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Electronic structure properties and BCS superconductivity in beta-pyrochlore oxides: KOs_2O_6

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 and to the rather large specific heat mass enhancement we obtain from the calculated density of states and the observed specific heat coefficient. An estimate of T_c within the framework of the BCS theory of superconductivity taking into account the possible effects of spin fluctuations arising from nesting yields the experimental value.

Transition metal (TM) oxides are of intrinsic interest in physics because of the very rich phenomenology they exhibit due to electron correlations, ranging from metalinsulator transitions to colossal magnetoresistance and high critical temperature superconductivity. TM oxide compounds with the pyrochlore structure have long been studied and have found many applications thanks to their diverse electronic properties, 1 but it is not until recently that superconductivity was found in one such a material, namely Cd 2 Re 2 O 7 . 2,3 Although its superconducting critical temperature turned out to be low (T c ≃1 K), it was an important discovery because it opened research in this area to a new class of materials. Very recently, superconductivity was reported in KOs 2 O 6 , 4,5 a so-called β-pyrochlore, with a T c of 9.6 K. More reports of superconductivity in the same family of compounds have followed at a rapid pace, with superconductivity being observed in RbOs 2 O 6 (T c = 6.3 K) 6,8 and in CsOs 2 O 6 (T c = 3.3 K), 10 adding to the interest in these materials.
The discovery of superconductivity in the βpyrochlores raises, of course, the question of the underlying mechanism. While the mechanism in Cd 2 Re 2 O 7 , an α-pyrochlore, 10 can be understood within the weakcoupling Bardeen, Cooper, and Schrieffer (BCS) theory of superconductivity, 11 Hiroi and co-workers have suggested 5 from the outset that KOs 2 O 6 is an unconventional superconductor, with the pairing mediated by spin fluctuations. 7 On the other hand, Batlogg and coworkers suggested 8 that RbOs 2 O 6 could be a conventional BCS-type superconductor, and recent pressure effects measurements appear to bring further support to their conclusions. 9 Given the close similarity between these two compounds, it seems unlikely that their superconductivity has a different origin. Clearly, a careful study of the electronic structure of these materials may shed light on the superconductivity mechanism.
In this work, we focus on KOs 2 O 6 and carry out a self-consistent first-principles density-functional calculation of its electronic structure, using a newly paral-lelized implementation 12 of the full-potential linearized augmented plane wave (FLAPW) method. 13 Our calculations are made within the Perdew, Burke, and Ernzerhof generalized gradient approximation (GGA) 14 to the exchange-correlation potential and include the spin-orbit coupling (SOC) term in the Hamiltonian. Our results show that the electronic structure near the Fermi energy (E F ) is dominated by strongly hybridized Os-5d and O-2p states. There is a van Hove singularity (vHS) very close to E F , leading to a relatively large density of states (DOS) at E F , and the Fermi surface shows strong nesting along several directions. These features could provide the scattering processes leading to the reported 4,5 anomalous temperature dependence of the resistivity above T c , and to the rather large specific heat mass enhancement we obtain from the calculated DOS and the measured low temperature specific heat coefficient. 5 We estimate the T c within the BCS framework and are able to obtain the experimental value if we take into account the possible effects of spin fluctuations arising from nesting. KOs 2 O 6 crystallizes in a cubic structure with space group F d3m, and has 18 atoms per unit cell: two K (8b), four Os (16c), and twelve O (48f ). 4 The structure has an internal parameter x, that fixes the position of the oxygen atoms. The experimental lattice constant recently given by Yonezawa and co-workers is a = 10.101Å, 5 but no value for the x parameter has yet been reported. We have optimized variationally both the lattice constant and the internal parameter, obtaining a = 10.298Å, which differs by 1.95% from the experimental value, and x = 0.316. The latter can be compared with the value of x = 0.315 recently reported for the related superconducting pyrochlore RbOs 2 O 6 by Brühwiler and collaborators. 8  for comparison with this material. The calculated energy bands along high symmetry directions in k-space, within 5 eV from E F , are shown in Fig. 1. There is a manifold of twelve states in the vicinity of E F , with a bandwidth of 3 eV, separated by relatively large energy gaps above and below. Two bands cross the Fermi energy: the lower band cuts the ΓX and W L lines and gives rise to a hole-like Fermi surface sheet in the form of a tubular network; the upper band crosses E F twice within the first Brillouin zone (cf. the ΓL, ΓX, and ΓK lines), giving rise to two Fermi surface sheets. As will be illustrated below, this results in an electronlike closed shell centered at the Γ point. Of importance is the existence of a vHS very close to E F , caused by a saddle point between the two sheets at ∼ 0.015 eV below the Fermi level, near the middle of the ΓL line.
The character of the bands near E F is further ana- lyzed by examining the density of states (DOS). The total DOS, shown in Fig. 2(a), exhibits a peak very close to E F , due to the vHS mentioned above. We point out that a precursor of the peak is already present, at a slightly lower energy, in a calculation without including spin-orbit coupling; the inclusion of the latter splits this peak, causes the saddle point, and pushes the higher peak closer to E F . In Fig. 2 Fig. 3, which presents a contour plot of the charge density in the (110) plane for states lying within 0.027 eV below E F . 18 An important finding is that the Fermi surface shows strong nesting, in particular the shell-like sheets. Contour plots of the eigenenergies for the corresponding band along two different planes centered at the Γ point are given in Fig. 4. Figure 4(a) shows the contour plot for k in the XΓX plane, with strong nesting occurring for k ≃ 0.4 4π/a and k ≃ 0.62 4π/a along the ΓK directions. Similarly, Fig. 4(b) shows a contour plot for k in the XΓK plane, also showing strong nesting, in particular for k ≃ 0.64 4π/a along ΓL directions.
Consider further some of the properties deduced from the electronic structure in relation to experiment. Firstly, the DOS at the Fermi level is relatively high, N (E F ) = 9.8 states/eV-unit cell, yielding a Sommerfeld electronic specific heat coefficient γ = 5.78 mJ/K 2 mol-Os. This is much lower than the experimental value of 19 mJ/K 2 mol-Os found by Hiroi and collaborators, 5 based on their estimation of the specific heat jump at T c and the BCS weak-coupling relation ∆C/γT c = 1.43. Thus, we find a very large specific heat mass enhancement γ exp /γ band ≃ 3.3, though it is still smaller than that of, e.g., Sr 2 RuO 4 , found to be 3.8 or more. 15,16 The above mass enhancement represents a rather high coupling constant, λ = 2.3. Given the relatively low T c of KOs 2 O 6 , it appears that, besides electron-phonon, there is a very important electronic many-body contribution to the specific heat mass enhancement. Interestingly, Hiroi and co-workers noted in particular the unusual behavior of the measured resistivity as a function of temperature. 4,5 Data corresponding to single crystals are greatly desirable to advance further in this direction. Let us just point out here that, under appropriate conditions, deviations from conventional Fermi liquid behavior in two-dimensional systems can be caused by both vHS's near the Fermi energy 19 and by Fermi surface nesting, 20 due to the increased phase space available for electron-electron scattering. 21 From the measured T c = 9.6 K, and using ∆ = 1.76k B T c , the superconducting gap is calculated to be ∆ = 1.456 meV. A calculation of the average Fermi velocity then allows us to readily estimate the BCS-Pippard coherence length ξ 0 =h v F /π∆. We find v F = 1.47 × 10 7 cm/s, which yields ξ 0 = 212Å. This is an order of magnitude larger than the reported Ginzburg-Landau coherence length ξ = 30Å, obtained from the estimated upper critical field H c2 . 5,22 Thus, the so-called dirty limit would apply. Indeed, writing ξ −1 = ξ −1 0 + l −1 , with l the mean free path, yields an estimated value of l = 35Å. This relatively short value suggests again that strong scattering processes play an important role in the electronic properties of this material.
We estimate the McMillan-Hopfield 23,24 electronphonon coupling constant λ ep , for which the constituent-weighted average can be written λ ep = i w i η i /M i ω 2 . 25 Here, the spherically averaged Hopfield parameters, η, were calculated in the crude rigid-muffin-tin approximation, with 26 is an electron-phonon matrix element. The quantities entering the latter are the logarithmic derivatives (D l ) and the partial wave amplitudes (φ l ), both evaluated at E F and at the muffin-tin radius (R); V is the oneelectron potential at R. The average phonon frequency can be estimated by ω 2 1/2 = 0.69 Θ D . There is currently no reported Θ D for KOs 2 O 6 , but Brühwiler and co-workers did such measurements for RbOs 2 O 6 . 8 Under the assumption that the values of Θ D fall in the same range for both materials, we take Θ D = 240 K, and obtain λ ep = 1.4. As expected, this value is well below the total λ obtained from the specific heat mass enhancement discussed above. 27 To estimate T c , 28 we use the Allen and Dynes modification of the McMillan equation, 23,29 including an effective electron-spin interaction coupling constant µ sp , 30 i.e., T c = ω 2 1/2 1.2 exp − 1.04(1 + λ ep + µ sp ) λ ep − (µ * + µ sp )(1 + 0.62λ ep ) .
(2) The so-called Coulomb pseudopotential µ * can be estimated from the DOS at E F , 30 and in our case is µ * = 0.09. On the other hand, there is no simple expression for µ sp . If µ sp = 0 we find T c = 18 K. Strong spin fluctuations, 31 with a coupling constant µ sp = 0.13 would yield the experimental value T c = 9.6 K. As an estimate, from our DOS we have calculated the Pauli paramagnetic susceptibility, finding χ band = 1.58 × 10 −4 cm 3 /mol. From Ref. 5, we estimate the experimental value at T = 0 to be χ exp ≃ 4 × 10 −4 cm 3 /mol, which yields an enhancement ratio χ exp /χ band = 2.53. Thus, strong spin fluctuations may indeed play an important role due to the significant nesting of the Fermi surface.