Energetics of hydrogen/lithium complexes in silicon analyzed using the Maxwell construction

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by  Andrew J. Morris

Energetics ofhydrogen/lithium complexes insiliconanalyzedusing the Maxwellconstruction
Andrew J. Morris∗,1 C. P. Grey,2 R. J. Needs,3 and Chris J. Pickard1
1DepartmentofPhysicsandAstronomy,UniversityCollegeLondon,GowerStreet,LondonWC1E6BT,UnitedKingdom
2DepartmentofChemistry,UniversityofCambridge, LensfieldRoad,CambridgeCB21EW,UnitedKingdom
3Theory of Condensed Matter Group, Cavendish Laboratory, University of Cambridge,
J. J. Thomson Avenue, Cambridge CB3 0HE, United Kingdom
(Dated:January25,2012)
Wehavestudiedhydrogen/lithiumcomplexesincrystallinesiliconusingdensity-functional-theorymethods
2 and the ab initiorandom structure searching (AIRSS)method for predicting structures. A method based on
1 the Maxwell construction and convex hull diagrams is introduced which gives a graphical representation of
0 the relativestabilitiesof point defects ina crystal and enables visualization of the changes instability when
2 thechemical potentials arealtered. Wehaveused thisapproach tostudylithiumand hydrogen impuritiesin
n silicon,whichmodelsaspectsoftheanodematerialintherecently-suggestedlithium-ionbatteries. Weshow
a thathydrogenmayplayaroleintheseanodes,findingthathydrogenatomsbindtothree-atomlithiumclusters
J insilicon,formingstable{H,3Li}and{2H,3Li}complexes,whilethe{H,2Li}complexisalmoststable.

4
2 PACSnumbers:61.05.-a,61.72.J-,82.47.Aa

]
i I. INTRODUCTION electrons,althoughwewillnotshowresultsforthiscasehere.
c
s Thisarticleisdividedintothefollowingsections,inSect.II,
- themodifiedMaxwellconstructionisintroducedtovisualize
l The propertiesof a material may be substantially affected
tr by the presenceof impurityatoms.1 One of the fundamental the energetics of the phase separation of defect complexes,
m andtheexampleofN/Ocomplexesinsiliconisusedtovali-
properties is the relative energies of different configurations
datethemethod.InSect.IIIthemethodisappliedtotheprob-
. oftheimpuritieswithinthematerial.Severaldifferentspecies
t lemofH/Lidefectcomplexesinsilicon.FinallySect.IVcon-
a of atom may be present, and the impurity atoms may have
m cludesthework.
strong interactions with each other and with the host mate-
- rial. In this paper we study the interactionsbetween lithium
d
andhydrogenimpuritiesincrystallinesilicon,whichisasys-
n II. CONVEXHULLFORDEFECTS
temofinterestforlithium-ionbatterytechnologies. Lithium-
o
c ion batteries are widely used in portable electronics and are
[ nowbeingemployedinthenextgenerationofhybridandall- Inthisarticlewedenoteadefectcomplexbylistingitscon-
electric vehicles.2 Recently there has been interest in using stituentatomsbetweenbraces,forexample,the{2H,3Li}de-
1
siliconanodes,whichhaveaveryhighvolumetricandgravi- fectcomplexcontainstwohydrogenandthreelithiumatoms.
v
0 metriccapacity.3–5 Crystallinesiliconisgenerallyusedinthe For simplicity we begin by considering a single impurity
4 batteryandthe firstcharge-dischargecycleinvolvesthe con- species a which forms a single impurity complex in a large
9 versionofcrystallinesilicontoanamorphousphase. sample of perfect bulk crystal. The formation energy of a
4 Pointdefectsmay forma varietyof complexescontaining complex{na a }containingna impurityatomsisgivenby
.
1 severalatomsofmorethanoneatomicspecies.Aninteresting
0 featureofourapproachistheuseofagraphicalrepresentation Ef =ED−na m a −EH, (1)
2 oftherelativestabilitiesofthedefectsbasedontheMaxwell
1 construction6,7 and the correspondingconvexhull. This is a where ED is the energy of the system including the defect
: complex,m a isthechemicalpotentialoftheimpurityspecies,
v standardprocedureusedinmaterialsscience(andelsewhere)
andE istheenergyofthehostcrystalwithoutdefects. The
i for studying phase separation, which we have applied to de- H
X energyofthehostcrystalcanbewrittenas
fectsincrystals. Theterm“convexhull”referstothefactthat
r
a thesecondlawofthermodynamicsrequiresthattheenergyper EH=(cid:229) nb m b , (2)
particle(orfreeenergyasappropriate)mustbeaconvexfunc-
b
tionoftherelativeconcentrationsoftheparticles.Appropriate
chemical potentials for the differentatomic species can nor- where there are nb atoms of the host crystal atoms of each
mallybeestimatedforaparticularsetofexternalconditions. species b which have chemical potentials m b . The complex
Wearealsointerestedinthechangesintherelativestabilities with the lowest value of Ef/na is the most stable and ones
ofdefectswhenthechemicalpotentialsoftheatomicspecies withhigherenergiesaremetastable.
arealtered.Chargeddefectsarealsoofinterestingeneral,and Eq.(1) can readily be extendedto a complexcontaininga
thesecanbeincludedbyintroducingachemicalpotentialfor largernumberof impurityspecies, andthe formationenergy
perimpurityatomisthen
Ef ED−(cid:229) a na m a −EH
E = = . (3)
∗Email:[email protected]. pa (cid:229) a na (cid:229) a na
2
Theconvexhulldiagramisconstructedbyplottingthefor- These energies are plotted on the diagram at the points
mationenergyperimpurityatomofeachcomplexagainstthe (E =0,C =0) and (0,1). The solid-red convex hull is then
pa N
fractionalconcentrationC ofimpurityspeciesi,where drawnasaseriesofstraight-linesegmentsbetween(0,0)and
i
(0,1)passingthroughthesquaresrepresentingtheenergiesper
n
i
Ci= (cid:229) a na . (4) impurityatom Epa in such a way thatnoneof the defectslie
belowthehull. Onlythecomplexeswhichlieonthehullare
Ifthecomplexescontainonlytwoatomicspecieswecansim- stable. The {O}, {N,3O}, {N,2O}, {2N,2O} and {2N,O}
ply plot E (C) on a graph as shown, for example, in Fig. defectsareallstable sincetheylie onthetie lineofthe con-
pa i
1. The solid red and blue lines in Fig. 1 show convex hulls vexhull,butthe{N,2O}complexisnotstableandtheenergy
whichconsistof straight-linesegmentsknownas “tie lines”. wouldbeloweredifthesedefectsdisproportionatedsuchthat
A higher-dimensionalhull is required if there are more than
twoimpurityspecies. Thecaseofthreeatomicspeciescanbe
represented on a “triangular plot” as shown, for example, in
Ref.8. Alternativelyonecanconsidertwo-dimensionalslices 4{N,2O}→2{N,3O}+{2N,2O}. (7)
ofthemulti-dimensionalhull. Electronscanalsobeincluded
asaspecies,sothatdefectcomplexesindifferentchargestates
ThedashedbluelineintheupperpanelofFig.1indicates
canbedescribed.
how the situation would be modified if the chemical poten-
Whenmorethanoneimpurityspeciesispresentitispossi-
tial for O is changed to the energy of the much more stable
ble for “phase separation” or “disproportionation” of defect
{4O}complex. Themodifiedcomplexhullisredrawnasthe
complexes to occur. For example, the lowest energy state
solid-blue line in the lower panel of Fig. 1. It is clear from
whenequalnumbersofimpurityspeciesAandBarepresent
eithertheupperorlowerpanelsthatthe{N,3O}defectisno
might consist of equal numbers of the defect complexes
longerstableunderthesecircumstancesandtheenergywould
{A,2B} and {A}, rather than the defect complex {2A,2B}.
belowerediftheydisproportionatedsuchthat
Wewillassumethatthedefectcomplexesdonotinteractwith
one another (except by a “chemical reaction”, forming new
complexeswhichcanbeincludedwithinthetheory)andthey 2{N,3O}→{4O}+{2N,2O}. (8)
are expectedto be distributedrandomlyoverthehostcrystal
ratherthanformingspatiallydistinctphasesasinthestandard
Theseresultsareentirelyconsistentwiththeanalysisreported
applicationto“phaseseparation”.WemayincludewithinE
pa in Ref. 10. The graphicalrepresentationis usefulbecause it
the contributions to the free energy from the vibrational en-
showswhichdefectsarestableforaparticularchoiceofchem-
tropyandtheconfigurationalentropyduetothedifferentde-
icalpotentials,anditalsoallowsaneasyvisualizationofthe
generateorientationsofacomplexatalatticesite. However,
effects of altering the chemical potentials. The convex hull
it is not possible to include the configurational entropy due
alsoshowstheamountbywhichtheenergyisloweredbythe
toarrangingdefectsatdifferentlatticesiteswithinE ,since
pa disproportionationofdefectcomplexes.
it depends on the concentrations of the impurity complexes.
HencetheMaxwellconstructionisexactonlyatzerotemper-
ature,andafullstatisticalmechanicaldescriptionisrequired
atfinitetemperatures. Thestandardplotofthedefectforma-
tionenergyagainstthechemicalpotentialfoundinthedefect B. Nativedefects
literature(see,forexample,Fig.2ofRef.9),suffersfromthe
samelimitation.
Our approach is easily extended to account for a crys-
tal containing n native vacancies V which form com-
v
A. ApplicationtoN/Ocomplexesinsilicon plexes{na a ,nb b ,nvV}ornI interstitialsIformingcomplexes
{na a ,nb b ,nII}. The native defects may be included within
this framework, not as new “impurities species”, but as a
WeuseN/Ocomplexesinsilicontoillustrateourapproach
change in the definition of the host material. The chemical
andoutlinetheadvantagesofthemethod.Fig.1showsacon-
potentialofanatomofthehostmaterialaddedatasurfaceof
vexhullcontainingN/O complexesfroma previousstudy.10
the crystal is equal to the cohesive energyof an atom in the
ThehullshownintheupperpanelofFig.1isconstructedas
perfectbulkcrystalsinceand,aslongasthesurfaceislarge,
follows. We choosethe chemicalpotentialsfor the N andO
thesurfaceenergyisunchangedbyaddinganatomata“kink
atomstobethesameasinourpreviousstudy.10Thechemical
site”onasurfaceterrace.Henceinthecasewherethevacancy
potentialforNisobtainedfromtheenergyofthe{2N}defect
isabsentpriortotheadditionofimpurities,m b isthechemical
as
potentialof each hostspecies b in the perfectcrystal. If the
m ({2N})= 1(E ({2N})−E ), (5) vacancyispresentpriortotheadditionofimpurities,m b isthe
N 2 D H chemicalpotentialofhostspeciesb inacrystalcontainingthe
vacancy.Notethatitisnotpossibletodescribemorethanone
whileforOitisobtainedfromtheenergyofthe{O}defectas
typeofvacancyinthisapproachunlesseachtypeistreatedas
m ({O})=E ({O})−E . (6) aseparatespecies. Thesameappliesforinterstitials.
O D H
3
moststablepositionforasinglelithiumimpuritytobeatthe
0.2 tetrahedral(T )site,whichispreferredtosubstitutiononasil-
{N,O} d
{3N,O} iconsite. Kimetal.17 performedadetailedstudyoftheelec-
{O} {2N}
tronicstructureofalithiumatominbulksilicon, concluding
a 0.0
p
E thatthepresenceoflithiumweakensthenearbySi–Sibonds.
m, 0.2 0.4 0.6 0.8
ato -0.2 {N,2O}
urity {N,3O} {2N,O} B. DFTcalculations
p -0.4
m
per i {4O} {2N,2O} icoWneushianvgeDprFeTdicatneddtshteruactburiensitioofrlaitnhdiuomm ismtrpuuctruitrieesseinarscihl--
eV {O} ing method (AIRSS).18,19 In this method randomly gener-
y, 0.4 {N,O} atedstructuresarerelaxedtoa minimumintheenergy. This
g
er approach has been successful in predicting the structures of
En point defects in semiconductors10,20 and ceramics,21 as well
n 0.2 {N,3O} {3N,O} as high-pressure phases of materials such as solid silane,18
o
ati {N,2O} hydrogen,22 and lithium.23 Each AIRSS run was performed
m {4O} {2N} ata fixedstoichiometry. Searcheswereperformedatseveral
or 0.0
F stoichiometriesand the results were analyzedusing the con-
0.2 0.4 0.6 0.8
vexhullconstruction.
-0.2 {2N,O} We used simulation cells containing 1, 2, 3 or 4 lithium
{{{222NNN,,,222OOO}}}
atoms, 1, 2, 3 or4 lithiumatomsplus1hydrogen,atomand
Fractional Concentration of N, C 1,2or3lithiumplus2hydrogenatoms. Theinitialstructures
N
werechoseninasimilarfashiontothoseinourearlierstudy
ofH/N/Odefects.10Asmallholewasmadeinthehostcrystal
FIG. 1: (Color online) Maxwell construction for N/O complexes
insilicon. Theupper panel shows the formation energy per impu- byremovingasiliconatomfromtheunitcellandplacingthe
rityatom(Epa)forvariouscomplexesusingthe{2N}and{O}de- appropriatenumberoflithiumatomsanda siliconatomran-
fects for the chemical potentials. The stable complexes, {N,3O}, domlywithinthehole. Theconfigurationswerethenrelaxed
{2N,2O}and{2N,O}arefoundontheconvexhull(solidred). The usingtheDFTforces. Someofthestartingconfigurationsre-
blue(dashed)lineisatie-linebetweenthevaluesofEpaforthe{4O} laxedtostructureinwhichtwoormoreseparatedefectswere
and{2N}complexes. IftheOchemicalpotentialischangedtothat present,andthesestructureswerediscarded.
ofthemorestable{4O}complex,thestablecomplexesbecomethose The plane-wave basis-set DFT code CASTEP24 was used
whichbothlieonthesolid-redconvexhullandalsoliebelowtheblue
to calculate ground-state total energies and optimize the ge-
dashedline. Thisconstructionshowsthestablecomplexeswhenthe
ometries of the candidate structures. We used “on-the-fly”
Ochemical potential ischanged totheenergy of {4O}, sinceonly
ultrasoft pseudopotentials25 and the PBE26 parameterization
thecomplexeswithformationenergiesbelowthetieline({2N,2O}
and{2N,O})arestable.Thisisconfirmedbyreplottingtheenergies of the generalized gradient approximation to the exchange-
(bluediamonds)inthelowerpanelusing{4O}asthechemicalpo- correlation functional. The initial searches were performed
tential. Onlytwo structures lieon the new convex hull (solid blue with simulation cells containing 32 silicon atoms plus im-
line). purity atoms, sampling the Brillouin zone using the 2×
2×2 multi-k-point generalization20,27,28 of the Baldereschi
scheme.29 The basis set contained all plane-waves with en-
III. STUDYOFH/LICOMPLEXESINSILICON ergiesupto300eV.
The lowest-energy structures from each search were then
A. HandLiinsilicon furtherrelaxedatahigherlevelofaccuracy.Weusedsimula-
tion cells containing 256 silicon atoms and a basis set with
Hydrogenisacommonimpurityinsiliconwhichbindstoa plane-waves up to 500eV, a harder Li pseudopotential and
varietyofdefectcomplexes,includingthose containingoxy- a standard 2×2×2 Monkhorst-Pack grid of k-points. The
gen and nitrogen.10 It is difficult to determine the concen- formation energies were then calculated using these struc-
tration of hydrogen in silicon samples although it is likely tures but with a larger 4×4×4 Monkhorst-Pack grid of k-
to be present at virtually every stage of manufacture and it points. We also performed some calculations in which we
maybeincorporatedatlevelsofuptoabout1020cm−1.11The allowed spin polarization to occur, but no significant effects
role of hydrogen impurities in semiconductors has been re- werefound.
viewed by Estreicher.12 In this article we present results for
H/Li impurities in bulk silicon. Both amorphous13,14 and
crystalline15phasesoftheLi-Sisystemhavebeenstudiedus- C. ResultsforH/Liinsilicon
ingdensity-functional-theory(DFT)methods,givingvoltages
which agree with experiment to within 0.1V. Wan et al.16 WemodeltheinitialstagesoftheincorporationofLiatoms
studied lithium defects in silicon using DFT and found the into bulk silicon as the formation of defect complexes with
4
Defect E (eV)
f
{3Li} 0.422
{H,3Li}∗ 0.103
{H,3Li} -0.139
{H,2Li}∗∗ 0.321
{H,2Li}∗ 0.301
{H,2Li} 0.101
{2H,3Li}∗∗ 0.114
{2H,3Li}∗ -0.010
{2H,3Li} -0.239
{H,Li}∗∗ 0.474 FIG.3: (Coloronline)Twoviewsofthe{H,3Li}defectinbulksili-
{H,Li}∗ 0.472 con.Siliconatomsareshowninyellow(lightgrey),lithiuminpurple
{H,Li} 0.401 (darkgrey)andhydrogeninwhite.Thisdefectispredictedtobesta-
{2H,2Li} -0.028 ble for 0<CH <0.4. The lithium atoms form a three-membered
{2H,Li}∗ 0.413 ringinavoidcreatedbybreakingaSi–Sibondandahydrogenatom
{2H,Li} -0.076 bondstothedisplacedsiliconatom.
TABLEI: Formationenergies,E ,ofvariousH/Licomplexesinsil-
f
icon. The chemical potential for H isobtained from the energy of
an H molecule at theT siteof bulk siliconand the chemical po-
2 d
tentialforLiisobtainedfromtheenergyofaLiatomattheT site.
d
Singleasterisks(*)anddoubleasterisks(**)denotethefirstandsec-
ondmetastabledefects,respectively,foragivendefectstoichiometry.
ThesameinformationisrepresentedgraphicallyinFig.2.
FIG.4: (Coloronline)The{2H,3Li}complexinbulksilicon. Sili-
conatomsareshowninyellow(lightgrey),lithiuminpurple(dark
grey) and hydrogen in white. This defect is predicted to be stable
pa 0.04 {H,2Li} for 0.25<CH <1.0. It is similar to the {H,3Li} defect shown in
E
m, Fig.3,exceptthattheadditionalhydrogenatomterminatesthedan-
o {H,3LI}* glingbondonthedisplacedsiliconatoms. Thelithiumatomsform
at athree-memberedringinthevoidcreatedbybreakingaSi–Sibond.
y 0.02 **
urit {2H,3Li} Ahydrogenatomsitsbetweenthesetwosiliconatoms.
p
m
er i {Li} 0.2 0.4 0.6 0.8 {H }
V p 0.00 2 varying concentrations of H and Li atoms. We assume that
e * the Li atoms are supplied from bulk body-centered-cubicLi
gy, {2H,3Li} {2H,2Li} metal. The {Li} defect in silicon has a formation energy of
ner 0.21 eV relative to bulk Li in accordance with the observed
n E -0.02 overpotential during Li insertion reactions in bulk silicon.32
atio {2H,Li} We assume that the Li atoms initially form {Li} defects in
m thebulksiliconandweinvestigatewhetherchemicalreactions
For -0.04 {H,3Li} between Li and H atomsin bulk silicon are energeticallyfa-
vorable. We choose the {Li} defect to define our chemical
{2H,3Li}
potentialbecause it is the lowest energydefect that we have
Fractional Concentration of H, CH foundinsiliconandbecausealithiumdefectofthesame(Td)
symmetry has been observed in electron paramagnetic reso-
FIG. 2: (Color online) Convex hull for Li/H complexes in silicon. nanceexperimentsinbulksilicon.30Ametastable{3Li}com-
Formation energy per impurity atom plotted against the fractional plexwithC symmetrywasalsofoundwithformationenergy
3v
concentrationCH ofhydrogen. Theplussigns(+)representdefect relativeto{Li}of0.14eVperatom,butthisistoohighinen-
complexes. ThechemicalpotentialsforLi({Li})andH({H2})are ergy to appear on Fig. 2. The hydrogen chemical potential
plotted at (0,0) and (0,1), respectively. Metastable complexes of a
was obtainedas in previousstudies10,20 from a H molecule
givenstoichiometryaredenotedbyasterisks(∗). Theformationen- 2
attheT siteinsilicon.Amorecompletemodelwouldrequire
ergiesofdefectcomplexesareindicatedbyblue(darkgrey)crosses d
andthepredictedstabledefectsbyred(mid-grey)crosses. Thered theinclusionofotherdefects,suchasvacanciesandvacancy
(mid-grey)linesshowtheformationenergyperimpurityatomofthe complexes.
(mixturesof)stabledefectspresentasafunctionofCH. Single-atomLidefectsarefavoredwhenonlyLiimpurities
arepresent,buttheconvexhullplottedinFig.2showsthatthe
additionofhydrogencausesthelithiumatomstocluster. For
5
0<C <0.25 the {H,3Li} defect forms and any additional We also studied H/Li complexesin silicon, which may be
H
lithium atomsoccupyT sites. The structure of the {H,3Li} relevant to understanding how the first few Li atoms enter
d
defect is picturedin Fig.3. Both the {H,3Li} and {2H,3Li} crystalline silicon. This extended the previous analysis of
complexes are present in the range 0.25<C <0.40. The lithiuminsiliconanodestoincludeitsinteractionwiththehy-
H
{2H,3Li}defect is present whenC >0.40, and any excess drogenthatisintroducedintothesiliconduringmanufacture.
H
hydrogen forms H molecules at the T site. The {H,3Li}, Whenonlylithiumimpurityatomsarepresentthemoststable
2 d
{2H,3Li}and{2H,Li}defectsresembleaChadiH∗ defect31 defectis{Li},butthepresenceofhydrogenatomscausesthe
2
with the lithium atoms close by. The metastable H∗ com- lithium atoms to form H/Li clusters containingthree lithium
2
plexfoundbyChadi31hasaformationenergyrelativetoaH atoms. We foundthat the {H,3Li}and {2H,3Li}complexes
2
moleculeattheT site of0.1eV perH atom.20 The{2H,Li} arestableandthatthe{2H,Li}complexisnearlystable. Hy-
d
defect could be present when thermal effects are taken into drogenpresentin the bulk favorsbondingcovalentlyto self-
account,duetoitscloseproximitytothetielineatC =0.66. interstitialdefects,20butoncethesilicondefectsaresaturated,
H
hydrogen will bind to lithium complexes, further stabilizing
them. ThecombinationofAIRSSandtheMaxwellconstruc-
IV. CONCLUSIONS tionandconvexhulldiagramsisapowerfultoolforanalyzing
theenergeticsofpointdefectsinmaterials.
WehaveusedconvexhulldiagramsbasedontheMaxwell
construction to visualize the energetics of defect complexes.
The graphical representation shows the stable defects for a
Acknowledgments
particular choice of chemical potentials and the relative sta-
bilitiesofthedefectcomplexes. Itallowsvisualizationofthe
changesinthestabilityofthedefectswhenthechemicalpo- WearegratefultoMarshallStoneham,TonyHarker,Dave
tentialsarechanged. Theapproachwasillustratedusingdata Bowler and Mike Gillan for fruitful discussions. This work
for N/O complexes in silicon from a previous study.10 The wassupportedbytheEngineeringandPhysicalSciencesRe-
methodcaneasilybeextendedtocrystalscontainingcharged searchCouncil(EPSRC)oftheU.K.Computationalresources
species, nativevacanciesor interstitials, whichis an area we were provided by the University College London Research
arecurrentlyinvestigating. Computingservice.
1 A.M.Stoneham, TheoryofDefectsinSolids: Electronicstruc- 496,25(2010).
tureofdefectsininsulatorsandsemiconductors(OxfordUniver- 16 W. Wan, Q. Zhang, Y. Cui, and E. Wang, J. Phys.: Condens.
sityPress,Oxford,1975). Matter22,415501(2010).
2 J.M.TarasconandM.Armand, Nature414,359,(2001). 17 H.Kim,K.E.Kweon,C.-Y.Chou,J.G.Ekerdt,andG.S.Hwang,
3 S.Lai, J.Electrochem.Soc.123,1196(1976). J.Phys.Chem.C114,17942(2010).
4 C.J.WenandR.A.Huggins, J.Sol.Stat.Chem.37,271(1981). 18 C.J.PickardandR.J.Needs,Phys.Rev.Lett.97,045504(2006).
5 W. J. Weydanz, M. Wohlfahrt-Mehrens and R. A. Huggins, J. 19 C. J. Pickard and R. J. Needs, J. Phys.: Condens. Matter 23,
PowerSources81-82,237(1999). 053201(2011).
6 J.W.Gibbs,GraphicalMethodsintheThermodynamicsofFlu- 20 A. J. Morris, C. J. Pickard, and R. J. Needs, Phys. Rev. B 78,
ids, Part 1 pp. 309-342, A Method of Geometrical Representa- 184102(2008).
tion of the Thermodynamic Properties of Substances by Means 21 J. Mulroue, A. J. Morris, and D. M. Duffy, Phys. Rev. B 84,
ofSurfaces,Part2pp.382-404, TransactionsoftheConnecticut 094118(2011).
AcademyofArtsandSciencesVol.II,1873. 22 C.J.PickardandR.J.Needs,NaturePhys.3,473(2007).
7 J. C. Maxwell, Theory of Heat, Longmans, Green and Co, pp. 23 C. J. Pickard and R. J. Needs, Phys. Rev. Lett. 102, 146401
195-208(1904)andScientificLettersandPapersofJ.C.Maxwell, (2009).
Vol.3,1874-1879, EditedbyP.M.Harman,(CambridgeUniver- 24 S.J.Clarketal., Z.Kristallogr.220,567(2005).
sityPress,Cambridge,2002). 25 D.Vanderbilt, Phys.Rev.B41,7892(1990).
8 S.P.Ong, L.Wang, B.Kang, and G.Ceder, Chem. Mater. 20, 26 J. P.Perdew, K.Burke, and M. Ernzerhof, Phys. Rev. Lett.77,
1798(2008). 3865(1996).
9 S.B.ZhangandJ.E.Northrup, Phys.Rev.Lett.67,2339(1991). 27 G.Rajagopal, R.J.Needs, S.Kenny, W.M.C.Foulkes, andA.
10 A. J. Morris, C. J. Pickard, and R. J. Needs, Phys. Rev. B 80, James, Phys.Rev.Lett.73,1959(1994).
144112(2009). 28 G.Rajagopal,R.J.Needs,A.James,S.D.Kenny,andW.M.C.
11 S.Pearton,M.Stavola,andJ.W.Corbett,AdvancedMaterials4, Foulkes, Phys.Rev.B51,10591(1995).
332(1992). 29 A.Baldereschi, Phys.Rev.B7,5212(1973).
12 S.K.Estreicher, Mater.Sci.Eng.B14,319(1995). 30 G.D.WatkinsandF.S.Ham, Phys.Rev.B1,4071(1970).
13 V.L.Chevrier andJ.R.Dahn, J.Electrochem.Soc.156, A454 31 D.J.Chadi, Appl.Phys.Lett.83,3710(2003).
(2009). 32 M.N.ObrovacandL.J.Krause, J.Electrochem.Soc.154A103
14 V.L.Chevrier andJ.R.Dahn, J.Electrochem.Soc.157, A392 (2007).
(2010).
15 V.L.Chevrier,J.Zwanziger, andJ.R.Dahn, J.AlloysCompd.