In a system ofNatoms the number of variables which describe the motion is 3 N(x, y and z compo-
nents). So there are also 3 Nphonons (there are as many phonons as primitive lattice vectors in the
first Brillouin zone). As a result the number of modes for phonons is conserved and furthermore there
is also conservation in the electron case. The number of quasiparticles is the same as the number of
free electrons. This quasipartcicles can have the same spin, charge and k vectors as the
electrons.
11.2 Particle like Quasiparticles
11.2.1 Electrons and Holes
11.2.2 Bogoliubov Quasiparticles
11.2.3 Polarons
A polaron is the simplest kind of quasiparticle which describes the interaction between electrons and
phonons. The combination of the electrons and the screening field around them defines the polaron.
In ionic crystals this effect is very important because the Coulomb-interactions between the ions and
the electrons are very high. Therefore in covalent crystals this effect just has a small impact because
the interactions between neutral atoms and electrons are very weak.
There are two types of polarons. The first one is calledlarge polaronand was described by Fröhlich:
This polaron looks like an electron which moves through the crystal with a particular effective mass.
The electron pushes the negative ions away and pulls on the positive ions (the whole system is still
moving). The polarization field is spread out and as a result the energy of the whole system does not
change dramatically. The second type aresmall polaronswhich were described by Holstein. This
type is similar to a mott insulator. The electrons get trapped from the ions and at high temperature
(thermal activation) the electrons hop from one place to another. At low temperature the electrons
are tunneling very slowly through the crystal.
11.2.4 Bipolarons
One way to describe superconductivity is via Bipolarons. Bipolarons consist of two polarons which
bind together. The reason is that they reduce the polarization field (sharing of the polarization field).
The bipolarons have two electrons with spin^12 but the total spin is integer so they are bosons.
11.2.5 Mott Wannier Excitons
11.3 Collective modes
11.3.1 Phonons
Phonons are a probably the most basic quasiparticle emerging in solids. They characterize the vibra-
tion of the lattice. More on phonons can be found in chapter 6.