1st Brillouin Zone

Everything within the lattice may be described as something within the 1st Brillouin zone of the reciprocal lattice, displaced by a vector G. This is how we get the "reflections" in the reduced zone of the energy- k plots seen in the notes and lecture.

Assumption of Weak Potential

There are a number of instances where the assumption of a weak potential gives an accurate description of electrons moving through a lattice, that is, the electrons behave almost as free electrons. There are two reasons why this is sometimes the case.
Firstly, the interaction of electrons with the ions is much stronger at small separations, the conduction electrons are separated from the ions by the core electrons that surround it, and so the interaction is weak. Secondly, in this region in which the conduction electrons are allowed, their mobility can have the effect of screening the fields of the ions, making the potential experienced by any given conduction electron very weak.

Photonic Crystals

Recently we covered Bloch's theorem and its application to the motion of electrons in a periodic potential, i.e. our crystal lattice of ions. The theorem however is not strictly contained to crystal structures, as it only discussed the translationally invariant Hamiltonians and the resulting form of wavefunctions.

An electromagnetic analogue of the above behaviour is found in optical nanostructures which exhibit a periodic potential, due to a periodic dielectric of the material. The behaviour of the EM field in the structure is found to mirror that of the electrons in the crystal structure.

Although we haven't yet covered it, the greatest similarity is between the Bloch treatment of semi-conductors and this photonic crystal. Analogous to the semi-conductor, forbidden energy zones or wavelength band gaps are formed. This gives rise to a range of behaviours such as inhibition of spontaneous emission and and high reflecting omni-directional mirrors.

Bloch lecture notes

You can download the most recent version of my lecture notes here.

Tutorial: Week 5

During tomorrow's tutorial we're going to look at some of the questions from "Tutorial 3: Crystal Structures and the Reciprocal lattice". To get the most benefit out of the tutorial please have a look at the questions beforehand if you have the chance.

Assignment 3: due Monday April 11

You can download the assignment here.

A scan of the relevant pages from Solid State Simulations is here.

Trying to Understand Bloch's Theorem "Oxford Dictionary"

A theorem relating to the quantum mechanics of crystals stating that the wave function ψ for an electron in a periodic potential has the form ψ(r) = exp(ik · r)U(r), where k is the wave vector, r is a position vector, and U(r) is a periodic function that satisfies U(r+R) = U(r), for all vectors R of the Bravais lattice of the crystal. Bloch's theorem is interpreted to mean that the wave function for an electron in a periodic potential is a plane wave modulated by a periodic function. This explains why a free-electron model has some success in describing the properties of certain metals although it is inadequate to give a quantitative description of the properties of most metals. Bloch's theorem was formulated by the German-born US physicist Felix Bloch (1905–83) in 1928.