Understanding & Solving Electron/Phonon Interaction Problems

In summary, the conversation discusses the need for a book with solved example problems for a calculation involving unperturbed electron and phonon Green functions, as well as electron/phonon interactions. The speaker has been able to prove particle conservation in the absence of interactions but is unsure how to do so in general. They mention a paper from 1994 that addresses the problem at zero temperature, but their work requires finite temperature treatment. The speaker also mentions Van Leeuwen's text as a potential resource, but there are differences in approach between different authors in the field. The expert summarizes the main points of the conversation and provides an explanation of the conservation laws in the electron-phonon case.
  • #1
bjnartowt
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TL;DR Summary
How to prove from non-equilibrium field-theory that the retarded, advanced, <, and > self-energies calculated using the self-consistent Born approximation give currents which conserve mass and conserve energy?
I am wondering if there is a book with solved example problems I can follow so that I can carry out the following calculation:

I have a set of unperturbed electron Green functions $g^{r0},g^{a0},g^{<0},g^{>0}$ and phonon Green functions $d^{r0},d^{a0},d^{<0},d^{>0}$ which are for a central region $C$ adjoined by a left and right reservoir $L,R$ each at a temperature $T_{L,R}$ and potential $\mu_{L,R}$. I use the self-consistent Born approximation to calculate $g^r,g^a,g^<,g^>$ and $d^r,d^a,d^<,d^>$ due to an electron/phonon interaction from three diagrams: one Hartree-like diagram and one Fock-like diagram for an electron colliding with a phonon and one polarization-bubble-like diagram for a phonon exciting a charge-neutral electron/hole pair. I get band-densities of number currents $N_{L,R}(\omega)$ and energy currents $E_{L,R}(\omega)$ for charge-carriers of energy $\hbar\omega$ from the equations $i\hbar(dN_{L,R}/dt)=[N_{L,R},H]$ and likewise for $E$.

I am able to prove that $N_L=N_R$ in the absence of interactions. However, I am not sure how to prove this in general.

Note that Annals of Physics 236, 1-42 (1994), Eq. (54) through Eq. (56) does treat this problem except for zero temperature. Finite temperature is critical for what I am doing.

(At this zero temperature case treated in Annals of Physics 236, 1-42 (1994), there are delta functions, and conservation of matter results from a fortuitous cancellation on integrating the band-density $N_{L,R}(\omega)$ to give the number-current $N_{L,R}$).

Van Leuwenn's text seems to give the material I need to solve this problem, but the authors seem to use a GW method. This is different from Hyldgaard et al, whose starting point seem to be Langreth. An earlier paper, Phys Rev B 46 11, published by roughly the same research group give their Eq. (76) and Fig. 1 as justification for current-conservation, which I am having trouble understanding.
 
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  • #2
Dear bjnartowt,

Very interesting question! There are several approaches to show if a considered many-body approximation is conserving. Personally, I find the easiest-to-understand method is to consider the time-dependent case, and specialize to the steady-state case afterwards. In this language, particle conservation means that the continuity equation is satisfied locally at every given time, and in steady-state with two terminals means ##I_L = I_R##. Leeuwen's approach is the one of Baym, and considers generating functionals for the self-energy, ##\Phi##-functionals. In the electronic case, ##\Phi = \Phi[g]## is a functional of the interacting electronic Green's function, but in your case, the functionals will be ##\Phi = \Phi[g,d]##. The electronic self-energy is then ##\Sigma(12) = \frac{\delta \Phi[g,d]}{\delta g(21)}##. Finite temperature is then included as a Matsubara track.

The conservation laws will follow from symmetries of the ##\Phi##-functional. Again, taking particle-number conservation as an example, this will follow from the symmetry (this is taken from memory, could be slightly off) ##\Phi[e^{\Lambda} g e^{-\Lambda},d] = \Phi[g,d]##. ##\Lambda## is a small quantity, and this symmetry (which is fulfilled in your case because all your electron loops are closed) gives the particle conservation (for the electrons!) you want, according to Baym's original argument.

The electron-phonon situation is very similar to the ##GW## approximation if you view the phonon as a screened interaction (electronic self-energy is gd). As such, the ideas from the electronic case applies also to the electron-phonon case.

I hope this helps a bit.
 
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1. What is the significance of electron/phonon interactions in materials?

Electron/phonon interactions play a crucial role in determining the physical properties of materials. These interactions affect the thermal, electrical, and optical properties of materials, making them important for various technological applications.

2. How do electron/phonon interactions affect the thermal conductivity of materials?

Electron/phonon interactions contribute to the thermal conductivity of materials by transferring heat energy through the lattice vibrations of phonons. This process is known as phonon scattering and can either increase or decrease the thermal conductivity, depending on the strength of the interactions.

3. Can electron/phonon interactions be controlled or manipulated?

Yes, electron/phonon interactions can be controlled and manipulated through various techniques such as doping, strain engineering, and surface modifications. These methods can alter the strength and nature of the interactions, leading to changes in the material's properties.

4. What are some common techniques used to study electron/phonon interactions?

Some common techniques used to study electron/phonon interactions include Raman spectroscopy, infrared spectroscopy, and inelastic neutron scattering. These methods allow scientists to probe the vibrations and energy levels of phonons and electrons, providing valuable insights into their interactions.

5. How can understanding and solving electron/phonon interaction problems benefit technology?

Understanding and solving electron/phonon interaction problems can lead to the development of materials with improved thermal, electrical, and optical properties. This can have a significant impact on various technologies, such as thermoelectrics, optoelectronics, and energy storage devices.

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