Microsoft Word - K-SC6.doc
April 2, 2007 
I. INTRODUCTION 
The study of non-Cu-based oxide superconductors has been 
extended during the last decade, aiming at understanding the role of 
electron correlations in the mechanism of superconductivity or 
searching for a novel pairing mechanism, hopefully to reach a 
higher Tc.  An interesting example recently found is a family of 
pyrochlore oxide superconductors.  The first discovered is 
α-pyrochlore rhenate Cd2Re2O7 with Tc = 1.0 K
1-3 and the second 
β-pyrochlore osmate AOs2O6 with Tc = 3.3, 6.3, and 9.6 K for A = 
Cs,4 Rb,5-7 and K,8 respectively.  They crystallize in the cubic 
pyrochlore structure of space group Fd-3m and commonly possess 
a 3D skeleton made of ReO6 or OsO6 octahedra.
9 The "pyrochlore" 
sublattice occupied by the transition metal ions is comprised of 
corner-sharing tetrahedra that are known to be highly frustrating for 
a localized spin system with antiferromagnetic nearest-neighbor 
interactions.  
A unique structural feature for the β-pyrochlores is that a 
relatively small A ion is located in an oversized atomic cage made 
of OsO6 octahedra, Fig. 1.  Due to this large size mismatch, the A 
atom can rattle in the cage.10 The rattling has been recognized 
recently as an interesting phenomenon for a class of compounds 
like filled skutterudites11 and Ge/Si clathrates12 and attracted many 
researchers, because it may suppress thermal conductivity leading 
to an enhanced thermoelectric efficiency.  On the other hand, the 
rattling is also intriguing from the viewpoint of lattice dynamics:  
it gives an almost localized mode even in a crystalline material and 
often exhibits unusual anharmonicity.11 Hence, it is considered that 
the rattling is a new type of low-lying excitations that potentially 
affects various properties in a crystal at low temperature. 
In the β pyrochlores, specific heat experiments found 
low-energy contributions that could be described approximately by 
the Einstein model and determined the Einstein temperature TE to 
be 70 K, 60 K, and 40 or 31 K for A = Cs, Rb, and K, 
respectively.13, 14  This tendency illustrates uniqueness of the 
rattling, because, to the contrary, one expects a higher frequency 
for a lighter atom in the case of conventional phonons.  Moreover, 
it was demonstrated that the specific heat  shows an unusual T5 
dependence at low temperature below 7 K for A = Cs and Rb, 
instead of a usual T3 dependence from a Debye-type phonon.13  
On one hand, structural refinements revealed large atomic 
displacement parameters at room temperature of 100Uiso = 2.48, 
4.26, and 7.35 Å2 for A = Cs, Rb, and K, respectively,10 and 3.41  
Å2 for Rb.15  Particularly, the value for K is enormous and may be 
the largest among rattlers so far known in related compounds.  
This trend over the β-pyrochlore series is ascribed to the fact that 
an available space for the A ion to move in a rather rigid cage 
increases with decreasing the ionic radius of the A ion.10  Kuneš et 
al. calculated an energy potential for each A ion and found in fact a 
large anharmonicity, that is, a deviation from a quadratic form 
expected for the harmonic oscillator approximation.16 Especially 
for the smallest K ion, they found 4 shallow potential minima 
locating away from the center (8b site) along the <111> directions 
pointing to the nearest K ions, as schematically shown in Fig. 1.  
The potential minima are so shallow that the K ion may not stop at 
one of them even at very low temperature. 
The electronic structures of α-Cd2Re2O7 and β-AOs2O6 have 
been calculated by first-principle density-functional methods, 
which reveal that a metallic conduction occurs in the (Re, Os)-O 
network:16-20 electronic states near the Fermi level originate from 
transition metal 5d and O 2p orbitals.  Although the overall shape 
of the density of states (DOS) is similar for the two pyrochlores, a 
difference in band filling may result in different properties; Re5+ 
for α-Cd2Re2O7 has two 5d electrons, while Os
5.5+ for β-AOs2O6 
has two and a half.  Moreover, a related α-pyrochlore Cd2Os2O7 
with Os5+ (5d3) exhibits a metal-to-insulator transition at 230 K.21, 
Various experiments have been carried out on the pyrochlore 
oxide superconductors to elucidate the mechanism of the 
superconductivity.  Most of the results obtained for α-Cd2Re2O7 
indicate that it is a weak-coupling BCS-type superconductor.23, 24  
In contrast, results on the β-pyrochlores are somewhat 
controversial. Although the Tc increases smoothly from Cs to K, 
the jump in specific heat at Tc, the upper critical field, and the 
Sommerfeld coefficient all exhibit a large enhancement toward K.25  
Thus, the K compound is always distinguished from the others.   
Pressure dependence of Tc was also studied for the two pyrochlores, 
showing a common feature: as pressure increases, Tc first increases, 
exhibits a broad maximum and goes to zero above a critical 
pressure that depends on the system, for example, about 6 GPa for 
the K compound.26-29 
On the symmetry of the superconducting gap for the 
β-pyrochlores, Rb-NMR and µSR experiments gave evidence for 
s-wave superconductivity for RbOs2O6.
30-32  In contrast, Arai et al. 
carried out K-NMR experiments and found no coherence peak in 
the relaxation rate below Tc for KOs2O6, which seemed to indicate 
unconventional superconductivity.32  However, their recent 
interpretation is that the absence of a coherence peak does not 
necessarily mean non-s pairing, because the relaxation rate probed 
by the K nuclei can be affected dominantly by strongly 
overdamped phonons.33  Kasahara et al. measured thermal 
conductivity using a KOs2O6 single crystal and concluded a full 
Extremely strong-coupling superconductivity and anomalous lattice properties in 
the β-pyrochlore oxide KOs2O6 
Z. Hiroi, S. Yonezawa, Y. Nagao and J. Yamaura 
Institute for Solid State Physics, University of Tokyo, Kashiwa, Chiba 277-8581, Japan 
Superconducting and normal-state properties of the β-pyrochlore oxide KOs2O6 are studied by means of 
thermodynamic and transport measurements.  It is shown that the superconductivity is of conventional s-wave 
type and lies in the extremely strong-coupling regime.  Specific heat and resistivity measurements reveal that 
there are characteristic low-energy phonons that give rise to unusual scattering of carriers due to strong 
electron-phonon interactions.  The entity of the low-energy phonons is ascribed to the heavy rattling of the K ion  
confined in an oversized cage made of OsO6 octahedra.  It is suggested that this electron-rattler coupling 
mediates the Cooper pairing, resulting in the extremely strong-coupling superconductivity. 
gap from the insensitivity of thermal conductivity to magnetic 
fields.34  Moreover, very recent photoemission spectroscopy 
(PES) experiments revealed the opening of a large isotropic gap 
below Tc.
35  On one hand, a µSR experiment claimed that the gap 
of KOs2O6 is anisotropic, or otherwise, there are two gaps.
Therefore, the pairing symmetry of the β-AOs2O6 superconductors 
may be of the conventional s wave, aside from minor aspects such 
as anisotropy or multi gaps, which means that the fundamental 
pairing mechanism is ascribed to phonons.  Then, an important 
question is what kind of phonons are relevant for the occurrence.  
To find out the reason of the observed singular appearance toward 
K in the series must be the key to understand interesting physics 
involved in this system. 
Previous studies have suffered poor quality of samples, 
because only polycrystalline samples were available.  The 
Sommerfeld coefficient γ was estimated from specific heat by 
extracting contributions from Os metal impurity to be 40 mJ K-2 
mol-1 for both Cs and Rb,13 and 34 mJ K-2 mol-1 for Rb.7  Recently, 
Brühwiler et al. obtained large values of γ = 76-110 mJ K-2 mol-1 
for KOs2O6 by collecting five dozen of tiny crystals.
14  However, 
there is an ambiguity in their values, because of uncertainty in their 
extrapolation method.  They also reported strong-coupling 
superconductivity with a coupling constant λep = 1.0-1.6.
Recently, we successfully prepared a large single crystal of 1 mm 
size for KOs2O6 and reported two intriguing phenomena: one is a 
sharp and huge peak in specific heat at Tp = 7.5-7.6 K below Tc, 
indicative of a first-order structural transition,37, 38 and the other is 
anisotropic flux pinning at low magnetic fields around 2 T.39  It 
was suggested that the former is associated with the rattling 
freedom of the K ion.  Moreover, anomalous concave-downward 
resistivity was observed down to Tc, suggesting a peculiar 
scattering mechanism of carriers. 
In this paper, we present specific heat, magnetization and 
resistivity measurements on the same high-quality single crystal of 
KOs2O6.  Reliable data on the superconducting and normal-state 
properties are obtained, which provide evidence for an extremely 
strong-coupling superconductivity realized in this compound.  We 
discuss the role of rattling vibrations of the K ion on the 
mechanism of the superconductivity. 
II. EXPERIMENTAL 
A. Sample preparation 
A high-quality single crystal was prepared and used for all the 
measurements in the present study, which was named KOs-729 
after the date of July 29, 2005 when the first experiment was 
performed on this crystal.  It was grown from a pellet containing 
an equimolar mixture of KOsO4 and Os metal in a sealed quartz 
tube at 723 K for 24 h.  Additional oxygen was supplied by using 
the thermal decomposition of AgO placed away from the pellet in 
the tube.  The KOsO4 powder had been prepared in advance from 
KO2 and Os metal in the presence of excess oxygen.  It was 
necessary to pay attention to avoid the formation of OsO4 in the 
course of preparation, which is volatile even at room temperature 
and highly toxic to eyes or nose.  After the reaction, several tiny 
crystals had grown on the surface of the pellet.   Although the 
mechanism of the crystal growth has not yet been understood 
clearly, probably it occurs through partial melting and the 
following reaction with a vapor phase.   
The KOs-729 crystal possesses a truncated octahedral shape 
with a large (111) facet as shown in Fig. 2 and is approximately 1.0 
× 0.7 × 0.3 mm3 in size and 1.302 mg in weight.  The high quality 
of the crystal has been demonstrated by a sharp peak at Tp in 
specific heat,38 which was absent in the previous polycrystalline 
samples or appeared as broad humps in our previous aggregate of 
tiny crystals37 or in five dozen of tiny crystals by Brühwiler et al.14  
Moreover, a dramatic angle dependence of flux-flow resistance was 
observed on this crystal, indicating that flux pinning is enhanced in 
magnetic fields along certain crystallographic directions such as 
[110], [001], and [112].39  This evidences the absence of domains 
in this relatively large crystal.  
A special care has been taken to keep the crystal always in a 
dry atmosphere, because it readily undergoes hydration in air, as 
reported in isostructural compounds such as KNbWO6.
40, 41  Once 
partial hydration takes place in KOs2O6, the second sharp anomaly 
in specific heat tends to collapse.  In contrast, the superconducting 
transition was robust, just slightly broadened after hydration. The 
hydration must be relatively slow in a single crystal compared with 
the case of polycrystalline samples, which may be the reason for 
the broad anomaly or the absence in previous samples.  
B. Physical-property measurements 
Both specific heat and electrical resistivity were measured in a 
temperature range between 300 K and 0.4 K and in magnetic fields 
of up to 14 T in a Quantum Design Physical Property Measurement 
System (PPMS) equipped with a 3He refrigerator.  The magnetic 
fields had been calibrated by measuring the magnetization of a 
standard Pd specimen and also by measuring the voltage of a Hall 
device (F.W. BELL, BHA-921).  Specific heat measurements 
FIG. 1. (Color online) Crystal structure of the β-pyrochlore 
oxide KOs2O6.  The K ion (big ball) is located in an oversized 
atomic cage made of OsO6 octahedra and can move along the 4 
<111> directions pointing to the neighboring K ions in adjacent 
cages. 
FIG. 2. (Color online) Photograph of the KOs-729 crystal used 
in the present study.  It possesses a truncated octahedral shape 
with 111 facets.  The approximate size is 1.0 × 0.7 × 0.3 mm3.  
Resistivity measurements were carried out with a current flow 
along the [-110] direction. 
were performed by the heat-relaxation method.  The KOs-729 
crystal was attached to an alumina platform by a small amount of 
Apiezon N grease.  In each measurement, heat capacity was 
obtained by fitting a heat relaxation curve recorded after a heat 
pulse giving a temperature rise of approximately 2%.  The heat 
capacity of an addendum had been measured in a separate run 
without a sample, and was subtracted from the data.  The 
measurements were done three times at each temperature with a 
scatter less than 0.3% at most. 
Resistivity measurements were carried out by the four-probe 
method with a current flow along the [-110] direction and magnetic 
fields along the [111], [110], [001] or [112] direction of the cubic 
crystal structure.  All the measurements were done at a current 
density of 1.5 A cm-2.  Magnetization was measured in magnetic 
fields up to 7 T in a Quantum Design magnetic property 
measurement system and also up to 14 T in PPMS.  The magnetic 
fields were applied approximately along the [111] or [-110] 
direction. 
III. RESULTS 
A. Superconducting properties 
1. Specific heat 
First of all, we analyze specific heat data in order to obtain a 
reliable value of the Sommerfeld coefficient γ.  There are two 
obstacles:  one is the large upper critical field Hc2 that is 
approximately 2 times greater than our experimental limit of 14 T.  
Brühwiler et al. reported γ = 76 (110) mJ K-2 mol-1 assuming µ0Hc2 
= 24 (35) T by an extrapolation method.14  Their values should be 
modified to γ ~ 100 mJ K-2 mol-1, because recent high magnetic 
field experiments revealed µ0Hc2 = 30.6 T or 33 T.
42, 43  
Nevertheless, there are still large ambiguity in their extrapolation 
method using specific heat data obtained only at H/Hc2 < 0.5.  The 
other difficulty comes from unusual lattice contributions in specific 
heat at low temperature and the existence of a sharp peak at Tp.  
Thus, it is not easy to extract the lattice contribution in a standard 
way used so far.  Here we carefully analyze specific heat data and 
reasonably divide them into electronic and lattice parts, from which 
a reliable value of γ is determined, and information on the 
superconducting gap is attained. 
Figure 3 shows the temperature dependence of specific heat of 
the KOs-729 crystal measured on cooling at zero field and in a 
magnetic field of 14 T applied along the [111] direction.  A 
superconducting transition at zero field takes place with a large 
jump, followed by a huge peak due to the second phase transition at 
Tp = 7.5 K. The entropy-conserving construction shown in the inset 
gives Tc = 9.60 K, ΔC/Tc = 201.2 mJ K
-2 mol-1, which is close to 
the values previously reported,14, 37 and a transition width (ΔTc) of 
0.3 K.  Tc is reduced to 5.2 K at 14 T, which is evident as a bump 
in the 14-T data shown in Fig. 3.  In contrast, Brühwiler et al. 
reported Tc = 6.2 K at 14 T, though the transition was not clearly 
observed in their specific heat data.  The C/T at zero field rapidly 
decreases to zero as T approaches absolute zero.  The absence of a 
residual T-linear contribution in specific heat indicates the high 
quality of the crystal.   
The specific heat of a crystal (C) is the sum of an electronic 
contribution (Ce) and an H-independent lattice contribution (Cl).  
The former becomes Cen for the normal state above Tc, which is 
taken as γT, and Ces for the superconducting state below Tc.  The γ 
is assumed to be T-independent, though it can not be the case for 
compounds with strong electron-phonon couplings.44  In the case 
of KOs2O6, Cl is large relative to Ce: for example, Cl is as large as 
~90% of the total C at just above Tc, as shown later.  In order to 
determine the value of γ, it is crucial to know the low-temperature 
form of Cl.  Since the minimum Tc attained at 14 T is 5.2 K, one 
has to estimate the Cl from the T dependence of the total C above 
~5.5 K.  Two terms in the harmonic-lattice approximation are 
often required for an adequate fit; Cl = β3T
3 + β5T
5.  The first term 
comes from a Debye-type acoustic phonon, and thus is dominant at 
low temperature, while the second term expresses a deviation at 
high temperature.  Actually, this approximation is valid for 
α-Cd2Re2O7, where β3 = 0.222 mJ K
-4 mol-1 and β5 = 2.70 × 10
mJ K-6 mol-1 are obtained by a fit to the data below 10 K.1  The 
Debye temperature ΘD is 458 K from the β3 value.  In strong 
contrast, it was found for two members of β-AOs2O6 that the T
term prevails in a wide temperature range; β5 = 14.2 × 10
-3 mJ K-6 
mol-1 below 5 K for CsOs2O6 and β5 = 30.2 × 10
-3 mJ K-6 mol-1 
below 7 K for RbOs2O6.
The C/T at H = 0 below 7 K shown in Fig. 3 is again plotted in 
two ways as functions of T2 and T4 in Fig. 4.  It is apparent from 
the T2 plot that possible T3 terms expected for ΘD = 458 K and 300 
K are negligibly small compared with the whole magnitude of 
specific heat, just as observed in other members.  On the other 
hand, in the T4 plot, there is distinct linear behavior at low 
temperature below 4 K, indicating that the C approaches 
asymptotically to T5 behavior as T → 0 with a large slope of 
0.3481(6) mJ K-6 mol-1.  It is reasonable to ascribe this T5 
contribution to the lattice, because Ces should decrease quickly as T 
→ 0.  Note that the value of the β5 for KOs2O6 is more than one 
order larger than those in other members.  At high temperatures 
above 4 K, a downward deviation from the initial T5 behavior is 
observed in Fig. 4b.  The temperature dependence of the 14-T 
data above 5.5 K, which is taken as γT + Cl, is also close to T
5, but 
with a smaller slope, which means that a single T5 term is not 
appropriate to describe the Cl in such a wide temperature range and 
also that an inclusion of higher order term of Tn is not helpful.  
Therefore, we adopt expediently an alternative empirical form to 
express this strange lattice contribution; Cl = β5T
5f(T), where f(T) = 
[1 + exp(1 - pT-q)]-1.  Since the f(T) is almost unity below a certain 
temperature and decreases gradually with increasing T, this Cl can 
reproduce T5 behavior at low temperatures and a weaker T 
FIG. 3. (Color online) Specific heat divided by 
temperature measured at zero field (circle) and a magnetic 
field of 14 T (triangle) applied along the [111] direction.  
The inset shows an enlargement of the superconducting 
transition with an entropy-conserving construction. 
dependence at high temperatures.  As shown in Fig. 4b, the 14-T 
data in the 5.5-7 K range can be fitted well by the function for a 
value of β5 fixed to the initial slope of 0.3481 mJ K
-6 mol-1 and a 
given value of γ; for example, p = 6.96(3) and q = 1.09(1) for γ = 
70 mJ K-2 mol-1. 
In order to determine the value of γ univocally, the entropy 
conservation is taken into account for the 14-T data, as shown in 
Fig. 5: since the normal-state specific heat expected for the case of 
Tc = 0 is given by (Cen + Cl) (dotted line in Fig. 5), the integration 
of [Cen + Cl - C(14 T)]/T should become zero due to entropy 
balance. It is shown in the inset to Fig. 5 that the integrated value 
changes almost linearly with γ and vanishes around γ = 70 mJ K-2 
mol-1.  Hence, one can determine the value of γ unambiguously.  
A certain ambiguity may arise from the assumed lattice function.  
However, since the temperature dependence of Cl is substantially 
weak in the T range of interest, a possible correction on the γ value 
must be minimal, say, less than 1 mJ K-2 mol-1. 
Next we determine Ces at zero field by subtracting the Cl 
estimated above.  The temperature dependence of Ces does have 
the BCS form, aexp(-Δ/kBTc), as shown in Fig. 6.  The energy gap 
Δ obtained by fitting is 22.5 K, which corresponds to 2Δ/kBTc = 
4.69, much larger than the BCS value of 3.53.  The above Ces at 
low temperature below 7 K is again plotted in Fig. 7 together with 
high-temperature Ces above 5.5 K, which is obtained by subtracting 
the 14-T data from the zero field data as Ces = C(0) - C(14 T).  
The two data sets obtained independently overlap well in the 5.5-7 
K range, assuring the validity of the above analyses.  Because of 
the existence of the second peak and its small shifts under magnetic 
fields, the data between 7 K and 8.3 K is to be excluded in the 
following discussion.  Taking γ = 70 mJ K-2 mol-1, the jump in 
specific heat at Tc, ΔC/γTc, reaches 2.87, much larger than 1.43 
expected for a weak-coupling superconductor, indicating that 
KOs2O6 lies in the strong-coupling regime.  Comparisons to other 
typical strong-coupling superconductors are made in section IV-B. 
Here we analyze the data based on the α model that was 
developed to provide a semi-empirical approximation to the 
thermodynamic properties of strong-coupled superconductors in a 
wide range of coupling strengths with a single adjustable parameter, 
α = Δ0/kBTc.
45 Recently, it was generalized to a multi-gap 
superconductor and successfully applied to the analyses on MgB2 
or Nb3Sn.
46, 47  Using the α model, the data in the vicinity of Tc is 
well reproduced, as shown in Fig. 7, and we obtain α = 2.50 
(2Δ0/kBTc = 5.00), slightly larger than the value obtained above 
from the temperature dependence of Ces.  One interesting point to 
be noted is that there is a significant deviation between the data and 
the fitting curve at intermediate temperatures, suggesting the 
existence of an additional structure in the gap. Presumably, this 
enhancement would be explained if one assumes the coexistence of 
another smaller gap (not so small as in MgB2, but intermediate).  
This possibility has been already pointed out in the previous µSR 
experiment.36  However, ambiguity associated with the second 
peak in the present data prevents us from further analyzing the data.  
This important issue will be revisited in future work, where the 
FIG. 6. (Color online) Temperature dependence of electronic 
specific heat measured at H = 0 for the superconducting state 
showing an exponential decrease at low temperature.  A magnitude 
of the gap obtained is 2Δ/kBTc = 4.69. 
FIG. 4. (Color online) Low-temperature specific heat below 7 K 
plotted as functions of T2 (a) and T4 (b).  The broken and dotted 
lines in (a) show calculated contributions from Debye T3 phonons of 
ΘD = 460 K and 300 K, respectively, which are much smaller than 
the experimental values.  The broken line in (b) is a linear fit to the 
zero-field data as T → 0, which gives a coefficient of the T5 term, β5 
= 0.3481(6) mJ K-6 mol-1.  The solid and dotted line is a fit to Cl = 
β5T5f(T).  See text for detail. 
FIG. 5. (Color online) Specific heat data same as shown in Fig. 3.  
The dotted line shows the estimated contribution of Cen + Cl, and the 
broken line represents Cl in the case of γ = 70 mJ K-2 mol-1.  The 
inset shows a change of entropy balance as a function of γ, from 
which the value of γ is decided to be 70 mJ K-2 mol-1.

  Superconducting and normal-state properties of the beta-pyrochlore oxide
KOs2O6 are studied by means of thermodynamic and transport measurements. It is
shown that the superconductivity is of conventional s-wave type and lies in the
extremely strong-coupling regime. Specific heat and resistivity measurements
reveal that there are characteristic low-energy phonons that give rise to
unusual scattering of carriers due to strong electron-phonon interactions. The
entity of the low-energy phonons is ascribed to the heavy rattling of the K ion
confined in an oversized cage made of OsO6 octahedra. It is suggested that this
electron-rattler coupling mediates the Cooper pairing, resulting in the
extremely strong-coupling superconductivity.

Microsoft Word - K-SC6.doc
April 2, 2007 
I. INTRODUCTION 
The study of non-Cu-based oxide superconductors has been 
extended during the last decade, aiming at understanding the role of 
electron correlations in the mechanism of superconductivity or 
searching for a novel pairing mechanism, hopefully to reach a 
higher Tc.  An interesting example recently found is a family of 
pyrochlore oxide superconductors.  The first discovered is 
α-pyrochlore rhenate Cd2Re2O7 with Tc = 1.0 K
1-3 and the second 
β-pyrochlore osmate AOs2O6 with Tc = 3.3, 6.3, and 9.6 K for A = 
Cs,4 Rb,5-7 and K,8 respectively.  They crystallize in the cubic 
pyrochlore structure of space group Fd-3m and commonly possess 
a 3D skeleton made of ReO6 or OsO6 octahedra.
9 The "pyrochlore" 
sublattice occupied by the transition metal ions is comprised of 
corner-sharing tetrahedra that are known to be highly frustrating for 
a localized spin system with antiferromagnetic nearest-neighbor 
interactions.  
A unique structural feature for the β-pyrochlores is that a 
relatively small A ion is located in an oversized atomic cage made 
of OsO6 octahedra, Fig. 1.  Due to this large size mismatch, the A 
atom can rattle in the cage.10 The rattling has been recognized 
recently as an interesting phenomenon for a class of compounds 
like filled skutterudites11 and Ge/Si clathrates12 and attracted many 
researchers, because it may suppress thermal conductivity leading 
to an enhanced thermoelectric efficiency.  On the other hand, the 
rattling is also intriguing from the viewpoint of lattice dynamics:  
it gives an almost localized mode even in a crystalline material and 
often exhibits unusual anharmonicity.11 Hence, it is considered that 
the rattling is a new type of low-lying excitations that potentially 
affects various properties in a crystal at low temperature. 
In the β pyrochlores, specific heat experiments found 
low-energy contributions that could be described approximately by 
the Einstein model and determined the Einstein temperature TE to 
be 70 K, 60 K, and 40 or 31 K for A = Cs, Rb, and K, 
respectively.13, 14  This tendency illustrates uniqueness of the 
rattling, because, to the contrary, one expects a higher frequency 
for a lighter atom in the case of conventional phonons.  Moreover, 
it was demonstrated that the specific heat  shows an unusual T5 
dependence at low temperature below 7 K for A = Cs and Rb, 
instead of a usual T3 dependence from a Debye-type phonon.13  
On one hand, structural refinements revealed large atomic 
displacement parameters at room temperature of 100Uiso = 2.48, 
4.26, and 7.35 Å2 for A = Cs, Rb, and K, respectively,10 and 3.41  
Å2 for Rb.15  Particularly, the value for K is enormous and may be 
the largest among rattlers so far known in related compounds.  
This trend over the β-pyrochlore series is ascribed to the fact that 
an available space for the A ion to move in a rather rigid cage 
increases with decreasing the ionic radius of the A ion.10  Kuneš et 
al. calculated an energy potential for each A ion and found in fact a 
large anharmonicity, that is, a deviation from a quadratic form 
expected for the harmonic oscillator approximation.16 Especially 
for the smallest K ion, they found 4 shallow potential minima 
locating away from the center (8b site) along the <111> directions 
pointing to the nearest K ions, as schematically shown in Fig. 1.  
The potential minima are so shallow that the K ion may not stop at 
one of them even at very low temperature. 
The electronic structures of α-Cd2Re2O7 and β-AOs2O6 have 
been calculated by first-principle density-functional methods, 
which reveal that a metallic conduction occurs in the (Re, Os)-O 
network:16-20 electronic states near the Fermi level originate from 
transition metal 5d and O 2p orbitals.  Although the overall shape 
of the density of states (DOS) is similar for the two pyrochlores, a 
difference in band filling may result in different properties; Re5+ 
for α-Cd2Re2O7 has two 5d electrons, while Os
5.5+ for β-AOs2O6 
has two and a half.  Moreover, a related α-pyrochlore Cd2Os2O7 
with Os5+ (5d3) exhibits a metal-to-insulator transition at 230 K.21, 
Various experiments have been carried out on the pyrochlore 
oxide superconductors to elucidate the mechanism of the 
superconductivity.  Most of the results obtained for α-Cd2Re2O7 
indicate that it is a weak-coupling BCS-type superconductor.23, 24  
In contrast, results on the β-pyrochlores are somewhat 
controversial. Although the Tc increases smoothly from Cs to K, 
the jump in specific heat at Tc, the upper critical field, and the 
Sommerfeld coefficient all exhibit a large enhancement toward K.25  
Thus, the K compound is always distinguished from the others.   
Pressure dependence of Tc was also studied for the two pyrochlores, 
showing a common feature: as pressure increases, Tc first increases, 
exhibits a broad maximum and goes to zero above a critical 
pressure that depends on the system, for example, about 6 GPa for 
the K compound.26-29 
On the symmetry of the superconducting gap for the 
β-pyrochlores, Rb-NMR and µSR experiments gave evidence for 
s-wave superconductivity for RbOs2O6.
30-32  In contrast, Arai et al. 
carried out K-NMR experiments and found no coherence peak in 
the relaxation rate below Tc for KOs2O6, which seemed to indicate 
unconventional superconductivity.32  However, their recent 
interpretation is that the absence of a coherence peak does not 
necessarily mean non-s pairing, because the relaxation rate probed 
by the K nuclei can be affected dominantly by strongly 
overdamped phonons.33  Kasahara et al. measured thermal 
conductivity using a KOs2O6 single crystal and concluded a full 
Extremely strong-coupling superconductivity and anomalous lattice properties in 
the β-pyrochlore oxide KOs2O6 
Z. Hiroi, S. Yonezawa, Y. Nagao and J. Yamaura 
Institute for Solid State Physics, University of Tokyo, Kashiwa, Chiba 277-8581, Japan 
Superconducting and normal-state properties of the β-pyrochlore oxide KOs2O6 are studied by means of 
thermodynamic and transport measurements.  It is shown that the superconductivity is of conventional s-wave 
type and lies in the extremely strong-coupling regime.  Specific heat and resistivity measurements reveal that 
there are characteristic low-energy phonons that give rise to unusual scattering of carriers due to strong 
electron-phonon interactions.  The entity of the low-energy phonons is ascribed to the heavy rattling of the K ion  
confined in an oversized cage made of OsO6 octahedra.  It is suggested that this electron-rattler coupling 
mediates the Cooper pairing, resulting in the extremely strong-coupling superconductivity. 
gap from the insensitivity of thermal conductivity to magnetic 
fields.34  Moreover, very recent photoemission spectroscopy 
(PES) experiments revealed the opening of a large isotropic gap 
below Tc.
35  On one hand, a µSR experiment claimed that the gap 
of KOs2O6 is anisotropic, or otherwise, there are two gaps.
Therefore, the pairing symmetry of the β-AOs2O6 superconductors 
may be of the conventional s wave, aside from minor aspects such 
as anisotropy or multi gaps, which means that the fundamental 
pairing mechanism is ascribed to phonons.  Then, an important 
question is what kind of phonons are relevant for the occurrence.  
To find out the reason of the observed singular appearance toward 
K in the series must be the key to understand interesting physics 
involved in this system. 
Previous studies have suffered poor quality of samples, 
because only polycrystalline samples were available.  The 
Sommerfeld coefficient γ was estimated from specific heat by 
extracting contributions from Os metal impurity to be 40 mJ K-2 
mol-1 for both Cs and Rb,13 and 34 mJ K-2 mol-1 for Rb.7  Recently, 
Brühwiler et al. obtained large values of γ = 76-110 mJ K-2 mol-1 
for KOs2O6 by collecting five dozen of tiny crystals.
14  However, 
there is an ambiguity in their values, because of uncertainty in their 
extrapolation method.  They also reported strong-coupling 
superconductivity with a coupling constant λep = 1.0-1.6.
Recently, we successfully prepared a large single crystal of 1 mm 
size for KOs2O6 and reported two intriguing phenomena: one is a 
sharp and huge peak in specific heat at Tp = 7.5-7.6 K below Tc, 
indicative of a first-order structural transition,37, 38 and the other is 
anisotropic flux pinning at low magnetic fields around 2 T.39  It 
was suggested that the former is associated with the rattling 
freedom of the K ion.  Moreover, anomalous concave-downward 
resistivity was observed down to Tc, suggesting a peculiar 
scattering mechanism of carriers. 
In this paper, we present specific heat, magnetization and 
resistivity measurements on the same high-quality single crystal of 
KOs2O6.  Reliable data on the superconducting and normal-state 
properties are obtained, which provide evidence for an extremely 
strong-coupling superconductivity realized in this compound.  We 
discuss the role of rattling vibrations of the K ion on the 
mechanism of the superconductivity. 
II. EXPERIMENTAL 
A. Sample preparation 
A high-quality single crystal was prepared and used for all the 
measurements in the present study, which was named KOs-729 
after the date of July 29, 2005 when the first experiment was 
performed on this crystal.  It was grown from a pellet containing 
an equimolar mixture of KOsO4 and Os metal in a sealed quartz 
tube at 723 K for 24 h.  Additional oxygen was supplied by using 
the thermal decomposition of AgO placed away from the pellet in 
the tube.  The KOsO4 powder had been prepared in advance from 
KO2 and Os metal in the presence of excess oxygen.  It was 
necessary to pay attention to avoid the formation of OsO4 in the 
course of preparation, which is volatile even at room temperature 
and highly toxic to eyes or nose.  After the reaction, several tiny 
crystals had grown on the surface of the pellet.   Although the 
mechanism of the crystal growth has not yet been understood 
clearly, probably it occurs through partial melting and the 
following reaction with a vapor phase.   
The KOs-729 crystal possesses a truncated octahedral shape 
with a large (111) facet as shown in Fig. 2 and is approximately 1.0 
× 0.7 × 0.3 mm3 in size and 1.302 mg in weight.  The high quality 
of the crystal has been demonstrated by a sharp peak at Tp in 
specific heat,38 which was absent in the previous polycrystalline 
samples or appeared as broad humps in our previous aggregate of 
tiny crystals37 or in five dozen of tiny crystals by Brühwiler et al.14  
Moreover, a dramatic angle dependence of flux-flow resistance was 
observed on this crystal, indicating that flux pinning is enhanced in 
magnetic fields along certain crystallographic directions such as 
[110], [001], and [112].39  This evidences the absence of domains 
in this relatively large crystal.  
A special care has been taken to keep the crystal always in a 
dry atmosphere, because it readily undergoes hydration in air, as 
reported in isostructural compounds such as KNbWO6.
40, 41  Once 
partial hydration takes place in KOs2O6, the second sharp anomaly 
in specific heat tends to collapse.  In contrast, the superconducting 
transition was robust, just slightly broadened after hydration. The 
hydration must be relatively slow in a single crystal compared with 
the case of polycrystalline samples, which may be the reason for 
the broad anomaly or the absence in previous samples.  
B. Physical-property measurements 
Both specific heat and electrical resistivity were measured in a 
temperature range between 300 K and 0.4 K and in magnetic fields 
of up to 14 T in a Quantum Design Physical Property Measurement 
System (PPMS) equipped with a 3He refrigerator.  The magnetic 
fields had been calibrated by measuring the magnetization of a 
standard Pd specimen and also by measuring the voltage of a Hall 
device (F.W. BELL, BHA-921).  Specific heat measurements 
FIG. 1. (Color online) Crystal structure of the β-pyrochlore 
oxide KOs2O6.  The K ion (big ball) is located in an oversized 
atomic cage made of OsO6 octahedra and can move along the 4 
<111> directions pointing to the neighboring K ions in adjacent 
cages. 
FIG. 2. (Color online) Photograph of the KOs-729 crystal used 
in the present study.  It possesses a truncated octahedral shape 
with 111 facets.  The approximate size is 1.0 × 0.7 × 0.3 mm3.  
Resistivity measurements were carried out with a current flow 
along the [-110] direction. 
were performed by the heat-relaxation method.  The KOs-729 
crystal was attached to an alumina platform by a small amount of 
Apiezon N grease.  In each measurement, heat capacity was 
obtained by fitting a heat relaxation curve recorded after a heat 
pulse giving a temperature rise of approximately 2%.  The heat 
capacity of an addendum had been measured in a separate run 
without a sample, and was subtracted from the data.  The 
measurements were done three times at each temperature with a 
scatter less than 0.3% at most. 
Resistivity measurements were carried out by the four-probe 
method with a current flow along the [-110] direction and magnetic 
fields along the [111], [110], [001] or [112] direction of the cubic 
crystal structure.  All the measurements were done at a current 
density of 1.5 A cm-2.  Magnetization was measured in magnetic 
fields up to 7 T in a Quantum Design magnetic property 
measurement system and also up to 14 T in PPMS.  The magnetic 
fields were applied approximately along the [111] or [-110] 
direction. 
III. RESULTS 
A. Superconducting properties 
1. Specific heat 
First of all, we analyze specific heat data in order to obtain a 
reliable value of the Sommerfeld coefficient γ.  There are two 
obstacles:  one is the large upper critical field Hc2 that is 
approximately 2 times greater than our experimental limit of 14 T.  
Brühwiler et al. reported γ = 76 (110) mJ K-2 mol-1 assuming µ0Hc2 
= 24 (35) T by an extrapolation method.14  Their values should be 
modified to γ ~ 100 mJ K-2 mol-1, because recent high magnetic 
field experiments revealed µ0Hc2 = 30.6 T or 33 T.
42, 43  
Nevertheless, there are still large ambiguity in their extrapolation 
method using specific heat data obtained only at H/Hc2 < 0.5.  The 
other difficulty comes from unusual lattice contributions in specific 
heat at low temperature and the existence of a sharp peak at Tp.  
Thus, it is not easy to extract the lattice contribution in a standard 
way used so far.  Here we carefully analyze specific heat data and 
reasonably divide them into electronic and lattice parts, from which 
a reliable value of γ is determined, and information on the 
superconducting gap is attained. 
Figure 3 shows the temperature dependence of specific heat of 
the KOs-729 crystal measured on cooling at zero field and in a 
magnetic field of 14 T applied along the [111] direction.  A 
superconducting transition at zero field takes place with a large 
jump, followed by a huge peak due to the second phase transition at 
Tp = 7.5 K. The entropy-conserving construction shown in the inset 
gives Tc = 9.60 K, ΔC/Tc = 201.2 mJ K
-2 mol-1, which is close to 
the values previously reported,14, 37 and a transition width (ΔTc) of 
0.3 K.  Tc is reduced to 5.2 K at 14 T, which is evident as a bump 
in the 14-T data shown in Fig. 3.  In contrast, Brühwiler et al. 
reported Tc = 6.2 K at 14 T, though the transition was not clearly 
observed in their specific heat data.  The C/T at zero field rapidly 
decreases to zero as T approaches absolute zero.  The absence of a 
residual T-linear contribution in specific heat indicates the high 
quality of the crystal.   
The specific heat of a crystal (C) is the sum of an electronic 
contribution (Ce) and an H-independent lattice contribution (Cl).  
The former becomes Cen for the normal state above Tc, which is 
taken as γT, and Ces for the superconducting state below Tc.  The γ 
is assumed to be T-independent, though it can not be the case for 
compounds with strong electron-phonon couplings.44  In the case 
of KOs2O6, Cl is large relative to Ce: for example, Cl is as large as 
~90% of the total C at just above Tc, as shown later.  In order to 
determine the value of γ, it is crucial to know the low-temperature 
form of Cl.  Since the minimum Tc attained at 14 T is 5.2 K, one 
has to estimate the Cl from the T dependence of the total C above 
~5.5 K.  Two terms in the harmonic-lattice approximation are 
often required for an adequate fit; Cl = β3T
3 + β5T
5.  The first term 
comes from a Debye-type acoustic phonon, and thus is dominant at 
low temperature, while the second term expresses a deviation at 
high temperature.  Actually, this approximation is valid for 
α-Cd2Re2O7, where β3 = 0.222 mJ K
-4 mol-1 and β5 = 2.70 × 10
mJ K-6 mol-1 are obtained by a fit to the data below 10 K.1  The 
Debye temperature ΘD is 458 K from the β3 value.  In strong 
contrast, it was found for two members of β-AOs2O6 that the T
term prevails in a wide temperature range; β5 = 14.2 × 10
-3 mJ K-6 
mol-1 below 5 K for CsOs2O6 and β5 = 30.2 × 10
-3 mJ K-6 mol-1 
below 7 K for RbOs2O6.
The C/T at H = 0 below 7 K shown in Fig. 3 is again plotted in 
two ways as functions of T2 and T4 in Fig. 4.  It is apparent from 
the T2 plot that possible T3 terms expected for ΘD = 458 K and 300 
K are negligibly small compared with the whole magnitude of 
specific heat, just as observed in other members.  On the other 
hand, in the T4 plot, there is distinct linear behavior at low 
temperature below 4 K, indicating that the C approaches 
asymptotically to T5 behavior as T → 0 with a large slope of 
0.3481(6) mJ K-6 mol-1.  It is reasonable to ascribe this T5 
contribution to the lattice, because Ces should decrease quickly as T 
→ 0.  Note that the value of the β5 for KOs2O6 is more than one 
order larger than those in other members.  At high temperatures 
above 4 K, a downward deviation from the initial T5 behavior is 
observed in Fig. 4b.  The temperature dependence of the 14-T 
data above 5.5 K, which is taken as γT + Cl, is also close to T
5, but 
with a smaller slope, which means that a single T5 term is not 
appropriate to describe the Cl in such a wide temperature range and 
also that an inclusion of higher order term of Tn is not helpful.  
Therefore, we adopt expediently an alternative empirical form to 
express this strange lattice contribution; Cl = β5T
5f(T), where f(T) = 
[1 + exp(1 - pT-q)]-1.  Since the f(T) is almost unity below a certain 
temperature and decreases gradually with increasing T, this Cl can 
reproduce T5 behavior at low temperatures and a weaker T 
FIG. 3. (Color online) Specific heat divided by 
temperature measured at zero field (circle) and a magnetic 
field of 14 T (triangle) applied along the [111] direction.  
The inset shows an enlargement of the superconducting 
transition with an entropy-conserving construction. 
dependence at high temperatures.  As shown in Fig. 4b, the 14-T 
data in the 5.5-7 K range can be fitted well by the function for a 
value of β5 fixed to the initial slope of 0.3481 mJ K
-6 mol-1 and a 
given value of γ; for example, p = 6.96(3) and q = 1.09(1) for γ = 
70 mJ K-2 mol-1. 
In order to determine the value of γ univocally, the entropy 
conservation is taken into account for the 14-T data, as shown in 
Fig. 5: since the normal-state specific heat expected for the case of 
Tc = 0 is given by (Cen + Cl) (dotted line in Fig. 5), the integration 
of [Cen + Cl - C(14 T)]/T should become zero due to entropy 
balance. It is shown in the inset to Fig. 5 that the integrated value 
changes almost linearly with γ and vanishes around γ = 70 mJ K-2 
mol-1.  Hence, one can determine the value of γ unambiguously.  
A certain ambiguity may arise from the assumed lattice function.  
However, since the temperature dependence of Cl is substantially 
weak in the T range of interest, a possible correction on the γ value 
must be minimal, say, less than 1 mJ K-2 mol-1. 
Next we determine Ces at zero field by subtracting the Cl 
estimated above.  The temperature dependence of Ces does have 
the BCS form, aexp(-Δ/kBTc), as shown in Fig. 6.  The energy gap 
Δ obtained by fitting is 22.5 K, which corresponds to 2Δ/kBTc = 
4.69, much larger than the BCS value of 3.53.  The above Ces at 
low temperature below 7 K is again plotted in Fig. 7 together with 
high-temperature Ces above 5.5 K, which is obtained by subtracting 
the 14-T data from the zero field data as Ces = C(0) - C(14 T).  
The two data sets obtained independently overlap well in the 5.5-7 
K range, assuring the validity of the above analyses.  Because of 
the existence of the second peak and its small shifts under magnetic 
fields, the data between 7 K and 8.3 K is to be excluded in the 
following discussion.  Taking γ = 70 mJ K-2 mol-1, the jump in 
specific heat at Tc, ΔC/γTc, reaches 2.87, much larger than 1.43 
expected for a weak-coupling superconductor, indicating that 
KOs2O6 lies in the strong-coupling regime.  Comparisons to other 
typical strong-coupling superconductors are made in section IV-B. 
Here we analyze the data based on the α model that was 
developed to provide a semi-empirical approximation to the 
thermodynamic properties of strong-coupled superconductors in a 
wide range of coupling strengths with a single adjustable parameter, 
α = Δ0/kBTc.
45 Recently, it was generalized to a multi-gap 
superconductor and successfully applied to the analyses on MgB2 
or Nb3Sn.
46, 47  Using the α model, the data in the vicinity of Tc is 
well reproduced, as shown in Fig. 7, and we obtain α = 2.50 
(2Δ0/kBTc = 5.00), slightly larger than the value obtained above 
from the temperature dependence of Ces.  One interesting point to 
be noted is that there is a significant deviation between the data and 
the fitting curve at intermediate temperatures, suggesting the 
existence of an additional structure in the gap. Presumably, this 
enhancement would be explained if one assumes the coexistence of 
another smaller gap (not so small as in MgB2, but intermediate).  
This possibility has been already pointed out in the previous µSR 
experiment.36  However, ambiguity associated with the second 
peak in the present data prevents us from further analyzing the data.  
This important issue will be revisited in future work, where the 
FIG. 6. (Color online) Temperature dependence of electronic 
specific heat measured at H = 0 for the superconducting state 
showing an exponential decrease at low temperature.  A magnitude 
of the gap obtained is 2Δ/kBTc = 4.69. 
FIG. 4. (Color online) Low-temperature specific heat below 7 K 
plotted as functions of T2 (a) and T4 (b).  The broken and dotted 
lines in (a) show calculated contributions from Debye T3 phonons of 
ΘD = 460 K and 300 K, respectively, which are much smaller than 
the experimental values.  The broken line in (b) is a linear fit to the 
zero-field data as T → 0, which gives a coefficient of the T5 term, β5 
= 0.3481(6) mJ K-6 mol-1.  The solid and dotted line is a fit to Cl = 
β5T5f(T).  See text for detail. 
FIG. 5. (Color online) Specific heat data same as shown in Fig. 3.  
The dotted line shows the estimated contribution of Cen + Cl, and the 
broken line represents Cl in the case of γ = 70 mJ K-2 mol-1.  The 
inset shows a change of entropy balance as a function of γ, from 
which the value of γ is decided to be 70 mJ K-2 mol-1.
