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## Electron-phonon coupling (EPC)

Moderators: mverstra, aromero

jerkov
Posts: 28
Joined: Tue Nov 26, 2019 6:02 pm

### Electron-phonon coupling (EPC)

Hi all,

I'm confused on where to start on an electron-phonon coupling (EPC) calculation. I'm working with a semiconductor, so it seems like the tutorial Eph (https://docs.abinit.org/tutorial/eph/) is not well suited for my problem. Also, this tutorial calculates the Eliashberg function (alpha^2F and lambda), whereas I want the matrix elements of the electron-phonon coupling g_mnv(k,q) that appear in the Hamiltonian H = isolated electronic problem + second quantized phonon problem + g_mnv(k,q)(fermion creation operator_k+q * fermion annihilation operator_k * (boson annihilation operator_q + boson creation operator_-q).

I've read in a review paper that this quantity can "be calculated using an alternative, variational formulation of density-functional perturbation theory (Gonze, Allan, and Teter, 1992; Gonze, 1995a, 1997; Gonze and Lee, 1997). A thorough discussion of the connection between the Sternheimer approach and the variational approach to DFPT is provided by Gonze (1995b)."

But I've looked through those papers and am unable to find any information on how to calculate the matrix elements. Is this something that I would have to calculate in my own post-process code? What elements of the anaddb output would I need to combine? Are there any examples?

***EDIT***
This paper (Liu, A. Y., & Quong, A. A. (1996). Linear-response calculation of electron-phonon coupling parameters. Physical Review B, 53(12), R7575–R7579. doi:10.1103/physrevb.53.r7575 ) references the matrix elements I'm talking about and says The electron-phonon matrix elements, g(nk,n'k',nu), are easily computed from the first-order change in the self-consistent potential." How do I set up a run that would allow me to take a derivative of the first-order changes in the self-consistent potential? What output files would I need and how would they be organized?
***************

Cheers,
J

gmatteo
Posts: 291
Joined: Sun Aug 16, 2009 5:40 pm

### Re: Electron-phonon coupling (EPC)

Also, this tutorial calculates the Eliashberg function (alpha^2F and lambda), whereas I want the matrix elements of the electron-phonon coupling g_mnv(k,q)
The anaddb implementation is mainly designed for metals (conventional superconductors or transport properties in the normal metallic state within the LOVA approximation to the Bolztmann equation).
The code reads e-ph matrix elements produced by the DFPT code (GKK or WF1) files.
I've read in a review paper that this quantity can "be calculated using an alternative, variational formulation of density-functional perturbation theory (Gonze, Allan, and Teter, 1992; Gonze, 1995a, 1997; Gonze and Lee, 1997).
Right, the e-ph matrix elements are essential a byproduct of the DFPT calculation. The DFPT code computes <k+q|H(1)|k> where H(1) is the first order (SCF) Hamiltonian induced by a single atomic displacement with wavevector q,
Anaddb reads these matrix elements and builds the e-ph matrix element <k+q| \Delta V_{q\nu} | k> in the phonon representation.
But I've looked through those papers and am unable to find any information on how to calculate the matrix elements. Is this something that I would have to calculate in my own post-process code? What elements of the anaddb output would I need to combine? Are there any examples?
The main question is "what quantity do you want to compute with these e-ph matrix elements" as there are several physical properties whose evaluation requires the knowledge of the e-ph matrix elements.
Note that we are gonna have a new implementation of EPH properties that is directly interfaced with Abinit.

Some of these features have been briefly discussed in https://www.sciencedirect.com/science/a ... 5519303741
(see the Section "Progress in electron–phonon calculations with Abinit"),
For a more detailed analysis and comparison between the new implementation and anaddb see
the more recent https://www.sciencedirect.com/science/a ... 5519303741

If you are interested in phonon-limited mobilities, I would suggest to have a look at
https://arxiv.org/pdf/2002.00630.pdf

Not all the features (and optimizations) are available in Abinit9.0, nevertheless one can compute the electron-phonon self-energy and phonon-induced lifetimes with the new implementation.
A new tutorial for E-Ph is being written and it will be released in the next version.

jerkov
Posts: 28
Joined: Tue Nov 26, 2019 6:02 pm

### Re: Electron-phonon coupling (EPC)

Hi Dr. Matteo,

Thanks for the thorough response. I checked out "The Abinit Project: Impact, environmen, and recent developments." From the description in Sect 3.3, it sounds like Abinit already does what I want: "To address these limitations, abinit v8 provides a new driver explicitly designed to compute the EPH matrix elements and related physical properties. A different philosophy is used, in which EPH matrix elements are computed directly starting from the basic ingredients, namely, the ground-state wavefunctions stored in the WFK file, and the first-order change of the Kohn-Sham (KS) potential produced by the DFPT code." I have v.8.10.2, so should have these capabilities.

I want to calculate the e-ph coupling constants themselves, as defined in eqn 1 in that Liu paper. They are also defined in eqn 4 of this paper (10.1103/PhysRevLett.55.837) where the pseudopotential total-energy method was introduced to calculate the e-ph coupling constants g_mnv(k,q).

It sounds like with the proper flags and control of databases, Abinit stores the matrix elements that you wrote <k+q| \Delta V_{q\nu} | k>. So my question is how do I access those? What would their units be? Does the calculation differ from the eph tutorial that is aimed at metals?

Cheers,
J