Itinerant in-plane magnetic fluctuations and many-body correlations in Na$_x$CoO$_2$

0.43 MB·English

by  M. M. Korshunov

Itinerant in-plane magnetic fluctuations and many-body correlations in Na CoO
x 2
M.M. Korshunov 1,2, I. Eremin 2,3, A. Shorikov 4, V.I. Anisimov 4, M. Renner 3, and W. Brenig 3
∗
1 L.V. Kirensky Institute of Physics, Siberian Branch of Russian Academy of Sciences, 660036 Krasnoyarsk, Russia
2 Max-Planck-Institut fu¨r Physik komplexer Systeme, D-01187 Dresden, Germany
3 Institute fu¨r Mathematische und Theoretische Physik,
TU Braunschweig, 38106 Braunschweig, Germany and
4 Institute of Metal Physics, Russian Academy of Sciences-Ural Division, 620041 Yekaterinburg GSP-170, Russia
7
(Dated: February 6, 2008)
0
0 Based on the ab-initio band structure for NaxCoO2 we derive the single-electron energies and
2 the effective tight-binding description for the t2g bands using projection procedure. Due to the
presenceofthenext-nearest-neighborhoppingsalocalminimumintheelectronicdispersioncloseto
n
a the Γ point of the first Brillouin zone forms. Correspondingly, in addition to a large Fermi surface
J an electron pocket close to the Γ point emerges at high doping concentrations. The latter yields
the new scattering channel resulting in a peak structure of the itinerant magnetic susceptibility at
0
smallmomenta. Thisindicatesdominantitinerantin-planeferromagneticfluctuationsabovecertain
3
critical concentration xm, in agreement with neutron scattering data. Below xm the magnetic
] susceptibility shows a tendency towards the antiferromagnetic fluctuations. We further analyze
l the many-body effects on the electronic and magnetic excitations using various approximations
e
- applicable for different U/t ratio.
r
t
s PACSnumbers: 74.70.-b;71.10.-w;75.40.Cx;31.15.Ar
.
t
a
m I. INTRODUCTION tered around the Γ point and to have mostly a char-
1g
- acter. Furthermore, a dispersion of the valence band is
d measuredwhichisonlyhalfofthatcalculatedwithinthe
n The recent discovery of the superconductivity in
LDA.Thisindicatestheimportanceoftheelectroniccor-
o hydrated lamellar cobaltate Na CoO yH O1 raised
c tremendous interest in the naturxe and2sy·mm2etry of the relations in NaxCoO2.
[ Shubnikov-de Haas effect measurements revealed two
superconductive pairing in these materials. The phase
well-defined frequencies in Na CoO , suggesting either
1 diagram of this compound, with varying electron dop- 0.3 2
v ingconcentrationxandwaterintercalationy,isrichand theexistenceofNasuperstructuresorthepresenceofthe
5 complicated;inadditiontosuperconductivity,itexhibits e′g pockets12. Lastpossibility was found to be incompat-
5 ible with existing specific heat data. Also, within the
magnetic and charge orders, and some other structural
7 transitions2,3,4,5. The parent compound, Na CoO , is LDA scheme the Na disorder was shown to destroy the
1 x 2 smalle pocketsinNa CoO becauseoftheirtendency
0 a quasi-two-dimensional system with Co in CoO2 layers ′g 2/3 2
7 forming a triangular lattice where the Co-Co in-plane towards the localization13.
0 distance is two times smaller than the inter-plane one. The hole pockets are absent in the LSDA+U
/ Na ions reside between the CoO layers and donate ad- (Local Spin Density Approximation + Hubbard U)
t 2
a ditional x electrons to the layer,loweringthe Co valence calculations14. However, in this approach the insulating
m from Co4+ (3d5 configuration) to Co3+ (3d6 configura- gap is formed by a splitting of the local single-electron
- tion) upon changing x from 0 (CoO ) to 1 (NaCoO ). states due to spin polarization, resulting in a spin polar-
d 2 2
The hole in the d-orbital occupies one of the t levels, izedFermisurfacewithanareatwiceaslargerasARPES
n 2g
which are lower than e levels by about 2 eV6. The de- observes.
o g
c generacyofthet2g levelsispartiallyliftedbythetrigonal The dynamical character of the strong electron corre-
v: crystal field distortion which splits the former into the lations has been taken into account within Dynamical
i higher lying a1g singlet and the lower two e′g states. MeanField Theory(DMFT) calculations15 and, surpris-
X
FirstprinciplesLDA(localdensityapproximation)and ingly has led to an enhancement of the area of the small
r LDA+U band structure calculations predict Na CoO Fermi surface pockets, in contrast to the experimental
a x 2
to have a large Fermi surface (FS) centered around the observations. At the same time, the use of the strong-
Γ = (0,0,0) point with mainly a character and six coupling Gutzwiller approximation within the multior-
1g
hole pockets near the K=(0,4π,0) points of the hexag- bital Hubbard model with fitting parameters16 yields an
3
onal Brillouin zone of mostly e character for a wide absence of the hole pockets at the Fermi surface. Ac-
′g
range of x6,7. At the same time, recent surface sensitive cording to these findings, the bands crossing the Fermi
Angle-Resolved Photo-Emission Spectroscopy (ARPES) surface have a1g character.
experiments8,9,10,11 revealsadopingdependentevolution Concerning the magnetic properties, LSDA predicts
oftheFermisurface,whichshowsnosignofholepockets Na CoO to have a weak intra-plane itinerant ferro-
x 2
for a wide range of Na concentrations, i.e. (0.3 x magnetic (FM) state for nearly all Na concentrations,
0.8). Instead, the Fermi surface is observed to b≤e cen≤- 0.3 x 0.717. On contrary, neutron scattering
≤ ≤
2
finds A-type antiferromagnetic order at T 22K 0.5 0.5
m
≈
with an inter(intra)-plane exchange constant J =
c(ab)
12( 6)meV and with ferromagnetic ordering within Co-
laye−r only for 0.75 x 0.918,19,20. 0 0
≤ ≤
In this paper we derive an effective low-energy model
V) V)
describingthebandscrossingtheFermilevelonthebasis e e
of the LDA band structure calculations. Due to the FS gy (-0.5 -0.5gy (
topology, inferred from LDA band structure, the mag- ner ner
E E
netic susceptibility χ (q,ω = 0) reveals two different
0
regimes. For x < 0.56 the susceptibility shows pro- -1 Co-d -1
a
1g
nouncedpeaksattheantiferromagnetic(AFM)wavevec- e’
g1
tor Q = 2π, 2π , 4π,0 resulting in a ten- e’g2
dencyAtFowMards nin(cid:16)-p3lan√e31(cid:17)20(cid:0) 3AFM(cid:1)oorder. For for x > -1.5Γ K M ΓA L H A 5 10-1.5
◦ PDOS (states/eV/atom)
0.56 the susceptibility is peaked at small momenta near
QFM =(0,0). Thisclearlydemonstratesthetendencyof FIG. 1: (color online) Calculated near-Fermi level LDA
the system towards an itinerant in-plane FM state. We band structure and partial density of states (PDOS) for
findthattheformationoftheelectronpocketaroundthe Na0.33CoO2. ThecontributionofCo-a1g statesisdenotedby
Γ point is crucial for the in-plane FM ordering at high the vertical broadening (in red) of the bands with thickness
doping concentrations. We further analyze the role of proportional to the weight of the contribution. The crosses
themany-bodyeffectscalculatedwithintheFluctuation- indicatethedispersionofthebandsobtainedbyprojectionon
Exchange (FLEX), Gutzwiller, and Hubbard-I approxi- the t2g orbitals. The horizontal line at zero energy denotes
theFermi level.
mations.
The paper is organized as follows. In Section II the
LDA band structure and tight-binding model parame-
ters derivation are described. The doping dependent
evolutionofthe magneticsusceptibility within the tight-
binding model is presented in Section III. The role of
strong electron correlations is analyzed in Section IV.
The last Section summarizes our study.
II. TIGHT-BINDING MODEL
FIG. 2: (color online) (a) Schematic crystal structure of the
Co-layerinNaxCoO2 withhoppingnotationswithinthefirst
The band structure of Na CoO was obtained
0.61 2 three coordination spheres (C.S.). (b) LDA-calculated Fermi
within the LDA21 in the framework of TB-LMTO- surface with cylindrical part (violet) havingmostly a1g char-
ASA (Tight Binding approachto the Linear Muffin-Tin- acter and six hole pockets (red) having mostly e′g character.
Orbitals using Atomic Sphere Approximation)22 com- kx and ky coordinates of the symmetry points are given in
putation scheme. This compound crystallizes at 12K unitsof 2π/a with a being thein-planelattice constant.
in the hexagonal structure (P6 /mmc symmetry group)
3
with a=2.83176˚Aand c=10.84312˚A23. A displacement
of Na atoms from its ideal sites 2d (1/3,2/3,3/4) on states,2s,2p,3dstatesofOand3s,3p,3dstatesofNaas
about 0.2˚Aare observed in defected cobaltates for both the valence states for the TB-LMTO-ASA computation
room and low temperatures. This is probably due to scheme. Theradiiofatomicspheresare1.99a.u. forCo,
the repulsion of the randomly distributed Na atoms, 1.61a.u. foroxygen,and2.68a.u. forNa. Twoclassesof
locally violating hexagonal symmetry23. In this study empty spheres (pseudo-atoms without core states) were
Na atoms were shifted back to the high symmetry 2d added in order to fill the unit cell volume.
sites. Oxygen was situated in the high-symmetry 2d- In order to find an appropriate basis the occupation
position (1/2,2/3,3/4). The obtained Co-O distance is matrixwasdiagonalizedanditseigenfunctionswereused
1.9066˚Awhich agrees well with experimentally observed as the new local orbitals. This procedure takes into ac-
one 1.9072(4)˚A23. This unit cell was used for all doping count the real distortion of the crystal structure. The
concentrations. The effect of the doping was taken into new orbitals are not pure trigonal a and e orbitals
1g ′g
account within the virtual crystal approximation where but we still use the former notations for the sake of sim-
each Co site has six nearestneighbor virtual atoms with plicity. 288 k-points in the whole Brillouin zone were
fractionalnumberofvalenceelectronsxandacorecharge used for the band structure calculations (12x12x2 mesh
10+x instead of randomly located Na. Note, that all for k , k , and k , correspondingly).
x y z
core states of the virtual atom are left unchanged and The bands crossing the Fermi level are shown in
corresponds to Na ones. We have chosen Co 4s,4p,3d (Fig. 1). One sees that they have mostly a character,
1g
3
consistent with previous LDA findings6. Note, the small tion are given by:
FSpocketsneartheKpointwithe -symmetrypresentat
′g
x=0.33 [see Fig. 2(b)] disappear for higher doping con- tˆ = 2tˆ cosk +2tˆ cosk +2tˆ cosk
k 1 2 2 3 3 1
centrations because of the corresponding bands sink be- + 2tˆ cos(k +k )+2tˆ cos(k +k )+2tˆ cos(k k )
4 1 3 5 2 1 6 1 2
lowtheFermilevel. Thedifferenceinthedispersionalong −
+ 2tˆ cos2k +2tˆ cos2k +2tˆ cos2k , (2)
K M and L H directions is due to non-negligible in- 7 2 8 3 9 1
− −
teractionbetweenCoO planes. AsmallgapbetweenCo-
3d and O-2p states at a2bout -1.25 eV presentfor x=0.61 where k1 = √23kx− 12ky, k2 =ky, k3 = √23kx+ 21ky.
disappears for x=0.33 due to the shift of the d-band to Note,the parametersdonotchangesignificantlyupon
lower energy upon decreasing number of electrons. changingthedopingconcentration. InFig.3weshowtwo
results of the rigid-band approximation with the Hamil-
Inthefollowingwerestrictourselvestothemodelwith
tonian(1)andthehoppingvaluesobtainedinLDAcalcu-
the in-plane hoppings inside CoO layer to describe the
2 lation for two different doping concentrations, x = 0.33
doping dependence of the itinerant in-plane magnetic
and x = 0.61 (see Table I). The doping concentration
order. Hence we neglect bonding-antibonding (bilayer)
used to calculate the chemical potential µ was fixed to
splitting present in the LDA-bands. This assumption
be x = 0.61 for both Hamiltonians. Although one finds
seems to be justified since the largest interlayer hopping
the pronounced differences in the dispersion around the
matrixelementisanorderofmagnitudesmallerthanthe
M-point, they are small around the FS. Since most of
intra-plane one (0.012 eV vs. 0.12 eV).
the physical quantities are determined by the states ly-
To construct the effective Hamiltonian and to derive ingclosetotheFermilevel,wecansafelyignorethesmall
the effective Co-Co hopping integrals tαβ for the t - differencesofthebandstructureanddescribethedoping
fg 2g
manifold we apply the projection procedure24,25. Here, evolutionoftheNaxCoO2bysimplyvaryingthechemical
potential. In the following we will use ab initio param-
(αβ) denotes a pair of orbitals, a , e or e . The
1g ′g1 ′g2
eters calculated for x = 0.33 and change the chemical
indices f and g correspond to the Co-sites on the tri-
potential to achieve different doping concentrations.
angular lattice. The obtained hoppings and the single-
electron energies are given in Table I for three difference Withintherigidbandapproximationthee′g holepock-
dopingconcentrations. Acomparisonbetweenthe bands ets arewell belowthe Fermi levelfor x 0.41. Mostim-
≥
obtainedusing projectionprocedureandthe LDA bands portanthowever,we find the local minimum of the band
is shown in Fig. 1 confirming the Co-t nature of the dispersion around the Γ point (see Fig. 3) to yield an
2g
near-Fermi level bands6,26. For simplicity we have enu- inner FS contour centered around this point. The area
merated site pairs with n = 0,1,2,..., tαβ tαβ (see of this electron FS pocket increases upon increasing the
fg → n doping concentration x. As we will show later, the main
Fig. 2(a) and the correspondence between in-plane vec-
reason for the local minimum around the Γ point is the
torsandindex n inTableI). Due to the C symmetryof
3 presence of the next-nearest-neighbor hopping integrals
the lattice, the following equalities apply: tαβ = tαβ ,
| 3 | | 1 | which enter our tight-binding dispersion. Although this
tαβ = tαβ , tαβ = tαβ . In addition tαβ = tαβ for
| 5 | | 4 | | 9 | | 7 | 1 2 minimum is not yetdirectly observedby ARPESexperi-
a a hoppings, which, however, does not hold for
1g 1g ments, note that the inner FS contour would reduce the
→
e orbitals. Since the hybridization between the a
′g1,2 1g total FS volume and therefore may explain why the vol-
and the e bands is not small, a simplified description
′g ume of the FS observed in ARPES so far is larger than
of the bands crossing the Fermi level in terms of the a1g it follows from Luttinger’s theorem28. Furthermore, an
bandonly(neglectinge bandandthecorrespondinghy-
′g emergenceofthispocketwouldinfluencetheHallconduc-
bridizations, see for example Ref.27) may lead to an in- tivity at high doping concentrations which is interesting
correctresultdue to a higher symmetry ofthe a1g-band. to check experimentally.
In summary, the free electron Hamiltonian for CoO - Note, the appearance of the inner contour of the FS
2
plane in a hole representation is given by: aroundtheΓpointforlargedopingconcentrationsisnot
unique to our calculations, previously it has been ob-
tained within the LDA calculations for a single Co-layer
H0 =− (εα−µ)nkασ− tαkβd†kασdkβσ. (1) per unit cell14.
kX,α,σ Xk,σ Xα,β
III. MAGNETIC SUSCEPTIBILITY
wheredkασ (d†kασ)istheannihilation(creation)operator
for the hole with momentum k, spin σ and orbital index
To analyze the possibility of the itinerant magnetism
α, nkασ = d†kασdkασ, and tαkβ is the Fourier transform we calculate the magnetic susceptibility χ0(q,ω = 0)
of the hopping matrix element, ǫα is the single-electron
based on the Hamiltonian H . The doping-dependent
0
energies, and µ is the chemical potential. Introducing evolution of the peaks in Reχ (q,0) is shown in Fig. 4.
0
matrix notations, tˆk αβ = tαkβ and tˆn αβ = tαnβ, the At x=0.41 the e′g bands are below the Fermi level, and
hoppings matrix el(cid:0)em(cid:1)ents in the mom(cid:0)ent(cid:1)um representa- theFShastheformoftheroundedhexagon. Itresultsin
4
TABLEI:Single-electronenergiesǫα(relativetoεa1g)andin-planehoppingintegralstαnβforNaxCoO2,wherex=0.33,0.61,0.7.
(all valuesare in eV)
in-planevector: (0, 1) (√3, 1) (√3, -1) (√3, 0) (√3, 3) (√3, -3) (0, 2) (√3, 1) (√3, -1)
2 2 2 2 2 2 2 2
α ǫα α β tαβ tαβ tαβ tαβ tαβ tαβ tαβ tαβ tαβ
1 2 3 4 5 6 7 8 9
→
x=0.33
a1g 0.000 a1g a1g 0.123 0.123 0.123 -0.022 -0.022 -0.021 -0.025 -0.025 -0.025
→
a1g →e′g1 -0.044 0.089 -0.044 0.010 0.010 -0.021 -0.021 0.042 -0.021
e′g1 -0.053 a1g →e′g2 -0.077 0.000 0.077 0.018 -0.018 0.000 -0.036 0.000 0.036
e′g1 →e′g1 -0.069 -0.005 -0.069 0.018 0.018 -0.026 -0.017 -0.085 -0.017
e′g2 -0.053 e′g1 →e′g2 0.037 0.000 -0.037 -0.026 0.026 0.000 -0.039 0.000 0.039
e′g2 →e′g2 -0.026 -0.090 -0.027 -0.011 -0.011 0.033 -0.062 0.006 -0.062
x=0.61
a1g 0.000 a1g a1g 0.110 0.110 0.110 -0.019 -0.019 -0.019 -0.023 -0.023 -0.023
→
a1g →e′g1 -0.050 0.100 -0.050 0.008 0.008 -0.016 -0.017 0.035 -0.017
e′g1 -0.028 a1g →e′g2 0.087 0.000 -0.087 -0.014 0.014 0.000 0.030 -0.000 -0.030
e′g1 →e′g1 -0.069 -0.031 -0.069 0.015 0.015 -0.022 -0.016 -0.076 -0.016
e′g2 -0.028 e′g1 →e′g2 -0.022 0.000 0.022 0.021 -0.021 0.000 0.035 0.000 -0.035
e′g2 →e′g2 -0.044 -0.081 -0.044 -0.009 -0.009 0.027 -0.056 0.005 -0.056
x=0.7
a1g 0.000 a1g a1g 0.105 0.105 0.105 -0.018 -0.018 -0.018 -0.022 -0.022 -0.022
→
a1g →e′g1 -0.052 0.104 -0.052 0.007 0.007 -0.015 -0.016 0.033 -0.016
e′g1 -0.019 a1g →e′g2 -0.090 0.000 -0.090 0.013 -0.013 0.000 -0.028 0.000 0.028
e′g1 →e′g1 -0.068 -0.039 -0.068 0.014 0.014 -0.020 -0.015 -0.073 -0.015
e′g2 -0.019 e′g1 →e′g2 0.016 0.000 -0.016 -0.020 0.020 0.000 -0.034 0.000 0.034
e′g2 →e′g2 -0.048 -0.077 -0.049 -0.009 -0.009 0.026 -0.054 0.005 -0.054
local minimum around the Γ point for the formation of
the scattering at small momenta was originally found in
Ref.27.
For large x the area of the inner FS contour increases
leading to a further decrease of the Q . Observe that
1
for x 0.88, the FS topology again changes yielding six
≈
distant FS contours that moves Q further to zero mo-
1
menta. Thescatteringatsmallmomentaseeninthebare
magnetic susceptibility for x > x is qualitatively con-
m
sistentwiththe ferromagneticorderingatQ =(0,0),
FIG. 3: (color online) Calculated tight-binding energy dis- FM
observed in the neutron scattering experiments18,19,20.
persion, the density of states (DOS), and the Fermi surface
for Na0.61CoO2 within therigid-band approximation with ab
initio parameters for x = 0.61 (the solid blue curve) and for
x=0.33 (red dashed curve). The horizontal (green) line de- IV. EFFECTS OF STRONG ELECTRON
notes thechemical potential µ for x=0.61. CORRELATIONS
It is important to understand the impact of electronic
a number of nesting wave vectors around the antiferro- correlationsonthemagneticinstabilitiesobtainedwithin
magnetic wave vector Q . The corresponding broad the rigid band approximation. Since obtained magnetic
AFM
peaks in the Reχ (q,0) appear around Q , indicat- susceptibility depends mostly on the topology of the FS
0 AFM
ingthetendencyoftheelectronicsystemtowardsan120 one expects that the behavior shown in Fig. 4 will be
◦
AFMSDWorderedstate29. Uponincreasingdoping,the valid even if one consider an RPA susceptibility with an
Fermilevelcrossesthe localminimumatthe Γ point,re- interaction term H taken into account, at least in the
int
sulting in an almost circle inner FS contour. As soon case if the only interaction is the on-site Hubbard repul-
as this change of the FS topology occurs, the scatter- sion U. The only difference would be a shift of the criti-
ing at the momentum Q is strongly suppressed at calconcentrationsx ,atwhichtheFStopologychanges
AFM m
x 0.56. Simultaneously, a new scattering vector,Q , andtendencytotheAFMorderchangestowardstheten-
m 1
at s≥mall momenta appears. Correspondingly, the mag- dency to the FM ordered state. Similar to Refs.16,30 we
netic susceptibility peaks at small momenta, indicating addthe on-site Coulombinteractionterms to Eq.(1). At
the tendency of the magnetic system towards itinerant present, it is not completely clear to which extent the
SDW order with small momenta. The relevance of the electronic correlations governs the low-energy properties
5
FIG.5: (coloronline)Aschematicpictureofthelocalatomic
states on Co and the single-particle excitations in NaxCoO2.
Here nh stands for number of holes, mi enumerates single-
particle excitations. The filling factor of the corresponding
state upon changing the doping concentration x is given in
squarebrackets.
aninfinitevalueofU. Thisapproximationcouldbejusti-
fiedby the largeratioofthe on-site Coulombinteraction
ontheCoO clusterU withrespecttothebandwidthW.
2
Inthe atomiclimit the locallow-energystatesonthe Co
sites are the vacuum state 0 and the single-occupied
| i
hole states aσ , e σ , e σ . The single-particle hole
1 2
| i | i | i
excitations and local atomic states are shown in Fig. 5.
Thesimplestwaytodescribethequasiparticleexcitations
betweenthesestatesistousetheprojectiveHubbardX-
operators that take the no-double occupancy constraint
into account automatically31: Xm Xp,q p q ,
f ↔ f ≡ | ih |
whereindexm (p,q)enumeratesquasiparticles. There
↔
isasimplecorrespondencebetweenthefermionic-likeX-
operators and single-electron creation-annihilation oper-
ators: d = γ (m)Xm, where γ (m) determines
fασ ασ f ασ
Pm
the partial weight of a quasiparticle m with spin σ and
orbital index α. In these notations the Hamiltonian of
the Hubbard model in the limit U has the form:
→∞
H =− (εp−µ)Xfp,p− tmfgm′Xfm†Xgm′. (3)
Xf,p fX=gmX,m′
6
To study a quasiparticle energy spectrum of the
FIG. 4: (color online) The contour plot of the real part of
system and its thermodynamics we use the Fourier
the magnetic susceptibility Reχ0(k,ω = 0) as a function of
themomentum in unitsof 2π/a (left), and theFermi surface transform of the two-time retarded Green func-
forcorrespondingdopingconcentrationx(right). Thearrows tion in the frequency representation, Gασ(k,E)
≡
tinedxtic.ate the scattering wave vectors Qi as described in the DDdkασ(cid:12)d†kασEEE. This can be rewritten as:
G (k,E(cid:12)) = γ (m)γ (m)Dmm′(k,E), where
ασ (cid:12) ασ β∗σ ′
mP,m′
inNaxCoO2duetomulti-orbitaleffectsinthiscompound Dmm′(k,E) = Xm Xm′† is the matrix Green
k k
whichcomplicatesthesituation. Therefore,inthefollow- DD (cid:12) EEE
function in the X-opera(cid:12)torsrepresentation.
ing we discuss three different approximations valid for (cid:12)
Using the diagram technique for Hubbard X-
different U/t ratio.
operators32,33 one obtains the generalized Dyson
equation34:
A. Hubbard-I approximation 1
Dˆ(k,E)=hGˆ−01(E)−Pˆ(k,E)tˆk+Σˆ(k,E)i− Pˆ(k,E).
To analyze the regime of strong electron correlations (4)
we projectthe doubly occupied states out andformulate Here, Gˆ−01(E) stands for the (exact) local Green func-
aneffectivemodelequivalenttotheHubbardmodelwith tion,Gm0 m′(E)=δmm′/[E−(εp−εq)]. IntheHubbard-
6
Hubbard-Iapproximationsonefindsthenarrowingofthe
bands with lowering the doping concentration x due to
doping dependence of the quasiparticle’s spectral weight
introduced by the strength operator Pˆ. However, the
dopingevolutionoftheFSisqualitativelysimilartothat
in the rigid-band picture. Namely, the bandwidth re-
duction and the spectral weight renormalization do not
change the topology of the FS. As a result, the presence
ofthestrongelectroniccorrelationswithinHubbard-Iap-
proximation do not change qualitatively our results for
the bare susceptibility. Quantitatively, the critical con-
centration x shifts towards higher values of the doping
m
and becomes x 0.68. The reason for this shift is the
m
≈
bandnarrowingandtherenormalizationofthequasipar-
ticle’s spectral weight, which enters the equation that
determines the position of the chemical potential µ.
Luttinger’s theorem, which holds for a perturbative
expansion of the Green’s function in terms of the inter-
actionstrengthisviolatedwithinthe Hubbard-Iapprox-
imation. This violation is due to the renormalization of
the spectral weight of the Green function by the occu-
pation factors in the strength operator in Eq. (4). This
is the reason why in spite of the e band narrowing the
′g
e hole pockets at the Fermi surface are still present at
′g
x=0.33.
B. Gutzwiller approximation
FIG. 6: (color online) Calculated band structure and The Gutzwiller approximation35,36,37 for the Hubbard
the Fermi surface topology for NaxCoO2 for x = model provides a good description for the correlated
0.33,0.47,0.58,0.68. Thedashed(red),solid(blue)anddash- metallic system. Its multiband generalization was for-
dotted (cyan) curves represent the results of the rigid-band, mulated in Ref.38. In this approach, the Hamiltonian
the Gutzwiller, and the Hubbard-I approximations, respec- describing the interacting system far from the metal-
tively. Thehorizontal(green)linedenotesthepositionofthe
insulator transition for U >>W
chemical potential µ.
H =H + U n n , (6)
0 α fα fα
↑ ↓
I approximation the self-energy Σˆ(k,E) is equal to zero Xf,α
and the strength operator Pˆ(k,E) is replaced by the with H being the free electron Hamiltonian (1), is re-
0
sum of the occupation factors, Pmm′(k,E) Pmm′ = placed by the effective non-interacting Hamiltonian:
→
δmm′hDXfp,pE+DXfq,qEi, m = m(p,q). Here ’h...i’ H = (εα+δεασ µ)n
stands for the usual thermodynamic average. Thus, one eff − − fασ
obtains: fX,α,σ
Dˆ(0)(k,E)=hGˆ−01−Pˆtˆki−1Pˆ. (5) − fX6=g,σXα,β t˜αfgβd†fασdgβσ+C. (7)
In the paramagnetic phase the occupation factors are: Here, t˜αfgβ = tαfgβ√qασ√qβσ is the renormalized hopping,
DXf0,0E = x, DXfaσ,aσE = 1−2x, DXfe1,2σ,e1,2σE = 0 bqαitσal=’s 1fi−llxninαgσ,fancαtσor=s,hxΨ=0|n1fασ|Ψn0i≡ishntfhαeσeiq0uiasttiohne oforr-
which yields the diagonal form of the strength operator, − ασ
Pˆ =diag 1+x,x,x . Therefore, the quasiparticle bands the chemical potential. δεασ aPαrσe the Lagrange multipli-
2
formed by(cid:0)the a1g (cid:1) a1g hoppings will be renormalized ers yielding the correlation induced shifts of the single-
→
by the (1 + x)/2 factor, while the quasiparticle bands electron energies. The constant C is determined from
formed by the e′g hopping elements will be renormalized the condition that the ground state energy is the same
by x. for both Hamiltonians
In Fig. 6 the quasiparticle spectrum, the DOS, and
theFSaredisplayedindifferentapproximations. Within Ψ H Ψ = Ψ H Ψ , (8)
0 eff 0 g g
h | | i h | | i
7
where Ψ is the wave function of the free electron sys- which sums all particle-hole(particle) ladder graphs for
0
| i
tem(7),and Ψ isthe Gutzwiller wavefunctionforthe the generating functional self-consistently valid for the
g
| i
Hamiltonian (6). intermediate strength of the correlations. The FLEX
The Lagrange multipliers are determined by minimiz- equations for the single-particle Green function G, the
ing the energy, self-energy Σ, the effective interaction V, the bare (χ0)
and renormalized spin (χs) and charge (χc) susceptibili-
hΨ0|Heff |Ψ0i = − (εα+δεασ−µ)hnfασi0 ties read
Xα,σ
− fX=g,σXα,βt˜αfgβDd†fασdgβσE0+C,(9) GΣk((ωωn)) == [Tωn−τVk+µ(ω−Σk(ωωn))G]−1(,ω ), ((1121))
6 k n N k−p n− m p m
with respect to the orbital filling factors n . Here C = Xp,m
ασ
δεασn , as determined from Eq. (8). This results 3 1
ασ V (ν ) = U2 χs(ν )+ χc(ν ) χ0(ν ) (,13)
αP,σ q m (cid:20)2 q m 2 q m − q m (cid:21)
in the following expression for the single-electron energy
T
renormalizations: χ0(ν ) = G (ω +ν )G (ω ), (14)
q m −N k+q n m k n
1 Xk,n
δεασ = 2(1−nασ)fX=g,βt˜αfgβDd†fασdgβσE0. (10) χs,c(ν ) = χ0q(νm) , (15)
6 q m 1 Uχ0(ν )
∓ q m
It is this energy shift that forces the e FS hole pockets
′g
tosinkbelowtheFermienergy16,whichisclearlyseenin where ω =iπT(2n+1) and ν =iπT(2m). In the last
n m
the doping-dependent evolution of the quasiparticle dis- equation the ’ ’ sign in the denominator corresponds to
persionandtheFSasobtainedwithinGutzwillerapprox- theχs(ν ),wh−ilethe’+’signcorrespondstotheχc(ν ).
q m q m
imation (Fig. 6). Although the narrowing of the bands We compute the Matsubara summations using ’almost
due to strong correlations is similar to the one found in real contour’ technique of Ref.40. I.e., the contour inte-
the Hubbard-I approximation, the FS obeys Luttinger’s gralsareperformedwithafiniteshiftiγ(0<γ <iTπ/2)
theorem. Note, in contrast to the Hubbard-I approxi- into the upper half-plane. All final results are analyti-
mation the relative positions of the t -bands are also callycontinuedfromω+iγ ontothe realaxisω+i0+ by
2g
renormalized by δεασ. Pad´e approximation. The following results are based on
Atthesametime,forx>0.4thetopologyoftheFSin FLEX solutions using a lattice of 64x64 sites with 4096
theGutzwillerapproximationisqualitativelythesameas equidistantω-pointsintheenergyrangeof[ 30,30]. The
−
in the rigid-band picture. The also yields similar results temperature has been kept at T =0.05τ, where τ is the
forthe baresusceptibility’sdopingdependencediscussed hopping amplitude to the nearest neighbors for the a
1g
in Section III. The only effect of the strong correlations band corresponding to ta1ga1g = ta1ga1g = ta1ga1g. The
1 2 3
for χ is the observed shift of the critical concentration Hubbard repulsion was set to U =8τ.
0
towardshighervalues,x 0.6. Thisisduetocombined Previously,theFLEXapproximationhasbeenapplied
m
≈
effectofthebandsnarrowingandthedopingdependence successfully to the study of superconductivity as well as
ofthea1g ande′g band’srelativepositions,determinedby spin and charge excitations in NaxCoO241,42. Comple-
the Eq. (10). mentary,wewillfocusonthequasiparticledispersionand
Note, for x < 0.4, due to different FS topology that studytheimpactofthemomentumandfrequencydepen-
occurs in the Gutzwiller approximation, the bare sus- dencies of the Σ(k,ω), and the role played by the next-
ceptibility differs from that obtained in Ref.29 where the nearest hopping integral, τ , corresponding to ta1ga1g =
′ 4
strong renormalization of the electronic bands removing ta1ga1g =ta1ga1g. The quasiparticle dispersionE , which
5 6 k
e′g pockets away from the FS was neglected. is determined from equation Ek τk+µ Σk(Ek)=0,
− −
is shown in Fig. 7 for τ = 0 and τ = 0.45, in units of
′ ′
−
τ. First, observe that the local minimum around the Γ
C. FLEX approximation point appears only if the next-nearest-neighbor hopping
τ isincludedwhichagreeswithourpreviousfindings. In
′
AcertaindisadvantageoftheGutzwillerandHubbard- addition, we obtain a pronounced mass enhancement of
I like approximations is that the dynamic character of theorderofunityattheFScrossings-theso-calledkink
electronic correlations is not taken into account within structure. This enhancement is due to low-energy spin
theseapproaches. Atthesametime,themomentumand fluctuations which are present in χs(ω)41.
q
frequency dependencies of the self-energy Σ(k,ω) play To shed more light onto the two-dimensionalspin cor-
a crucial role, in particular, for determining the low- relations, in Fig. 8 we display the static spin structure
energies excitations close to the Fermi level. In this sub- factor Reχs(ω = 0) from the FLEX for two different
k
section we focus on the a -band with nearest and next- doping concentrations. As doping increases from x = 0
1g
nearest hopping integrals only and employ the single- towards x = 0.35 the maximum in the spin susceptibil-
band Fluctuation Exchange approximation (FLEX)39 ity χs(ω = 0) moves towards the K-point of the first
k
8
FIG. 7: Quasiparticle dispersion Ek (in units of τ, relative FIG.8: Dopingdependenceofthestaticspinstructurefactor
to µ) within FLEX approximation for (a) τ′ = 0 and (b) Reχsk(ω =0) for (a) τ′ =0 and (b) τ′ =−0.45. Notice that
τ′= 0.45 and for two doping concentrations. forlargeU =8τ thecommensuratepeakatKpointisabsent
− at a verylow x.
BZ and develops into a sharp and commensurate peak
at Q and the incommensurate spin fluctuations are
AFM
suppressed. One may also note that the commensurate
peak is 60% larger for τ = 0.45 than for τ = 0.
′ ′
∼ −
Theseresultsareconsistentwiththoseobtainedinapre-
vioussections. Wefurthernoticesmoothevolutionofthe
quasiparticledynamicswithdopinginNa CoO showing
x 2
no sign of unusual behavior at x=0.
The frequency dependence of the imaginary part of
the quasiparticle self-energy, i.e. ImΣ , near the FS is
k
shown in Fig. 9. We find the self-energy to be nearly
isotropic along the FS with only a weak maximum oc-
curringintothe directionofthe commensuratespin fluc- FIG. 9: Frequency dependence of quasiparticle self-energy
tuations. NeartheFermienergytheself-energyisclearly ImΣk near the FS in direction Γ K for (a) τ′ = 0 and
− −
proportionalto ω2 at low energiesfor alldopings shown, (b) τ′=−0.45.
which is indicative of the normal Fermi-liquid behavior.
This is in sharp contrast to the FLEX analysis of the
V. CONCLUSION
Hubbardmodel onthe square lattice close to half-filling.
There one typically finds ’marginal’ Fermi-liquid behav-
ior with ImΣ ω over a wide range of frequencies43,44. Toconclude,wehavecalculatedthe dopingdependent
k
≈
Therefore,alongthislineoneistemptedtoconcludethat magnetic susceptibility in the tight-binding model with
the normal state of the superconducting cobaltates is ab-initio calculated parameters. We find, that at a crit-
moreofconventionalmetallicnaturethanintheHigh-T ical doping concentration, x , electron pocket develops
c m
cuprates. Thisisevenmoreso,ifonerealizesfromFig.9 ontheFSinthecenteroftheBrillouinzone. Forx<x ,
m
thatthequasiparticlescatteringratedisplaysitssmallest the system shows a tendency towards an 120 AFM or-
◦
curvature for x = 0.35, which implies the quasiparticles dered state, while for x > x a peak in the magnetic
m
to be rather well defined there. For lower x proximity susceptibility forms at small wave vectors indicating a
of the FS to the van Hove singularity (see flat region of strongtendency towardsanitinerantFS state. Within a
dispersionin Fig. 7) enhances both the absolute value of tight-bindingmodelwehaveestimatedx tobe approx-
m
ImΣ ω as well as the curvature. This effect is most imately0.56. Analyzingthe influence ofstrongCoulomb
k
∝
pronounced for τ =0. repulsion and the corresponding reduction of the band-
′
9
widthandthequasiparticlespectralweightinthestrong- displayaconventionalFermi-liquidtypeofenergydepen-
coupling Hubbard-I and Gutzwiller approximations, we dence. Wehavealsoshownthatthe staticspinstructure
have shown that the critical concentration changes to factor exhibits a large commensurate peak at wave vec-
x 0.68andx 0.6,respectively. Atthesametime, tor Q for doping concentrations of x 0.35. This
m m AFM
≈ ≈ ≈
the underlying physics of the formation of the itinerant response was found to be significantly enhanced by the
FM state remains the same. next-nearest-neighbor hopping, emphasizing its signifi-
Weneglectedthebonding-antibondingsplittingdueto cance.
the 3-dimensionalityinthe non-intercalatedcompounds.
This splitting was taken into account in Ref.41, where
within the FLEX approximation the single a -band
1g
Acknowledgments
Hubbard model was considered. The results obtained
alsosuggesta tendency to FMfluctuations for highdop-
ing concentrations. The presence of a local band mini- We would like to thank G. Bouzerar, P. Fulde, S.G.
mum aroundthe Γ point playeda crucialrole,similar to Ovchinnikov,N.B.Perkins,D.Singh,ZiqiangWang,and
our present study. V. Yushankhai for useful discussions, I. Mazin for criti-
To analyze the low-energy quasiparticle properties at calreadingofthe manuscript,andS.Borisenkofor shar-
low doping concentrations we have employed the single- ingwithustheexperimentalresultspriortopublication.
band Hubbard model within the FLEX approximation. M.M.K.acknowledgesupportformINTAS(YSGrant05-
We havefound a significantFS mass enhancementof or- 109-4891)and RFBR (Grants 06-02-16100,06-02-90537-
derunityduetoquasiparticlescatteringfromspinfluctu- BNTS). A.S. and V.I.A. acknowledge the financial sup-
ations. In contrast to the Hubbard model on the square portfromRFBR(Grants04-02-16096,06-02-81017),and
lattice we havefound the quasiparticlescattering rate to NWO (Grant 047.016.005).
∗ Electronic address: [email protected] (2006).
1 K. Takada, H. Sakurai, E. Takayama-Muromachi, F. 14 P. Zhang, W. Luo, M.L. Cohen, and S.G. Louie, Phys.
Izumi, R.A. Dilanian, and T. Sasaki, Nature 422, 53 Rev.Lett. 93, 236402 (2004).
(2003). 15 H.Ishida,M.D.Johannes,andA.Liebsch,Phys.Rev.Lett.
2 I. Terasaki, Y. Sasago, and K. Uchinokura, Phys. Rev. B 94, 196401 (2005).
56, R12685 (1997). 16 S. Zhou, M. Gao, H. Ding, P.A. Lee, and Z. Wang, Phys.
3 Y. Wang, N. Rogado, R.J. Cava, and N.P. Ong, Nature Rev.Lett. 94, 206401 (2005).
423, 425 (2003). 17 D.J. Singh, Phys. Rev.B 68, 020503(R) (2003).
4 M.L. Foo, Y. Wang, S. Watauchi, H.W. Zandbergen, T. 18 A.T. Boothroyd, R. Coldea, D.A. Tennant, D. Prab-
He,R.J.Cava,andN.P.Ong,Phys.Rev.Lett.92,247001 hakaran, L.M. Helme, and C.D. Frost, Phys. Rev. Lett.
(2004). 92, 197201 (2004).
5 B.C.Sales,R.Jin,K.A.Affholter,P.Khalifah,G.M.Veith, 19 S.P. Bayrakci, I. Mirebeau, P. Bourges, Y. Sidis, M. En-
and D.Mandrus, Phys. Rev.B 70, 174419 (2004). derle,J.Mesot,D.P.Chen,C.T.Lin,andB.Keimer,Phys.
6 D.J. Singh,Phys. Rev.B 61, 13397 (2000). Rev.Lett. 94, 157205 (2005).
7 K.-W.Lee, J.Kuneˇs,and W.E. Pickett,Phys.Rev.B 70, 20 L.M.Helme,A.T.Boothroyd,R.Coldea,D.Prabhakaran,
045104 (2004). D.A. Tennant, A. Hiess, and J. Kulda, Phys. Rev. Lett.
8 M.Z. Hasan, Y.-D. Chuang, D. Qian, Y.W. Li, Y. Kong, 94, 157206 (2005).
A. Kuprin, A.V. Fedorov, R. Kimmerling, E. Rotenberg, 21 W.KohnandL.J.Sham,Phys.Rev.A.140,1133(1965).
K.Rossnagel,Z.Hussain,H.Koh,N.S.Rogado,M.L.Foo, 22 O.K. Andersen and O. Jepsen, Phys. Rev. Lett. 53, 2571
and R.J. Cava, Phys. Rev.Lett. 92, 246402 (2004). (1984).
9 H.-B. Yang, S.-C. Wang, A.K.P. Sekharan, H. Matsui, S. 23 J.D. Jorgensen, M. Avdeev, D.G. Hinks, J.C. Burley, and
Souma, T. Sato, T. Takahashi, T. Takeuchi, J.C. Cam- S. Short,Phys. Rev.B. 68, 214517 (2003).
puzano, R. Jin, B.C. Sales, D. Mandrus, Z. Wang, and H. 24 N. Marzari and D. Vanderbilt, Phys. Rev. B. 56, 12847
Ding, Phys. Rev.Lett. 92, 246403 (2004). (1997).
10 H.-B. Yang, Z.-H. Pan, A.K.P. Sekharan, T. Sato, S. 25 V.I. Anisimov, D.E. Kondakov, A.V. Kozhevnikov, I.A.
Souma, T. Takahashi, R. Jin, B.C. Sales, D. Mandrus, Nekrasov, Z.V. Pchelkina, J.W. Allen, S.-K. Mo, H.-D.
A.V.Fedorov,Z.Wang,andH.Ding,Phys.Rev.Lett.95, Kim, P. Metcalf, S. Suga, A. Sekiyama, G. Keller, I.
146401 (2005). Leonov, X. Ren, and D. Vollhardt, Phys. Rev. B. 71,
11 D.Qian,D.Hsieh,L.Wray,Y.-D.Chuang,A.Fedorov,D. 125119 (2005).
Wu, J.L. Luo, N.L. Wang, L. Viciu, R.J. Cava, and M.Z. 26 M.D. Johannes, D.A. Papaconstantopoulos, D.J. Singh,
Hasan, Phys.Rev.Lett. 96, 216405 (2006). and M.J. Mehl, Europhys.Lett. 68, 433 (2004).
12 L.Balicas,J.G.Analytis,Y.J.Jo,K.Storr,H.Zandbergen, 27 K.Kuroki,S.Onari,Y.Tanaka,R.Arita,andT.Nojima,
Y.Xin,N.E.Hussey,F.C.Chou,andP.A.Lee,Phys.Rev. Phys. Rev.B 73, 184503 (2006).
Lett. 97, 126401 (2006). 28 S. Borisenko, privatecommunication.
13 D.J.SinghandD.Kasinathan,Phys.Rev.Lett.97,016404 29 M.D.Johannes,I.I.Mazin,D.J.Singh,andD.A.Papacon-
10
stantopoulos, Phys. Rev.Lett. 93, 097005 (2004). dens. Matter 9, 7343 (1997).
30 M.Indergand,Y.Yamashita,H.Kusunose,andM.Sigrist, 39 N.E. Bickers, D.J. Scalapino, and S.R. White, Phys. Rev.
Phys. Rev.B 71, 214414 (2005). Lett. 62, 961 (1989); J. Altmann, W. Brenig, and A.P.
31 J.C.Hubbard,Proc.Roy.Soc.LondonA277,237(1964). Kampf, Eur. Phys. J. B 18,429 (2000).
32 R.O. Zaitsev, Sov. Phys.JETP 41, 100 (1975). 40 J. Schmalian, M. Langer, S. Grabowski, and K.H. Benne-
33 Yu.IzumovandB.M.Letfullov,J.Phys.: Condens.Matter mann, Computer Phys.Comm. 93, 141 (1996).
3, 5373 (1991). 41 K.Kuroki,S.Ohkubo,T.Nojima,R.Arita,S.Onari,and
34 S.G.OvchinnikovandV.V.Val’kov,Hubbard Operators in Y. Tanaka, cond-mat/0610494 (unpublished); K. Kuroki,
the Theory of Strongly Correlated Electrons(ImperialCol- S.Onari,Y.Tanaka,R.Arita,andT.Nojima, Phys.Rev.
lege Press, London, 2004). B 73, 184503 (2006).
35 M.C. Gutzwiller, Phys. Rev. Lett. 10, 159 (1963); Phys. 42 Y. Yanase, M. Mochizuki, and M. Ogata, J. Phys. Soc.
Rev. B 134A, 923 (1964); Phys. Rev. B 137A, 1726 Jpn. 74, 2568 (2005), ibid.3351 (2005).
(1965). 43 J. Altmann,W. Brenig, and A.P. Kampf, Eur. Phys.J. B
36 F. Gebhard, Phys. Rev.B 41, 9452 (1990). 18,429 (2000).
37 G.KotliarandA.E.Ruckenstein,Phys.Rev.Lett.57,1362 44 S. Wermbter,Phys.Rev. B 55, R10149 (1997).
(1986).
38 J.Bu¨unemann,F.Gebhard,andW.Weber,J.Phys.: Con-