Anomalous superfluid density in quantum critical superconductors

Abstract

When a second-order magnetic phase transition is tuned to zero temperature by a nonthermal parameter, quantum fluctuations are critically enhanced, often leading to the emergence of unconventional superconductivity. In these "quantum critical" superconductors it has been widely reported that the normal-state properties above the superconducting transition temperature T(c) often exhibit anomalous non-Fermi liquid behaviors and enhanced electron correlations. However, the effect of these strong critical fluctuations on the superconducting condensate below T(c) is less well established. Here we report measurements of the magnetic penetration depth in heavy-fermion, iron-pnictide, and organic superconductors located close to antiferromagnetic quantum critical points, showing that the superfluid density in these nodal superconductors universally exhibits, unlike the expected T-linear dependence, an anomalous 3/2 power-law temperature dependence over a wide temperature range. We propose that this noninteger power law can be explained if a strong renormalization of effective Fermi velocity due to quantum fluctuations occurs only for momenta k close to the nodes in the superconducting energy gap Δ(k). We suggest that such "nodal criticality" may have an impact on low-energy properties of quantum critical superconductors.

Fig. 1.

Temperature dependence of the magnetic penetration depth in heavy-fermion superconductors near the antiferromagnetic QCP. (A) Low-temperature changes in the magnetic penetration depth Δλ(T) = λ(T) − λ(T = 0) of single crystals of Ce₂PdIn₈ and CeCoIn₅. The curves are shifted vertically for clarity and the data for CeCoIn₅ are multiplied by 3.5. Inset shows the ac susceptibility over the whole temperature range measured by the frequency shift of the tunnel diode oscillator, showing sharp superconducting transitions. The dashed line is an exponential temperature dependence expected for a fully gapped s-wave superconductor. (B) The same data plotted against (T/T_c)². The solid line represents a T² dependence. (C) The same data plotted against (T/T_c)^3/2. The solid line represents a T^3/2 dependence.

Fig. 2.

Universal T^3/2 dependence of superfluid density in unconventional superconductors in the vicinity of the antiferromagnetic order. (A) The normalized superfluid density ρ_s as a function of (T/T_c)^3/2 at low temperatures for Ce₂PdIn₈ and CeCoIn₅. The lines represent T^3/2 dependence. Inset shows the overall temperature dependence up to (T/T_c) = 1. (B) A similar plot for iron-pnictide superconductor BaFe₂(As_0.7P_0.3)₂ and organic superconductor κ-(ET)₂Cu[N(CN)₂]Br. The solid lines are the fits to T^3/2 dependence. The low-temperature data for BaFe₂(As_0.7P_0.3)₂ are vertically shifted for clarity.

Fig. 3.

Nodal quantum criticality in unconventional superconductors. (A) The momentum-dependent gap Δ(k) (whose magnitude is illustrated by thin lines with gray shading) opens on the Fermi surface (thick line) and has nodes (red circles) at certain directions. In -wave superconductors, for example, Δ(k) has strong in-plane anisotropy Δ₀ cos(2ϕ) as a function of azimuthal angle ϕ. In quasi-2D systems, the Fermi surface is approximated by a cylinder, and thus the gap has nodal lines perpendicular to the planes. At the nodes, the gap is zero and thus the quantum critical fluctuations may be present (red shading) on the ungapped Fermi surface. (B) The nodal quantum fluctuations lead to the momentum dependence of the renormalization in near the nodes (blue lines). (C) The angle dependence of the renormalized Fermi velocity relative to the unrenormalized one, v_F along the Fermi surface, assumed for calculations of the superfluid density in D. Near the nodes, we illustrate different cutoff levels, which model finite distances from the QCP or disorder. (D) Calculated normalized superfluid density as a function of (T/T_c)^3/2 with different cutoff levels, which explains the deviation from the T^3/2 dependence at very low temperatures. Inset is the full temperature dependence up to (T/T_c) = 1.