Impurity bound states in $d$-wave superconductors with subdominant order parameters

Single magnetic impurity induces intra-gap bound states in conventional $s$-wave superconductors (SCs) but, in $d$-wave SCs only virtual bound states can be induced. However, in small cuprate islands a fully gapped spectrum has recently been discovered [1,2]. In this work, we investigate the real bound states due to potential and magnetic impurities in the two candidate fully gapped states for this system: the topologically trivial $d+is$-wave state and the topologically non-trivial $d+id'$-wave (chiral $d$-wave state). Using the analytic T-matrix formalism and self-consistent numerical tight-binding lattice calculations, we show that potential and magnetic impurities create entirely different intra-gap bound states in $d+is$-wave and chiral $d$-wave SCs. Therefore, our results suggest that the bound states mainly depend on the subdominant order parameter. Considering that recent experiments have demonstrated an access to adjustable coupling $J$ [3], impurities thus offer an intriguing way to clearly distinguish between the chiral $d$-wave and topologically trivial $d+is$-wave state. [1] Gustafsson, David, et al., Nat. Nanotechnol. 8, 25 (2013). [2] Black-Schaffer, Annica M., et al., Phys. Rev. Lett. 110, 197001 (2013). [3] Hatter, Nino, et al. Nat., Comm. 6, 8988 (2015).