Download Density of states

Density of states
A Material is known to have a high density of states at the Fermi energy.
(a) What does this tell you about the electrical, thermal and optical properties of this material?
(b) Which of the following quasiparticles would you expect to observe in this material? (phonons, bipolarons,
excitons, polaritons, suface plasmons) Why?
(a): A high density of states at the fermi energy means that this material is a good electrical conductor. The
specific heat can be calculated via the internal energy.
Z ∞
Z ∞
E · D(E)
dE
E · D(E) · f (E)dE =
u(E, T ) =
(1)
−∞ 1 + exp E−µ
−∞
kB ·T
We know that cv =
du
dT .
So we get the following expression:
Z
∞
cv =
−∞
E · D(E) · (E − µ) · exp E−µ
kB T
dE
2
kB T 2 1 + exp E−µ
kb T
(2)
Hence we deal with a metal, we will have a good thermal conductor because of the phonon and electron
contribution. Light will get reflected out below the plasma frequency ωP .
(b): Due to fact that phonons describe lattice vibrations, it is possible to observe them in this material. They
can be measured with an EELS experiment or with Raman Spectroscopy.
A polaron describes a local polarisation of a crystal due to moving electrons. They are observable in materials
with a low electron density and describe a charge-phonon coupling. There are two different kinds of polarons,
the Fröhlich Polaron and the Holstein Polaron. The Fröhlich polarons describe large polarons, that means the
distortion is much larger than the lattice constant of the material so that a lot of atoms are involved. This
polarons typically form bands. The distortion of the holstein polarons is about the size of the lattice constant.
This small polarons appear in soft materials like organic semiconductors. The elecrons are localized, a charge
transport is possible via hopping or tunneling (not enough overlap of the distortions).
A bipolaron describes a bound pair of two polarons. One won’t expect the polarons to form bipolarons, simply
Figure 1: Distortion caused by a polaron in KCl[1]
because they are both electrons. But the energy win by sharing the distortion field is big enough to overcome
the Coulomb interaction. Bipolarons share the properties of bosons. Polarons and bipolarons do have a large
effect in ionic crystals, for neutral atoms the interaction with the electron is very weak, and can’t be observed.
1
Excitons are describing electron-hole pairs. There are 2 kinds of excitons, the Matt Wannier and the Frenkel
excitons. The distance between the electron and the hole is much higher than the lattice constant for the Mott
Wannier electrons. These excitons are typically found in semiconductor crystals with small energy gaps and
high dielectric constants. The Frenkel excitons have a much smaller distance between the hole and the electron.
The Frenkel excitons can be found in solid krypton, organic solar cells and organic light emitting diodes. For
our material electron-hole pairs would not exist because the density of states at the Fermi energy is not zero.
Figure 2: Mott Wannier exciton. Electron hole density is much higher than the lattice constsant[1]
A polariton describes the coupling between photons and phonons. To observe polaritons the material needs
at least 2 atoms per unit cell. This is required because we need an optical phonon branch to couple the photons
with. The acoustic branch is simply not possible because the dispersion relations only cross at k = 0 and
ω = 0. In ionic crystals it would be possible to observe them. In our case the photons wouldn’t be able to get
into the material because the electron density at the fermi energy is not zero and thus the photons would get
reflected out.
Figure 3: Photon and phonon dispersion relation. The photon dispersion relation is normally much steeper[2]
Plasmons describe the oscillation of a charge in a solid. One can think of a fixed positiv charge and electrons
that were displaced from their equilibrium state.
Surface plasmons describe the interaction between the surface electrons and an electric field (on a metaldielectric interface). They can be excited by electrons or photons. The charges on the surface respond to the
electric field of the light-wave, it tries to keep it out of the metal but in the dielectric is a finite electric field
and this wave can propagate along the surface of the metal. They can be observed in our material.
Figure 4: Surface plasmons. Electromagnetic wave propagating along the metal-dielectric interface[3]
2
References
[1] Introduction to Solid State Physics, Charles Kittel
[2] Advanced Solid State Physics, Lectures Summer 2012 http://lamp.tu-graz.ac.at/~hadley/ss2/
lectures12/june15/slide2.html
[3] Advanced Solid State Physics, Lectures Summer 2012 http://lamp.tu-graz.ac.at/~hadley/ss2/
lectures12/june13/slide14.html
3