Polarons

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Polarons

• Jakub Wyrodek 2732852

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Polarons 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).
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Creation and annihilation operators(1)

• For further analysis of polarons we have to
  introduce creation and annihilation operators.
• Creation operator
• annihilation operator
•  

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Creation and annihilation operators(2)
Further properties
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Where can we find polarons

• There are four different materials where polarons
  can be created
• Ionic crystals
• Non-polar covalent materials
• Metals
• Polymers

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Polarons 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.

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Polarons 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).

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Polarons 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.
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Polarons 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
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Polarons 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.
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Polarons in metals Fröhlich Model(1)

• To describe polarons in metals Herbert Fröhlich
  proposed such Hamiltonian for plane waves
  

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Polarons in metals Fröhlich Model(2)

• Electron are situated at spatial coordinate x,
  and experience pseudo potential
• from positively charged ions at lattice sites
  Ri

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Polarons 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.

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Polarons 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

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Polarons in metals Fröhlich Model(4)

• - denotes the polarization vectos.
• - eigen-frequencies for various phonon
  branches.
• - annihilation and creation
  operator for phonons.

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Polarons 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.

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Polarons 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

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Polarons 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.

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Polarons 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

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Polarons 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.