BerryPI: A software for studying polarization of crystalline solids with WIEN2k density functional all-electron package

doi:10.1016/j.cpc.2012.10.028

Computer Physics Communications

Volume 184, Issue 3, March 2013, Pages 647-651

https://doi.org/10.1016/j.cpc.2012.10.028 Get rights and content

Abstract

We present a module that enables computation of polarization using density functional theory based on the full potential linearized augmented plane wave package WIEN2k. The theoretical background of deriving microscopic polarization of materials using the modern theory of polarization (geometric Berry phase approach) is reviewed. The software is validated and then applied to determine spontaneous polarization and Born effective charges of several crystal structures, which are commonly studied theoretically and experimentally.

Introduction

First-principle microscopic theories, such as the density functional theory (DFT), play a major role in the development of parameter-free models that establish a relation between atomic structure and material properties using a minimum or no experimental input at all. Combined with the recent advances in high-performance computing, this development opens new opportunities in exploring novel materials and understanding their properties [1], [2]. In particular, the ability of DFT to capture microscopic polarization [3], [4], [5] enables calculation of the related material properties, such as spontaneous polarization and Born effective charge [4], permittivity [6], pyroelectric coefficient [7], and piezoelectric tensor [8], [9].

According to the modern theory of polarization, the polarization of a material is not a bulk property and it is ill defined [5], [7]. Instead, the material properties are related to the change of polarization $Δ P$ in response to an external perturbation [7] $Δ P = P^{(1)} - P^{(0)} = Ω^{- 1} \int d t \int_{cell} d r j (r, t),$ where $j (r, t)$ is the local transient current density resulting from a charge redistribution inside the bulk unit cell. The polarization of a given state consists of two components: ionic and electronic $P = P_{ion} + P_{el} .$ The calculation of the ionic contribution is straightforward; it is based on the position of atomic nuclei and the corresponding ionic charges [5]. The electronic part is related to the spatial distribution of the electron density [7], which can be expressed in terms of a geometric phase (Berry phase) [3], [10].

The calculation of polarization using the Berry phase is now implemented in major solid-state DFT packages, such as ABINIT [11] and VASP [12], which belong to the plane wave family. To the best of our knowledge, only one successful realization of the Berry phase approach using the all-electron full-potential linearized augmented plane wave (LAPW) method has been reported so far [8]. However, the package is not available for external users.

The purpose of this communication is to present a new software BerryPI that extends the capability of the popular all-electron full-potential DFT package WIEN2k [13] to calculation of polarization in solids using the Berry phase approach. BerryPI also relies on the WIEN2WANNIER [14] program in computing of overlap matrices as described below. As an example, we calculated the spontaneous polarization and Born effective charges of several perovskite, zinc-blende and rock-salt structures and compare the results with experimental data, pseudopotenial calculations and other DFT results reported in literature.

Section snippets

Method

We consider a periodic insulating crystal, which is represented by a unit cell with $N$ atoms and $M$ doubly-occupied bands (non-spin-polarized calculation is considered). It is assumed that the electronic ground state can be described by a single-particle mean-field Hamiltonian as in the density-functional theory. The eigenstates of this Hamiltonian are the Bloch functions $ψ_{n k} (r) = u_{n k} (r) e^{i k \cdot r},$ which are characterized by the band-index $n$ and the wave vector $k$ . The cell-periodic complex amplitude $u_{n k} ($

Program implementation

BerryPI is a Python script that controls the execution process according to the flow in Table 1. The script is invoked in the case directory after completing the standard WIEN2k self-consistency field cycle. The only input parameter required is the $k$ -mesh for Berry phase integration. The script determines the number of occupied bands $M$ , cell geometry, the ionic charges and their relative positions based on WIEN2k files. The electronic, ionic and total phases as well as the corresponding

Validation

First, we begin with calculation of polarization in the case where the outcome can be predicted exactly. For non-interacting (noble) atoms the net polarization is zero. Therefore the electronic and ionic polarization should cancel each other, $P_{ion} + P_{el} = 0$ . This property is used in order to test the accuracy of our calculations of polarization.

Two helium atoms were placed in a tetragonal cell as illustrated in Fig. 3. The cell dimensions $a = 10$ and $b = c = 5$ Bohr were chosen in order to prevent a

Applications

In the following, we provide two examples on calculation of the material properties related to polarization using BerryPI. The examples include modeling the spontaneous polarization of perovskite crystals and calculation of the Born effective charge of polar materials.

Conclusions

We presented a module that extends the capability of WIEN2k (all-electron density functional package) to calculation of polarization using the Berry phase approach. The accuracy of calculations was verified using a model of non-interacting noble atoms. We applied the approach to calculation of spontaneous polarization and Born effective charge of some well characterized perovskite crystals, sodium chloride and zinc-blende structures. Obtained results agree with alternative calculations and

Acknowledgments

The authors would like to thank Dr. Peter Blaha for stimulating discussions, critical reading of the manuscript and editorial suggestions. The authors also want to acknowledge funding provided by the Natural Sciences and Engineering Research Council of Canada under the Discovery Grant Program “Microscopic theory of high-field transport in disordered semiconductors”, the Ontario Ministry of Economic Development and Innovation under the Ontario Research Fund program and the Thunder Bay Community

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BerryPI: A software for studying polarization of crystalline solids with WIEN2k density functional all-electron package

Abstract

Introduction

Section snippets

Method

Program implementation

Validation

Applications

Conclusions

Acknowledgments

J. Phys. Chem. Solids

Comput. Mater. Sci.

Comput. Mater. Sci.

Comput. Phys. Comm.

Nature Mater.

Science

Phys. Rev. B

Phys. Rev. Lett.

Rev. Modern Phys.

Phys. Rev. Lett.

Theory of polarization: a modern approach

Phys. Rev. Lett.

Proc. R. Soc. Ser. A