# Transport properties from electron-phonon interaction

### This page gives hints on how to compute transport properties that are determined by the electron-phonon interaction (electrical resistivity, superconductivity, thermal conductivity) with the ABINIT package.

##### Copyright (C) 2016-2017 ABINIT group (MV)

Mentioned in topic_ElPhonInt, topic_TDepES, help_features#6.

### Table of content:

- 1. Introduction.
- 2. Related lesson(s) of the tutorial.
- 3. Related input variables.
- 4. Selected input files.
- 5. References.

### 1. Introduction.

Warning : this topic concerns metals only.

The calculation of bulk transport quantities (electrical and thermal resistivities - the part that is determined by the electron-phonon interaction) is possible using anaddb. Analogous quantities are obtained from the conducti post-processor, but due to electron-electron scattering, instead of electron-phonon.

A preliminary calculation of the derivatives of the wavefunctions with respect to k-vector must be carried out. After generating a GKK file (see topic_ElPhonInt), the Electron-Phonon Coupling (EPC) analysis is performed in anaddb, setting elphflag variable to 1. Most of the procedure is automatic, but can be lengthy if a large number of k-points is being used.

For the superconductivity calculations, The electron-phonon interaction is interpolated in reciprocal space, then integrated over the Fermi surface to give the Eliashberg function. Several quadrature methods are available. The default (telphint=1) is to use Gaussian weighting, with a width elphsmear. Another option is the improved tetrahedron [Bloechl1994a] (telphint=0). Finally (telphint=2), one can integrate a given set of electron bands, between ep_b_max and ep_b_min. The resulting integrated quantities are the Eliashberg function (in a file suffixed _A2F), and the EPC strength λ which is printed in the main output file.

The transport calculation is turned on by setting ifltransport to 1 in anaddb. The transport quantities depend on the Fermi velocity for each band, and the electronic band-dependence of the matrix elements must be preserved before integration, by setting ep_keepbands to 1. This increases the memory used, by the square of the number of bands crossing EF. The results are the transport Eliashberg function (in file _A2F_TR), the electrical resistivity (in file _RHO), and the thermal conductivity (in file _WTH).

It is also possible to consider the temperature dependence of the Fermi energy: cubic spline interpolation (ep_nspline) enables to linearly interpolate the transport arrays and reduce the memory usage. Besides setting the Fermi level with elph_fermie (in Hartree), it is also possible to specify the extra electrons per unit cell, (i.e., the doping concentration often expressed in cm-3) with ep_extrael.

Some details about the calculation of electron-phonon quantities in ABINIT and ANADDB can be found here.

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### 2. Related lesson(s) of the tutorial.

- The lesson on the electron-phonon interaction presents the use of the utility MRGKK and ANADDB to examine the electron-phonon interaction and the subsequent calculation of superconductivity temperature (for bulk systems).
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### 3. Related input variables.

Compulsory input variables:

... elphflag@anaddb [ELectron-PHonon FLAG]

Basic input variables:

... ep_keepbands@anaddb [Electron-Phonon KEEP dependence on electron BANDS]
... ifltransport@anaddb [IFLag for TRANSPORT]
... kptrlatt@anaddb [K PoinT Reciprocal LATTice]
... telphint@anaddb [Technique for ELectron-PHonon INTegration]

Useful input variables:

... a2fsmear@anaddb [Alpha2F SMEARing factor]
... elph_fermie@anaddb [ELectron-PHonon FERMI Energy]
... elphsmear@anaddb [ELectron-PHonon SMEARing factor]
... ep_b_max@anaddb [Electron Phonon integration Band MAXimum]
... ep_b_min@anaddb [Electron Phonon integration Band MINimum]
... ep_extrael@anaddb [Electron-Phonon EXTRA ELectrons]
... ep_nspline@anaddb [Electron Phonon Number for SPLINE interpolation]
... mustar@anaddb [MU STAR]
... prtfsurf@anaddb [PRinT the Fermi SURFace]
... prtvol@anaddb [PRinT VOLume]

Input variables for experts:

... band_gap@anaddb [BAND GAP]
... ep_nqpt@anaddb [Electron Phonon Number of Q PoinTs]
... kptrlatt_fine@anaddb [K PoinT Reciprocal LATTice for FINE grid]
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### 4. Selected input files.

WARNING : as of ABINITv8.6.x, the list of input files provided in the specific section of the topics Web pages is still to be reviewed/tuned. In some cases, it will be adequate, and in other cases, it might be incomplete, or perhaps even useless.
The user can find some related example input files in the ABINIT package in the directory /tests, or on the Web:

tests/v5/Input: t88.in t89.in t90.in t91.in t92.in t93.in t94.in t95.in t99.in

tests/v6/Input: t76.in t77.in t93.in t94.in t95.in

tests/v7/Input: t88.in t93.in t94.in
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### 5. References.

[Bloechl1994a] Peter E. Blöchl, O. Jepsen and O. K. Andersen, "Improved tetrahedron method for Brillouin-zone integrations", Phys. Rev. B 49, 16223–16233 (1994).
DOI: 10.1103/PhysRevB.49.16223.

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