Source code for pymatgen.analysis.ferroelectricity.polarization

# coding: utf-8
# Copyright (c) Pymatgen Development Team.
# Distributed under the terms of the MIT License.

r"""
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.
"""


from pymatgen.core.structure import Structure
from pymatgen.core.lattice import Lattice
import numpy as np

__author__ = "Tess Smidt"
__copyright__ = "Copyright 2017, The Materials Project"
__version__ = "1.0"
__email__ = "tsmidt@berkeley.edu"
__status__ = "Development"
__date__ = "April 15, 2017"


[docs]def zval_dict_from_potcar(potcar): """ Creates zval_dictionary for calculating the ionic polarization from Potcar object potcar: Potcar object """ zval_dict = {} for p in potcar: zval_dict.update({p.element: p.ZVAL}) return zval_dict
[docs]def calc_ionic(site, structure, zval): """ 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. """ norms = structure.lattice.lengths return np.multiply(norms, -site.frac_coords * zval)
[docs]def get_total_ionic_dipole(structure, zval_dict): """ 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. """ tot_ionic = [] for site in structure: zval = zval_dict[str(site.specie)] tot_ionic.append(calc_ionic(site, structure, zval)) return np.sum(tot_ionic, axis=0)
[docs]class PolarizationLattice(Structure): """ Why is a Lattice inheriting a structure? This is ridiculous. """
[docs] def get_nearest_site(self, coords, site, r=None): """ Given coords and a site, find closet site to coords. Args: coords (3x1 array): cartesian coords of center of sphere site: site to find closest to coords r: radius of sphere. Defaults to diagonal of unit cell Returns: Closest site and distance. """ index = self.index(site) if r is None: r = np.linalg.norm(np.sum(self.lattice.matrix, axis=0)) ns = self.get_sites_in_sphere(coords, r, include_index=True) # Get sites with identical index to site ns = [n for n in ns if n[2] == index] # Sort by distance to coords ns.sort(key=lambda x: x[1]) # Return PeriodicSite and distance of closest image return ns[0][0:2]
[docs]class Polarization: """ 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). """ def __init__(self, p_elecs, p_ions, structures, p_elecs_in_cartesian=True, p_ions_in_cartesian=False): """ 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. """ if len(p_elecs) != len(p_ions) or len(p_elecs) != len(structures): raise ValueError( "The number of electronic polarization and ionic polarization values must be equal.") if p_elecs_in_cartesian: p_elecs = np.array( [struct.lattice.get_vector_along_lattice_directions(p_elecs[i]) for i, struct in enumerate(structures)]) if p_ions_in_cartesian: p_ions = np.array( [struct.lattice.get_vector_along_lattice_directions(p_ions[i]) for i, struct in enumerate(structures)]) self.p_elecs = np.array(p_elecs) self.p_ions = np.array(p_ions) self.structures = structures
[docs] @classmethod def from_outcars_and_structures(cls, outcars, structures, calc_ionic_from_zval=False): """ 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 """ p_elecs = [] p_ions = [] for i, o in enumerate(outcars): p_elecs.append(o.p_elec) if calc_ionic_from_zval: p_ions.append( get_total_ionic_dipole(structures[i], o.zval_dict)) else: p_ions.append(o.p_ion) return cls(p_elecs, p_ions, structures)
[docs] def get_pelecs_and_pions(self, convert_to_muC_per_cm2=False): """ Get the electronic and ionic dipole moments / polarizations. convert_to_muC_per_cm2: Convert from electron * Angstroms to microCoulomb per centimeter**2 """ if not convert_to_muC_per_cm2: return self.p_elecs, self.p_ions if convert_to_muC_per_cm2: p_elecs = self.p_elecs.T p_ions = self.p_ions.T volumes = [s.lattice.volume for s in self.structures] e_to_muC = -1.6021766e-13 cm2_to_A2 = 1e16 units = 1.0 / np.array(volumes) units *= e_to_muC * cm2_to_A2 p_elecs = np.matmul(units, p_elecs) p_ions = np.matmul(units, p_ions) p_elecs, p_ions = p_elecs.T, p_ions.T return p_elecs, p_ions
[docs] def get_same_branch_polarization_data(self, convert_to_muC_per_cm2=True, all_in_polar=True): r""" 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 """ p_elec, p_ion = self.get_pelecs_and_pions() p_tot = p_elec + p_ion p_tot = np.array(p_tot) lattices = [s.lattice for s in self.structures] volumes = np.array([s.lattice.volume for s in self.structures]) L = len(p_elec) e_to_muC = -1.6021766e-13 cm2_to_A2 = 1e16 units = 1.0 / np.array(volumes) units *= e_to_muC * cm2_to_A2 # convert polarizations and lattice lengths prior to adjustment if convert_to_muC_per_cm2 and not all_in_polar: # Convert the total polarization p_tot = np.multiply(units.T[:, np.newaxis], p_tot) # adjust lattices for i in range(L): lattice = lattices[i] l = lattice.lengths a = lattice.angles lattices[i] = Lattice.from_parameters(*(np.array(l) * units.ravel()[i]), *a) # convert polarizations to polar lattice elif convert_to_muC_per_cm2 and all_in_polar: abc = [lattice.abc for lattice in lattices] abc = np.array(abc) # [N, 3] p_tot /= abc # e * Angstroms to e p_tot *= abc[-1] / volumes[-1] * e_to_muC * cm2_to_A2 # to muC / cm^2 for i in range(L): lattice = lattices[-1] # Use polar lattice l = lattice.lengths a = lattice.angles # Use polar units (volume) lattices[i] = Lattice.from_parameters(*(np.array(l) * units.ravel()[-1]), *a) d_structs = [] sites = [] for i in range(L): l = lattices[i] frac_coord = np.divide(np.array([p_tot[i]]), np.array([l.a, l.b, l.c])) d = PolarizationLattice(l, ["C"], [np.array(frac_coord).ravel()]) d_structs.append(d) site = d[0] if i == 0: # Adjust nonpolar polarization to be closest to zero. # This is compatible with both a polarization of zero or a half quantum. prev_site = [0, 0, 0] else: prev_site = sites[-1].coords new_site = d.get_nearest_site(prev_site, site) sites.append(new_site[0]) adjust_pol = [] for s, d in zip(sites, d_structs): l = d.lattice adjust_pol.append( np.multiply(s.frac_coords, np.array([l.a, l.b, l.c])).ravel()) adjust_pol = np.array(adjust_pol) return adjust_pol
[docs] def get_lattice_quanta(self, convert_to_muC_per_cm2=True, all_in_polar=True): """ Returns the dipole / polarization quanta along a, b, and c for all structures. """ lattices = [s.lattice for s in self.structures] volumes = np.array([s.lattice.volume for s in self.structures]) L = len(self.structures) e_to_muC = -1.6021766e-13 cm2_to_A2 = 1e16 units = 1.0 / np.array(volumes) units *= e_to_muC * cm2_to_A2 # convert polarizations and lattice lengths prior to adjustment if convert_to_muC_per_cm2 and not all_in_polar: # adjust lattices for i in range(L): lattice = lattices[i] l = lattice.lengths a = lattice.angles lattices[i] = Lattice.from_parameters(*(np.array(l) * units.ravel()[i]), *a) elif convert_to_muC_per_cm2 and all_in_polar: for i in range(L): lattice = lattices[-1] l = lattice.lengths a = lattice.angles lattices[i] = Lattice.from_parameters(*(np.array(l) * units.ravel()[-1]), *a) quanta = np.array([np.array(l.lengths) for l in lattices]) return quanta
[docs] def get_polarization_change(self, convert_to_muC_per_cm2=True, all_in_polar=True): """ Get difference between nonpolar and polar same branch polarization. """ tot = self.get_same_branch_polarization_data( convert_to_muC_per_cm2=convert_to_muC_per_cm2, all_in_polar=all_in_polar) # reshape to preserve backwards compatibility due to changes # when switching from np.matrix to np.array return (tot[-1] - tot[0]).reshape((1, 3))
[docs] def get_polarization_change_norm(self, convert_to_muC_per_cm2=True, all_in_polar=True): """ Get magnitude of difference between nonpolar and polar same branch polarization. """ polar = self.structures[-1] a, b, c = polar.lattice.matrix a, b, c = a / np.linalg.norm(a), b / np.linalg.norm( b), c / np.linalg.norm(c) P = self.get_polarization_change(convert_to_muC_per_cm2=convert_to_muC_per_cm2, all_in_polar=all_in_polar).ravel() P_norm = np.linalg.norm(a * P[0] + b * P[1] + c * P[2]) return P_norm
[docs] def same_branch_splines(self, convert_to_muC_per_cm2=True, all_in_polar=True): """ Fit splines to same branch polarization. This is used to assess any jumps in the same branch polarizaiton. """ from scipy.interpolate import UnivariateSpline tot = self.get_same_branch_polarization_data( convert_to_muC_per_cm2=convert_to_muC_per_cm2, all_in_polar=all_in_polar) L = tot.shape[0] try: sp_a = UnivariateSpline(range(L), tot[:, 0].ravel()) except Exception: sp_a = None try: sp_b = UnivariateSpline(range(L), tot[:, 1].ravel()) except Exception: sp_b = None try: sp_c = UnivariateSpline(range(L), tot[:, 2].ravel()) except Exception: sp_c = None return sp_a, sp_b, sp_c
[docs] def max_spline_jumps(self, convert_to_muC_per_cm2=True, all_in_polar=True): """ Get maximum difference between spline and same branch polarization data. """ tot = self.get_same_branch_polarization_data( convert_to_muC_per_cm2=convert_to_muC_per_cm2, all_in_polar=all_in_polar) sps = self.same_branch_splines(convert_to_muC_per_cm2=convert_to_muC_per_cm2, all_in_polar=all_in_polar) max_jumps = [None, None, None] for i, sp in enumerate(sps): if sp is not None: max_jumps[i] = max(tot[:, i].ravel() - sp(range(len(tot[:, i].ravel())))) return max_jumps
[docs] def smoothness(self, convert_to_muC_per_cm2=True, all_in_polar=True): """ Get rms average difference between spline and same branch polarization data. """ tot = self.get_same_branch_polarization_data( convert_to_muC_per_cm2=convert_to_muC_per_cm2, all_in_polar=all_in_polar) L = tot.shape[0] try: sp = self.same_branch_splines(convert_to_muC_per_cm2=convert_to_muC_per_cm2, all_in_polar=all_in_polar) except Exception: print("Something went wrong.") return None sp_latt = [sp[i](range(L)) for i in range(3)] diff = [sp_latt[i] - tot[:, i].ravel() for i in range(3)] rms = [np.sqrt(np.sum(np.square(diff[i])) / L) for i in range(3)] return rms
[docs]class EnergyTrend: """ Class for fitting trends to energies. """ def __init__(self, energies): """ :param energies: Energies """ self.energies = energies
[docs] def spline(self): """ Fit spline to energy trend data. """ from scipy.interpolate import UnivariateSpline sp = UnivariateSpline(range(len(self.energies)), self.energies, k=4) return sp
[docs] def smoothness(self): """ Get rms average difference between spline and energy trend. """ energies = self.energies try: sp = self.spline() except Exception: print("Energy spline failed.") return None spline_energies = sp(range(len(energies))) diff = spline_energies - energies rms = np.sqrt(np.sum(np.square(diff)) / len(energies)) return rms
[docs] def max_spline_jump(self): """ Get maximum difference between spline and energy trend. """ sp = self.spline() return max(self.energies - sp(range(len(self.energies))))
[docs] def endpoints_minima(self, slope_cutoff=5e-3): """ Test if spline endpoints are at minima for a given slope cutoff. """ energies = self.energies try: sp = self.spline() except Exception: print("Energy spline failed.") return None der = sp.derivative() der_energies = der(range(len(energies))) return {"polar": abs(der_energies[-1]) <= slope_cutoff, "nonpolar": abs(der_energies[0]) <= slope_cutoff}