Table Of ContentCorrelation-driven charge and spin fluctuations in LaCoO
3
M. Karolak,1 M. Izquierdo,2,3,4 S. L. Molodtsov,2,5,6 and A. I. Lichtenstein2,3
1Institut fu¨r Theoretische Physik und Astrophysik,
Universita¨t Wu¨rzburg, Am Hubland, 97074 Wu¨rzburg, Germany
2European XFEL GmbH, Albert-Einstein-Ring 19, 22761 Hamburg, Germany
3I. Institut fu¨r Theoretische Physik, Universita¨t Hamburg, Jungiusstraße 9, 20355 Hamburg, Germany
4Synchrotron Soleil, L’Orme des Merisiers St-Aubin, BP-48, 91192, Gif-sur-Yvette, France
5Institute of Experimental Physics, Technische Universit¨at Bergakademie Freiberg, 09599 Freiberg, Germany
5 6ITMO University, Kronverkskiy pr. 49, 197101 St. Petersburg, Russia
1
(Dated: January 15, 2015)
0
2 The spin transition in LaCoO3 has been investigated within the density-functional theory +
dynamical mean-field theory formalism using continuous time quantumMonte Carlo. Calculations
n
a on the experimental rhombohedral atomic structure with two Co sites per unit cell show that an
J independent treatment of the Co atoms results in a ground state with strong charge fluctuations
inducedbyelectroniccorrelations. Eachatomshowsacontributionfromeitherad5 orad7 statein
4
1 addition to themain d6 state. These states play a relevant role in thespin transition which can be
understoodasalowspin-highspin(LS-HS)transitionwithsignificantcontributions(∼10%)tothe
] LS and HS states of d5 and d7 states respectively. A thermodynamic analysis reveals a significant
l kinetic energy gain through introduction of charge fluctuations, which in addition to the potential
e
- energy reduction lowers thetotal energy of the system.
r
t
s PACSnumbers: 71.27.+a,71.30.+h,71.10.Fd,71.15.Mb
.
t
a
m The interpretationof the remarkably low temperature merically exact continuous-time quantum Monte Carlo
- (∼100K) spin transition in LaCoO3 (LCO) has been an (CT-QMC) [11]. This methodology allows the inclusion
d
intriguing research topic for decades (see Ref. [1] for a of local dynamical effects and temperature, which is not
n
review). A long standing debate regarding the origin of possiblewithintheinherentlyzerotemperatureDFTand
o
c the transition is still ongoing with low-spin (LS, t62ge0g) DFT+U. In both cases, a LS-HS transition scenario re-
[ to high spin (HS, t4 e2) and low-spin to intermediate sults. NoevidenceofISconfigurationwasobtained,even
2g g
spin (IS, t5 e1) models with concomitant orbital order- upon going beyond the d6 ionic picture [8]. Thus, in-
1 2g g
v ing competing [2, 3]. The interest in this system has cluding d7 and d8 charged states to describe the local
4 been recently boosted by the potential applications of dynamicsofthesystem,hasallowedtointerpretthespin
9 this material and its doped phases in various optimized, state transitions as an LS (with few HS ions) to LS-HS
2
environmental-friendly energy production domains [4]. short range ordered phase, with a subsequent melting of
3
FromthetheoreticalpointLCOaddressesoneofthemost the orderathigher temperatures. Atroomtemperature,
0
. challengingquestionsinsolidstatetheory: thetreatment when 50% LS-HS population is expected, a Co(LS)-O-
1 of strongly correlated materials in which HS (IS) with- Co(HS)arrangementisanticipated. Mostoftheabinitio
0
outlongrangemagneticorderis stabilized[1]. Although many-body calculations discussed so far have been per-
5
1 the addition of static local correlations to the density- formedassumingequivalentCo atoms,withexceptionof
: functionaltheory(DFT+U)[5]improvedthedescription a couple of DFT+U studies [12, 13]. In a groundbreak-
v
i of correlated materials with long range magnetic order, ing study on a two band Hubbard model within DMFT
X
the more advanced formalism combining DFT with dy- KuneˇsandKˇr´apek[14]showedthatchargeimbalancebe-
r namicalmean-field theory (DMFT) [6] has to be used to tween sites can occur on purely electronic grounds and
a
accuratelytreatsystemsinaparamagneticstatewithlo- possibly consititues an important piece in the LCO spin
cal moments, like LCO [7, 8]. Many-body calculations state transition puzzle.
on LCO have been performed using DFT+DMFT [9] In this letter, we bolster this proposalby reporting on
and the variational cluster approximation [10]. In the the first ab initio study of correlation driven charge and
firstcase,thecontributionfromLS,HSandISwasstud- spin fluctuations in LCO within the DFT+DMFT for-
ied and the spin state transition described as a smooth malismforinequivalentCoatoms. Ourresultsshowthat
crossover from the homogeneous LS state into a non- symmetry breaking of the Co sites creates a correlation-
homogeneous mixture of all three spin states. On the driven charge and spin fluctuation of purely electronic
other hand, the VCA calculations showed that only the origin.
LS and HS states have appreciable weight in the density
LaCoO isaperovskitesystemshowingasmalltetrag-
3
matrix over a wide temperature range [10]. The most
onal distortion of the CoO octahedra that varies with
6
recent attempts to settle the spin transition in LCO by
temperature,asdeterminedfromneutrondiffraction[15].
means of the DFT+DMFT approach [7, 8] used the nu- To describe it we use a fully ab initio approach start-
2
ingfromdensity-functionalcalculationsofLCOusingthe
experimentally determined structures for 5K, 300K and
650K. We use a rhombohedral unit cell containing two
formula units. The VASP code [16] with projector aug-
mented wave basis sets (PAW) [17, 18] and the Perdew-
Burke-Ernzerhof [19] functional was employed for DFT
calculations. A Wannier type construction using pro-
jected local orbitals as described in [20, 21] was applied
toconstructalocallowenergymodel,whichcontainsthe
on-site energies (crystal field splitting) and hoppings ex-
tractedfromDFT.Wefocushereonthe3dorbitalsofCo
exclusively, avoiding the possible ambiguities that arise
from the use of d+p orbital models in DFT+DMFT,
see e.g. [22, 23]. Since the system is not perfectly cubic
a straightforwardWannier constructionproducesa basis
that retains some on-site mixing between the Co t and
2g
e orbitals. This local t -e hybridization is mostly a
g 2g g
consequence of this specific choice of orbital representa-
tion, therefore we have performed a unitary transforma-
tion after the Wannier projection to minimize it. The
usualchoiceshere arearotationintothe so-called“crys-
tal field basis” or into a basis that renders the DFT oc-
cupancy matrix ρ = hcˆ†cˆ i diagonal on each atom, see
ij i j
e.g. Refs. 24 and 25. The data presented were obtained
using latter approach. We have, however, explored both FIG. 1. (color online) Phase diagram as a function of the
lattice structure temperature and the Hundsrule coupling J
approachesandfoundthattheyyieldsimilaroff-diagonal
for U = 3eV. The colored (shaded) parts illustrate different
elements in the frequency dependent DFT Green’s func-
regions in the phase diagram, white indicating LS and blue
tion G0(iω ) and the choice has no qualitative effect on
n (gray) HS. (a) shows the homogeneous phase exhibiting the
any conclusions drawn in what follows. LS, MS and HS states as indicated by the data points. MS
On the level of DFT we find a set of three orbitals refers to a mixture of LS and HS states. (b) shows the data
verycloseinenergy(twoeπ andonea ,thatwewillfor pointsand possible phase regions for thecalculation with in-
g 1g equivalent Co atoms. The colored (shaded) areas are again
brevity call t ) and two orbitals (e ) higher in energy
2g g only meant as an illustration, the checkerboard pattern now
by about ∆∼1.5–1.65eV.We solvean effective multior-
indicating the LS-HSphase where strong charge fluctuations
bitalHubbardmodelwithinDMFTusingahybridization are present.
expansionCT-QMCsolverusingdensity-densityinterac-
tion terms. To treat the two Co atoms in the unit cell
independentlyweemploytheso-calledinhomogeneousor
we employ the parametrization via the Slater integrals
realspaceDMFT [26], wherewe haveto solveanAnder-
[27] connected to the average direct and exchange cou-
sonImpuritymodelforeachcorrelatedatomαintheunit
plings U and J: F0 = U, J = 1/14(F2 + F4) and
cellandtheatomseffectivelyinteractviatheirrespective
F4 = 0.625F2. The aforementioned basis transforma-
baths of conduction electrons. The generalisedfive band
tion was applied here as well.
Hubbard model including the Coulomb interaction and
a double counting correction then contains the following One important element to properly describe the sys-
terms temwithinDFT+DMFTistheappropriatechoiceofthe
Coulombinteraction. Inthe caseofLCO,theproblemis
moredelicatesinceaftertheLDA+Ucalculationsthatin-
Hˆ =Hˆ0− X µDαCnˆαm,σ+12 XUασiσj′nˆαi,σnˆαj,σ′, (1) ttrhoadtuicteidstahestLrSon-IgSlymcoodrerlel[a3t],editehlaesctbreoennswyisdteemlybweiltihevaedn
α,m,σ α,i,j
σ,σ′ on-site Coulomb interaction of U = 8eV. Recently, con-
strained DFT results proposed a value of U = 6eV for
whereHˆ0isthe(Kohn-Sham)DFTHamiltonian,nˆαi,σ = the Co 3d shell in a d+p model [8]. Such high values
cˆ† cˆ and cˆ† is the creation operator of an elec- for the Coulomb interaction might be appropriate for a
αiσ αiσ αiσ
tron on site α in Wannier state i and spin σ. The dou- d+ p model, are however too large for d only calcula-
ble counting, ∝ µDC, amounts to a shift of the chemi- tions. Since we are not aware of any ab initio estimates
α
cal potential for the Co 3d shell and is determined self for the Co 3d shell only, measurements of the excitation
consistently. To obtain the Coulomb interaction matrix gap in LCO have been used as a guide. A gap size of
3
J(eV) Co (N ) d6 d5 d7 d5 d7 d6
3d S=0 1/2 1/2 3/2 3/2 2
0.60 1 (6.0) 93 4 3 - - -
2 (6.0) 93 4 3 - - -
0.75 1 (6.1) 81 3 15 - - -
2 (5.9) - - - 12 3 82
0.90 1 (6.0) - - - 3 7 87
2 (6.0) - - - 3 7 87
TABLE I. Most probable many-body configurations for the
300K structurewithtwoinequivalentCoatomsasafunction
of the Hunds coupling J obtained from the analysis of the
CT-QMCimaginarytimeevolution(in%). Numbersmissing
toorsumsexceeding100% are duetominorcontributionsof
other atomic states and rounding.
about 0.9eV was measured from photoemission and ab-
sorptionspectra[28],whilesmallervalues,0.6eV[29]and
0.3eV[30]wereobtainedfromopticalmeasurements. We
were able to obtain a charge gap ∼1 eV with a value of
U = 3eV, which will be used in all subsequent calcula-
tions.
Since the value of the Hunds rule coupling J is, for
a fixed U and crystal field splitting ∆, the critical pa-
rameter for the spin state transition [31] we have cal-
culated the system at different values of J. To ac-
count for the temperature, the calculations were per-
formedfortheexperimentalcrystallatticestructuresde-
termined at the temperatures 5K, 300K and 650K [15].
Inthe QMC solverwe usedthe calculationtemperatures
116K, 290K, 580K (equivalent to the inverse tempera-
tures β = 100,40,20eV−1) for these structures respec-
tively. FIG. 2. (color online) Orbitally resolved spectral functions
In a first approximation, the two Co atoms were con- forLCOforthe300Kcrystalstructureatsolvertemperature
of290K(β=40eV−1). (a)ThehomogeneousHS(J =0.9eV,
strained to be in the same charge and spin state. The
top) and LS spectra (J =0.6eV, bottom) with atomic states
calculations show that for the three crystal structures
given as an inset. (b) Results in the asymmetric Co configu-
and their respective crystal fields a spin state transi- ration for the values of J indicated. The largest and second
tion occurs at about the point where ∆ ∼ 2J, i.e. for largest many-bodycontributionsin theLS-HSordered phase
J ∼ 0.75−0.8eV. In Fig. 1a we have plotted our data with J = 0.75 eV are again given for a simplified octahedral
for U = 3eV as a function of the ”lattice temperature” crystal field.
and J. We can see that there is a crossover region, that
we call mixed spin (MS), between the homogeneous LS
(white region) and HS (blue region) phases. The tran- tween the two ions develop. This can happen sponta-
sition in the MS region is governed by an increased ad- neouslyvianoiseintroducedbytheQMCprocedure,but
mixture of the HS contribution to the LS state. The we have also introduced a small difference in the levels
hybridization expansion CT-QMC solver allows for an (µDC-µDC =0.02eV)in the firstDMFT iterationto ren-
1 2
analysis of the local eigenstates contributing to the par- derthetwoatomsexplicitlyinequivalent[32]. Morecon-
titionfunctionduringtheimaginarytimeevolution. The cretely,acontributionofd5 andd7 statesto the nominal
states observed here are almost pure LS or HS (> 90% d6 average charge develops as a function of J. Thus, at
probability), i.e. d6 and d6 with no contribution smallJ thetwoatomsbothconvergetotheLSconfigura-
S=0 S=2
from any IS (d6 ) states. Small (∼ 3%) contributions tion as before, but with increasing J charge fluctuations
S=1
of d5 and d7 are found in the LS and HS states between the two atoms occur, see the LS-HS region in
S=1/2 S=3/2
respectively. Fig. 1b. Regarding the spin configuration in the LS-HS
Inasecondstep,theconstraintofequivalentCoatoms phase one atom will be in a predominantly LS and the
inthe unit cellwasremoved. The calculationsshowthat other in a predominantly HS state. This is accompanied
within this assumption, strong charge fluctuations be- bythe occupanciesoftherespectiveCo3dshellstodevi-
4
ate from N =6.0, see Tab. I. Consequently, the QMC (a) 6 6 6 7 6 5
3d d d d +d d +d
S=0 S=2 S=0 S=1/2 S=2 S=3/2
partitionfunctionshowssizeable contributionsofd7
on atom 1 and d5 on atom 2, respectively. OS=th1/e2r 0.9 ∝ Uσii,-σ ∝ Uσij,-σ ∝ Uσij,σ
S=3/2
theoretical results and the interpretation of experimen- 0.6
tal data [2, 33] indicate that such a state exists in LCO
0.3
at room temperature. From Fig. 1 one can see that the
spintransitioncanbe studiedasafunctionoftheHunds 0
coupling J and of the temperature. This implies, that 〈ˆni,σ ˆni,-σ〉 〈ˆni,σ ˆnj,-σ〉 〈ˆni,σ ˆnj,σ〉
the transitioncanbe drivenonly byelectronic means,as p(k) (b) Ekin(d6S=0) = -0.29 eV d6S6=0
shownbypreviousmodelcalculations[34]. Inthefollow- 6 d
E (d ) = -0.52 eV S=2
ing,thedetailedelectronicstructureconfigurationwillbe 0.3 Ekin(d6S=2+d7 ) = -0.67 eV d6S=0+d7S=1/2
investigatedasafunctionofJ assumingthe300Kcrystal 0.2 Ekin(d6S=0+d5S=1/2) = -0.66 eV d6S=2+d5S=3/2
structure, which exhibits all relevant features present in kin S=2 S=3/2
the whole temperature range. 0.1
The evolution of the Co 3d spectra as a function of J 0
0 2 4 6 8 10 12 14 16
is given in Fig. 2. The orbitally resolved spectral func-
k
tion(obtainedviamaximumentropy[35]fromthe QMC
Green’s function) is shown for the homogeneous case for FIG. 3. (color online) Analysis of the total energy contri-
the LS and HS states of the 300K crystal structure in butions for the 300K structure. (a) Orbitally averaged dou-
2a. The strongest overall changes are visible for the un- ble occupancies in the homogeneous and inhomogeneous so-
occupiedpartof the spectra,suggestingthat experimen- lutions(emptyandsolidsymbolsrespectively)alongwiththe
corresponding Coulomb interaction term. The potential en-
tal techniques addressing those states will be relevant
ergy (obtained from the double occupancies) is lower in the
to understand the system in more detail [36]. The low-
LS-HSphase as compared to the homogeneous LS-LS phase,
spin state in Fig. 2a is closest to the DFT solution, the
see text. (b) Histogram of the expansion order for 290K
strongestmodification is the rigid upwardshift of the eg (β = 40eV−1) of the QMC diagrams contributing on one
bands and, as a consequence, the gap opening between atom to the fermionic trace for the homogeneous (shaded)
t and e states. This is in accordance with combined and the LS-HS phase (solid symbols). Its average is propor-
2g g
DFT andcluster calculations[37]as wellasrecentQMC tional to the kinetic energy. Numerical values correspond to
thehistograms shown.
[8]. The formation of local moments in the higher tem-
perature HS states leads to the appearance of additional
features in the spectrum. As a result the gap changes
its character from t2g −eg to t2g −t2g with incoherent The HS atom is then in a state of predominantly d6S=2
t2g excitations on both gap edges. The occupied parts character with a contribution of a d5 state. The LS
S=3/2
of the spectra exhibit a transfer of spectral weight away atomonthe otherhandisnowina d6 withanadmix-
S=0
fromthe strongt2g excitationpeak towardshigher bind- tureofd7 ,whichishigherinenergycomparedtothe
ing energies as the LS to HS crossover commences, see S=1/2
pure d6 LS state. Balancing the energy loss of the LS
Ref. [8] and references therein. The spectral function S=0
atomwiththeenergygainbroughtaboutbytheHSatom
for the asymmetric Co configuration is displayed in Fig.
the mixed LS-HS arrangement constitutes a net gain in
2b for the LS-LS (J = 0.6eV), the LS-HS (J = 0.75eV)
potential energy. The total energies can be analyzed in
andtheHS-HS(J =0.9eV)arrangements. Again,asthe
detailusingFig. 3. Thebulkofthepotentialenergygain
transition from LS-LS to HS-HS commences the first t
2g can be understood as a reduction of the on-site double
excitationpeak is largestfor the LS-LSstate, reducedin
occupancieshnˆ nˆ i,seeterms∝Uσ,−σ inFig. 3a. In
the LS-HS and even more so in the HS-HS state. Also, i,σ i,−σ ii
the homogeneous LS state these terms are close to their
the progressive reduction of the gap expected from ex-
maximumof1forthe t andzeroforthe e states,thus
perimental studies is reproduced [33, 37, 38]. 2g g
givingtheaveragevalueof0.6perorbital,whiledropping
Thetendencyofthesystemtointroducechargefluctu- toabout0.2perorbitalintheHSstate(emptysymbolsin
ations canbe understoodby analyzing the total energies Fig. 3a). In the LS-LS state both atoms contribute with
within DMFT. The kinetic energy is accessible within ∼0.6 U per orbital, while in the LS-HS state one atom
ii
QMC as E =−Thki, where T is the temperature and contributes ∼ 0.6 U , the other only ∼ 0.2 U (solid
kin ii ii
hki is the mean order of the QMC histogram, whereas symbols in Fig. 3a). The net gain due to the contri-
thepotentialenergycanbeobtainedfromthedoubleoc- butions of the “anti-Hund” double occupancies ∝Uσ,−σ
cupancy as 21Pα,i,j,σ,σ′Uασiσj′hnˆαi,σnˆαj,σ′i [39, 40]. As a and the energy loss due to the “Hund” terms ∝Uiσj,σijal-
consequence of the charge fluctuations the “critical” J most cancel. Using the proper numerical values for the
for the transition on one atom is lowered (via the d5 ad- double occupancies and interaction matrices we obtain
mixture) allowing to introduce one HS site into the cell. the difference ∆E =ELS−HS−ELS−LS ≈−2.4eV as-
pot pot pot
5
suming J = 0.7eV. At large enough J the Hunds rule fruitful discussions. M. I. acknowledges BMBF pro-
energy overcomes the crystal field and the HS-HS state posal05K12GU2forfinancialsupport. Financialsupport
becomes accessible. bytheDeutscheForschungsgemeinschaft(DFG)through
TheQMChistogram(∼−E )showninFig. 3billus- SFB 668 is gratefully acknowledged. M.K. acknowledges
kin
trates beautifully the appearance of charge fluctuations support from the DFG via FOR1162.
and can be interpreted using a reasoning similar to Ref.
[41]. The QMC histogramfor the pure LSandHS states
shows the usual one peak structure, HS being more itin-
erant and exhibiting a shoulder in addition to the prin-
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