Superfluid Density, Penetration Depth, Condensate Density

TL;DR

The paper addresses how to meaningfully define and relate three interconnected quantities in superconductors—the unitless superfluid density $\rho_s$, the penetration-depth-based response, and the scalar condensate density ${\cal N}_s$—by tracing their historical development from London to GL to BCS formalisms. It emphasizes that, in conventional superconductors, the penetration depth is governed by band-structure properties via the Drude plasma frequency and $N(0)$, with $\frac{c^2}{\lambda^2_{band}(0)}=\Omega_p^2=4\pi e^2 N(0) v_F^2$, while the true superfluid density $\rho_s$ tracks the temperature dependence of $\lambda$ and thus the gap. The work introduces and quantifies the condensate density ${\cal N}_s$ as the density of superconducting electrons, deriving its zero-temperature relation ${\cal N}^{BCS}_s(0) \approx N(0)\Delta_0$ and linking its thermodynamics to the energy gain of pairing via $\delta {\cal G}(T)$. It further discusses how optical sum rules and far-IR spectroscopy separate normal and condensate contributions, illustrating how $\Omega_p^*$ and the fractions $w_n(T)$, $w_s(T)$ describe spectral weight transfer in the superconducting state. The findings provide a framework for comparing conventional and exotic superconductors, clarifying when band-structure parameters set the scale for $\lambda$ and $n_s$, and offering a path to interpret the observed $T_c$–$\rho_s$ correlations in cuprates and related materials.

Abstract

Fascination with the concept of superconducting (SC) {\it superfluid density} $ρ_s$ has persisted since the beginning of superconductivity theory, with numerical values of an actual density rarely provided. Over time $ρ_s$, addressed mostly in cuprate and following high temperature superconductors, has become synonymous with the normalized (unitless) inverse square of the magnetic penetration depth $λ_L$ (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 $λ_{band}$ 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 $λ_{band}$ and ${\cal N}_s$ are provided for a few conventional SCs.