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.

Lecture Audio

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.

Lecture Audio

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.

Lecture Audio

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.

Lecture Audio

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.

Lecture Audio

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.

Lecture Audio

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.

Lecture Audio