Final Review 1

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

Lecture 26: Landau Levels

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 "Landau levels" of the electrons.

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Lecture 25: Vortices

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

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

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Lecture 23: Superconductivity

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're there, they're very hard to disturb (that's quantum stability), and in this phase of matter, electrons can carry current without energy loss.

We show a video of a magnet levitating over a superconductor (called the Meissner effect), available at
www.fys.uio.no/super/levitation


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Lecture 22: Antiferromagnets

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:
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Lecture 21: Mean Field Approach to Ferromagnetism

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 act like a permanent magnet. But you can magnetize it, by rubbing it with a permanent magnet. As you do so, you align domains. We used the Barkhausen experiment to hear the domains flip! We wound a pickup coil (lots of wire loops) around the object to be magnetized, and hooked the wire up to a speaker. You can find out more about this setup at www.simscience.org.

We also passed around magnets of various strengths. The weakest magnets were transition metal based (like iron), because the individual magnetic moments are weak. The strongest moment was neodymium-based. Neodymium (Nd) has a large magnetic moment, because it has unfilled f-shells. These and other "rare earth" magnets are surprisingly strong, and pinch your finders if you're not careful! You can buy your own rare earth magnets to play with at Edmund Scientifics.

Then we discussed the mean field theory of ferromagnetism. Mean fields aren't cruel.
What we mean is "average", in the sense that each spin in the system feels an average, effective field due to its neighboring spins. This modifies our equations for magnetization, and we're able to show using this "mean field theory" that when ferromagnets form as temperature is lowered from the disordered paramagnetic phase, the magnetization rises continuously.


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Lecture 20: Spin Waves are the Goldstone Modes of Ferromagnets

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 spin waves. These are waves of precession of the magnetic moments.

Lecture 19: Pauli Paramagnetism and Intro to Ferromagnets

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 begin to study ferromagnets -- these are your refrigerator magnets.

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Lecture 18: Paramagnetism and Diamagnetism

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 penetrate) due to screening currents which come from the atomic cores.

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Lecture 17: Magnetization of Paramagnets

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.

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Lecture 16: Paragmagnetism

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 origin of the magnetic moments (they come from the electron's spin and orbital angular momentum), and calculate the magnetization that results when a magnetic field is applied to the solid.

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Lecture 15: Continuity Equations

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 they can screen a stray charge that may be produced by, say, thermal fluctuations. We highlight 2 applications of semiconductors: LED's and solar cells.
We discuss a bit the impending energy challenge, referring to a
talk by Nobel Laureate Richard Smalley. Sadly, he has
passed away.


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Lecture 14: Band Bending

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 equation. Why do we say the bands bend? When any 2 substances are in contact and achieve equilibrium, they trade particles until the chemical potentials are equal. The same thing happens for the holes and electrons in the p-n junction -- particles diffuse across the junction until the chemical potentials are equal. This causes the built-in voltage, and the conduction and valence bands "bend" in response.

p-n Junctions are in Ch. 17 of Kittel, pgs. 503-513.
Even better, see Vol III, Ch. 14 of the Feynman Lectures.

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Lecture 13: p-n Junctions

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 the strength of the permanent electric field that happens at the interface. This permanent electric field produces a real live voltage in the material. Can you use it to run a light bulb?

Due to technical difficulties this year, I post last year's lecture:

Lecture Audio

Lecture 12: Semiconductors

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 or less free inside the material, but act as though it has a different mass.) We look at the physics of how the dopant atoms "ionize" to contribute carriers to the semiconductor.
There's also a way to measure whether the semiconductor's main carriers are holes or electrons, using the Hall Effect.

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Lecture 11: Metals, Insulators, and Semiconductors

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 to excite electrons across the gap into the next highest energy band. But semiconductors have lower band gaps (about 1eV), so that at room temperature, there are many electrons excited into the next band. The missing electrons in the lower band are called holes. Holes aren't real particles, they're just missing electrons -- but we can treat them as if they were real particles with positive charge. We also introduce how to dope semiconductors into n-type and p-type semiconductors.

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Lecture 10: Tight Binding Approximation

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 atoms together, and electrons will begin to hop from one atom to the next. We treat this hopping as the quantum mechanical perturbation. Remarkably, this approach also gives a bandstructure similar to the case where we started from free electrons! Both limits have a little bit of truth to them, and some combination of these 2 effects happens in real materials.

As a special bonus, we play with the excellent program "Atom in a Box", available at www.daugerresearch.com. (Mac only.) We use the program to illustrate the difference between a stationary state (a state with a pure energy -- i.e. an eigenstate of the energy), and a nonstationary state (a state with a mixed energy -- one that adds 2 eigenstates with different energies.) Be sure and play with the program -- you'll see that the nonstationary states have a "wobbly' probability density. That is the sense in which they are not stationary.

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Lecture 09: Bloch's Theorem

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'll hear about eigenstates and eigenvalues, and how in a crystal, the eigenstates of energy can be simultaneously diagonalized with the eigenstates of translation in the crystal. These give the states mentioned above, where the electrons are delocalized throughout the whole crystal.

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

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 (that is, pretend the Coulomb attraction between electron and lattice is very weak compared to the Fermi energy of the free electron system). We find this causes band gaps to open in the electronic energies at the Brillouin zone boundaries. These are caused by the formation of standing electronic waves due to reflections in the crystal.

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Lecture 7: Conductivity

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 solid.) We derive the electrical conductivity (the response to an electric potential gradient), and the thermal conductivity (the response to a thermal gradient) for the free electron picture. The famous Wiedemann-Franz ratio relates the two, and remarkably can be expressed in terms of fundamental constants. That's right -- measure 2 things about a metal, divide them, and you'll get an answer only dependent on fundamental constants. Solids are amazing.

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Lecture 6: Debye Approximation and Free Electron Model

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 the Coulomb potential of the crystalline lattice. Rather, they have only kinetic energy, and only "interact" through the Pauli Exclusion Principle. Remarkably, this gives a good starting point for describing the behavior of real metals.

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Lecture 5: Heat Capacity

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:

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Lecture 4: Diatomic Chain

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 harmonic oscillator eigenstates actually connect back to oscillatory motion is at:
http://www.fi.uib.no/AMOS/MOV/HO/

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Lecture 3: Reciprocal Lattice

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: Since crystals break the continuous translational symmetry of free space, there is a corresponding Goldstone mode (the phonons, or vibrations of the crystal), which has the important property that the frequency goes continuously to zero.

Be sure and play with the Solid State Simulations package by Cornell, available at: http://www.ruph.cornell.edu/sss/sss.html

Demos: Today, we used the "Bravais" SSS program in class.
Visual Aid: The squishy crystal model -- actual balls and springs.

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Lecture 2: Bravais Lattices

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 patterns are beautiful. Be sure and google Penrose tiles.

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Lecture 1: The Failure of Reductionism

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 paradigm for gaining knowledge in condensed matter/solid state physics. Emergence is the idea that when many particles get together, new phenomena appear which are not encoded in the microscopic laws, and in fact are independent of the microscopic laws. For example, all solids are hard, regardless of what atoms are in them. Likewise, if we changed the microscopic laws, by changing, say, the shape of the Coulomb potential, we'd still get solids that are hard. That means hardness (and many properties like it) is not caused by the microscopic laws, but rather is caused by deeper physics. It turns out in fact that it's related to symmetry.

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