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 ${\ensuremath{\zeta}}_{1}$ and ${\ensuremath{\zeta}}_{2}$, 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 ${\ensuremath{\zeta}}_{1}$ and ${\ensuremath{\zeta}}_{2}$ values. The temperature (T) and angular (\ensuremath{\theta}) dependence of the resulting fluctuation magnetization M(\ensuremath{\theta},T) is calculated.In a weak magnetic field B [B${B}_{0}$=${\ensuremath{\varphi}}_{0}$/(${\mathrm{sv}}_{F}$${\ensuremath{\tau}}_{\ensuremath{\varphi}}$), where ${\ensuremath{\varphi}}_{0}$, ${v}_{F}$, and ${\ensuremath{\tau}}_{\ensuremath{\varphi}}$ are the flux quantum, intralayer Fermi velocity, and phase coherence lifetime, respectively], the susceptibility ${\ensuremath{\chi}}_{\ensuremath{\alpha}\ensuremath{\beta}}$ is diagonal in the crystal representation, resulting in an ordinary (anisotropic mass) \ensuremath{\theta} dependence at fixed T near ${T}_{c}$. At very high fields in very high-quality single crystals (${\ensuremath{\omega}}_{c}$${\ensuremath{\tau}}_{\ensuremath{\varphi}}$\ensuremath{\gg}1, where ${\ensuremath{\omega}}_{c}$ is the pair cyclotron frequency), M is best evaluated in the field representation. The component ${M}_{\mathrm{B}}$(\ensuremath{\theta},T) exhibits anomalous oscillations in its \ensuremath{\theta} 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 \ensuremath{\pi}s/a, where a is the intralayer oxygen site repeat distance, or equal to \ensuremath{\pi}[1+(B/${B}_{0}$)sin\ensuremath{\theta}].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 ${\ensuremath{\tau}}_{\ensuremath{\varphi}}$${k}_{B}$${T}_{c}$/\ensuremath{\Elzxh}\ensuremath{\gtrsim}\ensuremath{\gtrsim}\ensuremath{\gtrsim}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 ${\ensuremath{\chi}}_{\mathrm{cc}}$(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 ${T}_{0}$(\ensuremath{\theta}) depends strongly upon \ensuremath{\theta} in the vicinity of \ensuremath{\pi}/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 \ensuremath{\theta}=\ensuremath{\pi}/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.