Cooper-pair formation by anharmonic rattling modes in the β-pyrochlore superconductor KOs2O6

We study the influence of anharmonic rattling phonons in the β-pyrochlore superconductor KOs2O6 using the strong-coupling Eliashberg approach. In particular, by 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 effective temperature-dependent α2F(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 the context of the recent experimental data.


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Einstein modes and that the T -linear coefficient of the specific heat γ = 70 mJ K −2 mol −1 (see [7,8]) is strongly enhanced over the value obtained from band structure calculations [9,10]. The latter show that conduction states comprise nearly half-filled (5/12) t 2g bands with a bare value of γ b = 17 mJ K −2 mol −1 . It is a matter of dispute how much of the enhancement is due to lattice vibrations or electronic correlations [7,9].
The electronic structure calculations have indicated that these anomalies may be partly due to highly anharmonic low-frequency rattling motion of the alkali ions inside an oversized cage formed by the Os and O ions [9]. Furthermore, this is consistent with the x-ray observation of anomalously large atomic displacement for the K ions [11], and the low-frequency phonon structures seen in photoemission spectra [12]. Moreover, recent NMR data [13] have indicated that the relaxation at the K sites is entirely caused by fluctuations of the electric field gradient ascribed to the highly anharmonic low-frequency dispersionless oscillation (rattling) of K ions in the cage. Recently, Dahm and Ueda [14] have developed a phenomenological model to describe the influence of the anharmonic phonons on the NMR relaxation and the electrical resistivity.
Various experiments have been carried out in KOs 2 O 6 to elucidate the mechanism of superconductivity and gap symmetry. Most importantly, thermal conductivity [15] and laser photoemission spectroscopy (PES) [12] measurements have revealed an almost isotropic s-wave superconducting gap. Furthermore, the absence of the coherence peak in the K-NMR relaxation rate expected for the isotropic gap can be explained as due to strongly overdamped phonons [13]. Simultaneously, penetration depth measurements by µSR [16] and scanning tunneling experiments [17] indicate a sizeable anisotropic component admixture, approximately 30% according to the latter. Alternatively, a two-gap model for the two Fermi surface sheets [9] has been proposed [16,17], also because the pressure dependence was found to be opposite to that of a single band Bardeen-Cooper-Schrieffer (BCS) model. As a whole the experimental results point to a phonon-dominated mechanism for Cooper pairing in β-pyrochlores, even in KOs 2 O 6 .
Nevertheless, the large value of the electron-phonon coupling constant λ ep ≈ 2.38, the presence of the low-energy rattling phonon modes and their influence on the electronic properties, and finally, the first-order structural phase transition at T p < T c [8] make an investigation of the superconductivity in KOs 2 O 6 very interesting. In particular, the novel aspects of the strong-coupling Eliashberg theory in the presence of coupling between conduction electrons and anharmonic rattling phonons require a separate analysis. Previously, an attempt to consider the effect of the temperature-independent rattling phonons has been made in [18] in relation to CsOs 2 O 6 .
In this work, we analyze the superconductivity in KOs 2 O 6 using the standard Eliashberg formalism. We neglect the sizeable anisotropic gap admixture for simplicity. The exotic character of superconductivity in KOs 2 O 6 is due to a large contribution from a strongly anharmonic local rattling mode, usually not included in a quasi-harmonic treatment of lattice vibrations and their renormalization by the coupling with conduction electrons. We find that the phonon spectrum can be modeled by the two Lorentzians peaked at the energies of the two Einstein modes, ω E1 and ω E2 , representing the lowest energy rattling phonons. An additional contribution arises from the Debye frequency at ω D . To explain the superconducting transition temperature in KOs 2 O 6 , we have employed the mean-field analysis of the temperature dependence of the lowest Einstein mode proposed previously [14]. We show that its energy should decrease with temperatures down to T p < T c and then jump to a higher frequency 3 indicating the first-order structural phase transition. Solving the nonlinear Eliashberg equations below T c with temperature-dependent electron-phonon coupling function, α 2 F(ω), we compare our results with those from various experiments.
On the real frequency axis the finite temperature Eliashberg equations for the superconducting gap (ω, T ) and the renormalization function Z (ω, T ) are given by the following equation [19]: where and is the generalized electron-phonon coupling function averaged over the Fermi surface. We assume that there are three contributions to the Eliashberg function. The last one arises from the usual dispersive acoustic phonons limited by the Debye energy T D = 325 K. The first two contributions are due to local low-energy rattling phonons whose energies are designated by ω E1 and ω E2 . The latter requires special consideration. It has been shown recently that the low-energy phonons ascribed to the heavy rattling of the K ions confined in an oversized cage made of OsO 6 octahedra is responsible for the unusual scattering processes in KOs 2 O 6 [8]. In particular, it has been assumed that contrary to the RbOs 2 O 6 and CsOs 2 O 6 cases where a single Einstein mode due to rattling motion is enough, two modes exist athω E1 = 22 K andhω E2 = 61 K in KOs 2 O 6 corresponding to the first two excited energy levels of the corresponding anharmonic potential. Simultaneously, the meanfield description of the local alkali-ion anharmonic motion has been developed in [14]. Starting from the standard anharmonic Hamiltonian where x, p and M are the spatial coordinate, momentum and mass of the alkali ion, respectively, and a, b are constants with b > 0, one finds that according to ab initio calculations [9] for KOs 2 O 6 , the quadratic term becomes negative, i.e. a < 0, which results in a double well potential. We note that in reality the double wells are not in the plane [9]. Treating the Hamiltonian (5) in the self-consistent quasi-harmonic approximation, Dahm and Ueda [14] have found that the oscillation of potassium ions can be described by an effective harmonic oscillation with effective low-energy frequency that now depends on temperature. In particular, it decreases monotonically upon decreasing temperature 5 . At low enough temperatures (T ω 0 ), this 4 frequency becomes nearly temperature independent. Obviously this model requires a certain modification when applied to KOs 2 O 6 . In particular, although this effective model works well at high temperatures, it fails to explain the occurrence of the first-order phase transitions at temperature T p < T c and the occurrence of the second Einstein frequency at higher energy. Therefore, for modeling the contribution of the rattling K ions to the Eliashberg function, we adopt two Einstein-like modes centered at ω E1 and ω E2 . Furthermore, to take into account the upper-lying energy levels of the shallow potential we use the self-consistent quasi-harmonic approximation for the lowest mode ω E1 that results in its temperature dependence. Most importantly, before entering the temperature-independent regime, the effective rattling phonon frequency shows a discontinuous jump toward higher frequency that will result in a specific heat anomaly at T p . The increase of the ω E1 at T p is in accordance with recent experiments showing no structural changes below the phase transition [8,13,20]. Indeed, below T p the K ions seem to be stabilized in their equilibrium positions, which corresponds to the increase of ω E1 .
To find the exact shift of the lowest frequency at T p , and also the approximate positions of the rattling mode frequencies, we simulated the specific heat data around T p following the model proposed by Hiroi et al [8] with temperature-dependent ω E1 . The contribution from the two frequencies representing the rattling modes is given by the following equation [8]: where a = 0.24 and ω Ei with i = 1, 2 are the corresponding Einstein phonons. As mentioned above, for the sake of simplicity we assume that the higher mode is temperature independent and the lower one follows the mean-field temperature dependence that originates from integrating out the upper-lying energy levels with temperature. Generally, for the contribution of temperature-dependent Einstein frequency one finds where R is the gas constant. The last term is absent for the upper temperature-independent frequency, ω E2 . The results of the fit are shown in figure 1. The change of the specific heat at T p = 7.5 K is reproduced by assuming a jump of the lower Einstein frequency from hω E1 = 27.4 K tohω E1 = 30 K at this temperature with ω E2 being constant. Above T p the lower frequency mode is temperature dependent and the overall behavior of ω E1 (T ) is shown in figure 2. Note that the strong temperature dependence of the lowest rattling phonon mode frequency has been recently observed by Raman scattering [20,21] and inelastic neutron scattering experiments [22]. We further note that the increase of the lower frequency at T p is consistent with the exact diagonalization studies performed recently [23]. In addition to the anharmonic temperature dependence of the rattling phonon mode that modifies the usual approximation for the Eliashberg function, the conduction electrons couple to the rattling phonons, further renormalizing their energy positions and introducing an extra damping. In particular, the spectral function of the low-energy rattling phonon frequency is given by  Calculated temperature dependence of the lower frequency ω E1 of the rattling phonon as obtained using the quasi-harmonic approximation [14].
The parameter of the model bh/M 2 ω 3 0 (T = 300 K) = 0.25 has been fixed by assuming the room temperature position of the rattling phonon frequency around 6 meV as found by neutron scattering experiments [24]. At T p the effective frequency demonstrates a discontinuous jump to a higher energy (as shown in the inset) whose position is determined by the absolute value of the specific heat jump shown in figure 1.
where 0 is an anharmonic phonon damping rate. The real part of the rattling phonon self-energy leads to a renormalized phonon frequency, Following a previous estimation [14], we have used −Re (ω) = 1 meV and 0 = 0.25 meV. It is important to remember that the electron-phonon coupling parameter α 2 (T ) ∼ c/ω E1 (T ) with c = 1.23 meV 2 is also temperature dependent.
We further use i = ω i /5, c E2 = 0.767, and c p = 0.3. The (green) dashed-dotted curve refers to the lower energy rattling mode contribution to α 2 F(ω) with parameters as described in the text. In figure 3, we show the Eliashberg function α 2 i F i ( ) including all three contributions for T = 0.5 K, i.e. well below T p . In contrast to the usual Eliashberg theory, the α 2 F(ω) spectrum and the coupling constant λ = 2 +∞ −∞ d α 2 F( )/ are now temperature dependent. Setting µ * = 0.091 and solving equations (1)-(3) in the linearized limit, we find the superconducting transition temperature T c = 9.6 K. In figure 4, we show the results of the solution of the Eliashberg equations for T = 0.5 K. One finds the typical behavior of a strong-coupling superconductor. In particular, Re (ω) shows three peaks at the energy of the rattling phonons and at ω D . For larger energies, it becomes negative, reflecting the effective repulsion for ω > ω D . 6 Due to the presence of the low-energy rattling phonons one obtains a strong renormalization of the quasi-particle mass and we also find that 2 0 /k B T c ≈ 5.0. The latter is in good agreement with the experimentally observed value [12].
One of the important consequences of the temperature-dependent phonon spectrum is the slightly anomalous behavior of the superconducting gap as a function of temperature. In figure 5, we show the temperature dependence of the superconducting gap. One finds a kink at T = T p due a discontinuous change of the phonon frequency and the corresponding coupling constant. In particular, due to the slight increase of ω E1 below T p the electron-phonon coupling constant decreases, which results in a decrease of the superconducting gap with respect to its value without the structural phase transition. At the same time, overall we find relatively small modification of the superconducting properties due to the structural phase transition. This is consistent with experimental observations [8], which show a weak connection between the structural transition and superconductivity in KOs 2 O 6 . The most important result is that due to the temperature-dependent low-energy mode, the superconductivity is enhanced in KOs 2 O 6 . In particular, the dimensionless coupling constant λ(T ) increases with decreasing temperature due to a decrease in ω E1 . Further, it allows us to understand qualitatively why T c in KOs 2 O 6 is higher than that in the Rb and Cs counterparts since the characteristic energies of the rattling modes effectively lie at higher energies in these compounds. At the same time, using simply different values of ω E1 and ω E2 for all three compounds [21] one does not get the exact values of T c without further modification of the µ * value. We also find that the electron-phonon coupling strength changes from λ(T > T p ) = 1.7 toward λ(T < T p ) = 1.6 and is somewhat lower than estimated previously [8]. This is because a simple application of the McMillan-Dynes formula for estimating λ is probably questionable in KOs 2 O 6 due to the temperature dependent rattling mode, which does not exist in the simple Eliashberg theory. Our obtained results, although 8 showing an anomaly at T p , cannot explain fully the photoemission data of [12] as is evident from figure 5. On the other hand, the tunneling data of [17] give almost no temperature dependence of (T ) for T < 6 K; this is in much better agreement with the theoretical curve in figure 5.
Our calculation shows that the superconducting properties of KOs 2 O 6 should be sensitive to the external pressure. Due to the proximity to the first-order phase transition, the application of pressure may result in a modification of the electron-phonon coupling strength. Such an unusual behavior has indeed been recently found experimentally [25] although a detailed understanding both theoretically and experimentally is still required.
In summary, we have investigated the superconducting properties of the β-pyrochlore compound KOs 2 O 6 based on the strong-coupling Eliashberg approach. 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 discuss the consequence of this first-order phase transition for the anomalous temperature dependence of the superconducting gap. In particular, we have found that the superconducting gap as a function of temperature is anomalous, reflecting the temperature dependence of the Eliashberg function.