Nernst effect and its thickness dependence in superconducting NbN films

Superconducting thin films and layered crystals display a Nernst signal generated by short-lived Cooper pairs above their critical temperature. Several experimental studies have broadly verified the standard theory invoking Gaussian fluctuations of a two-dimensional (2D) superconducting order parameter. Here, we present a study of the Nernst effect in granular NbN thin films with a thickness varying from 4 to 30 nm, exceeding the short superconducting coherence length and putting the system in the three-dimensional (3D) limit. We find that the Nernst conductivity decreases linearly with reduced temperature (${\ensuremath{\alpha}}_{xy}\ensuremath{\propto}\frac{T\ensuremath{-}{T}_{c}}{{T}_{c}}$), but the amplitude of ${\ensuremath{\alpha}}_{xy}$ scales with thickness. While the temperature dependence corresponds to what is expected in a 2D picture, scaling with thickness corresponds to a 3D picture. We argue that this behavior indicates a $2+1\mathrm{D}$ situation, in which the relevant coherence length along the thickness of the film has no temperature dependence. We find no visible discontinuity in the temperature dependence of the Nernst conductivity across ${\mathrm{T}}_{c}$. Explaining how the response of the superconducting vortices evolves to the one above the critical temperature of short-lived Cooper pairs emerges as a challenge to the theory.