Protection of Unconventional Superconductivity from Disorder

TL;DR

The paper tackles why unconventional superconductors with a sign-changing gap are typically sensitive to disorder and reveals that symmetry-enforced zeros of Bloch weights on kagome and Lieb lattices can protect compensated pairing, yielding unusually weak $T_{ ext{c}}$ suppression. By combining group-theoretical analysis of Bloch states with Abrikosov-Gor'kov disorder theory, it shows that order parameters transforming trivially under the impurity-site symmetry can evade strong pair-breaking when Bloch weights are anisotropic across sublattices, as illustrated in kagome and Lieb lattices and contrasted with square/honeycomb lattices. The authors derive explicit expressions for the disorder self-energy and $T_{ ext{c}}$ suppression, and demonstrate that sublattice-selective impurity effects and nonuniform Bloch weights can suppress pair-breaking channels, eliminating in-gap impurity states in these systems. These findings point to material and engineered platforms, such as kagome $A$V$_3$Sb$_5$ compounds and inverse-Lieb systems, where disorder-robust unconventional superconductivity could enable higher effective $T_{ ext{c}}$ and novel spectroscopic signatures.

Abstract

Unconventional superconductivity is a desirable state of matter due to its potential for high transition temperatures $T_{\mathrm{c}}$ and associated favorable superconducting properties. However, the sign-changing nature of the order parameter of unconventional superconductors renders their condensates fragile to disorder, an inevitability in real materials. We uncover the generic properties of electronic band structures and associated Bloch weights able to support robust unconventional superconductivity. We demonstrate this property in several case studies of the kagome and Lieb lattices, showing how unconventional superconductors exhibit unusually weak $T_{\mathrm{c}}$ suppression by disorder, despite featuring fully compensated sign-changing order parameters. We contrast these results with those for unconventional superconductivity on the square and honeycomb lattices, which are unable to protect the condensates from disorder. Finally, we discuss material candidates for which this effect may be realized.

Figures (4)

• Figure 1: Schematic illustration of $T_{\mathrm{c}}$ suppression by disorder. Cooper pair electrons, in purple, scatter off an impurity, in black, with the resulting $T_{\mathrm{c}}$ suppression indicated by the blue line in the insets. The nature of the superconducting condensate is illustrated by the underlying blue and orange regions. In (a), a conventional, non-sign-changing order parameter on the square lattice is shown. Here, the non-compensated nature of the gap results in robust superconductivity. In contrast, (b) shows a compensated order parameter on the square lattice leading to fragile superconductivity. A compensated order on the Lieb lattice is shown in (c). Here, the anisotropy of the Bloch weights reduces scattering along sign-changing directions making the superconductivity more robust to disorder, reminiscent of the conventional case.
• Figure 2: Electronic structure of the Lieb lattice (a) Energy bands of the Lieb lattice tight-binding model with nearest-neighbor hopping $t = 1$ and varying next-nearest-neighbor hopping $t'$, plotted along $\Gamma-\mathrm{M}-\mathrm{X}-\Gamma$. The inset defines the three sites of the Lieb lattice. (b) The sublattice weight $|u_{\alpha}^{n}(\bm{k})|^{2}$ for each sublattice A, B, and C, and each band plotted in the BZ. The gray lines illustrate $|u_{\alpha}^{n}(\bm{k})|^{2} = 0.1$ contours for $t' = 0$ (opaque) to $t' = 0.4$ (faint). The symmetry-enforced nodes of the Bloch weights at high-symmetry points are shown in pink, and the Fermi surface for $\mu = 2.025$ is shown in black.
• Figure 3: Calculated $T_{\mathrm{c}}$-suppression for the honeycomb, kagome, and Lieb lattices. The black curve shows the solution of the AG equation $\ln(\frac{T_{\mathrm{c}}}{T_{\mathrm{c}0}}) = \psi(\frac{1}{2}) - \psi(\frac{1}{2} + \frac{\hbar}{4\pi k_{\mathrm{B}}T_{\mathrm{c}}}\frac{1}{\tau})$mineev:1999, with $\psi$ the digamma function. Unconventional superconductivity on the honeycomb lattice (a) exhibits the standard behavior of $T_{\mathrm{c}}$ versus scattering rate. In contrast, some of the unconventional order parameters on the kagome (b) and Lieb (c) lattices are remarkably robust to disorder, as seen by the orange curves with diamond markers in panels (b) and (c). The parameters used to generate each figure are listed in the SM SM.
• Figure 4: Origin of robust superconductivity. The approximate anomalous Green function at sublattice A (a) and B (b) of the Lieb lattice with on-site $d_{x^{2}-y^{2}}$-wave superconductivity. On the 2c Wyckoff position (sublattices A and C), the anomalous Green function for a $d_{x^{2}-y^{2}}$-wave order averages to a finite value.