOSFET works?
--electrons and holes in semiconductors, doping of semiconductors
--p-n junctions
--resistivity
10. What do we need to know and understand to make a MOSFET?
--thermal oxidation of silicon
--lithography and photoresist
--etching of oxide (dielectrics), metals, and semiconductors
--diffusion (and ion implantation) of dopants
--testing, packaging
1.
2.
3.
4.
Wednesday, September 2, 2015
Agenda
Professor Ken Springer may (?) talk to the class on Wednesday September 9 about
Writing and Lab Reports
Review of HW #1: Odai Abdelqader
JaegerLithographyCh2r1 powerpoint slides (on blackboard)
JaegerOxidationCh03r2 powerpoint slides (on blackboard)
Monday, September 7, 2015 (no class)
No Agenda
Wednesday, September 9, 2015
Agenda
1. HW #2 discussion: Camille Aucoin; Lab 1 discussion: Rami Abdelqader
2. JaegerLithographyCh2r1 powerpoint slides (on blackboard)
3. JaegerOxidationCh03r2 powerpoint slides (on blackboard)
Monday, September 14, 2015
Agenda
1. JaegerLithographyCh2r1 powerpoint slides (on Blackboard) (S
2. JaegerOxidationCh03r2 powerpoint slides (on Blackboard)
3. Jaeger_diffusionCh04r4 powerpoint slides (on Blackboard)
Wednesday, September 16, 2015
Agenda
1. HW#3 discussion: Andrew Campbell; Lab 2 discussion: Roderick Brogan
2. JaegerOxidationCh03r2 powerpoint slides (on Blackboard)
3. Jaeger_diffusionCh04r4 powerpoint slides (on Blackboard)
Homework Assignment #1
(due Tuesday, September 1st, 2015)
1. In your own words, state the format requirements for homework assignments. What
will happen if homework is turned in that does not satisfy the format requirements?
2. How many homework assignments must be completed and turned in to pass this
course? How many laboratory reports must be completed and turned in to pass this
course?
3. Read Chapter 1 of Introduction to Microelectronic Fabrication. How many inches is a)
100 mm? b) 200 mm? c) 300 mm? d) 400 mm? How many millimeters equal 12
inches?
4. How many complete 1mm x 1mm die can fit on a 100 mm wafer? How many complete 1
cm x 1 cm die can fit on a 300 mm wafer?
5. Determine the cost per good die for a wafer processing cost of $50,000 and a yield of 95%
for a) a 1mm x 1 mm die size and wafer size of 100 mm? b) for a 2 cm x 2 cm die size and
a wafer size of 300 mm?
6. What is the chemical make up of photoresist? What safety precautions should be used
with photoresist? How and why is photoresist used?
7. Assume that you have a thousand dollars that accrues interest of 10%/time period over
15 time periods. (How many weeks are in this semester?) What is the accrued value of
the thousand dollars after 15 time periods? How much better off will you be if you work
10% harder each week of this semester?
8. Write a summary of the article “On Physics Education in Brazil,” from the book Surely
You're Joking, Mr. Feynman! (Adventures of a Curious Character), by Ralph Leighton and Edward
Hutchings, 1985. This article can be found on Blackboard under Course Documents or on numerous
sites online.
Homework Assignment #2
(due Tuesday, September 8th, 2015)
1. Evaluate the following integrals:
For problems 2 and 3 recall that the del operator is given by
2. Calculate the divergence (
3. Calculate the curl (
) of the following vectors:
) of the following functions:
4. Below is a list of Maxwell’s Equations in differential form:
One of those equations states that the divergence of the electric displacement vector
D(x,y,z,t) is equal to the charge density (x,y,z,t)—this equation is known as Gauss’s
Law for the electric displacement vector. For problems in space where there is only a
variation in the x direction, show that this equation reduces to the derivative of the
electric field E(x) with respect to x is equal to the charge density divided by the electric
permittivity:
5. Show that the units of ∇ • 𝐷 = 𝜌 are the same on both sides of the equation.
6. The speed of light c = 1/ m oe o , where e o is the permittivity of free space and m o is the
permeability of free space. Show that the units of 1/ m oe o are length/time (or meters/sec).
7. (20 points) What is the oxide thickness on the Si wafer that you are processing? At what
temperature was the oxide be deposited? How long is the deposition process expected to take?
Was the oxide be deposited by a wet or dry process? Using the formulas in your text book
(Jaeger), calculate the theoretical thickness of the first oxide layer grown in the lab. Check your
calculations with the figures in the text book showing oxide thickness as a function of time for
various temperatures.
More about problem #1: See step 1-2 in the traveler, the initial oxidation (Offline)
1000 °C in steam, ~ 5 Hours-target 10,000Å (1 µm). This means that the <100> wafer is put in
the furnace for a 1000 C (wet) for ~ 5 hours. The formula for oxide thickness as a function of
time X(t) is Eq. 3.9 in Jaeger. The oxidation time t is ~ 5 hours. To use Eq. 3.9, you need to
convert Centigrade to Kelvin and you need to find A and B. You can find A and B from Table
4.1 if you know the wafer orientation (100 for this class) and you know if the oxide is deposited
with steam (wet) or dry. So for this problem, you will use an activation energy of 2.05 eV and a
diffusion coefficient of 9.7 x 107 µm/hr to find B/A. That is, B/A = Doexp(-Ea/kT) = (9.7 x 107
µm/hr)exp(-2.05 eV/kT), where T is in Kelvin and k is Boltzman’s constant. After you calculate
B/A, look at Figure 3.5 and see if your calculation agrees with the appropriate curve.
Next you need to find B. From Table 3.1, B = Doexp(-Ea/kT), where the activation energy is
0.78 eV and Do is 386 µm2/hr. After you calculate B, look at Figure 3.4 and see if your
calculation agrees with the curve for H20.
The sentence under Eq. 3.8 in the text explains how to calculate . Explain why for this
particular calculation you can assume no initial oxide thickness. (Note that a freshly cleaned Si
wafer will develop a very thin (~1 to 4 nm) layer of oxide just from being exposed to air.)
Check your final answer for the field oxide thickness with Fig. 3.6.
8. (20 points) In a later laboratory session, the gate oxide for the MOSFET device will be grown
offline. What is the expected thickness of the gate oxide? Will the gate oxide be grown by a
wet or dry process? Why? Using the time and temperature in the MOSFET traveler, calculate
the gate oxide thickness. Check your calculations with the figures in the text book showing oxide
thickness as a function of time for various temperatures. For best theoretical agreement with
experiment, an initial oxide thickness of 25 nm should be used in the calculation—even though
the initial thickness in reality is likely closer to 2.5 nm…
Homework Assignment #3
(due Tuesday, September 15th, 2015)
1. (30 pts) During the processing of your wafer, there are three times that the wafer
is oxidized: a) step 1.2 (in the traveler) which results in a thick field oxide of
about 1 micron; b) a second oxide growth during the boron diffusion (step 2.7 in
the traveler), which provides about 0.5 microns of oxide in the source and drain
region; and c) a final oxide growth for the gate (step 3-17 in the traveler) which
provides about 0.08 microns of oxide in the gate region. Using the time and
temperatures given in the three different oxide growths, calculate the total
thickness of the oxide that surrounds the source and drain regions. Note: The
oxide surrounding the source and drain regions is not ever etched away—and only
becomes thicker with each successive oxide growth that occurs. Check your
numerical calculations with the appropriate figures in the text.
Here is one way to do this problem:
i) You have calculated the oxide thickness outside the source and drain after step 1.2 in
the traveler in HW#2.
ii) Use the (correctly calculated!) thickness (~ 1 micron) of the oxide calculated in HW
#2 for the value of Xi in calculating the total thickness of the oxide that surrounds the
source and drain regions after step 2.7. In other words, use Xi ~ 1.0 micron in the
equation
t=
Xi2
Xi
. Of course, you need to use the values for B and B/A that are appropriate
+
B ( B A)
for the oxide growth of step 2.7. Once you have , then you can use
0.5
éì
ù
ü
B
Xo ( t ) = 0.5A êí1+ 4 2 ( t + t )ý -1ú to find the total thickness of the oxide outside the source and gate
þ
A
êëî
úû
(after step 2.7), again using the appropriate values of B and B/A.
iii) To complete this problem, you need the total thickness of the oxide after step 3-17. Now you need
to calculate  again, but this time you need to use the total thickness Xo that you calculated in step ii) for
Xi. And of course you need to use the appropriate values of B and B/A in this second calculation. The
result of this step is the answer for the total oxide thickness outside the source and drain. You will see
that this answer is different from the answer you would get by (wrongly!) adding the thickness of the
first, second and third oxide growths.
2. In calculating the thickness of thermal oxides, there are two different activation energies and
two different diffusion terms for B and B/A, suggesting that two different physical processes are
involved. In Table 3-1 (Jaeger), we see that for one of the coefficients, the activation energies
and diffusion coefficients do not depend on the orientation of the wafer, while for the other
coefficient, the activation energies and diffusion coefficient do depend on the wafer orientation.
What are some possible mechanisms that might explain the dependence or lack of dependence on
wafer orientation? (Note: the number of Si atoms/cm2 at the surface of a silicon wafer depends
on the orientation
3. Read the article by Michael Riordan called “The Silicon Dioxide Solution, (How physicist
Jean Hoerni built the bridge from the transistor to the integrated circuit),” IEEE Spectrum,
December, 2007 (available online at http://spectrum.ieee.org/semiconductors/design/the-silicondioxide-solution or on Blackboard under Course Documents as SiO2_history.pdf).
Write a summary of this article in WORD that includes the advantages and repercusions of
Hoerni’s idea of leaving the oxide on the wafer.
Homework Assignment #4
(due Tuesday, September 22nd, 2015)
For Problems 1 through 4, review Chapter 4 of the (Jaeger) textbook and the slides on diffusion,
which are posted on Blackboard.
1. Step 2-4 in the MOSFET process traveler is (mostly) a constant source diffusion. For this
case, the solution to Eq. 4.3 in the textbook is given by Eq. 4.4. Calculate the theoretical
diffusion depth of step 2-4 using formulas and graphs in the textbook. Assume that the
surface concentration of this diffusion is equal to the solid solubility limit, which can be obtained
from Fig. 4.6 in the text. You will need to know the background doping of your wafer. The
resistivity of the n-doped substrates used this semester will be obtained in a later lab, but when
we ordered the substrates, we specified that the resistivity be between 1 and 10 ohm-cm. From
past measurements, the resistivity is between 5 and 6 ohm-cm. Using 5.5 ohm-cm as the
resistivity, you can find the background doping from Fig. 4.8 in the text. Now you can set the
constant background doping (NB) equal to the impurity concentration N(x) which is given by Eq.
4.4. The value xj that makes N(xj) = NB is the location of the metalurgical junction or the
location of the p-n junction. Note that you have to calculate a diffusion coefficient D using
Table 1 from the text. Equation 4.4 contains a function called the complimentary error function,
which is plotted in Fig. 4.4.
2. Step 2-7 in the MOSFET process traveler is a limited-source diffusion. For this case, the
solution to Eq. 4.3 in the (Jaeger) textbook is given by Eq. 4.6. Calculate the theoretical depth
of this step using formulas and graphs in the textbook. A limited-source diffusion assumes
an impulse function representing the total number of boron atoms/cm2 (deposited in step 2-4 of
the MOSFET process traveler) at the silicon surface. The magnitude of this impulse is related to
the total dose (Q) and is given by Eq. 4.5 in the text.
In problem 1 you determined the resistivity and the background doping of your wafer.
Now you can set the constant background doping (NB) equal to the impurity concentration N(x)
which is given by Eq. 4.6. The value xj that makes N(xj) = NB is called the metalurgical junction
or the location of the p-n junction. Note that you have to calculate a diffusion coefficient D
using Table 1 from the text. Equation 4.6 has a Gaussian dependence, which is also shown in
Fig. 4.4.
3. For this problem it helps to read section 4.2.3 in Jaeger. Calculate the Dt products for the
diffusions in Problem 1 and Problem 2 and compare them. What can you say about the final
profile and final depth of the junction after the diffusion steps have been performed according to
the traveler. (Slide 19 of the Diffusion power point slides also addresses this issue.)
4. Show that the expression on the extreme right hand side of Eq. 4.5 in the (Jaeger) text has
units of atoms/cm2.
Homework Assignment #5
(due Tuesday, September 29th, 2015)
1. Read the article “The Accidental Entrepreneur,” by Gordon Moore. Write a summary of this
article in WORD.
2. Calculate the number of Si atoms per cubic centimeter. You can do this by considering the
diamond structure (Fig. 1-8 of the "DiamondLatticeAndSilicon.pdf " notes) as the unit cell. The
lattice constant of silicon is 5.43 Angstroms at room temperature (see Fig. 1-8 again). You also
need to know how many Si atoms are in the diamond unit cell. With a little (or considerable)
thought, you will find that each corner atom contributes 1/8th of an atom/cell (why?). And each
face atom contributes (1/8th?, 1/4th? or ½?) of an atom/cell. And each interior atom contributes
1 atom/unit cell. With this information, you can now calculate the number of Si atoms per cubic
centimeter.
3. Calculate the density of silicon. Hopefully you still have your first year college chemistry
textbook. If so, you will find that
Density = (# of atoms/cm3)*(number of grams/mole)/(number of atoms/mole)
The first term is what you calculated in Problem 1. You can find the number of grams/mole
from looking at Si in the periodic table. The number of atoms/mole is Avagodro's number. If
you do everything right, you should get an answer of about 2 grams/cm3.
4. Find the number of Si atoms/cm2 on the (100) surface of silicon. (Read the appropriate
chapters of Streetman for an understanding of the nomenclature for specifying crystal surfaces
and directions.)
5. Find the number of Si atoms/cm2 on the (110) surface of silicon. (Again, read the appropriate
chapters of Streetman for an understanding of the nomenclature for specifying crystal surfaces
and directions.)
6. Show that the complimentary error function solution to the diffusion equation satisfies the
required boundary conditions. (Note: for problems 6 and 7, see the powerpoint slides on
diffusion.)
7. Show that the gaussian function solution to the diffusion equation satisfies the required
boundary conditions.