Scanning Tunneling Spectroscopy in the Superconducting State and Vortex Cores of the β-Pyrochlore KOs_ {2} O_ {6}

Abstract

We performed the first scanning tunneling spectroscopy measurements on the pyrochlore super-conductor KOs2O6 (Tc = 9.6 K) in both zero magnetic field and the vortex state at several temper-atures above 1.95 K. This material presents atomically flat surfaces, yielding spatially homogeneous spectra which reveal fully-gapped superconductivity with a gap anisotropy of 30%. Measurements performed at fields of 2 and 6 T display a hexagonal Abrikosov flux line lattice. From the shape of the vortex cores, we extract a coherence length of 31–4 A, in agreement with the value derived from the upper critical field Hc2. We observe a reduction in size of the vortex cores (and hence the coher-ence length) with increasing field which is consistent with the unexpectedly high and unsaturated upper critical field reported. The discovery of superconductivity in the β-pyrochlore osmate compounds AOs 2 O 6 (A = K, Rb, Cs) [1] has high-lighted the question of the origin of superconductivity in classes of materials which possess geometrical frustra-tion [2, 3]. Interest has been predominantly focused on the highest-T c compound KOs 2 O 6 which presents many striking characteristics. In particular, the absence of in-version symmetry in its crystal structure [4] raises the question of its Cooper pair symmetry and the possibility of spin singlet-triplet mixing [5, 6]. The pyrochlore osmate compound KOs 2 O 6 displays a critical temperature T c = 9.6 K, the largest in its class of materials (CsOs 2 O 6 and RbOs 2 O 6 which differ only by the nature of the alkali ion have T c s of 3.3 and 6.3 K re-spectively). Although band structure calculations show that the K ion does not influence the density of states (DOS) at the Fermi level [7, 8], it seems to affect sev-eral key properties [9]. In particular, the first order phase transition revealed by specific heat measurements in magnetic fields at the temperature T p ≈ 7.5 K has been ascribed to a "freezing" of its rattling motion [10]. The negative curvature of the resistivity as a function of temperature also indicates a large electron-phonon scattering [11]. Specific heat measurements [12] sug-gest the coexistence of strong electron correlations and strong electron-phonon coupling, two generally antago-nistic phenomena with respect to the superconducting pairing symmetry. The nature of the symmetry remains a controversial subject in the literature. NMR [13] and µSR [14] data suggest anisotropic gap functions with nodes whereas thermal conductivity experiments [15] fa-vor a fully-gapped state.

Full text

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arXiv:0704.0529v1 [cond-mat.supr-con] 4 Apr 2007
Scanning Tunneling Spectroscopy in the Superconducting State and Vortex Cores of
the β-pyrochlore KOs2O6
C. Dubois,∗G. Santi, I. Cuttat, C. Berthod, N. Jenkins, A. P. Petrovi´ c, A. A. Manuel, and Ø. Fischer
DPMC-MaNEP, Universit´ e de Gen` eve, Quai Ernest-Ansermet 24, 1211 Gen` eve 4, Switzerland
S. M. Kazakov, Z. Bukowski, and J. Karpinski
Laboratory for Solid State Physics ETHZ, CH-8093 Z¨ urich, Switzerland
(Dated: February 1, 2008)
We performed the first scanning tunneling spectroscopy measurements on the pyrochlore super-
conductor KOs2O6 (Tc = 9.6 K) in both zero magnetic field and the vortex state at several temper-
atures above 1.95 K. This material presents atomically flat surfaces, yielding spatially homogeneous
spectra which reveal fully-gapped superconductivity with a gap anisotropy of 30%. Measurements
performed at fields of 2 and 6 T display a hexagonal Abrikosov flux line lattice. From the shape of
the vortex cores, we extract a coherence length of 31–40˚ A, in agreement with the value derived from
the upper critical field Hc2. We observe a reduction in size of the vortex cores (and hence the coher-
ence length) with increasing field which is consistent with the unexpectedly high and unsaturated
upper critical field reported.
PACS numbers: 74.70.Dd, 74.50.+r, 74.25.Qt
The discovery of superconductivity in the β-pyrochlore
osmate compounds AOs2O6(A = K, Rb, Cs) [1] has high-
lighted the question of the origin of superconductivity
in classes of materials which possess geometrical frustra-
tion [2, 3]. Interest has been predominantly focused on
the highest-Tccompound KOs2O6which presents many
striking characteristics. In particular, the absence of in-
version symmetry in its crystal structure [4] raises the
question of its Cooper pair symmetry and the possibility
of spin singlet-triplet mixing [5, 6].
The pyrochlore osmate compound KOs2O6displays a
critical temperature Tc= 9.6 K, the largest in its class of
materials (CsOs2O6 and RbOs2O6 which differ only by
the nature of the alkali ion have Tcs of 3.3 and 6.3 K re-
spectively). Although band structure calculations show
that the K ion does not influence the density of states
(DOS) at the Fermi level [7, 8], it seems to affect sev-
eral key properties [9]. In particular, the first order
phase transition revealed by specific heat measurements
in magnetic fields at the temperature Tp ≈ 7.5 K has
been ascribed to a “freezing” of its rattling motion [10].
The negative curvature of the resistivity as a function
of temperature also indicates a large electron-phonon
scattering [11].Specific heat measurements [12] sug-
gest the coexistence of strong electron correlations and
strong electron-phonon coupling, two generally antago-
nistic phenomena with respect to the superconducting
pairing symmetry. The nature of the symmetry remains
a controversial subject in the literature. NMR [13] and
µSR [14] data suggest anisotropic gap functions with
nodes whereas thermal conductivity experiments [15] fa-
vor a fully-gapped state.
The peculiar behavior of KOs2O6 is demonstrated
by its upper critical magnetic field Hc2, whose tem-
perature dependence is linear down to sub-Kelvin tem-
peratures and whose amplitude is above the Clogston
limit [16].One possible interpretation is the occur-
rence of spin-triplet superconductivity driven by spin-
orbit coupling [5, 6]. Alternatively, it has also been sug-
gested that this behavior can be explained by the peculiar
topology of the Fermi surface (FS) sheets of KOs2O6,
assuming that superconductivity occurs mainly on the
closed sheet [16].
The understanding of the physics of this compound
would greatly benefit from a detailed knowledge of the
local density of states (LDOS). Scanning Tunneling Spec-
troscopy (STS) is an ideal tool for this, particularly
since it allows one to map the vortices in real space and
also access the normal state below Tc by probing their
cores [17, 18, 19, 20]. In this Letter we present a detailed
STS study of KOs2O6single crystals, including the first
vortex imaging in this material.
The KOs2O6single crystals were grown from Os and
KO2in oxygen-filled quartz ampoules. Their dimensions
are around 0.3 × 0.3 × 0.3 mm3. The details of their
chemical properties as well as their growth conditions can
be found in Ref. 4. AC susceptibility measurements show
a very sharp superconducting transition (∆Tc= 0.35 K).
Our measurements are carried out using a home-built low
temperature scanning tunneling microscope featuring a
compact nanopositioning stage [21] to target the small-
sized crystals. Electrochemically etched iridium tips are
used for STS measurements on as-grown single crystal
surfaces and the differential conductivity was measured
using a standard AC lock-in technique.
The surface topography of as-grown samples (Fig. 1a)
reveals atomically flat regions speckled with small corru-
gated islands a few˚ Angstr¨ oms high whose spectroscopic
characteristics are noisy and not superconducting (thus
restraining our field of view for spectroscopic imaging).

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0

50100
x (nm)
150200
0
50
100
150
200
−6
−4
−2
0
2
4
6
y (nm)
Height z (˚ A)
−5 −4 −3 −2 −1
Bias voltage V (mV)
012345
0
5
10
15
20
25
30
100 (˚ A)
Conductance (shifted, arb. units)
(a) (b)
FIG. 1: (a) Large-scale topography of KOs2O6 (T = 2 K,
Rt = 60 MΩ); the box shows the measurement area for the
vortex maps. (b) Spectroscopic trace along a 100˚ A path
taken on an atomically flat region with one spectrum every
1˚ A. The spectra show raw data offset vertically for clarity
(T = 2 K, Rt = 20 MΩ).
The large flat regions display highly homogeneous super-
conducting spectra (Fig. 1b), which were perfectly repro-
ducible over the timescale of our experiments (4 months).
We have checked that the spectra obtained by varying
the tunnel resistance Rtall collapse onto a single curve,
thus confirming true vacuum tunneling conditions. We
have also verified that the numerical derivative of the
tunnel current with respect to the voltage gives the same
spectroscopic signature as the dI/dV lock-in signal. We
stress that all measurements presented in this paper are
raw data.
The lack of inversion symmetry in this compound to-
gether with several experimental findings raises the ques-
tion of the symmetry of the gap function. In order to
clarify this point, we have fitted our data to several sym-
metry models, focusing on the question of the presence
or absence of nodes and the amplitude of any possible
gap anisotropy. We therefore considered three scenarii
with an approximate angular dependence of the gap, i.e.
an isotropic s-wave (∆0), a d-wave (∆cos2φ) with nodes
and an “anisotropic” s-wave (∆0+ ∆sinφ) which has
the same angular dependence as the s-p-wave singlet-
triplet mixed state [6]. We do not take the real topol-
ogy of the FS [7] into account, since it comprises two
3D Fermi sheets and is hence unlikely to have any sig-
nificant effect on the gap structure. For an anisotropic
gap, ∆(φ), the quasiparticle DOS is given by N(ω) ∝
|Re[?(ω+iΓ)/?(ω + iΓ)2− |∆(φ)|2?φ]| where Γ is a phe-
nomenological scattering rate. In addition, we included
broadenings due to the experimental temperature and
the lock-in in our fits. The results are presented in Fig. 2.
The d-wave model can be rejected at this stage since its
zero bias conductance (ZBC) is larger than in experi-
ment (increasing Γ in the model can only increase the
ZBC). The differences between symmetries appear much
more clearly in the second derivative spectrum (d2I/dV2,
Fig. 2d) which is not surprising as it emphasizes the varia-
tions of the DOS on a small energy scale and is very sensi-
tive to the model parameters (in contrast with the dI/dV
−4 −3 −2−101234
−2
−1
0
1
2
V (mV)
d2I/dV2


Experiment
anisotropic s−wave
s−wave
d−wave
−2 −1012
0
0.5
1
1.5
V (mV)
dI/dV (norm.)
−505
0
0.5
1
1.5
2
2.5
3
3.5
V (mV)
dI/dV (normalized)
1.95 K
3.10 K
4.00 K
5.10 K
6.00 K
9.00 K
10.00 K
anisotropic
s-wave
1.09
0.40
0.05
3.58
(meV)
∆0
∆
Γ
2∆
kBTc
sd
-1.22
-
0.12
2.93 3.66
1.52
0
(a) (b)
(c)
T = 1.95 K
T = 1.95 K
(d)
FIG. 2:
(a) Normalized dI/dV spectra at different temperatures from
1.95 to 10 K (spectra are offset vertically for clarity). (b) Pa-
rameters for the different theoretical models. (c) Comparison
of the experimental spectrum at low temperature and low en-
ergy with the different theoretical models; the color codes are
explained in (d). (d) Same as (c) for the second derivative
d2I/dV2.
Experimental and theoretical tunneling spectra.
curve). The best fit is clearly given by the “anisotropic”
s-wave model with an anisotropy of around 30%. With
respect to the singlet-triplet mixed state, we note that
we do not see any evidence in our data for a second co-
herence peak arising from spin-orbit splitting. Since the
3D nature of both sheets implies that tunneling takes
place in both of them, the absence of a second peak also
rules out the possibility of two different isotropic gaps
on separate FS sheets. Our results would however be
compatible with multiband superconductivity with two
(overlapping) anisotropic gaps. Finally, we see no signa-
ture of a normal-normal tunneling channel in our junc-
tion, suggesting that all electrons involved in the tunnel-
ing process come from the superconducting condensate.
To investigate the temperature evolution of the quasi-
particle DOS, we acquired tunneling conductance spec-
tra at different temperatures between 1.95 K and 10 K
(Fig. 2a). The closure of the gap at the bulk Tcshows
that we are probing the bulk properties of KOs2O6. This

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−6−4 −2
bias voltage V (mV)
0246
0
5
10
15
20
25
Conductance (shifted, arb. units)
−6 −4 −2
bias voltage V (mV)
024 60
5
10
15
20
25




(a)(b)
H = 2 T
H = 6 T
FIG. 3: Spectroscopic traces at T = 2 K across vortices for
a field of 2 T (a) and 6 T (b). The spectra at the vortex
centers are highlighted in red. The spatial variation of the
conductance is shown in the corresponding insets.
is further confirmed by the fact that similar spectra were
also obtained on freshly cleaved surfaces. The totally
flat conductance spectra at higher temperature show no
support for a pseudogap in the DOS above Tc, imply-
ing that the steep decrease in the 1/(T1T) curve around
16 K in NMR data [13] must have a different origin. The
spectra taken between 6 and 9 K (not shown) were very
noisy. This could be explained by the proximity to the
first order transition at Tp≃ 7.5 K [10].
The BCS coupling ratio 2∆max/kBTc inferred from
our measured gaps and critical temperature is about 3.6
for the anisotropic s-wave case, a value slightly smaller
than that reported from specific heat measurements [12].
However, we stress that STS is a direct probe of the su-
perconducting gap. Our findings therefore lead us to the
conclusion that KOs2O6is fully gapped with a significant
anisotropy of around 30%.
We now focus on measurements performed in an ap-
plied magnetic field. In the vortex cores whose radial size
is roughly given by the coherence length ξ, superconduc-
tivity is suppressed leading to a drastic change in the
LDOS which can be measured by STM. Our measure-
ments were performed for two fields, 2 and 6 T, over the
particularly flat region of about 60 × 60 nm2(Fig. 1a).
Each measurement was taken at 2 K with a typical ac-
quisition time of 40 hours.
The results are presented in Figs 3 and 4. The vor-
tex maps (insets of Fig. 3 and Figs 4a and
the ZBC normalized to the conductance at 6 meV. Fig. 3
displays the spectra taken along traces passing through
vortex cores for each of the two fields considered. The
suppression of superconductivity and its effect on the
conductance in a vortex core can clearly be seen. The
vortex maps show a roughly hexagonal vortex lattice
with vortex spacings d = 352 ± 17˚ A and 216 ± 21˚ A
at 2 and 6 T respectively, in agreement with the spacings
d =
nal lattice [22], i.e. 345˚ A and 199˚ A. We ascribe the
4b) show
?2Φ0/H√3?1/2expected for an Abrikosov hexago-
variations in the core shapes and the deviation from a
perfectly hexagonal lattice to vortex pinning. In partic-
ular, the vortex identified by the arrow in Fig. 4 appears
to be split. We attribute this to the vortex oscillating
between two pinning centers during the measurement, a
situation which has been seen in other compounds [23].
One should also note that the islands (surface defects)
at the border of the measurement area (Fig. 1) could
influence the vortex core shapes and positions.
In order to estimate the coherence length ξ from our
measurements, we now consider the spatial dependence of
the ZBC. Due to the proximity of the vortices, we model
the LDOS as a superposition of isolated vortex LDOS
which can be expressed as N(ω,r) =?
En) + |vn(r)|2δ(ω + En), where ψn(r) = (un(r),vn(r))
is the wave function of the nth vortex core state and
En its energy.An approximate solution for the iso-
lated vortex was given long ago [24] in which the ra-
dial dependence of each ψn(r) consists of a rapidly os-
cillating n-dependent Bessel function multiplied by a
cosh−1/π(r/ξ) envelope common to all states. We there-
fore construct a phenomenological model for our 2D ZBC
maps, σ(ω = 0,r) ∝ N(ω = 0,r), by retaining the slowly
varying parts of the wave functions alone, i.e.
n|un(r)|2δ(ω −
σ(ω = 0,r) = σ0+ Λ
?
i
?
cosh|r − ri|
ξ
?−2
π
(1)
where σ0= 0.13 is the residual normalized conductance
at zero bias in the absence of field (Fig. 2c), Λ a scaling
factor, ξ the coherence length and the sum runs over all
the vortices with positions riin the map. Using (1), we
fitted ri and ξ over the entire map for each field, thus
considering all imaged vortices to determine ξ.
The results from the 2D fits are presented in Fig. 4c
and d in map format and along traces selected to pass
through vortex cores in Fig. 4e and f. The traces help
to visualize the spatial extent of the vortices and assess
the (extremely high) quality of the 2D fits. We first ob-
serve that the normalized ZBC between the vortices is
slightly enhanced at H = 2 T but increases strongly at
H = 6 T with respect to the value at zero-field (Fig. 2c),
indicating a significant core overlap. From our data taken
at T = 2 K, we obtain ξ = 35 ± 3˚ A and 45 ± 7˚ A at
H = 6 and 2 T respectively (the uncertainties are esti-
mated from the spread of the results obtained on several
maps: two for 6 T and three for 2 T). Using Ginzburg-
Landau theory, we extrapolate the corresponding T = 0
values as ξ = 31±3 and 40±6˚ A respectively, consistent
with the value derived from Hc2. Furthermore, our re-
sults indicate that the vortex size decreases with increas-
ing field and, although at the limit of the error bars, we
believe this trend to be genuine. In addition, this finding
is consistent with the abnormally large Hc2: if the vor-
tices become smaller as the field increases, the material
can accommodate more vortices before the breakdown

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020
Distance (nm)
4060
0.4
0.6
0.8
1

(f)
020
Distance (nm)
4060
0.2
0.4
0.6
0.8
1
ZBC (normalized)
(e)
x (nm)


0

1020304050
0
10
20
30
40
50
(d)
x (nm)
y (nm)

0

10203040 5060
0
10
20
30
40
50
60(c)



0
10
20
30
40
50(b)
y (nm)


0
10
20
30
40
50
60
(a)
H = 2 T
H = 6 T
0 0.20.40.6 0.81
FIG. 4: (a), (b) Experimental ZBC maps (T = 2 K) normal-
ized to the background conductance at 2 and 6 T respectively
with corresponding fits (c), (d); large values (red) correspond
to normal regions (i.e. vortex cores) and low values (blue) to
superconducting (gapped) regions. (e), (f) Experimental ZBC
profiles across vortex centers together with the corresponding
profiles from the 2D fits (red lines).
of superconductivity, leading to a higher upper critical
field. This correlates with the observed temperature de-
pendence of the upper critical field.
We find that the spectra at the vortex centers are flat
for both fields (Fig. 3), showing the presence of localized
quasiparticle states in the vortex cores. However, our
spectra show no excess spectral weight at or close to zero
bias and thus no ZBCP which is the generally expected
signature of vortex core states. The absence of a ZBCP
is at first glance striking considering the large mean free
path ℓ ≈ 200 nm ≫ ξ in KOs2O6[15]. In fact, this ab-
sence is common to many non-cuprate superconductors,
the only known exceptions being 2H-NbSe2[17, 25, 26]
and YNi2B2C [27]. Although no definitive theory cur-
rently exists to explain such an absence, a possible ex-
planation assumes that the scattering rate is strongly en-
hanced in the vortex cores. This interpretation is sup-
ported by our numerical solutions of the Bogoliubov-de
Gennes equations for a single vortex with an r-dependent
scattering rate Γ. Furthermore, these simulations show a
radial dependence of the LDOS which is fully consistent
with (1).
In conclusion, we have presented the first scanning tun-
neling spectroscopic measurements on superconducting
KOs2O6. The fitted spectra demonstrate that KOs2O6
is a fully-gapped superconductor with an anisotropy of
around 30%, possibly resulting from a s-p singlet-triplet
mixed state allowed by the lack of inversion symme-
try. We have imaged hexagonal vortex lattices matching
Abrikosov’s prediction for 2 and 6 T fields. Using Caroli-
de Gennes-Matricon theory we extract a field-dependent
coherence length of 31–40˚ A, in good agreement with the
thermodynamic estimate from Hc2. The absence of a zero
bias conductance peak, the apparent field dependence of
ξ and the precise radial dependence of the LDOS all call
for deeper exploration.
We acknowledge T. Jarlborg, M. Decroux, I. Maggio-
Aprile and P. Legendre for valuable discussions and thank
P.E. Bisson, L. Stark and M. Lancon for technical sup-
port. This work was supported by the Swiss National
Science Foundation through the NCCR MaNEP.
∗Electronic address: duboisc@mit.edu
[1] S. Yonezawa, Y. Muraoka, Y. Matsushita and Z. Hiroi,
J. Phys.:Condens. Matter 16, L9 (2004); ibid, J. Phys.
Soc. Jpn 73, 819 (2004); S. Yonezawa, Y. Muraoka and
Z. Hiroi, J. Phys. Soc. Jpn 73, 1655 (2004).
[2] P. W. Anderson, Mater. Res. Bull. 8, 153 (1973).
[3] H. Aoki, J. Phys.: Condens. Matter 16, V1 (2004).
[4] G. Schuck, S. Kazakov, K. Rogacki, N. Zhigadlo, and
J. Karpinski, Phys. Rev. B 73, 144506 (2006).
[5] P. A. Frigeri, D. F. Agterberg, A. Koga, and M. Sigrist,
Phys. Rev. Lett. 92, 097001 (2004); ibid, Phys. Rev. Lett.
93, 099903 (2004).
[6] N. Hayashi, Y. Kato, P. A. Frigeri, K. Wakabayashi, and
M. Sigrist, Physica C 437-38, 96 (2006).
[7] J. Kuneˇ s, T. Jeong, and W. E. Pickett, Phys. Rev. B 70,
174510 (2004).
[8] R. Saniz, J. Medvedeva, L.-H. Ye, T. Shishidou, and
A. Freeman, Phys. Rev. B 70, 100505(R) (2004).
[9] J. Kuneˇ s and W. E. Pickett, Phys. Rev. B 74, 094302
(2006).
[10] Z. Hiroi,S.Yonezawa,
mat/0607064, to be published in the Proceedings of
HFM2006 (J.Phys.: Condens. Matter) (2006).
[11] Z. Hiroi, S. Yonezawa, J. Yamaura, T. Muramatsu, and
Y. Muraoka, J. Phys. Soc. Jpn. 74, 1682 (2005).
[12] M. Br¨ uhwiler, S. Kazakov, J. Karpinski, and B. Batlogg,
Phys. Rev. B 73, 094518 (2006).
[13] K. Arai, J. Kikuchi,
S. Yonezawa, Y. Muraoka, and Z. Hiroi, Physica B 359-
361, 488 (2005).
[14] A. Koda,W. Higemoto,
R. Kadono, S. Yonezawa, Y. Muraoka, and Z. Hiroi, J.
andJ.Yamaura,cond-
K. Kodama,M. Takigawa,
K. Ohishi,S. R. Saha,

Page 5

5
Phys. Soc. Jpn. 74, 1678 (2005).
[15] Y. Kasahara, Y. Shimono, T. Shibauchi, Y. Matsuda,
S. Yonezawa, Y. Muraoka, and Z. Hiroi, Phys. Rev. Lett.
96, 247004 (2006).
[16] T. Shibauchi, L. Krusin-Elbaum, Y. Kasahara, Y. Shi-
mono, Y. Matsuda, R. D. McDonald, C. H. Mielke,
S. Yonezawa, Z. Hiroi, M. Arai, et al., Phys. Rev. B 74,
220506 (2006).
[17] H. Hess, R. Robinson, R. Dynes, J. J. Valles, , and
J. Waszczak, Phys. Rev. Lett. 62, 214 (1989).
[18] Y. DeWilde, M. Iavarone, U. Welp, V. Metlushko,
A. Koshelev, I. Aranson, G. Crabtree, and P. Canfield,
Phys. Rev. Lett. 78, 4273 (1997).
[19] M. Eskildsen, M. Kugler, S. Tanaka, J. Jun, S. Kazakov,
J. Karpinski, and Ø. Fischer, Phys. Rev. Lett. 89, 187003
(2002).
[20] N. Bergeal, V. Dubost, Y. Noat, W. Sacks, D. Roditchev,
N. Emery, C. H´ erold, J.-F. Marˆ ech´ e, P. Lagrange, and
G. Loupias, Phys. Rev. Lett. 97, 077003 (2006).
[21] C. Dubois, P. E. Bisson, S. Reymond, A. A. Manuel, and
Ø. Fischer, Rev. Sci. Instrum. 77, 043712 (2006).
[22] A. A. Abrikosov, Sov. Phys.-JETP 5, 1174 (1957).
[23] B. Hoogenboom, M. Kugler, B. Revaz, I. Maggio-Aprile,
Ø. Fischer, and C. Renner, Phys. Rev. B 62, 9179 (2000).
[24] C. Caroli, P. de Gennes, and J. Matricon, Physics Letters
9, 307 (1964).
[25] F. Gygi and M. Schluter, Phys. Rev. B 41, 822 (1990).
[26] C. Renner, A. D. Kent, P. Niedermann, Ø. Fischer, and
F. L´ evy, Phys. Rev. Lett. 67, 1650 (1991).
[27] H. Nishimori, K. Uchiyama, S. Kaneko, A. Tokura,
H. Takeya, K. Hirata, and N. Nishida, J. Phys. Soc. Jpn.
73, 3247 (2004).