Modeling the electrical conductivity in BaTiO3 on the basis of first-principles calculations

0.17 MB·English

by  Paul Erhart

publishedinJournalofAppliedPhysics104,044315(2008),doi:10.1063/1.2956327
ModelingtheelectricalconductivityinBaTiO onthebasisoffirst-principlescalculations
3
PaulErhart1,2 andKarstenAlbe2
1)LawrenceLivermoreNationalLaboratory,Chemistry,Materials,Earth,andLifeSciencesDirectorate,
Livermore,California,94551
2)TechnischeUniversita¨tDarmstadt,Institutfu¨rMaterialwissenschaft,
64287Darmstadt,Germany
(Dated:19January2012)
Thedependenceoftheelectricalconductivityontheoxygenpartialpressureiscalculatedfortheprototypicalperovskite
2 BaTiØ3 based on data obtained from first-principles calculations within density functional theory. The equilibrium
1 point defect concentrationsare obtained via a self-consistent determination of the electron chemical potential. This
0 allows to derive charge carrier concentrations for a given temperature and chemical environmentand eventually the
2 electrial conductivity. The calculations are in excellent agreement with experimental data if an accidental acceptor
n dopantlevelof1017cm−3 isassumed. Itisshownthatdoublychargedoxygenvacanciesareaccountableforthehigh-
a temperature n-type conduction under oxygen-poorconditions. The high-temperaturep-type conduction observed at
J largeoxygenpressuresisduetobariumvacanciesandtitanium-oxygendi-vacanciesunderTiandBa-richconditions,

8 respectively. Finally, the connectionbetween the presentapproachand the mass-action law approachto pointdefect
1 thermodynamicsisdiscussed.

]
i PACSnumbers:61.72.Ji71.15.Mb71.55.-i77.84.Dy
c
s
-
l I. INTRODUCTION In the presentcontributionwe demonstrateforthe case of
r
mt BaTiØ3 how theoretical data obtained from first-principles
calculations5canbeusedtoderivethedependenceoftheelec-
Pointdefectscontrolthefunctionalpropertiesofsemicon-
. tricalconductivityontheoxygenpartialpressure.Theelectri-
t
a ductors and many insulators, but are usually difficult to as- calconductivityisatechnologicallyhighlyrelevantproperty,
m sess experimentally. If bulk properties like conductivity or
the understanding of which is at the very foundation of de-
diffusivityaremeasured,itisnecessarytointroducemodelas-
- vicetechnology.Thesuccessfulmodelingofthispropertyde-
d sumptionsinordertorelatethemeasuredmacroscopicquan-
scribedinthisworkprovidesthebasisforfutureworkwhich
n tities to microscopic point defect properties. In contrast, lo-
o should address, e.g., the role of kinetic effects, extrinsic de-
cal probes such as electron spin resonance or positron anni-
c fects,ordefectassociation.
hilation spectroscopy can provide very specific information
[
onpointdefectstructuresbutareusuallyrestrictedtocertain Bariumtitanateisaprototypicalferroelectricmaterialwith

1 electronicconfigurations(unpairedspins) ortypesofdefects a paraelectric-ferroelectric transition temperature of 393K.
v (open volumes, vacancies). In general, only by combining Its most important technological application is in thin-film
53 severalexperimentalprobesaconsistentdescriptionofthede- capacitors.8 In addition BaTiO3 servesas an end memberin
8 fectchemistryinagivenmaterialcanbeobtained.1Tocompli- several lead-free ferroelectric alloys,9 and is used—often in
3 cate thingsfurther, the correlationbetween defectproperties combinationwithSrTiO3—toobtaintunableRFdevices.10,11
. and any experimentally measured response is typically indi- Because of its technological importance it has been exten-
1
rectandoftenpronetoambiguities. sively investigated, and a reliable as well as extensive data
0
baseisavailable(seee.g.,Refs.12–20). BaTiO istherefore
2 Computationalmodelingtechniquesevolvedrapidlyinre- 3
1 notonly a very interesting material for theoreticalinvestiga-
cent years, in particular in the realm of first-principles cal-
: tions,butitprovidesalsoanexcellenttestbedforcarryingout
v culations. Amongthese methodsschemesbased on density-
a stringentcomparison between calculated and experimental
i functional theory (DFT) are extremely popular as they have
X data.
becomeincreasinglyreliableandhavebeenshowntobecapa-
r
bleofpredictingvariousmaterialsproperties.First-principles In the following we first introduce the thermodynamic
a
modelingisparticularlyattractivewithregardtopointdefects. framework and the relevant equations of semiconductor
Itallowstoobtaindetailedinformationaboutthermodynamic physics. Combiningtheseequationsweareabletodetermine
and kinetic properties (formation energies and volumes, mi- self-consistently for a given temperature and a given chem-
grationbarriersandentropies)aswellastheelectronicstruc- ical environment(1) the electron chemical potential, (2) the
turewhich—atthislevelofdetail—arenotavailablethrough pointdefectconcentrations,(3)the chargecarrierconcentra-
experimentaltechniques(seee.g.,Refs.2–5). However,only tions,andeventually(4)theelectricalconductivity.Insection
inveryfewcasescalculationsofpointdefectpropertieshave Sect.IIItheresultsofthecalculationsarecomparedtohigh-
beenemployedtoderivemacroscopicallymeasurablequanti- temperatureexperimentaldataandthedependenceofthecon-
tiessuchasconductivitiesordiffusivities(seee.g.,Refs.6and ductivityonthe oxygenpartialpressureis analyzedin terms
7). It is however instrumental to develop these connections of the point defect equilibria in the material. The model is
betweencalculationandexperimentinordertoverifytheun- subsequentlyemployedto extrapolatethe materialsbehavior
derlyingmethodsandtoestablishtheirpredictivepower. tolowertemperatures,whereexperimentalmeasurementsare
1
nolongeravailable. Theintrinsicchargecarrierconcentrationsareobtainedby
integratingthenumberofunoccupiedstatesuptothevalence
band maximum (VBM) and the number of occupied states
II. THERMODYNAMICFORMALISM abovetheconductionbandminimum(CBM),
∞
A. Gibbsfreeenergyofpointdefectformation ne(µe)= D(E)f(E,µe)dE (5a)
Z
CBM
VBM
The Gibbs free energy of formation of a point defect in n (µ )= D(E)[1−f(E,µ )]dE, (5b)
h e e
charge state q can be consistently derived from thermody- Z−∞
namic principles and depends on the chemical potentials µi where f(E,µ ) = {1 + exp[(E−µ )/k T]}−1 is the
oftheconstituents(“thechemicalenvironment”)andtheelec- e e B
Fermi-Diracdistribution.
tronchemicalpotentialµ asfollows21,22
e The concentrations of point defect induced carriers are
obtained by summing the concentrations of acceptors and
∆Gd =(G −G )− ∆n µ +q(E +µ ),
def host j j VBM e donors,
Xj
(1) donors ∆Gd
n = eq c0exp − i (6a)
D i i (cid:18) k T∗(cid:19)
whereGdef andGhost aretheGibbsfreeenergiesofthesys- Xi B
temwithandwithoutthedefect,respectively. Thedifference acceptors ∆Gd
in the numberof atomsoftype i betweenthese two systems n = eq c0exp − i , (6b)
A i i (cid:18) k T∗(cid:19)
isdenoted∆nj andthesumrunsovertheelementspresentin Xi B
the system. It is convenientto separate the chemical poten-
whereeistheunitofchargeandq isthechargestateofde-
tialintothechemicalpotentialofthegroundstateµ0 andthe i
j fect i. Additional charge carriers contributed by dopants or
variationrelativetothegroundstatechemicalpotential∆µ ,
j impurities(“accidentaldopants”)canbesimplyaddedton
i.e. A
andn ,respectively.Theirconcentrationsaregivenby
D
µj =µ0j +∆µj. (2) neDxt =eqDc0D,ext[1−f(EG−ED,µe)] (7)
Finally, the position of the valence band, EVBM, defines the neAxt =eqAc0A,ext[f(EA,µe)], (8)
referenceoftheenergyscalefortheelectronchemicalpoten-
where E and E are the donor and acceptor equilibrium
tial,µ . D A
e transitionlevelsmeasuredwithrespecttotheconductionband
Knowledge of the defect formation energy allows to cal-
minimum(CBM)andthevalencebandmaximum(VBM),re-
culate the defect concentrationas a function of temperature, spectively, and c0,ext and c0,ext are the impurity concentra-
whichisgivenby D A
tions.
Mathematically, it is possible that there is more than one
∆Gd
cd =c0exp − i , (3) solution of the chargeneutrality condition Eq. (4). This can
i i (cid:18) k T (cid:19)
B occur e.g., if some defect concentrations change as a func-
wherec0 denotesthenumberofsitesavailablefordefectson tionoftemperaturewhereassomeothersareheldconstant.In
i suchacasethesolutionwiththelowestGibbsfreeenergyis
therespectivesublattice pervolume(e.g.,thedensityofbar-
selected.ThedifferenceoftheGibbsfreeenergywithrespect
iumsitesinthecaseofbariumvacancies).
to the equivalentdefect-freereferencesystem is obtainedby
summing the formationenergies of all defects in the system
minustheconfigurationalentropy,
B. Chargeneutralitycondition: theintrinsicelectron
chemicalpotential
∆G≈ N ∆Gd−k T lnΩ, (9)
i i B
Xi
The electron chemical potential µ , which appears in
e whereN denotesthe numberof defectsof typei and Ω de-
Eq.(1),isactuallynotafreeparameterbutfixedbythecharge i
notes the number of possible configurations(compare chap-
neutralitycondition1,
ter3.3ofRef.1).
n +n =n +n , (4)
e A h D
which links the concentration of intrinsic electrons n and C. Electricalconductivity
e
holes n to the concentration of charge carriers induced by
h
acceptorsnA and donorsnD. As discussed in the following Theelectricalconductivityisobtainedbysummingoverall
eachterminEq.(4)isexponentiallydependentontheelectron mobilechargecarryingspecies1
chemicalpotentialµ . FindingasolutionofEq.(4)therefore
e
yields the intrinsic (self-consistent) chemical potential for a σ = Biciqie (10)
giventemperatureandchemicalenvironment(Sect.IID). Xi
2
whereBiarethemobilities,ciaretheconcentrationspervol- ∆µBa (eV) Ba−richer
ume, and qi are the charge numbers. Typically one distin- −18 −15 −12 −9 −6 −3 0
guishestheelectronicandioniccontributionsσ =σ +σ . 0
el ion
AccordingtoEq.(10)theformerissimply B A
TiO −3
2
σ =B n e+B n e (11)
el e e h h
−6
electrons holes er
wthhaetrtheencehaarngdennehuatrraeligtyivce|onn{dbziyt}ioEnqs(|.4)({5izs)fu}unlfidlelredth.eChcoarngsetrcaainr-t O−richer −9 Ti−rich
riermobilitiessubsumethecontributionsofallpossiblescat- C −12
teringmechanisms—mostimportantlydefectsandphonons— BaO V)
asonrdtatoreetxhpereerfimoreenvtaelryddatiaffiicnuslttetaod.caFlcourlathtee. eAletcptrreosnenmtowbeilriety- D −15µ (eTi
∆
weusetheexpressiongiveninRef.14whichisafittosingle
−18
crystaldatafromSeuterusingtheexpressiongivenbyIhrig23
cm2K3/2 0.021eV
B =8080 ·T−3/2·exp − . (12) FIG.1. PhasediagramforcubicbariumtitanateatzeroKelvinas
e s (cid:20) k T (cid:21) determinedfromdensityfunctionaltheorycalculations(Ref.5).The
B
areaconfinedbetweenpointsA,B,CandDisthechemicalstability
For the hole mobility we follow Ref. 14 and assume Bh ≈ rangeofBaTiO3.ThechemicalpotentialsconfinedtothelinesA–D
Be/2. andB–CarereferredtoasBaandTi-richinthetext,respectively.
In order to obtain the ionic conductivity one can use
the Einstein-Smoluchowski relation B = eD /k T to re-
i i B
place the defect mobility with the defect diffusivity D =
D0exp(−∆Gmi /kBT)whichyields i wFuhretrheer∆cHonfs[tBraaiTntisØr3e]suislttfhreomforthmeatfioornmeantieorngyofocfoBmapTeitOin3g.
σ = qie2D0ci exp −∆Gmi . (13) phases,namely
ion
k T (cid:18) k T (cid:19)
Xi B B
∆µ +∆µ ≤∆H [BaØ] (15a)
Ba Ø f
Forcubiccrystalsthe pre-factorisD0 = 6Γ0a20 where a0 is ∆µTi+2∆µØ ≤∆Hf[TiØ2]. (15b)
thelatticeconstantandΓ0istheattemptfrequency.Thelatter
canbeapproximatedbythelowestopticalphononfrequency
whichyields24 Γ0 ≈ 5THzandD0 ≈ 10−3cm/s2. Themi- Ifallofthese restrictionsareincluded,oneobtainsthe static
phase diagram for T=0 K depicted in Fig. 1. The outer tri-
grationenergiesforintrinsicvacancieshavebeenreportedin
angle follows from condition (14) while the lines separating
Ref. 5. Since both the migration entropy and the migration
theBaTiO ,TiO andBaOphasesresultfromEqs.(15). The
volumeareaboutafactorofmagnitudesmallerthanthe for- 3 2
grayshadedareaisthe(zeroKelvin)stabilityrangeofBaTiO
mation entropy and volume, they can be safely neglected in 3
the presentcase, i.e., we canassume ∆Gm ≈ ∆Em. Using withrespecttoBaOandTiO2.
i i
these data it is found that for the present material the ionic Experimentally, the way to control the thermodynamic
contribution at elevated temperaturesis about four orders of boundaryconditionsistouseeitherBaOorTiO2 excessdur-
magnitudesmallerthantheelectroniccontribution.Inthefol- ingmaterialsprocessing,andtovarytheoxygenpartialpres-
lowingwethereforeconsidertheelectronicpartonly. sure, pØ2, during processing and measurements. Adjusting
the excessof either Ba or Ticorrespondsto constrainingthe
accessiblerangeofchemicalpotentialstothelinesA–D(Ba-
D. Phasestability:limitationsonthechemicalpotentials richlimit,equilibriumbetweenBaOandBaTiO )orB–C(Ti-
3
rich limit, equilibriumbetween TiO and BaTiO ) in Fig. 1.
2 3
The chemical potentials, µ = µ0 + ∆µ , which appear Varyingtheoxygenpartialpressure(i.e.,totheoxygenchem-
j j j
in Eq. (1), are subject to severalthermodynamicconstraints. icalpotential)isequivalenttomovingalongtheselineswhere
First,theycannotbecomemorepositivethanthechemicalpo- theextremalpointsAandBonone,andCandDontheother
tential of the reference phase, i.e. ∆µj ≤ 0, where the ref- sidecorrespondtometal-rich(lowpØ2)andoxygen-rich(high
erence phase for oxygen is the Ø2 molecule, for barium the pØ2)conditions,respectively.
body-centeredcubic crystal, and for titaniumthe hexagonal- WhiletheformationenergiescalculatedinRef.5aregiven
close packed crystal. If any chemical potential reaches its as a function of the chemical potentials, experimentally the
upperlimit, the respectiveelementalgroundstate phase pre- conductivityis measured as a function of the oxygen partial
cipitates. Second, the chemicalpotentialsofthe constituting pressure. Inordertocomparetheconductivitiesascalculated
elementsarecoupledbytherequirementthat5 fordifferentchemicalpotentialswith experimentaldata, one
mustthereforeconvertbetweentheoxygenchemicalpotential
∆µBa+∆µTi+3∆µØ =∆Hf[BaTiØ3], (14) andtheoxygenpartialpressure.Thetwoquantitiesarerelated
3
accordingto25,26
(a)
−1/6
µ (T,p )=µ (T,p0)+ 1k T ln pØ2 (16) 100
Ø Ø2 Ø 2 B (cid:18) p0 (cid:19)
−1m) 10−1 1273 K
where p0 denotes the reference pressure. We choose the c 1473 K
Ø2 1
isolatedoxygendimermoleculeasthezeroKelvinreference −Ω 10−2
ismtaeten,taµl0Ød(a0taKa,npd0)fo=llo21wEinØg2.RFeof.r2c5onwsiestuesnecythweiethxptehreimexepnetar-l vity ( 10−3 +1/4
tvearlTmuheinefeopdrhtEaesmØe2pde=iraag−trua5rme.1d6ienepVeFn/igdd.eimn1ceeirsoasfntrdµicØtht(leyTe,vxpap0Ølie2dr)imo(nRelneyfta.al2tly7z)de.reo- Conducti 10−4 ne−x1t /=4 1018 cm−3 1073 K +1/6
A
temperature. At finite temperatures the construction would 10−5 next = 1017 cm−3
A
have to be based on the free energies of formation instead. next = 0 cm−3 (pure)
The major effect arises from the differencesbetween the vi- 10−6 A
brationalentropiesbetweenthevariousrelevantphases. With (b)
−1/6
theexceptionofoxygenallthese phasesarecrystalline. The 100
entropiesofcrystallinesolidsare,however,muchsmallerthan
1273 K
theentropiesofgases. Bythefarthemostimportanttermis 10−1
therefore the change of the free energy of the oxygen reser- −1) 1473 K
m
voir, which is properlytaken into accountvia Eq. (16). It is c 10−2
1
thereforeadmissibletousethephasediagramestablishedhere −
Ω
alsoatfinitetemperatures. y ( 10−3
vit −1/4
ucti 10−4 1073 K
d
E. Summaryofalgorithm n
o
C 10−5 Daniels and Härdtl (1976)
In summary computing the conductivity proceeds as fol- Eror and Smyth (1978)
lows: (i) The electron chemicalpotential is self-consistently 10−6 Chan, Sharma and Smyth (1981)
determinedasdescribedinSect.IIBforafixedsetofatomic Song and Yoo (1999)
chemicalpotentials[seeSect.IID].(ii)Theconcentrationsof 10−7 10−20 10−15 10−10 10−5 100
theintrinsicchargecarriersandtheintrinsicdefectsareeval-
Oxygen partial pressure (atm)
uated using Eqs. (5) and (6). (iii) The conductivityis calcu-
lated as described in Sect. IIC and the oxygenchemical po-
FIG. 2. (Color online) (a) Calculated conductivity as a function
tentialisconvertedtoanoxygenpartialpressureaccordingto of oxygen chemical potential along the lineA–D (i.e., for barium-
Eq.(16). Inthefollowingweexplicitlyassumethatthemate- richconditions) inFig.1. Theresultsintheabsence of any impu-
rialisalwaysabletoreachequilibrium,whichisareasonable rities(nA = 0cm−3) areshown by dotted lines; solidand dashed
assumption at elevated temperatures. Further computational lines correspond to acceptor doping levels of nA = 1017cm−3
detailsaregivenintheappendix. and nA = 1018cm−3, respectively. (b) Comparison of calcu-
lated (thick solid lines) and experimentally measured conductivity
curves(thinlinesandsymbols). ExperimentaldatafromRefs.12–
15.Forthecalculationsin(b)anaccidentalacceptordopinglevelof
III. RESULTSANDDISCUSSION
nA =1017cm−3wasadopted.
A. Equilibriumconductivityatelevatedtemperatures
inFig.2. Allcurvesdisplaytheshapecharacteristicforatran-
Wehaveimplementedthemodeldescribedaboveandused sition fromn-type(negativeslope)to p-type(positiveslope)
the formation energies from Ref. 5. A band gap of EG = conduction.
3.0eV was employed which is 0.4eV smaller than the zero
Kelvin-extrapolatedbandgapmimickingtheshrinkingofthe
band gap with increasing temperature. The value of 3.0eV
1. Undopedmaterial
liesbetweenthevalueobtainedbytemperaturescalingofthe
bandgapreportedbyWemple,28 whichyieldsapproximately
2.8eV at 1400K, and the values for the band gap discussed Firstweconsideranideallypurematerial(next = 0cm−3)
A
by Chan et al.14, which range between 3.0eV and 3.4eV. under Ba-rich conditions (i.e., for chemical potentials along
Both undoped and weakly (“accidentally”) doped materials A–DinFig.1)forwhichthethindottedlinesinFig.2areob-
(next =1017cm−3and1018cm−3)wereconsidered.Thecal- tained. Throughoutthen-typeregiona slopeof−1/6isob-
A
culatedequilibriumconductivityasafunctionoftemperature, served[comparethedashedlinesegmentsinFig.2(a)],which
impurityconcentrationandoxygenpartialpressureis shown changesto+1/6inthep-typeregion. Analysisofthedefect
4
1019
(a) pure, 1073 K, Ba−rich 3.0
1018 n V) pure 1473 K
−3cm) 11001167 h ential (e 22..05 nnAeAexxtt == 11001178 ccmm−−33 1607733 K K
centrations ( 111000111345 [VTi−VO]2− VnOe2+ chemical pot 11..05 at high T at 0 K
Con 1012 V 4− ctron 0.5 d gap d gap decreasing T
Ti e n n
1011 El 0.0 ba ba increasing nAext
1010
10−20 10−15 10−10 10−5 100 105
1019
(b) next = 1017 cm−3, 107 3 K, Ba−rich Oxygen partial pressure (atm)
1018 A
3) 1017 VO2+ nh FwIiGth.b4o.th(tCemolpoerroantulirneea)nVdairmiaptiuornityofctohneceenletrcatrtioonn.chTehmeiccuarlvpeostemnotivael
−cm 1016 ne downward, i.e. tomorep-typeconditions, withdecreasingtemper-
ns ( 1015 aintudriecaatestwheelrleadsucinticorneaosfinthgeibmapnudrigtyapcaotnhciegnhtrearttieomnsp.erTahtueregsr.ay bars
o
ntrati 1014 [VTi−VO]2−
ce 1013 In the p-type region (V − V )′′ di-vacancies dominate,
n Ti Ø
Co 1012 leading to a slope of +1/6. The latter can be derived using
VTi4− the mass-action law approach as follows. The point defect
1011 reaction
1010
10−20 10−15 10−10 10−5 100 105 1 ′′ ·
BaØ+ Ø2 ↔(VTi−VØ) +2h +BaTiØ3
Oxygen partial pressure (atm) 2
leadsto
FIG. 3. (Color online) Charge carrier and defect concentrations
f1o0r17(ac)mp−u3re).and (b) accidentally acceptor doped material (neAxt = KII =[(VTi−VØ)′′]n2hpØ−21/2 (19)
whichusingthesimplifiedchargeneutralitycondition[(V −
Ti
concentrations[Fig.3(a)]showsthatinthen-typeregiondou- VØ)′′]=2nhyields
blychargedoxygenvacanciesarethedominantdefectswhich
givesrisetoaslopeof−1/6.Thisslopecanalsobederivedif n ∝p+1/6. (20)
h Ø2
onetreatsthepointdefectequilibriainthematerialusingthe
mass-actionlawapproach.14,29 Startingfromthepointdefect For clarification it should be pointed out that if the con-
reaction ductivityisplottednotagainsttheoxygenpartialpressurebut
againsttheoxygenchemicalpotentialtheminimaalloccurat
1
ØØ ↔VØ··+2e′+ 2Ø2, tihneFsigam.2eicshtehmusicmalepreoltyenaticaol.nsTehqeuegnracdeuoafltshheiftteomfptehreatmurineidmea-
oneobtainsamass-actionlawwhichlinkstheoxygenpartial pendenceof the relation between the chemical potentialand
pressure,p totheconcentrationofdoublychargedoxygen theoxygenpartialpressuredescribedbyEq.(16).
Ø2
vacancies[V··],andtheconcentrationofelectronsn The self-consistentlydetermined electron chemicalpoten-
Ø e
tial is shown in Fig. 4. At low oxygenpartialpressures it is
K =[V··]n2p1/2. (17) located in the upper half of the band gap corresponding to
I Ø e Ø2 n-type material whereas for larger oxygen partial pressures
ApplicationoftheBrouwerapproximation(onedefectdomi- theelectronchemicalpotentialresidesinthelowerhalfofthe
natesthechargeneutralitycondition,Eq.(4))fortheoxygen bandgap.Figure4showsthatforanideallypurematerialthe
vacancies[V··]=2n thenyields pressure dependenceof the electron chemical potential does
Ø e
notchangewithtemperature. Thisobservationcontrastswith
n ∝p−1/6. (18) thetemperatureinducedshiftoftheminimumoftheconduc-
e Ø2 tivitycurves[Fig.2(a)].
whichbecauseofEq.(11)leadstothesameslopeinthecon- Thetemperatureandpressuredependenceoftheconductiv-
ductivity. ity as well as the electron chemicalpotential obtainedunder
5
Ti-richconditions(i.e.,forchemicalpotentialsalongtheline to the remarkable finding that for a certain range of oxygen
B–C in Fig. 1) very closely resemble the results under Ba- partialpressures,theelectronchemicalpotentialmovesfrom
richconditions. Inthen-typeregiondoublychargedoxygen theuppertothelowerhalfofthebandgapasthetemperature
vacanciesareagaintheprimarydefects. Inthep-typeregion, islowered,indicatingatransitionfrontop-typeconduction.
however,doublychargedbariumvacanciesdominate.Follow- If the dopant concentration is further raised (thin dotted
ingasimilarderivationasfortheVTi−VØ di-vacancies(see lines in Fig. 2, neAxt = 1018cm−3) extrinsic acceptors dom-
e.g.,Ref.14)oneagainobtainsaslopeof+1/6. inateovertheentirerangeofchemicalpotentialsandaslope
Asdemonstratedabovetheuseofthemass-actionlawand of −1/4 (+1/4) is obtained throughout the n-type (p-type)
the Brouwer approximationallow to deducethe slope of the region. Thetemperaturedependenceoftheelectronchemical
curvesforsituationsinwhichonedefectdominates. Suchan potentialcurves(Fig.4)isevenmorepronouncedthaninthe
approachis,however,boundtofailinanytransitionregionor caseofnext =1017cm−3.
A
inregionswhereseveralpointdefectshavesimilarconcentra- TheentiresituationisverysimilarifTi-richconditionsare
tions. This is for example the case near the crossings of the imposed, the major differencebeing again the occurrenceof
linesinFig.3. Thislimitationisavoidedbyusingthefullap- barium vacancies instead of V −V di-vacanciesin the p-
Ti Ø
proachoutlinedinSect.IIwhichfurthermoredoesnotrequire typeregion.
anypresumptionswithregardtotheprevalenceofanypartic-
ular defect reaction. In addition it allows to obtain the con-
centrationsofsecondarypointdefectswhichalthoughtheydo B. Comparisonwithexperiment
notaffectthechargecarrierconcentrationsstillcanimpactthe
electronicpropertiesthroughcarrierscatteringandtrapping.
Theconductivityofbothnominallyundopedaswellasin-
tentionally doped barium titanate has been repeatedly mea-
sured as a function of oxygen partial pressure and at ele-
2. Weaklyacceptordopedmaterial vated temperatures.8,12–15,30,31 These studies provide a com-
prehensive data set for comparing our calculations with ex-
If under Ba-rich conditions a low concentration of accep- periment.InFig.2(b)theresultsofseveralmeasurementsare
torsispresentinthematerial,oneobtainstheboldsolidlines plottedtogetherwiththecurvescalculatedforadopinglevel
in Fig. 2 (neAxt = 1017cm−3). For low oxygen partial pres- ofneAxt = 1017cm−3. Theagreementis verygood. Thecal-
suresandhightemperaturestheyhaveaslopeof−1/6inthe culationsreproducethen-type/p-typetransition,thetempera-
n-type region just as in the undoped material. Again this is turedependenceofthe positionof theminimaaswellas the
due to positively charged oxygen vacancies as illustrated in changesintheslopes.
Fig. 3(b). As the temperature is lowered and/or the oxygen In order to explain the experimentally observed transition
partial pressure rises, a transition to a slope of −1/4 is ob- within the n-type regionfrom a slope of −1/6to a slope of
served. As shown in Fig. 3(b) this change corresponds to −1/4,twodifferentmodelshavebeendiscussed:(i)Themost
the onset of extrinsic behavior, i.e. the dominant source for earlystudiesproposedthetransitiontoberelatedtoachange
holesarenolongerintrinsicbutextrinsicdefects. Inthiscase of the charge state of the oxygen vacancy.12 The measure-
thesimplifiedchargeneutralityconditionreads[V··] = 2next mentscould be reproducedusing a modelin which the oxy-
Ø A
whichif insertedinto(17) alsoleadsto a slopeof +1/4. At genvacancy+1/+2transitionlevelislocatedabout1.3eV
higher oxygen partial pressures and higher temperatures the belowtheCBMandthusveryclosetothecenteroftheband
slope changes to +1/6—as in the ideal case—indicating in- gap.(ii)Moststudies(seee.g.,Refs.13,14,31,and32),how-
trinsicbehavioranddominanceofV −V di-vacancies. In ever,assumethateveninthemostcarefullypreparedsamples
Ti Ø
contrast, at lower temperatures a slope of +1/4 is observed a backgroundconcentrationof “accidental”acceptorimpuri-
whichisconsistentwithextrinsicbehavior.14 ties is present which gives rise to the transition between the
As shown in Fig. 4 the electron chemical potential as a slopes.
function of the oxygen partial pressure again displays a re- ThepresentcalculationsinconjunctionwiththeDFTdata
duction of the slope for pressures &10−3atm which results from Ref. 5 providevery strong evidence for the second ex-
fromthecouplingoftheconcentrationsofintrinsicholesand planation. In order to obtain furthersupportfor this picture,
extrinsicdefectswhichareofsimilarmagnitudeinthisrange. we artificially pushed the +2/+1 transition level, which is
Forevenlargeroxygenpartialpressures[outsidetherangeof locatedjust0.05eVbelowtheconductionbandminimum,to-
Fig.2butvisibleinFig.3(b)],theconcentrationofV −V wardthethemiddleofthebandgapbyreducingtheformation
Ti Ø
di-vacanciesexceedstheconcentrationofextrinsicacceptors energy of the singly charged oxygen vacancy. The thus ob-
andtheslopeoftheconductivitycurvesrevertsto+1/6. The tainedconductivitycurvesdoindeeddisplayatransitionfrom
transitionfromn-typetop-typeconductioncorrelateswitha σ ∝ p−1/6 toσ ∝ p−1/4. However,inordertoreproduceat
Ø2 Ø2
significantvariationoftheelectronchemicalpotentialoverthe least approximately the experimental data the formation en-
bandgap. Incontrasttothe caseofanideallypurematerial, ergyofV· hadtobereducedbyabout1eVwhichissignifi-
Ø
material which contains extrinsic defects exhibits a marked cantlylargerthantheerrorbaroftheDFTcalculations.
temperaturedependenceoftheelectronchemicalpotentialvs Thep-typeregionwithinwhichtheslopeispositiveisdom-
pressure curves. As the temperature is reduced the electron inatedbyacceptordefects.Sinceexperimentallyoneobserves
chemicalpotentialcurvesarepusheddownwards,whichleads a slope of about +1/6 in this region, it has been widely as-
6
1500 regions within which a certain defect prevails. The concen-
(a) Ti−rich
1400 tration of the dominantdefects is shown by the dashed lines
1300 1018 cm−3 1018 cm−3 alongwhichtheconcentrationisconstant.
K) The diagrams can be read as follows: Assume a mate-
e ( 1200 rialissynthesizedat1400KunderBa-richconditionsandan
atur 1100 oxygen partial pressure of 10−2atm [point A in Fig. 5(b)].
mper 1000 1T0h1e6mcmat−er3i.alUcnodnetraitnhseasnynatchceespistocroinmdiptuiornitsytchoendcoemntirnaatinotndoef-
e 900
T fect is the oxygenvacancyin chargestate +2. The material
V 2+
800 O is subsequently annealed at a temperature of 1200K while
V 2− 1015 cm−3 keeping the oxygen partial pressure at 10−2atm [point B in
700 Ba
Fig.5(b)]. Undertheseconditionsthedominantdefectisthe
1500 V −V di-vacancy. If the material is cooled below about
(b) Ba−rich Ti Ø
1400 A 1000Kthedottedlinein Fig. 5(b)indicatesthattheconcen-
trationofV −V di-vacanciesfallsbelow1016cm−3[point
K) 1300 1018 cm−3 C in Fig. 5T(ib)]. ØTherefore, at temperatures below 1000K
e ( 1200 B theaccidentalacceptordopantsdominatethe chargeequilib-
atur 1100 riumandthematerialdisplaysextrinsicbehavior[pointDin
per 1000 C Fig. 5(b)]. It is important to point out that this analysis is
m D strictlyvalidonlyinthermodynamicequilibrium.
e 900
T
V 2+
800 O 1015 cm−3
700 [VTi−VO]2− VTi4− IV. SUMMARYANDCONCLUSIONS
10−15 10−10 10−5 100 105
Oxygen partial pressure (atm) By combining thermodynamic considerations and several
basic relations of semiconductor physics we have obtained
FIG.5. (Color online) “Phase diagram” illustratingthe prevalent a concise scheme for the modeling of electrical conductivi-
defectsasafunctionoftemperatureandoxygenpartialtemperature. ties on the basis of first-principles calculations. Compared
Thethicksolidlinesseparatetheregionswithinwhichtheindicated to “classical” defectmodelswhichare based onmass-action
defectsdominate. Thedottedanddashedlinesconnectpointsalong lawstoconnectdefectconcentrations,thepresentschemere-
whichtheconcentrationofthedominantdefectisconstant(dash-dot- quiresaminimumnumberofapproximations.Inparticular,it
dot: 1018cm−3,dash-dot: 1017cm−3,dotted: 1016cm−3,dashed: doesnotrelyonanyassumptionswithregardtotheprevalence
1015cm−3).
ofanyparticulardefectreactionordefect.
We have applied this scheme to BaTiO , which is impor-
3
tantbothfromthetechnologicalandthefundamentalperspec-
sumedthatbariumvacanciesinchargestate−2areresponsi-
tive, using a complete set of thermodynamic data on intrin-
ble for this behavior. If one considers single vacancies only
sicpointdefectsobtainedfromdensity-functionaltheorycal-
thealternativeintrinsicacceptordefectwouldbethetitanium
culations. A numerical algorithm was implemented for the
vacancy,whichoccursinchargestate−4andthuswouldlead
self-consistentdeterminationof theelectronchemicalpoten-
to a slope of +1/5. The discussion above, however, shows
tial,whichenabledanextensiveanalysisofthedependenceof
that at least under Ba-rich conditionsthe dominant defect is
theelectricalconductivityonchemicalenvironmentaswellas
the V − V di-vacancy, which—equivalent to the barium
Ti Ø temperature.
vacancy—givesrise to a slope of +1/6. Thus, on the basis
In agreement with earlier experimental studies our analy-
of the conductivitycurvesalone the intrinsic acceptordefect
sishasshownthatthen-typeconductivity,whichisobserved
cannotbedeterminedunambiguously.
underlowoxygenpartialpressures,isduetodoublycharged
oxygen vacancies. The p-type region, which is observed at
largeroxygenpartialpressure,iscausedbybariumvacancies
C. Equilibriumdefectconcentrations and V −V di-vacancies under Ti and Ba-rich conditions,
Ti Ø
respectively. It needs to be stressed that since both of these
Thesuccessfulvalidationofourcalculationsthroughcom- defectsoccurinchargestate−2andthereforeleadtoaslope
parison with experimentaldata demonstrates the capacity of of +1/6 in the conductivity vs oxygen partial pressure plot,
DFT calculations and allows to use the present calculations theycannotbedistinguishedonthe basisoftheconductivity
for obtaining a more detailed picture of the thermodynami- curvesalone.
cal behavior of point defects in this material. We can thus Ourapproachfurthermoreallowsustodeterminetheevo-
determine the prevalent intrinsic point defects depending on lution of the defect concentrations under “true” equilibrium
temperatureandoxygenpartialpressure.Theresultsobtained conditions,i.e.intheabsenceofanykineticbarriers.Wehave
foranideallypurematerialareshownforBaandTi-richcon- employedthispossibilitytoestablishapointdefect“phasedi-
ditions in Fig. 5. In this figure thick solid lines confine the agram”whichdisplaysthedominantintrinsicpointdefectas
7
afunctionoftemperatureandchemicalenvironment. ∆E = ∆Eerr, ∆E = 0. Thischoicehasneitheranim-
CB G VB
The successful application of the scheme outlined in the pactonthelocationoftheequilibriumtransitionlevelsnoron
present paper demonstrates the predictive power of first- theconclusionsinRef.5. Thevaluesof∆E and∆E do,
VB CB
principlescalculations. It also constitutes the stepping stone however,affecttheabsolutevaluesoftheformationenergies
for future work which should address the effects of kinetic andarethereforeimportantinthepresentwork.
barriersandimplementamorecomplextreatmentofextrinsic Byexplicitcalculationonecanshowthattheeffectofshift-
defects. ing ∆E versus ∆E is equivalent to rigidly shifting the
VB CB
conductivity curves in Fig. 2 (details below) along the pres-
sureaxis. Neithertheshape,theslopes,northemagnitudeof
ACKNOWLEDGMENTS thesecurvesareaffected.Wehavethereforedecidedtoadjust
the ratio of ∆E and ∆E such that the minimum of the
VB CB
This project was funded by the Sonderforschungsbere- conductivity at the highest temperature considered (1473K)
ich 595 “Fatigue in functional materials” of the Deutsche islocatedatthesameoxygenpartialpressureasintheexper-
Forschungsgemeinschaft. iments. Thefinalvaluesare∆EVB = 0.45∆EGerr = 0.76,eV
and∆E = 0.55∆Eerr = 0.92eV, whichare actuallyofa
CB G
veryreasonablemagnitude, consideringthata rule of thumb
formanysemiconductorsisaratioof2:1for∆E :∆E . A
AppendixA:Computationaldetails CB VB
shiftof 0.1eVin eitherdirectionamountsto a shift in along
the pressureaxisby10±2atm. Alldata discussedin the fol-
1. Generalremarks
lowing was obtained using the values for ∆E and ∆E
VB CB
quotedabove.
The density functional theory (DFT) calculations5 from
which we obtain our input data provide values for the ener-
giesofformationE 33. Inordertoobtainthefreeenergiesof
i
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9