Table Of ContentCation- and vacancy-ordering in Li CoO
x 2
C. Wolverton and Alex Zunger
National Renewable Energy Laboratory, Golden, CO 80401
(February 1, 2008)
8
9 Using a combination of first-principles total energies, a cluster expansion technique, and Monte
9 Carlosimulations,wehavestudiedtheLi/CoorderinginLiCoO2 andLi-vacancy/Coorderinginthe
1 2CoO2. Wefind: (i)Agroundstatesearchofthespaceofsubstitutionalcationconfigurationsyields
n the CuPt structure as the lowest-energy state in the octahedral system LiCoO2 (and 2CoO2), in
a agreementwiththeexperimentallyobservedphase. (ii)Finitetemperaturecalculationspredictthat
J the solid-state order-disorder transitions for LiCoO2 and 2CoO2 occur at temperatures ( 5100 K
∼
2 and 4400K,respectively)muchhigherthanmelting,thusmakingthesetransitionsexperimentally
1 inacc∼essible. (iii) TheenergyofthereactionEtot(σ,LiCoO2) Etot(σ,2CoO2) Etot(Li,bcc)gives
− −
the average battery voltage V of a LixCoO2/Li cell for the cathode in the structure σ. Searching
] the space of configurations σ for large average voltages, we find that σ=CuPt (a monolayer 111
i h i
c superlattice)hasahighvoltage(V=3.78V),butthatthiscouldbeincreasedbycationrandomization
s
- (V=3.99 V), partial disordering (V=3.86 V), or by forming a 2-layer Li2Co2O4 superlattice along
l 111 (V=4.90 V).
r h i
t
m
.
t
a
m
I. INTRODUCTION ertiesofLiCoO2(xLi=1),butalso(b)thevacancy/Coor-
d- dering (different sites for 2 and Co) in 2CoO2 (xLi=0).
n MuchliketheABC2 semiconductors(A,B =Al,Ga,In A third type of ordering in these materials, vacancy/Li
o and C = N,P,As, or Sb) exhibit cation ordering in ordering in Lix21−xCoO2 (0 ≤ xLi ≤ 1), is not treated
[c a tetrahedrally-coordinated network1, the LiMO2 ox- here.
ides (M=3d transition metal)2,3 form a similar series Our calculation proceeds in three steps: (1) Total en-
1 of structures based on the octahedrally-coordinatednet- ergy calculations: We calculate the T=0 total energy
v work with anions (O) on one fcc sublattice and cations of a set of (not necessarily stable) ordered structures
0
(Li and M) on the other (Fig. 1). Cation arrange- via the full potential, all-electron linearized augmented
11 ments in isovalent (III-III-V) or heterovalent (I-III-VI) plane wave method (LAPW)15,16 with all atomic posi-
1 semiconductor alloys have been observed1 in the disor- tions fully relaxed via quantum mechanical forces. We
0 dered,CuAu-type (CA), CuPt-type(CP),andchalcopy- then map those energies onto a (2) cluster expansion
8 rite(CH)structures(bottomrowofFig. 1),whilecation (CE).17,18,19,20,21,22ThisexpansionisageneralizedIsing-
9 arrangements in the oxides have been observed2,3 in the like expression for the energy of an arbitrary substitu-
t/ disordered, CP, CH, D4, and Y2 structures (top row of tional cation arrangement. Once the coefficients of the
a Fig. 1). Ab-initio total energy calculations1 have shown expansion are known, the Ising-like expression may be
m
that in the tetrahedrally-coordinated III-V semiconduc- easily evaluated for any cation configuration. Thus, one
- tor alloys, the CuPt structure is the least stable [due to can calculate (via first-principles) the total energy of a
d
n thefactthatitrepresentsastackingalongtheelastically few cation arrangements, but then effectively search the
o hard (111) direction], while the chalcopyrite structure is space of 2N configurations (where N is typically <∼104).
c moststable(itpossessesboththelowestelectrostaticand Specifically, the cluster expansionmay be used to search
v: strainenergies). Similarstudieshavebeenperformedfor the entire configurational space for stable ground state
i the octahedrally-coordinated networks of the spin alloy structures, where one can obtain low energy, but oth-
X Mn↑S-Mn↓S and the lead chalcogenides.4 In this paper, erwise unsuspected states (i.e., states which are not in-
r weexaminetheenergeticsandthermodynamicsofcation cluded in the set of calculated total energies). Having
a
orderingtendencies in the octahedralLiCoO2 oxide,and obtained such a general and computationally simple pa-
compare to the tetrahedral semiconductor case which is rameterization of the configuration energy, we subject it
wellstudied. TheLiCoO2compoundisusedasacathode to(3)Monte Carlo simulated annealing23 toextendfirst-
material in rechargeable Li batteries.5,6,7,8,9,10,11,12,13,14 principles calculations (at zero temperature) to finite-
WhenLiisde-intercalatedfromthecompound,itcreates temperatures, thus obtaining order-disorder transition
avacancy(denoted2)thatcanbepositionedindifferent temperatures and thermodynamic functions.
latticelocations. Hence,wewillexaminenotonly(a)the We find for LiCoO2 the following:
Li/Cocationordering(differentsitesforLiandCo)prop- (a)Agroundstatesearchofthespaceofsubstitutional
1
SuperlaCatio (ZincbleTetrahe (RocksOctahe
repFrIeGse.n1t.tCheatcioantioanrsr,anwgheimletticeetnhntseignretyetaratohmedsraalrnde)e(stedralhmeicaonniodnusc.toSrh)oawnndaorcetathhalt)eeddralrnaalm(eosxiodfe)thneetcwaotirokns.coTnhfiegbulraactkionans,dtwhehistteruacttoumres
itself (bold lines indicate superlattice planes), and the equivalent superlattice of cations. NDote: Some of the structures are
shown as only a portion of the complete unit cell. is
o
- r
-- d
e
cation configurations yields the CuPt structure as the ble,similartotheheterovarelenttetrahedralI-III-VI2 (e.g.,
groundstate in the octahedralLiCoO2 system, in agree- CuInSe2) systems24, butdis the opposite situation from
ment with the well-established experimentally observed the isovalent tetrahedral III-V systems such as GaInP2,
phase.5 We find that this re(1sult holds even if the CuPt in which bulk ordered comCpounds are unstable.
sthtreucCtEurepaisranmotetienrcilzuadtieodn.i[111]nTt,1) alohhee sCeutPotfecnaetriogniesstursuecdtutroefiits tah(ee)drTahleLirCeloaOti2vesyosrtdeemr(CP)iosfuPt-tyqsutrituectduiffraelreennterfgroiems itnhethteetorac--
the least stable bulk structunre in tetrahedral ABC2. hedral cases: E(CuPt) <pE(CH) < E (CA) in both
(b) Finite temperature cgalculations predict that the LiCoO2 and 2CoO2, comepared with E(CH) < E(CA)
solid-state order-disorder transition for LiCoO2 occurs <E (CuPt),universallyfoundinthelattice-mismatched
at temperatures (∼5100 K) much higher than melting, tetrahedral systems (Fig. 2).
thus making this transition-experimentally inaccessible. D
In contrast, order-disorder--transitions in isovalent tetra- 4
hedral ABC2 systems are <∼ 1000 K.1 The addition of II. METHODS OF CALCULATION
Li vacancies lowers this transition to ∼4400 K; but, this
transition temperature is still too high to be observed. We use
( C
Ttishhanutostt,httehhoeerbmsfienoridvteyedntademmisiopcraedlrleay[001]rteusd1,1) alortea(brcoleac,lkcsbuaullattt)iiopsnhosansdleyeomsftoLanbisCitlroizaOetd2e citshoenasscissloutcssitaoetfredaenxwpIiastnihnsgitoh-lneik(seCi(CA)teEexuAu-ty)opfrteeascnshinoidniqeauinle,l1wa7th,t1ii9cc,h2e0,(e2fa1cc,c2h,2icwnahttiihocihns
kinetically. ng case), and the pseudo-spinpevariable Si is given the value
(c) The energy of intercalation reaction +1(−1)ifanA(B)atomisassignedtositei. Withinthis
Etot(σ,LiCoO2) − Etot(σ,2(2CoO2) − Etot(Li,bcc) gives description, the energy of aChny configuration σ of cations
tthhee caavtehraogdee binatttheerystvroulcttau[20gree,2) aVσ,6ofthausLipxrCovoOid2in/gLiacmelelafnors can be written as:19 (CHalco
forpredictionofbatteryin1]telorcalationvoltagesfromfirst- ECE(σ)=)pyDfJfΠf(σ), (1)
principles energetics. Searchnging the space of configura- Xfrit
tionsσforlargeaveragevoltages,wefindthatσ=CuPt[a e
monolayerh111isuperlattice]hasahighvoltage(V=3.78 wheref isafigurecomprisedofseverallatticesites(pairs,
(
2
V), but that this could b[e i,2ncreased by cation random- triplets,etc.),Df isthenumberoffiguresperlatticesite,
ibzyatfioornm(iVng=a3.929-laVy)e,rpLair2tCi110]aol2) aloOdi4sosrudpeerrinlagtt(iVce=a3l.o8n6gVh)1,1o1ri JisfaisfutnhcetiIosningd-elifikneedintaesraY2actpiornodfuocrtthoveefirgtuhreefifg,uarnedfΠoff
n
(V=4.90 V). g the variables Si, averaged over all symmetry equivalent
(d) Ordered cation arrangements in LiCoO2 are sta- figures of lattice sites. This expression incorporates the
(
2
[,2
31) a 2 W
1]lo 2
n
g
(
2
,2
)
a
lo
n
g
TABLEI. Lattice averaged spin productsΠ (σ) (for a few figures f) of thecation configurations σ shown in Fig. 1.
f
Π (σ)
f
Interaction Figure D AC+BC CP D4 Y2 CH CA W2 V2 Z2 Random
f
f x=1/2
J0 Empty 1 1 1 1 1 1 1 1 1 1 1
J1 Point 0 0 0 0 0 0 0 0 0 0 0
J2 NN Pair 6 1 0 0 0 -1/3 -1/3 -1/6 1/2 1/3 0
K2 2NN Pair 3 1 -1 -1 -1/3 1/3 1 0 0 1/3 0
L2 3NN Pair 12 1 0 0 0 1/3 -1/3 0 0 -1/3 0
M2 4NN Pair 6 1 1 1 -1/3 -1/3 1 0 0 -1/3 0
N2 5NN Pair 12 1 0 0 0 -1/3 -1/3 1/6 -1/2 1/3 0
O2 6NN Pair 4 1 -1 -1 1 -1 1 0 0 -1 0
P2 7NN Pair 24 1 0 0 0 1/3 -1/3 0 0 -1/3 0
Q2 8NN Pair 3 1 1 1 1 1 1 -1 -1 1 0
R2 9NN Pair 6 1 0 0 0 -1/3 -1/3 -1/6 1/2 1/3 0
S2 10NN Pair 12 1 0 0 0 -1/3 -1/3 1/6 -1/2 1/3 0
J4 Tetrahedron 2 1 -1 1 -1 1 1 0 0 1/3 0
effects ofatomic relaxation(indeed,one doesnotrequire will move from each CoO unit to fill the hole in the LiO
thatthecationssitpreciselyattheideallatticepositions, unit, thus creating normal octet bonds. These “charge
butmerelythatthere isa one-to-onecorrespondencebe- compensated” end-point compounds (LiO)∗ and (CoO)∗
tweenlatticesitesandatomicpositions).25 Wedetermine willhavealowerenergythanthe nominalLiO andCoO.
{Jf} by fitting ECE(σ) of Nσ structures to LDA total Our calculations thus consider only charge compensated
energies ELDA(σ), given the matrices {Πf(σ)} for these structures. UsingtheprocedureofWeietal.24intreating
structures. TableIgivesthevaluesofthelattice-averaged heterovalent alloys, the conventional, high-energy “end-
spin products Π (σ) and degeneracies D for the struc- point”compoundsLiO+CoOarenotincludedinthe CE
f f
tures in Fig. 1. The values of Π (σ) sometimes take on becausetheyarenotchargecompensated. OurCEcould
f
interestingdegeneracies: Forexample,CAandCHdiffer be used to predict the energies of (LiO)∗ + (CoO)∗,
only by Π (σ) for figures beyond the nearest neighbor, and we will see that this energy is indeed lower than
f
andthe CuPtandD4 structureshave3,26 equivalentpair that of nominal LiO+CoO. We only include the eight
correlation functions for all pair separations. Also, all (LiO)n(CoO)n compounds shown in Fig. 1 in our fit.
odd-bodycorrelationfunctionsarezeroforanystructure These cation orderings correspond to both the observed
whichpossessesA→B(orSˆ →−Sˆ)symmetry,suchas structures in the LiMO2 series (CuPt, D4, Y2, and CH)
i i
all those shown in Fig. 1 This means that CuPt and D4 aswellasothercationarrangementsnotobservedinthis
possess not only equivalent pair correlations, but three- series (CuAu,W2,V2, and Z2). For substitutional order-
body correlations as well. Thus, in terms of Eq. (1), all ing problems, it is possible to enumerate all configura-
terms in the expansion which correspond to one-, two-, tions up to a given unit cell size.28 The set of cation
and three-body figures do not distinguish between CuPt configurations considered here includes all of the possi-
and D4, and thus the first cluster correlation which can ble equiatomic structures with unit cell size up to eight
break the degeneracy between these two structures is in atoms.
the four-body terms. In spite of the similarities between The set of twelve figures f retained in the expansion
the CuPt and D4 structures, the former is a superlat- is the “empty” figure, the first ten neighbor pairs, and
tice alongthe [111]directionandhence isrhombohedral, the nearest neighbor tetrahedron. In fitting the LDA
while the latter is not a superlattice and is cubic (see total energies to the cluster expansion, we include a La-
Fig. 1). Thus, the CuPt structure has one extra struc- grange multiplier with the constraint that the pair in-
turaldegreeoffreedom(namely a c/aratio)thatthe D4 teractions should be as smooth as possible in recipro-
structure does not have.27 cal space. This technique (more fully explained in Ref.
We use N = 8 configurations in the fitting proce- 29) allows one to retain more figures in the expansion
σ
dure. These are shown in Fig. 1. The choice of end- than total energies, and also requires the pair interac-
point configurationsrequires some discussion. The nom- tions to be as convergent as possible in real space. Al-
inal end point configurations, LiO and CoO in the NaCl though more sophisticated versions of the cluster expan-
structure, do not obey the octet rule, as LiO has 7 va- sion approach29 are available when one requires extreme
lence electrons/formulaunit, while CoOhas (in addition accuracyandhasaccessto alargedatabaseofstructural
to its filled t2g shell) 9 valence electrons/formula. As energies, we use the simple real-space expansion of Eq.
a result, these nominal structures have a very high en- (1) with the Lagrange multiplier. The use of this simple
ergy. In the 1:1 structures (LiO) (CoO) , an electron formispredicatedontheassumptionthatifwespecialize
n n
3
TABLE II. Predicted structural information of observed LiCoO2 phases. Where available, experimental data and other
calculated resultsareshown. CuPt-ABCandCuPt-AAArefertoCuPtconfigurationsofcations(alternately stackedLi/Co
or2/Co close-packedlayers)inABC... orAAA... typestackings,respectively. For2CoO2,CuPt-ABCandCuPt-AAAare
isostructuralwith CdCl2 andCdI2,respectively. BulkmoduliforLiCoO2 (CuPt)and2CoO2 (CuPt)werecalculated (present
work) to be2.4 and 2.8 Mbar, respectively.
Compound Method a (˚A) c(˚A) Li-O (˚A) Co-O (˚A) Veq (˚A3)
LiCoO2 (CuPt - ABC) Expt. a 2.82 14.04 2.07 1.94 32.23
FLAPW (present work) 2.81 13.60 2.08 1.90 31.2
Pseudopot. b 2.93 13.2 2.10 1.96 32.71
2CoO2 (CuPt - ABC) FLAPW (present work) 2.78 12.13 1.85 26.9
Pseudopot. b 2.88 12.26 1.90 29.36
LiCoO2 (CuPt - AAA) FLAPW (present work) 2.79 4.74 2.11 1.90 32.0
2CoO2 (CuPt - AAA) Expt. c 2.822 4.29 1.91 29.6
FLAPW (present work) 2.80 4.01 1.85 27.1
LiCoO2 (D4) Expt. d 8.002 2.06 1.95 32.0
FLAPW (present work) 7.90 2.05 1.91 30.7
2CoO2 (D4) FLAPW (present work) 3.85 1.85 28.5
aRef. 13
bRef. 36
cRef. 11
dRef. 40
tofixedcomposition(e.g.,eitherxLi=1orxLi=0)theex- hydrostatic deformation. The resulting CE will reveal
pansionconvergesquicklywith a smallnumber ofterms. Li/Co ordering tendencies at xLi=1.
For our CE, the error in the input energies used in the (b) Formation enthalpies for different 2/Co arrange-
fit is a negligible amount, < 1 meV/formula unit. To ments σ on the fcc lattice:
obtain some idea of the errors involved in the CE pre-
dictions, we have removed some structures and figures ∆Hf(σ,2CoO2)=Etot(σ,2CoO2)+Etot(Li,bcc)
from the fitting process, and examined the resulting er- −Etot(LiO,B1)−Etot(CoO,B1) (3)
rors: Removing the CuPt (CH) structure and the four-
body tetrahedronfigure fromthe fit produces an11 (49) where Etot(Li,bcc) is the total energy of Li in the bcc
meV/formulauniterrorintheenergyofCuPt(CH),neg- structure with lattice constant obtained from total en-
ligiblechangesintheotherfittedenergies,andanenergy ergy minimization. The resulting CE will reveal 2/Co
oftherandomcationarrangementwhichchangesbyonly ordering tendencies at xLi=0.
1 (7) meV/formula unit. The magnitude of these errors (c)TheLibatteryintercalationreactionenergyfordif-
is quite small in terms of the energetic scales of cation ferentLi/Co(and2/Co)arrangementsonthefcclattice:
ordering(∼1000-2000meV/formulaunit)andLiinterca-
lation (∼4000 meV/formula unit). ∆Hreact(σ)=Etot(σ,LiCoO2)−Etot(σ,2CoO2)
The expression of Eq. (1) can be applied to differ- −Etot(Li,bcc) (4)
ent ordering problems, with a separate expansion con-
structedforeachsituation. Here,weconstructthreesep- ∆Hreact is the energy gained upon complete de-
arate cluster expansions to describe three different types intercalationofLifromLiCoO2,relativetoLimetal,and
of structural energetics: is simply the difference between Eq. (2) and Eq. (3). If
(a) Formation enthalpies for different Li/Co arrange- one assumes that the Li is removed without a change of
ments σ on the fcc lattice: the cathode structure σ (a topotactic reaction),
∆Hf(σ,LiCoO2)=Etot(σ,LiCoO2)−Etot(LiO,B1) LiCoO2(σ)→Li1−xCoO2(σ)+xLi++xe−, (5)
−Etot(CoO,B1) (2)
then (see, e.g., Ref. 6) the reaction energy ∆Hreact of
where the last two terms refer to LiO and CoO in the Eq. (4) is equal to the integral of the (zero temperature
NaCl (B1) structure with lattice constants obtained by andpressure)opencircuitvoltageV ofaLixCoO2/Licell
minimizing the respective total energies with respect to between Li compositions xLi=0 and xLi=1:
4
.
rate semi-coreenergy window as opposedto treating the
Co 3p as a core state; negligible differences were found,
and thus the latter was used in all the calculations de-
Tetrahedral Octahedral
scribed below. Brillouin-zoneintegrations are performed
GaInP2 CdZnTe2 CuInSe2 LiCoO2 oCoO2 using the equivalent k-point sampling method, using k-
(III-III-V2) (II-II-VI2) (I-III-VI2) (I-III-VI2) (0-III-VI2) points for each structure corresponding to the same 28
(6x6x6)specialk-points for the fcc structure.31 All total
)
C
B CP energies are optimized with respect to volume as well as
(
E all cell-internal and -external coordinates. Convergence
CP
-) CA testsoftheenergydifferences(withrespecttobasisfunc-
AC Rand CA tion cutoff, k-point sampling, and muffin-tin radii) indi-
E( CH Rand cate that the total energy differences are converged to
-) 2 PS PCSH PS PS PS within ∼0.01-0.02eV/formula unit.
ABC » » » anSdp2inCpooOla2riinzetdhceaClcuuPlatticoantsiownearrerpaenrgfeomrmenedtifnorbLotiChofOer2-
= E( CP CA CA romagnetic (FM) and anti-ferromagnetic (AFM) geome-
m RCaAnd Rand tries. For LiCoO2, both the FM and AFM calculations
Hfor CH Rand ceovnerv,efrogred2CtooOth2e,bnootnh-mFMagnaentdicAsFolMuticoanlcu(µlaCtoio=n0s).shHowowed-
D
CH a weakly magnetic solution(µCo ∼0.45for both FM and
AFM)withthetotalenergyoftheFM(AFM)statebeing
CH 14(∼0)meV/formulaunitbelowthenon-magneticstate.
CP CP Because spin polarization only has a small effect on the
energyofthesecompounds,thecalculationsbelowforthe
energetics of cation ordering in LiCoO2 and 2CoO2 are
non-magnetic (NM). NM, FM, and AFM (with the ob-
servedalternating[111]layersofspins)calculationswere
also performed for CoO, with the AFM solution being
FIG. 2. Formation energies of cation ordering in tetrahe-
dralandoctahedralABC2 compoundsinstructuresshownin lowest in energy, and hence used here.32
Fig. 1. For the GaInP2 and CdZnTe2 compounds, the en- Having obtained the coefficients {Jf} of the CE of
ergyscalewasmultipliedby5forvisualclarity. TheGaInP2, Eq. (1) for the three types (a)-(c) of ordering reac-
CdZnTe2, and CuInSe2 energetics were taken from Refs. 38, tions, we subjected ECE(σ) to a Monte Carlo simu-
39, 24, respectively. The LiCoO2 and 2CoO2 energetics are lated annealing method for treating the configurational
from the present work. “PS” represents the energy of a thermodynamics.23 A system size of 163 = 4096 atoms
phase-separatedmixtureofAC+BCrocksalt (zincblende)bi- (with periodic boundary conditions) was used in all cal-
nariesintheoctahedral(tetrahedral)systems. “Rand”isthe culations. Monte Carlo simulations were performed in
energy of a phase in which cations are distributed randomly the canonical ensemble with the transition temperatures
(i.e., with no correlations) on theirsublattice. beingcalculatedfromthediscontinuityintheinternalen-
ergyasa function oftemperature,andthe groundstates
1 determined from the simulation at a temperature where
∆Hreact(σ)=−F dxV(σ,x)=−V(σ) (6) allconfigurationalchangesprovedtobeenergeticallyun-
Z
0 favorable. This gives: (i) the T=0 K ground state struc-
tures (from a simulation of a finite size cell initially at
where F is the Faraday constant. Hence, |∆Hreact| is
high temperature, and subsequently slowly cooled to a
simply the intercalationvoltageaveraged overLicompo-
lowtemperaturewhereallconfigurationalchangesproved
sition. We note that the intercalation voltage calculated
to be energetically unfavorable), (ii) the pair correlation
in this manner is a bulk, thermodynamic quantity and
functions or atomic short-range order present in the dis-
does not contain contributions from the cathode surface
orderedalloy,and(iii)the order-disordertransitiontem-
or from kinetic phenomena.
perature, T .
The total energies needed for Eqs. (2)-(4) have been c
obtainedusingthefirst-principlesfull-potentialLAPW15
method. In the LAPW calculations, we used the ex-
III. T =0 FORMATION ENERGIES
change correlation of Ceperley and Alder as parameter-
ized by Perdew and Zunger.30 LAPW sphere radii were
chosen to be 2.0, 2.0, and 1.3 a.u. for Li, Co, and O, A. Energetics of Li/Co ordering in LiCoO2
respectively. A well converged basis set was used, corre-
sponding to an energy cutoff of 25.5 Ry (RKmax=6.57). The formation energies [Eq. (2)] of LiCoO2 in various
Tests were performed placing the Co 3p levels in a sepa- cationarrangementsaregiveninTableIIIandcalculated
5
TABLE III. FLAPW calculated formation energies (eV/formula unit) of various cation arrangements in LiCoO2 and
2CoO2+Li(bcc): ∆Hf(σ,LiCoO2), ∆Hf(σ,2CoO2), (formation energies of σ) and ∆Hreact(σ) (average intercalation volt-
age of LiCoO2 relative to Li) are defined in Eqs. (2)-(4). Veq is the equilibrium volume(˚A3/formula unit) of LiCoO2, and δV
is the change in volume upon Li extraction [i.e., Veq(LiCoO2) - Veq(2CoO2)]. All energies of various ordered, disordered, and
partially-orderedcation arrangementsinLiCoO2 arefrom clusterexpansions(CE)ofFLAPWenergetics, andaredescribedin
thetext.
LiCoO2 2CoO2+Li(bcc)
Cation Structure ∆Hf(σ) ∆Hf(σ) ∆Hreact Veq δV
CuPt -3.38 +0.40 -3.78 31.3 4.3
D4 -3.37 +0.54 -3.91 30.5 1.9
Y2 -3.07 +0.80 -3.87 31.4 1.9
CH -2.84 +0.64 -3.48 30.8 3.2
W2 -2.82 +0.94 -3.76 30.6 3.1
CuAu -2.23 +1.65 -3.88 29.5 3.8
V2 -2.02 +2.88 -4.90 31.3 1.0
Z2 -2.13 +2.38 -4.51 31.5 2.2
Random(η=0,SRO=0) -2.68 +1.31 -3.99
Disordered(η=0,SRO=0) -2.95 +0.91 -3.86
6
CuPt(η=0.88,SRO=0) -3.22 +0.60 -3.82
D4(η=0.88,SRO=0) -3.21 +0.71 -3.92
structuralpropertiesareshowninTableII. Wenotethat not distort in any preferred direction, and consequently,
theD4structureisonlyslightlyhigherinenergythanthe 2CoO2 (D4) does not relax as much as CuPt.
CuPt structure. This competition is interesting because (3) The CuPt structure of 2CoO2 (isostructural
LiCoO2 has been synthesized in the D4 structure by with CdCl2) has an ABC... stacking of the cation
solution growth at low temperature.3,7,8,9,10,33,34,35 (Al- planes. However, recent electrochemical measurements
though there was initially some discussion in the litera- of Amatucci et al.11 have succeeded in completely de-
tureaboutthislow-temperaturesynthesizedphasebeing intercalating Li from LiCoO2, forming a 2CoO2 struc-
CuPt with imperfect long-range order7, it is now estab- ture which is isostructural with CdI2, with the stacking
lishedthatthisphaseisD4(or“D4-like”).3,33,9,10,34)The ofplanesinanAAA... arrangement(seeFig. 3)whichwe
near degeneracy of the calculated energies of the CuPt call “CuPt (AAA)”. These two structure are not related
andD4structuresissimplyaconsequenceoftheiridenti- to one another by substitutionaldegreesof freedom,and
cal pair and three body correlations Π (σ) noted above. thus are not describable by a single cluster expansion.
f
We will see that the four-body interaction J4 which dis- To examine these non-substitutional degree of freedom,
tinguishesthesestructuresisquitesmall,consistentwith we have performed total energy calculations of 2CoO2
the small energy difference between CuPt and D4. in both the CuPt and CuPt (AAA) structures (CdI2).
Consistent with the observations of Amatucci et al.11,
we find that the 2CoO2 in the AAA stacking is lower
B. Energetics of 2/Co ordering in 2CoO2 in energy than the CuPt structure by ∼0.05eV/formula
unit.
The formation energies [Eq. (3)] of 2CoO2 in various (4)WefindthatLiCoO2 intheCuPt(AAA)structure
2/Co arrangements are also given in Table III. These (Fig. 3)ishigherinenergythantheCuPtstructure(with
configurationscorrespondto various arrangementsofCo ABC stacking) by ∼0.15 eV/formula unit, in agreement
and 2. We note the following: with the fact that the observed CuPt ground state in
(1)Therelativeorderofenergeticsissimilarin2CoO2 LiCoO2 has ABC stacking.
as in LiCoO2. There is only one qualitative difference:
CH drops in energy significantly upon extraction of Li,
andislowerinenergythantheY2structure,whereasthe C. Effect of cation arrangement on average voltages
reverse is true for LiCoO2.
(2) The separation in energy between CuPt and D4 Table III gives the calculated reaction energies given
increases in 2CoO2 compared to LiCoO2, due to the in Eq. (4) for each of the cation arrangementsσ studied
symmetry of the phases: Upon extraction of Li in the here. The average voltages for all cation arrangements
rhombohedral CuPt structure, the c/a ratio decreases considered are in the ∼4 V range. In particular, the av-
significantly,providingasignificantsourceofenergylow- eragevoltage for LiCoO2 in the CuPt structure (3.78 V)
eringfor2CoO2 -CuPt. D4, onthe otherhand,is nota isinreasonableagreementwithmeasuredvalues(4.0-4.2
layered superlattice in any direction and has cubic sym- V)5,12,13,11 and pseudopotential calculations (3.75 V).36
metry. Hence, the cell parameters of 2CoO2 (D4) can- Some configurations like CH, show a marked relaxation
6
LiCoO2 (CuPt) - ABC oCoO2 (CdCl2) - ABC Cluster interactions in LiCoO
2
A
C
(a) (b)
BCAB extraLcition LiCoO Ordering2 J(R) (meV/cation) COlurdseterirningg J(k) (meV/cation) a b d g
A
Stacking
sequence
(cationAAAA layers) LiCoO2 - AAA Li oCoO2 (CdI2) - AAA CoO Ordering2 J(R) (meV/cation) COlurdseterir(nicng)g J(k) (meV/cation) a b d g (d)
insertion
A
A O ) 2 (e) (f)
catFioIGn.p3Ala.neDsiffinerLeniCtosOta2ckainndg2aCrroaOng2e.mTehnteswohfitcel,osbelapckacaknedd DCoO ) - H ( Co2 J(R) (meV/cation) COlurdseterirningg J(k) (meV/cation) a b d g
grey atoms represent Li, Co, and O, respectively. Clock- H (Li
wise from the top left, the structures are: (i) The stable D G K L G X W G
LiCoO2 (CuPt) phase(equivalenttothephaseshown inFig. Interatomic Distance R/alat
1 but from a point of view which emphasizes the layered na-
FIG.4. Pair interaction energies J and J(k) in both real
ture of the compound) with close packed cation planes in f
tAt(hhisBeeoCsLAt.ri.BC.ucCotOs.ut..2aracslkftcirwncugicst.tthuac(rCkieid,iin)CwgTli.2th)h(eifcio)lorobmsTseeehrdpevaef2crdkoCem2doOCceo2ax(OttCiro2auncPptphitnl)aagsnseteLrs“iuCicfntruuoPamrnet t([(((hldfea)e)f)]]t.c,)alIuanansnndtddree(rbaraeel)vxc]es,iprppaeaarnngoceesceira,ogilnnpiesstospesoraiofctcfieaevnl2ea(etr/rviigCogainhloeutse)voo.srofdlIoLtnefairt/giJenCergaoicinnitonidroLdin2ceisCarCtiaonoerOOgea2i2snph[[rL(o(eecwif)C)enroaarOfnneoddd2r
(AAA)” (isostructural with CdI2) which corresponds to an tendency for unlike atoms (“ordering”)f and negative values
AAA... stacking of close packed cation planes. (iv) A hypo-
indicate a tendency for like atoms (“clustering”). The labels
tthioentiocaflLLiiiCnotoO2thsetr2ucCtouOre2(“CCduIP2t)(pAhAasAe.)”,formedfrominser- pαo,iβnt,sγi,narnedcipδrioncdailcaspteacteh,ew21h[i1c1h0]c,or12r[e1s1p1o]n,d12[t2o01t]h,eaonrdde[0ri0n0g]
waves of the Y2, (CuPt and D4), CH, and phase separated
ofthe2CoO2phase,andhenceshowasignificantlylower structures, respectively.
voltage than the other configurations. Thus, as also has
been pointed out by previous authors14,36, we find that
IV. CLUSTER EXPANSIONS OF Li/Co AND
first-principlescalculationscanprovidepredictionsinter- 2/Co ORDERING AND ∆Hreact
calation energies and hence, battery voltages.
Aninterestingaspectoftheeffectofcationorderingon
We can now use the set of first-principles calculated
averagevoltageis thatLiCoO2 inthe V2structure hasa
energetics described in Section III to determine a set of
muchhigheraveragevoltagethanCuPt. Thisincreasein
interaction coefficients of the cluster expansion [Eq. (1)]
voltageisofinterestbecauseV2isa(LiO)2(CoO)2 (111)
for cation ordering. We have constructed three cluster
superlattice, whereas CuPt is a (LiO)1(CoO)1 (111) su-
expansionsofthree differenttypes ofordering: (i) Li/Co
perlattice. Ifoneexchangeseveryotherpairofcationsin
the CuPt layered sequence, the V2 layering is obtained. ordering in LiCoO2, (ii) 2/Co ordering in 2CoO2, and
(iii) cation ordering effects on the average intercalation
Thus,V2isjustCuPtwithanti-sitesontwooutofevery
voltage (see below).
four layers. This suggests that anti-site defects LiCo and
The pair interactions J found from our CE fits are
CoLi in the LiCoO2 CuPt structure, while energetically f
shown in Fig. 4, both as real-space pairs J(|R −R |)
verycostly,shouldincreasethevoltageofthecompound. i j
and as the lattice Fourier transform in reciprocal space
J(k). We see that the pair interactions in real space
are decaying with distance quite quickly, indicating con-
7
vergence of the expansion. In reciprocal space, the pair A. Ground States
interactions in Fig. 4 show some interesting properties:
Minima in J(k) indicate wavevectorswhere composition The simulated annealing algorithm finds the CuPt
waves are likely to form low energy structures. The four structure as the low temperature state. In Table III,
body interactions J4 found from our CE fits are much we simply note that this structure was the lowest in en-
smaller than the pair interactions (e.g., J2) with the ra- ergy of the eight structures calculated by LAPW. But,
tio J4/J2=0.004,0.02,and 0.04for the types of ordering the simulated annealing prediction of the ground state
(i)-(iii) above. demonstratesthat CuPt is alsothe lowestenergyconfig-
For Li/Co ordering in LiCoO2, the three local minima uration out of an astronomical number of possible con-
of J(k) are located at three wavevectors(shown by bold figurations (without symmetry, there are ∼ 2N possible
arrows in Fig. 4): L-point 21(111), W-point 21(201), and configurations that the algorithm could explore, where
near the K-point 12(110). (Additionally, an extremely N=4096). For our cluster expansionof 2CoO2, the sim-
shallow minima occurs between the Γ and X points.) ulatedannealing algorithmalsofinds CuPtas the lowest
These three wavevectorsare the compositionwavesused energy substitutional configuration. As we have already
to build all of the structures in the LiMO2 series: CuPt shown above, non-substitutional configurations are even
and D4 [12(111)], CH [12(201)], and Y2 [21(110)]. The lower in energy for the 2CoO2 system (e.g., the CdI2
global minimum of J(k) occurs at the L-point [1(111)], structure).
2
the composition wave used to construct the CuPt and By combining the simulated annealing algorithm with
D4 structures. Thus,weanticipate thatthe pairinterac- the cluster expansion of average voltage, one can search
tions J(k) for cation ordering in other LiMO2 systems forthecationconfigurationwithmaximumvoltage. This
(for other transition metals M) is likely to be qualita- search yields a phase separated (LiO+CoO) configura-
tively similar to that of Fig. 4 with changes in the rel- tion (5.8 V). De-intercalating Li from this configuration
ative minima at these three points. For 2/Co ordering would correspond to the artificial case of: LiO → Li +
in 2CoO2, the minima in J(k) occur at the same points 2O(fcc). It is interesting to note that other authors36
as in the case of Li/Co ordering in LiCoO2, indicating by very different means have also arrived at the conclu-
that the relative ordering tendencies are similar in the sion that this admittedly artificial case corresponds to
two systems. the theoreticalmaximum voltage in LiMO2 compounds.
For the cluster expansion of average intercalation en-
ergy,theminimumofJ(k)occursattheΓpoint,theori-
ginofreciprocalspace(alsoshownbyaboldarrowinFig. B. Order-Disorder Transitions
4, δ), indicating that phase separation into LiO+CoO
shouldproduce a low∆Hreact, andhence a highvoltage. For LiCoO2, the order-disordertransitionbetween the
Once the interactions {Jf} are obtained, Eq. (1) pro- low-temperature CuPt phase and the high-temperature
vides an efficient parameterization of the energy of any disordered phase is predicted to occur at ∼5100 K (Fig.
configuration. Applications of this cluster expansion pa- 5), well above the melting point of this material. (Note
rameterization which we now discuss include a search of thatthecalculationsinthispaperareallsolid-state,and
configurationspace(viaaMonteCarlosimulatedanneal- thus do not consider the liquid phase.) Antaya et al.34
ing algorithm) for ground state structures, which need report a disordered rocksalt phase of LiCoO2, grown by
not necessarily be included in the input set, thus open- laser ablation deposition at 150◦ C, whereas growth at
ing the possibility of discoveringunsuspected low energy higher temperatures results in either the D4 or CuPt
states. One can also perform Monte Carlo simulations phases. Our calculations indicate that the observed34
at finite temperatures to assess the thermodynamic and disorderedrocksaltphaseofLiCoO2isnotthermodynam-
order-disorderproperties of the system. Finally, one can ically stable,butisratheronlystabilizedkinetically,con-
easilycalculatetheenergeticsofdisorderedandpartially sistent with the fact that the disordered phase can only
ordered cation arrangements in these systems. be grown at low temperatures. By performing a simula-
As pointed out previously, the cluster expansions use tion of2/Co ordering 2CoO2 at finite temperatures, we
asinputonlychargecompensatedcompounds,andthere- wereabletoascertaintheeffectofvacanciesontheorder-
forecanbeusedtopredicttheenergiesofchargecompen- disordertransitiontemperature. Uponcompleteremoval
sated(LiO)∗ +(CoO)∗. We findfromourCEofLiCoO2 of Li, the order-disorder transition of 2CoO2 drops to
that (LiO)∗ + (CoO)∗ is 0.79 eV/formula unit lower ∼4400K,stillmuchtoohightobeexperimentallyacces-
in energy than the nominal, non-charge-compensated sible. Thus, the addition of Li vacancies is not likely to
LiO+CoO. Similarly, the CE of 2CoO2 predicts that make the disordered rocksalt phase thermodynamically
(2O)∗ + (CoO)∗ is 0.84 eV below the non-compensated accessible to experiments (although this phase is still ki-
compounds. netically accessible).
At temperatures below 2000 K, the CuPt structure
is predicted to be completely ordered. Thus, at any
growth temperatures where thermodynamic equilibrium
8
400 are examples of phases which are not directly accessible
(a) LiCoO to first-principles calculations, but may be accessed via
350 2
the cluster expansion. We show the cluster expansion
ation) 300 D E(T) D(Risoocrkdsearelt)d eneRragnedtiocms oaflsloeyv:erTalhseupcherpfehcatslyesrianndToamblealIlIoIy: is a phase
c
V/ 250 Tc in which Li and Co atoms (or 2 and Co for 2CoO2) are
e
m distributed on their cation fcc sublattice with no atom-
(Pt 200 O(CrduePret)d atototmhecoenrrtehlaaltpioynabseTtw→een∞c.atTiohneseitneesr.gTyhoifstchoirsrersapnodnodms
u 150
C
E phase is easily computed from the cluster expansion of
E - 100 Eq. (1), since the absence of atomic correlations leads
to the simple values Π = 0, and thus the energy of
f
50
the random alloy is given by ECE(Random) = J0. The
0 energies of random cation arrangements in LiCoO2 and
2CoO2areshowninTableIII. Theorderingenergyofan
orderedcompoundσistheenergyrequiredtoconstructσ
6
(b) fromtherandomcationarrangement: δEord(σ)=E(σ)−
T
c E(Random). From Table III, we can see that for both
s) 5 C (T) LiCoO2 and 2CoO2, all ordered cation configurations
unit p considered have δEord <0, except for CA, V2, and Z2.
b. 4 Partial Short-Range Order: Because Antaya et al.34
ar report the existence of some degree of CuPt-type or-
(
at 3 dering in their disordered phase, we have also com-
He puted (Table III) the energetics of a disordered rock-
cific 2 sdaelrt (pShRaOse).wSitRhOsoims eadfiengirteee-teomf pateoramtiucresheoffretc-rta,nagnedoris-
e
p characterized in real space by the pair correlation func-
S 1
tions Π0,n 6= 0 for the nth atomic shell. Thus, the
SRO parameters,Π0,n measure the extent to which spa-
0
tial correlations exist in disordered alloys. The SRO
2000 4000 6000 8000
parameters used to compute the energetics of for the
Temperature (K)
first ten neighbor shells were obtained from a Monte
FIG.5. Monte Carlo calculated (a) energy relative to the Carlo simulation of the LiCoO2 disordered alloy just
above the order-disorder transition (in parentheses are
T=0 energy of the CuPt structure and (b) heat capacity as
the values for fully ordered CuPt or D4): −0.06(0.0),
functionsoftemperatureforLiCoO2. Thetransitionbetween
−0.27(−1.0), +0.03(0.0), +0.12(+1.0), +0.02(0.0),
the CuPt and disordered (rocksalt) LiCoO2 can be seen at
−0.07(-1.0), −0.02(0.0), +0.10(1.0), −0.01(0.0), and
5100 K.
∼ −0.01(0.0). Note that the energetic effect of SRO is
to significantly lower the energy of the random phase in
is achieved, the CuPt phase should form with a long- both LiCoO2 and 2CoO2 by 0.27 and 0.40 eV/formula
range order (LRO) parameter of nearly unity. Thus, unit, respectively.
anti-site defects LiCo or CoLi are probably not formed Partial Long-Range Order: There have also been re-
under conditions of thermodynamic equilibrium. Also, ports of long-rangeorderedLiCoO2 (either CuPt or D4)
since CuPt is completely ordered by 2000K, even the with small quantities of Li on the Co sites, or vice versa.
D4 structure is not stabilized by thermodynamic fac- This amounts to a CuPt or D4 phase with partial long-
tors(i.e.,thermalfluctuationsinenergyaresmallerthan range order (LRO). If the LRO parameter η=1, then all
the CuPt-D4 energy difference for temperatures of in- Li and Co atoms reside completely on their own sublat-
terest). However, the D4 structure has been observed in tice and LRO is perfect. However, for states of partial
low-temperaturesolutiongrownandlaserablation-grown LRO, η < 1, and there is an amount 1−η of intermixing
2
samples, which are probably not equilibrium phases. betweensublattices. Forsimplicity,weassumethatthere
are no short-range correlations between the intermixed
atoms. In Table III, we showthe energeticsof CuPtand
C. Properties of Disordered and Partially Ordered D4 structures with LRO parameter η=0.88, correspond-
Cation Arrangements
ing to 6% of Li on the Co sites, and vice versa. The
LiCoO2 energiesofCuPtandD4 arebothraisedby 0.16
Using the CE, we can compute the energetics of any eV/formulaunitrelativetotheη=1fullyorderedphases,
cation arrangement such as random alloys or any disor- while the correspondingincreases for 2CoO2 is 0.20 and
dered (short-range or long-rangeordered) phases. These
9
0.17 eV/formula unit. 2CoO2, compared with E(CH) < E(CA) < E (CuPt),
The cluster expansion of voltage can also be used to universally found in the lattice-mismatched tetrahedral
predicttheaveragevoltagesofconfigurationsnotdirectly systems. Also, in the octahedral systems, the ordering
accessible to first-principles calculations (Table III). In energyδEord <0forbothCuPtandCH,whileδEord <0
particular, we see that the random alloy (3.99 V) is pre- for CH in the tetrahedral systems.
dicted to have a higher averagevoltage than the ordered TheCuPtstructureispreferredinoctahedralnetworks
CuPtphase(3.78V).Sincethisphasehasbeenproduced duetostrainenergyarguments: ForanoctahedralABC2
bylaserablation,34 itwouldbeinterestingtomeasureits system which has distinct equilibrium A−C and B−C
electrochemicalproperties,in order to compare with our bond lengths, the CuPt structure has the property that
predictions. The increase in voltage due to disorder is the cell-internal distortion of the anions (C) accommo-
significantly reduced when one considers the disordered dates any equilibrium A−C and B −C bond lengths,
phasewithSROdescribedabove(3.86V).Thus,itispos- and maintains all A−C bond lengths equal to one an-
sible that experimentally, the voltage of the disordered other (and similarly for B−C). The D4 structure also
LiCoO2 relative to CuPt could be used to indirectly de- possesses this optimal structural relaxation. This opti-
termine the amount of short-range order in the sample. mal bond-length accommodation is interesting in light
Also in Table III are the voltages of partially long-range of the fact that the cation CuPt structure in tetrahedral
orderedCuPtandD4phaseswith6%oftheCoatomson systems, when relaxed, possesses two equilibrium A−C
theLisites(η=0.88). Eventhissmallamountofanti-site bonds (and similarly for B−C) as opposedto the single
defectsincreasesthevoltagesofCuPtandD4by0.05and A−C bond in the octahedral case. The distinction be-
0.01 V, respectively. Note that for either LRO or SRO, tweenthe two A−C bonds intetrahedral CuPtis due to
thequalitativeeffectofdisorderingisthesame: Disorder thefactthatsomeoftheA−C bondsinthisstructureare
raisesthe energyof2CoO2 morethanLiCoO2,andthus along the direction of cell-internal distortion and other
raises the average voltage. A−C bonds are perpendicular to this direction. Thus,
whenCatomsarerelaxed,thesetwotypesofA−C bonds
adopt different bond lengths. However,in the octahedral
V. ENERGETICS OF OCTAHEDRAL VS. CuPtstructure,noneoftheA−C bondsareeitheralong
TETRAHEDRAL ABC2 NETWORKS orperpendiculartothedistortiondirectionoftheanions,
butratherallA−C bondsareatequivalentanglestothis
We now compare our results for cationorderingin the direction. Thus, when the anions relax, all A−C bonds
octahedral LiCoO2 and 2CoO2 systems with the well- are distorted by equal amounts. The calculated equilib-
studied cases of cation ordering in isovalent and het- riumbondlengthsinallcationarrangementsforLiCoO2
erovalent tetrahedral ABC2 systems. In the (heterova- are given in Table IV. One can see from this table that
lent)octahedralLiMO2systems,orderedcationarrange- CuPt and D4 are the only structures for which there is
ments are stable with respect to LiO+MO.This ordered only one type of symmetry-inequivalent Li-O and Co-
compound stability is qualitatively similar to heterova- O bond. Structures with Li-O and Co-O bonds equal to
lent tetrahedral systems, such as CuInSe2, where cation oneanother(e.g.,CH,CA)areenergeticallyunfavorable.
ordered phases are stable relative to decomposition into In tetrahedral systems, the configuration which possess
CuSe+InSe zincblende binaries. Just as in the LiCoO2 the optimum structural geometry, analogous to CuPt in
case,thebinariescorrespondtocompounds(rocksaltLiO octahedral coordination, is the CH structure, which is
and CoO and zincblende CuSe and InSe) which do not the lowest energy ordered compound in size mismatched
satisfy the octet rule, and hence are relatively high en- semiconductor alloys. This strain energy argument can
ergy configurations. The high energy of the constituents also explain the relative stability of CuPt, CH, and CA
“exposes” the ordered compounds (which do satisfy the in octahedral vs. tetrahedral systems: Using a simple
octet rule) as stable in these I-III-VI2 systems. The sta- valence force field which includes only energetic effects
bility of ordered compounds in octahedral LiMO2 sys- duetostrain,oneobtainsthecorrectorderofthesethree
tems is, however, in contrast to the isovalent tetrahe- structuresforbothoctahedralandtetrahedralsystemsas
dral AIIIBIIICV systems, in which bulk ordered com- comparedwithLAPW.37,38,39 Oneshouldnote,however,
2
pounds are unstable with respect to AC+BC. In these that in the LiMO2 series, there are systems other than
III-III-V2 systems, the phase-separated state is the low M=Co which possess ground states other than CuPt,
energygroundstate,andorderedcompoundswhichhave e.g.,theCHandY2structures. Thus,clearlystrain-only
been observedhavebeen shown1 to be a resultofa com- argumentsdonotexplainthetotalityoforderingtenden-
bination of epitaxial strain and surface-reconstruction- cies in these compounds, as other effects must dominate
induced ordering. in some systems.
Therelativeorderofenergiesoforderedcompoundsin Anotherdistinctionbetweentheorderingtendenciesof
the octahedralLiCoO2 and 2CoO2 systems is also quite the octahedralLiCoO2 systemwith thoseofthe tetrahe-
different from the (isovalent or heterovalent) tetrahedral dralsystems is in the energy scale. In Fig. 2, the energy
cases: E(CuPt)<E(CH)<E (CA)inbothLiCoO2and scale of the tetrahedral systems is multiplied by a factor
of5,andisstillsmallerthantheoctahedralenergyscale.
10
