# pymatgen.analysis.ferroelectricity.polarization module¶

This module contains classes useful for analyzing ferroelectric candidates. The Polarization class can recover the spontaneous polarization using multiple calculations along a nonpolar to polar ferroelectric distortion. The EnergyTrend class is useful for assessing the trend in energy across the distortion.

See Nicola Spaldin’s “A beginner’s guide to the modern theory of polarization” (https://arxiv.org/abs/1202.1831) for an introduction to crystal polarization.

VASP reports dipole moment values (used to derive polarization) along Cartesian directions (see pead.F around line 970 in the VASP source to confirm this). However, it is most convenient to perform the adjustments necessary to recover a same branch polarization by expressing the polarization along lattice directions. For this reason, calc_ionic calculates ionic contributions to the polarization along lattice directions. We provide the means to convert Cartesian direction polarizations to lattice direction polarizations in the Polarization class.

We recommend using our calc_ionic function for calculating the ionic polarization rather than the values from OUTCAR. We find that the ionic dipole moment reported in OUTCAR differ from the naive calculation of \sum_i Z_i r_i where i is the index of the atom, Z_i is the ZVAL from the pseudopotential file, and r is the distance in Angstroms along the lattice vectors. Note, this difference is not simply due to VASP using Cartesian directions and calc_ionic using lattice direction but rather how the ionic polarization is computed. Compare calc_ionic to VASP SUBROUTINE POINT_CHARGE_DIPOL in dipol.F in the VASP source to see the differences. We are able to recover a smooth same branch polarization more frequently using the naive calculation in calc_ionic than using the ionic dipole moment reported in the OUTCAR.

Some defintions of terms used in the comments below:

A polar structure belongs to a polar space group. A polar space group has a one of the 10 polar point group:

(1, 2, m, mm2, 4, 4mm, 3, 3m, 6, 6m)

Being nonpolar is not equivalent to being centrosymmetric (having inversion symmetry). For example, any space group with point group 222 is nonpolar but not centrosymmetric.

By symmetry the polarization of a nonpolar material modulo the quantum of polarization can only be zero or 1/2. We use a nonpolar structure to help determine the spontaneous polarization because it serves as a reference point.

class EnergyTrend(energies)[source]

Bases: object

Class for fitting trends to energies.

Parameters

energies – Energies

endpoints_minima(slope_cutoff=0.005)[source]

Test if spline endpoints are at minima for a given slope cutoff.

max_spline_jump()[source]

Get maximum difference between spline and energy trend.

smoothness()[source]

Get rms average difference between spline and energy trend.

spline()[source]

Fit spline to energy trend data.

class Polarization(p_elecs, p_ions, structures, p_elecs_in_cartesian=True, p_ions_in_cartesian=False)[source]

Bases: object

Class for recovering the same branch polarization for a set of polarization calculations along the nonpolar - polar distortion path of a ferroelectric.

p_elecs, p_ions, and structures lists should be given in order of nonpolar to polar! For example, the structures returned from:

nonpolar.interpolate(polar,interpolate_lattices=True)

if nonpolar is the nonpolar Structure and polar is the polar structure.

It is assumed that the electronic and ionic dipole moment values are given in electron Angstroms along the three lattice directions (a,b,c).

p_elecs: np.array of electronic contribution to the polarization with shape [N, 3] p_ions: np.array of ionic contribution to the polarization with shape [N, 3] p_elecs_in_cartesian: whether p_elecs is along Cartesian directions (rather than lattice directions).

Default is True because that is the convention for VASP.

p_ions_in_cartesian: whether p_ions is along Cartesian directions (rather than lattice directions).

Default is False because calc_ionic (which we recommend using for calculating the ionic contribution to the polarization) uses lattice directions.

classmethod from_outcars_and_structures(outcars, structures, calc_ionic_from_zval=False)[source]

Create Polarization object from list of Outcars and Structures in order of nonpolar to polar.

Note, we recommend calculating the ionic dipole moment using calc_ionic than using the values in Outcar (see module comments). To do this set calc_ionic_from_zval = True

get_lattice_quanta(convert_to_muC_per_cm2=True, all_in_polar=True)[source]

Returns the dipole / polarization quanta along a, b, and c for all structures.

get_pelecs_and_pions(convert_to_muC_per_cm2=False)[source]

Get the electronic and ionic dipole moments / polarizations.

convert_to_muC_per_cm2: Convert from electron * Angstroms to microCoulomb

per centimeter**2

get_polarization_change(convert_to_muC_per_cm2=True, all_in_polar=True)[source]

Get difference between nonpolar and polar same branch polarization.

get_polarization_change_norm(convert_to_muC_per_cm2=True, all_in_polar=True)[source]

Get magnitude of difference between nonpolar and polar same branch polarization.

get_same_branch_polarization_data(convert_to_muC_per_cm2=True, all_in_polar=True)[source]

Get same branch dipole moment (convert_to_muC_per_cm2=False) or polarization for given polarization data (convert_to_muC_per_cm2=True).

Polarization is a lattice vector, meaning it is only defined modulo the quantum of polarization:

P = P_0 + \sum_i \frac{n_i e R_i}{\Omega}

where n_i is an integer, e is the charge of the electron in microCoulombs, R_i is a lattice vector, and \Omega is the unit cell volume in cm**3 (giving polarization units of microCoulomb per centimeter**2).

The quantum of the dipole moment in electron Angstroms (as given by VASP) is:

\sum_i n_i e R_i

where e, the electron charge, is 1 and R_i is a lattice vector, and n_i is an integer.

Given N polarization calculations in order from nonpolar to polar, this algorithm minimizes the distance between adjacent polarization images. To do this, it constructs a polarization lattice for each polarization calculation using the pymatgen.core.structure class and calls the get_nearest_site method to find the image of a given polarization lattice vector that is closest to the previous polarization lattice vector image.

Note, using convert_to_muC_per_cm2=True and all_in_polar=True calculates the “proper polarization” (meaning the change in polarization does not depend on the choice of polarization branch) while convert_to_muC_per_cm2=True and all_in_polar=False calculates the “improper polarization” (meaning the change in polarization does depend on the choice of branch). As one might guess from the names. We recommend calculating the “proper polarization”.

convert_to_muC_per_cm2: convert polarization from electron * Angstroms to

microCoulomb per centimeter**2

all_in_polar: convert polarization to be in polar (final structure) polarization lattice

max_spline_jumps(convert_to_muC_per_cm2=True, all_in_polar=True)[source]

Get maximum difference between spline and same branch polarization data.

same_branch_splines(convert_to_muC_per_cm2=True, all_in_polar=True)[source]

Fit splines to same branch polarization. This is used to assess any jumps in the same branch polarizaiton.

smoothness(convert_to_muC_per_cm2=True, all_in_polar=True)[source]

Get rms average difference between spline and same branch polarization data.

class PolarizationLattice(lattice: Union[List, numpy.ndarray, pymatgen.core.lattice.Lattice], species: Sequence[Union[str, pymatgen.core.periodic_table.Element, pymatgen.core.periodic_table.Specie, pymatgen.core.periodic_table.DummySpecie, pymatgen.core.composition.Composition]], coords: Sequence[Sequence[float]], charge: float = None, validate_proximity: bool = False, to_unit_cell: bool = False, coords_are_cartesian: bool = False, site_properties: dict = None)[source]

Why is a Lattice inheriting a structure? This is ridiculous.

Create a periodic structure.

Parameters
• lattice – The lattice, either as a pymatgen.core.lattice.Lattice or simply as any 2D array. Each row should correspond to a lattice vector. E.g., [[10,0,0], [20,10,0], [0,0,30]] specifies a lattice with lattice vectors [10,0,0], [20,10,0] and [0,0,30].

• species

List of species on each site. Can take in flexible input, including:

1. A sequence of element / specie specified either as string symbols, e.g. [“Li”, “Fe2+”, “P”, …] or atomic numbers, e.g., (3, 56, …) or actual Element or Specie objects.

2. List of dict of elements/species and occupancies, e.g., [{“Fe” : 0.5, “Mn”:0.5}, …]. This allows the setup of disordered structures.

• coords (Nx3 array) – list of fractional/cartesian coordinates of each species.

• charge (int) – overall charge of the structure. Defaults to behavior in SiteCollection where total charge is the sum of the oxidation states.

• validate_proximity (bool) – Whether to check if there are sites that are less than 0.01 Ang apart. Defaults to False.

• to_unit_cell (bool) – Whether to map all sites into the unit cell, i.e., fractional coords between 0 and 1. Defaults to False.

• coords_are_cartesian (bool) – Set to True if you are providing coordinates in cartesian coordinates. Defaults to False.

• site_properties (dict) – Properties associated with the sites as a dict of sequences, e.g., {“magmom”:[5,5,5,5]}. The sequences have to be the same length as the atomic species and fractional_coords. Defaults to None for no properties.

get_nearest_site(coords, site, r=None)[source]

Given coords and a site, find closet site to coords. :param coords: cartesian coords of center of sphere :type coords: 3x1 array :param site: site to find closest to coords :param r: radius of sphere. Defaults to diagonal of unit cell

Returns

Closest site and distance.

calc_ionic(site, structure, zval)[source]

Calculate the ionic dipole moment using ZVAL from pseudopotential

site: PeriodicSite structure: Structure zval: Charge value for ion (ZVAL for VASP pseudopotential)

Returns polarization in electron Angstroms.

get_total_ionic_dipole(structure, zval_dict)[source]

Get the total ionic dipole moment for a structure.

structure: pymatgen Structure zval_dict: specie, zval dictionary pairs center (np.array with shape [3,1]) : dipole center used by VASP tiny (float) : tolerance for determining boundary of calculation.

zval_dict_from_potcar(potcar)[source]

Creates zval_dictionary for calculating the ionic polarization from Potcar object

potcar: Potcar object