Effective dimensionality crossover for superconducting surfaces in parallel magnetic fields

The superconducting fluctuations above ${T}_{c}$ in the presence of a magnetic field parallel to the surface are known to be one dimensional (1D) in the bulk and 2D near the surface. The crossover between these two behaviors is studied as a function of the surface boundary condition on the order parameter. Similar effects can be obtained due to the proximity of an additional surface, in the case of a film, as found by Thompson. The parameter governing the crossover is an effective mass related to the dependence of the eigenvalue of the Ginzburg-Landau equation on the distance from the surface. Predictions for the surface specific heat and static impedance and their crossovers from 1D to 2D behaviors are obtained. For a film with a "critical" thickness of $\ensuremath{\sim}1.8{\ensuremath{\xi}}_{T}$, Gaussian exponents of 5/4 for the divergences of the above quantities are obtained. This corresponds to an effective dimensionality of 3/2, and is due to the fact that the dependence of the Ginzburg-Landau free energy on one of the momentum components becomes quartic. The same happens more generally in other $k$-space instabilities (Lifschitz multicritical points). The statement about the above effective change in dimensionality is also proved to order $\ensuremath{\epsilon}=4\ensuremath{-}d$ in a renormalization-group calculation.

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