How is the Electron-Phonon Interaction Hamiltonian derived?

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In summary, there are multiple books that provide derivations of the Electron-Phonon Interaction Hamiltonian, such as "Methods of Quantum Field Theory in Statistical Physics" by Abrikosov, Gorkov and Dzyaloshinski and "Theoretical Solid State Physics" by Haug Volume II. Other resources include "Introduction to Solid State Physics" by Charles Kittel and a book by Bir and Pikus on symmetry and strain-induced effects in semiconductors. Additionally, Frohlich's work on the Hamiltonian approach is also useful.
  • #1
quantumlaser
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Does anyone have a good derivation of the Electron-Phonon Interaction Hamiltonian? I've found a few in various books (specifically B. Ridley and others that cite him) and on the internet, but all of them seem to skip some non-trivial steps or introduce quantities that I've never encountered.
On that note, how does one define creation/annihilation operators for phonons? They have no mass (they're not really even particles), so the operators must be different than the standard ones used in the QHO solution. Is the mass the mass of the atoms in the lattice, or maybe some reduced mass?. Thanks.
 
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  • #2
quantumlaser said:
Does anyone have a good derivation of the Electron-Phonon Interaction Hamiltonian? I've found a few in various books (specifically B. Ridley and others that cite him) and on the internet, but all of them seem to skip some non-trivial steps or introduce quantities that I've never encountered.
On that note, how does one define creation/annihilation operators for phonons? They have no mass (they're not really even particles), so the operators must be different than the standard ones used in the QHO solution. Is the mass the mass of the atoms in the lattice, or maybe some reduced mass?. Thanks.

The derivation is in many books. For example, "Methods of Quantum Field Theory in Statistical Physics" by Abrikosov, Gorkov and Dzyaloshinski.
 
  • #3
There's another derivation in 'Theoretical Solid State Physics' by Haug Volume II. Not sure if it has the steps you need but there is a large section devoted to it.

Can I enquire what's you're interest?
 
  • #4
gareth said:
Can I enquire what's you're interest?

I'm trying to study electron-lattice interaction in multiple quantum well structures. Many papers start with an electron-phonon Hamiltonian, but I'm not sure how to derive it. I'm a graduate student in physics, but my background in solid state is fairly weak since I recently switched fields.
 
  • #5
Electron-phonon interaction was considered as ealy as
Bloch theory of metals in 1920 years. The calculations of electron-phonon amplitude You can find for example in:
Sommerfeld A., Bethe H. A., Handbuchder Physik, 2e
Auf1 24(2), 333 (1933).
But it was not hamiltonian approach.

Frohlich showed that hamiltonian writings has some advantages
of E-PH interaction
Frohlich H., Proc. Roy. Soc., 215A, 291 (1952).

The best methodical (to my mind) approach is in the book by Haken:
Quantum Field Theory of Solids: An Introduction (Paperback)
by H. Haken (Author), i has only russian edition of 1980y

See also Appendix J in:
CHARLES KITTEL Introduction to Solid State Physics SEVENTH EDITION (1996).
 
  • #6
Look for a book by Bir and Pikus, it is amongst the best at working through the electron-phonon interactions and their derivations. My advisor made me work it from cover to cover before he'd let me go to work for him on a dissertation topic.
 
  • #7
Which book do you mean, exactly? I could only find one:

"Symmetry and Strain-induced Effects in Semiconductors"

Is that the one you are talking about?
 
  • #8
olgranpappy said:
Which book do you mean, exactly? I could only find one:

"Symmetry and Strain-induced Effects in Semiconductors"

Is that the one you are talking about?

Yes... It was so long ago, I forgot the exact title and when I looked for it today along with a couple of other things (like my copy of my dissertation) they are packed away in some box in my basement maybe never to see the light of day again...
 

Related to How is the Electron-Phonon Interaction Hamiltonian derived?

1. How does electron-lattice interaction affect the properties of materials?

The electron-lattice interaction, also known as electron-phonon interaction, plays a crucial role in determining the properties of materials. This interaction can affect various properties such as conductivity, thermal conductivity, and magnetic properties. It can also lead to phenomena such as superconductivity and ferroelectricity.

2. What is the mechanism behind electron-lattice interaction?

The electron-lattice interaction is a result of the coupling between electrons and lattice vibrations, also known as phonons. This coupling occurs due to the electric field generated by the movement of electrons, which in turn affects the positions of the atoms in the lattice. This creates a feedback loop between the electrons and lattice vibrations, leading to the interaction.

3. How does temperature affect the strength of electron-lattice interaction?

The strength of electron-lattice interaction is directly proportional to temperature. As the temperature increases, the thermal energy also increases, resulting in more vigorous lattice vibrations. This leads to a stronger coupling between electrons and lattice vibrations, resulting in a stronger electron-lattice interaction.

4. What role does electron-lattice interaction play in the formation of energy bands in materials?

The electron-lattice interaction is responsible for the formation of energy bands in materials. The coupling between electrons and lattice vibrations leads to the creation of new energy states, resulting in the formation of energy bands. These energy bands determine the electronic and optical properties of a material.

5. How does the strength of the electron-lattice interaction vary in different materials?

The strength of electron-lattice interaction varies in different materials based on their crystal structure and chemical composition. Materials with a higher number of atoms in the unit cell and higher atomic masses tend to have a stronger electron-lattice interaction due to the increased number of lattice vibrations. Additionally, the presence of impurities and defects can also affect the strength of this interaction.

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