How is the Electron-Phonon Interaction Hamiltonian derived?

[quantumlaser]

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.

 

[olgranpappy]

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.

 

[gareth]

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?

 

[quantumlaser]

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.

 

[Minich]

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, 2^e
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).

 

[Dr Transport]

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.

 

[olgranpappy]

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?

 

[Dr Transport]

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...