Journal ofMaterials Chemistry A

PAPER

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Understanding th

Center for Computationally Assisted Scien

University, Fargo, ND 58108, USA. E-mail: k

Cite this: J. Mater. Chem. A, 2014, 2,18271

Received 9th August 2014Accepted 5th September 2014

DOI: 10.1039/c4ta04116j

www.rsc.org/MaterialsA

This journal is © The Royal Society of C

e electronic and ionic conductionand lithium over-stoichiometry in LiMn2O4 spinel

Khang Hoang*

We report a first-principles study of defect thermodynamics and transport in spinel-type lithiummanganese

oxide LiMn2O4, an important lithium-ion battery electrodematerial, using density-functional theory and the

Heyd–Scuseria–Ernzerhof screened hybrid functional. We find that intrinsic point defects in LiMn2O4 have

low formation energies and hence can occur with high concentrations. The electronic conduction

proceeds via hopping of small polarons and the ionic conduction occurs via lithium vacancy and/or

interstitialcy migration mechanisms. The total conductivity is dominated by the electronic contribution.

LiMn2O4 is found to be prone to lithium over-stoichiometry, i.e., lithium excess at the manganese sites,

and Mn3+/Mn4+ disorder. Other defects such as manganese antisites and vacancies and lithium

interstitials may also occur in LiMn2O4 samples. In light of our results, we discuss possible implications of

the defects on the electrochemical properties and provide explanations for the experimental

observations and guidelines for defect-controlled synthesis and defect characterization.

1 Introduction

Spinel-type LiMn2O4, a mixed-valent compound containingboth Mn3+ andMn4+ ions, has been considered as an alternativeto layered LiCoO2 for lithium-ion battery electrodes as manga-nese is inexpensive and environmentally benign compared tocobalt in the layered oxide.1,2 The material crystallizes in thecubic crystal structure of space group Fd�3m at room tempera-ture but transforms into an orthorhombic or tetragonal struc-ture at lower temperatures. In the cubic phase, the Li+ ions stayat the tetrahedral 8a sites of the cubic close-packed oxygenarray, whereas the Mn3+ and Mn4+ ions randomly occupy theoctahedral 16d sites. In the orthorhombic or tetragonal phase,the Mn3+/Mn4+ arrangement is believed to be charge-ordered.3–9

However, truly stoichiometric LiMn2O4 is hard to obtain inexperiments and intrinsic electronic and ionic defects appear tooccur at multiple lattice sites.9–15 In fact, LiMn2O4 samples areoen made lithium over-stoichiometric (i.e., Li-excess), inten-tionally or unintentionally.13–22 The total bulk conductivity ofLiMn2O4 has also been reported and thought to be predomi-nantly from hopping of polarons.22–24 A deeper understanding ofthese properties and observations clearly requires a detailedunderstanding of the defect thermodynamics and transport.Such an understanding is currently lacking.

Computational studies of LiMn2O4 have focused mainly onthe bulk properties and lithium and small polaron migra-tion,25–30 except for some studies of defect energetics byAmmundsen et al.30 using interatomic potential simulations

ce and Technology, North Dakota State

hang.hoang@ndsu.edu

hemistry 2014

and Koyama et al.31 using density-functional theory (DFT)within the local density approximation (LDA).32 First-principlescalculations based on DFT have been proven to be a powerfultool for addressing electronic and atomistic processes in solids.A comprehensive and systematic DFT study of intrinsic pointdefects in a battery electrode material, for example, can providea detailed picture of the defect formation and migration andinvaluable insights into the electrochemical performance.33,34

For LiMn2O4, such a study is quite challenging, partly becausestandard DFT calculations using LDA or the generalized-gradient approximation (GGA)35 fail to produce the correctphysics of even the host compound. LDA/GGA calculationscarried out by Mishra and Ceder,36 e.g., showed that LiMn2O4 isa metal with a Mn oxidation state of +3.5, which is in contrast towhat is known about LiMn2O4 as a material with a nite bandgap and mixed Mn3+ and Mn4+ ions. The GGA + U method,37,38

an extension of GGA, can give a reasonable electronic structure.However, since one oen has to assume that the transitionmetal has the same Hubbard U value in different chemicalenvironments, the transferability of GGA + U results across thecompounds is low, making defect calculations becomeinaccurate.

In this article, we present for the rst time a comprehensivestudy of electronic and ionic defects in LiMn2O4 using a hybridHartree–Fock/DFT method. In particular, we used the Heyd–Scuseria–Ernzerhof (HSE06)39,40 screened hybrid functionalwhere all orbitals are treated on the same footing. The atomicand electronic structure and phase stability of the hostcompound and the structure and energetics of all possibleintrinsic point defects were investigated; the migration ofselected defects was also explored. We nd that defects in

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LiMn2O4 have low calculated formation energies and hence canoccur with high concentrations, and lithium antisites are thedominant ionic defect. On the basis of our results, we discussthe Mn3+/Mn4+ disorder, electronic and ionic conduction, deli-thiation and lithiation mechanisms, lithium over-stoichiom-etry, and possible implications on the electrochemicalproperties. Ultimately, our work provides explanations for theexperimental observations, guidelines for defect characteriza-tion and defect-controlled synthesis, and insights for rationaldesign of LiMn2O4-based electrode materials with improvedelectrochemical performance.

2 Methodology2.1 Computational details

Our calculations were based on DFT, using the HSE06 hybridfunctional,39,40 the projector augmented wave method,41,42 and aplane-wave basis set, as implemented in the Vienna Ab InitioSimulation Package (VASP).43–45 Point defects were treatedwithin the supercell approach, in which a defect is included in anite volume of the host material and this structure is period-ically repeated. For bulk and defect calculations, wemainly usedsupercells of LiMn2O4 containing 56 atoms per cell; integrationsover the Brillouin zone were carried out using a 2 � 2 � 2Monkhorst–Pack k-point mesh.46 A denser, G-centered 4 � 4 � 4k-point mesh was used in calculations to produce the electronicdensity of states. The plane-wave basis-set cutoff was set to500 eV. Convergence with respect to self-consistent iterationswas assumed when the total energy difference between cycleswas less than 10�4 eV and the residual forces were less than0.01 eV A�1. In the defect calculations, which were performedwith spin polarization and the ferromagnetic spin congura-tion, the lattice parameters were xed to the calculated bulkvalues but all the internal coordinates were fully relaxed.

2.2 Defect formation energies

The key quantities that determine the properties of a defect arethe migration barrier and formation energy. In our calculations,the former is calculated by using climbing-image nudgedelastic-band (NEB) method;47 the latter is computed using thetotal energies from DFT calculations. Following the approachdescribed in ref. 34 and references therein, the formationenergy of a defect X in charge state q is dened as

EfðXqÞ ¼ Etot

�Xq�� EtotðbulkÞ �

Xi

nimi þ qðEv þ meÞ þ Dq;

(1)

where Etot(Xq) and Etot(bulk) are, respectively, the total energies

of a supercell containing the defect X and of a supercell of theperfect bulk material; mi is the atomic chemical potential ofspecies i (and is referenced to bulk metals or O2 molecules at0 K), and ni indicates the number of atoms of species i that havebeen added (ni > 0) or removed (ni < 0) to form the defect. me isthe electronic chemical potential, referenced to the valence-band maximum in the bulk (Ev). D

q is the correction term toalign the electrostatic potentials of the bulk and defect

18272 | J. Mater. Chem. A, 2014, 2, 18271–18280

supercells and to account for nite-cell-size effects on the totalenergies of charged defects.48 To correct for the nite-sizeeffects, we adopted the approach of Freysoldt et al.,49,50 in whichDq was determined using a calculated static dielectric constantof 11.02 for LiMn2O4. The dielectric constant was computedfollowing the procedure described in ref. 34 according to whichthe electronic contribution (4.78) was obtained in HSE06calculations whereas the ionic contribution (6.24) was obtainedin GGA + U with U¼ 4.84 eV for Mn, taken as an average value ofMn3+ (4.64 eV) and Mn4+ (5.04 eV).51

In eqn (1), the atomic chemical potentials mi can beemployed to describe experimental conditions and are subjectto various thermodynamic constraints.34,48 The stability ofLiMn2O4, for example, requires

mLi + 2mMn + 4mO ¼ DHf (LiMn2O4), (2)

where DHf is the formation enthalpy. There are otherconstraints imposed by competing Li–Mn–O phases. By takinginto account all these thermodynamic constraints, one candetermine the range of Li, Mn, and O chemical potential valuesin which the host compound LiMn2O4 is thermodynamicallystable. The oxygen chemical potential, mO, can also be related tothe temperatures and pressures through standard thermody-namic expressions for O2 gas.33 Finally, the electronic chemicalpotential me, i.e., the Fermi level, is not a free parameter butsubject to the charge neutrality condition that involves allpossible intrinsic defects and any impurities in the material.34,48

The concentration of a defect at temperature T is related toits formation energy through the expression48

c ¼ NsitesNconfig exp

�Ef

kBT

!; (3)

where Nsites is the number of high-symmetry sites in the latticeper unit volume on which the defect can be incorporated, Ncong

is the number of equivalent congurations (per site), and kB isthe Boltzmann constant. Strictly speaking, this expression isonly valid in thermodynamic equilibrium. Materials synthesis,on the other hand, may not be an equilibrium process.However, even in that case the use of the equilibrium expressioncan still be justied if the synthesis conditions are close enoughto equilibrium. Besides, as discussed in ref. 48, the use of eqn(3) does not require that all aspects of the process have to be inequilibrium. What is important is that the relevant defects aremobile enough to allow for equilibration at the temperatures ofinterest. It emerges from eqn (3) that defects with low formationenergies will easily form and occur in high concentrations.

3 Results3.1 Bulk properties

We began with a cubic supercell of LiMn2O4 (space group Fd�3m,experimental lattice parameter a ¼ 8.24 A),52 consisting of 8 Liatoms at the 8a sites, 16 Mn atoms at the 16d sites, and 32 Oatoms at the 32e sites; the interstitial 16c sites are le empty.Aer structural optimization, this cubic cell transforms into atetragonally distorted cell with a¼ c¼ 8.34 A and b¼ 8.11 A; see

This journal is © The Royal Society of Chemistry 2014

Fig. 1 Supercell model for spinel-type LiMn2O4 after structural opti-mization. Large gray spheres are Li, medium blue (yellow) spheres areMn3+ (Mn4+), and small red spheres are O. Fig. 2 Electronic density of states of LiMn2O4 in antiferromagnetic

(blue curves) and ferromagnetic (red curves) spin configurations. Thezero of energy is set to the highest occupied state.

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Fig. 1. The ordering of Mn3+/Mn4+ is visible in the relaxedstructure. If viewed along the [101] direction, the atomicarrangement follows an A–B–C–. pattern with layer A con-sisting of Mn3+ chains, B of Li+ chains and Mn3+/Mn4+-alter-nating chains, and C of Mn4+ chains. The Mn ions are stable inhigh-spin states with calculated magnetic moments of 3.78 mB

(Mn3+) and 3.04 mB (Mn4+). TheMnO6 unit has either six Mn4+–Obonds with similar bond lengths (1.85–1.93 A) or four shortMn3+–O bonds (1.92–1.98 A) and two long Mn3+–O bonds(2.20 A). The local distortions as seen in Fig. 1 are thus due toJahn–Teller effects associated with the Mn3+ ions.

Among several different Mn3+/Mn4+ arrangements we inves-tigated, the described model is found to have the lowest energy.It is lower in energy than the second-lowest energy Mn3+/Mn4+

arrangement by 0.06 eV per formula unit (f.u.), and the LiMn2O4

supercell where every Mn ion has an oxidation state of +3.5 by0.73 eV per f.u. Our results are thus consistent with experi-mental reports showing a transformation into a tetragonal ororthorhombic phase at low temperatures associated withcharge ordering.3–9 We use this structural model for furtherstudies of the bulk properties and for defect calculations (seebelow). Since the above mentioned global and local distortionsare relatively small, the atomic positions in this model will benominally referred to using the Wyckoff positions of the cubicstructure.

Fig. 2 shows the total electronic density of states of LiMn2O4.An analysis of the wavefunctions shows that the valence-bandmaximum (VBM) predominantly consists of the 3d states fromthe Mn3+ sites, whereas the conduction-band minimum (CBM)are predominantly the 3d states from the Mn4+ sites. The Li 2sstate is high up in the conduction band, indicating that Lidonates its electron to the lattice and becomes Li+. LiMn2O4

thus can be regarded nominally as an ordered arrangement ofLi+, Mn3+, Mn4+, and O2� units. The calculated band gaps are2.12 and 2.42 eV for the ferromagnetic (FM) and antiferro-magnetic (AFM) congurations, respectively. In addition to thespin congurations, we nd that the calculated band gap of

This journal is © The Royal Society of Chemistry 2014

LiMn2O4 also depends on the Mn3+/Mn4+ arrangement. Forexample, our calculations using a smaller, 14-atom cell ofLiMn2O4, which also relaxes to a tetragonal structure but with adifferent Mn3+/Mn4+ arrangement, give band gaps of 1.77 and1.92 eV for the FM and AFM congurations.

Experimentally, Raja et al.53 reported an optical band gap of1.43 eV from ultraviolet-visible spectroscopy for nanocrystallineLiMn2O4 powders with a nominal composition of Li0.88Mn2O4.Kushida and Kuriyama,54 on the other hand, observed twooptical absorption peaks associated with d–d transitions atabout 1.63 and 2.00 eV in LiMn2O4 thin lms on silica glass. Thediscrepancies in the experimental values suggest that the bandgap value is sensitive to the quality of the LiMn2O4 samples,which in turn depends on the synthesis conditions. This obvi-ously complicates the comparison between the calculated andmeasured bulk properties.

3.2 Chemical-potential diagram

Fig. 3 shows the atomic chemical-potential diagram forLiMn2O4, constructed by exploring all possible Li–Mn–O phasesavailable in the Materials Project database.55 The stabilityregion of LiMn2O4 in the (mLi, mMn) plane is dened by Mn2O3,Li2MnO3, and Li5Mn7O16. The calculated formation enthalpiesat 0 K of tetragonal LiMn2O4, orthorhombic Mn2O3, monoclinicLi2MnO3, and orthorhombic Li5Mn7O16 are, respectively,�13.89 eV, �10.09 eV, �12.30 eV, and �54.39 eV per f.u. Forcomparison, the experimental formation enthalpy of LiMn2O4

at 298 K is �14.31 eV per f.u.56 Points A, B, and C representthree-phase equilibria associated with LiMn2O4. Point A, forexample, is an equilibrium between LiMn2O4, Mn2O3, andLi5Mn7O16. The presence of these equilibria is consistent withthe fact that LiMn2O4 samples oen contain Mn2O3 and/orLi2MnO3 as impurity phases.17,57,58 Strobel et al.12 also reportedthat annealing LiMn2O4 under oxygen pressures in the range0.2–5 atm at 450 �C resulted inMn atoms being expelled in form

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Fig. 3 Chemical-potential diagram at 0 K for LiMn2O4. Only the O2

gas phase and the Li–Mn–O phases that define the stability region,here shown as a shaded triangle, are included.

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of Mn2O3. Li5Mn7O16, which can be rewritten as Li1+aMn2�aO4

(a ¼ 0.25), is also closely related to LiMn2O4. In fact, in theLi–Mn–O phase diagram, it is located on the tie-line betweenLiMn2O4 (a ¼ 0) where the average Mn oxidation state is +3.5and Li4Mn5O12 (a ¼ 0.33) where all Mn ions have the oxidationstate of +4.

3.3 Defect structure and energetics

Fig. 4 shows the calculated formation energies of low-energydefects in LiMn2O4, obtained at point B in the chemical-potential diagram. These defects include hole (h+) and electron(h�) polarons, lithium vacancies (VLi), interstitials (Lii), andantisites (LiMn), and manganese vacancies (VMn) and antisites

Fig. 4 Calculated formation energies of intrinsic point defects inLiMn2O4, plotted as a function of the Fermi level. The energies areobtained at point B in Fig. 3. In the absence of extrinsic chargedimpurities, the Fermi level is at me ¼ minte , where charge neutrality ismaintained.

18274 | J. Mater. Chem. A, 2014, 2, 18271–18280

(MnLi). The formation energies are plotted as a function of theFermi level me, with me varies from the VBM to CBM. Asmentioned earlier, the actual position of the Fermi level of thesystem is determined by the charge neutrality condition. In theabsence of electrically active impurities that can shi the Fermi-level position or when such impurities occur in much lowerconcentrations than charged intrinsic defects, the Fermi level isat minte , determined only by the intrinsic defects.33,34 With thechosen set of the atomic chemical potentials, minte is at 0.98 eV,exclusively dened by small hole polarons (h+) and negativelycharged lithium antisites

�Li�Mn

�. We nd that intrinsic point

defects in LiMn2O4 have very low formation energies and hencecan occur with high concentrations. Polarons and chargedlithium and manganese antisites have positive calculatedformation energies only near midgap. Positively chargedlithium interstitials

�Liþi�also have a negative formation energy

near the VBM. Before discussing the implications of theseresults, let us describe the defects in more detail.

Small polarons. A hole (or electron) polaron is a quasiparticleformed by the hole (electron) and its self-induced local latticedistortion. The creation of h+ involves removing an electronfrom the VBM which is predominantly Mn3+ 3d states. Thisresults in a Mn4+ ion at one of the Mn3+ sites, i.e., a localizedhole. The local lattice geometry near the newly formed Mn4+ ionis slightly distorted with the six neighboring O atoms movingtoward the Mn4+. The average Mn–O bond length at the Mn4+

site is 1.92 A and the Jahn–Teller distortion vanishes at this site.The formation of h�, on the other hand, corresponds to addingan electron to the CBM, which is predominantly Mn4+ 3d states,resulting in a Mn3+ ion at one of the Mn4+ sites, i.e., a localizedelectron. The local geometry near the newly formed Mn3+ ion isalso distorted, but with the neighboring O atoms slightlymoving away from Mn3+. At this Mn3+ site, there are four shortand two longMn–O bonds with the average bond lengths of 1.93and 2.17 A, respectively. Since the distortion is limitedmainly tothe neighboring O atoms of the resulted Mn3+ or Mn4+ ion,these polarons can be regarded as small polarons.59

We nd that the calculated formation energy of the polaronsis as low as 0.32 eV (h+) or 0.47 eV (h�), in which h+ is alwaysenergetically more favorable. It should be noted that these areadditionally formed polarons, i.e., they are considered as“defects” as compared to the perfect bulk material. The self-trapping energies of h+ and h� are 0.82 and 1.02 eV, respec-tively, dened as the difference between the formation energy ofthe free hole or electron and that of the hole or electronpolaron.34 With these high self-trapping energies, the polaronsare very stable in LiMn2O4. This is not surprising, given theconsideration that half of the Mn sites in the host compoundcan be regarded as being stable as hole polarons (Mn4+) in ahypothetical all-Mn3+ LiMn2O4 and the other half can beregarded as electron polarons (Mn3+) in a hypothetical all-Mn4+

LiMn2O4.Vacancies and interstitials. The formation of V�

Li involvesremoving a Li+ ion, which causes negligible disturbance in thelattice. V0

Li is, on the other hand, created by removing a Li atom,which is in fact a Li+ ion and an electron from a neighboring Mnatom. This results in a void at the site of the removed Li+ and a

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Fig. 5 Structure of Li0Mn, the intrinsic ionic defect with the lowestformation energy in LiMn2O4. This neutral defect is a complex of onenegatively charged antisite Li2�Mn (large dark gray sphere) and two smallhole polarons h+ (medium light yellow spheres).

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Mn4+ at a neighboring Mn site (originally a Mn3+ ion). V0Li is thus

not the neutral charge state of a lithium vacancy but a complexof V�

Li and h+. This defect complex has a binding energy of 0.60eV with respect to V�

Li and h+. Defects such as V�Li , as well as h

+

and h�, are regarded as elementary defects; the structure andenergetics of other defects, e.g., V0

Li, can be interpreted in termsof these basic building blocks. We nd that the formationenergy of V0

Li is always lower than that of V�Li . For lithium

interstitials, Liþi is created by adding a Li+ ion. The defect isenergetically most favorable when combining with another Li+

ion from an 8a site to form a Li–Li dumbbell centered at the 8asite. This is in contrast to what has been commonly assumedthat lithium interstitials are most stable at the 16c sites. Theenergy in the dumbbell conguration is lower than that atthe 16c site by at least 0.15 eV. Finally, Li0i , created by adding aLi atom, is a complex of Liþi and h� with a binding energy of0.52 eV.

Among the manganese vacancies, V3�Mn, i.e., the removal of a

Mn3+ ion, is the elementary defect. Other defects such as V2�Mn,

V�Mn, or V

0Mn are complexes of V3�

Mn and, respectively, one, two, orthree h+. V0

Mn is found to be the lowest-energy manganesevacancy conguration, suggesting that it is more favorable toform a manganese vacancy when it is surrounded by Mn4+ ions.The removal of manganese from the Mn3+ site costs less energythan from the Mn4+ site; the formation energy difference isabout 0.20 eV or higher. Regarding the oxygen vacancies, theremoval of an O2� ion does not lead to a void formed by theremoved ion, oen denoted as V2þ

O , but a complex of V2þO and a

hole–electron polaron pair (h+–h�), hereaer denoted as V2þO* .

Clearly, V2þO is not stable as a single point defect, and its

formation is associated with some Mn3+/Mn4+ disorder. Otheroxygen vacancies such as Vþ

O or V0O are complexes of V2þ

O and oneor two h�. We nd that the oxygen vacancies have much higherformation energies than other intrinsic defects; their lowestvalue is 1.49 eV for V0

O at point C in the chemical-potentialdiagram. We also investigated manganese and oxygen intersti-tials and found that they all have very high formation energies(about 3 eV or higher), suggesting that these interstitials are notlikely to form in the material.

Antisite defects. Lithium antisites LiMn are created byreplacing Mn at a Mn site with Li. Li2�Mn, i.e., Li

+ substitutingMn3+, is an elementary defect. Other antisites such as Li�Mn orLi0Mnare complexes of Li2�Mn and, respectively, one or two h+.Among all possible ionic defects in LiMn2O4, Li0Mn has thelowest formation energy, as low as 0.11 eV. Fig. 5 shows thestructure of Li0Mn where Li2�Mn is clearly seen surrounded by sixMn4+ ions (i.e., h+) and six Li+ ions. It should be noted that thetwo h+ in the Li0Mn complex are created together with the Li2�Mn, inaddition to those Mn4+ ions already present in bulk LiMn2O4.We nd that the energy cost for ion substitution at the Mn3+ siteis lower than at the Mn4+ site; the energy difference is 0.26 eVor higher. Manganese antisites MnLi are created in a similarway by replacing Li at a Li (tetrahedral) site with Mn. Mnþ

Li is anelementary defect, in which the Mn ion is stable as high-spin Mn2+ with a calculated magnetic moment of 4.45 mB.Other manganese antisites such as Mn0

Li or Mn2þLi are

complexes of MnþLi and h� or h+. For comparison, in layered

This journal is © The Royal Society of Chemistry 2014

LiMO2 (M ¼ Ni, Co) the transition metal is also found to bestable as high-spin M2+ at the Li (octahedral) site.34

Defect complexes. In addition to the above defects, weexplicitly investigated hole–electron polaron pairs (h+–h�),antisite defect pairs (MnLi–LiMn), and lithium Frenkel pair(Liþi � V�

Li ). The hole–electron polaron pair is formed byswitching the positions of a Mn3+ and its neighboring Mn4+ ion.Aer structural relaxations, the pair distance is 2.93 A. Thisdefect pair has a formation energy of 0.37 eV and a binding of0.54 eV. The antisite pair is created by switching the positions ofa Li atom and its neighboring Mn atom. This ultimately resultsin a Mnþ

Li � Li2�Mn � hþ complex, in which the distance from themanganese antisite to the lithium antisite is 2.92 A and thatfrom the lithium antisite to the hole polaron is 3.47 A. Thiscomplex has a formation energy of 0.63 eV and a binding of 1.76eV. Finally, the lithium Frenkel pair is created by moving a Li+

ion away from an 8a site to form a Li–Li dumbbell with anotherLi+ ion. This results in a V�

Li at the 8a site and a Liþi . Aerrelaxations, the distance between the vacancy and the center ofthe dumbbell is about 4.0 A (the pair is unstable towardrecombination at shorter distances). This pair has a formationenergy of 1.85 eV and a binding energy of 0.30 eV. It should benoted that the formation energies of these three defectcomplexes are independent of the chemical potentials.

3.4 Defect migration

Fig. 6 shows the migration barriers (Em) for the hole and elec-tron polarons and lithium vacancies and interstitials inLiMn2O4. All the migration barrier calculations were carried outwith the G point only. The migration of a polaron between twopositions q1 and q2 can be described by the transfer of thelattice distortion over a one-dimensional Born–Oppenheimersurface.60 We estimate the energy barrier by computing theenergies of a set of cell congurations linearly interpolatedbetween q1 and q2 and identify the energy maximum. Themigration barrier of h+ and h� is found to be 0.46 eV. For

J. Mater. Chem. A, 2014, 2, 18271–18280 | 18275

Fig. 6 Migration barriers of the small hole (h+) and electron (h�)polarons and lithium vacancies ðV�

Li Þ and interstitials ðLiþi Þ in long-range charge-ordered LiMn2O4.

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comparison, Ouyang et al.27 reported a migration barrier of0.35 eV for polarons in LiMn2O4, obtained in GGA + Ucalculations with U ¼ 4.5 eV.

For lithium vacancies V�Li , we nd two distinct migration

paths with barriers of 0.19 eV (path V1) and 0.47 eV (path V2),calculated using the NEB method.47 Both paths involve movinga Li+ ion from an 8a site to the vacancy (an empty 8a site)through an interstitial 16c site. Here, moving a Li+ ion in onedirection is equivalent to V�

Li migrating in the opposite direc-tion. The migration bottleneck is a Mn ring at the 16c site,consisting of six Mn ions in the plane perpendicular to themigration path. In path V1, the Mn ring has four Mn4+ ions andtwo Mn3+ ions, whereas in path V2 it has two Mn4+ ions and fourMn3+ ions. We have also considered situations in which Li+ ionsmigrate through a Mn ring that consists of three Mn4+ ions andthree Mn3+ ions and nd migration barriers of 0.47–0.57 eV. Anexample of such situations is when the h+ component of the V0

Li

complex is kept xed while the V�Li component of the complex is

migrating.Lithium interstitials Liþi migrate through an interstitialcy

mechanism involving concerted motion of three lithium ions:two ions of the Li–Li dumbbell and one ion that is next to thedumbbell. We nd barriers of 0.12 eV for the migration path(hereaer called path I1) that goes through Mn rings all con-sisting of four Mn4+ ions and two Mn3+ ions, and 0.49 eV for thepath (path I2) that goes through at least one Mn ring thatconsists of two Mn4+ ions and four Mn3+ ions.

For comparison, Xu and Meng29 obtained from GGA + Ucalculations with U ¼ 4.84 eV lithium migration barriers of�0.2–0.4 eV associated with Mn4+-rich rings,�0.6 eV with Mn3+-rich rings, and �0.8 eV with Mn rings that have equal numbersof Mn4+ and Mn3+ ions. The migrating species in their calcu-lations could be V0

Li, instead of V�Li like in our calculations.

However, it is not clear from their work how the two compo-nents of the V0

Li complex migrate relative to each other. We notethat, in our calculations, a lower bound on themigration barrier

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of a defect complex, e.g., V0Li or Li

0i , can be estimated by taking

the higher of the migration barriers of its constituents.61

We nd that the tetragonal distortion has minor effects onthe migration barriers. For example, our calculations usingcubic supercells that have the same volume and Mn3+/Mn4+

arrangement as the tetragonal supercells give V�Li barriers of

0.20 eV and 0.45 eV for paths V1 and V2, respectively, which arealmost identical to the values reported earlier. However, in thepresence of Mn3+/Mn4+ disorder and other lattice defects, e.g.,in lithium over-stoichiometric Li[Mn2�aLia]O4, the lithium ionsare not likely to encounter Mn rings that are all Mn4+-rich forthe whole diffusion length, and the overall migration barrierwill therefore be determined by the higher-barrier segment. As aresult, the hole and electron polarons and lithium vacanciesand interstitials, except the lithium at the octahedral 16d site,all have an estimated migration barrier of about 0.5 eV.

Finally, we investigated the migration of the Li+ ion that isassociated with the lithium antisite Li0Mn, cf. Fig. 5. The energybarrier for Li+ migration from the octahedral 16d site to one ofthe six neighboring Li tetrahedral 8a sites through a vacancymechanism is found to be 0.6–2.0 eV. With this higher migra-tion barrier, the ion is trapped at the 16d site and expected notbe deintercalated during charging.

4 Discussion

We list in Table 1 the calculated formation energies of relevantpoint defects in LiMn2O4 for three different sets of the atomicchemical potentials, corresponding to different sets of theexperimental conditions. The chemical potential of oxygen, mO,is �0.15 eV, �0.42 eV, and�0.69 eV, respectively, at points A, B,and C in the chemical-potential diagram, cf. Fig. 3. mO can becontrolled by controlling temperature and pressure and/oroxygen reducing agents. Lower mO values are usually associatedwith higher temperatures and/or lower oxygen partial pressuresand/or the presence of oxygen reducing agents. For each set ofthe atomic chemical potentials, the formation energy values areobtained at the respective Fermi-level position minte . We nd thatminte is at 0.98–1.11 eV, which is always away from both the VBMand CBM. Most of the defects have a calculated formationenergy of 1.0 eV or lower, at least under certain conditions. Theycan therefore occur in the material with high concentrations,e.g., during synthesis. These defects, except the mobile onessuch as the polarons and lithium vacancies, are expected to gettrapped when the material is cooled to room temperature. Wealso nd that the formation energy of the polaron pair h+–h� islow, only 0.37 eV, indicating that LiMn2O4 is prone toMn3+/Mn4+ disorder. MnLi–LiMn also has a low formationenergy, which suggests the presence of cation mixing (seefurther discussions in sections 4.3 and 4.4).

4.1 Electronic and ionic conduction

Strictly speaking, each ionic defect in LiMn2O4 has only onestable charge state, which is also called the elementary defect;oxygen vacancies do not even have any conguration that isstable as a single point defect, as mentioned earlier. Removing

This journal is © The Royal Society of Chemistry 2014

Table 1 Formation energies (Ef) and binding energies (Eb) of intrinsicdefects in LiMn2O4. The formation energies are obtained at points A, B,and C in the chemical-potential diagram

Defect

Ef (eV)

Constituents Eb (eV)A B C

h+ 0.32 0.32 0.45h� 0.59 0.59 0.47h+ � h� 0.37 0.37 0.37 h+ + h� 0.54V�

Li 0.92 1.28 1.33V0

Li 0.64 1.00 1.18 V�Li þ hþ 0.60

Liþi 1.23 0.87 0.82

Li0i 1.30 0.94 0.76 Liþi þ h� 0.52

Liþi � V�Li 1.85 1.85 1.85 Liþi þ V�

Li 0.30

Li2�Mn1.30 1.30 1.31

Li�Mn 0.32 0.32 0.46 Li2�Mn þ hþ 1.30

Li0Mn0.11 0.11 0.38 Li2�Mn þ 2hþ 1.82

MnþLi 0.77 0.77 0.62

Mn0Li 0.92 0.92 0.65 Mnþ

Li þ h� 0.44Mn2þ

Li 1.02 1.02 1.00 MnþLi þ hþ 0.07

MnLi–LiMn 0.63 0.63 0.63 MnþLi þ Li2�Mn þ hþ 1.76

V3�Mn 3.11 3.47 3.51

V2�Mn 1.80 2.16 2.34 V3�

Mn þ hþ 1.63V�

Mn 1.03 1.39 1.71 V3�Mn þ 2hþ 2.72

V0Mn 0.80 1.16 1.61 V3�

Mn þ 3hþ 3.27

V2þO*

2.42 2.15 2.15 V2þO þ hþ þ h�

VþO 2.15 1.88 1.75 V2þ

O þ h�

V0O 2.03 1.76 1.49 V2þ

O þ 2h�

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(adding) electrons from (to) these elementary defects alwaysresults in defect complexes consisting of the elementary defectsand small hole (electron) polarons. Besides, several positivelyand negatively charged defects have positive formation energiesonly near midgap, cf. Fig. 4, making them perfect charge-compensators. Any attempt to deliberately shi the Fermi levelof the system from minte to the VBM or CBM will result in thecharged defects having negative formation energies, i.e., theintrinsic defects will form spontaneously and counteract theeffects of shiing.33,34 Clearly, intrinsic point defects in LiMn2O4

cannot act as sources of band-like electrons and holes, and thematerial cannot be made n-type or p-type. The electronicconduction therefore proceeds via hopping of small hole andelectron polarons. Regarding the ionic conduction, lithium ionsare the current-carrying species, which migrate via vacancy and/or interstitialcy mechanisms.

The activation energies for electronic and ionic conductioncan be estimated from the formation energies and migrationbarriers of the current-carrying defects, Ea ¼ Ef + Em. As dis-cussed earlier, except for paths V1 and I1 which are unlikely tobe realized in Mn3+/Mn4+-disordered LiMn2O4, the barriers forthe polarons and lithium ions are basically similar, �0.5 eV.Therefore, the relative contribution of a defect or migrationmechanism to the total conductivity is determined exclusivelyby the defect's concentration. If the defect is predominantlyathermal, as it is the case for hole and electron polarons inLiMn2O4 and lithium vacancies in Li-decient or partially

This journal is © The Royal Society of Chemistry 2014

delithiated Li1�xMn2O4, the activation energy includes only themigration part, i.e., Ea¼ Em.34 The electronic activation energy isthus �0.5 eV, i.e., the barrier for polarons. In partially deli-thiated Li1�xMn2O4, the ionic activation energy is also �0.5 eV,i.e., the barrier for lithium vacancies. In stoichiometricLiMn2O4, however, lithium vacancies and/or interstitials have tobe thermally activated; the ionic activation energy includes boththe migration (�0.5 eV) and formation (cf. Table 1) parts, whichis estimated to be as low as �1.1 eV (1.3 eV) for the lithiumvacancy (interstitialcy) mechanism. Clearly, the total conduc-tivity is dominated by the electronic contribution.

Experimentally, Fang and Chung24 reported activation ener-gies of 0.43 eV and 0.38 eV for electronic conduction in LiMn2O4

below and above room temperature, respectively. Other authorsreported values of 0.40–0.44 eV.22,23 Regarding lithium diffu-sion, Verhoeven et al.62 obtained an activation energy of 0.5 �0.1 eV in the temperature range 345–400 K from 7Li NMRexperiments on Li[Mn1.96Li0.04]O4. Takai et al.63 reported acti-vation energies of 0.52 eV and 1.11 eV for lithium diffusion inLiMn2O4 below and above 600 �C, extracted from tracer diffu-sion coefficients measured by neutron radiography. The lower-temperature values are very close to our estimated migrationbarrier. The value 1.11 eV at high temperatures could indicatethat the system is in the intrinsic region where the activationenergy includes both the formation and migration parts.

4.2 Delithiation and lithiation

The structure of the lithium vacancy V0Li in a battery electrode

material oen provides direct information on the delithiationmechanism. In LiMn2O4, V0

Li indicates that for each Li atomremoved from LiMn2O4 electrodes during delithiation, thematerial is le with one negatively charged lithium vacancy V�

Li

and one hole polaron h+; i.e., the extraction of Li is associatedwith the oxidation of Mn3+ to Mn4+. The deintercalationvoltage64 associated with the extraction of the rst lithium, i.e.,the creation of V0

Li, is 4.29 V. The partially delithiated compo-sition can be written as Li1�xMn2O4 (here we ignore the pre-existing intrinsic defects which will be discussed later). Thelithium interstitial Li0i , on the other hand, provides informationon the lithiation mechanism. For each Li atom inserted intoLiMn2O4 electrodes during lithiation, the material receives onepositively lithium interstitial Liþi and one electron polaron h�;i.e., the Li insertion is associated with the reduction of Mn4+ toMn3+. The partially lithiated composition can be written asLi1+xMn2O4 (not to be confused with lithium over-stoichio-metric Li1+aMn2�aO4 where the Li excess replaces Mn at the 16dsites). Since there are no band-like carriers, h+ and h� are theelectronic charge carriers in the delithiation and lithiationprocesses. Also, it is important to note that polarons andlithium vacancies (interstitials) created from delithiation (lith-iation) are not thermal defects.

4.3 Lithium over-stoichiometry

Among all possible ionic defects in LiMn2O4, the lithiumantisite Li0Mn is dominant. The defect has a formation energy of0.11–0.38 eV, depending the specic set of the atomic

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chemical potentials. For comparison, the calculated formationenergy of the lithium antisite in layered LiCoO2 and LiNiO2 isin the range of 0.92–2.73 eV ðLi0CoÞ or 0.68–1.11 eV ðLi0NiÞ.34 Thelow energy of Li0Mn can partially be ascribed to the smalldifference in the ionic radii of Li+ (0.76 A) and high-spin Mn3+

(0.65 A); for reference, the Shannon ionic radii of six-foldcoordinated, low-spin Co3+ and Ni3+ are 0.55 A and 0.56 A,respectively.65 Given Li0Mn with that low formation energy, thesynthesis of LiMn2O4 under equilibrium or near-equilibriumconditions is expected to result in a lithium over-stoichio-metric compound with the composition Li1+aMn2�aO4 or,more explicitly, Li[Mn2�aLia]O4 or Li+[Mn1�3a

3+Mn1+2a4+Lia

+]O4

2�. In this composition, each negatively charged lithiumantisite Li2�Mn is charge-compensated by two hole polarons h+,and the average Mn oxidation state is higher than +3.5; i.e.,Mn4+ is slightly more favorable than Mn3+. Since the Li+ ion atthe octahedral 16d site gets trapped due to its lower mobility,it is unlikely to be deintercalated during charging. Besides, Li[Mn2�aLia]O4 has only (1 � 3a) Mn3+ ions for the oxidationreactions. As a result, there will be residual lithium in the fullydelithiated compound, both at the 16d and 8a sites, i.e.,Li3a

+[Mn2�a4+Lia

+]O42�. The theoretical capacity will therefore

decrease from 148 mA h g�1 to 148(1 � 3a) mA h g�1.Our results for Li0Mn thus explain why LiMn2O4 samples are

oen lithium over-stoichiometric. Martinez et al.13 found about90% of the lithium ions at the tetrahedral 8a sites and 10% atthe octahedral 16d sites, conrming that the Li excess replacesMn at the 16d sites. Xia and Yoshio21 seemed to indicate thatstoichiometric LiMn2O4 electrodes are unstable toward thelithium over-stoichiometric ones. We note that, as a conse-quence of the lithium over-stoichiometry and the likely randomdistribution of Li0Mn, the transformation from the Mn3+/Mn4+-disordered, cubic phase to a long-range ordered, tetragonal/orthorhombic phase may not be realized in practice, even at lowtemperatures. In fact, there have been reports of the absence ofthe long-range charge order associated with the Jahn–Tellereffect in lithium over-stoichiometric Li1+aMn2�aO4.15,19 Kama-zawa et al.,19 for example, observed only short-range charged-order in Li1.1Mn1.9O4. Regarding the electrochemical perfor-mance, Li1+aMn2�aO4 samples have been reported to show asignicantly enhanced cycling stability compared to stoichio-metric LiMn2O, although at the expense of capacity.20,21 Theenhancement has been attributed mainly to the suppression ofthe Jahn–Teller effect on deep discharge. However, the presenceof the residual lithium in the delithiated compound, i.e.,Li3a

+[Mn2�a4+Lia

+]O42�, can also help improve the cycling

stability, unlike in LiMn2O4 where the complete extraction oflithium results in unstable l-MnO2 electrodes.20

4.4 Other possible defects

The manganese antisite MnþLi can occur in LiMn2O4, given its

formation energy of only 0.62–0.77 eV. For comparison, theformation energy of CoþLi in LiCoO2 is 0.55–2.08 eV and that ofNiþLi in LiNiO2 is 0.53–0.96 eV.34 Mnþ

Li can be created togetherwith Li2�Mn and h+, i.e., in form of MnLi–LiMn with a formationenergy of 0.63 eV, or with h�, i.e., in form of Mn0

Li with a

18278 | J. Mater. Chem. A, 2014, 2, 18271–18280

formation energy of 0.65–0.92 eV, cf. Table 1. Manganese anti-sites have also been observed in experiments. Bjork et al.11

reported that 9% of the lithium ions at the tetrahedral 8a siteswere substituted by Mn2+ ions in high-temperature synthesis.This is consistent with our results showing that Mnþ

Li has thelowest formation energy at point C which corresponds to a lowmO value. If created in form of Mn0

Li, the co-generation of h� willresult in an increase in the amount of the Jahn–Teller distortedMn3+ ions. Besides, we speculate that manganese antisites mayalso act as nucleation sites for the formation of impurity phasesduring electrochemical cycling.

Next, with a formation energy of 0.76–1.23 eV, lithiuminterstitials can also occur in the material, e.g., when synthe-sized under conditions near point C in the chemical-potentialdiagram, cf. Fig. 3. Liþi can be created together with h�, i.e., inform of Li0i which results in the composition Li1+aMn2O4

(assuming no other defects). In this composition, the averageMn oxidation state is <3.5. This defect is unlikely to formthrough the lithium Frenkel pair mechanism, i.e., in combi-nation with V�

Li , because the Liþi � V�Li pair is either unstable or

high in energy. Experimentally, Berg et al.17 reported that, intheir Li1+aMn2�aO4 (a ¼ 0.14) samples, lithium ions occupyboth the 8a sites with 100% occupancy and the 16c sites with7.0% occupancy, and manganese ions occupy the 16d sites with93.0% occupancy. In light of our results for the lithium inter-stitials, it would be interesting to re-examine the samples andsee if some of the lithium is really stable at the 16c sites. Theresults of Berg et al.may also suggest the presence of Lii and VMn

in form of a neutral Lii–VMn complex. However, we nd that thiscomplex has a high formation energy (1.93–2.20 eV), indicatingthat it is not likely to occur under equilibrium or near-equilib-rium synthesis conditions.

Manganese vacancies can form when synthesized at lowertemperatures, e.g., at point A in Fig. 3 where the formationenergy of V0

Mn is just 0.80 eV, cf. Table 1. Gummow et al.20

reported the presence of vacancies on both the 8a and 16d sitesin Li1�aMn2�2aO4 (0 < a # 0.11) synthesized at temperaturesbetween 400 and 600 �C, although they also acknowledged thatsamples with a precise, predetermined composition were diffi-cult to prepare. The cycling stability of this compound wasfound to be inferior to that of Li1+aMn2�aO4.20 Finally, with amuch higher formation energy (1.49–2.03 eV), oxygen vacanciesare expected not to occur inside the material. This is consistentwith experiments where no oxygen vacancies have beenfound.11,14 We note that oxygen vacancies may still occur at thesurface or interface where the lattice environment is less con-strained than in the bulk. The formation of manganese vacan-cies, as well as lithium interstitials and manganese antisites, isalso expected to be energetically more favorable at the surface orinterface.

Apparently, manganese antisites and vacancies and lithiuminterstitials can lead to inferior cycling stability and henceshould be avoided. One can tune the synthesis conditions toreduce these defects in LiMn2O4 samples. From our resultssummarized in Table 1, the best compromise could be tosynthesize the material under the conditions near point B in thechemical-potential diagram where there is a three-phase

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equilibrium between LiMn2O4, Li2MnO3 and Li5Mn7O16, cf.Fig. 3. The formation energy of the manganese antisites underthese conditions is, however, still quite low (0.77 eV). Furtherreduction of the defects may thus require partially ion substi-tution that can signicantly change the chemical environmentand hence the defect landscape.

5 Conclusions

We have carried out a DFT study of the bulk properties anddefect thermodynamics and transport in spinel-type LiMn2O4,using the HSE06 screened hybrid density functional. We ndthat the tetragonal distortion of cubic LiMn2O4 during struc-tural optimizations at 0 K is associated with charge ordering.The compound is found to be thermodynamically stable and itsstability region in the Li–Mn–O phase diagram is dened by theMn2O3, Li2MnO3, and Li5Mn7O16 phases.

Intrinsic electronic and ionic defects in LiMn2O4 can formwith high concentrations. Several charged defects have positiveformation energies only in a region near midgap, making themperfect charge-compensators. The defects cannot act as sourcesof band-like carriers and the material cannot be doped n- orp-type. The electronic conduction proceeds via hopping of thesmall hole and electrons polarons and the ionic conductionoccurs via lithium vacancy and/or interstitialcy migrationmechanisms. The total bulk conductivity is found to bepredominantly from the electronic contribution. An analysis ofthe structure of lithium vacancies and interstitials shows thatlithium extraction (insertion) is associated with the oxidation(reduction) reaction at the Mn site.

Among the intrinsic ionic defects, lithium antisites are thedominant defect with a very low calculated formation energy. Thislow energy is ascribed to the small ionic radius difference betweenLi+ and high-spin Mn3+. The formation energy of the hole–elec-tron polaron pair is also very low. Our results thus indicate thatLiMn2O4 is prone to lithium over-stoichiometry and Mn3+/Mn4+

disorder. In the lithium over-stoichiometric compound, there isresidual lithium that is not deintercalated during charging andcan help improve the cycling stability. Other defects such asmanganese antisites and vacancies and lithium interstitials canalso occur, under certain experimental conditions, but with lowerconcentrations than the lithium antisites. An elimination of themanganese antisites may require signicant changes to thechemical environment, e.g., through ion substitution.

Acknowledgements

This work was supported by the U.S. Department of Energy(Grant no. DE-SC0001717) and the Center for ComputationallyAssisted Science and Technology (CCAST) at North Dakota StateUniversity.

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