# Temperature dependence of the electronic structure

### This page gives hints on how to calculate the temperature dependence of the electronic structure with the ABINIT package.

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

Mentioned in topic_ElPhonInt, 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.

The electronic structure changes with temperature. In most materials, such changes are mainly driven by the electron-phonon interaction, which is also present at zero Kelvin, inducing the so-called zero-point motion renormalization (ZPR) of the eigenvalues. These effects can be computed thanks to the Allen-Heine-Cardona (AHC) theory [Allen1976],[Allen1981],[Allen1983], which is based on diagrammatic method of many-body perturbation theory. An extension to the standard AHC theory also gives access to the electronic lifetime and decay rates. These physical properties are available from ABINIT since v7.10.4.

The AHC formalism and the implemented equations can be found in [Ponce2014a]. An extended verification and validation study (also versus other first-principle codes) of the ABINIT implementation can be found in [Ponce2014]. The AHC implementation can be used with any XC functional working with the response-function (RF) part of the code, and requires the use of norm-conserving pseudopotentials. NetCDF support is mandatory.

The AHC implementation in ABINIT is built on a Sternheimer approach to efficiently compute the sum over highly energetic bands appearing in the AHC equations [Gonze2011]. Such behavior is controlled by the input variable ieig2rf.

The k-point convergence can be strongly improved by restoring the charge neutrality through the reading of the Born effective charge and dielectric tensor (controlled by the input variable getddb). More information on the importance of charge neutrality fulfillment can be found in [Ponce2015]. The value of elph2_imagden sets the imaginary shifts used to smooth numerical instabilities in the denominator of the sum-over-states expression.

We have checked that the implementation correctly holds for arbitrarily small elph2_imagden parameters, [Ponce2015]. The input variable smdelta triggers the calculation of the electronic lifetime and the value of the smearing delta function can be specified through esmear.

A double grid can be used to speed-up the calculations with getwfkfine or irdwfkfine. The variable getgam_eig2nkq gives the contribution at Γ so that the Debye-Waller term can be computed. This variable is only relevant for calculations of AHC using the abinit program only. It is nonetheless recommended to use the provided python post-processing script temperature_para.py with its module rf_mods.py in the directory scripts/post_processing/ to allow for more flexibility. The python scripts support multi-threading.

The following steps are required to perform an AHC calculation:

- Perform a response function calculation at q=Γ with electric field perturbation.
- Perform phonon calculations and produce the EPC for a large set of wavevectors q, reading the Born effective charge and dielectric tensor with getddb.
- Gather and compute the impact of the electron-phonon coupling on the electronic eigenenergies using the temperature_para.py python script.

The outputs of the script are provided in text and NetCDF format to allow for later reading inside ABINIT. This could be used in the future developments of ABINIT to compute temperature-dependent optical properties for example.

For the temperature dependence of the Fermi energy, see topic_ElPhonTransport.
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### 2. Related lesson(s) of the tutorial.

- A lesson has been developed on the temperature dependence of the electronic structure.
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### 3. Related input variables.

Compulsory input variables:

... ieig2rf [Integer for second-order EIGenvalues from Response-Function]
... smdelta [SMeared DELTA function]

Basic input variables:

... bdeigrf [BanD for second-order EIGenvalues from Response-Function]
... elph2_imagden [ELectron-PHonon interaction at 2nd order : IMAGinary shift of the DENominator]

Useful input variables:

... esmear [Eigenvalue SMEARing]
... getddb [GET the DDB from ...]
... getgam_eig2nkq [GET the GAMma phonon data EIG2NKQ from dataset]
... getwfkfine [GET the fine grid wavefunctions from _WFK file]
... irdwfkfine [Integer that governs the ReaDing of the grid _WFK file on the 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: t28.in

tests/v6/Input: t37.in t40.in t50.in t53.in t57.in t59.in t69.in

tests/v7/Input: t50.in t51.in t52.in t53.in t54.in t55.in t56.in t57.in t58.in t59.in
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### 5. References.

[Allen1976] P. B. Allen and V. Heine, "Theory of the temperature dependence of electronic band structures", J. of Phys. C: Solid State Physics 9, 2305 (1976).
[Allen1981] P. B. Allen and M. Cardona, "Theory of the temperature dependence of the direct gap of germanium", Phys. Rev. B 23, 1495–1505 (1981).
[Allen1983] P. B. Allen and M. Cardona, "Temperature dependence of the direct gap of Si and Ge", Phys. Rev. B 27, 4760–4769 (1983).
[Gonze2011] X. Gonze, P. Boulanger and M. Côté, "Theoretical approaches to the temperature and zero-point motion effects on the electronic band structure.", Annalen der Physik 523, 168 (2011).
[Ponce2014] S. Poncé, G. Antonius, P. Boulanger, E. Cannuccia, A. Marini, M. Côté and X. Gonze, "Verification of first-principles codes: Comparison of total energies, phonon frequencies, electron–phonon coupling and zero-point motion correction to the gap between ABINIT and QE/Yambo", Comp. Mat. Sci. 83, 341 - 348 (2014).
[Ponce2014a] S. Poncé, G. Antonius, Y. Gillet, P. Boulanger, J. Laflamme Janssen, A. Marini, M. Côté and X. Gonze, "Temperature dependence of electronic eigenenergies in the adiabatic harmonic approximation", Phys. Rev. B 90, 214304 (2014).
[Ponce2015] S. Poncé, Y. Gillet, J. Laflamme Janssen, A. Marini, M. Verstraete and X. Gonze, "Temperature dependence of the electronic structure of semiconductors and insulators", J. Chem. Phys. 143, 102813 (2015).

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