Phenomenological model of the copper oxide superconductors

We propose a phenomenological model for the copper oxide superconductors in which one complex s-wave order parameter (OP) is associated with each of the N conducting layers per unit cell, with N-1 equal spacings d and one different spacing d'; the c axis repeat distance s=d'+(N-1)d. The layers are coupled by Josephson-like tunneling, with parameters ζ₁ and ζ₂, respectively. The Gaussian fluctuation free energy is diagonalized, yielding N distinct Tc values. Just above the highest Tc, the fluctuations are usually dominated by the three-dimensional (3D) regime of a single cellular OP. In the 2D regime further above Tc, more of the OP's contribute to the fluctuations, their relative contributions depending upon the ζ₁ and ζ₂ values. The temperature (T) and angular ({θ}) dependence of the resulting fluctuation magnetization M({θ},T) is calculated.In a weak magnetic field B [BB₀=φ₀/(svF{{τ}}_{{φ}}$), where ${{φ}}₀$, ${v}F$, and ${{τ}}_{{φ}}$ are the flux quantum, intralayer Fermi velocity, and phase coherence lifetime, respectively], the susceptibility ${{χ}}_{{α}{β}}is diagonal in the crystal representation, resulting in an ordinary (anisotropic mass) θ dependence at fixed T near{T}c$. At very high fields in very high-quality single crystals (${{ω}}cτ_φ{}1, where ωc is the pair cyclotron frequency), M is best evaluated in the field representation. The component MB({θ},T) exhibits anomalous oscillations in its {θ} dependence, arising from degenerate multiple minima in the pair potential, provided that the effective high-momentum cutoff qc* is sufficiently large. In this field regime, qc* is either on the order of {π}s/a, where a is the intralayer oxygen site repeat distance, or equal to {π}[1+(B/B₀)sin{θ}].The oscillations are similar to de Haas--van Alphen oscillations in the B dependence of the normal state M. They are broadened by local, clean-limit dynamic effects but should be observable for τ_φ{k}BTc/{}{}{}{}1 and N=2, which we argue is the case in the best samples. In addition, the fluctuation specific heat (FSH) and Aslamazov-Larkin conductivity are calculated for B=0, including dynamic effects. For arbitrary N, the FSH above Tc is found to be proportional to χcc(T)/T. For fits of the theoretical FSH to experimental data, it is necessary to modify the mean-field expressions for TTc, such that the entropy of the superconducting transition remains zero. An example of such a modification is given. All fluctuation quantities exhibit dimensional crossover (DCR) from 3D behavior near Tc to 2D behavior further from Tc. For the fluctuation diamagnetism the DCR temperature T₀({θ}) depends strongly upon {θ} in the vicinity of {π}/2, where it diverges. Away from Tc dynamic effects are found to be important and can be so strong as to mimic DCR behavior, even for {θ}={π}/2, for which DCR should not occur. The dynamic effects make quantitative corrections to the fluctuation quantities for T as close as 1 K to Tc. For N=2 detailed plots of the above fluctuation quantities for a range of the microscopic parameters are presented.