# Tailoring Native Defects in LiFePO4: Insights from First-Principles
Calculations

Khang Hoang  Center for Computational Materials Science, Naval Research
Laboratory, Washington, DC 20375, USA  School of Physics, Astronomy, and
Computational Sciences, George Mason University, Fairfax, Virginia 22030, USA
Michelle Johannes<EMAIL_ADDRESS>Center for Computational
Materials Science, Naval Research Laboratory, Washington, DC 20375, USA

###### Abstract

We report first-principles density-functional theory studies of native point
defects and defect complexes in olivine-type LiFePO4, a promising candidate
for rechargeable Li-ion battery electrodes. The defects are characterized by
their formation energies which are calculated within the GGA+$U$ framework. We
find that native point defects are charged, and each defect is stable in one
charge state only. Removing electrons from the stable defects always generates
defect complexes containing small hole polarons. Defect formation energies,
hence concentrations, and defect energy landscapes are all sensitive to the
choice of atomic chemical potentials which represent experimental conditions.
One can, therefore, suppress or enhance certain native defects in LiFePO4 via
tuning the synthesis conditions. Based on our results, we provide insights on
how to obtain samples in experiments with tailored defect concentrations for
targeted applications. We also discuss the mechanisms for ionic and electronic
conduction in LiFePO4 and suggest strategies for enhancing the electrical
conductivity.

lithium iron phosphate, defects, first-principles, polaron, ionic conduction

##  I Introduction

Olivine-type LiFePO4 is a promising candidate for rechargeable Li-ion battery
electrodes.Padhi et al. (1997) The material is known for its structural and
chemical stabilities, high intercalation voltage ($\sim$3.5 V relative to
lithium metal), high theoretical discharge capacity (170 mAh/g), environmental
friendliness, and potentially low costs.Ellis et al. (2010); Manthiram (2011)
The major drawback of LiFePO4 is poor ionic and electronic conduction (with an
electrical conductivity of about 10-9 S/cm at 298 K)Delacourt et al. (2005)
that limits its applicability to devices. While the conduction can be improved
by, e.g., making LiFePO4 nanoparticles and coating with conductive
carbon,Huang et al. (2001); Ellis et al. (2007) the high processing cost
associated with the manufacturing of carbon-coated LiFePO4 nanoparticles may
make it less competitive than other materials. Another approach is to dope
LiFePO4 with aliovalent impurities (Mg, Ti, Zr, Nb) which was reported to have
enhanced the conductivity by eight orders of magnitude.Chung et al. (2002) The
role of these dopants in the conductivity enhancement, however, remains
controversial.Ravet et al. (2003); Wagemaker et al. (2008) A better
understanding of aliovalent doping, and also better solutions for improving
the performance, first requires a deeper understanding of the fundamental
properties, especially those associated with native defects, which is
currently not available. First-principles density-functional theory (DFT)
studies of native point defects and defect complexes in LiFePO4 can help
address these issues.

It is now generally accepted that LiFePO4 is an insulating, large band-gap
material in which electronic conduction proceeds via hopping of small hole
polarons.Zhou et al. (2004); Maxisch et al. (2006); Ellis et al. (2006);
Zaghib et al. (2007) These polarons may be coupled to other defects such as
lithium vacancies.Maxisch et al. (2006); Ellis et al. (2006) Iron antisites
(FeLi) have also been reported to be present in LiFePO4 samples.Yang et al.
(2002); Maier and Amin (2008); Chen et al. (2008); Chung et al. (2008); Axmann
et al. (2009); Chung et al. (2010) This native defect is believed to be
responsible for the loss of electrochemical activity in LiFePO4 due to the
blockage of lithium channels caused by its low mobility.Axmann et al. (2009);
Chung et al. (2010) Clearly, native defects have strong effects on the
material’s performance. Experimental reports on the defects have, however,
painted different pictures. Some authors reported evidence of some iron and
lithium atoms exchanging sites and forming the antisite pair FeLi-LiFe,Chung
et al. (2008, 2010) while others determined that FeLi is formed in association
with lithium vacancies ($V_{\rm{Li}}$).Maier and Amin (2008); Axmann et al.
(2009) These conflicting reports suggest that the results may be sensitive to
the actual synthesis conditions, and indicate that a better understanding of
the formation of native defects in LiFePO4 is needed in order to produce
samples with controlled defect concentrations.

Computational studies of native defects in LiFePO4 and related compounds have
been reported by several research groups.Maxisch et al. (2006); Morgan et al.
(2004); Islam et al. (2005); Fisher et al. (2008); Adams (2010); Malik et al.
(2010) Notably, Maxisch et al. studied the migration of small hole polarons in
LiFePO4 using first-principles calculations where the polarons were created
both in the absence and in the presence of lithium vacancies.Maxisch et al.
(2006) The first systematic study of native defects in LiFePO4 was, however,
carried out by Islam et al. using interatomic-potential simulations where they
found the antisite pair FeLi-LiFe to be energetically most favorable.Islam et
al. (2005); Fisher et al. (2008) Based on results of first-principles
calculations, Malik et al. recently came to a similar conclusion about the
antisite pair.Malik et al. (2010) Although these studies have provided
valuable information on the native defects in LiFePO4, they have three major
limitations. First, studies that make use of interatomic potentials may not
well describe all the defects in LiFePO4. Second, these studies seem to have
focused on neutral defect complexes and did not explicitly report the
structure and energetics of native point defects as individuals. Third, and
most importantly, none of these previous studies have thoroughly investigated
the dependence of defect formation energies and hence defect concentrations on
the atomic chemical potentials which represent experimental conditions during
synthesis.

We herein report our first-principles studies of the structure, energetics,
and migration of native point defects and defect complexes in LiFePO4. We find
that defect formation is sensitive to the synthesis conditions. Native defects
can occur in the material with high concentrations and therefore are expected
to have important implications for ionic and electronic conduction. We will
show how conflicting experimental data on the native defects can be reconciled
under our results and provide general guidelines for producing samples with
tailored defect concentrations. Comparison with previous theoretical works
will be made where appropriate. In the following, we provide technical details
of the calculations and present the theoretical approach. Next, we discuss the
structural and electronic properties of LiFePO4 which form the basis for our
discussion of the formation of native defects in the material. We then present
results of the first-principles calculations for native point defects and
defect complexes, focusing on their formation energies and migration barriers,
and discuss the dependence of defect formation energies on the atomic chemical
potentials. Based on our results, we discuss the implications of native
defects on ionic and electronic conduction, and suggest strategies for
enhancing the electrical conductivity. Finally, we end this Article with some
important conclusions.

##  II Methodology

Computational Details. Our calculations were based on density-functional
theory within the GGA+$U$ framework,Anisimov et al. (1991, 1993);
Liechtenstein et al. (1995) which is an extension of the generalized-gradient
approximation (GGA),Perdew et al. (1996) and the projector augmented wave
method,Blöchl (1994); Kresse and Joubert (1999) as implemented in the VASP
code.Kresse and Hafner (1993); Kresse and Furthmüller (1996); Kresse and
Furthm ller (1996) In this work, we used $U$=5.30 eV and $J$=1.00 eV for iron
in all the calculations (except otherwise noted), i.e., the effective
interaction parameter $U$$-$$J$=4.30 eV (hereafter $U$$-$$J$ will be referred
to as $U$ for simplicity). This value of $U$ is the averaged value based on
those Zhou et al. calculated self-consistently for iron in LiFePO4 (i.e.,
Fe2+: $U$=3.71 eV) and in FePO4 (i.e., Fe3+: $U$=5.90 eV), which has been
shown to correctly reproduce the experimental intercalation potential of
LiFePO4.Zhou et al. (2004) It is known that the results obtained within
GGA+$U$ depend on the value of $U$. However, we have checked the $U$
dependence in our calculations and find that the physics of what we are
presenting is insensitive to the $U$ value for 3.71 eV $\leq U\leq$ 5.90 eV.

Calculations for bulk olivine-type LiFePO4 (orthorhombic $Pnma$; 28 atoms/unit
cell) were performed using a 4$\times$7$\times$9 Monkhorst-Pack
$\mathbf{k}$-point mesh.Monkhorst and Pack (1976) For defect calculations, we
used a (1$\times$2$\times$2) supercell, which corresponds to 112 atoms/cell,
and a 2$\times$2$\times$2 $\mathbf{k}$-point mesh. The plane-wave basis-set
cutoff was set to 400 eV. Convergence with respect to self-consistent
iterations was assumed when the total energy difference between cycles was
less than 10-4 eV and the residual forces were less than 0.01 eV/Å. In the
defect calculations, the lattice parameters were fixed to the calculated bulk
values, but all the internal coordinates were fully relaxed. The migration of
selected defects in LiFePO4 was studied using the climbing-image nudged
elastic-band method (NEB).Henkelman et al. (2000) All calculations were
performed with spin polarization and, unless otherwise noted, the
antiferromagnetic spin configuration of LiFePO4 was used.Rousse et al. (2003)

Defect Formation Energies. Throughout this Article, we employ defect formation
energies to characterize different native point defects and defect complexes
in LiFePO4. The formation energy of a defect is a crucial factor in
determining its concentration. In thermal equilibrium, the concentration of
the defect X at temperature $T$ can be obtained via the relationVan de Walle
and Neugebauer (2004); Janotti and Van de Walle (2009)

$c(\mathrm{X})=N_{\mathrm{sites}}N_{\mathrm{config}}\mathrm{exp}[-E^{f}(\mathrm{X})/k_{B}T],$
(1)

where $N_{\mathrm{sites}}$ is the number of high-symmetry sites in the lattice
per unit volume on which the defect can be incorporated, and
$N_{\mathrm{config}}$ is the number of equivalent configurations (per site).
Note that the energy in Eq. (1) is, in principle, a free energy; however, the
entropy and volume terms are often neglected because they are negligible at
relevant experimental conditions.Janotti and Van de Walle (2009) It emerges
from Eq. (1) that defects with low formation energies will easily form and
occur in high concentrations.

The formation energy of a defect X in charge state $q$ is defined asVan de
Walle and Neugebauer (2004)

$\displaystyle
E^{f}({\mathrm{X}}^{q})=E_{\mathrm{tot}}({\mathrm{X}}^{q})-E_{\mathrm{tot}}({\mathrm{bulk}})-\sum_{i}{n_{i}\mu_{i}}$
$\displaystyle+q(E_{\mathrm{v}}+\Delta V+\epsilon_{F}),$ (2)

where $E_{\mathrm{tot}}(\mathrm{X}^{q})$ and $E_{\mathrm{tot}}(\mathrm{bulk})$
are, respectively, the total energies of a supercell containing the defect X
and of a supercell of the perfect bulk material; $\mu_{i}$ is the atomic
chemical potential of species $i$ (and is referenced to the standard state),
and $n_{i}$ denotes the number of atoms of species $i$ that have been added
($n_{i}$$>$0) or removed ($n_{i}$$<$0) to form the defect. $\epsilon_{F}$ is
the electron chemical potential, i.e., the Fermi level, referenced to the
valence-band maximum in the bulk ($E_{\mathrm{v}}$). $\Delta V$ is the
“potential alignment” term, i.e., the shift in the band positions due to the
presence of the charged defect and the neutralizing background, obtained by
aligning the average electrostatic potential in regions far away from the
defect to the bulk value.Van de Walle and Neugebauer (2004) Note that we
denote defect X in charge state $q$ as Xq. For example, Fe${}_{\rm Li}^{+}$
indicates that defect FeLi occurs with charge $q$=+1, which is equivalent to
Fe${}_{\rm Li}^{\bullet}$ in the Kröger-Vink notation. For a brief discussion
on the use of notations, see, e.g., Ref.Van de Walle and Janotti (2010).

Chemical Potentials. The atomic chemical potentials $\mu_{i}$ are variables
and can be chosen to represent experimental conditions. $\mu_{i}$ can, in
principle, be related to temperatures and pressures via standard thermodynamic
expressions. The chemical potential for O2 in oxygen gas, for example, is
given byOng et al. (2008)

$\mu_{\mathrm{O}_{2}}(T,p)=\mu_{\mathrm{O}_{2}}(T,p_{\circ})+kT{\rm
ln}\frac{p}{p_{\circ}},$ (3)

where $p$ and $p_{\circ}$ are, respectively, the partial pressure and
reference partial pressure of oxygen; $k$ is Boltzmann’s constant. This
expression allows us to calculate $\mu_{\mathrm{O}_{2}}(T,p)$ if we know the
temperature dependence of $\mu_{\mathrm{O}_{2}}(T,p_{\circ})$ at a particular
pressure $p_{\circ}$. In this work, we choose the reference state of
$\mu_{\mathrm{O}_{2}}(T,p)$ to be the total energy of an isolated O2 molecule
($E_{{\rm O}_{2}}^{\rm tot}$).Not

The value of $\mu_{i}$ is subject to various thermodynamic limits. For
LiFePO4, the stability condition requires that

$\mu_{\rm Li}+\mu_{\rm Fe}+\mu_{\rm P}+2\mu_{{\rm O}_{2}}=\Delta H^{f}({\rm
LiFePO}_{4}),$ (4)

where $\Delta H^{f}$ is the formation enthalpy. This condition places a lower
bound on the value of $\mu_{i}$. Additionally, one needs to avoid
precipitating bulk Li, Fe, and P, or forming O2 gas. This sets an upper bound
on the chemical potentials: $\mu_{i}$$\leq$0.Van de Walle and Neugebauer
(2004) There are, however, further constraints imposed by other competing Li-
Fe-P-O2 phases which usually place stronger bounds on $\mu_{i}$. For example,
in order to avoid the formation of Li3PO4,

$3\mu_{\rm Li}+\mu_{\rm P}+2\mu_{{\rm O}_{2}}\leq\Delta H^{f}({\rm Li}_{3}{\rm
PO}_{4}).$ (5)

After taking into account the constraints imposed by all possible competing
phases, one can define the chemical potential range of Li, Fe, and O2 that
stabilizes LiFePO4 which is, in fact, bound in a polyhedron in the three-
dimensional ($\mu_{\rm Li}$, $\mu_{\rm Fe}$, $\mu_{{\rm O}_{2}}$) space. For a
given point in the polyhedron, one can determine the remaining variable
$\mu_{\rm P}$ via Eq. (4). In this work, the formation enthalpies of all
different Li-Fe-P-O2 phases are taken from Ong et al.Ong et al. (2008) who
have computed the energies using a methodology similar to ours. For example,
the calculated formation enthalpy of LiFePO4 at $T$=0 K (with respect to its
constituents) is $-$18.853 eV per formula unit (f.u.),Ong et al. (2008) almost
identical to that (of $-$18.882 eV/f.u.) obtained in our calculations. Ong et
al. have also calculated the phase diagrams of the quaternary Li-Fe-P-O2
system at 0 K that involve all possible phases between Li, Fe, P, and O2.
These phase diagrams show LiFePO4 is stable over a range of the oxygen
chemical potential values, from $-$11.52 eV (where the first Fe2+-containing
phase appears) to $-$8.25 eV (the last of the Fe2+-containing phosphates being
reduced).Ong et al. (2008) This corresponds to $\mu_{{\rm O}_{2}}$ ranging
from $-$3.03 to $-$8.25 eV with respect to our chosen reference ($E_{{\rm
O}_{2}}^{\rm tot}$).

Figure 1: Chemical-potential diagram for LiFePO4 at $\mu_{{\rm
O}_{2}}$=$-$4.59 eV. The $\mu_{{\rm O}_{2}}$ axis extends out of the page.
Only phases that can be in equilibrium with LiFePO4 are included and the lines
delineating these phases define the stability region of LiFePO4, here shown as
a shaded polygon.

Figure 1 shows the slice of the ($\mu_{\rm Li}$, $\mu_{\rm Fe}$, $\mu_{{\rm
O}_{2}}$) polyhedron in the $\mu_{{\rm O}_{2}}$=$-$4.59 eV plane, constructed
with the calculated formation enthalpies (taken from Ref.Ong et al. (2008))
for different Li-Fe-P-O2 phases. The shaded area (marked by Points A, B, C, D,
and E) shows the range of $\mu_{\rm Li}$ and $\mu_{\rm Fe}$ values where
LiFePO4 is stable. Point A, for example, corresponds to equilibrium of LiFePO4
with Fe2O3 and Fe3(PO4)2. At this point in the chemical-potential diagram, the
system is close to forming Fe-containing secondary phases (i.e., Fe2O3 and
Fe3(PO4)2) and far from forming Li-containing secondary phases. This can be
considered as representing a “Li-deficient” environment. Similarly, Point D
can be considered as representing a “Li-excess” environment, where the system
is close to forming Li-containing secondary phases (i.e., Li4P2O7 and Li3PO4).
Note that “Li-deficient” and “Li-excess” environments in this sense do not
necessarily mean that $\mu_{\rm Li}$ in the latter is higher than in the
former, as seen in Fig. 1. Reasonable choices of the atomic chemical
potentials should be those that ensure the stability of the host compound. In
the next sections we will present our calculated formation energies for
various native defects in LiFePO4 and discuss how these defects are formed
under different experimental conditions.

Defect Complexes. Native point defects in LiFePO4 may not stay isolated but
could instead agglomerate and form defect complexes. For a complex XY
consisting of X and Y, its binding energy $E_{b}$ can be calculated using the
formation energy of the complex and those of its constituentsVan de Walle and
Neugebauer (2004)

$E_{b}=E^{f}({\rm X})+E^{f}({\rm Y})-E^{f}({\rm XY}),$ (6)

where the relation is defined such that a positive binding energy corresponds
to a stable, bound defect complex. Having a positive binding energy, however,
does not mean that the complex will readily form. For example, under thermal
equilibrium, the binding energy $E_{b}$ needs to be greater than the larger of
$E^{f}(\rm X)$ and $E^{f}(\rm Y)$ in order for the complex to have higher
concentration than its constituents.Van de Walle and Neugebauer (2004) For
further discussions on the formation of defect complexes, see, e.g., Ref.Van
de Walle and Neugebauer (2004).

##  III Bulk Properties

Before presenting our results for native defects in LiFePO4, let us discuss
some basic properties of the pristine compound. Olivine-type LiFePO4 was
reported to crystallize in the orthorhombic space group $Pnma$ with
$a$=10.3377(5), $b$=6.0112(2), and $c$=4.6950(2) Å.Rousse et al. (2003) The
compound can be regarded as an ordered arrangement of Li+, Fe2+, and (PO4)3-
units. Li+ forms Li channels along the $b$-axis whereas Fe2+ stays at the
center of a slightly distorted FeO6 octahedron (interwoven with PO4
tetrahedra). This simple bonding picture will be very useful when interpreting
the structure and energetics of native defects in LiFePO4. The calculated
lattice parameters are $a$=10.461, $b$=6.061, and $c$=4.752 Å, in satisfactory
agreement with the experimental values. The calculated values are slightly
larger than the experimental ones as expected since it is well known that GGA
tends to overestimate the lattice parameters. The calculated magnetic moment
for iron (Fe2+) is 3.76 $\mu_{\rm B}$, comparable to the experimental value of
4.19(5) $\mu_{\rm B}$ at 2 K.Rousse et al. (2003)

Figure 2: Electronic density of states (DOS) of LiFePO4 in (a)
antiferromagnetic (AFM) and (b) ferromagnetic (FM) spin configurations. The
zero of the energy is set to the highest occupied state.

Figure 2 shows the total electronic density of states of LiFePO4 in
antiferromagnetic (AFM) and ferromagnetic (FM) spin configurations. An
analysis of the wavefunctions shows that, in both configurations, the valence-
band maximum (VBM) and conduction-band minimum (CBM) are Fe 3$d$ states.
Between the highly localized Fe $d$ states just below the Fermi level (at 0
eV) and the lower valence band (which consists predominantly of O 2$p$ and Fe
3$d$ states) there is an energy gap of about 0.40 eV (AFM). The Li 2$s$ state
is high up in the conduction band, suggesting that Li donates its electron to
the lattice and becomes Li+. There is strong mixing between P 3$p$ and O 2$p$
states, indicating covalent bonding within the (PO4)3- unit. The calculated
band gap is 3.62 and 3.58 eV for AFM and FM spin configurations, respectively,
in agreement with previously reported value (of 3.7 eV).Zhou et al. (2004)
Experimentally, LiFePO4 has been reported to have a band gap of about
3.8$-$4.0 eV, obtained from diffuse reflectance measurements.Zhou et al.
(2004); Zaghib et al. (2007) The compound is therefore an insulating, large
band-gap material.

In the GGA+$U$ framework, the electronic structure can depend on the $U$
value. Indeed, we find that the calculated band gap of LiFePO4 is 3.20 and
4.00 eV in the AFM spin configuration for $U$=3.71 and 5.90 eV, respectively,
compared to 3.62 eV obtained in calculations using $U$=4.30 eV mentioned
earlier. The energy gap between the highest valence band (Fe 3$d$ states) and
the lower valence band (predominantly O 2$p$ and Fe 3$d$ states) is also
larger for smaller $U$ value: 0.58 and 0.20 eV for $U$=3.71 and 5.90 eV,
respectively. However, our GGA+$U$ calculations show that the electronic
structure near the band gap region is not sensitive to the choice of $U$
value, for $U$ lying within the range from 3.71 to 5.90 eV. As we illustrate
in the next section, knowing the structural and electronic properties,
especially the nature of the electronic states near the VBM and CBM, is
essential in understanding the formation of native defects in LiFePO4.

##  IV Formation of Native Defects

In insulating, large band-gap materials such as LiFePO4, native point defects
are expected to exist in charged states other than neutral, and charge
neutrality requires that defects with opposite charge states coexist in equal
concentrations.Peles and Van de Walle (2007); Hoang and Van de Walle (2009);
Wilson-Short et al. (2009) We therefore investigated various native defects in
LiFePO4 in all possible charge states. These defects include hole polarons
(hereafter denoted as $p^{+}$), lithium vacancies ($V_{\rm Li}$) and
interstitials (Lii), iron antisites (FeLi), lithium antisites (LiFe), iron
vacancies ($V_{\rm Fe}$), and PO4 vacancies ($V_{{\rm PO}_{4}}$). We also
considered defect complexes that consist of certain point defects such as
FeLi-$V_{\rm Li}$ (a complex of FeLi and $V_{\rm Li}$), FeLi-LiFe (a complex
of FeLi and LiFe), and 2FeLi-$V_{\rm Fe}$ (a complex of two FeLi and one
$V_{\rm Fe}$).

Figure 3: Calculated formation energies of native point defects and defect
complexes in LiFePO4, plotted as a function of Fermi level with respect to the
VBM. The energies are obtained at Point A in the chemical-potential diagram
for $\mu_{{\rm O}_{2}}$=$-$4.59 eV (cf. Fig. 1), representing equilibrium with
Fe2O3 and Fe3(PO4)2.

Figure 3 shows the calculated formation energies of relevant native point
defects and defect complexes in LiFePO4 for a representative oxygen chemical
potential value, $\mu_{{\rm O}_{2}}$=$-$4.59 eV, and $\mu_{{\rm Li}}$=$-$2.85
eV, $\mu_{{\rm Fe}}$=$-$2.18 eV, and $\mu_{{\rm P}}$=$-$4.64 eV. This set of
atomic chemical potentials corresponds to Point A in Fig. 1, representing the
limiting case (Li-deficient) where Fe2O3, Fe3(PO4)2, and LiFePO4 are in
equilibrium. The slope in the formation energy plots indicates the charge
state. Positive slope indicates that the defect is positively charged,
negative slope indicates the defect is negatively charged. With the chosen set
of atomic chemical potentials, the positively charged iron antisite Fe${}_{\rm
Li}^{+}$ and negatively charged lithium vacancy ($V_{\rm Li}^{-}$) have the
lowest formation energies among the charged point defects for a wide range of
Fermi-level values. While there are different charged point defects coexisting
in the system with different concentrations, the ones with the lowest
formation energies have the highest concentrations and are dominant.Peles and
Van de Walle (2007); Hoang and Van de Walle (2009); Wilson-Short et al. (2009)
Figure 3 indicates that, in the absence of electrically active impurities that
can affect the Fermi-level position, or when such impurities occur in much
lower concentrations than charged native defects, the Fermi level will be
pinned at $\epsilon_{F}$=1.06 eV, where the formation energies and hence,
approximately, the concentrations of Fe${}_{\rm Li}^{+}$ and $V_{\rm Li}^{-}$
are equal. Also, charged native defects have positive formation energies only
near $\epsilon_{F}$=1.06 eV. Therefore, any attempt to deliberately shift the
Fermi level far away from this position and closer to the VBM or CBM, e.g.,
via doping with acceptors or donors, will result in positively or negatively
charged native defects having negative formation energies, i.e., the native
defects will form spontaneously and counteract the effects of doping.Van de
Walle and Neugebauer (2004); Janotti and Van de Walle (2009); Van de Walle et
al. (2010); Catlow, C. R. A.; Sokol, A. A.; Walsh, A. (2011) This indicates
that LiFePO4 cannot be doped $p$-type or $n$-type. In the following, we
analyze in detail the structure and energetics of the native defects. The
dependence of defect formation energies on the choice of atomic chemical
potentials will be discussed in the next section.

Small Hole Polarons. The creation of a free positively charged (hole) polaron
$p^{+}$ (i.e., $p^{+}$ in the absence of other defects or extrinsic
impurities) involves removing one electron from the LiFePO4 supercell
(hereafter referred to as “the system”). This results in the formation of a
Fe3+ site in the system. The calculated magnetic moment at this (Fe3+) site is
4.28 $\mu_{\rm B}$, compared to 3.76 $\mu_{\rm B}$ at other iron (Fe2+) sites.
The local geometry near the Fe3+ site is slightly distorted with the
neighboring O atoms moving toward Fe3+; the average Fe-O bond length is 2.07
Å, compared to 2.18 Å of the other Fe-O bonds. Note that in pristine FePO4,
the delithiated phase of LiFePO4, the calculated magnetic moment is 4.29
$\mu_{\rm B}$ at the iron (Fe3+) sites, and the calculated average Fe-O bond
length is 2.06 Å. This indicates that a hole (created by removing an electron
from the system) has been successfully stabilized at one of the iron sites and
the lattice geometry is locally distorted, giving rise to a hole polaron in
LiFePO4. Since the local distortion is found to be mostly limited to the
neighboring O atoms of the Fe3+ site, this hole polaron is considered as small
polaron where the hole is “self-trapped” in its own potential.Shluger and
Stoneham (1993); Stoneham et al. (2007) The formation of free hole polarons in
LiFePO4 is necessarily related to the rather strong interaction between Fe
3$d$ and O $p$ states, and the fact that the VBM consists predominantly of the
highly localized $d$ states.

We have investigated the migration path of $p^{+}$ and estimated the energy
barrier using the NEB method.Henkelman et al. (2000) The migration of $p^{+}$
involves an electron and its associated lattice distortion being transferred
from a Fe2+ site to a neighboring Fe3+ site. Since spin conservation is
required in this process, we carried out our calculations not using the
ground-state AFM structure of LiFePO4 but the FM one where all the spins are
aligned in the same direction. We calculated the migration path by sampling
the atomic positions between ground-state configurations. For those
configurations other than ground-state ones, the atomic positions were kept
fixed and only electron density was relaxed self-consistently, similar to the
method presented in Ref.Maxisch et al. (2006). The migration barrier is the
energy difference between the highest-energy configuration and the ground
state. We find that the migration barrier of $p^{+}$ is 0.25 eV between the
two nearest Fe sites approximately in the $b$-$c$ plane, which is comparable
to that (0.22 eV) reported by in Ref.Maxisch et al. (2006).

Figure 4: Defects in LiFePO4: (a) $V_{\rm Li}^{0}$ can be regarded as a
complex of $V_{\rm Li}^{-}$ (represented by the empty sphere) and hole polaron
$p^{+}$ (i.e., Fe3+; decorated with the square of the wavefunctions of the
lowest unoccupied state in the electronic structure of LiFePO4 in the presence
of $V_{\rm Li}^{0}$); (b) Fe${}_{\rm Li}^{+}$-$V_{\rm Li}^{-}$, a complex of
Fe${}_{\rm Li}^{+}$ and $V_{\rm Li}^{-}$; and (c) Fe${}_{\rm
Li}^{+}$-Li${}_{\rm Fe}^{-}$, a complex of Fe${}_{\rm Li}^{+}$ and Li${}_{\rm
Fe}^{-}$. Large (gray) spheres are Li, medium (blue) spheres Fe, small
(yellow) spheres P, and smaller (red) spheres O.

Vacancies and Interstitials. Negatively charged lithium vacancies ($V_{\rm
Li}^{-}$) are created by removing a Li+ ion from the system. Since, in
LiFePO4, Li donates one electron to the lattice one expects that the removal
of Li+ causes only a small disturbance in the system. Indeed we see that
lattice relaxations around the void formed by the removed Li+ are negligible.
The energy needed to form $V_{\rm Li}^{-}$ should also be small, consistent
with our results in Fig. 3. $V_{\rm Li}^{0}$, on the other hand, is created by
removing a Li atom (i.e., Li+ and an electron) from the system. This leads to
the formation of a void (at the site of the removed Li+) and an Fe3+ (formed
by the removed electron) at the neighboring Fe site. Similar to the free hole
polaron, the neighboring O atoms of the Fe3+ site in $V_{\rm Li}^{0}$ also
move toward Fe3+, with the average Fe-O distance being 2.07 Å. The calculated
magnetic moment is 4.29 $\mu_{\rm B}$ at the Fe3+ site, equal to that at the
Fe3+ site in the case of a free polaron. $V_{\rm Li}^{0}$, therefore, should
be regarded as a complex of $V_{\rm Li}^{-}$ and $p^{+}$, with the two defects
being 3.26 Å apart. Figure 4(a) shows the structure of $V_{\rm Li}^{0}$. The
binding energy of $V_{\rm Li}^{0}$ is 0.34 eV (with respect to $V_{\rm
Li}^{-}$ and $p^{+}$). Note that this value is 0.42 eV in our calculations
using (1$\times$3$\times$3) supercells which have 252 atoms/cell. Our
estimated binding energy is thus comparable to that of 0.39 and about 0.50 eV
reported by Fisher et al.Fisher et al. (2008) and Maxisch et al.,Maxisch et
al. (2006) respectively. For lithium interstitials, the stable defect is
Li${}_{i}^{+}$, created by adding Li+ into the system. Other charge states of
$V_{\rm Li}$ and Lii are not included in Fig. 3 because they either have too
high energies to be relevant or are unstable.

The migration path of $V_{\rm Li}^{-}$ is calculated by moving a Li+ unit from
a nearby lattice site into the vacancy. The energy barrier for $V_{\rm
Li}^{-}$ is estimated to be 0.32 eV along the $b$-axis and 2.27 eV along the
$c$-axis. This suggests that, in the absence of other native defects and
extrinsic impurities, lithium diffusion in LiFePO4 is highly one-dimensional
along the Li channels ($b$-axis) because the energy barrier to cross between
the channels is too high. The migration path of $V_{\rm Li}^{-}$ is, however,
not a straight line but a curved path along the $b$-axis. Our results are thus
in general agreement with previously reported theoretical studiesMorgan et al.
(2004); Islam et al. (2005); Fisher et al. (2008); Adams (2010) and
experimental observation.Nishimura et al. (2008) The estimated energy barriers
for the migration of $V_{\rm Li}^{-}$ along the $b$ and $c$ axes are lower
than those (0.55 and 2.89 eV, respectively) reported by Islam et al.Islam et
al. (2005) obtained from calculations using inter-atomic potentials, but
closer to those (0.27 and about 2.50 eV) reported by Morgan et al.Morgan et
al. (2004) obtained in GGA calculations with smaller supercells. For $V_{\rm
Li}^{0}$, a complex of $p^{+}$ and $V_{\rm Li}^{-}$, one can estimate the
lower bound of the migration barrier by taking the higher of the migration
energies of the constituents,Wilson-Short et al. (2009) which is 0.32 eV
(along the $b$-axis), the value for $V_{\rm Li}^{-}$.

Other possible vacancies in LiFePO4 are those associated with Fe2+ and (PO4)3-
units. The creation of $V_{\rm Fe}^{2-}$ corresponds to removing Fe2+ from the
system. We find that this negatively charged defect causes significant
relaxations in the lattice geometry. The neighboring Li+ ions move toward the
defect, resulting in the Li channels being bent near $V_{\rm Fe}^{2-}$ where
Li+ ions are displaced up to 0.27 Å from their original positions. $V_{\rm
Fe}^{-}$ can be considered as a complex of $V_{\rm Fe}^{2-}$ and $p^{+}$ with
the distance between the two defects being 3.81 Å. $V_{\rm Fe}^{3-}$, on the
other hand, corresponds to removing Fe2+ but leaving an electron in the
system. This defect can be regarded as a complex of $V_{\rm Fe}^{2-}$ and a
negatively charged (electron) polaron (hereafter denoted as $p^{-}$). At the
Fe site where the electron polaron resides, which is 7.68 Å from the vacancy,
the calculated magnetic moment is 2.86 $\mu_{\rm B}$; the average Fe-O
distance is 2.30 Å, which is larger than that associated with other Fe sites
(2.18 Å). Finally, $V_{{\rm PO}_{4}}$ is stable as $V_{{\rm PO}_{4}}^{3+}$ as
expected. This positively charged defect corresponds to removing the whole
(PO4)3- unit from the system. With the chosen set of atomic chemical
potentials, $V_{\rm Fe}^{-}$ and $V_{{\rm PO}_{4}}^{3+}$ have very high
formation energies (2.33 eV and 3.56 eV, respectively, at $\epsilon_{F}$=1.06
eV) and are therefore not included in Fig. 3.

Antisite Defects. Lithium antisites LiFe are created by replacing Fe at an Fe
site with Li. Li${}_{\rm Fe}^{-}$ can be considered as replacing Fe2+ with
Li+. Due to the Coulombic interaction, the two nearest Li+ ion neighbors of
Li${}_{\rm Fe}^{-}$ are pulled closer to the negatively charged defect with
the distance being 3.25 Å, compared to 3.32 Å of the equivalent bond in
pristine LiFePO4. Li${}_{\rm Fe}^{0}$, on the other hand, can be regarded as a
complex of Li${}_{\rm Fe}^{-}$ and $p^{+}$ with the distance between the two
defects being 3.98 Å. The binding energy of Li${}_{\rm Fe}^{0}$ (with respect
to Li${}_{\rm Fe}^{-}$ and $p^{+}$) is 0.30 eV. Similarly, one can replace Li
at an Li site with Fe, which creates an iron antisite FeLi. Fe${}_{\rm
Li}^{+}$ corresponds to replacing Li+ with Fe2+, whereas Fe${}_{\rm Li}^{2+}$
can be regarded as a complex of Fe${}_{\rm Li}^{+}$ and $p^{+}$. For
Fe${}_{\rm Li}^{0}$, which corresponds to replacing one Li+ with Fe2+ and
adding an extra electron to the system, the extra electron is stabilized at
the substituting Fe atom, where the calculated magnetic moment is 2.95
$\mu_{\rm B}$. One might also regard Fe${}_{\rm Li}^{0}$ as a complex of
Fe${}_{\rm Li}^{+}$ and $p^{-}$, but in this case the two defects stay at the
same lattice site. With the chosen set of chemical potentials, Fe${}_{\rm
Li}^{0}$ has a very high formation energy (2.04 eV) and is therefore not
included in Fig. 3. Again, other native defects that are not included here are
unstable or have too high formation energies to be relevant.

Defect Complexes. From the above analyses, it is clear that defects such as
$p^{+}$ ($p^{-}$), $V_{\rm Li}^{-}$, $V_{\rm Fe}^{2-}$, Fe${}_{\rm Li}^{+}$,
Li${}_{\rm Fe}^{-}$, and $V_{{\rm PO}_{4}}^{3+}$ can be considered as
elementary native defects in LiFePO4, i.e., the structure and energetics of
other native defects can be interpreted in terms of these basic building
blocks. This is similar to what has been observed in complex hydrides.Wilson-
Short et al. (2009) These elementary defects (except the free polarons) are,
in fact, point defects that are formed by adding and/or removing only Li+,
Fe2+, and (PO4)3- units. They have low formation energies (cf. Fig. 3) because
the addition/removal of these units causes the least disturbance to the
system, which is consistent with the simple bonding picture for LiFePO4
presented in the previous section. The identification of the elementary native
defects, therefore, not only helps us gain a deeper understanding of the
structure and energetics of the defects in LiFePO4 but also has important
implications. For example, one should treat the migration of defects such as
$V_{\rm Li}^{0}$ as that of a $V_{\rm Li}^{-}$ and $p^{+}$ complex with a
finite binding energy, rather than as a single point defect.

In addition to the defect complexes that involve $p^{+}$ and $p^{-}$ such as
$V_{\rm Li}^{0}$, $V_{\rm Fe}^{-}$, $V_{\rm Fe}^{3-}$, Li${}_{\rm Fe}^{0}$,
Fe${}_{\rm Li}^{0}$, and Fe${}_{\rm Li}^{2+}$ described above, we also
considered those consisting of $V_{\rm Li}^{-}$, Fe${}_{\rm Li}^{+}$,
Li${}_{\rm Fe}^{-}$, and $V_{\rm Fe}^{2-}$ such as Fe${}_{\rm
Li}^{+}$-Li${}_{\rm Fe}^{-}$, Fe${}_{\rm Li}^{+}$-$V_{\rm Li}^{-}$, and
2Fe${}_{\rm Li}^{+}$-$V_{\rm Fe}^{2-}$. Figure 4(b) shows the structure of
Fe${}_{\rm Li}^{+}$-$V_{\rm Li}^{-}$. The distance between Fe${}_{\rm Li}^{+}$
and $V_{\rm Li}^{-}$ is 2.96 Å (along the $b$-axis), compared to 3.03 Å
between the two Li sites in pristine LiFePO4. We find that this complex has a
formation energy of 0.36$-$0.56 eV for reasonable choices of atomic chemical
potentials, and a binding energy of 0.49 eV. With such a relatively high
binding energy, even higher than the formation energy of isolated Fe${}_{\rm
Li}^{+}$ and $V_{\rm Li}^{-}$ (0.42 eV at $\epsilon_{F}$=1.06 eV, cf. Fig. 3),
Fe${}_{\rm Li}^{+}$-$V_{\rm Li}^{-}$ is expected to occur with a concentration
larger than either of its constituents under thermal equilibrium conditions
during synthesis.Van de Walle and Neugebauer (2004) In Fe${}_{\rm
Li}^{+}$-$V_{\rm Li}^{-}$, the energy barrier for migrating Fe${}_{\rm
Li}^{+}$ to $V_{\rm Li}^{-}$ is about 0.74 eV, comparable to that (0.70 eV)
reported by Fisher et al.Fisher et al. (2008) This value is twice as high as
the migration barrier of $V_{\rm Li}^{-}$, indicating Fe${}_{\rm Li}^{+}$ has
low mobility.

Figure 4(c) shows the structure of Fe${}_{\rm Li}^{+}$-Li${}_{\rm Fe}^{-}$.
This antisite pair has a formation energy of 0.51 eV. This value is
independent of the choice of chemical potentials because the chemical
potential term in the formation energy formula cancels out, cf. Eq. (II).
Fe${}_{\rm Li}^{+}$-Li${}_{\rm Fe}^{-}$ has a binding energy of 0.44 eV; the
distance between Fe${}_{\rm Li}^{+}$ and Li${}_{\rm Fe}^{-}$ is 3.45 Å,
compared to 3.32 Å between the lithium and iron sites. Finally, we find that
2Fe${}_{\rm Li}^{+}$-$V_{\rm Fe}^{2-}$ has a formation energy of 1.47$-$1.67
eV for reasonable choices of the atomic chemical potentials, and a binding
energy of 1.25 eV. With this high formation energy, the complex is unlikely to
form in LiFePO4, and is therefore not included in Fig. 3. Note that the
formation energies of Fe${}_{\rm Li}^{+}$-$V_{\rm Li}^{-}$ and 2Fe${}_{\rm
Li}^{+}$-$V_{\rm Fe}^{2-}$ have the same dependence on the atomic chemical
potentials (both contain the term $-\mu_{\rm Fe}+2\mu_{\rm Li}$) and, hence,
the same dependence on $\mu_{{\rm O}_{2}}$. For any given set of chemical
potentials, the formation energy of 2Fe${}_{\rm Li}^{+}$-$V_{\rm Fe}^{2-}$ is
higher than that of Fe${}_{\rm Li}^{+}$-$V_{\rm Li}^{-}$ by 1.11 eV. We also
considered possible lithium and iron Frenkel pairs (i.e., interstitial-vacancy
pairs), but these pairs are unstable toward recombination, probably because
there is no energy barrier or too small of a barrier between the vacancy and
the interstitial.

The above mentioned neutral defect complexes have also been studied by other
research groups using either interatomic-potential simulationsIslam et al.
(2005); Fisher et al. (2008) or first-principles DFT calculations.Malik et al.
(2010) Islam et al. found that Fe${}_{\rm Li}^{+}$-Li${}_{\rm Fe}^{-}$ has a
formation energy of 0.74 eV (or 1.13 eV if the two defects in the pair are
considered as isolated defects) and a binding energy of 0.40 eV, and is
energetically most favorable among possible native defects.Islam et al. (2005)
The reported formation energy is, however, higher than our calculated value by
0.23 eV. This difference may be due to the different methods used in the
calculations. Fisher et al. reported a formation energy of 3.13 eV for
Fe${}_{\rm Li}^{+}$-$V_{\rm Li}^{-}$,Fisher et al. (2008) which is much higher
than our calculated value. Note, however, that Fisher et al. assumed the
reaction ${\rm FeO}+2{\rm Li}_{\rm{Li}}^{0}$$\rightarrow$${\rm Fe}_{\rm
Li}^{+}+V_{\rm Li}^{-}+{\rm Li}_{2}{\rm O}$ for the formation of Fe${}_{\rm
Li}^{+}$-$V_{\rm Li}^{-}$ which implies that LiFePO4 is in equilibrium with
FeO and Li2O. This scenario is unlikely to occur, as indicated in the Li-Fe-
P-O2 phase diagrams calculated by Ong et al.,Ong et al. (2008) where
equilibrium between these phases has never been observed. This may also be the
reason that the formation energy of $V_{\rm Li}^{0}$ reported by the same
authors (4.41 eV)Fisher et al. (2008) is much higher than our calculated
values.

Based on first-principles calculations, Malik et al. reported a formation
energy of 0.515$-$0.550 eV for the antisite pair Fe${}_{\rm
Li}^{+}$-Li${}_{\rm Fe}^{-}$,Malik et al. (2010) which is very close to our
calculated value (0.51 eV). For 2Fe${}_{\rm Li}^{+}$-$V_{\rm Fe}^{2-}$, the
formation energy was reported to be of about 1.60$-$1.70 eV for $\mu_{{\rm
O}_{2}}$ ranging from $-$3.03 to $-$8.21 eV,Malik et al. (2010) which is also
comparable to our results. Malik et al., however, obtained a much higher
formation energy for Fe${}_{\rm Li}^{+}$-$V_{\rm Li}^{-}$, from about 3.60 to
5.10 eV for the same range of $\mu_{{\rm O}_{2}}$ values.Malik et al. (2010)
This energy is much higher than that obtained in our calculations (0.36$-$0.56
eV). Although we have no explanation for this discrepancy, we observe that the
calculated formation energies of 2Fe${}_{\rm Li}^{+}$-$V_{\rm Fe}^{2-}$ and
Fe${}_{\rm Li}^{+}$-$V_{\rm Li}^{-}$ in Malik et al.’s work have distinct
$\mu_{{\rm O}_{2}}$-dependencies (see Figure S1 in the Supporting Information
of Ref. Malik et al. (2010)), instead of having the same dependence on
$\mu_{{\rm O}_{2}}$ as we discussed above, indicating their scheme of
accounting for the atomic chemical potentials differs from the standard
procedure.

##  V Tailoring Defect Concentrations

It is important to note that the energy landscape presented in Fig. 3 may
change as one changes the atomic chemical potentials, i.e., synthesis
conditions. The calculated formation energies are a function of four variables
$\mu_{\rm Li}$, $\mu_{\rm Fe}$, $\mu_{\rm P}$, and $\mu_{{\rm O}_{2}}$, which
in turn depend on each other and vary within the established constraints. A
change in one variable leads to changes in the other three. In the following
discussions, we focus on two “knobs” that can be used to experimentally tailor
the formation energy and hence the concentration of different native defects
in LiFePO4, and suppress or enhance certain defects for targeted applications.
One is $\mu_{{\rm O}_{2}}$, which can be controlled by controlling temperature
and pressure and/or oxygen reducing agents. Lower $\mu_{{\rm O}_{2}}$ values
represent the so-called “more reducing environments,” which are usually
associated with higher temperatures and/or lower oxygen partial pressures
and/or the presence of oxygen reducing agents; whereas higher $\mu_{{\rm
O}_{2}}$ values represent “less reducing environments.”Ong et al. (2008) The
other is the degree of lithium off-stoichiometry with respect to LiFePO4
exhibited through the tendency toward formation of Li-containing or Fe-
containing secondary phases in the synthesis of LiFePO4. As discussed
previously, in the environments to which we refer as Li-excess (Li-deficient),
the system is close to forming Li-containing (Fe-containing) secondary phases.

Figure 5: Calculated formation energies of native point defects and defect
complexes in LiFePO4, plotted as a function of Fermi level with respect to the
VBM. The energies are obtained at $\mu_{{\rm O}_{2}}$=$-$3.03 eV, and
equilibrium with Fe2O3 and Fe7(PO4)6 is assumed.

Varying the Atomic Chemical Potentials. Let us assume, for example, Li-
deficient environments and vary $\mu_{{\rm O}_{2}}$ from $-$3.03 (where
LiFePO4 first starts to form) to $-$8.25 eV (where it ceases to form).Ong et
al. (2008) This amounts to choosing different cuts along the $\mu_{{\rm
O}_{2}}$ axis in Fig. 1 to give different two-dimensional polygons of LiFePO4
stability. Figure 5 shows the calculated formation energies for $\mu_{{\rm
O}_{2}}$=$-$3.03 eV, assuming equilibrium with Fe2O3 and Fe7(PO4)6 (i.e., Li-
deficient) which gives rise to $\mu_{\rm Li}$=$-$3.41, $\mu_{\rm Fe}$=$-$3.35,
and $\mu_{\rm P}$=$-$6.03 eV. Figure 5 clearly shows changes in the energy
landscape of the defects, compared to Fig. 3. The lowest energy point defects
that determine the Fermi-level position are now $p^{+}$ and $V_{\rm Li}^{-}$.
Near $\epsilon_{F}$=0.59 eV where $p^{+}$ and $V_{\rm Li}^{-}$ have equal
formation energies, $V_{\rm Li}^{0}$ also has the lowest energy. This
indicates that, under high $\mu_{{\rm O}_{2}}$ and Li-deficient environments,
$p^{+}$ and $V_{\rm Li}^{-}$ are the dominant native point defects in LiFePO4
and are likely to exist in the form of the neutral complex $V_{\rm Li}^{0}$.
Note that, with the chosen set of atomic chemical potentials, Li${}_{\rm
Fe}^{-}$ also has a low formation energy, very close to that of $V_{\rm
Li}^{-}$, indicating the presence of a relatively high concentration of
Li${}_{\rm Fe}^{-}$. Similar to $V_{\rm Li}^{-}$, Li${}_{\rm Fe}^{-}$ can
combine with $p^{+}$ to form Li${}_{\rm Fe}^{0}$. However, since Li${}_{\rm
Fe}^{0}$ has a higher formation energy and a smaller binding energy than
$V_{\rm Li}^{0}$ , only a small portion of Li${}_{\rm Fe}^{-}$ is expected to
be stable in form of Li${}_{\rm Fe}^{0}$ under thermal equilibrium conditions.
Iron vacancies have the lowest energies in a wide range of the Fermi-level
values as expected, given the very low iron chemical potential.

Figure 6: Calculated formation energies of native point defects and defect
complexes in LiFePO4, plotted as a function of Fermi level with respect to the
VBM. The energies are obtained at $\mu_{{\rm O}_{2}}$=$-$8.21 eV, and
equilibrium with Fe2P and Fe3P is assumed.

Figure 6 shows the calculated formation energies for $\mu_{{\rm
O}_{2}}$=$-$8.21 eV. The formation energies are obtained by assuming
equilibrium with Fe2P and Fe3P (i.e., Li-deficient) which gives rise to
$\mu_{\rm Li}$=$-$1.80, $\mu_{\rm Fe}$=$-$0.24, and $\mu_{\rm P}$=$-$0.39 eV.
We find that Fe${}_{\rm Li}^{+}$ and Li${}_{\rm{Fe}}^{-}$ are now the dominant
native point defects, pinning the Fermi level at $\epsilon_{F}$=2.00 eV. The
complex Fe${}_{\rm Li}^{+}$-Li${}_{\rm{Fe}}^{-}$ has a binding energy of 0.44
eV, comparable to the formation energies of Fe${}_{\rm Li}^{+}$ and
Li${}_{\rm{Fe}}^{-}$ (which are both 0.48 eV at $\epsilon_{F}$=2.00 eV). This
suggests that Fe${}_{\rm Li}^{+}$ and Li${}_{\rm{Fe}}^{-}$ are likely to exist
both in the form of Fe${}_{\rm Li}^{+}$-Li${}_{\rm{Fe}}^{-}$, but also as
isolated point defects. With this set of atomic chemical potentials, we find
that $V_{{\rm PO}_{4}}^{3+}$ has the lowest formation energy near the VBM, and
Fe${}_{\rm Li}^{0}$ has a formation energy of 1.15 eV that, while very high,
is lower than $V_{\rm Li}^{0}$ ($1.93$ eV) and Li${}_{\rm Fe}^{0}$ ($1.98$
eV), which were found to have lower formation energies under different
conditions (cf. Figures 3 and 5).

We also investigated the dependence of defect formation energies on
$\mu_{\rm{Li}}$ (and $\mu_{\rm{Fe}}$), i.e., Li-deficiency versus Li-excess,
for a given $\mu_{{\rm O}_{2}}$ value. For $\mu_{{\rm O}_{2}}$=$-4.59$ eV, for
example, the results obtained at Points B and C in Fig. 1 show energy
landscapes that are similar to that at Point A (Li-deficient), namely
Fe${}_{\rm Li}^{+}$ and $V_{\rm Li}^{-}$ are the dominant native point defects
in LiFePO4 and likely to exist in form of Fe${}_{\rm Li}^{+}$-$V_{\rm
Li}^{-}$. At Point D, where LiFePO4 is in equilibrium with Li4P2O7 and Li3PO4
(Li-excess), we find instead that Fe${}_{\rm Li}^{+}$ and Li${}_{\rm Fe}^{-}$
are energetically most favorable, and are likely to exist as Fe${}_{\rm
Li}^{+}$-Li${}_{\rm{Fe}}^{-}$. The calculated formation energy of $p^{+}$ is
only slightly higher than that of Fe${}_{\rm Li}^{+}$, indicating a coexisting
high concentration of $p^{+}$. The hole polarons in this case are expected to
exist as isolated defects under thermal equilibrium since Li${}_{\rm Fe}^{0}$
has a relatively high formation energy (0.66 eV) and a small binding energy
(0.30 eV). Point E gives results that are similar to those at Point D. In
contrast, when we choose $\mu_{{\rm O}_{2}}$=$-8.21$ eV, we find that
Fe${}_{\rm Li}^{+}$ and Li${}_{\rm Fe}^{-}$ are the most energetically
favorable defects regardless of the choice of phase-equilibrium conditions.

Table 1: Calculated formation energies ($E^{f}$) and migration barriers
($E_{m}$) of the most relevant native point defects and defect complexes in
LiFePO4. (1)$-$(8) are the equilibrium conditions; see text. Binding energies
($E_{b}$) of the defect complexes (with respect to their isolated
constituents) are given in the last column. The formation energy of
2Fe${}_{\rm{Li}}^{+}$-$V_{\rm{Fe}}^{2-}$ is high and thus the complex is not
likely to form, but is also given here for comparison.

Defect |  $E^{f}$ (eV) |  $E_{m}$ (eV) | Constituents |  $E_{b}$(eV)
---|---|---|---|---
| (1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) |  |  |
$p^{+}$ | 0.33 | 0.32 | 0.54 | 0.40 | 0.80 | 0.49 | 1.74 | 1.72 | 0.25 |  |
Fe${}_{\rm{Li}}^{+}$ | 0.57 | 0.63 | 0.43 | 0.55 | 0.42 | 0.48 | 0.48 | 0.48 |  |  |
$V_{\rm{Li}}^{-}$ | 0.33 | 0.41 | 0.43 | 0.47 | 0.42 | 0.56 | 0.53 | 0.55 | 0.32 |  |
Li${}_{\rm{Fe}}^{-}$ | 0.39 | 0.32 | 0.52 | 0.40 | 0.53 | 0.48 | 0.48 | 0.48 |  |  |
$V_{\rm{Li}}^{0}$ | 0.32 | 0.38 | 0.62 | 0.52 | 0.88 | 0.70 | 1.93 | 1.95 | 0.32 _a_ |  $V_{\rm{Li}}^{-}$ \+ $p^{+}$ | 0.34, 0.42 _b_
Li${}_{\rm{Fe}}^{0}$ | 0.42 | 0.34 | 0.76 | 0.50 | 1.03 | 0.66 | 1.92 | 1.90 |  | Li${}_{\rm{Fe}}^{-}$ \+ $p^{+}$ | 0.30
Fe${}_{\rm{Li}}^{+}$-$V_{\rm{Li}}^{-}$ | 0.41 | 0.55 | 0.37 | 0.53 | 0.36 | 0.55 | 0.52 | 0.56 |  | Fe${}_{\rm{Li}}^{+}$ \+ $V_{\rm{Li}}^{-}$ | 0.49
Fe${}_{\rm{Li}}^{+}$-Li${}_{\rm{Fe}}^{-}$ | 0.51 | 0.51 | 0.51 | 0.51 | 0.51 | 0.51 | 0.51 | 0.51 |  | Fe${}_{\rm{Li}}^{+}$ \+ Li${}_{\rm{Fe}}^{-}$ | 0.44
2Fe${}_{\rm{Li}}^{+}$-$V_{\rm{Fe}}^{2-}$ | 1.52 | 1.66 | 1.48 | 1.64 | 1.47 | 1.66 | 1.63 | 1.67 |  | 2Fe${}_{\rm{Li}}^{+}$ \+ $V_{\rm{Fe}}^{2-}$ | 1.25

_a_ Lower bound, estimated by considering $V_{\rm{Li}}^{0}$ as a complex of
$V_{\rm{Li}}^{-}$ and $p^{+}$ and taking the higher of the migration energies
of the constituents. _b_ The value obtained in calculations using larger
supercells (252 atoms/cell).

Identifying the General Trends. We list in Table 1 the formation energies of
the most relevant native point defects and defect complexes in LiFePO4,
migration barriers of selected point defects, and binding energies of the
defect complexes. The chemical potentials are chosen with representative
$\mu_{{\rm O}_{2}}$ values and Li-deficient and Li-excess environments to
reflect different experimental conditions. Specifically, these conditions
represent equilibrium of LiFePO4 with (1) Fe2O3 and Fe7(PO4)6 and (2)
Li3Fe2(PO4)3 and Li3PO4, for $\mu_{{\rm O}_{2}}$=$-$3.03 eV; (3) Fe2O3 and
Fe3(PO4)2 and (4) Li4P2O7 and Li3PO4, for $\mu_{{\rm O}_{2}}$=$-$3.89 eV; (5)
Fe2O3 and Fe3(PO4)2 (i.e., Point A in Fig. 1) and (6) Li4P2O7 and Li3PO4
(i.e., Point D in Fig. 1), for $\mu_{{\rm O}_{2}}$=$-$4.59 eV; (7) Fe3P and
Fe2P and (8) Li3PO4 and Fe3P, for $\mu_{{\rm O}_{2}}$=$-$8.21 eV. Conditions
(1), (3), (5), and (7) represent Li-deficient environments, whereas (2), (4),
and (6) represent Li-excess. Under each condition, the formation energies for
charged defects are taken at the Fermi-level position determined by relevant
charged point defects: (1) $\epsilon_{F}$=0.59 eV (where $p^{+}$ and
$V_{\rm{Li}}^{-}$ have equal formation energies), (2) $\epsilon_{F}$=0.58 eV
($p^{+}$ and Li${}_{\rm{Fe}}^{-}$), (3) $\epsilon_{F}$=0.79 eV (Fe${}_{\rm
Li}^{+}$ and $V_{\rm{Li}}^{-}$), (4) $\epsilon_{F}$=0.66 eV ($p^{+}$ and
Li${}_{\rm{Fe}}^{-}$), (5) $\epsilon_{F}$=1.06 eV (Fe${}_{\rm Li}^{+}$ and
$V_{\rm{Li}}^{-}$), (6) $\epsilon_{F}$=0.74 eV (Fe${}_{\rm Li}^{+}$ and
Li${}_{\rm{Fe}}^{-}$), (7) $\epsilon_{F}$=2.00 eV (Fe${}_{\rm Li}^{+}$ and
Li${}_{\rm{Fe}}^{-}$), and (8) $\epsilon_{F}$=1.98 eV (Fe${}_{\rm Li}^{+}$ and
Li${}_{\rm{Fe}}^{-}$). The results for other $\mu_{{\rm O}_{2}}$ values are
not included in Table 1 because they give results that are similar to those
presented here. For example, the energy landscapes for $\mu_{{\rm
O}_{2}}$=$-$3.25 eV and $\mu_{{\rm O}_{2}}$=$-7.59$ eV are similar to those
for $\mu_{{\rm O}_{2}}$=$-$3.03 eV and $\mu_{{\rm O}_{2}}$=$-4.59$ eV,
respectively.

Figure 7: Calculated formation energies of the low-energy positively and
negatively charged point defects (i.e., $p^{+}$, Fe${}_{\rm{Li}}^{+}$,
$V_{\rm{Li}}^{-}$, and Li${}_{\rm{Fe}}^{-}$) and their neutral defect
complexes, plotted as a function of Fermi level with respect to the VBM, under
different conditions: $\mu_{{\rm O}_{2}}$=$-$3.03, $-$3.89, $-$4.59, and
$-$8.21 eV, and Li-deficient and Li-excess environments. Panels (a)$-$(h)
correspond to conditions (1)$-$(8), see text. Only the complex that consists
of the lowest-energy negatively and positively charged point defects are
included.

In order to help capture the most general trends in the energy landscape of
native defects in LiFePO4 in going from high to low $\mu_{{\rm O}_{2}}$ values
and from Li-deficient to Li-excess environments, we plot in Fig. 7 the
calculated formation energies of the most relevant native point defects and
their lowest-energy complexes obtained under conditions (1)$-$(8). We find
that, at a given Fermi-level position $\epsilon_{F}$, the formation energy of
Fe${}_{\rm{Li}}^{+}$ decreases as $\mu_{{\rm O}_{2}}$ decreases. This is
because $\mu_{\rm Fe}$ increases more rapidly than $\mu_{\rm Li}$ does as
$\mu_{{\rm O}_{2}}$ decreases from $-$3.03 eV to $-$8.21 eV. The formation
energy of $p^{+}$, on the other hand, is independent of the choice of atomic
chemical potentials, and depends only on $\epsilon_{F}$. At high $\mu_{{\rm
O}_{2}}$ values, $p^{+}$ is lower in energy than Fe${}_{\rm{Li}}^{+}$, but
then the two defects switch orders before $\mu_{{\rm O}_{2}}$ reaches $-$3.89
eV (under Li-deficient environments) or $-$4.59 eV (Li-excess). The formation
energy of the dominant positive defects, $p^{+}$ and Fe${}_{\rm{Li}}^{+}$,
differs by as much as 1.3 eV for some sets of atomic chemical potentials. On
the contrary, the dominant negative defects, $V_{\rm{Li}}^{-}$ and
Li${}_{\rm{Fe}}^{-}$, have comparable formation energies throughout the range
of conditions. The largest formation energy difference between the two defects
is just 0.2 eV. The formation energy of Li${}_{\rm{Fe}}^{-}$ is slightly lower
than that of $V_{\rm{Li}}^{-}$ under Li-excess environments; whereas it is
slightly higher under Li-deficient environments, except near $\mu_{{\rm
O}_{2}}$=$-$8.21 eV where $V_{\rm{Li}}^{-}$ is higher than
Li${}_{\rm{Fe}}^{-}$. Both $V_{\rm{Li}}^{-}$ and Li${}_{\rm{Fe}}^{-}$ have
their formation energies increased as $\mu_{{\rm O}_{2}}$ decreases. In going
from high to low $\mu_{{\rm O}_{2}}$ values, the changes in the formation
energy of Fe${}_{\rm{Li}}^{+}$ and that of $V_{\rm{Li}}^{-}$ and
Li${}_{\rm{Fe}}^{-}$ leads to a shift of the Fermi level from about 0.6 eV
above the VBM to 2.0 eV above the VBM. The variation in the calculated
formation energy of $p^{+}$ as seen in Table 1 is a result of this shift.

Under Li-deficient environments, we find that Fe${}_{\rm{Li}}^{+}$ and
$V_{\rm{Li}}^{-}$ are energetically most favorable over a wide range of
$\mu_{{\rm O}_{2}}$ values, from $-$3.89 to $-$7.59 eV, and are likely to
exist in form of the neutral complex Fe${}_{\rm{Li}}^{+}$-$V_{\rm{Li}}^{-}$.
At the higher end in the range of $\mu_{{\rm O}_{2}}$ values, from $-$3.03 to
$-$3.25 eV, $p^{+}$ and $V_{\rm{Li}}^{-}$ are the most favorable, and are
likely to exist in form of the complex $V_{\rm{Li}}^{0}$. Finally, only at the
lowest end of the $\mu_{{\rm O}_{2}}$ values, the most favorable point defects
are Fe${}_{\rm{Li}}^{+}$ and Li${}_{\rm{Fe}}^{-}$, which may exist in form of
the neutral complex Fe${}_{\rm{Li}}^{+}$-Li${}_{\rm{Fe}}^{-}$. Under Li-excess
environments, Li${}_{\rm{Fe}}^{-}$ dominates the negatively charged point
defects in the whole range of $\mu_{{\rm O}_{2}}$ values. This makes $p^{+}$
and Li${}_{\rm{Fe}}^{-}$ the most favorable point defects for $\mu_{{\rm
O}_{2}}$ ranging from $-$3.03 to $-$3.89 eV, and Fe${}_{\rm{Li}}^{+}$ and
Li${}_{\rm{Fe}}^{-}$ the most favorable defects for $\mu_{{\rm O}_{2}}$
ranging from $-$4.59 to $-$8.21 eV. Note that, although the formation energy
difference between $V_{\rm{Li}}^{-}$ and Li${}_{\rm{Fe}}^{-}$ is small (less
than 0.2 eV), the difference in their concentrations can still be significant,
as indicated by the exponential dependence in Eq. (1).

Overall, we find that the calculated formation energies of the dominant native
point defects are low, from about 0.3 to 0.5 eV for $\mu_{{\rm O}_{2}}$ from
$-$3.03 to $-$8.21 eV (cf. Table 1). With such low formation energies, the
defects will easily form and occur with high concentrations. The dominant
defects may be different, however, if one changes the experimental conditions
during synthesis, as discussed above. This is consistent with the reported
experimental data showing the presence of various native defects in LiFePO4
samples.Ellis et al. (2006); Zaghib et al. (2007); Yang et al. (2002); Maier
and Amin (2008); Chen et al. (2008); Chung et al. (2008); Axmann et al.
(2009); Chung et al. (2010) We note that there are several limitations
inherent in our calculations. The first set of limitations comes from standard
methodological uncertainties contained in the calculated formation enthalpies
and phase diagrams as discussed in Ref.Ong et al. (2008). The second set comes
from the calculation of defect formation energies using supercell models where
supercell finite-size effects are expected.Van de Walle and Neugebauer (2004)
Since applying approximations indiscriminately in an attempt to correct for
finite-size effects tends to “overshoot” and makes the energies even less
accurate,Van de Walle and Neugebauer (2004) we did not include any corrections
pertaining to such effects in our defect calculations. A proper treatment of
finite-size effects, however if applied, will lead to an increase in the
calculated formation energy of the charged native point defects, and hence the
binding energy of the neutral defect complexes. In spite of the limitations,
the general trends discussed above should still hold true. And, our results
therefore can serve as guidelines for tailoring the defect concentrations in
LiFePO4, and suppressing or enhancing certain defects for targeted
applications.

##  VI Electronic and Ionic Conduction

Strictly speaking, lithium vacancies are only stable as $V_{\rm{Li}}^{-}$, and
$V_{\rm{Li}}^{0}$ (which is, in fact, a complex of $V_{\rm{Li}}^{-}$ and
$p^{+}$) cannot be considered the vacancies’s neutral charge state. Likewise,
lithium antisites, iron antisites, and iron vacancies also have one stable
charge state only and occur as, respectively, Li${}_{\rm{Fe}}^{-}$,
Fe${}_{\rm{Li}}^{+}$, and $V_{\rm{Fe}}^{2-}$. Removing/adding electrons
from/to these stable point defects always results in defect complexes
consisting of the point defects and small hole/electron polarons, as presented
in the previous sections. The fact that small polarons can be stabilized, both
in the absence and in the presence of other native defects is necessarily
related to the electronic structure of LiFePO4 where the VBM and CBM consist
predominantly of the highly localized Fe 3$d$ states. Combined with the fact
that charged native defects have negative formation energies near the VBM and
CBM (cf. Figures 3, 5, and 6), our results therefore indicate that native
defects in LiFePO4 cannot act as sources of band-like hole and electron
conductivities. These defects will, however, act as compensating centers in
donor-like doping (for the negatively charged defects) or acceptor-like doping
(the positively charged defects). The electronic conduction in LiFePO4 thus
occurs via hopping of small hole polarons. This mechanism, in fact, has been
proposed for LiFePO4 in several previous works.Zhou et al. (2004); Maxisch et
al. (2006); Ellis et al. (2006); Zaghib et al. (2007) Zaghib et al.Zaghib et
al. (2007) found experimental evidence of intra-atomic Fe2+$-$Fe3+ transitions
in the optical spectrum of LiFePO4, which indirectly confirms the formation of
small hole polarons. The activation energy for electronic conductivity in
LiFePO4 was estimated to be 0.65 eV,Zaghib et al. (2007) comparable to that of
0.55$-$0.78 eV reported by several other experimental groups.Delacourt et al.
(2005); Ellis et al. (2006); Amin et al. (2008)

In order to compare our results with the measured activation energies, let us
assume two scenarios for hole polaron hopping in LiFePO4. In the first
scenario, we assume self-diffusion of free $p^{+}$ defects. The activation
energy $E_{a}$ for this process is calculated as the summation of the
formation energy and migration barrier of $p^{+}$, where the former is
associated with the intrinsic concentration and the latter with the
mobility,Balluffi et al. (2005)

$E_{a}=E^{f}(p^{+})+E_{m}(p^{+}),$ (7)

which gives $E_{a}$=0.57 eV, if $E^{f}(p^{+})$ is taken under the most
favorable condition where $p^{+}$ has the lowest formation energy (cf. Table
1). In the second scenario, we assume that $p^{+}$ and $V_{\rm{Li}}^{-}$ are
formed via the formation of the neutral complex $V_{\rm{Li}}^{0}$, similar to
cases where defects are created via a Frenkel or Schottky mechanism.Balluffi
et al. (2005) At high temperatures, the activation energy for the diffusion of
$p^{+}$ can be calculated as

$E_{a}=\frac{1}{2}E^{f}(V_{\rm{Li}}^{0})+E_{m}(p^{+}),$ (8)

which results in $E_{a}$=0.41 eV, assuming the condition where
$V_{\rm{Li}}^{0}$ has the lowest formation energy (cf. Table 1). The lower
bound of the activation energy for polaron conductivity is therefore
0.41$-$0.57 eV. This range of $E_{a}$ values is comparable to that obtained in
experiments.Delacourt et al. (2005); Ellis et al. (2006); Zaghib et al.
(2007); Amin et al. (2008)

Among the native defects, $V_{\rm{Li}}^{-}$ is the most plausible candidate
for ionic conduction in LiFePO4, because of its low formation energy and high
mobility. Using formulae similar to Eqs. (7) and (8), we estimate the lower
bound of the activation energy for self-diffusion of $V_{\rm{Li}}^{-}$ along
the $b$-axis to be 0.48$-$0.65 eV. The reported experimental value is 0.54
eVLi et al. (2008) or 0.62$-$0.74 eV,Amin et al. (2008) obtained from ionic
conductivity measurements carried out on LiFePO4 single crystals, which is
comparable to our calculated value. The diffusion of $V_{\rm{Li}}^{-}$,
however, may be impeded by other native defects or extrinsic impurities that
have lower mobility. The presence of Fe${}_{\rm Li}^{+}$ in LiFePO4 has been
reported and the defect is believed to reduce the electrochemical performance
of the material by blocking the lithium channels.Yang et al. (2002); Maier and
Amin (2008); Chen et al. (2008); Chung et al. (2008); Axmann et al. (2009);
Chung et al. (2010) Indeed, our results also show that Fe${}_{\rm Li}^{+}$
occurs with a high concentration under various conditions (cf. Table 1) and
has low mobility. Whether Fe${}_{\rm Li}^{+}$ is stable in form of the neutral
complex Fe${}_{\rm{Li}}^{+}$-$V_{\rm{Li}}^{-}$ or
Fe${}_{\rm{Li}}^{+}$-Li${}_{\rm{Fe}}^{-}$ as suggested in several experimental
works,Maier and Amin (2008); Chung et al. (2008); Axmann et al. (2009); Chung
et al. (2010) however, depends on the specific conditions under which the
samples are produced.

What is most fascinating about our results is that one can suppress
Fe${}_{\rm{Li}}^{+}$ by adjusting suitable experimental conditions during
synthesis. In fact, $p^{+}$-rich and Fe${}_{\rm Li}^{+}$-free LiFePO4 samples,
which are believed to be desirable for high intrinsic electronic and ionic
conductivities, can be produced if one maintains high $\mu_{{\rm O}_{2}}$
values (cf. Fig. 7). Of course, $\mu_{{\rm O}_{2}}$ should not be so high that
LiFePO4 becomes unstable toward forming secondary phases. Although LiFePO4
cannot be doped $p$-type or $n$-type as discussed earlier, the incorporation
of suitable electrically active impurities in the material can enhance the
electronic (ionic) conductivity via increasing the concentration of $p^{+}$
($V_{\rm{Li}}^{-}$). These impurities, if present in the samples with a
concentration higher than that of the charged native defects, can shift the
Fermi level,Peles and Van de Walle (2007); Hoang and Van de Walle (2009);
Wilson-Short et al. (2009) and hence lower the formation energy of either
$p^{+}$ or $V_{\rm{Li}}^{-}$. A decrease in the formation energy of $p^{+}$,
however, may result in an increase in the formation energy of
$V_{\rm{Li}}^{-}$ and vice versa. For example, impurities with positive
effective charges (i.e., donor-like doping) may shift the Fermi level to the
right (cf. Fig. 7), resulting in an increased (decreased) formation energy of
$p^{+}$ ($V_{\rm{Li}}^{-}$). Impurities with negative effective charges (i.e.,
acceptor-like doping), on the other hand, may produce the opposite effects,
namely, decreasing (increasing) the formation energy of $p^{+}$
($V_{\rm{Li}}^{-}$). An enhancement in both electronic and ionic
conductivities would, therefore, require a delicate combination of defect-
controlled synthesis, doping with suitable electrically active impurities, and
post-synthesis treatments. An example of the latter would be thermal treatment
which, in fact, has been reported to cause lithium loss in LiFePO4 and lower
the activation energy of the electrical conductivity.Amin and Maier (2008)

##  VII Conclusion

In summary, we have carried out comprehensive first-principles studies of
native point defects and defect complexes in LiFePO4. We find that lithium
vacancies, lithium antisites, iron antisites, and iron vacancies each have one
stable charge state only and occur as, respectively, $V_{\rm{Li}}^{-}$,
Li${}_{\rm{Fe}}^{-}$, Fe${}_{\rm{Li}}^{+}$, and $V_{\rm{Fe}}^{2-}$. The
removal/addition of electrons from/to these stable native point defects does
not result in a transition to other charge states of the same defects, but
instead generates small hole/electron polarons. The fact that small polarons
can be stabilized, both in the presence and in the absence of other native
defects, is necessarily related to the electronic structure of LiFePO4. Our
analysis thus indicates that native defects in the material cannot act as
sources of band-like electron and hole conductivities, and the electronic
conduction, in fact, proceeds via hopping of small hole polarons ($p^{+}$).
The ionic conduction, on the other hand, occurs via diffusion of lithium
vacancies.

Among all possible native defects, $p^{+}$, $V_{\rm{Li}}^{-}$,
Li${}_{\rm{Fe}}^{-}$, and Fe${}_{\rm{Li}}^{+}$ are found to have low formation
energies and are hence expected to occur in LiFePO4 with high concentrations.
The dominant point defects in the samples are likely to exist in forms of a
neutral defect complex such as $V_{\rm{Li}}^{0}$, Li${}_{\rm{Fe}}^{0}$,
Fe${}_{\rm{Li}}^{+}$-$V_{\rm{Li}}^{-}$, or
Fe${}_{\rm{Li}}^{+}$-Li${}_{\rm{Fe}}^{-}$. The energy landscape of these
defects is, however, sensitive to the choice of atomic chemical potentials
which represent experimental conditions during synthesis. This explains the
conflicting experimental data on defect formation in LiFePO4. Our results also
raise the necessity of having prior knowledge of the native defects in LiFePO4
samples before any useful interpretations of the measured transport data can
be made. We suggest that one can suppress or enhance certain native defects in
LiFePO4 via tuning the experimental conditions during synthesis, and thereby
produce samples with tailored defect concentrations for optimal performance.
The electrical conductivity may be enhanced through increase of hole polaron
and lithium vacancy concentrations via a combination of defect-controlled
synthesis, incorporation of suitable electrically active impurities that can
shift the Fermi level, and post-synthesis treatments.

###### Acknowledgements.

The authors acknowledge helpful discussions with S. C. Erwin and J. Allen, and
the use of computing facilities at the DoD HPC Centers. K.H. was supported by
Naval Research Laboratory through Grant No. NRL-N00173-08-G001, and M.J. by
the Office of Naval Research.

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