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 1 and 2, 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 OPs contribute to the fluctuations, their relative contributions depending upon the 1 and 2 values. The temperature (T) and angular () dependence of the resulting fluctuation magnetization M(,T) is calculated. In a weak magnetic field B [B<B0=0/(svF), where 0, vF, and are the flux quantum, intralayer Fermi velocity, and phase coherence lifetime, respectively], the susceptibility is diagonal in the crystal representation, resulting in an ordinary (anisotropic mass) dependence at fixed T near Tc. At very high fields in very high-quality single crystals (c1, where c is the pair cyclotron frequency), M is best evaluated in the field representation. The component MB(T) exhibits anomalous oscillations in its 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 s/a, where a is the intralayer oxygen site repeat distance, or equal to [1+(B/B0)sin]. 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 kBTc/Latin small letter h with stroke1 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 cc(T)/T. For fits of the theoretical FSH to experimental data, it is necessary to modify the mean-field expressions for T<Tc, 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() depends strongly upon in the vicinity of /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 =/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. © 1990 The American Physical Society.
