Set of supplementary notes.

CONTENTS 7
Outline
“Any new discovery contains the germ of a new industry.”
(J. J. Thomson)
In this course, we will take a tentative first step towards working with electrons in solids. This
requires skills you have acquired in other courses, in particular in electromagnetism, quantum
mechanics and statistical physics.
The course is structured as follows: roughly the first half of Lent term consists of a progression
of more and more sophisticated models, each fixing the deficiencies of its predecessor. We
will start by considering electrons in insulators (Lorentz oscillator model) and metals (Drude
model) by very intuitive classical approaches, which get quite a number of things right. We
will then introduce quantum statistics (Sommerfeld model) to correct those things which go
most spectacularly wrong in the classical approach. We will then introduce the crystalline
lattice, which the Sommerfeld model does not take into account, and ponder its symmetry
properties and its vibrations. Arguably the toughest point of the course is the introduction
of Bloch’s theorem and the calculation of electronic energy levels (or band structure) in the
presence of a periodic potential caused by the lattice atoms. This is done in two ways, (i)
using the nearly free electron gas approach, and (ii) using linear combination of atomic orbitals
(or tight binding), which is technically easier. Having reached this point, it is downhill again,
considering band structures of real materials and how band structures can be determined experimentally.
Lent term will conclude with an introduction to how all of this can be applied to
build semiconductor-based devices.
The models discussed up to this point in the course neglect correlations between the electrons:
having computed the single-electron energy eigenstates of a solid, we then fill up these
states with the available electrons, ignoring the mutual interaction between the electrons. We
will apply these models in order to understand semiconductor-based electronic devices such as
p-n junction based diodes, field effect transistors and solar cells.
Because the mobile electrons are strongly diluted in semiconductors, neglecting the effects
of the mutual repulsion between the electrons can be justified. This can be very different
in metals, however, which can be regarded as dense electron liquids. We find that in many
metals, the strong electron-electron interaction can cause new collective phenomena. The most
prominent examples are magnetism, of which there are many varieties, and superconductivity,
but there are many other possible forms of electronic quantum self-organisation. We will take a
first step towards understanding the driving force behind such collective states by considering
charge and spin instabilities in metals, and we will discuss a very general conceptual framework
for working with strongly interacting fermionic systems, in many ways the standard model of
condensed matter physics, namely Landau’s Fermi liquid theory.