Interplay of frustration, magnetism, charge ordering, and covalency in a model of Na0.5CoO2

0.14 MB·English

by  Jaime Merino

Interplay of frustration, magnetism, charge ordering, and covalency in the ionic
Hubbard model on the triangular lattice at three-quarters filling
Jaime Merino,1 B. J. Powell,2 and Ross H. McKenzie2
1Departamento de F´ısica Te´orica de la Materia Condensada,
Universidad Aut´onoma de Madrid, Madrid 28049, Spain
9 2Centre for Organic Photonics and Electronics, School of Physical Sciences,
0 University of Queensland, Brisbane 4072, Australia
0 (Dated: January 19, 2009)
2
We investigate the ionic Hubbard model on a triangular lattice at three-quarters filling. This

n model displays a subtle interplay between metallic and insulating phases and between charge and
a magnetic order. We find crossovers between Mott, charge transfer and covalent insulators and
J
magnetic orderwith large momentsthat persist evenwhen the charge transfer isweak. Wediscuss

9 ourfindings in thecontext of recent experiments on thelayered cobaltates A0.5CoO2 (A=K,Na).
1
PACSnumbers: 71.10.Fd,71.15.-m,71.27.+a
]
l
e
- The competition between metallic and insulating a the site energy. We specialise to the case with two
r
t states in strongly correlated materials leads to many sublattices, A (ǫ =∆/2) and B (ǫ =−∆/2), consisting
s i i
. novelbehaviours. The Mottinsulatoroccurswhenasin- ofalternatingrows,withdifferentsiteenergiesonthetwo
t
a gle bandis half-filledandthe on-siteCoulombrepulsion, sublattices (c.f., Fig. 15 of Ref. 10). This is the lattice
m
U, is much larger than hopping integral, t. A menagerie relevanttoA CoO wherethedifferenceinsiteenergies
0.5 2
- of strongly correlated states is found when a system is results from the ordering of the A-atoms [11, 12, 13].
d
driven away from the Mott insulating state, either by Twolimitsofmodel(1)at3/4-fillingmaybeeasilyun-
n
o doping, as in the cuprates [1], or reducing U/t, as in the derstood. For non-interacting electrons, U =0, a metal-
c organics[2]. Geometricfrustrationcausesyetmorenovel lic state occurs for all ∆ as at least one-bandcrosses the
[ physicsinMottsystems[2]. Thereforetheobservationof Fermi energy. In the atomic limit t = 0, and U > ∆

2 strongly correlated phases in the triangular lattice com- oneexpectsachargetransferinsulatorwithachargegap
v pounds A CoO , where A is K or Na [3], has created of about ∆ whereas for U < ∆ a Mott insulator with
0.5 2
5 intense interest. charge gap of U occurs. However, realistic parametriza-
2
Animportantmodelforinvestigatinginsulatingstates tion of A CoO materials imply U ≫ ∆ and ∆ ∼ |t|
0 x 2
in correlated materials is the ionic Hubbard model. On [14]; we will show below that in this parameter regime
4
. ahalf-filledsquarelatticethis modeldisplaysacrossover the model show very different behaviour from either of
8
betweenMottandbandinsulatingstateswhichhasbeen thelimitsdiscussedabove. Thisinterestingregimeneeds
0
8 analyzedwithquantumMonteCarlo(QMC)[4],dynam- to be analyzed using non-perturbative and/or numeri-
0 icalmeanfieldtheory(DMFT)anditsclusterextensions cal techniques. Thus, we have performed Lanczos diag-
: [5]. However, except for the case of one dimension [6], onalization calculations on 18 site clusters with periodic
v
i thismodelhasnotbeenstudiedawayfromhalf-filling[7] boundary conditions.
X
and/or on geometrically frustrated lattices. In Fig. 1 we plot the charge transfer, n −n as a
B A
r In this Letter we study the ionic Hubbard model on function of ∆/|t| for several values of U. We also plot
a
a triangular lattice at three-quarter filling. This Hamil- n −n in two analytically tractable limits: the non-
B A
tonian displays a subtle interplay between metallic and interacting limit, U = 0 [15]; and the strong coupling
insulating phases and charge and magnetic order. It has limitU ≫∆≫|t|[16]. Severalinterestingeffectscanbe
regimesanalogoustoMott,chargetransfer[8],andcova- observedinthiscalculation. Firstly,thesignoftstrongly
lent insulators [9]. The study of this model is motivated effects the degree of charge transfer on the triangular
in part by our recent proposal [10] that it is an effective lattice. Secondly,chargetransferdependsonlyweaklyon
low-energy Hamiltonian for Na CoO , at values of x at U. Thirdly, regardless of the sign of t or the magnitude
x 2
which ordering of the sodium ions occurs. of U, the charge transfer increases rather slowly as ∆
The Hamiltonian for the ionic Hubbard model is increases.
The chargegap,i.e., the difference inthe chemicalpo-
H =−t X c†iσcjσ +UXni↑ni↓+Xǫiniσ, (1) tentials for electrons and holes, is ∆c ≡ E0(N + 1) +
hijiσ i iσ E0(N −1)−2E0(N), where E0(N) is the ground state
energyforN electrons. We plotthe variationof∆ with
c
where c(†) anihilates (creates an electron with spin σ at ∆ for various values of U in Fig. 2. ∆ vanishes for
iσ c
sitei,tisthehoppingintegral,U istheeffectiveCoulomb U = 0, however finite size effects mean that we cannot
repulsion between electrons on the same site, and ǫ is accurately calculate ∆ for small ∆. ∆ = ∆ for t = 0
i c c
2
1 lattice is half-filled and hence becomes a Mott insulator.
t>0
If there were no hybridisation between that chains, one
0.8 t<0 would find a metallic state for any finite charge trans-
fer from the B-sites to the A-sites (self doping), even for
0.6 U ≫ |t| as the A-chains are now electron-doped Mott
A
n insulators and the B-chains are hole-doped band insula-
-B
n 0.4 tors. However, Fig. 2 shows that the insulating regime
tb, t<0
of the model extends far beyond the well understood
U=0, t<0
n −n = 1 regime. This is because the real space in-
0.2 B A
terpretation is incorrect as hybridization between A and
B chains is substantial. For |t|∼∆≪U the system can
0
0 2 4 6 8 10 remain insulating with a small gap [O(t)]. This state is
∆/|t|
analogous to a covalent insulator [9].
One expects that for ∆=0 the ground state is metal-
FIG. 1: (Color online) Variation of the charge transfer with
licastherethesystemis3/4-filled. However,asmallbut
∆ for different Coulomb repulsion energies. Lanczos results
finite∆=0+ leadstoastronglynestedFermisurfacefor
onan18-siteclusterfor: U/t =0(stars),15(opensymbols)
and 100 (filled symbols) are| |shown. The U = 0 results are t>0 whereas for t<0 the Fermi surface rather feature-
in excellent agreement with the infinite lattice tight-binding less. Thus, rather different behaviors might be expected
(tb) result (dashed-dotted line) [15]. Strong coupling results for different signs of t even at weak coupling. At large
(dashed lines) are also shown for comparison. U our exact diagonalization results suggest that a gap
may be present even for a small value of ∆/t. However
finite-size effects, inherent to the method, mean that it
8
is not possible to resolve whether a gap opens at ∆ = 0
U=100|t|, t>0
7 U=100|t|, t<0 or at some finite value of ∆.
U=15|t|, t>0 To test this covalent insulator interpretation in the
6
U=15|t|, t<0 ∆ ∼ |t| and large U regime we have also calculated the
5 spectral density, A(ω), c.f., Fig. 3. There are three dis-
∆/|t|c4 tinct contributions to the A(ω): at low energies there is
a lower Hubbard band; just below the chemical poten-
3
tial (ω =µ) is a weakly correlated band; and just above
2 ω = µ is the upper Hubbard band. Furthermore, the
large energy separation, much larger than the expected
1
U = 15|t|, between the lower and upper Hubbard bands
0 isduetoanupward(downward)shiftoftheupper(lower)
0 2 4 6 8 10
∆/|t| Hubbard bands due to the strong hybridization. In con-
trast,inthestrongcouplinglimitA(ω)hasamuchlarger
gap, O(∆), between the contributions from the weakly
FIG. 2: (Color online) Variation of the charge gap, ∆c, with correlated band and the upper Hubbard band.
∆ for U/t =15 and 100 (the charge gap is zero for U =0).
| | Themagneticmomentassociatedwiththepossiblean-
The tendencytowards an insulating state is greater for t<0
tiferromagnetism, m = (3hSzSzi)1/2, where ν = A or
than for t> 0. Dashed lines show the strong coupling limit: ν i j
U >> ∆ >> t. Note that although strong correlations are B andSz = 1(n −n ),is evaluatedbetweentwonext-
i 2 i↑ i↓
essential for creation of thecharge gap they are not required nearestneighborsontheνsublatticeatthecenterofclus-
for the charge transfer, c.f.. Fig. 1. Note that, for U >> ∆ ter (to reduce finite size effects [18]). Fig. 4 shows that
withU large,thechargegapisrobustagainst thevalueofU.
m increaseswith∆andissubstantiallyenhancedbyU,
A
whereas m is always small. This is in marked contrast
B
toaspindensitywave,aspredictedbyHartree-Fockcal-
and U ≫∆; this result is reminiscent of a charge trans- culationswherethemagneticmomentisfarsmallerthan
fer insulator [8]. Both perturbative [17] and numerical that experimentally observed [19].
results show that the charge gap depends on the sign of We now turn to discuss the consequences of our re-
t due to the different magnetic and electronic properties sults for understanding experiments; for simplicity and
arising from the geometrical frustration of the triangu- concreteness we focus on Na CoO . The x = 0.5 mate-
x 2
lar lattice. In contrast, on a square lattice, ∆c, does not rials have remarkably different properties from those on
depend on the sign of t. other values of x [20, 21]. Above 51 K the intralayer re-
Inthelimit, ∆≫U ≫|t|,theAandBsublatticesare sistivity of Na CoO is weakly temperature dependent
0.5 2
wellseparatedinenergy;theBsitesaredoublyoccupied with values of a few mΩcm [20] characteristic of a bad
(i.e.,theB-sublatticeisabandinsulator)andtheAsub- metal [10]. Below 51 K the resistivity increases, consis-
3
0.2 tent with a small gap opening (∼10 meV) [20]. Thus
A-sites a (bad) metal-insulator transition occurs at 51 K. The
B-sites insulating state of Na CoO has a number of counter-
0.5 2
intuitive properties,notthe leastofwhichis the absence
µ
= of strong charge ordering. NMR observes no charge or-
ω
)
ω( 0.1 dering up to a resolution of nB − nA < 0.4 [22, 23],
A
LHB while neutron crystallography suggest n −n ≃ 0.12
B A
[11]. Thus the insulating state is not the simple charge-
UHB
transfer-like state predicted by (1) in the strong cou-
∆=10|t| pling limit. Na0.5CoO2 develops a commensurate mag-
netic order below 88 K [22, 24]. A large magnetic mo-
0
ment [m = 0.26(2)µ per magnetic Co ion] is observed
B
in spite of the weak charge order [note that classically
ω) 0.05 m<(nB −nA)µB/2]. Above 100 K the optical conduc-
A( ∆=|t| tivity [21]showsno evidence ofaDrude peak, consistent
LHB withabadmetal. Intheinsulatingphasespectralweight
UHB
islostbelow∼10meV,consistentwiththegapseeninthe
0
-10 -5 0 5 10 15 20 25 dcconductivityandapeakemergesat∼20meV,whichis
ω/|t| toosharpandtoolowenergytocorrespondtoaHubbard
band [21]. ARPES shows that the highest energy occu-
piedstatesare∼10meVbelowtheFermienergy[25]. No
FIG.3: (Coloronline)Evolutionfrom achargetransferinsu- equivalentinsulatingstateisseeninthemisfitcobaltates
lator to a covalent insulator. The energy dependence of the [22], which supports the contention that Na-ordering is
spectral density, A(ω), is shown for two different parameter
vital for understanding the insulating state.
regimes of the model with t<0 and U =15t. The spectral
| | Various theories have been proposed to explain these
density in the upper panel (∆ = 10t) is that characteris-
tic of a charge transfer insulator [9] t|h|ere is a single weakly intriguing experiments. Lee et al. [26] have performed
correlated band largely associated with the B sites and ly- LDA+U calculations, which include Na-ordering, but
ing between lower (LHB) and upper (UHB) Hubbard bands not strong correlations. Other groups [27] have studied
(separated by U) that are largely associated with the A strongly correlated models that include the Coulomb in-
∼
sites and ∆c ∼ ∆. In contrast, the lower panel (∆ = |t|) teraction with neighbouring sites, but neglect the effects
showsaspectraldensitycharacteristicof acovalentinsulator
of Na-ordering. Marianetti and Kotliar [28] have also
[9]: thereareonlysmalldifferencebetweenAandBsites,the
studied the Hamiltonians proposed in [10] for x = 0.3
separation of the LHBand UHB is >U, and ∆c t.
∼| | and 0.7.
0.6 In order to compare our results with experiments on
0.15
Na CoO we need to restrict ourselves to the relevant
0.5 2
0.5 B 0.1 parameter values: t<0 and |t|∼∆≪U [14]. This cor-
m
0.05 respondswiththeregimeofthethreequartersfilledionic
0.4
Hubbardmodelthatisboththemostinterestingandthe
0
0 1 2 most difficult to study via exact diagonalisationbecause
mA0.3 ∆/|t|
of the deleterious finite size effects. Nevertheless we pro-
pose that in Na CoO the insulating state is analogous
0.2 0.5 2
toacovalentinsulator. Thisexplainsawiderangeofex-
0.1 U=2|t| periments. The peak observedat ω ∼30 meV in the op-
U=15|t|
ticalconductivity[21],isinterpretedasthetransferofan
U=100|t|
0 electron from the weakly correlatedband to form a dou-
0 1 2 3 4 5 6 7 8
∆/|t| blon in the strongly correlated band. The weak charge
transfer (n −n = 0.1−0.3; c.f., Fig. 1) is caused by
B A
thestronghybridisationbetweentheAandBsublattices
FIG. 4: (Color online) The magnetic moment as a function
and is consistent with the value extracted from crystal-
of ∆ for t<0 and various U. The magnetic moment on the
lographic experiments (0.12 [11]) and the bounds from
A sites, mA, (main panel) is strongly enhanced when U
≫ NMR(<0.4[22]). Thelargemoment(0.1-0.2µ ;c.f.,Fig.
t. The inset shows the moment on the B sublattice. mB B
i|s|much smaller, reduced by ∆, and only weakly dependent 4) is comparable to the moment found by neutron scat-
on U. These results demonstrate that, for large U and ∆ > tering(0.26µ [24])andresultsfromthe electronsinthe
B
t, theelectrons on the A-sublattice are much more strongl∼y strongly correlated band, i.e., the single spin hybridised
| |
correlated than those on the B-sublattice despite the small betweentheAandBsublattices. Finitesizeeffectsmean
charge transfer (see Fig. 1).
thatwecannotaccuratelycalculatethechargegapinthis
4
regime. However, we propose that the experimental sys- rows of singly occupied A-sites. In the strong coupling
tem corresponds to a parameter range where the gap is limit, U ∆ t, virtual hopping processes lead to
≫ ≫ | |
small, ∆ < O(|t|), consistent with the gap, ∼7-10 meV magnetic interactions between electrons in A-sites. Eq.
c
(1)reducestoaHeisenbergmodelonanrectangularlat-
[20, 25], seen experimentally in ARPES and resistivity.
This is consistent with the expectation that ∆c → 0 as tice: H =JP{ij}(cid:2)Si·Sj− 41(cid:3)+J⊥P[ij](cid:2)Si·Sj− 14(cid:3),
∆/|t| → 0. Accurately calculating ∆c for small ∆/|t| where J = 4Ut2 − 8∆t23 and J⊥ = 1∆6t24[U1 + 2∆1+U + 21∆]. J
and large U, and hence further testing our hypothesis, results from the usual superexchange antiferromagnetic
couplingand‘ring’exchangeprocessarounda3-sitepla-
therefore remains an important theoretical challenge.
quette[6].Inourpreviouspaper[10],theO(t3)termwas
WethankH.Alloul,Y.S.Lee,andR.Singhforhelpful
neglected. J becomes negative for t >0 and ∆ < √2Ut
discussions. J.M. acknowledges financial support from
leading to a ferromagnetic interaction. In contra∼st J is
⊥
the Ramo´n y Cajal program, MEC (CTQ2005-09385-
always antiferromagnetic.
C03-03). B.J.P.wastherecipientofanARCQueenEliz- [17] The lowest order correction to the charge gap, in the
abeth II Fellowship (DP0878523). R.H.M. was the re- strong-coupling limit, comes from the kinetic energy
cipient of an ARC Professorial Fellowship (DP0877875). of a hole (doublon) propagating along the B(A)-chains
Some of the numerics were performed on the APAC na- whenextracting(adding)anelectrontothezerothorder
ground state configuration. Using degenerate perturba-
tional facility.
tion theory we find that the gap is given, to O(t3), by
∆c =∆−2|t|+4t2/∆−8t3/∆2+E1t−DJ, where E1t−DJ is
theenergyofaholeinahalf-filledt-J chainwithJ given
by theexpression in Ref.16.Et−J isgiven bytheBethe
1D
ansatzexpression[32]forJ >0,whileEt−J = 2t J/2
[1] Forarecentreview,see,P.A.Lee,N.Nagaosa,andX.-G. 1D − | |−
for J <0.
Wen,Rev.Mod. Phys.78, 17 (2006). [18] A. Sandvik,Phys.Rev.B 56, 11678 (1997).
[2] Forarecentreview,see,B.J.PowellandR.H.McKenzie,
[19] A Hartree-Fock analysis of model (1) with a non-zero ∆
J. Phys.: Condens. Matter 18, R827, (2006).
leads tosmall magneticmomentsforchargegapsconsis-
[3] N.P.Ong and R.J. Cava, Science 305, 52 (2004).
tent with experiments. Alternatively, t<0 and U =4t
[4] K.Bouadim et al., Phys. Rev.B 76, 085112 (2007). | |
leads to substantial magnetic moments: m 0.3 consis-
[5] S. S. Kancharla and E. Dagotto, Phys. Rev. Lett. 98, ∼
tentwithexperiments,butagapof9t,abouttwoorders
016402 (2007). | |
of magnitude larger than theexperimental gap.
[6] K.Penc et al., Phys. Rev.B 54, 4056 (1996). [20] M. L. Foo et al.,Phys. Rev.Lett. 92, 247001 (2004).
[7] K. Penc and F. Mila, Phys. Rev. B 54, 4056 (1994); K. [21] N.L.Wanget al.,Phys.Rev.Lett.93, 147403 (2004); J.
Pencand W. Stephan,Phys.Rev.B 62, 12707 (2000). Hwang et al., Phys. Rev. B 72, 024549 (2005); S. Lupi
[8] J. Zaanen, G.A. Sawatzky,and J. W. Allen, Phys.Rev. et al.,Phys.Rev. B 72, 024550 (2005).
Lett.55, 419 (1985). [22] J. Bobroff et al., Phys.Rev.B 76, 100407 (2007);
[9] D.D. Sarma, J. Sol. State Chem. 88, 45 (1990).
[23] NMR relaxation rates for the two different cobalt states
[10] J. Merino, B.J. Powell, and R.H. McKenzie, Phys. Rev.
differed by a factor of 3 suggesting that the sites had
B 73, 235107 (2006). a different valence [F. N∼ing et al., Phys. Rev. Lett. 93,
[11] A.J.Williamsetal.,Phys.Rev.B73,134401(2006);D.
237201 (2004)]. Measurements of the electric field gra-
Argyriou et al., Phys.Rev.B 76, 134506 (2007).
dient by zero-field NMR implied that there was a small
[12] P. Zhang et al., Phys. Rev. B 71, 153102 (2005); M.
chargeorderingalongtheCo(1)chains(Bsublattice)[F.
Roger et al.,Nature 445, 631 (2007). Ning et al., Phys. Rev.Lett. 100, 26407 (2008)].
[13] H.W. Zandbergen et al.,Rev.B 70, 024101 (2004). [24] G.Gasparovicetal.,Phys.Rev.Lett.96,464403(2006);
[14] Atomisticcalculationsgivet=-(0.08-0.14)eV,U=2.5-2.8 M. Yokoiet al., J. Phys.Soc. Jpn.74, 1800 (2005).
eV, and ∆=0.03-0.16 eV [12, 29]. Further, CoO2, which [25] D. Qian et al.,Phys. Rev.Lett. 96, 046407 (2006).
is described by model (1) at half filling with ∆ = 0, is [26] K.-W. Lee et al., Phys. Rev. Lett. 94, 026403 (2005);
astrongly correlated metal ratherthanaMott insulator ibid. 96, 096403 (2006).
a[30M].oTtthiinsssuulgagtoerstosnUt/h|te|t<∼ria1n2g−ula1r5,latthteicceri[t1i0c,al31v]a.lue for [27] SC.hoZyh,ouD.aGndalaZn.kWis,anagn,dibPi.d.P9h8il,lip2s2,64P0h2ys(.20R0e7v).;BT.7P5,.
[15] Thetight-bindingbandstructureofmodel(1)consistsof
073103 (2007).
two bands due to hybridization between A and B-sites. [28] C.A.MarianettiandG.Kotliar,ibid.98,176405(2007).
At 3/4-filling, for t > 0 and any non-zero ∆, the lower [29] D.J. Singh, Phys. Rev. B 61, 13397 (2000); K.W. Lee,
band is filled and the upperband is half-filled, while for J. Kune˘s, and W.E. Pickett, ibid. 70, 045104 (2004); P.
[16] Itn<th0etshterolnowgecroubpalnindgislimfililtedwefofirn|d∆,|fr>∼om0.6d4etg.enerateper- Zhangetal.,Phys.Rev.Lett.93,236402(2004);S.Lan-
dron and M-B. Lepetit, arXiv:0805.3121.
1tu−rb8a(tt/io∆n)2th+e1o6r(yt,/t∆h)a3t+, fOor((eti/t∆he)r4)s.iFgnorotf=t,0ntBhe−grnoAun=d [[3301]] CM..dCeaVpoanuelxeettaal.l,.,PPhhyyss..RReevv..BLe6t3t,.09885,1204464(20020(12)0.07).
state of model (1) is a charge ordered insulator consist- [32] M. Ogata and H. Shiba, Phys.Rev.B 41 2326 (1990).
ing of rows of doubly occupied B-sites alternating with