Title: Polarons
1Polarons
2Polarons Definition
- The Polaron is a quasiparticle composed of an
electron and polarization field. The slowly
moving electron can interact with lattice ions,
by decreasing the long range forces between them.
We can observe that, as a lattice polarization
deformation. - The process is schematically shown on Fig.1
Fig.1 Basics of polaron creation(1).
3Creation and annihilation operators(1)
- For further analysis of polarons we have to
introduce creation and annihilation operators. - Creation operator
- annihilation operator
- Â
4Creation and annihilation operators(2)
Further properties
5Where can we find polarons
- There are four different materials where polarons
can be created - Ionic crystals
- Non-polar covalent materials
- Metals
- Polymers
6Polarons in metals(1)
- If we want to make proper calculations we need
to choose the right material. In my case it will
be metal. - There are two main models to discuss
- a)The Fröhlich Large Polaron model -
- b)The Holstein Small Polaron.
7Polarons in metals The Fröhlich Large Polaron
model.
- Interaction between electrons and lattice
deformation, results in composite entities of
electrons surrounded by clouds of virtual
phonons. Such entities can be described as large
polaron(spatially extended) or constraint in
space small polaron. - Large polarons are itinerant entities(Fröhlich
Model), while small have the ability to be self
trapped in form of localized states(The Holstien
Small Polaron).
8Polarons in metals Fröhlich Model-Basics
This two pictures show the basics of Fröhlich
Model where wave vector for
the absorption of phonon and
for the emission.
9Polarons in metals Fröhlich Model-Basics
In the first picture the electrons in Bloch state
k have finite life time. They are scattered to k
positions. Thats the fundamentals of metals
resistivity.
Electrons with eigen values and positions can be
modified by movement in the latticeThey gaini
polarization cloud. We call it poloron
10Polarons in metals Fröhlich Model-Basics
When electron emits phonon which can be next
absorbed by another electron. This electron-
electron interaction can be positive what leads
us to superconductivity.
The phonon- properties and effective frequency
are normalized by electron-phonon interaction.
Especially for long time phonon absorption by
electron we can obtain electron-hole creation.
11Polarons in metals Fröhlich Model(1)
- To describe polarons in metals Herbert Fröhlich
proposed such Hamiltonian for plane waves
12Polarons in metals Fröhlich Model(2)
- Electron are situated at spatial coordinate x,
and experience pseudo potential - from positively charged ions at lattice sites
Ri
13Polarons in metals Fröhlich Model(3)
- The dynamical displacement of the lattice by
electron movement can be now described as - Where
- is an equilibrium positions of ions.
14Polarons in metals Fröhlich Model(3)
- We describe as an ion- ion
interaction, which is being assumed in form of
Coulomb potential. - The term denotes the kinetic energy of the
ions on site i. - Now we can write the lattice displacement in more
complicated form
15Polarons in metals Fröhlich Model(4)
- - denotes the polarization vectos.
- - eigen-frequencies for various phonon
branches. - - annihilation and creation
operator for phonons.
16Polarons in metals Fröhlich Model(5)
- So now we can write a Hamiltonian using equation
for lattice displacements - Where and it is
the charge density fluctuation operator for
electrons. -
17Polarons in metals Fröhlich Model(6)
- Modes that are being created from ion-ion
interaction are practically dispersion-less ionic
plasma modes with frequency - But we have to consider also the electron charge
fluctuation, what leads to normalization for - a)Longitudinal acoustic branch with phonon freq.
( - Fermi velocity). - b)Dressed electron-phonon coupling
18Polarons in metals Fröhlich Model(7)
- b)Dressed electron-phonon coupling
- -wave dependent dielectric constant.
- - Coulomb screened renormalization of
bare ion potential. - -spectral weight renormalization for
electron quasi-particle spectrum.
19Polarons in metals Fröhlich Model(8)
- From the last equations we can see that the
Hamiltonian is in very general form. There is a
possibility to take under account weak coupling
limit, for only one phonon accompanying electron.
It s the lowest perturbative approach - Leading to total wave vector
20Polarons in metals Fröhlich Model(9)
- The equation of the total wave vector gives us an
entrance to theory of superconductivity. - It is caused by pairing of electrons. The passage
of first electron polarize the lattice and this
polarization is reabsorbed by another electron.
Hamiltonian describing pairing is given - For interaction is
attractive, leading to instability of the
surface, what creates superconducting ground
state.