MAT 200B. Electronic and Atomic Structure of Materials. Winter 2009

Course info:

Instructor         Prof. Nicola Spaldin, Room 3014 MRL x7920, nicola@mrl.ucsb.edu
Office Hours    Tuesday 1.00 - 2.00 pm, Thursday 4.00 - 5.00 pm

Class Time       Tuesday and Thursday 2-3.45 pm
Place               1335 EII

Discussion Session       Wednesday 9 - 10 am Place: EII 1335
This is an optional discussion session in which Casey and Andrew will work through practice problems with you that will be related to your homework and exam. assignments.

TAs                 Casey Holder, coholder@umail.ucsb.edu    Office Hours: Friday 3 - 4 pm Location: EII 2515

                       Andrew Dittmore, dittmore@engineering.ucsb.edu,      Office Hours: Monday 3 - 4 pm Location: Bldg 489 Rm 133

Recommended Books:

J.D. Livingston, Electronic properties of engineering materials
C. Kittel, Introduction to Solid State Physics
P. A. Cox, The Electronic Structure and Chemistry of Solids
N. Ashcroft and D. Mermin, Solid State Physics
R. Hoffmann, Solids and Surfaces, A Chemist's View of Bonding in Extended Structures
H. Ibach and H. Luth, Solid-State Physics, An Introduction to Principles of Materials Science
N.J.B. Green, Quantum mechanics 1: Foundations
P.A. Cox, Introduction to Quantum Theory and Atomic Structure
J. N. Israelachvili, Intermolecular and Surface Forces


Syllabus:

The free electron model.
Electrical and thermal conductivity. Application of the Fermi-Dirac distribution to the free-electron gas. Electronic specific heat. Motion in magnetic fields. Basic assumptions of the FEM. Failures of the FEM

Review of Fundamental Concepts of Quantum Mechanics.
Bohr atom. Quantum numbers. The Schrodinger equation.

Electron levels in a periodic potential.
Weak periodic potential. Tight binding/Molecular orbital theory. Other methods for calculating band structure. Electron dynamics. The Fermi Surface. The Peierls distortion. Qualitative understanding of band width and band gaps

Classification of Solids.
Spatial distribution of valence electrons. Molecular solids: Coordinated polymeric structures; Fullerenes and fullerides. Ionic solids. Covalent solids. Metallic solids: Simple metals, Transition metals. Hydrogen-bonded solids

Role of electronic structure in atomic bonding.
Primary and secondary bonds. Equilibrium separation. Bond energy. Bond stiffness. The notion of bond valence. Bonding/nonbonding electrons; core vs. valence electrons. Relationship between spectroscopy (energy levels), structure, and properties

Cohesion/Cohesive Energy.
Noble gases, Lennard-Jones. Ionic crystals, Madelung. Covalent crystals. Metals. Intermolecular electrostatic forces. Polymers. Colloids. Solutions.

Semiconductors.
Band structure: s-p bonding, hybrid orbitals. Carrier statistics. Intrinsic and extrinsic semiconductors. p-n junctions. Organic semiconductors

Surfaces, interfaces, and junction effects.
Work functions. Contact potentials. Thermionic emission. Electronic surface levels

Transition-metal compounds.
d electrons. Correlation between electronic structure and magnetism. Crystal fields. Jahn-Teller effects. Diamagnetism. Hund's rules. Paramagnetism. Exchange. The Hubbard Model. Local Moments. The Kondo effect

The noncrystalline state.
Short-range order. The glass transition and free volume. Pair distribution functions. Models: Hard sphere; Random walk; Network; Fractal

The liquid crystalline state.
Structural classes: Nematic; Twisted nematic; Smectic; Columnar. Descriptors. Applications

Interactions in colloids and soft materials.
Interactions of molecules in free space. The kBT criterion for gauging the strength of an interaction. Classification of forces. Dispersion forces, van der Waals. Repulsive potentials. Total intermolecular pair potentials. Screening and DLVO theory. The hydrophobic effect. Entropic effects. The hydrogen bond.


Grading:

Your grade is based on your performance in three components of the course: homework sets (40%), in-class mid-term exam. (30%) and in-class final exam. (30%).

Homework.
Homework sets will be assigned each Thursday or Friday and will be due in class the following Tuesday. To protect your TAs, late homework will not be accepted. Your lowest homework score will not be counted towards your grade to accomodate illness, travel, unexpected circumstances etc. RULES: You are encouraged to work together on finding methods to solve the homework problems but the final write-up must be your own. Identical solutions, even with the variable names changed, will not be accepted even if you were in the same study group. You are also encouraged to use other sources (books other than those in the recommended list, journal articles, web-sites, etc.) in preparing your solutions, but you may not reproduce text or figures from them verbatim, and the sources must be acknowledged.

Mid-term
The mid-term will be a closed book exam. during our regular class session on THURSDAY FEBRUARY 4th. You are allowed to bring in one 8.5 x 11" single-sided page of hand-written (or typed by yourself) notes. Material covered in class up to and including Thursday January 31st will be included on the mid-term.

Final.
The final will be a closed-book exam. and will be held both on Friday March 12th in the late afternoon in a location to be determined, and on Wednesday March 17th from 4 - 7 pm in our usual classroom. You should attend one or other final exam. session of your choice. You may bring in one 8.5 x 11" page of notes with writing on both sides.
 
 

Week by Week Outline and Announcements:

Week 1. Jan 4 - 8

We're going to start with a review of the behavior of metals and try to describe their behavior using Classical Free Electron Theory. We'll see that this does rather well in many cases, but also that it sometimes breaks down. This will motivate us to introduce the Schrodinger Equation so that we can develop the Quantum Mechanical Free Electron Theory (also known as the Free Electron Fermi Gas) and extend this to band theory. I particularly like the following treatments of this topic:

Livingston, Chapter 1 (Classical) and 12 (Quantum Mechanical)
Ashcroft and Mermin, Chapters 1, 2 and 3
Ibach and Luth, Chapter 6

If you don't have any familiarity with quantum mechanics, I'd also suggest that you read through the small book by Cox: Introduction to Quantum Theory and Atomic Structure. This is quite gentle bed-time reading.

Here is Homework 1, due at the beginning of class on Tuesday January 12th.

Here are the solutions.


Week 2. Jan 11 - 15

This week we'll use our knowledge of the Schrodinger equation that we developed last week to derive expressions for measurable quantities of the free electron Fermi gas. Then we'll compare those predictions with experiment. We'll find that, although the Free electron Fermi gas works for many quantities, we need to make one more extension - band theory - to fully describe the behavior of metals.

We'll then make a brief introduction to band theory. We'll find that, in order to proceed, we need a more detailed knowledge of atomic structure. So we'll solve the Schrodinger equation for a one-electron atom, then extend our results to develop a familiarity with the periodic table.

Many introductory quantum mechanics books describe the solution of the Schrodinger equation for a hydrogen atom, then make a qualitative extension to many-electron atoms. One that is quite good (although goes into more detail than we will need) is

K.S. Krane, Modern Physics, Wiley, 1983, Chapters 7 and 8

Here is the problem that will be covered in discussion this week.

Here is Homework 2, due at the beginning of class on Tuesday January 19th.

Here are the solutions.


Week 3. Jan 18 - 22

Now that we understand how the atomic orbitals are arranged in atoms, we'll start putting atoms together to form bonds. We'll use a method called linear combination of atomic orbitals (LCAO), also sometimes called tight-binding, to see how the energy levels are modified from the atomic levels first in diatomic molecules then in solids.

We'll find that the isolated atomic orbitals overlap to form bands of energy levels in solids, and we'll look at how the properties and arrangement of the atomic orbitals determine the band structure of the solids.

Here the solid state chemistry text books have the best description. I particularly like (in this order):

Cox, Electronic structure and chemistry of solids, Chapter 4
Hoffmann, pages 3 - 35 (you can skip the surfaces part)
Livingston, Chapters 10 and 11

Here is the problem that will be covered in discussion this week.

Here is Homework 3, due in class on Tuesday January 26th

Here are the solutions


Week 4. Jan 25 - 29

We've got quite good at being able to work out the band structure - the E versus k diagrams for given arrangements of atoms (mostly we've focussed on a 1D chain!).

Our next major task is to understand how and why the atoms in solids arrange themselves in the way that they do. We'll review the different types of bonds (ionic, covalent, metallic, van der Waals) and show how these lead to different crystal structures. We'll also need to learn some conventions and definitions for describing crystal lattices. This will give our brains a bit of a rest from all of the new concepts from last week and put us in a position to work out band structures for realistic materials!

We'll also start looking at examples of where band structures are useful. In particular we'll use band structures to explain why some materials are metallic, whereas others are insulating or semiconducting. For the metals, we'll explain differences in mobilities, and finally be able to explain the knotty problem of why some metals have positive Hall coefficients.

I like the following references:

Ashcroft and Mermin, Chapter 4
Ibach and Luth, Chapters 1 and 2

Here is the problem that will be covered in discussion this week.

Here is Homework 4, due in class on Tuesday Feb. 2nd

Here are the solutions


Week 5. Feb 1 - Feb 5

My office hours this week are cancelled since I'm traveling.

Discussion this week will be in our regular Tuesday class time; Casey will work through last year's mid-term exam which you can download here .

On Wednesday from 9-10am Andrew will hold office hours in our usual classroon.

Remember that the mid-term exam. is on Thursday during our regular class time.


Week 6. Feb 9 - Feb 13

We'll continue our discussion of realistic band structures, seeing how we can explain them using LCAO thoery, and what information we can extract from them. For semiconductors, we'll look at the relationship between band structure and optical properties. Then we'll show how to understand a certain type of distortion, the Peierls distortion, in terms of the electronic band structure of the material. This will be our first encounter of how the electronic band structure directly determines the physical structure.

Then we'll start a new topic: the formal discussion of the reciprocal lattice (this is the crystal lattice in "k-" or reciprocal-space), and the concept of the Brillouin zone.

The reciprocal lattice and Brillouin Zone concepts are nicely discussed in Appendix B of Cox, as well as in Chapter 5 of Ashcroft and Mermin.

Here is Homework 6, due in class on Tuesday February 16th. (I'm labeling the homeworks by week so there is no Homework 5.)

Here are the solutions to questions 1 and 2 and to to question 3.


Week 7. Feb 15 - Feb 19
We'll start this week by looking at an alternative theory, the nearly free electron model, that also produces band gaps in solids. This one is kind of the "opposite" of LCAO theory, in that it starts with free electrons and subjects them to a weak periodic potential. But the end result, the production of band gaps, is similar.

The nearly free electron model was discussed in Chapter 4 of Cox that we have already used; I also like Chapter 13 of Livingston.

Then we'll start discussing the properties of semiconductors. After a brief review of what we already know, we'll look at the effect on the electronic properties of making semiconductors very small, such as in quantum dots or quantum wells. This will involve a bit more quantum mechanics, and we'll work formally through the solution to the Schrodinger equation for the "particle in a box" problem.

Then, we'll show how to work out the numbers of electrons and holes available to contribute to electrical conduction from a knowledge of the band structure. This will allow us to work out the electrical conductivity and the Fermi energy (which will be very important later when we start making junctions between different semiconductors). We'll introduce the concept of extrinsic semiconductors, and show how chemical dopants can drastically modify their properties.

We'll be following Chapter 12 of Ibach and Luth for the semiconductors component of the class.

Here is the problem that will be covered in discussion this week.

Here is Homework 7, due in class on Tuesday February 23rd.

Here are the solutions


Week 8. Feb 22 - Feb 26

Next we'll build on the formalism that we used to work out the carrier concentrations and Fermi energy in intrinsic semiconductors, and extend it to extrinsic semiconductors. In particular, we need to know how the carrier concentrations and Fermi energy vary with the number and concentration of impurity atoms.

Then we'll work through one example of a semiconductor device, the p-n junction, in some detail. In addition to being important in its own right, this is the first time that we will think about the interesting phenomena that arise at interfaces between materials, even those that are structurally very similar.

We'll be continuing with Chapter 12 of Ibach and Luth, as well as the additional notes on p-n junctions.

Here is the problem that will be covered in discussion this week.

Here is Homework 8, due in class on Tuesday March 2nd.

Here are the solutions


Week 9. March 1 - 5

Next we will move onto another class of material systems that are distinguished from the semiconductors by the importance of d electrons in determining their behavior: The transition metals and transition metal compounds.

We'll start with a discussion of the transition metals. We'll review how the d electrons appear in the electronic structure and determine their effect on the behavior. In particular, our goal is to explain why things like Fe, Co and Ni are ferromagnetic (that is they show a spontaneous and field-switchable magnetization) while things like Na are not.

Then we'll move on to transition metal compounds. Here we will need to understand how the energetics of the d electrons are affected by the surrounding anion environment. We will learn about crystal field and Jahn Teller effects and, just like when we talked about the Peirls distortion, show how the electronic band structure influences the structural distortions in a compound.

The best reference for this week is Cox Chapters 3 (pages 68 - end) and Chapter 5 (pages 134 - 137 and 145 - end).

Here is the problem that will be covered in discussion this week.

Here is Homework 9, due in class on Tuesday March 9th, or Thursday March 11th if you hosted prospective graduate students.

Here are the solutions


Week 10. March 8 - 12.

First we'll finish our discussion of transition metal oxides.

This week we'll look at situations where band theory breaks down and is unable to predict the properties of materials accurately. In particular we'll discuss the Mott transition, in which systems which band theory predicts should be metallic become insulating because of the explicit Coulomb interactions between electrons.