How is the Electron-Phonon Interaction Hamiltonian derived?

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Discussion Overview

The discussion centers around the derivation of the Electron-Phonon Interaction Hamiltonian, exploring various sources and methods for understanding this concept within the context of solid-state physics. Participants express interest in both theoretical and practical aspects of electron-phonon interactions, particularly in relation to quantum well structures.

Discussion Character

  • Exploratory
  • Technical explanation
  • Homework-related
  • Debate/contested

Main Points Raised

  • Some participants seek a clear derivation of the Electron-Phonon Interaction Hamiltonian, noting that existing resources often skip important steps or introduce unfamiliar quantities.
  • Questions arise regarding the definition of creation and annihilation operators for phonons, particularly due to their lack of mass and how this affects their treatment compared to standard quantum harmonic oscillator solutions.
  • References to various texts are provided, including "Methods of Quantum Field Theory in Statistical Physics" by Abrikosov, Gorkov, and Dzyaloshinski, and "Theoretical Solid State Physics" by Haug, suggesting these may contain relevant derivations.
  • Historical context is introduced, mentioning early considerations of electron-phonon interactions in Bloch's theory and later developments by Frohlich, highlighting the evolution of the Hamiltonian approach.
  • Recommendations for further reading include works by Bir and Pikus, as well as Haken's "Quantum Field Theory of Solids," indicating these texts may offer thorough explanations of electron-phonon interactions.
  • Participants express varying levels of familiarity with solid-state physics, with some indicating a need for foundational knowledge due to recent transitions into the field.

Areas of Agreement / Disagreement

Participants generally agree on the complexity of deriving the Electron-Phonon Interaction Hamiltonian and the need for comprehensive resources. However, there is no consensus on a single derivation method or the best reference material, as multiple sources are suggested and some participants express uncertainty about their adequacy.

Contextual Notes

Some discussions highlight limitations in existing literature, such as missing steps in derivations or the introduction of unfamiliar concepts, which may affect understanding. Additionally, the varying levels of expertise among participants may influence the clarity of the discussion.

Who May Find This Useful

This discussion may be useful for graduate students and researchers in physics, particularly those focusing on solid-state physics, electron-phonon interactions, and related theoretical frameworks.

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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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.
 
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?
 
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.
 
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).
 
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.
 
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?
 
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...
 

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