We study the influence of the rattling phonons on superconductivity in $β$-pyrochlore KOs$_{2}$O$_{6}$ compound based on the strong-coupling Eliashberg approach. In particular, analyzing the specific heat data we find that the rattling phonon frequency changes discontinuously at the critical temperature of the first order phase transition. Solving the strong-coupling Eliashberg equations with temperature dependent $α^{2}F(ω)$, we investigate the consequence of this first order phase transition for the anomalous temperature dependence of the superconducting gap. We discuss our results in context of the recent experimental data.
The superconducting transition of the ?-pyrochlore oxide KOs2O6 with Tc = 9.60 K is studied by resistivity measurements under various magnetic fields using a high-quality single crystal. The reentrant behavior of superconductivity is observed near Tc in low magnetic fields below 2 T. The recovered resistance probably due to flux flow exhibits an anomalous angle dependence, indicating that flux pinning is enhanced in magnetic fields along certain crystallographic directions such as [110], [001] and [112]. It is suggested that there is an intrinsic pinning mechanism coming from the specific crys…
Magnetic penetration depth ($λ$) in a pyrochlore superconductor KOs$_2$O$_6$ was studied by means of muon spin rotation (\msr). The temperature dependence of transverse spin relaxation rate, which is proportional to $λ^{-2}$, exhibits a considerable deviation from that expected for the empirical two-fluid model ($\propto T^4$), rather obeying a $T^2$-tendency. Furthermore, $λ$ apparently increases with increasing applied field, suggesting the presence of nonlinear and nonlocal effects. These findings strongly suggest that the superconducting order parameter in KOs$_2$O$_6$ is anisotropic.
We report nuclear magnetic resonance studies on the beta-pyrochlore oxide superconductor KOs2O6. The nuclear relaxation at the K sites is entirely caused by fluctuations of electric field gradient, which we ascribe to highly anharmonic low frequency oscillation (rattling) of K ions. A phenomenological analysis shows a crossover from overdamped to underdamped behavior of the rattling phonons with decreasing temperature and its sudden sharpening below the superconducting transition temperature Tc. Absence of the Hebel-Slichter peak in the relaxation rate at the O sites below Tc also indicates st…
We report the single-crystal X-ray analysis of the structure of the pyrochlore oxide superconductor KOs$_2$O$_6$. The structure was identified as the $β$-pyrochlore structure with space group $Fd\bar{3}m$ and lattice constant $a$ = 10.089(2)~Åat 300 K: the K atom is located at the 8$b$ site, not at the 16$d$ site as in conventional pyrochlore oxides. We found an anomalously large atomic displacement parameter $U_\mathrm{iso}$ = 0.0735(8)~Å$^2$ at 300 K for the K cation, which suggests that the K cation weakly bound to an oversized Os$_{12}$O$_{18}$ cage exhibits intensive rattling, as recently…
To elucidate the nature of the superconducting ground state of the geometrically frustrated pyrochlore KOs2O6 (Tc=9.6K), the thermal conductivity was measured down to low temperatures (~Tc/100). We found that the quasiparticle mean free path is strikingly enhanced below a transition at Tp=7.5K, indicating enormous electron inelastic scattering in the normal state. In a magnetic field the conduction at T ->0K is nearly constant up to ~0.4Hc2, in contrast with the rapid growth expected for superconductors with an anisotropic gap. This unambiguously indicates a fully gapped superconductivity, in …
Electronic, magnetic, and dynamical properties of the new superconducting $β$-pyrochlore KOs$_2$O$_6$ and related RbOs$_2$O$_6$ and CsOs$_2$O$_6$ compounds are calculated and compared with experiment and contrasted with structurally related spinel pyrochlores. The calculated susceptibility Stoner enhancement (110%) and thermal mass enhancement $λ$ = 2.5-3 reflect moderate but perhaps important Coulomb correlations. The K$^{+}$ ion optic mode is found to be unstable, allowing large excursions of 0.5-0.6 Åfrom its ideal site of the K ion along $<111>$ directions. This dynamical mode is much less…
We investigate with a microscopic model anharmonic K-cation oscillation observed by neutron experiments in beta-pyrochlore superconductor KOs2O6, which also shows a mysterious first-order structural transition at Tp=7.5 K. We have identified a set of microscopic model parameters that successfully reproduce the observed temperature dependence and the superconducting transition temperature. Considering changes in the parameters at Tp, we can explain puzzling experimental results about electron-phonon coupling and neutron data. Our analysis demonstrates that the first-order transition is multipol…
Resistivity and specific heat have been measured on a single crystalline sample of the beta-pyrochlore oxide superconductor, KOs2O6. It is found that a second peak in specific heat, which may evidence an unknown phase transition, appears around Tp ~ 7.5 K below the superconducting transition temperature Tc = 9.53 K. Applying magnetic fields up to 14 T, Tc is reduced gradually down to 7.1 K, while Tp is raised a little and becomes even higher than Tc at 14 T, which implies that the second anomaly is not associated directly with the superconductivity. It is demonstrated, however, that there is s…
To assess electron correlation and electron-phonon coupling in the recently discovered beta-pyrochlores KOs2O6 and RbOs2O6, we have performed specific heat measurements in magnetic fields up to 14 T. We present data from high quality single crystalline KOs2O6, showing that KOs2O6 is a strong coupling superconductor with a coupling parameter lambda_ep \approx 1.0 to 1.6 (RbOs2O6: lambda_ep \approx 1). The estimated Sommerfeld coefficient of KOs2O6, gamma=76 to 110 mJ/(mol K^2), is twice that of RbOs2O6 [gamma=44 mJ/(mol K^2)]. Using strong-coupling corrections, we extract useful thermodynamic p…
The entire temperature dependence of the upper critical field $H_{\rm c2}$ in the $β$-pyrochlore KOs$_2$O$_6$ is obtained from high-field resistivity and magnetic measurements. Both techniques identically give $H_{\rm c2}(T \simeq 0 {\rm K})$ not only surprisingly high ($\sim 33$ T), but also the approach to it unusually temperature-\emph{linear} all the way below $T_{\rm c}$ (= 9.6 K). We show that, while $H_{\rm c2}(0)$ exceeds a simple spin-singlet paramagnetic limit $H_{\rm P}$, it is well below an $H_{\rm P}$ enhanced due to the missing spatial inversion symmetry reported recently in KOs$…
Microwave penetration depth $λ$ and surface resistance at 27 GHz are measured in high quality crystals of KOs$_2$O$_6$. Firm evidence for fully-gapped superconductivity is provided from $λ(T)$. Below the second transition at $T_{\rm p}\sim 8$ K, the superfluid density shows a step-like change with a suppression of effective critical temperature $T_{\rm c}$. Concurrently, the extracted quasiparticle scattering time shows a steep enhancement, indicating a strong coupling between the anomalous rattling motion of K ions and quasiparticles. The results imply that the rattling phonons help to enhanc…
Superconductivity in the potassium osmium oxide KOs2O6 crystallizing in the beta-pyrochlore structure is studied by means of electrical resistivity, magnetic susceptibility and specific heat. It is the second superconductor in the family of pyrochlore oxides, following the alfa-type pyrochlore oxide Cd2Re2O7 which is believed to be a conventional s-wave superconductor. The superconducting transition temperature Tc of KOs2O6 is 9.6 K, almost one order higher than the Tc = 1.0 K of Cd2Re2O7. Moreover, the superconductivity of KOs2O6 is remarkably robust under high magnetic fields, with a large u…
Another phase transition that is probably of first order is found in the beta-pyrochlore oxide superconductor KOs2O6 with a superconducting transition temperature Tc of 9.6 K. It takes place at Tp=7.5 K in the superconducting state in a zero magnetic field. By applying magnetic fields of up to 140 kOe, the Tc gradually decreased to 5.2 K, while Tp changed little, eventually breaking through the Hc2 line at approximately 65 kOe in the H-T diagram. Both the normal-state resistivity and Hc2 change slightly but significantly across the second phase transition. It is suggested that the transition i…
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 i…
A miniature Hall sensor array was used to detect magnetic induction locally in the vortex states of the $β$-pyrochlore superconductor KOs$_2$O$_6$. Below the first-order transition at $T_{\rm p}\sim 8$ K, which is associated with a change in the rattling motion of K ions, the lower critical field and the remanent magnetization both show a distinct decrease, suggesting that the electron-phonon coupling is weakened below the transition. At high magnetic fields, the local induction shows an unexpectedly large jump at $T_{\rm p}$ whose sign changes with position inside the sample. Our results demo…
We report a first-principles density-functional calculation of the electronic structure and properties of the recently discovered superconducting beta-pyrochlore oxide KOs_2O_6. We find that the electronic structure near the Fermi energy E_F is dominated by strongly hybridized Os-5d and O-2p states. A van Hove singularity very close to E_F leads to a relatively large density of states at E_F, and the Fermi surface exhibits strong nesting along several directions. These features could provide the scattering processes leading to the observed anomalous temperature dependence of the resistivity an…
Rattling-induced superconductivity in the β-pyrochlore oxide KOs2O6 is investigated under high pressure up to 5 GPa. Resistivity measurements in a high-quality single crystal reveal a gradual decrease in the superconducting transition temperature Tc from 9.7 K at 1.0 GPa to 6.5 K at 3.5 GPa, followed by a sudden drop to 3.3 K at 3.6 GPa. Powder X-ray diffraction experiments show a structural transition from cubic to monoclinic or triclinic at a similar pressure. The sudden drop in Tc is ascribed to this structural tran-sition, by which an enhancement in Tc due to a strong electron-rattler inte…
A new modification of KOs2O6, the representative of a new structural type (Pearson symbol aP18, a=5.5668(1)A, b=6.4519(2)A, c=7.2356(2)A, space group P-1, no.2) was synthesized employing high pressure technique. Its structure was determined by single-crystal X-ray diffraction. The structure can be described as two OsO6 octahedral chains relating to each other through inversion and forming big voids with K atoms inside. Quantum chemical calculations were performed on the novel compound and structurally related cubic compound. High-pressure X-ray study showed that cubic KOs2O6 phase was stable u…
Recently new pyrochlore oxides superconductors AOs2O6 (A=K,Rb,Cs) were found and we measured the pressure dependence of magnetization up to 1.2 GPa in order to deduce the pressure effect of Tc in the three beta-pyrochlore oxides. It is found that the initial pressure dependence of Tc is positive for all the compounds. Only KOs2O6 exhibits a saturation in Tc at 0.56 GPa and the downturn at higher pressure
Single crystals of KOs2O6 have been grown in a sealed quartz ampoule. Detailed single crystal X-ray diffraction studies at room temperature show Bragg peaks that violate Fd-3m symmetry. With a comparative structure refinement the structure is identified as non-centrosymmetric (F-43m). Compared to the ideal beta-pyrochlore lattice (Fd-3m), both Os tetrahedral and O octahedral network exhibit breathing mode like volume changes accompanied by strong anisotropic character of the K channels. The crystals show metallic conductivity and a sharp transition to the superconducting state at Tc = 9.65 K. …
The discovery of a new beta-pyrochlore oxide superconductor CsOs2O6 with Tc = 3.3 K is reported. It is the third superconductor in the family of beta-pyrochlore oxides, following KOs2O6 with Tc = 9.6 K and RbOs2O6 with Tc = 6.3 K. The Tc of this series decreases with increasing the ionic radius of alkaline metal ions, imposing negative chemical pressure upon the Os pyrochlore lattice.
We report the discovery of a new pyrochlore oxide superconductor RbOs2O6. The compound crystallizes in the same beta-pyrochlore structure as the recently discovered superconductor KOs2O6, where Os atoms form a corner-sharing tetrahedral network called the pyrochlore lattice with Rb or K atoms in the cage. Resistivity, magnetic susceptibility and specific heat measurements on polycrystalline samples evidence a bulk superconductivity with Tc = 6.3 K.
We report the basic thermodynamic properties of the new geometrically frustrated beta-pyrochlore bulk superconductor RbOs2O6 with a critical temperature Tc = 6.4 K. Specific heat measurements are performed in magnetic fields up to 12 T. The electronic density of states at the Fermi level in the normal state results in gamma = (33.7 \pm 0.2) mJ/mol_f.u./K^2. In the superconducting state, the specific heat follows conventional BCS-type behavior down to 1 K, i.e. over three orders of magnitude in specific heat data. The upper critical field slope at Tc is 1.2 T/K, corresponding to a Maki-paramete…
Inelastic neutron scattering and neutron powder diffraction experiments were carried out to investigate a localized mode, proposed from various bulk measurements, in the beta-pyrochlore AOs2O6 (A=K, Rb, Cs). The localized mode was identified in all the three compounds as well as another beta-pyrochlore CsW2O6. The anharmonicity of the mode is weak in RbOs2O6 and CsOs2O6 but substantial in KOs2O6.
The superconducting properties of two beta-pyrochlore oxides, CsOs2O6 and RbOs2O6, are studied by thermodynamic and transport measurements using high-quality single crystals. It is shown that the character of superconductivity changes systematically from weak coupling for CsOs2O6 to moderately strong coupling for RbOs2O6, and finally to extremely strong coupling with BCS-type superconductivity for KOs2O6, with increasing Tc. Strong-coupling correction analyses of the superconducting properties reveal that a low-energy rattling mode of the alkali metal ions is responsible for the mechanism of t…
ß-pyrochore KOs2O6, which shows superconductivity below ~ 9.7K, has been converted into KxOs2O6 (x < 2/3 - 1/2) electrochemically to show spin glass-like behavior below ~ 6.1K. Room temperature sample surface potential versus charge transfer scan indicates that there are at least two two-phase regions for x between 1 and 0.5. Rattling model of superconductivity for the title compound has been examined using electrochemical potassium de-intercalation. The significant reduction of superconducting volume fraction due to minor potassium reduction suggests the importance of defect and phase coheren…
We study the interplay between superconductivity and non-Fermi liquid behavior of a Fermi surface coupled to a massless $SU(N)$ matrix boson near the quantum critical point. The presence of thermal infrared singularities in both the fermionic self-energy and the gap equation invalidates the Eliashberg approximation, and makes the quantum-critical pairing problem qualitatively different from that at zero temperature. Taking the large $N$ limit, we solve the gap equation beyond the Eliashberg approximation, and obtain the superconducting temperature $T_c$ as a function of $N$. Our results show a…
Using the Eliashberg theory of superconductivity we have examined several properties of a model in which electrons are coupled only to rattling phonon modes represented by a sharp peak in the electron-phonon coupling function. Our choice of parameters was guided by experiments on $β$-pyrochlore oxide superconductor KOs$_{2}$Os$_{6}$. We have calculated the temperature dependence of the superconducting gap edge, the quasiparticle decay rate, the NMR relaxation rate assuming that the coupling between the nuclear spins and the conduction electrons is via a contact hyperfine interaction which woul…
We report on a study of the superconducting order parameter thermodynamic fluctuations in $\mathrm{YBa}_2\mathrm{Cu}_3\mathrm{O}_{7 - δ}$, $\mathrm{Bi}_{2}\mathrm{Sr}_{2}\mathrm{CaCu}_{2}\mathrm{O}_{8+ δ}$ and $\mathrm{KOs}_2\mathrm{O}_6$ compounds. A non-perturbative technique within the framework of the renormalized Gaussian approach is proposed. The essential features are reported (analytically and numerically) through Ginzburg-Landau (GL) model-based calculations which take into account both the dimension and the microscopic parameters of the system. By presenting a self-consistent approac…
In order to clarify the mechanism of the enhancement of superconducting transition temperature $T_{\rm c}$ due to anharmonic local oscillation of a guest ion in a cage composed of host atoms, i.e., {\it rattling}, we analyze the anharmonic Holstein model by applying the Migdal-Eliashberg theory. From the evaluation of the normal-state electron-phonon coupling constant, it is found that the strong coupling state is developed, when the bottom of a potential for the guest ion becomes wide and flat. Then, $T_{\rm c}$ is enhanced with the increase of the anharmonicity in the potential, although $T_…
We show that a random interacting model exhibits solvable non-Fermi liquid behavior and exotic pairing behavior. This model, dubbed as the Yukawa-SYK model, describes the random Yukawa coupling between $M$ quantum dots each hosting $N$ flavors of fermions and $N^2$ bosons that self-tunes to criticality at low energies. The diagrammatic expansion is controlled by $1/MN$, and the results become exact in a large-$M$, large-$N$ limit. We find that pairing only develops within a region of the $(M,N)$ plane --- even though the pairing interaction is strongly attractive, the incoherence of the fermio…
We investigate three dimensional anharmonic phonons in tetrahedral symmetry and superconductivity mediated by these phonons. Three dimensional anharmonic phonon spectra are calculated directly by solving Schrödinger equation and the superconducting transition temperature is determined by using the theory of strong coupling superconductivity assuming an isotropic gap function. With increasing the third order anharmonicity $b$ of the tetrahedral ion potential, we find a crossover in the energy spectrum to a quantum tunneling regime. We obtain strongly enhanced transition temperatures around the …
Generation of new quadratic coefficients within Ginzburg-Landau theory: Applications for specific heat calculations in various high-temperature superconductors
