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
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
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
gsections.6While these e′
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
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
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
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
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.
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
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
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
gcarriers. In any case,
g)=0.3 eV˚ A.
gband maximum and EF.
gFermi surface, as
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-
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′
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′
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
gbands below EF that
gband is above EF
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