The unconventional temperature variation of the static susceptibility $\ensuremath{\chi}(T)$ that has been discovered in various copper oxide superconductors is explained in terms of a model density of states that has a step shape at an energy threshold ${E}_{0}$ along with a logarithmic Van Hove singularity at the same ${E}_{0}$. Calculations of $\ensuremath{\chi}(T)$ and the Knight shift above the superconducting transition temperature ${T}_{c}$ yield good fits to the YBCO, BSCCO, and LSCO data by adjusting only the Fermi energy $\ensuremath{\mu}$ in correspondence to the oxygen or Sr content, respectively. When $\ensuremath{\mu}$ is right on or slightly below the Van Hove singularity, an upturn in $\ensuremath{\chi}$ occurs as the temperature $T$ is lowered. By contrast, when $\ensuremath{\mu}$ is slightly above the threshold energy ${E}_{0}$, a downturn in $\ensuremath{\chi}$ is achieved as $T$ is lowered. A correlation of these phenomena with experimental data provides insight into the proximity of the Van Hove singularity to $\ensuremath{\mu}$ in several cuprate superconductors. The YBCO and TBCO cuprates with the higher ${T}_{c}$ values exhibit a nearly constant susceptibility that suggests a Fermi energy well removed from the Van Hove singularity. The sensitivity of ${T}_{c}$ as well as the susceptibility to chemical changes may provide tests of electronic mechanisms of electron pairing as well as the BCS theory.
We have measured the temperature dependence of both the surface resistance and the change of the penetration depth in two optimized epitaxial $c$-axis oriented YBa${}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{7\ensuremath{-}\ensuremath{\delta}}$ (YBCO) films at 87 GHz by incorporating each film as an end plate in a cylindrical copper cavity. A high frequency is used in order to increase losses in the superconducting samples relative to the losses in the copper cavity. It is found that our measuring frequency is of a magnitude comparable to the relevant low-temperature scattering rates, so that the real part of the conductivity would be expected to display significant frequency dependence. The two films investigated were both 350 nm thick, but prepared by different techniques. The experimental results are compared to weak-coupling $d$- and $s$-wave models of superconductivity which incorporate both inelastic and elastic scattering, with the latter forming a small part of the total scattering. The sizable surface resistance at low temperatures and the approximately linear temperature variation can be accounted for without subtracting an extrinsic residual surface resistance, if $d$- or anisotropic $s$-wave order parameters with nearly vanishing Fermi surface averages and scattering phase shifts close to 0.4$\ensuremath{\pi}$ are assumed. Large low-temperature losses are obtained theoretically in spite of the fact that order parameter amplitudes must be in the range of $2{\ensuremath{\Delta}}_{0}{(0)/k}_{B}{T}_{c}=6.0--7.5$, considerably larger than the corresponding weak-coupling values, in order to describe the data at higher temperatures. When inelastic scattering is represented by a phenomenological temperature-dependent scattering rate, a quantitative fit to the experimental data for both the surface resistance and the penetration depth of YBCO over the whole measured temperature range from 4.2 to 145 K can be obtained within a single model. Some discrepancy between theory and experiment remains near the transition temperature where fluctuation effects, not treated in this paper, are clearly visible. While very different parameter sets can be found that would fit the real part of the conductivity, having to explain both penetration depth and surface resistance puts severe constraints on the available parameter space. A description of the inelastic scattering on the basis of spin fluctuation exchange within the nested Fermi-liquid model with full frequency dependence taken into account still gives reasonable fits to the data, even though only a single parameter, fixed by the normal-state resistivity, is involved. For $s$-wave states, whose Fermi surface average is a sizable fraction of the order-parameter amplitude, scattering rates drop well below the experimental frequency at sufficiently low temperatures for the whole range of scattering phase shifts. Thermally excited quasiparticles still present then act as a nearly ideally conducting system which results in losses too low to be compatible with the experimental observations.
In the theoretical analyses of impurity effects in superconductors the assumption is usually made that all quantities, except for the Green functions, are slowly varying functions of energy. When this so-called Fermi Surface Restricted Approximation is combined with the assumption that impurities can be represented by delta-function potentials of arbitrary strength, many reasonable looking results can be obtained. The agreement with experiments is not entirely satisfactory and one reason for this might be the assumption that the impurity potential has zero range. The generalization to finite range potentials appears to be straightforward, independent of the strength of the potential. However, the selfenergy resulting from scattering off finite range impurities of infinite strength such as hard spheres, diverges in this approximation at frequencies much larger than the gap amplitude! To track down the source of this unacceptable result we consider the normal state. The elementary results for scattering off a hard sphere, including the result that even an infinitely strong delta-function potential does not lead to scattering at all in systems of two and more dimensions, are recovered only when the energy dependencies of all quantities involved are properly taken into account. To obtain resonant scattering, believed to be important for the creation of mid-gap states, the range of the potential is almost as important as its strength.
Li et al. found that the critical current density [Formula: see text] across atomically clean c-axis twist junctions of Bi 2 Sr 2 CaCu 2 O 8+δ is the same as that of the constituent single crystal, [Formula: see text], independent of the twist angle ϕ 0 , even at and below T c . We investigated theoretically if a d x 2 -y 2 -wave order parameter might twist by mixing in d xy -wave components, but found that such mixing cannot possibly explain the data near to T c . Combined with group theoretical arguments, we then conclude that the order parameter contains at least a substantial s-wave component, but does not contain any purported d x 2 -y 2 -wave compoenent, except possibly below a second, unobserved phase transition. By studying tunneling models, we further conclude that the intrinsic c-axis Josephson tunneling in Bi 2 Sr 2 CaCu 2 O 8+δ is likely to be most incoherent. We also propose a c-axis junction version of a tricrystal experiment, which does not rely upon expensive substrates.
