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arXiv:cond-mat/0604002v1 [cond-mat.supr-con] 31 Mar 2006

Destruction of the small Fermi surfaces in NaxCoO2by Na disorder

D.J. Singh1and Deepa Kasinathan2

1Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge,TN 37831-6032 and

2Department of Physics, University of California Davis, Davis, CA 95616

(Dated: February 6, 2008)

We show using density functional calculations that the small e′

destroyed by Na disorder. This provides a means to resolve the prediction of these sections in band

structure calculations with their non-observation in angle resolved photoemission experiments.

gFermi surfaces in NaxCoO2 are

PACS numbers:

The layered oxide, NaxCoO2, has attracted consid-

erable interest because of its enhanced thermoelectric

properties,1and because, when hydrated the material

becomes a superconductor, possibly with unconventional

pairing.2,3. NaxCoO2consists of triangular sheets of Co

ions, with nominal d electron count, 5+x.

coordinated by distorted edge sharing octahedra, made

from triangular sheets of O ions above and below the Co

sheets. These CoO2 tri-layers are stacked in an alter-

nating fashion to form prismatic sites, which contain the

Na.

Establishing the electronic structure is requisite for un-

derstanding of both the thermoelectric behavior and the

superconductivity. Local density approximation (LDA)

band structure calculations,4,5show a Fermi level that

lies near the top of a narrow manifold of t2gbands, which

is separated by a gap from the higher lying eg derived

states. With the actual rhombohedral Co site symmetry,

the three t2gorbitals are further divided into an agand

two degenerate egsymmetry orbitals, denoted e′

tinguish them from the higher lying egmanifold defined

by the primary crystal field splitting. Significantly, local

density approximation band structure calculations show

the presence of two sheets of Fermi surface over a wide

range of x. These are a large ag derived hole cylinder

around Γ and six small hole sections along the hexagonal

Γ - K lines. The calculated Fermi surfaces of the hy-

drated superconducting compound similarly show both

ag and small e′

small, and therefore are not expected to contribute sub-

stantially to the conduction, they contribute strongly to

the density of states and should be very important for

scattering, magnetic susceptibility and other properties

because of the two dimensional nature of the material

and their heavy mass.7,8

In contrast to the band structure calculations, angle

resolved photoemission (ARPES) experiments by sev-

eral groups at various doping levels observe the large ag

cylinder but do not find the small e′

Early on,4it was noted that although metallic, NaxCoO2

had a small band width compared to plausible values

for the on-site Coulomb repulsion, U. One explanation

that has been advanced for the absence of the e′

tions is that they are destroyed by correlation effects.

LDA+U calculations, which incorporate Coulomb repul-

sion in a static mean field like way, have indeed shown

These are

gto dis-

gsections.6While these e′

gsections are

gsections.9,10,11,12,13

gsec-

that the small sections are removed for reasonable values

of U.7,14However, the LDA+U approximation also fa-

vors insulating, charge ordered ground states.7Removal

of the small sections is also found in the large U limit

Gutzwiller approximation.15However, in metals, fluctu-

ations, including charge and orbital fluctuations are fa-

vored by kinetic energy considerations and the presence

of e′

gsections would open more degrees of freedom for

fluctuations. Dynamical mean field calculations, which

incorporate physics like the LDA+U approximation, but

in a beyond mean field manner that incorporates fluc-

tuations, yielded the opposite tendency to the LDA+U

approximation, i.e an enlargement of the e′

the LDA.16Low temperature specific heat measurements

on superconducting hydrated NaxCoO2 show two dis-

tinct energy gaps,17which is difficult to understand if

there is only one, simple Fermi surface. Thus, the dis-

crepancy between the ARPES and LDA Fermi surfaces

remains a puzzle. Here we resolve this puzzle by showing

that the small sections are destroyed by scattering due

to the Na disorder inherent in the structural chemistry

of NaxCoO2.

As mentioned, the Na ions are contained in O prisms

defined by the O triangular lattice. There are two types

of sites: Those directly above a Co ion, denoted NaCo

here, and those above the holes in the Co sheet, denoted

Nah(these are directly above the O atoms in the O layer

on the opposite side of the Co sheet). There is one NaCo

and one Nahsite per Co, with a total Na filling of x per

Co, or x/2 per site. However, the number of available

sites is smaller since occupation of an NaCo next to an

occupied Nahsite or vice versa would be highly unfavor-

able due to the short Na - Na distance of 1.64˚ A that

would result. In fact, local structures that within a given

Na layer have Na of a given type (NaCo or Nah) hav-

ing Na neighbors of the same type are favored. Also Na

in adjacent layers tend not to coordinate the same Co

ion.18,19,20

To investigate the effect of Na ion ordering, we focus on

the x = 2/3 composition and perform electronic structure

calculations for two supercells with different Na order-

ings and Co sites. Specifically, we report results for two

√3×√3 cells with lattice parameters a=2.84˚ A, c=10.81

˚ A. Results are reported for the ideal structure, with api-

cal O height zO=0.0864 (Ref. 5), but with different Na

orderings, and for fully relaxed atomic positions. Each

gpockets over

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FIG. 1: (color online) Structure of two supercells, A (left) and

B (right). Na ions are indicated by the large blue spheres, Co

by the mid-size green spheres, and O by the small red spheres.

-0.1

-0.05

0

0.05

0.1

0.15

0.2

M

Γ

E(eV)

-0.1

-0.05

0

0.05

0.1

0.15

0.2

LA

E(eV)

-0.1

-0.05

0

0.05

0.1

0.15

0.2

M

Γ

E(eV)

-0.1

-0.05

0

0.05

0.1

0.15

0.2

LA

E(eV)

FIG. 2: (color online) Band dispersions along the Γ-M and

A-L directions for the two relaxed supercells: A (top) and B

(bottom). The different line types are for the three Γ-M or

A-L directions, separated by 120◦, which are non-equivalent

in the relaxed supercells.

supercell contains 6 Co and 12 Na sites in two layers.

The two supercells, as shown in Fig. 1 are: (A) with one

Na layer on the NaCosites and the other on the Nahsites

(both with two Na atoms per layer); and (B) with both

Na layers on the Nahsites.

The calculations were done within the local den-

sity approximation with the general potential linearized

augmented planewave (LAPW) method including local

orbitals.21,22,23Well converged basis sets, of approxi-

mately 2600 LAPW functions plus local orbitals, were

used, with LAPW sphere radii of 1.96 a0for the Na and

Co and 1.52 a0for O.

The calculated band structures near the Fermi energy,

EF, for the relaxed cells are shown in Fig. 2, along the

supercell Γ-M lines. Due to the

riodicity, these are folded Γ-K directions in the original

hexagonal zone, and therefore intersect the small pock-

√3 ×√3 supercell pe-

ets. The band structures obtained for the two supercells

are very similar to each other and to the virtual crystal

band structure.5In particular, at this doping level there

is sufficient kz dispersion that only the outer of the two

ag and e′

gpairs of bands produce Fermi surfaces in the

kz = 0 plane, while both ag and neither e′

duce Fermi surfaces at kz = 1/2. This differs in detail

from virtual crystal calculations,5where both e′

reach maxima above EF in the kz = 1/2 plane. This

difference in the e′

g, but not the agtopology, when using

real Na ions instead of the virtual crystal approximation

is an indication that variations in the Na potential due

to disorder could localize the e′

the maximum of the agband is 0.17 eV above EF, while

the e′

gmaximum is only 0.01 eV above EF. Besides the

positions of the band maxima, there is also a large differ-

ence in band velocity along the Γ-M line, v(ag)=1.4 eV

˚ A, v(e′

The second indication of localization of the e′

comes from the Co and O 1s core level positions. These

measure the variation of the local electrostatic potential

in the supercells. Supercell A has two different types of

Co, the four Co coordinated by a Na in the prism im-

mediately above or below and the two without. In the

unrelaxed cell those with Na are 0.033 eV higher than

the Co without. This variation is higher than the en-

ergy difference between the e′

Secondly, while supercell B has all Co equivalently coor-

dinated, the O sites differ in coordination in both super-

cells. The variation in O core level position is 0.06 eV in

both A and B. This variation may be significant because

the bands are in fact hybridized Co-O bands. The vari-

ations in core level position are largely preserved in the

relaxed cells, and structural inhomogeneities in the Co-O

bond lengths and angles are introduced, which will also

contribute to scattering. The variation of Co 1s levels

is reduced to 0.029 eV in supercell A, i.e. only slightly

smaller than in the unrelaxed cell, and still larger than

the e′

gband maximum relative to EF, while the variation

in the O 1s positions increases to 0.07 eV.25Thus the po-

tential variations due to Na disorder are strong enough

to localize the egstates.26

The relaxation in supercell A is larger than in supercell

B. This is because the Co and nearby NaCoions are both

positive and move apart by 0.02˚ A, while the nearest

neighbor Na-O distance contracts by between 0.006˚ A to

0.01˚ A, depending on the site. There are also distortions

of the O cages. Measured by the difference between the

shortest and longest Co-O distance for a given Co site,

these range from 0.01˚ A(2/3 of the Co in B) to 0.015˚ A

(for the Co with NaConeighbors in A).

In any case, the results show that scattering due to

Na disorder is strong enough to localize the e′

at x = 2/3. The actual position of the e′

mum relative to EF is a function of doping, as is the Na

distribution. The linear size of the e′

predicted by band structure calculations varies from zero

at high x to approximately 0.18% of the dimension of the

gbands pro-

gbands

gcarriers. In any case,

g)=0.3 eV˚ A.

gcarriers

gband maximum and EF.

gcarriers

gband maxi-

gFermi surface, as

2

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zone at x ∼ 0.3.5,8At x=0.3, the e′

eV above EF. Observation of this Fermi surface would

require a mean free path for e′

verse of this reciprocal space size. At x ∼ 2/3 this would

imply ℓeg′ > 50˚ A, where ℓeg′ is the mean free path for

the heavy carriers on the small Fermi surfaces. In met-

als with Fermi surfaces having similar orbital character

(here Co t2g), the constant scattering time approxima-

tion is reasonable. Then ℓ=vFτ, where τ is a scatter-

ing time, vF is the Fermi velocity on a given section of

Fermi surface and ℓ is the mean free path for carriers on

that section. Taking the ratio of the velocities for the

two sheets along the Γ-M line, the implication is that a

mean free path ℓag∼ 250˚ A, for the main Fermi surface

would be needed in order for the small section to be seen.

Note that the small sections are elliptical in shape, elon-

gated along Γ-M, so the criterion in the tangential direc-

tion would be the same as the velocity is larger, and the

size smaller, in proportion. Within kinetic transport the-

ory, the conductivity is given by σ ∝ N(EF)v2

N(EF) is the density of states at EF and vF is the aver-

age Fermi velocity in the direction in which the conduc-

tivity is measured. Thus the large difference in the Fermi

velocity between the large and small sections means that

the conductivity is dominated by the large section, re-

gardless of whether the small sections are present or not.

Thus the measured resistivity is, for practical purposes,

governed by the transport in the agsection. Depending

on the doping level, NaxCoO2samples of the type used

in photoemission have been reported to have ratios of

the resistivity at 300K to the residual resistivity in the

range of 20 - 30,1,27and resistivity saturation at or be-

low ∼ 600K. These values are inconsistent with a mean

free path at low temperature as long as 250˚ A. We note

that although the small sections become larger as x is

lowered, they do so more slowly than in a rigid band pic-

ture and the large ratio of the vF on the two surfaces is

gmaximum lies 0.1

gcarriers of at least the in-

Fτ, where

maintained.5,8

Very recent high resolution photoemission work11

shows dispersions of the agand e′

have a noticeable anti-crossing along Γ-K, for which the

one electron e′

gand ag bands have different symmetries

and would not mix. The observed mixing is strong. Fur-

thermore, a non-dispersive spectral weight is seen just

below EF in the region where the e′

in band calculations. Both of these features are natural

if scattering strong enough to localize the e′

assumed.

To summarize we find that disorder in the Na layer

of NaxCoO2produces sufficiently strong potential varia-

tions to localize the e′

gFermi surface pockets at least for

high x. Other scattering mechanisms and defects will en-

hance this tendency. We argue that the non-observation

of the small pockets and other features seen in recent pho-

toemission experiments can be qualitatively understood

within this framework, specifically that the carriers as-

sociated with the small pocket are localized due to scat-

tering related to Na and other disorder. In the hydrated

superconducting compound, each Na is coordinated with

four H2O molecules, is far from the CoO2sheets and the

Na is in a more ordered pattern with respect to the Co

atoms. One may speculate that this leads to a sufficient

reduction in the potential scattering due to the Na, and

that this could lead to the re-appearance of the e′

tions.

We are grateful for helpful discussions with H. Ding,

M.Z. Hasan, D. Mandrus, R. Jin, I.I. Mazin, M.D. Jo-

hannes and W.E. Pickett.

sored by the Division of Materials Sciences and Engineer-

ing, Office of Basic Energy Sciences, U.S. Department of

Energy, under contract DE-AC05-00OR22725 with Oak

Ridge National Laboratory, managed and operated by

UT-Battelle, LLC. Work at UC Davis supported by DOE

grant DE-FG03-01ER45876.

gbands below EF that

gband is above EF

gpockets is

gsec-

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