Chirality Detection through Vortex Bound States in ($d+id'$)-Wave Superconductor

We present a method for detecting the chirality $\chi$ of a ($d_{zx}+i \chi d_{yz}$)-wave superconductor through the analysis of the local density of states (LDOS) at the vortex core. Employing the quasiclassical Eilenberger theory, we examine the LDOS in a semi-infinite superconductor with a quantum vortex penetrating the surface perpendicularly. We show that $\mathrm{sgn}[\chi]$ changes completely the LDOS at the core-surface intersection. Remarkably, the difference between LDOS for the $\chi = 1$ and $\chi = -1$ states becomes more prominent when the surface is dirtier, meaning that one does not need to pay close attention to the surface quality of the sample. The difference between these two states arises from the symmetry of the subdominant Cooper pairs induced at the core-surface intersection: whether the subdominant $s$-wave Cooper pairs are present or not. Due to the unique nature of this phenomenon in the ($d_{zx}+i \chi d_{yz}$)-wave superconductor, one can potentially demonstrate the realization of the ($d_{zx}+i \chi d_{yz}$)-wave superconductivity and determine its chirality by, for instance, through scanning tunnel spectroscopy experiments.