arXiv:cond-mat/0405072 v1 4 May 2004
Dimensionality of superconductivity in the infinite-layer
high-temperature cuprate
Sr
0
.
9
M
0
.
1
CuO
2
(M = La, Gd)
V. S. Zapf,
1
N.-C. Yeh,
1
A. D. Beyer,
1
C. R. Hughes,
1
C. H. Mielke,
2
N. Harrison,
2
M. S. Park,
3
K. H. Kim,
3
S.-I. Lee,
3
1
Department of Physics, California Institute of Technology
, Pasadena, CA
2
National High Magnetic Field Laboratory, Los Alamos, NM
3
Department of Physics, Pohang University of Science and Tec
hnology, Pohang, Korea
(Dated: October 20, 2004)
The high magnetic field phase diagram of the electron-doped i
nfinite layer high-temperature
superconducting (high-
T
c
) compound Sr
0
.
9
La
0
.
1
CuO
2
was probed by means of penetration depth
and magnetization measurements in pulsed fields to 60 T. An an
isotropy ratio of 8 was detected for
the upper critical fields with
H
parallel (
H
ab
c
2
) and perpendicular (
H
c
c
2
) to the CuO
2
planes, with
H
ab
c
2
extrapolating to near the Pauli paramagnetic limit of 160 T.
The longer superconducting coherence
length than the lattice constant along the c-axis indicates
that the orbital degrees of freedom of
the pairing wavefunction are three dimensional. By contras
t, low-field magnetization and specific
heat measurements of Sr
0
.
9
Gd
0
.
1
CuO
2
indicate a coexistence of bulk s-wave superconductivity wi
th
large moment Gd paramagnetism close to the CuO
2
planes, suggesting a strong confinement of the
spin degrees of freedom of the Cooper pair to the CuO
2
planes. The region between
H
ab
c
2
and the
irreversibility line in the magnetization,
H
ab
irr
, is anomalously large for an electron-doped high-
T
c
cuprate, suggesting the existence of additional quantum flu
ctuations perhaps due to a competing
spin-density wave order.
PACS numbers: 74.25.Dw,74.25.Op,74.72,Dn,74,25.Bt,74.
25.Fy,74.25.Ha
Keywords: cuprate, superconductivity, infinite layer, com
peting order, upper critical field, Gd substitution
In the high-
T
c
cuprate superconductors, anisotropy has
been suggested to play an important role in the super-
conducting pairing mechanism and the elevated
T
c
in
both experimental and theoretical work [1, 2]. It is
surprising therefore to find superconductivity (SC) with
T
c
= 43 K in the optimal electron-doped infinite-layer
cuprates Sr
0
.
9
M
0
.
1
CuO
2
(M = La, Gd), which exhibit
only a 16% difference between the
a
and
c
tetragonal
lattice parameters. The structure of Sr
0
.
9
M
0
.
1
CuO
2
is
the most basic among all high-
T
c
cuprates, consisting
entirely of CuO
2
sheets separated by rare-earth (RE)
ions with tetragonal lattice parameters
c
= 3
.
41
̊
A
and
a
= 3
.
95
̊
A.[3] The recent success in producing
high-quality polycrystalline samples of the infinite-laye
r
cuprates with no observable impurity phases [3] has en-
gendered a renewed interest in these compounds. X-ray
near-edge absorption spectroscopy indicate electron dop-
ing, [4] and bulk SC has been verified by powdered mag-
netization (
M
) measurements [5] and specific heat (
C
)
measurements (data presented later in this work). Sev-
eral recent studies of these high purity polycrystalline
samples suggest three-dimensional (3D) superconductiv-
ity in Sr
0
.
9
La
0
.
1
CuO
2
. Scanning tunnel spectroscopy
(STS) measurements [6] indicate an unconventional but
isotropic s-wave superconducting gap with no pseudogap
at zero field. The s-wave symmetry of the gap is also sup-
ported by specific heat measurements and Cu-site substi-
tution studies,[6, 7, 8] although it may be contradicted
by NMR measurements.[9] Kim et al [5] estimated the
c-axis coherence length (
ξ
c
) from a Hao-Clem analysis
[10] of the reversible magnetization of grain-aligned poly
-
crystal, and found that
ξ
c
exceeds the spacing between
the CuO
2
planes, indicating 3D superconductivity. On
the other hand, they also find significant anisotropy be-
tween magnetic fields
H
≤
5 T oriented parallel and per-
pendicular to the CuO
2
planes, with an anisotropy ratio
γ
=
ξ
c
/ξ
ab
=
H
ab
c
2
/H
c
c
2
= 9
.
3, which is larger than
γ
= 5
observed in YBa
2
Cu
3
O
7
−
δ
although much smaller than
γ
= 55 observed in optimally doped Bi
2
Sr
2
CaCu
2
O
8
−
δ
.
[5, 11, 12, 13] It is interesting to note that the only ma-
jor crystallographic difference between the a-b and the
c directions in Sr
0
.
9
La
0
.
1
CuO
2
is the presence of oxygen
in the a-b plane, which allows coupling of adjacent Cu
spins and has been implicated as the cause of antiferro-
magntic ordering or spin fluctuations in other members
of the high-
T
c
cuprate family, as well as a possible mech-
anism for superconducting pairing. The importance of
the CuO
2
planes to the SC in Sr
0
.
9
M
0
.
1
CuO
2
is further
supported by the fact that Ni substitution on the Cu site
rapidly suppresses
T
c
whereas out-of-plane Gd substitu-
tion on the Sr site leaves
T
c
unchanged. [7, 14]
In this work we determine the upper critical field
H
c
2
and the irreversibility field
H
irr
of Sr
0
.
9
La
0
.
1
CuO
2
by
means of magnetization and penetration depth measure-
ments in pulsed magnetic fields up to 60 T in order
to directly investigate the degree of upper critical field
anisotropy and the role of vortex fluctuations. We also
present specific heat (
C
) and magnetization (
M
) mea-
surements in low DC fields to 6 T as a function of tem-
perature (
T
) of Sr
0
.
9
Gd
0
.
1
CuO
2
, confirming the bulk co-
existence of Gd paramagnetism (PM) and SC. Our re-
sults suggest strong confinement of the spin pairing wave
function to the CuO
2
planes and significant field-induced
superconducting fluctuations.
Noncrystalline samples of Sr
0
.
9
La
0
.
1
CuO
2
and
Sr
0
.
9
Gd
0
.
1
CuO
2
were prepared under high pressures as
described previously. [3] Magnetization measurements
in pulsed magnetic fields were performed at the National
2
FIG. 1: a) Change in resonant frequency ∆
f
of the TDO tank
circuit relative to the normal state of Sr
0
.
9
La
0
.
1
CuO
2
as a
function of magnetic field
H
of Sr
0
.
9
La
0
.
1
CuO
2
polycrystal at
various temperatures
T
. The estimated change in penetration
depth
λ
is indicated on the right axis. Inset: ∆
f
as a function
of
T
at zero field where
T
c
= 43
K
. b) Derivative of
λ
with
H
for various
T
, with arrows indicating
H
kink
.
High Magnetic Field Laboratory (NHMFL) in Los
Alamos, NM in a
3
He refrigerator in a 50 T magnet
using a compensated coil. The sample consisted of four
pieces of polycrystalline Sr
0
.
9
La
0
.
1
CuO
2
with a total
mass of 4.8 mg to maximize signal and minimize heating.
The irreversibility field
H
irr
was identified from the onset
of reversibility in the
M
(
H
) loops. The penetration
depth of Sr
0
.
9
La
0
.
1
CuO
2
was determined by measuring
the frequency shift ∆
f
of a tunnel diode oscillator
(TDO) resonant tank circuit with the sample contained
in one of the component inductors. [15] A ten turn 0.7
mm diameter aluminum coil was tightly wound around
the sample with a filling factor of greater than 90%,
with the coil axis oriented perpendicular to the pulsed
field. To maintain temperature stability, the sample
was thermally anchored to a sapphire plate and placed
in
3
He exchange gas. Small changes in the resonant
frequency can be related to changes in the penetration
depth ∆
λ
by ∆
λ
=
−
R
2
r
s
∆
f
f
0
, where
R
is the radius of the
coil and
r
s
is the radius of the sample. [15] In our case,
R
∼
r
s
= 0
.
7 mm and the reference frequency
f
0
∼
60
MHz such that ∆
f
= (0.16 MHz/
μ
m)∆
λ
.
The frequency shift ∆
f
relative to the normal state
and the corresponding ∆
λ
of Sr
0
.
9
La
0
.
1
CuO
2
are shown
in Fig. 1a as a function of
H
. The inset shows the
T
-dependence of ∆
f
in zero magnetic field. The nor-
mal state resonant frequency
f
that is reached with in-
creasing field can only be determined for
T
≥
30 K; for
lower
T
the sample remains superconducting to 60 T so
∆
f
is estimated. The frequency shift of the empty coil
0
10
20
30
40
50
0
10
20
30
40
H(T)
T(K)
Sr
0.9
La
0.1
CuO
2
H
kink
= H
c2
c
H
c2
ab
H
irr
ab
H
irr
from Ref. 3
FIG. 2: Field
H
vs temperature
T
phase diagram of
Sr
0
.
9
La
0
.
1
CuO
2
.
H
ab
c
2
: onset of SC where ∆
f
exceeds 5 kHz
in penetration depth measurements. Open circles and open
diamonds indicate the 10 kHz and 20 kHz onsets, respectively
.
H
ab
irr
: onset of irreversibility in pulsed field
M
vs
H
measure-
ments.
H
kink
: change in slope of
λ
(
H
), identified with
H
c
c
2
(see text). Open squares: onset of irreversibility in
M
vs
T
measurements by Jung et al. [3] Lines are guides to the eye.
has been subtracted from all data. In applied fields, the
SC transition is very broad, which can be attributed to
the large anisotropy in
H
c
2
of the randomly orientated
grains in the polycrystal.[5] By contrast, the SC tran-
sition as a function of
T
at
H
= 0 is very sharp, in-
dicating a high quality sample. Therefore, the onset of
diamagnetism with decreasing
H
in the
λ
(
H
) data can
be identified with the largest
H
c
2
,
H
ab
c
2
for fields in the
CuO
2
planes. The onset is defined as
H
where ∆
f >
5
kHz (∆
λ >
30 nm), just above the noise of the exper-
iment. Different onset criteria have only minor effects
on the determination of
H
ab
c
2
, as shown in Fig. 2 where
H
ab
c
2
using an onset criteria of 10 kHz and 20 kHz are
shown as open circles and diamonds, respectively. The
upper critical field
H
ab
c
2
is linear in
T
up to the
H
= 60
T maximum of the experiment, and extrapolates to 153
T at zero temperature. (Although if we assume the typ-
ical Werthamer-Helfand-Hohenberg curvature for an or-
bitally limited superconductor, [16] 153 T would consti-
tute an upper limit for
H
ab
c
2
). The linearly extrapolated
value of 153 T is close to the s-wave Pauli paramagnetic
limit of
H
p
c
2
=
∆
√
2
μ
B
= 159 T, where ∆ = 13 meV
has been determined independently from STS data. [6]
This raises the possibility of spin-limited superconduc-
tivity for
H
in the plane, which has also been observed
in YBa
2
Cu
3
O
7
−
δ
.[17]
Determination of the critical fields for
H
along the c-
axis,
H
c
c
2
, from these data on noncrystalline samples is
more difficult. However, at fields below the onset of dia-
magnetism we do observe a significant change in slope of
the
λ
(
H
) data, indicated as
H
kink
in Fig. 1b. The
H
kink
vs
T
curve is shown in Fig. 2 and extrapolates to 12 T at
zero temperature. This value of
H
kink
(
T
= 0) is close to
H
c
c
2
= 14 T determined from a Hao-Clem analysis men-
tioned previously. [5] We therefore associate
H
kink
with
H
c
c
2
. In
λ
(
H
) measurements of a noncrystalline sample,
3
a change in slope near
H
c
c
2
could be expected since the
number of grains in the polycrystal that are supercon-
ducting varies with
H
for
H
c
c
2
< H < H
ab
c
2
, whereas for
H < H
c
c
2
the entire sample is superconducting, yield-
ing different
H
dependencies of the flux expulsion in
these two regions. Our data yield an anisotropy ratio
γ
=
dH
ab
c
2
dT
/
H
c
c
2
dT
∼
8, roughly in agreement with
γ
= 9
.
3
determined from low field studies. [5]
In penetration depth measurements of single crys-
talline organic superconductors for
H
along the conduct-
ing planes, a kink below
H
c
2
for fields has been asso-
ciated with the vortex melting transition.[15] However,
in this work we have determined the vortex dynamics
separately by means of magnetization measurements in
pulsed fields, and we find a significant difference be-
tween the onset of irreversibility in
M
(
H
),
H
irr
, and
H
kink
as shown in Fig. 2. Following similar arguments
for the
λ
(
H
) measurements, we note that
H
ab
irr
> H
c
irr
in cuprate superconductors, therefore we assign the on-
set of irreversibility for polycrystalline Sr
0
.
9
La
0
.
1
CuO
2
to
H
ab
irr
. Although we can’t rule out the possibility that
H
kink
might be associated with a vortex phase trans-
formation, the fact that
H
kink
(
T
→
0) saturates, and
H
kink
(
T
→
0)
<< H
ab
c
2
(
T
→
0) indicates that
H
kink
(
T
)
is unlikely caused by a thermally-induced vortex melt-
ing transition for
H
||
ab
. Future pulsed-field measure-
ments on grain-aligned or epitaxial thin film samples will
be necessary to conclusively determine whether
H
kink
(
T
)
obtained in this work may be identified with
H
c
c
2
(
T
).
The region between
H
ab
irr
and
H
ab
c
2
in the phase dia-
gram in Fig. 2 is significantly larger than is observed
in other electron-doped high-
T
c
compounds where
H
c
2
typically tracks
H
irr
. [2, 18] It is particularly surpris-
ing that
H
ab
irr
(
T
→
0)
∼
45 T is much smaller than
H
ab
c
2
(
T
→
0)
∼
150 T. In hole-doped cuprates, a large
separation between
H
irr
and
H
c
2
is often observed and
is generally referred to as a vortex-liquid phase due to
thermally induced fluctuations. The large discrepancy
between
H
ab
irr
and
H
ab
c
2
in e-doped Sr
0
.
9
La
0
.
1
CuO
2
even to
T
→
0 suggests the presence of field-enhanced SC fluc-
tuations in Sr
0
.
9
La
0
.
1
CuO
2
. Enhanced SC fluctuations
down to very low
T
may be consistent with a scenario
of SC coexisting with a competing order, such as a spin-
density wave (SDW) near a quantum critical point. [19]
In particular, antiferromagnetic spin fluctuations associ
-
ated with the competing SDW can be enhanced by ex-
ternal fields. [19, 20, 21] The conjecture of a competing
order in the SC state of Sr
0
.
9
La
0
.
1
CuO
2
is also consistent
with our recent STS studies, [22, 23] where we observe
the emergence of a second energy gap with increasing
tunnelling current upon the closing of the SC gap. In con-
trast to the SC gap, the current-induced gap is not spa-
tially uniform probably due to interactions of the SDW
with charge disorder. [21] We note that experimental
evidence for coexistence of a SDW with cuprate super-
conductivity has been found in other high-
T
c
compounds.
[24, 25, 26, 27, 28]
To further investigate the dimensionality of the super-
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
-1
0
1
2
3
4
5
6
M (emu/g)
H (T)
Sr
0.9
Gd
0.1
CuO
2
0
5
20
30
40
50
60
M (emu/g)
T(K)
2 T
0.5 T
5 T
FIG. 3: Magnetization
M
vs
H
of polycrystalline
Sr
0
.
9
Gd
0
.
1
CuO
2
at
T
= 5 K. Inset shows field-cooled (closed
symbols) and zero-field-cooled (open symbols)
M
vs
T
data
at
H
= 5 T, 2 T, and 0.5 T.
conductivity, low-field
C
(
T
) and
M
(
T, H
) of polycrys-
talline Sr
0
.
9
Gd
0
.
1
CuO
2
were measured with
H
up to 6
T and
T
down to 1.8 K in a Quantum Design Physical
Properties Measurement System and a SQUID magne-
tometer, respectively. In contrast to in-plane Ni substi-
tution on the Cu site, out of plane Gd substitution on
the Sr site does not suppress
T
c
. In fact, the Gd ions
exhibit local moment paramagnetism (PM) that coexists
with SC to low
T
. [14] Figure 3 shows
M
(
H
) at
T
= 5
K, and zero field/field cooled magnetization curves as a
function of
T
in the inset. The
M
(
H
) curve in the main
figure can be viewed as a superposition of a SC hysteresis
curve and a Brillouin function resulting from the PM of
Gd. Further proof for this coexistence is evident in the
inset, which shows a large positive magnetization asso-
ciated with Gd paramagnetism, but nevertheless signifi-
cant hysteresis between zero-field-cooled and field-cooled
curves, indicating superconductivity.
In Fig. 4, the magnitude of the paramagnetic contri-
bution from the Gd ions is investigated quantitatively.
The main figure shows
C
(
T
) at
H
= 0 and
H
= 6 T.
The
H
= 0 data is fit by a
T
3
dependence to model
the phonon contribution, (the electronic contribution at
these temperature can be neglected). For
H
= 6 T,
C
(
T
)
can be fit by the same
T
3
dependence as the
H
= 0 data,
plus an additional contribution from the Gd paramag-
netic moments, derived from mean field theory assuming
the Hund’s rule
J
= 7
/
2 moment, and one Gd ion for
every ten unit cells. The fit is remarkable, considering
that there are no fitting parameters. In the upper inset
of Fig. 4, 1
/χ
is plotted as function of
T
, where the line
is a Curie-Weiss fit with
μ
eff
= 8
.
2
μ
B
, which is close to
the Hund’s rule moment of 7.6
μ
B
and the typically ob-
served Gd moment of 8
μ
B
. Thus we can conclude that
all of the Gd ions in the sample are paramagnetic and
coexist with SC down to 1.8 K, despite the close prox-
imity of the Gd ions to the CuO
2
planes (1.7
̊
A). This is
evidence for a strong confinement of the superconduct-
ing singlet spin pair wave function to the CuO
2
planes.
On the other hand, the c-axis superconducting coherence
4
0
10
20
30
40
0
100
200
300
0
1
2
3
4
5
6
5
10
15
20
25
30
C (mJ/mol K
2
)
T(K)
Sr
0.9
Gd
0.1
CuO
2
H = 6 T
H = 0 T
μ
eff
= 8.2
μ
B
χ
-1
(mol/cm
3
)
T (K)
∆
C/T vs T
0
1
2
3
4
40
42
44
46
FIG. 4: Main figure: specific heat
C
vs
T
at
H
= 0 and
6 T. Lines are fits to a field-independent
T
3
phonon term
plus a contribution from Gd paramagetism, assuming a Gd
total momentum of
J
= 7
/
2. Upper inset: inverse magnetic
susceptibility 1
/χ
vs
T
with
H
= 100 Oe, fit by a Curie-Weiss
law above
T
c
= 43 K with
μ
eff
= 8
.
2
μ
B
and
θ
CW
=
−
27 K.
Lower inset: electronic specific heat ∆
C
vs
T
showing the
superconducting transition.
length
ξ
c
= 5
.
2
̊
A is longer than the spacing between the
Gd ion and the CuO
2
planes (1.7
̊
A), and also exceeds the
interplane distance, implying 3D SC. The notion of 3D
SC is corroborated by our STS studies, [6] which probe
the charge degrees of freedom and reveal an isotropic
s-wave superconducting gap. The apparent problem of
3D isotropic s-wave SC coexisting with strong Gd local
moments less than 1.7
̊
A from the CuO
2
planes can be
resolved by considering the spin and charge (orbital) de-
grees of freedom of the Cooper pairs separately. Whereas
the singlet spin pairing is confined to the CuO
2
planes,
the orbital pair wave function could still overlap adjacent
CuO
2
planes, resulting in 3D SC for all values of
T < T
c
and
H < H
c
2
. The bulk nature of the SC is evident in
the lower inset of Fig. 4, which shows the electronic con-
tribution to
C
plotted as ∆
C/T
vs
T
. The peak near
43 K is associated with
T
c
, and the ratio ∆
C/γT
c
is 2.9,
assuming a Sommerfeld coefficient [8] of
γ
= 1
.
2 mJ/mol
K
2
.
In conclusion, a large upper critical field anisotropy ra-
tio
γ
= 8 has been inferred from penetration depth mea-
surements of Sr
0
.
9
La
0
.
1
CuO
2
in pulsed fields, despite the
nearly cubic crystal structure. The in plane upper criti-
cal field
H
ab
c
2
extrapolates close to the Pauli paramagnetic
limit
H
P
c
2
= 159 T, suggesting possible spin limiting for
this orientation, as has been observed in YBa
2
Cu
3
O
7
−
δ
.
[17] There is a large separation between
H
ab
c
2
and the irre-
versibility field
H
ab
irr
, which extends down to
T
→
0, and
is abnormal for electron doped high-
T
c
cuprates. This
suggests the existence of field-induced superconducting
spin fluctuations perhaps due to a competing SDW. In
spite of the significant anisotropy in the upper critical
fields,
ξ
c
is longer than the spacing between CuO
2
planes,
indicating three-dimensionality of the orbital wave func-
tion. The low-field thermodynamic measurements of
M
(
T
) and
C
(
T
) for Sr
0
.
9
Gd
0
.
1
CuO
2
polycrystals indi-
cate a coexistence of bulk SC with Gd paramagnetism,
with the full
J
= 7
/
2 Hund’s rule moment despite the
close proximity of Gd atoms to the CuO
2
planes, and a
T
c
of 43 K in both Sr
0
.
9
Gd
0
.
1
CuO
2
and Sr
0
.
9
La
0
.
1
CuO
2
.
This can interpreted in terms of a strong confinement of
the spin degrees of freedom of the Cooper pairs to the
CuO
2
planes, whereas the orbital wave functions over-
lap adjacent CuO
2
planes, and exhibit isotropic s-wave
symmetry as determined by STS measurements. [6]
This work was supported by the National Science
Foundation under Grant No. DMR-0103045 and DMR-
0405088, and the National High Magnetic Field Labora-
tory at Los Alamos, NM. V.Z. acknowledges support by
the Caltech Millikan Postdoctoral Fellowship program.
[1] See, e.g. P. W. Anderson, ”The Theory of Superconduc-
tivity in High-T
c
cuprates,” and references therein.
[2] M. B. Maple, E. D. Bauer, V. S. Zapf, and J. Wos-
nitza, ’Survey of Important Experimental Results’ in
”Unconventional Superconductivity in Novel Materials,”
Springer Verlag, to be published, and references therein.
[3] C. U. Jung
et al.
, Physica C
366
, 299 (2002).
[4] R. S. Liu
et al.
, Solid State Comm.
118
, 367 (2001).
[5] M.-S. Kim
et al.
, Phys. Rev. B
66
, 214509 (2002).
[6] C.-T. Chen
et al.
, Phys. Rev. Lett.
88
, 227002 (2002).
[7] C. U. Jung
et al.
, Phys. Rev. B
65
, 172501 (2002).
[8] Z. Y. Liu
et al.
, to appear in Europhys. Lett. (2004);
available at cond-mat/0306238 (unpublished).
[9] G. V. M. Williams
et al.
, Phys. Rev. B
65
, 224520 (2002).
[10] Z. Hao
et al.
, Phys. Rev. B
43
, 2844 (1991).
[11] M.-S. Kim
et al.
, Solid State Comm.
123
, 17 (2002).
[12] D. E. Farrell
et al.
, Phys. Rev. Lett
63
, 782 (1989).
[13] D. E. Farrell
et al.
, Phys. Rev. Lett.
61
, 2805 (1988).
[14] C. U. Jung, J. Y. Kim, S.-I. Lee, and M.-S. Kim, Physica
C
391
, 319 (2003).
[15] C. Mielke
et al.
, J. Phys.: Condens. Matter
13
, 8325
(2001).
[16] N. R. Werthamer, E. Helfand, and P. C. Hohenberg,
Phys. Rev. B
147
, 295 (1966).
[17] J. L. OBrien
et al.
, Phys. Rev. B
61
, 1584 (2000).
[18] P. Fournier and R. L. Greene, Phys. Rev. B
68
, 094507
(2003).
[19] E. Demler, S. Sachdev, and Y. Zhang, Phys. Rev. Lett.
87
, 067202 (2001).
[20] S. Sachdev and S. C. Zhang, Science
295
, 452 (2002).
[21] C.-T. Chen and N.-C. Yeh, Phys. Rev. B
68
, 220505(R)
(2003).
[22] C.-T. Chen
et al.
(unpublished).
[23] N. C. Yeh
et al.
, to appear in Physica C (2004).
[24] Y. S. Lee
et al.
, Phys. Rev. B
60
, 3643 (1999).
[25] H. J. Kang
et al.
, Nature
423
, 522 (2003).
[26] M. Matsuura
et al.
, Phys. Rev. B
68
, 144503 (2003).
[27] B. Lake
et al.
, Nature
415
, 299 (2002).
[28] H. A. Mook
et al.
, Phys. Rev. B
66
, 144513 (2002).