University of Minnesota
School of Physics & Astronomy

Phys 5701.001

Solid-State Physics for Engineers and Scientists

Final Exam
modified 16-May-2018 at 6:28PM by Paul Crowell

The final exam is posted below. It will be due by 5:00 PM, Friday, May 11. The exam is three hours, closed book. There are a bunch of sketches to make, and a ruler is therefore essential. A calculator is useful.

I will check my email reasonably regularly (except Saturday evening), in case there are problems.

To hand in the exam,

If you complete the exam before Thursday at 5:00, you can hand it in at my office or slip under the office door. I will take all exams handed in before that time and give them to the grader.


1. Scan your answers to a single PDF file.
2. Submit at the following link: Moodle Link for Final

If there is an issue with submission, please contact Yuting Wang directly at I will be traveling next Friday and will not be in good email contact.


Final Exam Solutions | Download posted 16-May-2018 at 6:28PM
Final Exam | Download posted 8-May-2018 at 9:52PM

Problem Sets
modified 3-May-2018 at 9:54PM by Paul Crowell
Problem Set 10 | Download posted 25-Apr-2018 at 8:12PM
Problem Set 9 | Download posted 14-Apr-2018 at 9:01PM
Problem Set 8 | Download posted 2-Apr-2018 at 11:07PM
Problem Set 7 | Download posted 22-Mar-2018 at 7:28PM
Problem Set 6 | Download posted 10-Mar-2018 at 1:40PM
Problem Set 5 | Download posted 25-Feb-2018 at 12:20AM
Problem Set 4 | Download posted 13-Feb-2018 at 4:24PM
Problem Set 3 | Download posted 5-Feb-2018 at 7:52PM
Problem Set 2 | Download posted 2-Feb-2018 at 12:21AM
Problem Set 1 | Download posted 16-Jan-2018 at 6:05PM
Problem Set Solutions
modified 3-May-2018 at 9:54PM by Paul Crowell

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Lecture Slides
modified 3-May-2018 at 10:30PM by Paul Crowell

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Week 15 and Final Exam
modified 30-Apr-2018 at 2:22PM by Paul Crowell

This week's lecture will discuss optical effects in solids. There is no additional reading (see Week 14). I will note, however, an article of direct relevance (on excitons in quasi-2D semiconductors) in this month's Reviews of Modern Physics:

The last problem set is due Thursday, May 3rd.

The final exam, which is equal in weight to a third mid-term, will be available this Friday evening and will be due the following Friday at 5 PM (instructions for handing in to be determined). I am attaching last year's final, which was similar in format. This year's will be shorter (probably 5 problems). Note that last year I covered magnetism and plasmons, but this year we did not.

Final and solutions from 2017 | Download posted 30-Apr-2018 at 1:55PM

Problem Set 10
modified 25-Apr-2018 at 8:12PM by Paul Crowell

These problems address (mostly) topics in nanostructures. There is one question on optics. It will be due Thursday, May 3.

Problem Set 10 | Download posted 25-Apr-2018 at 8:11PM

Quiz 2 and Solutions
modified 25-Apr-2018 at 3:58PM by Paul Crowell

You can take up to three hours, although the quiz is not nearly that long! There are no extensive calculations. As always, clear explanations of your reasoning are very important. The quiz is closed book, but you can use a calculator, and a rule will be useful for one problem.

If you are confused by a question, stop and send me an email. I will try to check my email regularly on Saturday and Sunday.

Quiz 2 Solutions | Download posted 25-Apr-2018 at 3:58PM
Quiz 2 | Download posted 21-Apr-2018 at 4:08PM

Week 14
posted 23-Apr-2018 at 8:24PM by Paul Crowell


Kittel, Chapter 15 (Optical Processes and Excitons). The section on Kramers-Kronig is a bit mathematical. I will discuss the overall philosophy, which is extremely important, in class, without drowning in mathematics.

Other important points in this chapter: excitons, Raman scattering, and various electron spectroscopies, about which I will have less to day. You can skip the section on electron-hole droplets.

Ashcroft and Mermin does not have a good chapter on optics. A fantastic reference, which is on the web as an online book form a University of Minnesota IP address, is Yu and Cardona, "Fundamentals of Semiconductors":

Chapters 6 and 7 cover optics. They are admittedly more advanced than Kittel, but one can still learn a lot without picking up every detail.

I will post the last problem set (due Thursday, May 3), in the next couple days.

Week 13
modified 16-Apr-2018 at 12:26PM by Paul Crowell

Reading: Kittel, Chapters 17 and 18 (Surfaces and Interfaces, Nanostructures)

These cover a broad range of topics, mostly at a superficial level. Some material may refer to quantum mechanics you do not know, particularly the sections on electrons in a magnetic field (e.g., quantum Hall effect). My goal here is simply to expose you to the phenomenology, and I do not expect you to understand it all. If you find something too heavy-going, just skip it.

I am posting Chapter 4 of Davies, which is a compendium on the solutions of the Schrodinger equation for various forms of quantum confinement. This is good as a reference. It is essentially "just math," but I will emphasize some points in class.

Problem Set 9 (due Thursday 4/19) is now posted. Note that I posted a revised version on Saturday. The revised version contains a hint on Problem 3 and makes clear that you need to know the donor density for Problem 1(a).

I will post Quiz 2 on Thursday.

Davies, Chapter 4
modified 14-Apr-2018 at 9:14PM by Paul Crowell

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Quiz 2
modified 10-Apr-2018 at 6:16PM by Paul Crowell

Quiz 2 will be available Thursday, April 19th and will be due the following Tuesday. It will be a timed take-home like the first quiz. It will cover material up through Problem Set 8.

I am attaching Quiz 2 from last year and solutions.

Quiz 2 from 2017 | Download posted 10-Apr-2018 at 6:16PM

Week 12
modified 6-Apr-2018 at 8:10PM by Paul Crowell

We will continue with semiconductors, moving onto semiconductor heterostructures. There is going to be some material in Tuesday's lecture (and on Problem Set 8) that will be unfamiliar to those who have not had statistical mechanics, but I will do my best. In the end, recognizing the role of the Boltzmann factor and whether or not the energy scale is set by the gap or the donor (or acceptor) binding energy is going to be sufficient.

I am attaching pages from Ashcroft and Mermin as well as Davies (the chapter on heterostructures) that you should read over the next week. I will do the p-n junction on Thursday. Given the composition of the class, I am going to do more optics this year, and I will come up with some readings for the following week.

I apologize for some of the very uncreative problems on Problem Set 8. Spend your time thinking about the Anderson localization problem.

modified 6-Apr-2018 at 8:10PM by Paul Crowell

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Week 11
modified 2-Apr-2018 at 11:08PM by Paul Crowell

As a reminder, I will hold an office hour on Monday, April 2, from 1:30 - 3:15. Problem Set 7 is due Tuesday. Problem Set 8 is now posted. It will be due Thursday, April 12.

This week, we will go back to Chapter 8 (semiconductors), but first we will conclude our discussion of transport in the "semi-classical model" and its application to metals (particularly in the presence of a magnetic field). To this end, it will be helpful to read through p. 194 by Tuesday's lecture.

Note the key fact: \hbar dk/dt = F, where F is the external force (e.g. the Lorentz force). This may puzzle you, because \hbar k is NOT the electron momentum, and so this does not appear to be consistent with F=ma at first glance. The truth comes out on p. 193. The electron is acted on by both the external force and the periodic potential. When both the lattice and the electrons are included (see Eqs. 8.15 and 8.16), then \hbar k is a momentum (the crystal momentum).

The semiclassical model leads to some reasonably intuitive phenomena such as magneto-oscillations, which are really just an extension of cyclotron motion to periodic systems, and some non-intuitive ones, such as Bloch oscillations (p. 217). In the latter case, a constant electric field leads to oscillations in the group velocity. If you understand Bloch oscillations, then you understand crystal momentum.

I am adding below a "simple" derivation of the expression (Kittel 9.37) for the period of magneto-oscillations. In class,I will just write down this expression, but for Fermi surfaces with a circular cross-section, it is possible to derive it accepting only the fact that the energies must be quantized in units of the cyclotron energy.

Magneto-oscillations | Download posted 1-Apr-2018 at 12:46PM

Week 10
modified 29-Mar-2018 at 12:57AM by Paul Crowell

We will do metals (Chapter 9) before semiconductors (Chapter 8). Read Chapter 9, particularly 221 - 242. The remainder of the chapter (after p. 242) may be heavy sailing. Do not worry about it. I will return to the topic of electrons in a magnetic field in a couple weeks. If you wish, you can skip this section for now.

Pages 232 - 240 cover various approaches to calculating band structure. Among these, tight binding is by far the most important, and is the only one I will discuss in any detail.

Problem Set 7 is posted. It will probably be due Tuesday, April 3. I will have an office hour from 1:30 - 3:15 on Monday, April 2nd.

Week 9
modified 9-Mar-2018 at 3:10PM by Paul Crowell

I am assuming that you will read the section on magneto-transport and the Hall effect (pp. 152 - 155), which I did not get to in lecture. We will start with Chapter 7 on March 20. This chapter is in a sense the crux of the class. What happens to electronic states (originally planes waves in an empty box) when we "turn on" the periodic potential? Although this is a difficult problem to solve analytically (except in some special cases), there turn out to be some global rules (such as Bloch's theorem) that allow us to make sense of what is going on. My goal is for you to understand why gaps form, the relation to symmetry, and how this impacts the behavior of real materials.

Problem Set 6 is now posted and will be due on March 27th.

Week 8
modified 1-Mar-2018 at 9:31PM by Paul Crowell

We will continue with Chapter 6. In all likelihood I will not start Chapter 7 until after spring break, but read pp. 163 - 167 by Thursday, March 8.

The solutions to Quiz 1 are now posted.

Professor Valls will lecture on Tuesday. The topics will be thermodynamics of the electron gas (e.g. specific heat) and transport in the relaxation time approximation.

I will return on Thursday (I hope) and do magnetotransport (a first pass), thermal transport, and perhaps an aside on optical properties of metals.

Problem Set 5 is due Thursday, March 8.

Quiz 1 is posted here.
modified 9-Mar-2018 at 6:48PM by Paul Crowell

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Week 7
modified 25-Feb-2018 at 8:17PM by Paul Crowell

Tuesday's lecture will cover some more topics in quantum mechanics, some of which appear in Kittel's Chapter 6, including spin, Fermi-Dirac statistics and the Fermi-Dirac distribution. I will then continue with a discussion of Dirac notation and matrix mechanics.

Thursday's lecture will cover most of the remainder of Chapter 6, through approx. p. 152.

Problem Set 5 is posted. It will be due Thursday, March 8.

My office hour this week will start at 3:15, and I will shift it from 3:15 - 5:00.

Week 6
modified 17-Feb-2018 at 12:37PM by Paul Crowell

Problem Set 4 is posted and will be due Friday, February 23rd, by 5 PM. You may either put the problem set under my office door (PAN 220) or upload a SINGLE PDF file to the class Moodle Page:

The lecture on Tuesday, February 20 will cover material in the second part of Chapter 5, including the generalization of the phonon density of states, van Hove singularities, anharmonicity, and thermal transport. We will start Chapter 6 (electrons) the week of February 27.

There will not be a lecture on Thursday, February 22nd.

I cannot overemphasize the importance of the density of states, and how we start with a density of states in k-space, which is always (L/2\pi)^d, (where d is the dimensionality) and, with knowledge of the dispersion relation E vs. k, calculate the density of states with respect to energy (or frequency). Depending on the type of particles we are dealing with (phonons, electrons, etc.) the dispersion relation and hence the form of the density of states with respect to energy will vary.

In contrast, the details of the Debye calculation are not important. If you have not had statistical mechanics, do not worry about it. You should know that the acoustic phonon contribution to the heat capacity scales as (T/\Theta_D)^d (where d is the dimensionality). We will discuss the contribution from optical phonons on Tuesday.

Quiz 1
modified 14-Feb-2018 at 11:43AM by Paul Crowell

The first quiz will be available (by download from the web site) next Thursday. It will be similar in format to the quiz from last year, which is posted here. This year's quiz will only cover material up through Chapter 5 (phonons).

I will probably give you 2 hours, but I am inclined to make it closed book (so that I can ask easier questions!). You can take it when you wish and hand it in on Tuesday, February 27.

There will not be a lecture on Thursday, February 22nd.

A&M Periodic Table (goes with quiz) | Download posted 14-Feb-2018 at 11:43AM
Quiz 1 from 2017 | Download posted 13-Feb-2018 at 4:30PM

Week 5
modified 14-Feb-2018 at 10:06AM by Paul Crowell

Problem Set 4 is posted and will be due Friday, February 23rd, by 5 PM. You may either put the problem set under my office door (PAN 220) or upload a SINGLE PDF file to the class Moodle Page:

Reading: Kittel Chapters 4 and 5. Analogous chapters in Ashcroft and Mermin are Chapters 22 - 24.

Tuesday's lecture: Phonon dispersion relations, optical and acoustic modes, phonons and momentum

Thursday's lecture: Phonons: thermal properties

Problem Set 3 is due Thursday

Week 4
modified 8-Feb-2018 at 2:34PM by Paul Crowell

Reading: Kittel, Chapter 4. This material is covered (much more thoroughly) in Ashcroft and Mermin, Chapter 22. A&M have four chapters on phonons. Neutron scattering gets its own chapter (24) as do anharmonic effects (25).

Tuesday's Lecture: We will spend the first few minutes discussing a couple points from the problem set. After that, I will turn to topics in quantum mechanics (the PHYS 5012 students can leave at that point).

Thursday's Lecture: I will conclude the brief review of chemical bonding and then start on phonons.

Problem Set 3 will be posted shortly. It will be due Thursday, February 15th.

Week 3
modified 2-Feb-2018 at 12:21AM by Paul Crowell

Problem Set 2 is posted. It will be due Tuesday, February 6th.

Because on Thursday I mutilated the discussion of the Miller indices (hkl) in terms of the shortest reciprocal lattice vector perpendicular to a set of lattice planes, I am attaching a discussion here.

Both lectures this week will be on "core material" (i.e., no special quantum mechanics lecture). I will probably start the material in Chapter 3 of Kittel (crystal binding) in the second half of Thursday's lecture. I will have relatively little to say about this beyond what is in the text. For those looking for more, I recommend Chapters 19 and 20 of Ashcroft and Mermin.

Note on Superlattice Problem | Download posted 2-Feb-2018 at 12:21AM
Structure Factor and Intensity of the Diffraction Peaks | Download posted 30-Jan-2018 at 11:17PM
Miller indices and the Bragg condition | Download posted 27-Jan-2018 at 12:12PM

Office Hours
modified 18-Jan-2018 at 7:19PM by Paul Crowell

My office hours will be Wednesday 2:45 - 4:30 and by appointment.

Week 2
posted 18-Jan-2018 at 7:11PM by Paul Crowell

The lecture on Tuesday, January 23 will continue the "primer on quantum mechanics" from last Tuesday. I will cover Hilbert space, Dirac notation, operators, expectation values, and the matrix formulation of quantum mechanics. I will probably not get to spin.

The lecture on Thursday, January 25th will continue our discussion of crystal structure, with the concept of reciprocal space and the basics of diffraction.

Problem Set 1 will be due on Thursday, January 25th. Note that the last problem (on diffraction) does not have to be handed in this Thursday. It is on there to give you a head start on next week.

Week 1
posted 16-Jan-2018 at 6:01PM by Paul Crowell

Reading: Kittel, Chapters 1 and 2
Alternatives: Simon, Chapters 12 and 13, Ashcroft and Mermin (A&M), Chapters 4 - 7

We will spend the first two weeks on "structure and symmetry," with an emphasis on the important crystalline systems and the notion of reciprocal space. I am not going to emphasize the mathematical aspects of symmetry, such as point groups and space groups. If you are interested, Ashcroft and Mermin address these (slightly) in their Chapter 7.

The first problem set is posted. Because we are interleaving the first couple lectures on quantum mechanics, I may adjust the due date, but for the moment assume it is due Thursday, January 25.