Local observation of linear-$T$ superfluid density in chiral $d$-wave superconductor candidate URu$_2$Si$_2$

The heavy fermion superconductor URu$_2$Si$_2$ has been extensively studied to reveal the order parameters of the enigmatic hidden order phase (HO) ($T_{HO}$ = 17.5 K) and the coexisting unconventional superconducting phase (SC) ($T_c$ = 1.5 K). To observe the local superfluid density in URu$_2$Si$_2$, we used a scanning SQUID microscope to apply a low magnetic field locally at the micrometer scale and measured the susceptibility response. [1] 

Ferromagnetic (FM) domains coexisting with the superconducting phase were observed in this sample. 

However, we measured the susceptibility as a function of the distance between the sample and the SQUID far from these FM domains, estimating the London penetration depth at each temperature. 

Performing this at three locations, we obtained similar results showing a linear temperature dependence of the superfluid density. This suggests the presence of line nodes in the superconducting gap of URu$_2$Si$_2$.

Imaging homogeneous single-phase superfluid density in chiral superconductor candidate UTe$_2$

UTe$_2$ is a newly discovered odd-parity superconductor. In some samples, spontaneous time-reversal symmetry breaking and multiple superconducting transitions have been observed at ambient pressure, suggesting the emergence of a chiral superconducting state. However, with the ability to produce high-quality samples, heat capacity measurements have indicated that the multiple superconducting transitions near zero magnetic field at ambient pressure are due to sample inhomogeneity. Therefore, an accurate understanding of the superconducting properties of UTe$_2$ requires measurements on uniform samples. At the time of our study, there had been no reports of studies on UTe$_2$ using scanning magnetic microscopes.

We utilized a scanning SQUID microscope to observe the local superfluid density in UTe$_2$. [2] The experiment used a cleaved surface of high-quality UTe$_2$ single crystals. Observing the local susceptibility (superfluid response), we evaluated sample inhomogeneity on a micron scale and estimated the local superconducting transition temperature $T_c$. A $T_c$ increase of 30 mK was observed near the crystal edge, but uniform $T_c$ was maintained away from the edge, with no evidence of the expected multiple superconducting transitions near $T_c$. From the susceptibility observed locally in this highly uniform region, we successfully estimated the temperature variation of the superfluid density. This temperature variation of the superfluid density cannot be explained by a superconducting gap model with point nodes along the $b$-axis. Still, it is consistent with a model with a slightly opened gap or point nodes along the $a$-axis on a quasi-two-dimensional Fermi surface.

Anomalous temperature dependence of superfluid density in quasi-2D superconductor Pd$_x$ErTe$_3$

The superfluid density $n_s$, a crucial parameter indicating the stiffness of the superconducting order parameter, typically shows a mild increase in its temperature derivative near the critical temperature $T_c$ for conventional 3D BCS(Bardeen-Cooper-Schrieffer) superconductors. Exceptionally, in two-dimensional systems thinner than the coherence length, a sharp temperature derivative $dn_s(T)/dT|{T\rightarrow T_c}$ is expected. This is because thermal fluctuations are stronger in low-dimensional systems, reducing $T_c$ from the BCS theory value to the Berezinskii-Kosterlitz-Thouless (BKT) transition temperature. 

However, we observed a significant increase in this derivative $dn_s(T)/dT|_{T\rightarrow T_c}$ for our quasi-2D samples, suggesting the influence of quantum phase fluctuations and thermal fluctuations. [3] Using scanning SQUID susceptometry, we examined the local superfluid responses in quasi-2D layered superconductors, Pd$_x$ErTe$_3$, materials known for their disordered charge density waves. These responses reveal a homogeneous superfluid density that markedly increases near $T_c$, deviating from the expected behavior in conventional 3D BCS superconductors. Our findings align with simulations from the quantum rotor model, indicating that quantum phase fluctuations are crucial in suppressing the critical temperature. In this model, Cooper pairs exist at all temperature ranges (considering only the low-temperature region well below the mean-field transition temperature $T_{MF}$). The Hamiltonian can be written using the number of Cooper pairs $n_j$, phase $\theta_j$, local capacitance (effective mass) $C$, and interaction with the nearest neighbor site $J$ on the site $j$ of a two-dimensional square lattice system:

$H = \sum_{j}{\frac{n_j^2}{2C}} - J\sum_{<i,j>}{\cos{(\theta_i-\theta_j)}}$

These results underscore the utility of temperature-dependent superfluid density in exploring quantum phase fluctuations in quasi-2D superconductors.