physics 281, fall semester, 2011 principles of solid state ...msg.byu.edu/281/281pack.pdf ·...

PHYSICS 281, FALL SEMESTER, 2011

PRINCIPLES OF SOLID STATE PHYSICS

Instructor: Gus Hart, N249 ESC, 422-7444e-mail: [email protected] home page: msg.byu.edu/281.php

Textbook: Solid State Physics, by Harold T. Stokes.

Identification Number: Each of you will receive a personal identification number forthis course. The purpose of this number is to protect your privacy. You will put thisnumber on all homework assignments, labs, and exams. These items will be returned toyou sorted by this number in the bins outside N375 ESC. Most of you will receive this IDnumber by e-mail before the first day of classes. If you have not received it by the first dayof classes, you can obtain your identification number over the internet. Go to our coursehome page and click on the link “Obtain your class ID number.”

Reading: Reading assignments are shown on the course schedule. You should completethis assignment before coming to class.

Warm-Up Exercises: There will be a warm-up exercise for each reading assignment.You should work the exercise after completing the reading assignment. The exercises willbe available at our course homepage on the internet. You will submit your responses toeach exercise over the internet. Each exercise will be due at 11:00 am, two hours beforeclass.

You will not be allowed to make up a missed warm-up exercise for any reason. However,three of your scores with the most missed points will be changed to perfect scores. Warm-up exercises are worth 5% of your final grade.

Quizzes: Multiple-choice quiz questions will be given throughout each lecture. You willanswer the questions electronically using a transmitter available at the bookstore. On thereverse side of the transmitter is an alphanumeric ID code for your transmitter. Go toour course homepage on the internet and register your transmitter ID number. We needthis information in order to give you credit for quizzes. If you have a used clicker with anillegible ID code, come see me as soon as possible.

You will not be allowed to make up a missed quiz for any reason (tardy, absent, forgottransmitter, etc.). However, three of your daily scores with the most missed points will bechanged to perfect scores. Quizzes are worth 5% of your final grade.

Homework: Homework assignments will be submitted over the internet. You will findlinks on the course home page. These assignments are due at 12:50 pm (a few minutesbefore class) on the days indicated on the schedule. You will find additional informationabout these homework problems on a sheet in this packet. You are encouraged to work ingroups. Homework is worth 20% of your final grade.

Labs: The labs will be set up in S415 ESC. Each of these labs will require approximately20 minutes to complete. You may work alone or in pairs. You will schedule your time foreach lab on a sign-up sheet which will be passed around in class on the first day of the lab.The labs are due at 12:50 pm (before class) on the days indicated on the schedule. Theyare to be submitted through the Physics 281 slots of the homework bins outside N375ESC. Late labs will receive half credit. After the labs have been graded and returned, youmay correct your mistakes and resubmit them. You will then receive half credit for anyadditional points you earn by making the corrections. These retries are due at 1:00 pmon the days indicated on the schedule. No labs will be accepted after the retry due date.Note: Lab 3 is handed in in two parts. The first part will be handed back to you beforethe next part is due. Do not wait to hand in the first part because the second part is onthe same page. Labs are worth 5% of your final grade.

Midterm exams: Midterm exams will be given in the Testing Center in the GrantBuilding (2nd floor). They will each be available on the days indicated on the schedule.Midterm exams are worth 45% of your final grade.

Final Exam: The final exam will cover the subject material of the entire course. Itwill be open-book and open notebook, and it will be timed. You will be allowed 3 hoursto complete it. It will be available at the Testing Center during the entire week of finalexams. The final exam is worth 20% of your final grade.

Final Grades:

A 90.0% B+ 84.0% C+ 72.0% D 50.0%A− 87.0% B 78.0% C 60.0%

B− 75.0%

Honor Code Standards: In keeping with the principles of the BYU Honor Code, stu-dents are expected to be honest in all of their academic work. Academic honesty means,most fundamentally, that any work you present as your own must in fact be your ownwork and not that of another. Violations of this principle may result in a failing grade inthe course and additional disciplinary action by the university.

Students are also expected to adhere to the Dress and Grooming Standards. Adherencedemonstrates respect for yourself and others and ensures an effective learning and workingenvironment. It is the university’s expectation, and my own expectation in class, thateach student will abide by all Honor Code standards. Please call the Honor Code Officeat 422-2847 if you have questions about those standards.

Preventing Sexual Discrimination or Harassment: Sexual discrimination or ha-rassment (including student-to-student harassment) is prohibited both by the law and byBrigham Young University policy. If you feel you are being subjected to sexual discrimi-nation or harassment, please bring your concerns to the professor. Alternatively, you maylodge a complaint with the Equal Employment Office (D-240C ASB) or with the HonorCode Office (4440 WSC).

Students with Disabilities: If you have a disability that may affect your performancein this course, you should get in touch with the office of Services for Students with Dis-abilities (1520 WSC). This office can evaluate your disability and assist the professor inarranging for reasonable accommodations.

Known Errors in Textbook

p. 20, paragraph before Problem 2-12: 47 should be 49 (appears twice).p. 51, Problem 6-11: answer should be 3.64 cm/s.p. 58, Problem 7-5: answer should be 3.13 eV.p. 81, column 1, line 1: Fig. 10-4 should be Fig. 10-3.p. 81, Eqs. (10-8) and (10-9): lower limit to integrals should be Ec.p. 82, Eq. (10-13): upper limit to integral should be Ev.p. 82, Problem 10-6: answer should be 3.09× 1025 m−3.p. 84, column 2, line 1: 8× 1010 should be 1.0× 1011.p. 84, column 2, line 20: 3000 should be 4300.p. 97, column 1, line 2: V −kBT should be Va −kBT/e.

PHYSICS 281 COURSE SCHEDULE, FALL SEMESTER, 2011

Date Read Quizzes Homework Due Labs Exams

1st try Retries

1 Mon, Aug 292 Wed, Aug 31 1-1,2,3,4,5,6,7,8 13 Fri, Sep 02 1-9,10,11,12,13 2 1 #1 begins

4 Mon, Sep 05 Labor Day5 Wed, Sep 07 1-14,15,16; 2-1,2,3,4 3 2 16 Fri, Sep 09 2-5 4 3 2

7 Mon, Sep 12 2-6 5 4 3 #1 due#2 begins

8 Wed, Sep 14 2-7 6 5 49 Fri, Sep 16 3-1,2,3 7 5

10 Mon, Sep 19 3-4 8 6 #2 due#3A begins

11 Wed, Sep 21 3-5,6 9 6 #3A due12 Fri, Sep 23 Review 7 #3B begins

13 Mon, Sep 26 Review #1 starts 2:0014 Wed, Sep 28 4-1,2,3,4,5 10 7 #2 retry due ends Thursday noon15 Fri, Sep 30 5-1,2 11 8

16 Mon, Oct 03 5-3 12 9 8 #3B due17 Wed, Oct 05 5-4,5 13 10 918 Fri, Oct 07 6-1,2,3 14 11 10 #3 retry due

19 Mon, Oct 10 6-4,5,6 15 1120 Wed, Oct 12 7-1,2,3 16 1221 Fri, Oct 14 7-4,5 17 12

22 Mon, Oct 17 7-6 18 1323 Wed, Oct 19 8-1,2,3 19 14 1324 Fri, Oct 21 No class

25 Mon, Oct 24 8-4,5,6 20 1426 Wed, Oct 26 9-1,2,3,4 21 1527 Fri, Oct 28 9-5,6,7,8,9 22 16 15

28 Mon, Oct 31 Review 17 16 #2 starts 2:0029 Wed, Nov 02 10-1,2,3 23 17 ends noon Thursday30 Fri, Nov 04 10-4 24 18

31 Mon, Nov 07 10-5 25 1832 Wed, Nov 09 10-6,7,8 26 19 #4 begins33 Fri, Nov 11 10-9; 11-1,2,3,4,5,6 27 20 19

34 Mon, Nov 14 11-7,8,9; 12-1 28 21 2035 Wed, Nov 16 12-2,3,4 29 22 21 #4 due36 Fri, Nov 18 12-5,6,7 30 22

37 Mon, Nov 21 12-8, 13-1,2,3,4,5,6 31 23 #5 begins38 Tue, Nov 22 No class39 Wed, Nov 23 Thanksgiving40 Fri, Nov 25 Thanksgiving

41 Mon, Nov 28 13-7,11 32 24 23 #4 retry due42 Wed, Nov 30 13-8,9,10 33 24 #5 due43 Fri, Dec 02 Review 25

44 Mon, Dec 05 Review 25 #3 starts 2:0045 Wed, Dec 07 Review for final #5 retry due ends Thursday noon

Homework, Physics 281, Fall Semester, 2011

The homework problems for this course are on the following pages in this packet. Problems1-1 through 1-3 belong to assignment 1, problems 2-1 through 2-5 belong to assignment2, etc. Each of you will do the problems using different data, resulting in answers thatare different from those of other students. Blanks are left in the problems where you canwrite your data. Your data for the entire semester is available over the internet. Go toour course home page, click on “Online Homework” and then “Homework Data Sheet”.If you do not have a course identification number yet, you must obtain one before yourhomework data sheet will be available. Go to our course home page and click on “Obtainyour Class ID Number.”

At the end of the homework problems, there is information about the answers. You aregiven a range of possible values for each answer, along with the units, if any. For example400, 800 J means that your answer will lie between 400 and 800 J. These numbers alsoindicate the accuracy to which you must calculate the answer. This is simply the numberof digits shown. For example 400, 800 J means that the answer must be given to thenearest 1 J. As another example, 15.0, 60.0 N means that the answer must be givento the nearest 0.1 N. In some cases, the accuracy is indicated explicitly. For example,32000, 39000 ±100 km means the answer must be given to the nearest 100 km. Do notround off intermediate answers.

After working the problems, submit your answers over the internet. Go to our course homepage and click on “Online Homework” and then click on the assignment number. Fill in theanswers as indicated. Do not put units on your answer. Use the units indicated. If a verylarge or very small value needs to be written in scientific notation, indicate the exponentof 10 with an “e”. For example, 3.00×108 would be written 3.00e8, and 1.6×10−19 wouldbe written 1.6e-19. Do not put any spaces or commas in the number. Give answers onlyfor problems you have worked. Do not guess the answer.

Your homework assignment will be graded during class time. You may see your score overthe internet by going to the course home page and clicking on “Online Homework” andthen on “Homework Status.” You will see your score and also the correct answers for anyproblems you missed.

You will have 5 tries for each problem. The first try is due at 12:45 pm on the daysindicated on the course schedule. All retries are due at 12:45 pm on the next day classmeets, as indicated on the course schedule. After each try, a new set of data will appear atthe bottom of the homework status page. Use this new data for the next try. You only needto resubmit answers you missed in the previous try. Retries will be graded immediatelywhen submitted so that, if you want, you can continue to work on the problems you missedon the retry.

You will receive 5 points for each problem done correctly the first try, 4 points the secondtry, 3 points the third try, 2 points the fourth try, and 1 point the fifth try. You will receiveno points for a problem until it is done correctly.

You can change any answer you submitted for the first try by resubmitting that answerbefore it is due. You only need to resubmit the answer you want to change. Leave the otheranswers blank, and the computer will automatically use previously submitted answers.

Multiple choice questions are worth 2 points each. You do not get a second try on thesequestions.

If the first try is late, each completed problem on any of the subsequent tries will bededucted one point. These deducted points become “late points.” Also, each late multiplechoice question will be deducted one point. Any points generated by tries submitted afterthe retries due date will also become late points. You will receive full credit for late pointson the three assignments with the most late points. You will receive half credit for allother late points. You will always receive at least partial credit for all late work (until theend of the semester).

Physics 281 Homework Problems, Fall 2011

1-1. Consider a crystal with a bcc structure and lattice parameter a = [01] A.

(a) Find the distance between adjacent atoms in the [100] direction. (b) Repeat for the

[110] direction. (c) Repeat for the [111] direction. (d) Find the distance between the

(100) planes. (e) Repeat for the (110) planes. (See Problem 1-1 in textbook.)


Find the diameter of the atoms. (Consider the atoms to be spheres which touch each

other.) (See Problem 1-2 in textbook.)

1-3. Consider a crystal with an fcc structure and lattice parameter a = [03] A.

(a) Find the distance between adjacent atoms in the [100] direction. (b) Repeat for the

[110] direction. (c) Repeat for the [111] direction. (d) Find the distance between the

(100) planes. (e) Repeat for the (110) planes. (See Problem 1-4 in textbook.)


Find the diameter of the atoms. (Consider the atoms to be spheres which touch each

other.) (See Problem 1-5 in textbook.)


Find the volume of a primitive unit cell. (See Problem 1-3 in textbook.)

2-2. Suppose barium (Ba) had an fcc structure with a lattice parameter

a = [02] A. What would be its density? (See Problem 1-6 in textbook.)

2-3. Consider some crystal which has a bcc structure at room temperature. As we raise the

temperature, it suddenly changes to an fcc structure. (For example, this happens in iron

at 910C.) When the structure changes from bcc to fcc, the density increases by

[03] %. By what percentage does the nearest-neighbor distance between the

atoms change? (See Problem 1-7 in textbook.)


Find the volume of a primitive unit cell. (See Problem 1-8 in textbook.)

3-1. Consider a box of volume [01] m3 filled with balls of diameter

[02] mm. (a) If we pack the balls in an sc lattice, how many can we get into

the box? (b) Repeat for bcc lattice. (c) Repeat for fcc lattice. (See Problem 1-10 in

textbook.)

3-2. Consider a crystal with the sodium chloride structure and lattice parameter

a = [03] A. Find the distance between nearest-neighbor atoms. (See

Problem 1-11 in textbook.)

3-3. Suppose that zinc sulfide (ZnS) had the sodium chloride structure. If its density were

[04] g/cm3, calculate its lattice parameter a. (See Problem 1-12 in textbook.)

3-4. Consider the two-dimensional crystal shown in the

figure. The and • symbols represent two different

kinds of atoms. (a) Are all atoms in equivalent

positions? (b) Are all • atoms in equivalent

positions? (c) Draw a “conventional” unit cell

which is rectangular in shape and has a atom on

each corner. (d) How many atoms are in this unit

cell? (e) How many • atoms? (f) Draw a

Wigner-Seitz primitive unit cell centered on a

atom. Warning: make your drawing carefully!

Turn in parts (c) and (f) of this problem on paper.

Use the sheet of paper on the following page. (See


3-5. Consider the two-dimensional crystal in the

figure. There is only one kind of atom in the

crystal, represented by the symbol • in the

figure. (a) Are all atoms in equivalent positions?

(b) Draw a primitive unit cell. (It does not

necessarily need to be a Wigner-Seitz cell.)

(c) How many atoms are in this cell? Warning:

make your drawing carefully! Turn in part (b) of

this problem on paper. Use the sheet of paper

on the following page. (See Problem 1-15 in

textbook.)

Physics 281 Identification numberHomework Set 3

Score: out of 4 points

Draw diagrams on this sheet of paper and submit it through the Physics 281 slots of thehomework bins outside N375 ESC. This part of the assignment is due at class time onthe same day that the rest of the assignment is due. Late papers will receive half credit.

3-4 (c).

3-4 (f).

3-5 (b).

4-1. Consider a crystal with the cesium chloride structure and lattice parameter

a = [01] A. Find the volume of a primitive unit cell. (See Problem 1-16 in

textbook.)

4-2. Consider a crystal with the diamond structure and lattice parameter

a = [02] A. Find the volume of a primitive unit cell. (See Problem 1-19 in

textbook.)

4-3. Consider a crystal with the diamond structure and lattice parameter

a = [03] A. Find the diameter of the atom. (Consider the atoms to be spheres

that touch each other. (See Problem 1-20 in textbook.)

4-4. Consider a crystal with the diamond structure containing 1 mole of atoms. (1 mole is

equal to Avogadro’s number. See Appendix 13 in the textbook.) It is found that the

energy required to separate all of the atoms from each other to an infinite distance apart

is [04] kJ (called the cohesive energy of the crystal). Find the binding energy

(cohesive energy per bond) of a single covalent bond between a pair of nearest neighbor

atoms. (See Problem 1-24 in textbook.)

5-1. Two speakers are located 21.8 m apart on the stage of an auditorium. A listener is seated

25.3 m from one speaker and [01] m from the other. We drive the two

speakers at a frequency which we sweep. (a) If the two speakers are in phase, find the

lowest frequency at which the waves constructively interfere at the listener so that he

hears maximum intensity. (b) Find the lowest frequency at which the waves destructively

interfere at the listener so that he hears minimum intensity. The speed of sound is

343 m/s. (See Problem 2-3 in textbook.)

5-2. Consider a beam of laser light (wavelength λ = 633 nm) aimed at a screen. If we place a

diffraction grating into the path of the laser beam, new spots appear on the screen, one

above and one below the original spot. If the new spots are [02] m from the

original spot and if the distance between the grating and the screen is 1.21 m, find the

distance between the slits in the grating. (See Problem 2-4 in textbook.)

5-3. Consider a set of crystal planes which are separated by [03] A. If we use

x rays of wavelength λ = 1.542 A, find the smallest Bragg angle for reflection from these

planes. (See Problem 2-6 in textbook.)

5-4. When some crystal is heated from 0C to 100C, its lattice parameter a increases by

[04] % due to thermal expansion. If we observe an x-ray reflection at a Bragg

angle θ = 25.8 at 0C, by how much will θ change when the sample is heated to 100C?

(You do not need to know λ or d to solve this problem.) (See Problem 2-9 in textbook.)

5-5. Consider planes shown for the two-dimensional

square lattice in the figure. If

a = [05] A and we use x rays of

wavelength 1.542 A, find the smallest Bragg angle

at which we would observe reflections from these

planes. (See Problem 2-10 in textbook.) a

a


(a) Find the smallest Bragg angle for reflection of x rays (λ = 1.542 A) from the (100)

planes. (b) Repeat for the (110) planes. (See Problem 2-11 in textbook.)

6-2. Consider a crystal with the sodium chloride structure and lattice parameter

a = [02] A. (a) Using x rays of wavelength λ = 1.542 A, find the smallest

Bragg angle for reflections from the (100) planes. (b) Repeat for the (110) planes. (See


6-3. An x-ray diffraction pattern from a powdered sample of lithium (Li) shows 10 lines. The

wavelength of the x rays is 1.542 A. Identify the planes for the Bragg reflections at the

following Bragg angles: (a) 18.2, (b) 26.1, (c) 32.7, (d) 38.5, (e) 44.2, (f) 49.7,

(g) 55.5, (h) 61.8, (i) 80. (See Problem 2-18 in textbook.)

6-4. (a) Consider x-ray diffraction in an fcc lattice. Which planes will reflect the x rays at the

smallest Bragg angle? (b) Which planes will reflect the x rays at the next smallest Bragg

angle? (c) Repeat part (a) for a bcc lattice. (d) Repeat part (b) for a bcc lattice. This is

a multiple choice problem. (Hint: Consider Bragg’s law in reciprocal space. The smallest

Bragg angles will occur for the shortest reciprocal lattice vectors G. (See Problem 2-19

in textbook.)

7-1. Consider diffraction of x rays from some crystal with the sodium chloride structure. We

find that the “first” diffraction line (the smallest Bragg angle) is at θ = [01] ,

using x rays of wavelength 1.542 A. (a) Find the lattice parameter a of the crystal. If we

now apply enough pressure, we find that the structure suddenly changes to the cesium

chloride structure. (This actually happens in NaCl at 300,000 atm.) This change in

structure also causes the x-ray diffraction pattern to change, and we find that the first

diffraction line suddenly decreases by 1.2. (b) Find the new lattice parameter a. (See


7-2. Extra credit. Consider a pair of ions, one with a positive charge +e and the other with a

negative charge −e. The net force between the two ions separated by a distance r is

given by

F = − e2

4πε0r2+ B exp(−r/R),

where the first term is the electrostatic force (Coulomb’s Law) and the second term is the

short-range repulsive force. The parameters in the repulsive force for these two ions are

B = 5.451× 10−6 N and R = [02] A. (a) Find the distance r = r0 between the

two ions when they are at equilibrium (F = 0). Use the values of e and ε0 exactly as

given in Appendix 13. Next consider the harmonic approximation for the interaction

between the two ions. (b) Find the spring constant α for the interaction between the two

ions. (Find the value of dF/dr at the point where F = 0, as in Fig. 3-1 in the textbook.

Use the results of part (a).) At room temperature, the pair of ions has a thermal energy

equal to 0.025 eV. (c) Find the amplitude of the oscillation using the harmonic

approximation. Next consider oscillations beyond the harmonic approximation. Use

Eq. (3-26) in the textbook to find an expression for the potential energy of the two ions.

If their thermal energy is 0.025 eV, find (d) the minimum value of r and (e) the

maximum value of r as these two ions oscillate back and forth. (Do not use the harmonic

approximation here.) (f) By what percentage is the average of these two distances

greater than r0 (the distance between the two ions when F = 0)? Use the result from

part (a). (See Problems 1-21, 3-1, 3-12, and 3-13 in textbook.)

8-1. Consider a wire made of copper (Cu). If the wire is [01] mm in diameter and

carries 10 A of current, find the drift velocity of the conduction electrons. (See


8-2. Consider a metal with a bcc structure, lattice parameter a = [02] A,

valence Z = 1, and electrical conductivity σ = [03] /Ω ·m. Find the average

time between collisions of a conduction electron in this metal. (See Problem 4-4 in

textbook.)

8-3. Consider a metal with an fcc structure, lattice parameter a = [04] A, and

valence Z = 1. Calculate the Hall coefficient. Use the classical model. (See Problem 4-6

in textbook.)

8-4. Consider a slab of metal 0.118 mm thick, 1.52 mm wide, and 8.33 mm long. The Hall

coefficient for this metal is [05] m3/C. (a) If we drive a current of

[06] A down the length of the slab, what is the current density J? (b) If we

then put the slab in a magnetic field B = 0.93 T with the field perpendicular to the

1.52 mm × 8.33 mm face, what is the magnitude of the Hall field EH that will be

produced? (c) What magnitude of Hall voltage will we observe across the width of the

slab? (See Problem 4-7 in textbook.)

8-5. Consider a field B = [07] T. Use the classical model to find the wavelength of

radiation at the cyclotron resonance in a metal placed in that field. (See Problem 4-8 in

textbook.)

9-1. Consider visible light of wavelength [01] nm. Find the (a) energy and

(b) momentum of the photons in this light wave. (See Problem 5-1 in textbook.)

9-2. Consider a collision between a photon and an electron. The electron is initially at rest.

After the collision, the photon emerges in a direction 90 from its original direction. See

figure below. Note that the figure is two-dimensional with the x axis pointing toward the

right and the y axis pointing toward the top of the page. The wavelength of the photon

before the collision is [02] A (an x ray), and its wavelength after the collision

is 0.0243 A longer.

(a) Find the energy hωi of the photon before the collision.

(b) Find the energy hωf of the photon after the collision.

(c) From conservation of energy, find the velocity v of the electron after the collision.

(d) Find the magnitude of the momentum hki of the photon before the collision.

(e) Find the magnitude of the momentum hkf of the photon after the collision.

(f) From conservation of momentum, find the x component (px) of the electron’s

momentum after the collision.

(g) Find the y component (py) of the electron’s momentum after the collision.

(h) Find the magnitude p of the electron’s momentum after the collision. (If you divide

this result by the mass m of the electron, it should agree with the answer to part (c) to

two significant figures. Agreement is only approximate because we have neglected

relativistic effects. The electron is moving at a few percent of the speed of light.)

(i) Find the angle θ between the direction of the electron’s velocity after the collision and

the direction of the photon before the collision. (See Problem 5-2 in textbook.)

electron after collision:energy Emomentum p

incident photon:energy hωi

momentum hkiphoton after collision:

energy hωf

momentum hkf

θ

90

10-1. Suppose that in Fig. 5-2 in the textbook, there were a peak at

Ef = [01] J. (Note that Ei = 5.60× 10−21 J.)

(a) Find the frequency ω of the phonons absorbed by the neutrons in that peak.

(b) Find the x component of the wave vector k of those phonons. Be sure to use the

coordinate system in Fig. 5-3 in the textbook. Also, assume the directions of the incident

neutrons and the position of the detector is the same as in the figure.

(c) Find the y component of the wave vector k of those phonons.

(d) Find the x component of the equivalent wave vector k inside the first Brillouin zone.

(e) Find the y component of the equivalent wave vector k inside the first Brillouin zone.

(f) Find the magnitude of the equivalent wave vector k inside the first Brillouin zone.

(g) Find the magnitude of the angle between the x axis (the [100] direction) and the

equivalent wave vector k inside the first Brillouin zone. (See Problem 5-7 in textbook.)

11-1. Consider a phonon in CsCl. If its wave vector is

k = ([01] A−1) ı + ([02] A−1) , find the (a) x component,

(b) y component, and (c) z component of the wave vector of this phonon inside the first

Brillouin zone.

Next consider a phonon in Na. If its wave vector is

k = ([03] A−1) ı + ([04] A−1) + ([05] A−1) k, find the

(d) x component, (e) y component, and (f) z component of the wave vector of this

phonon inside the first Brillouin zone. (See Problem 5-8 in textbook.)

11-2. Suppose we obtained peaks similar to those in Fig. 5-6 in the textbook for Brillouin

scattering in a certain crystal with an index of refraction n = 1.38. The wavelength of

the incident light (in vacuum) is λ = [06] A, and one of the peaks lies

∆ω = [07] s−1 above the central peak. (a) Did the photons in this peak

absorb or emit a phonon? (b) If the incident photons travel in the [110] direction and the

scattered photons travel in the [110] direction (1 means −1), find the direction of the

wave vector k of the phonon. (c) Find the magnitude k of the wave vector of the phonon.

(d) Find the frequency ω of the phonon. (d) Find the speed of the phonons. (This is the

speed of sound in the crystal.) (See Problems 5-10 and 5-11 in textbook.)

11-3. Find the de Broglie wavelength of an electron which has a kinetic energy equal to

[08] eV. (See Problem 5-13 in textbook.)

11-4. Consider diffraction of electrons from the (100) planes in aluminum (Al). Find the

minimum energy (in units of eV) of the electrons that would permit us to observe

reflection at a Bragg angle of [09] . (See Problem 5-14 in textbook.)

12-1. Consider an electron in a “box” which is [01] cm long. Using Eq. (6-27), find

the smallest velocity the electron can have. (See Problem 6-11 in textbook.)

12-2. Consider an energy barrier between two metals. The height of the barrier is 6.372 eV,

and its width is 35.9 A. If 6.3× 1020 conduction electrons, each with kinetic energy

[02] eV, approach this barrier, how many of them will be able to tunnel

through to the other side? (See Problem 6-13 in textbook.)

13-1. Consider a metal with a bcc structure, lattice parameter a = [01] A, and

valence Z = 1. Find the (a) Fermi energy and (b) Fermi velocity of the conduction

electrons. (See Problem 7-5 in textbook.)

13-2. (a) How far below the Fermi level will we find states which are [02] % occupied

by electrons at 283 K? (b) How far above the Fermi level will we find states which are

[03] % occupied by electrons at 283 K? (See Problem 7-7 in textbook.)

14-1. Consider a metal with an fcc structure, lattice parameter a = [01] A, valence

Z = 1, and electrical conductivity σ = [02] /Ω·m. (a) Find the radius of

the Fermi surface. If an electric field E = 1.00 V/m is applied, causing current to flow,

find (b) the drift velocity of the conduction electrons and (c) the displacement ∆k of the

Fermi surface. (d) Find the average distance ` which a conduction electron near the

Fermi surface travels between collisions. (See Problems 7-11 and 7-12 in textbook.)

15-1. Consider a metal with a bcc structure and lattice parameter a = [01] A. Find

the volume of the first Brillouin zone. (See Problem 8-7 in textbook.)

15-2. Consider a piece of sodium metal (Na) of mass [02] g. (a) How many electron

states are in each band? (b) How many conduction electrons are there? (See Problem 8-8

in textbook.)

16-1. Consider a conduction electron in copper (Cu). The wave vector of the electron points in

the [100] direction. The magnitude of its wave vector is k = [01] A−1. We

apply an electric field E = 1.00 V/m in a negative [100] direction. If the electron could

travel without hindrance (no collisions), how long would it take to reach the boundary of

the first Brillouin zone? You may use Fig. 8-11 on p. 68 in the textbook to determine

how far the electron must travel in k space to reach the boundary of the first Brillouin

zone. (See Problem 9-2 in textbook.)

17-1. A cyclotron resonance is observed in some metal at 8900 MHz for a field

B = [01] T. Find the effective mass of the detected electron in terms of the

mass m of a free electron. (See Problem 9-10 in textbook.)

18-1. Find the density of electrons in the CB in a crystal of pure silicon (Si) at

[01] K. For Si, m∗n = 1.09m in the CB, and m∗

p = 1.15m in the VB, where

m is the mass of a free electron. Caution: Be sure to use conversion factors and physical

constants exactly as found in the appendices of the textbook. The answer is very

sensitive to small changes in these numbers. (See Problem 10-8 in textbook.)

19-1. Consider a sample of n-type silicon (Si) with Nd = [01] m−3. Find (a) n

and (b) p at 300 K. Use ni = 1.0× 1016 m−3 for Si at 300 K. (c) What fraction of total

atoms in the crystal are donors? (See Problems 10-9 and 10-10 in textbook.)

19-2. Consider a sample of n-type silicon (Si) at 300 K with Nd = [02] m−3.

Using Eq. (10-11), calculate how far the Fermi energy is below the CB. You may use the

value of Nc from Problem 10-4 in the textbook. (See Problem 10-11 in textbook.)

19-3. Consider a Si crystal doped with both donors (Nd = [03] m−3) and

acceptors (Na = [04] m−3). Find the final values of (a) n and (b) p at

300 K. Use ni = 1.0× 1016 m−3 for Si at 300 K. (See Problem 10-14 in textbook.)

20-1. Consider a sample of n-type silicon (Si) with Nd = [01] m−3. Calculate

the conductivity of the Si crystal at 300 K due to (a) the electrons, (b) the holes, and

(c) the electrons and holes together. Use ni = 1.0× 1016 m−3 for Si at 300 K. (See


20-2. A 5.2-mm cube of n-type germanium (Ge) at 300 K passes a current of

[02] mA when 12.4 mV is applied between two of its parallel faces. Find the

density of electrons in the conduction band. (See Problem 10-18 in textbook.)

20-3. Calculate the Hall coefficient RH for an n-type Si crystal with

Nd = [03] m−3. (See Problem 10-19 in textbook.)

20-4. Consider a piece of some n-type semiconductor 54 mm long, 4.3 mm wide, and

0.97 mm thick. If we apply 0.25 V across the length, we find that it conducts a current of

[04] mA. If we then put it in a magnetic field of 0.63 Tesla, perpendicular to

its 54 mm-by-4.3 mm face, we find that the Hall voltage across its width is

[05] mV. Calculate (a) the mobility of the conduction electrons and (b) the

electron density in the CB. (See Problem 10-20 in textbook.)

21-1. The figure shows the band structure of GaAs (see Fig. 10-10 in the

textbook). Which band contains the “light” holes? (By “light”

holes, we mean the holes with less effective mass.) Answer (a) or

(b), as shown in the figure. (See Problem 10-21 in textbook.)

(a)(b)

21-2. Consider a p -n junction in silicon (Si) with Nd = [01] m−3 in the

n-type Si and Na = [01] m−3 in the p -type Si. (Note that Nd = Na.)

(a) Calculate the contact potential φ at 300 K. Use ni = 1.0× 1016 m−3 for Si at 300 K.

(b) Find the width of the depletion layer (εr = 12 in Si). (c) Find the magnitude of the

electric field at the center of the depletion layer. (See Problems 11-4, 11-9, and 11-10 in

textbook.)

22-1. (a) If I0 = [01] µA for some p -n junction, find the current at 300 K if it is

forward biased with [02] V. (b) Find the current if it is reverse biased with

[02] V. (See Problem 11-12 in textbook.)

23-1. If an electron absorbs a photon of wavelength [01] nm, how far does the

electron move in k-space? (See Problem 12-3 in textbook.)

23-2. Consider a semiconductor with Eg = [02] eV. What is the maximum

wavelength of light that will excite electrons from the VB into the CB? (See


23-3. An LED made of some semiconductor produces light of wavelength [03] nm.

What is the width of the energy gap in this semiconductor? (See Problem 12-7 in

textbook.)

24-1. Suppose that total population inversion could be achieved in a ruby laser. If half of the

electrons in E2 could then drop to E1 in 30 ns, what would be the average power of the

resulting laser pulse over this time interval? Assume that the ruby crystal is a cylinder

5.00 cm long and [01] cm in diameter and that one Al in every

[02] has been replaced by a Cr. The density of Al2O3 is 3.7 g/cm3. Use

E2 − E1 = 1.786 eV in Fig. 12-21 in the textbook. (See Problem 12-9 in textbook.)

24-2. For lead (Pb), we find that Tc = 7.193 K and λ0 = 390 A. Find the penetration depth λ

at (a) T = [03] K, and (b) T = [04] K. (See Problem 13-1 in

textbook.)

24-3. (a) Find the critical field in lead (Pb) at T = [05] K.

(b) How much current can a lead (Pb) wire, 1.0 mm in diameter, carry in its

superconducting state at that temperature? (See Problems 13-2 and 13-3 in textbook.)

25-1. Consider a superconductor with an energy gap 2∆ = [01] eV. Find the

maximum wavelength of electromagnetic radiation which will be absorbed by this

superconductor. (See Problem 13-7 in textbook.)

Answers to Homework Problems, Physics 281, Fall Semester, 2011

1-1a. 3.00, 4.00 A1-1b. 4.00, 6.00 A1-1c. 2.50, 3.50 A1-1d. 1.50, 2.00 A1-1e. 2.00, 3.00 A1-2. 3.00, 4.50 A1-3a. 4.00, 5.00 A1-3b. 2.50, 4.00 A1-3c. 6.00, 9.00 A1-3d. 2.00, 2.50 A1-3e. 1.40, 1.80 A1-4. 2.50, 4.00 A2-1. 10.0, 40.0 A3

2-2. 5.0, 9.9 g/cm3

2-3. 2.00, 2.50%2-4. 15.0, 35.0 A3

3-1a. 1.00× 108, 9.90× 108

3-1b. 1.00× 108, 9.90× 108

3-1c. 1.00× 108, 9.90× 108

3-2. 2.00, 2.50 A3-3. 4.00, 5.50 A4-1. 20.0, 70.0 A3

4-2. 15.0, 35.0 A3

4-3. 1.70, 2.20 A4-4. 2.50, 4.00 eV5-1a. 50, 99 Hz5-1b. 20, 70 Hz5-2. 2.0, 4.0 µm5-3. 14.0, 25.0

5-4. −0.020, −0.090

5-5. 20.0, 40.0

6-1a. 20.0, 40.0

6-1b. 15.0, 25.0

6-2a. 15.0, 25.0

6-2b. 20.0, 40.0

7-1a. 3.50, 5.50 A7-1b. 2.00, 3.50 A7-2a. 2.0000, 2.7000 A7-2b. 70, 130 N/m7-2c. 0.0700, 0.1100 A7-2d. 2.0000, 2.7000 A7-2e. 2.0000, 2.7000 A

7-2f. 0.20, 0.30%8-1. 0.20, 0.80 mm/s8-2. 1.0× 10−14, 5.0× 10−14 s8-3. −0.40× 10−10, −0.99× 10−10 m3/C8-4a. 6.0× 106, 9.0× 106 A/m2

8-4b. 2.0× 10−4, 8.0× 10−4 V/m8-4c. 0.40, 0.99 µV8-5. 1.10, 1.80 cm9-1a. 2.00, 2.80 eV9-1b. 1.00× 10−27, 1.50× 10−27 kg·m/s9-2a. 6.00, 9.90 keV9-2b. 6.00, 9.90 keV9-2c. 5.0× 106, 8.0× 106 m/s9-2d. 3.00× 10−24, 6.00× 10−24 kg·m/s9-2e. 3.00× 10−24, 6.00× 10−24 kg·m/s9-2f. 3.00× 10−24, 6.00× 10−24 kg·m/s9-2g. 3.00× 10−24, 6.00× 10−24 kg·m/s9-2h. 4.00× 10−24, 8.00× 10−24 kg·m/s9-2i. 44.0, 45.0

10-1a. 1.0× 1013, 4.0× 1013 s−1

10-1b. 4.20, 4.80 A−1

10-1c. −0.30, −0.80 A−1

10-1d. 0.70, 1.30 A−1

10-1e. −0.30, −0.80 A−1

10-1f. 1.10, 1.30 A−1

10-1g. 15.0, 45.0

11-1a. 0.00, 0.80 A−1

11-1b. 0.00, −0.80 A−1

11-1c. 0, 0 A−1

11-1d. 0.00, −0.40 A−1

11-1e. 0.00, 0.40 A−1

11-1f. 0.70, 1.10 A−1

11-2c. 2.00× 10−3, 2.50× 10−3 A−1

11-2d. 2.00× 1011, 4.00× 1011 s−1

11-2e. 8000, 20000 m/s11-3. 1.20, 1.50 A11-4. 50, 140 eV12-1. 1.80, 3.10 cm/s12-2. 1.0× 1010, 5.0× 1010

13-1a. 1.5, 2.5 eV13-1b. 7.5× 105, 9.0× 105 m/s13-2a. 0.050, 0.099 eV

13-2b. 0.050, 0.099 eV14-1a. 1.00× 1010, 1.30× 1010 m−1

14-1b. 1.00× 10−3, 3.00× 10−3 m/s14-1c. 10.0, 30.0 m−1

14-1d. 100, 200 A15-1. 2.00, 8.00 A−3

15-2a. 5.00× 1022, 8.00× 1022

15-2b. 2.50× 1022, 4.00× 1022

16-1. 1.5, 5.0 µs17-1. 0.40, 0.99m18-1. 1.5× 1016, 9.0× 1016 m−3

19-1a. 1.00× 1021, 2.00× 1021 m−3

19-1b. 5.0× 1010, 9.0× 1010 m−3

19-1c. 2.00× 10−8, 4.00× 10−8

19-2. 0.200, 0.300 eV19-3a. 4.0× 1010, 9.9× 1010 m−3

19-3b. 1.00× 1021, 2.50× 1021 m−3

20-1a. 20.0, 50.0 /Ω·m20-1b. 3.0× 10−10, 7.0× 10−10 /Ω·m20-1c. 20.0, 50.0 /Ω·m20-2. 2.0× 1020, 9.9× 1020 m−3

20-3. −3.0× 10−3, −6.0× 10−3 m3/C20-4a. 0.40, 0.80 m2/V·s20-4b. 3.0× 1021, 8.0× 1021 m−3

21-2a. 0.600, 0.650 V21-2b. 1.00, 1.50 µm21-2c. 1.00× 106, 1.50× 106 V/m22-1a. 10.0, 20.0 mA22-1b. −10.0, −20.0 µA23-1. 1.00× 10−3, 1.30× 10−3 A−1

23-2. 1200, 1600± 10 nm23-3. 2.00, 2.50 eV24-1. 400, 990± 10 MW24-2a. 400, 450 A24-2b. 600, 999 A24-3a. 0.0200, 0.0400 T24-3b. 60, 99 A25-1. 1.00, 1.60 mm

Physics 281 Identification numberLab #1Crystal Model

In this lab, you will build a model of the conventional unit cell of the face-centered-cubic lattice (see figure below). You may work in groups, but each student must buildhis/her own model. Whereas the other labs are done in S415 ESC, you can take your kithome and do this lab anywhere you want. Answer the questions and turn this sheet in (tothe 281 box outside room N375 ESC). (Keep the model. It’s yours.)

You will use 14 pingpong balls, which will be provided to you. Glue them togetherso that nearest-neighbor balls touch each other. Super glue works well. Super glue will beprovided IF you share a tube with at least 3 others, and IF you promise to put today’scopy of the Daily Universe underneath so you don’t deface BYU furniture with drops ofglue.

The following questions are instructive as you build your model:How many (111) planes cut through the conventional fcc cell?Inside a (111) plane of an fcc crystal, how are the atoms arranged?What is the shortest distance between two atoms in a (111) plane (in terms of the

cube edge, a)?

I have built the model described above.

Signed

Physics 281 Identification numberLab #2Reciprocal Lattice

In this lab, you will use diffraction techniques to determine the reciprocal lattice of atwo-dimensional “crystal.” The crystal is actually a set of black dots printed on transparentfilm. Instead of x rays, we will use a He-Ne laser with wavelength 632 nm.

Shine the laser beam through the center of the slide containing the two-dimensionalcrystal. You will observe on the screen a pattern of spots forming a square lattice. Theserepresent the reciprocal lattice of the crystal. The figure below shows the relationshipbetween the spots and the reciprocal lattice.

k1

crystal

k2G

screen

center spot

P

L

k1 is the wave vector of the incident laser beam. Most of the intensity of this beam isundiffracted and produces a bright center spot on the screen. The other spots on thescreen are caused by diffraction. k2 is a wave vector pointing toward one of the diffractedspots. The reciprocal lattice vector associated with this spot is given by G = k2 − k1

and is approximately parallel in direction to and proportional in length to the vector Pbetween the spots on the screen. For this reason, the pattern of spots on the screen isa projection of the reciprocal lattice onto real space and is a scaled version of the actualreciprocal lattice. Since the pattern of spots forms a square lattice, the reciprocal latticeis square also.

In the figure, we can see from similar triangles that G/k = P/L, where k is themagnitude of the wave vectors k1 and k2, and L is the distance between the slide (crystal)and the screen. From measurements on the screen, determine the lattice parameter b ofthe reciprocal lattice. Then determine the lattice parameter a of the direct lattice in realspace. This is the spacing of dots on the slide (distance between “atoms” in the crystal).

k:

L:

Lattice parameter of the lattice of dots on screen:

Lattice parameter b of the reciprocal lattice:

Lattice parameter a of the direct lattice:

Physics 281Lab #3Lattice Vibrations

In this lab, you will investigate lattice vibrations in one-dimensional lattices us-ing a simple model: weights connected by springs. Two lattice models will be set up:(1) a monatomic lattice of steel weights and (2) a diatomic lattice of alternating steel andaluminum weights. The steel weights each have a mass m1 = 25.4 g, and the aluminumweights each have a mass m2 = 10.1 g. Adjacent weights are separated, center to center,by 8.7 cm. This means that a = 8.7 cm in the monatomic lattice and a = 17.4 cm in thediatomic lattice. The springs have a spring constant α = 22 N/m.

Part A

Calculate from theory the values of ω for each of the values of k shown in Part A onpage 2 of this lab. Use Eq. (3-9) in the textbook for the monatomic lattice, and use Eq. (3-18) for the diatomic lattice. Be sure to use SI units and to use radians when evaluatingsin(ka/2). Record your results in the space provided. Carefully plot these values on thegraphs on page 3 of this lab, and draw a smooth curve through each set of points. Turnin pages 2 and 3 to be graded.

Part B

In the lab, you will measure points on the dispersion curves by exciting standingwaves in the “lattices.” These excitations are driven by an electromagnet at one end ofthe lattice. The frequency of the excitation is controlled by an oscillator.

You are given a list of some of the standing waves which can be easily excited in themodels. For each standing wave, the period T of the oscillation is given. Calculate thefrequency f = 1/T of the oscillation and record it in the space provided. Also calculatethe angular frequency ω = 2πf and record it. (Do these calculations before coming to thelab.)

For each standing wave, find the wavelength. This is most easily done by counting thenumber n of antinodes (points of maximum amplitude on the wave) across the length L ofthe lattice. This will give you the number of half-wavelengths so that λ = 2L/n. For somestanding waves, it will be difficult to see the antinodes clearly. It is much easier to seethe nodes (points of zero amplitude on the wave). Between each pair of adjacent nodes,there is an antinode. Also, you will notice that not all nodes are exactly at the positionof a weight. Some are at points on the springs between the weights. Look closely at thesprings, and you will be able to find these.

Count the number n of antinodes and record it. Calculate the wavelength λ = 2L/n(use L = 1.79 m) and the wave number k = 2π/λ and record it. Finally plot the point ω(k)on the graph. Make these points distinct so that they can be easily distinguished fromthe points used in drawing the curves. Don’t draw a line through the data points. I onlywant to see the line through the points in part A. The data points ought to fall close tothe curves you drew in part A. If they don’t, then either you drew the curves wrong oryou are measuring the data incorrectly. You need to rectify the problem.

Physics 281 Identification numberLab #3, page 2Lattice Vibrations

Part A. Calculations from Theory (Don’t forget to draw the graph.)

Monatomic Lattice Diatomic Lattice Diatomic LatticeAcoustic Branch Optical Branch

k (m−1) ω (s−1) k (m−1) ω (s−1) k (m−1) ω (s−1)

0 0 05 5 5

10 10 29.0 10 72.4

15 35.7 15 1520 18 18253036

Part B. Experimental Data

Standing waves in the monatomic lattice

T (ms) f (Hz) ω (s−1) n λ (m) k (m−1)

465216120.3109.9

Standing waves in the diatomic lattice

T (ms) f (Hz) ω (s−1) n λ (m) k (m−1)

56039018715295

Lab #3, page 3 Monatomic Lattice

ω (s−1)

80

70

60

50

40

30

20

10

00 10 20 30 40

k (m−1)

Diatomic Lattice

ω (s−1)

80

70

60

50

40

30

20

10

00 5 10 15 20

k (m−1)

Physics 281 Identification numberLab #4Hall Effect in a Semiconductor

In this lab, you will observe the Hall effect in a doped semiconductor. The dimensionsof the sample are length ` = 4.5 mm, width w =2.0 mm, and thickness d = 0.10 mm. Thepower supply should already be connected as shown in the figure. Turn on the power supplyand set it at 20 V. The current I through the sample can be determined by measuring thevoltage drop across the 3000-Ω resistor. The voltage V across the sample can be measuredwith a voltmeter between points A and D. You can now calculate the resistance R = V/Iand the conductivity σ = `/wdR. Record I, V , R, and σ below.

PowerSupply

+

−3000 Ω

A

B

CD

`

w

Next measure the voltage across the width of the sample, between points B and C.Insert the sample between the poles of the magnet. Repeat this with the field in thereversed direction. You will notice that the voltage is not zero when the sample is out ofthe magnet. This is not a Hall voltage, but simply some of the voltage drop across thelength of the sample which you pick up because of a slight misalignment of the connectionsat points B and C. You can zero out this contribution by pushing the “relative” buttonon the voltmeter. Now the voltage measured with the sample in the field is the true Hallvoltage. If you reverse the direction of the field, the sign of that voltage should change,but the magnitude should remain the same.

Record the Hall voltage VH below. The magnitude of the magnetic field B will be givento you in the lab. From geometry, you can now calculate the current density J , the Hallfield EH , and finally, the Hall coefficient RH . Using σ, you can also calculate the mobility µof the carriers. Are the carriers electrons or holes (n-type or p-type semiconductor)?Calculate the carrier concentration (n or p).

I = J =V = VH =R = EH =σ = RH =B = µ =n-type or p-type semiconductor?

density of impurity atoms (n or p) =

Physics 281 Identification numberLab #5Light Emitting Diode

Light emitting diodes (LED’s) are commonly made of the semiconductor alloyGaAs0.6P0.4. In this lab, you will use an optical measurement to obtain the value ofthe band-gap energy in this material.

When the p-n junction of an LED is forward biased, electrons and holes recombineand emit photons of energy hω equal to the band-gap energy Eg. Thus, a measurementof the wavelength of the emitted light provides a very direct measurement of Eg. Youwill measure the wavelength λ of the light from an LED, using a diffraction grating with5000 lines/inch. View the first diffraction maximum on either side of the central maximumand obtain the angle θ, as shown in the figure below. (The angle θ can be accuratelydetermined using a spectroscope, which will be provided in the lab.)

LED slitdiffraction

grating

θ

Calculate the distance d between lines on the grating. (There are 5000 lines/inch onthe grating. Ignore any values for d which may be printed on the apparatus.) Calculatethe wavelength (in nm) using λ = d sin θ. Also calculate the band-gap energy Eg (in eV).Record your data and calculations below.

d = θ = λ = Eg =

Physics 281Sample Final Exam

1. How many lattice points are contained in a primitive unit cell of an fcc lattice? (a) 1,(b) 2, (c) 3, or (d) 4.

2. The Bravais lattice of silicon is (a) diamond, (b) fcc, (c) cubic, (d) a mess.

3. Consider x-ray diffraction from some set of planes in a crystal. If we use x rays of shorterwavelength, the Bragg angle will (a) increase, (b) decrease, or (c) remain the same.

4. The reciprocal lattice of sodium chloride (NaCl) is (a) sc, (b) bcc, or (c) fcc.

5. In a bcc lattice, which of the following planes are spaced furthest apart? (a) (100),(b) (110), or (c) (111).

6. Consider the lattice wave in the monatomic simple-cubic crystal shown on the last pageof this exam. The wave is (a) transverse with k = 0, (b) transverse with k = π/a,(c) longitudinal with k = 0, or (d) longitudinal with k = π/a.

7. Consider a photon and an acoustic phonon with the same energy. The crystal momentumof the phonon is (a) greater than, (b) less than, or (c) the same as that of the photon.

8. Consider a neutron and an electron of the same kinetic energy. The de Broglie wavelengthof the neutron is (a) greater than, (b) less than, or (c) the same as that of the electron.

9. Consider Bragg diffraction of neutrons from some crystal. If we increase the energy of theneutrons, the Bragg angles will (a) increase, (b) decrease, or (c) remain the same.

10. The average kinetic energy per conduction electron in a metal at room temperature isgenerally (a) much greater than, (b) much less than, or (c) about the same as kBT .

11. Consider an electron state at an energy slightly below the Fermi energy. If we increase thetemperature, the probability that this state will be occupied (a) increases, (b) decreases,or (c) remains the same. Consider the Fermi energy to be independent of temperature.

12. For a real metal, the density of electron states in ~k-space is generally greatest (a) at thecenter of the first Brillouin zone, (b) near the edge of the zone, (c) somewhere between thecenter and edge of the zone, or (d) the same everywhere in the zone.

13. The volume of the first Brillouin zone in Si is (a) 2π3a−3, (b) 8π3a−3, (c) 32π3a−3, or(d) 64π3a−3.

14. The volume of the first Brillouin zone in lithium (Li) is (a) greater than, (b) less than, or(c) equal to that in sodium (Na).

15. A crystal with an even number of electrons per primitive unit cell (a) must be (b) may be,or (c) cannot be a metal.

Physics 281Sample Final Exam, p. 2

16. If we increase the temperature, the average time between collisions of a conduction electronin a metal will (a) increase, (b) decrease, or (c) remain the same.

17. An electron in an energy state near the top of an energy band usually has an effective masswhich is (a) positive, (b) negative, (c) infinite, or (d) zero.

18. At room temperature, the number of electrons in the conduction band in pure germa-nium (Ge) is (a) more than, (b) less than, (c) equal to that in pure silicon (Si). Considerthe samples of Ge and Si to have the same volume.

19. If we dope Si with Al, the number of electrons in the conduction band will (a) increase,(b) decrease, or (c) remain the same.

20. In an n-type semiconductor, the Fermi energy is generally (a) above or (b) below themidpoint of the energy gap between the valence and conduction bands.

21. If we raise the temperature of an extrinsic semiconductor, the conductivity will (a) increase,(b) decrease, or (c) remain the same. Assume that the semiconductor is still extrinsic afterthe temperature is raised.

22. Consider a current I flowing through a semiconductor in a magnetic field B as shown inthe figure on the last page of this exam. We find that the voltage at point P1 is positivewith respect to the voltage at point P2. This semiconductor is (a) n-type or (b) p-type.

23. If we reverse-bias a p-n junction in a semiconductor, the width of the depletion layer(a) increases, (b) decreases, or (c) remains the same.

24. When a p-n junction is forward-biased, the magnitude of the generation current is(a) greater than, (b) less than, or (c) equal to the magnitude of the recombination current.

25. The minimum energy in the conduction band of silicon (Si) occurs at k ∼= 0.99 A−1

. Theactual momentum of the electrons in those states is (a) much greater than hk, (b) muchless than hk, or (c) approximately equal to hk.


Problem 6:

ka

Problem 22:


Problem 6:

ka

Problem 22:

I

I

B

P1

P2

+

!

I

I

B

P1

P2

+

−

physics 281, fall semester, 2011 principles of solid state ...msg.byu.edu/281/281pack.pdf ·...

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