pymatgen.analysis.defects.thermodynamics — pymatgen 2020.7.3 documentation

# Source code for pymatgen.analysis.defects.thermodynamics

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

    """
    Defect thermodynamics, such as defect phase diagrams, etc.
    """

    import logging
    import numpy as np
    from monty.json import MSONable
    from scipy.spatial import HalfspaceIntersection
    from scipy.optimize import bisect
    from itertools import chain

    from pymatgen.electronic_structure.dos import FermiDos
    from pymatgen.analysis.defects.core import DefectEntry
    from pymatgen.analysis.structure_matcher import PointDefectComparator

    import matplotlib.pyplot as plt
    import matplotlib.cm as cm

    __author__ = "Danny Broberg, Shyam Dwaraknath"
    __copyright__ = "Copyright 2018, The Materials Project"
    __version__ = "1.0"
    __maintainer__ = "Shyam Dwaraknath"
    __email__ = "shyamd@lbl.gov"
    __status__ = "Development"
    __date__ = "Mar 15, 2018"

    logger = logging.getLogger(__name__)

    class DefectPhaseDiagram(MSONable):
        """
        This is similar to a PhaseDiagram object in pymatgen,
        but has ability to do quick analysis of defect formation energies
        when fed DefectEntry objects.

        uses many of the capabilities from PyCDT's DefectsAnalyzer class...

        This class is able to get:
            a) stability of charge states for a given defect,
            b) list of all formation ens
            c) transition levels in the gap
        """

        def __init__(self, entries, vbm, band_gap, filter_compatible=True, metadata=None):
            """
            Args:
                dentries ([DefectEntry]): A list of DefectEntry objects
                vbm (float): Valence Band energy to use for all defect entries.
                    NOTE if using band shifting-type correction then this VBM
                    should still be that of the GGA calculation
                    (the bandedgeshifting_correction accounts for shift's
                    contribution to formation energy).
                band_gap (float): Band gap to use for all defect entries.
                    NOTE if using band shifting-type correction then this gap
                    should still be that of the Hybrid calculation you are shifting to.
                filter_compatible (bool): Whether to consider entries which were ruled
                    incompatible by the DefectComaptibility class. Note this must be set to False
                    if you desire a suggestion for larger supercell sizes.
                    Default is True (to omit calculations which have "is_compatible"=False in
                        DefectEntry'sparameters)
                metadata (dict): Dictionary of metadata to store with the PhaseDiagram. Has
                    no impact on calculations
            """
            self.vbm = vbm
            self.band_gap = band_gap
            self.filter_compatible = filter_compatible

            if filter_compatible:
                self.entries = [e for e in entries if e.parameters.get("is_compatible", True)]
            else:
                self.entries = entries

            for ent_ind, ent in enumerate(self.entries):
                if 'vbm' not in ent.parameters.keys() or ent.parameters['vbm'] != vbm:
                    logger.info("Entry {} did not have vbm equal to given DefectPhaseDiagram value."
                                " Manually overriding.".format(ent.name))
                    new_ent = ent.copy()
                    new_ent.parameters['vbm'] = vbm
                    self.entries[ent_ind] = new_ent

            self.metadata = metadata or {}
            self.find_stable_charges()

        def as_dict(self):
            """
            Returns:
                Json-serializable dict representation of DefectPhaseDiagram
            """
            d = {"@module": self.__class__.__module__,
                 "@class": self.__class__.__name__,
                 "entries": [entry.as_dict() for entry in self.entries],
                 "vbm": self.vbm,
                 "band_gap": self.band_gap,
                 "filter_compatible": self.filter_compatible,
                 "metadata": self.metadata}
            return d

        @classmethod
        def from_dict(cls, d):
            """
            Reconstitute a DefectPhaseDiagram object from a dict representation created using
            as_dict().

            Args:
                d (dict): dict representation of DefectPhaseDiagram.

            Returns:
                DefectPhaseDiagram object
            """
            entries = [DefectEntry.from_dict(entry_dict) for entry_dict in d.get("entries")]
            vbm = d["vbm"]
            band_gap = d["band_gap"]
            filter_compatible = d.get("filter_compatible", True)
            metadata = d.get("metadata", {})
            if 'entry_id' in d.keys() and 'entry_id' not in metadata:
                metadata['entry_id'] = d['entry_id']

            return cls(entries, vbm, band_gap, filter_compatible=filter_compatible,
                       metadata=metadata)

        def find_stable_charges(self):
            """
            Sets the stable charges and transition states for a series of
            defect entries. This function uses scipy's HalfspaceInterection
            to oncstruct the polygons corresponding to defect stability as
            a function of the Fermi-level. The Halfspace Intersection
            constructs N-dimensional hyperplanes, in this case N=2,  based
            on the equation of defect formation energy with considering chemical
            potentials:
                E_form = E_0^{Corrected} + Q_{defect}*(E_{VBM} + E_{Fermi})

            Extra hyperplanes are constructed to bound this space so that
            the algorithm can actually find enclosed region.

            This code was modeled after the Halfspace Intersection code for
            the Pourbaix Diagram
            """

            def similar_defects(entryset):
                """
                Used for grouping similar defects of different charges
                Can distinguish identical defects even if they are not in same position
                """
                pdc = PointDefectComparator(check_charge=False, check_primitive_cell=True,
                                            check_lattice_scale=False)
                grp_def_sets = []
                grp_def_indices = []
                for ent_ind, ent in enumerate(entryset):
                    # TODO: more pythonic way of grouping entry sets with PointDefectComparator.
                    # this is currently most time intensive part of DefectPhaseDiagram
                    matched_ind = None
                    for grp_ind, defgrp in enumerate(grp_def_sets):
                        if pdc.are_equal(ent.defect, defgrp[0].defect):
                            matched_ind = grp_ind
                            break
                    if matched_ind is not None:
                        grp_def_sets[matched_ind].append(ent.copy())
                        grp_def_indices[matched_ind].append(ent_ind)
                    else:
                        grp_def_sets.append([ent.copy()])
                        grp_def_indices.append([ent_ind])

                return zip(grp_def_sets, grp_def_indices)

            # Limits for search
            # E_fermi = { -1 eV to band gap+1}
            # E_formation = { (min(Eform) - 30) to (max(Eform) + 30)}
            all_eform = [one_def.formation_energy(fermi_level=self.band_gap / 2.) for one_def in self.entries]
            min_y_lim = min(all_eform) - 30
            max_y_lim = max(all_eform) + 30
            limits = [[-1, self.band_gap + 1], [min_y_lim, max_y_lim]]

            stable_entries = {}
            finished_charges = {}
            transition_level_map = {}

            # Grouping by defect types
            for defects, index_list in similar_defects(self.entries):
                defects = list(defects)

                # prepping coefficient matrix for half-space intersection
                # [-Q, 1, -1*(E_form+Q*VBM)] -> -Q*E_fermi+E+-1*(E_form+Q*VBM) <= 0  where E_fermi and E are the variables
                # in the hyperplanes
                hyperplanes = np.array(
                    [[-1.0 * entry.charge, 1, -1.0 * (entry.energy + entry.charge * self.vbm)] for entry in defects])

                border_hyperplanes = [[-1, 0, limits[0][0]], [1, 0, -1 * limits[0][1]], [0, -1, limits[1][0]],
                                      [0, 1, -1 * limits[1][1]]]
                hs_hyperplanes = np.vstack([hyperplanes, border_hyperplanes])

                interior_point = [self.band_gap / 2, min(all_eform) - 1.]

                hs_ints = HalfspaceIntersection(hs_hyperplanes, np.array(interior_point))

                # Group the intersections and coresponding facets
                ints_and_facets = zip(hs_ints.intersections, hs_ints.dual_facets)
                # Only inlcude the facets corresponding to entries, not the boundaries
                total_entries = len(defects)
                ints_and_facets = filter(lambda int_and_facet: all(np.array(int_and_facet[1]) < total_entries),
                                         ints_and_facets)
                # sort based on transition level
                ints_and_facets = list(sorted(ints_and_facets, key=lambda int_and_facet: int_and_facet[0][0]))

                # log a defect name for tracking (using full index list to avoid naming
                # in-equivalent defects with same name)
                str_index_list = [str(ind) for ind in sorted(index_list)]
                track_name = defects[0].name + "@" + str("-".join(str_index_list))

                if len(ints_and_facets):
                    # Unpack into lists
                    _, facets = zip(*ints_and_facets)
                    # Map of transition level: charge states

                    transition_level_map[track_name] = {
                        intersection[0]: [defects[i].charge for i in facet]
                        for intersection, facet in ints_and_facets
                    }

                    stable_entries[track_name] = list(set([defects[i] for dual in facets for i in dual]))

                    finished_charges[track_name] = [defect.charge for defect in defects]
                else:
                    # if ints_and_facets is empty, then there is likely only one defect...
                    if len(defects) != 1:
                        # confirm formation energies dominant for one defect over other identical defects
                        name_set = [one_def.name + '_chg' + str(one_def.charge) for one_def in defects]
                        vb_list = [one_def.formation_energy(fermi_level=limits[0][0]) for one_def in defects]
                        cb_list = [one_def.formation_energy(fermi_level=limits[0][1]) for one_def in defects]

                        vbm_def_index = vb_list.index(min(vb_list))
                        name_stable_below_vbm = name_set[vbm_def_index]
                        cbm_def_index = cb_list.index(min(cb_list))
                        name_stable_above_cbm = name_set[cbm_def_index]

                        if name_stable_below_vbm != name_stable_above_cbm:
                            raise ValueError("HalfSpace identified only one stable charge out of list: {}\n"
                                             "But {} is stable below vbm and {} is "
                                             "stable above cbm.\nList of VBM formation energies: {}\n"
                                             "List of CBM formation energies: {}"
                                             "".format(name_set, name_stable_below_vbm, name_stable_above_cbm,
                                                       vb_list, cb_list))
                        else:
                            logger.info("{} is only stable defect out of {}".format(name_stable_below_vbm, name_set))
                            transition_level_map[track_name] = {}
                            stable_entries[track_name] = list([defects[vbm_def_index]])
                            finished_charges[track_name] = [one_def.charge for one_def in defects]
                    else:
                        transition_level_map[track_name] = {}

                        stable_entries[track_name] = list([defects[0]])

                        finished_charges[track_name] = [defects[0].charge]

            self.transition_level_map = transition_level_map
            self.transition_levels = {
                defect_name: list(defect_tls.keys())
                for defect_name, defect_tls in transition_level_map.items()
            }
            self.stable_entries = stable_entries
            self.finished_charges = finished_charges
            self.stable_charges = {
                defect_name: [entry.charge for entry in entries]
                for defect_name, entries in stable_entries.items()
            }

        @property
        def defect_types(self):
            """
            List types of defects existing in the DefectPhaseDiagram
            """
            return list(self.finished_charges.keys())

        @property
        def all_stable_entries(self):
            """
            List all stable entries (defect+charge) in the DefectPhaseDiagram
            """
            return set(chain.from_iterable(self.stable_entries.values()))

        @property
        def all_unstable_entries(self):
            """
            List all unstable entries (defect+charge) in the DefectPhaseDiagram
            """
            all_stable_entries = self.all_stable_entries
            return [e for e in self.entries if e not in all_stable_entries]

        def defect_concentrations(self, chemical_potentials, temperature=300, fermi_level=0.):
            """
            Give list of all concentrations at specified efermi in the DefectPhaseDiagram
            args:
                chemical_potentials = {Element: number} is dict of chemical potentials to provide formation energies for
                temperature = temperature to produce concentrations from
                fermi_level: (float) is fermi level relative to valence band maximum
                    Default efermi = 0 = VBM energy
            returns:
                list of dictionaries of defect concentrations
            """
            concentrations = []
            for dfct in self.all_stable_entries:
                concentrations.append({
                    'conc':
                        dfct.defect_concentration(
                            chemical_potentials=chemical_potentials, temperature=temperature, fermi_level=fermi_level),
                    'name':
                        dfct.name,
                    'charge':
                        dfct.charge
                })

            return concentrations

        def suggest_charges(self, tolerance=0.1):
            """
            Suggest possible charges for defects to compute based on proximity
            of known transitions from entires to VBM and CBM

            Args:
                tolerance (float): tolerance with respect to the VBM and CBM to
                        `          continue to compute new charges
            """
            recommendations = {}

            for def_type in self.defect_types:
                test_charges = np.arange(
                    np.min(self.stable_charges[def_type]) - 1,
                    np.max(self.stable_charges[def_type]) + 2)
                test_charges = [charge for charge in test_charges if charge not in self.finished_charges[def_type]]

                if len(self.transition_level_map[def_type].keys()):
                    # More positive charges will shift the minimum transition level down
                    # Max charge is limited by this if its transition level is close to VBM
                    min_tl = min(self.transition_level_map[def_type].keys())
                    if min_tl < tolerance:
                        max_charge = max(self.transition_level_map[def_type][min_tl])
                        test_charges = [charge for charge in test_charges if charge < max_charge]

                    # More negative charges will shift the maximum transition level up
                    # Minimum charge is limited by this if transition level is near CBM
                    max_tl = max(self.transition_level_map[def_type].keys())
                    if max_tl > (self.band_gap - tolerance):
                        min_charge = min(self.transition_level_map[def_type][max_tl])
                        test_charges = [charge for charge in test_charges if charge > min_charge]
                else:
                    test_charges = [charge for charge in test_charges if charge not in self.stable_charges[def_type]]

                recommendations[def_type] = test_charges

            return recommendations

        def suggest_larger_supercells(self, tolerance=0.1):
            """
            Suggest larger supercells for different defect+chg combinations based on use of
            compatibility analysis. Does this for any charged defects which have is_compatible = False,
            and the defect+chg formation energy is stable at fermi levels within the band gap.

            NOTE: Requires self.filter_compatible = False
            Args:
                tolerance (float): tolerance with respect to the VBM and CBM for considering
                                   larger supercells for a given charge
            """
            if self.filter_compatible:
                raise ValueError("Cannot suggest larger supercells if filter_compatible is True.")

            recommendations = {}

            for def_type in self.defect_types:
                template_entry = self.stable_entries[def_type][0].copy()
                defect_indices = [int(def_ind) for def_ind in def_type.split('@')[-1].split('-')]

                for charge in self.finished_charges[def_type]:
                    chg_defect = template_entry.defect.copy()
                    chg_defect.set_charge(charge)

                    for entry_index in defect_indices:
                        entry = self.entries[entry_index]
                        if entry.charge == charge:
                            break

                    if entry.parameters.get("is_compatible", True):
                        continue
                    else:
                        # consider if transition level is within
                        # tolerance of band edges
                        suggest_bigger_supercell = True
                        for tl, chgset in self.transition_level_map.items():
                            sorted_chgset = list(chgset)
                            sorted_chgset.sort(reverse=True)
                            if charge == sorted_chgset[0] and tl < tolerance:
                                suggest_bigger_supercell = False
                            elif charge == sorted_chgset[1] and tl > (self.band_gap - tolerance):
                                suggest_bigger_supercell = False

                    if suggest_bigger_supercell:
                        if def_type not in recommendations:
                            recommendations[def_type] = []
                        recommendations[def_type].append(charge)

            return recommendations

        def solve_for_fermi_energy(self, temperature, chemical_potentials, bulk_dos):
            """
            Solve for the Fermi energy self-consistently as a function of T
            Observations are Defect concentrations, electron and hole conc
            Args:
                temperature: Temperature to equilibrate fermi energies for
                chemical_potentials: dict of chemical potentials to use for calculation fermi level
                bulk_dos: bulk system dos (pymatgen Dos object)
            Returns:
                Fermi energy dictated by charge neutrality
            """

            fdos = FermiDos(bulk_dos, bandgap=self.band_gap)
            _, fdos_vbm = fdos.get_cbm_vbm()

            def _get_total_q(ef):
                qd_tot = sum([
                    d['charge'] * d['conc']
                    for d in self.defect_concentrations(
                        chemical_potentials=chemical_potentials, temperature=temperature, fermi_level=ef)
                ])
                qd_tot += fdos.get_doping(fermi_level=ef + fdos_vbm, temperature=temperature)
                return qd_tot

            return bisect(_get_total_q, -1., self.band_gap + 1.)

        def solve_for_non_equilibrium_fermi_energy(self, temperature, quench_temperature,
                                                   chemical_potentials, bulk_dos):
            """
            Solve for the Fermi energy after quenching in the defect concentrations at a higher
            temperature (the quench temperature),
            as outlined in P. Canepa et al (2017) Chemistry of Materials (doi: 10.1021/acs.chemmater.7b02909)

            Args:
                temperature: Temperature to equilibrate fermi energy at after quenching in defects
                quench_temperature: Temperature to equilibrate defect concentrations at (higher temperature)
                chemical_potentials: dict of chemical potentials to use for calculation fermi level
                bulk_dos: bulk system dos (pymatgen Dos object)
            Returns:
                Fermi energy dictated by charge neutrality with respect to frozen in defect concentrations
            """

            high_temp_fermi_level = self.solve_for_fermi_energy(quench_temperature, chemical_potentials,
                                                                bulk_dos)
            fixed_defect_charge = sum([
                d['charge'] * d['conc']
                for d in self.defect_concentrations(
                    chemical_potentials=chemical_potentials, temperature=quench_temperature,
                    fermi_level=high_temp_fermi_level)
            ])

            fdos = FermiDos(bulk_dos, bandgap=self.band_gap)
            _, fdos_vbm = fdos.get_cbm_vbm()

            def _get_total_q(ef):
                qd_tot = fixed_defect_charge
                qd_tot += fdos.get_doping(fermi_level=ef + fdos_vbm, temperature=temperature)
                return qd_tot

            return bisect(_get_total_q, -1., self.band_gap + 1.)

            return

        def get_dopability_limits(self, chemical_potentials):
            """
            Find Dopability limits for a given chemical potential.
            This is defined by the defect formation energies which first cross zero
            in formation energies.
            This determine bounds on the fermi level.

            Does this by computing formation energy for every stable defect with non-zero charge.
            If the formation energy value changes sign on either side of the band gap, then
            compute the fermi level value where the formation energy is zero
            (formation energies are lines and basic algebra shows: x_crossing = x1 - (y1 / q)
            for fermi level, x1, producing formation energy y1)

            Args:
                chemical_potentials: dict of chemical potentials to use for calculation fermi level
            Returns:
                 lower dopability limit, upper dopability limit
                (returns None if no limit exists for upper or lower i.e. no negative defect
                crossing before +/- 20 of band edges OR defect formation energies are entirely zero)
            """
            min_fl_range = -20.
            max_fl_range = self.band_gap + 20.

            lower_lim = None
            upper_lim = None
            for def_entry in self.all_stable_entries:
                min_fl_formen = def_entry.formation_energy(chemical_potentials=chemical_potentials,
                                                           fermi_level=min_fl_range)
                max_fl_formen = def_entry.formation_energy(chemical_potentials=chemical_potentials,
                                                           fermi_level=max_fl_range)

                if min_fl_formen < 0. and max_fl_formen < 0.:
                    logger.error("Formation energy is negative through entire gap for entry {} q={}."
                                 " Cannot return dopability limits.".format(def_entry.name, def_entry.charge))
                    return None, None
                elif np.sign(min_fl_formen) != np.sign(max_fl_formen):
                    x_crossing = min_fl_range - (min_fl_formen / def_entry.charge)
                    if min_fl_formen < 0.:
                        if lower_lim is None or lower_lim < x_crossing:
                            lower_lim = x_crossing
                    else:
                        if upper_lim is None or upper_lim > x_crossing:
                            upper_lim = x_crossing

            return lower_lim, upper_lim

        def plot(self, mu_elts=None, xlim=None, ylim=None, ax_fontsize=1.3, lg_fontsize=1.,
                 lg_position=None, fermi_level=None, title=None, saved=False):
            """
            Produce defect Formation energy vs Fermi energy plot
            Args:
                mu_elts:
                    a dictionnary of {Element:value} giving the chemical
                    potential of each element
                xlim:
                    Tuple (min,max) giving the range of the x (fermi energy) axis
                ylim:
                    Tuple (min,max) giving the range for the formation energy axis
                ax_fontsize:
                    float  multiplier to change axis label fontsize
                lg_fontsize:
                    float  multiplier to change legend label fontsize
                lg_position:
                    Tuple (horizontal-position, vertical-position) giving the position
                    to place the legend.
                    Example: (0.5,-0.75) will likely put it below the x-axis.
                saved:

            Returns:
                a matplotlib object

            """
            if xlim is None:
                xlim = (-0.5, self.band_gap + 0.5)
            xy = {}
            lower_cap = -100.
            upper_cap = 100.
            y_range_vals = []  # for finding max/min values on y-axis based on x-limits
            for defnom, def_tl in self.transition_level_map.items():
                xy[defnom] = [[], []]
                if def_tl:
                    org_x = list(def_tl.keys())  # list of transition levels
                    org_x.sort()  # sorted with lowest first

                    # establish lower x-bound
                    first_charge = max(def_tl[org_x[0]])
                    for chg_ent in self.stable_entries[defnom]:
                        if chg_ent.charge == first_charge:
                            form_en = chg_ent.formation_energy(chemical_potentials=mu_elts,
                                                               fermi_level=lower_cap)
                            fe_left = chg_ent.formation_energy(chemical_potentials=mu_elts,
                                                               fermi_level=xlim[0])

                    xy[defnom][0].append(lower_cap)
                    xy[defnom][1].append(form_en)
                    y_range_vals.append(fe_left)

                    # iterate over stable charge state transitions
                    for fl in org_x:
                        charge = max(def_tl[fl])
                        for chg_ent in self.stable_entries[defnom]:
                            if chg_ent.charge == charge:
                                form_en = chg_ent.formation_energy(chemical_potentials=mu_elts,
                                                                   fermi_level=fl)
                        xy[defnom][0].append(fl)
                        xy[defnom][1].append(form_en)
                        y_range_vals.append(form_en)

                    # establish upper x-bound
                    last_charge = min(def_tl[org_x[-1]])
                    for chg_ent in self.stable_entries[defnom]:
                        if chg_ent.charge == last_charge:
                            form_en = chg_ent.formation_energy(chemical_potentials=mu_elts,
                                                               fermi_level=upper_cap)
                            fe_right = chg_ent.formation_energy(chemical_potentials=mu_elts,
                                                                fermi_level=xlim[1])
                    xy[defnom][0].append(upper_cap)
                    xy[defnom][1].append(form_en)
                    y_range_vals.append(fe_right)
                else:
                    # no transition - just one stable charge
                    chg_ent = self.stable_entries[defnom][0]
                    for x_extrem in [lower_cap, upper_cap]:
                        xy[defnom][0].append(x_extrem)
                        xy[defnom][1].append(chg_ent.formation_energy(chemical_potentials=mu_elts,
                                                                      fermi_level=x_extrem)
                                             )
                    for x_window in xlim:
                        y_range_vals.append(chg_ent.formation_energy(chemical_potentials=mu_elts,
                                                                     fermi_level=x_window)
                                            )

            if ylim is None:
                window = max(y_range_vals) - min(y_range_vals)
                spacer = 0.1 * window
                ylim = (min(y_range_vals) - spacer, max(y_range_vals) + spacer)

            if len(xy) <= 8:
                colors = cm.Dark2(np.linspace(0, 1, len(xy)))
            else:
                colors = cm.gist_rainbow(np.linspace(0, 1, len(xy)))

            plt.figure()
            plt.clf()
            width = 12
            # plot formation energy lines
            for_legend = []
            for cnt, defnom in enumerate(xy.keys()):
                plt.plot(xy[defnom][0], xy[defnom][1], linewidth=3, color=colors[cnt])
                for_legend.append(self.stable_entries[defnom][0].copy())

            # plot transtition levels
            for cnt, defnom in enumerate(xy.keys()):
                x_trans, y_trans = [], []
                for x_val, chargeset in self.transition_level_map[defnom].items():
                    x_trans.append(x_val)
                    for chg_ent in self.stable_entries[defnom]:
                        if chg_ent.charge == chargeset[0]:
                            form_en = chg_ent.formation_energy(chemical_potentials=mu_elts,
                                                               fermi_level=x_val)
                    y_trans.append(form_en)
                if len(x_trans):
                    plt.plot(x_trans, y_trans, marker='*', color=colors[cnt], markersize=12, fillstyle='full')

            # get latex-like legend titles
            legends_txt = []
            for dfct in for_legend:
                flds = dfct.name.split('_')
                if 'Vac' == flds[0]:
                    base = '$Vac'
                    sub_str = '_{' + flds[1] + '}$'
                elif 'Sub' == flds[0]:
                    flds = dfct.name.split('_')
                    base = '$' + flds[1]
                    sub_str = '_{' + flds[3] + '}$'
                elif 'Int' == flds[0]:
                    base = '$' + flds[1]
                    sub_str = '_{inter}$'
                else:
                    base = dfct.name
                    sub_str = ''

                legends_txt.append(base + sub_str)

            if not lg_position:
                plt.legend(legends_txt, fontsize=lg_fontsize * width, loc=0)
            else:
                plt.legend(legends_txt, fontsize=lg_fontsize * width, ncol=3,
                           loc='lower center', bbox_to_anchor=lg_position)

            plt.ylim(ylim)
            plt.xlim(xlim)

            plt.plot([xlim[0], xlim[1]], [0, 0], 'k-')  # black dashed line for Eformation = 0
            plt.axvline(x=0.0, linestyle='--', color='k', linewidth=3)  # black dashed lines for gap edges
            plt.axvline(x=self.band_gap, linestyle='--', color='k',
                        linewidth=3)

            if fermi_level is not None:
                plt.axvline(x=fermi_level, linestyle='-.', color='k', linewidth=2)  # smaller dashed lines for gap edges

            plt.xlabel("Fermi energy (eV)", size=ax_fontsize * width)
            plt.ylabel("Defect Formation\nEnergy (eV)", size=ax_fontsize * width)
            if title:
                plt.title("{}".format(title), size=ax_fontsize * width)

            if saved:
                plt.savefig(str(title) + "FreyplnravgPlot.pdf")
            else:
                return plt