# Berry phase computation of electric polarisation and finite electric field

### This page gives hints on how to compute the polarisation and take into account a finite homogeneous electric field with the ABINIT package.

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

Mentioned in topic_Phonons, help_features#2.5.

### 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 effect of an homogeneous static electric field on an insulator may be treated in ABINIT from two perspectives. One is perturbative, and yields the susceptibility in the form of the second derivative of the total energy with respect to the electric field, at zero field strength (see DFPT).

ABINIT can also be used to compute the effect of an electric field of finite amplitude, using techniques from the Modern Theory of Polarization [Resta1994],[Nunes2001],[Souza2002]. The latter is based on the notion of "Berry phase". In this approach, the total energy to minimize includes the contribution due to the interaction of the external electric field with the material polarization PTot, as follows:

E = E0 - ΩPTot.E, where E0 is the usual ground state energy obtained from Kohn-Sham DFT in the absence of the external field E, PTot is the polarization, made up of an ionic contribution and an electronic contribution, and Ω the volume of the unit cell.

Some details of the implementation of The Modern Theory of Polarization in ABINIT are given in [the 2016 ABINIT publication].

In the NCPP case, the electric field has no additional contribution to the Hellmann-Feynman forces, because the electronic states do not depend explicitly on ionic position [Souza2002]. In the PAW case however, as the projectors do depend on ion location, an additional force and additional stresses terms arise [Zwanziger2012].

The generalisation to fixed D-field or fixed reduced fields are also available, as described in M. Stengel, N.A. Spaldin and D. Vanderbilt, Nat. Phys. 5,304 (2009).

The polarization and finite electric field calculation in ABINIT is accessed through the variables berryopt and efield. In addition, displacement fields and mixed boundary conditions (a mix of electric field and displacement field) can be computed as well.
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### 2. Related lesson(s) of the tutorial.

- The lesson on polarization and finite electric field deals with the computation of the polarization of an insulator (e.g. ferroelectric, or dielectric material) thanks to the Berry phase approach, and also presents the computation of materials properties in the presence of a finite electric field (also thanks to the Berry phase approach).
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### 3. Related input variables.

Compulsory input variables:

... berryopt [BERRY phase OPTions]

Basic input variables:

... bdberry [BanD limits for BERRY phase]
... dfield [Displacement FIELD]
... efield [Electric FIELD]
... jfielddir [electric/displacement FIELD DIRection]
... kberry [K wavevectors for BERRY phase computation]
... nberry [Number of BERRY phase computations]

Useful input variables:

... berrysav [BERRY SAVe]
... ddamp [electric Displacement field DAMPing parameter]
... maxestep [MAXimum Electric field STEP]
... polcen [POLarization for CENtrosymmetric geometry]
... red_dfield [REDuced Displacement FIELD]
... red_efield [REDuced Electric FIELD]
... red_efieldbar [REDuced Electric FIELD BAR]

Input variables for experts:

... berrystep [BERRY phase : multiple STEP]
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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/paral/Input: t07.in

tests/v2/Input: t81.in

tests/v3/Input: t03.in t05.in

tests/v4/Input: t55.in t66.in t72.in t75.in t78.in t80.in

tests/v5/Input: t112.in t113.in t23.in

tests/v6/Input: t121.in t122.in t123.in t124.in t125.in t126.in t43.in

tests/v7/Input: t03.in
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### 5. References.

[Nunes2001] R. W. Nunes and X. Gonze, "Berry-phase treatment of the homogeneous electric field perturbation in insulators", Phys. Rev. B 63, 155107 (2001).
[Resta1994] R. Resta, "Macroscopic polarization in crystalline dielectrics: the geometric phase approach", Rev. Mod. Phys. 66, 899–915 (1994).
[Souza2002] I. Souza, J. Íñiguez and D. Vanderbilt, "First-Principles Approach to Insulators in Finite Electric Fields", Phys. Rev. Lett. 89, 117602 (2002).
[Zwanziger2012] J.W. Zwanziger, J. Galbraith, Y. Kipouros, M. Torrent, M. Giantomassi and X. Gonze, "Finite homogeneous electric fields in the projector augmented wave formalism: Applications to linear and nonlinear response", Comput. Mater. Sci. 58, 113–118 (2012).

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