Direct numerical demonstration of sign-preserving quasiparticle interference via an impurity inside a vortex core in an unconventional superconductor

We perform large-scale self-consistent numerical calculations, solving the Bogoliubov-de Gennes (BdG) equations in the magnetic field together with random impurities to directly demonstrate the typical quasiparticle interference (QPI) in the presence of vortices as observed by scanning tunneling microscopy and spectroscopy experiments in unconventional superconductors. The calculations reveal that the vortex itself never works as a scatter, causing the QPI pattern, but the vortex core containing impurity brings about the enhancement of the sign-preserving QPI peaks. Its origin is Andreev-bound states distorted by impurity, and all the measurement findings are consistently explained by the scenario based on the numerical results.

Figure 1

(a) A spatial profile of the $d_{x^2-y^2}$-wave order-parameter amplitude $|{\Delta }_d\left({\mathit{r}}_i\right)|$. (b) A $\mathit{q}$-space profile of $Z(\mathit{q},E)$, which is obtained by Fourier transforming the quantity $Z(\mathit{r},\omega )\equiv N(\mathit{r},\omega )/N(\mathit{r},-\omega )$ at the energy $\omega =0.04t$. The system size is $160\times 160$ square grids with 10 randomly distributed impurities without vortex.

Figure 2

Left panel: Local densities of states far from a vortex core and inside a vortex core without any impurities. Inset: A spatial profile of the $d_{x^2-y^2}$-wave order-parameter amplitude $|{\Delta }_d\left({\mathit{r}}_i\right)|$. Right panel: A $\mathit{q}$-space profile of $Z(\mathit{q},E)$. For the definition of $Z(\mathit{q},E)$ and the system size, see the caption of Fig. 1.

Figure 3

(a) and (c) Spatial profiles of the $d_{x^2-y^2}$-wave order-parameter amplitude $|{\Delta }_d\left({\mathit{r}}_i\right)|$. (b) and (d) $\mathit{q}$-space profiles of $Z(\mathit{q},E)$. For the definition $Z(\mathit{q},E)$, see the caption of Fig. 1. We consider the $160\times 160$ square lattice system with 10 randomly distributed impurities with vortices. (a) and (b) Any impurities are not located inside vortices (“unpinned vortex”). (c) and (d) One impurity is located inside a vortex (“pinned vortex').

Figure 4

Left panel: A $\mathit{k}$-space profile of the Fermi surface (red solid curve) in the square lattice model with $\mu =-1.5t$ and the white and shaded areas represent signs of $d$-wave superconducting gap on $\mathit{k}$ space. The arrows denote typical sign-preserving scattering $\mathit{q}$ vectors (see the text). Right panel: A $\mathit{q}$-space profile of the sign preserving (+) and reversing ($-$) on the superconducting gap.