Solid State Physics

Final Review 1

April 27, 2006 17:34

This is the first of a 2-part review for the final exam. HREF="http://128.210.157.22:1013/Boilercast/2006/Spring/PHYS545/0101/PHYS545_2006_04_27_0900.mp3">Lecture Audio

A metal in a magnetic field has its Fermi sea sectioned into onion-like layers, shaped like cylinders. These are Landau levels, due to the harmonic oscillator motion of electrons moving in circular orbits in a magnetic field. (They're only "circular" for free electrons, and can have funny shapes for electrons in a real metal.) We show a very easy way to spot the quantum harmonic oscillator in this type of problem. The quantum nature of the harmonic oscillator leads to the quantized "Land...

Lecture 25: Vortices

April 20, 2006 17:29

There are many more phases of matter than solid, liquid, and gas. Superconductivity is a different phase of matter, and superconductors in the vortex state are yet again another phase of matter. We study vortices today, what they are, and how they happen. HREF="http://128.210.157.22:1013/Boilercast/2006/Spring/PHYS545/0101/PHYS545_2006_04_20_0900.mp3">Lecture Audio

Lecture 24: Condensation Energy

April 18, 2006 17:27

When superconductors go superconducting, the energy gain is called the condensation energy. Lecture Audio

Lecture 23: Superconductivity

April 13, 2006 14:01

The quantum stability of a superconductor ensures that electrons can carry current perfectly, without losing energy. There are 2 ingredients to this physics: 1. Electrons pair into "composite bosons"; 2. The bosonic pairs all fall into the same lowest energy wavefunction (called Bose condensation.) Since bosons don't obey the Pauli exclusion principle, they can all occupy the same wavefunction macroscopically -- that's right, you might get 10^23 bosons in the same wavefunction. Once they...

Lecture 22: Antiferromagnets

April 11, 2006 18:50 - 11.1 MB

We finish off the low temperature corrections to the magnetization in a ferromagnet due to spin wave excitations, and also calculate the energy and heat capacity of spin waves. Now, on to antiferromagnets, where neighboring spins are antialigned. We derive the susceptibility, and the spin wave dispersion. Due to technical difficulties, I post last year's audio: Lecture Audio

Lecture 21: Mean Field Approach to Ferromagnetism

April 06, 2006 18:50

We started off today with a demonstration of Barkhausen Noise in ferromagnets. (Your refrigerator magnets are ferromagnets.) If you've ever used a permanent magnet to magnetize a paperclip, you know that not all magnetic materials have a discernible north and south pole. Rather, as with paperclips, many ferromagnets have instead a "domain structure" -- there are many regions in the paperclip which are magnetized, but the many domains point in different directions, and the paperclip doesn't ...

Lecture 20: Spin Waves are the Goldstone Modes of Ferromagnets

April 04, 2006 18:48

Ferromagnets spontaneously break a continuous symmetry -- that is, when the net magnetization develops, it must choose a particular direction to point. But raise the temperature to disorder this, then lower it again, and -- surprise! -- the magnetization will now form in a different direction. You already know that when a continuous symmetry (here, the rotational symmetry) is broken, the system has Goldstone modes. (See Lectures 1 and 3.) The Goldstone modes of a ferromagnet are called sp...

Lecture 19: Pauli Paramagnetism and Intro to Ferromagnets

March 30, 2006 19:46

How many electrons get polarized when you apply a magnetic field to a metal? Is it all the electrons inside the Fermi surface? It turns out that only a small fraction of the electrons are able to respond -- most are stuck deep inside the Fermi surface, and the Pauli exclusion principle does not allow the spins to flip in response to the magnetic field. This is Pauli paramagnetism, and we derive the corresponding magnetic susceptibility (how easy it is to magnetize something). We also ...

Lecture 18: Paramagnetism and Diamagnetism

March 28, 2006 19:45

Magnetic moments in a solid come from the electronic spin, and also its orbital angular momentum. We review how the orbital angular momentum contributes to the magnetic moment. We also use Atom in a Box by Dauger Research www.daugerresearch.com to show how this net angular momentum can arise from adding, say, p orbitals together in the right way. We also show how diamagnetism arises from atomic cores. Every material is weakly diamagnetic (meaning it resists having a magnetic field pene...

Lecture 17: Magnetization of Paramagnets

March 21, 2006 19:42

Paramagnets have magnetic moments whose directions fluctuate wildly with temperature. But, if you apply an external magnetic field, you can align the moments, and the paramagnet develops a net magnetization. Turn the external field off, and the paramagnet loses its magnetization. We calculate the Curie susceptibility -- how easy it is to magnetize a paramagnet by applying a net magnetic field. Lecture Audio

Lecture 16: Paragmagnetism

March 09, 2006 16:08

There are many flavors of magnetism in solids. You're probably most familiar with ferromagnets (like your refrigerator magnets). In these materials, tiny atomic current loops (atomic electromagnets) align in order to create one larger magnet. What we talk about today is the case where the magnetic moments are too far apart to communicate how to align with each other. Rather, the moments point any which way with temperature, which is referred to as a paramagnetic phase. We discuss the ori...

Lecture 15: Continuity Equations

March 07, 2006 15:03

We derive the Einstein relations, which connect the conductivity with the diffusion coefficient. This is far more exciting than it sounds, because it's a consequence of the far-reaching fluctuation-dissipation theorem. Another instance of this theorem happens with Brownian motion, and the applet we used in class can be found at http://galileo.phys.virginia.edu/classes/109N/more_stuff/Applets/brownian/brownian.html We also derive the continuity equations in a semiconductor, and see how fast...

Lecture 14: Band Bending

March 02, 2006 14:52

We answer that question: can you use a p-n junction to run a light bulb? More about the p-n junction: thermal equilibrium, and recombination of carriers. When a voltage is applied to a p-n junction, large currents flow if the junction is "forward biased", but if you switch the sign of the applied voltage, the current response is very small. You can use this to build a rectifier. We also discuss band bending, and calculate the form of the voltage across the junction using Poisson's equ...

Lecture 13: p-n Junctions

February 28, 2006 14:31 - 10.5 MB

We talk more about holes today. They don't really exist, you know! But when only a few electrons are missing from the valence band, it's so much more convenient to describe only the missing states that the fictional particles we call "holes" are a very useful concept. We talk more about their mass, velocity, momentum, and other properties. Then we discuss the p-n junction, where a semiconductor surface is donor-doped on, say, the right, and acceptor-doped on, say, the left. We calculate ...

Lecture 12: Semiconductors

February 23, 2006 14:24

Today is all about semiconductors. We talk about how to dope them. Donor atoms "donate" electrons into the conduction band, giving n-type semiconductors, with mostly electrons carrying current. Acceptor atoms "accept" atoms from the valence band, leaveng holes there. These are p-type semiconductors. We also discuss the effective mass of the electrons and holes in a band, and how to calculate it. (It changes based on the curvature of the band -- that's right, the electron might act more ...

Lecture 11: Metals, Insulators, and Semiconductors

February 21, 2006 18:24

Electronic energy levels in simple crystalline solids have a bandstructure to them. (Bandstructure is just energy vs. wavevector or momentum.) Depending on the filling of the bands, the material can either become a metal, insulator, or semiconductor. Metals have partially filled bands. Insulators and semiconductors have a filled band at zero temperature, with an energy gap to the next band. Good insulators have such a large gap (about 5eV or more) that even room temperature is not enough...

Lecture 10: Tight Binding Approximation

February 17, 2006 03:02

We solve for the electronic states in a 1D crystal in the "tight binding" approximation. Rather than starting from the box of free electrons and adding the lattice in slowly (i.e. as a quantum mechanical perturbation), we work from the other limit today. This time, we'll assume the electrons are far from free, rather they're tightly bound to each atom. Start with a 1D cystal where the atoms are infinitely far apart, and we know the ground state of each electron. Now, slowly bring the ato...

Lecture 09: Bloch's Theorem

February 15, 2006 02:56

Have you ever wondered how electrons can sneak through a metal and conduct electricity with all those atoms in the way? It's Bloch's theorem. The electrons organize themselves into the right quantum mechanical states that automatically take into account the periodicity of the crystal. Electrons in a metal are shared by each atom in a type of molecular bond that extends over the whole crystal. These are the states which carry current. This lecture is heavy on the quantum mechanics -- you'...

Lecture 8: Wiedeman-Franz Ratio and Electrons in a Lattice

February 09, 2006 16:52

We give some intuition today about when you should expect the Wiedemann-Franz ratio (which relates the electrical to the thermal conductivity in a metal) to hold, and when you should expect a deviation from the ratio we calculated for free electrons. (The ratio holds at low and high temperatures, but deviates from the free electron picture in between.) We also introduce a crystalline lattice into our free electron box today. We'll do this through adding the lattice in perturbatively (tha...

Lecture 7: Conductivity

February 07, 2006 16:47

Today, we derive the electronic heat capacity in metals. This gives a contribution to the heat capacity that is linear in temperature. Phonons gave a T^3 dependence, and so this can distinguish the 2 contributions to the specific heat. We also discuss how to measure the occupied density of states through X-ray measurements, as well as the effective mass of an electron inside of a solid. (The effective mass is one important way that we correct the free electron picture for use in a real s...

Lecture 6: Debye Approximation and Free Electron Model

February 02, 2006 15:48

The Debye approximation is a way of calculating phonon properties. Here's the approximation: 1. Pretend the phonon dispersion is linear. 2. Set a high frequency cutoff ωD = Debye frequency that gets the total number of modes in the system correct. That's it -- now you're guaranteed to get both the low temperature and high temperature limits of the heat capacity correct. We also start the free electron model. Free means the electrons do not interact with each other's Coulomb potentials, or t...

Lecture 5: Heat Capacity

January 31, 2006 15:53 - 24.9 MB

We define the heat capacity, and calculate the phonon heat capacity in the high and low temperature limits. We also introduce the density of states. Technical difficulties meant that this lecture did not get recorded this year. In its place, I post last year's lecture 5: Lecture Audio

Lecture 4: Diatomic Chain

January 19, 2006 16:12

We discuss generalities of phonon spectra. These include: frequency goes to zero at the reciprocal lattice vectors; group velocity goes to zero at the zone edge; frequency goes linear in k for small frequency; all physical modes are contained in the first Brillouin zone. We derive the dispersion relation for a linear chain of 2 distinct atom types. We discuss the quantization of phonons, and embark on a lightning fast review of the harmonic oscillator. A great java app showing how the h...

Lecture 3: Reciprocal Lattice

January 17, 2006 15:53

We review lattice planes, and talk about how to construct the corresponding Miller indices. We define the reciprocal lattice: Think of this as the Fourier wavevectors of the original lattice. It turns out that the reciprocal lattice of a Bravais lattice is itself a Bravais lattice. We define the first Brillouin zone. We calculate the dispersion (frequency vs. wavevector) for phonons in a 1D crystal in the harmonic approximation, and encounter our first instance of Goldstone's theorem: S...

Lecture 2: Bravais Lattices

January 13, 2006 02:58

A lattice is a regular arrangement of an infinite set of points in space. A Bravais lattice is one where every point looks the same as every other point. You can build any lattice from a Bravais lattice by "decorating" it, in which case we call it a lattice with a basis. We show how to construct the Wigner-Seitz cell, a particular type of unit cell. Roger Penrose, mathematician, came up with a way to tile space that has (in a manner of speaking) five fold symmetry, and never repeats. The ...

Lecture 1: The Failure of Reductionism

January 10, 2006 16:36

Reductionism is the idea that by breaking things into their smallest constituents, we will learn all about them. For example, we might want to learn about solids by breaking them into atoms, then learn about the atoms by breaking them into the constituent electrons and nuclei, and so on. But reductionism is merely a philosophy handed down to us by the Greeks -- is it really correct? New ideas in the field point toward the failure of reductionism, and lead to "emergence" as a better paradig...