Superfluid Density, Penetration Depth, Condensate Density

TL;DR

This paper reviews the concept of superfluid density in superconductors, clarifies its relation to penetration depth and condensate density, and provides numerical estimates and theoretical insights, especially within BCS theory.

Contribution

It offers a comprehensive analysis connecting superfluid density, penetration depth, and condensate density, with updated theoretical perspectives and numerical values for conventional superconductors.

Findings

Superfluid density is closely related to the inverse square of penetration depth.

BCS theory distinguishes between superfluid density and particle density at the Fermi surface.

Numerical estimates for superfluid parameters are provided for selected conventional superconductors.

Abstract

Fascination with the concept of superconducting (SC) {\it superfluid density} $\rho_s$ has persisted since the beginning of superconductivity theory, with numerical values of an actual density rarely provided. Over time $\rho_s$, addressed mostly in cuprate and following high temperature superconductors, has become synonymous with the normalized (unitless) inverse square of the magnetic penetration depth $\lambda_L$ (the London expression, with superfluid density denoted $n_s$), with interest primarily on its temperature $T$ dependence that is expected to reflect the T-dependence of the SC gap amplitude and gap symmetry. In conventional superconductors, generalized expressions from the London penetration depth via Ginzburg-Landau theory, then to BCS theory provide updated pictures of the supercurrent density-vector potential relationship. The BCS value $\lambda_{band}$ is distinct from any particle density, instead involving particle availability at the Fermi surface and Fermi velocity as the determining factors, thus providing a basis for a more fundamental theory and understanding of what is being probed in penetration depth studies. The number density of superconducting electrons ${\cal N}_s(T$=0) -- the scalar SC {\it condensate density} -- is provided, first from a phenomenological estimate but then supported by BCS theory. A straightforward relation connecting ${\cal N}_s(0)$ to the density of dynamically transporting carriers in the normal state at $T_c$ is obtained. Numerical values of relevant material parameters including $\lambda_{band}$ and ${\cal N}_s$ are provided for a few conventional SCs.