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 ${\zeta }_1$ and ${\zeta }_2$, respectively. The Gaussian fluctuation free energy is diagonalized, yielding N distinct $T_c$ values. Just above the highest $T_c$, the fluctuations are usually dominated by the three-dimensional (3D) regime of a single cellular OP. In the 2D regime further above $T_c$, more of the OP’s contribute to the fluctuations, their relative contributions depending upon the ${\zeta }_1$ and ${\zeta }_2$ values. The temperature (T) and angular (θ) dependence of the resulting fluctuation magnetization M(θ,T) is calculated.

In a weak magnetic field B [B<$B_0$=${\phi }_0$/(${\mathrm{sv}}_F$${\tau }_{\phi }$), where ${\phi }_0$, $v_F$, and ${\tau }_{\phi }$ are the flux quantum, intralayer Fermi velocity, and phase coherence lifetime, respectively], the susceptibility ${\chi }_{\alpha \beta }$ is diagonal in the crystal representation, resulting in an ordinary (anisotropic mass) θ dependence at fixed T near $T_c$. At very high fields in very high-quality single crystals (${\omega }_c$${\tau }_{\phi }$≫1, where ${\omega }_c$ is the pair cyclotron frequency), M is best evaluated in the field representation. The component $M_{\mathrm{B}}$(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 $q_c^{\mathrm{*}}$ is sufficiently large. In this field regime, $q_c^{\mathrm{*}}$ is either on the order of πs/a, where a is the intralayer oxygen site repeat distance, or equal to π[1+(B/$B_0$)sinθ].

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 ${\tau }_{\phi }$$k_B$$T_c$/ħ≳≳≳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 $T_c$ is found to be proportional to ${\chi }_{\mathrm{cc}}$(T)/T. For fits of the theoretical FSH to experimental data, it is necessary to modify the mean-field expressions for T<$T_c$, 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 $T_c$ to 2D behavior further from $T_c$. For the fluctuation diamagnetism the DCR temperature $T_0$(θ) depends strongly upon θ in the vicinity of π/2, where it diverges. Away from $T_c$ 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 $T_c$. For N=2 detailed plots of the above fluctuation quantities for a range of the microscopic parameters are presented.

• Received 8 August 1989

DOI:https://doi.org/10.1103/PhysRevB.41.2073