Scanning tunneling spectroscopic (STS) studies of the bulk magnetic doping effects on the surface state of Bi$_{2}$Se$_{3}^{\ast }$

We report STS studies of MBE-grown undoped and Cr-doped Bi$_{2}$Se$_{3}$ bi-layers on InP (111) and as a function of the updoped layer thickness and the Cr-doping level ($x)$. Our studies reveal \textit{gapless} Dirac spectra at all temperatures ($T)$ for samples with an undoped top layer larger than 5 QLs, implying that the interlayer magnetic correlation length $\xi_{\bot }$ is \textless $\sim$ 5-QL. For samples with an undoped top layer smaller than 5 QLs, STS reveals \textit{gapped} spectra at $T$ \textless $T_{c}=$ (260 $\pm$ 20) K. The gap is spatially inhomogeneous and increases with decreasing $T$, reaching an $x$-independent maximum $\Delta =$ (0.8 $\pm$ 0.2) eV at $T$ \textless \textless $T_{c}$. Further, the gap inhomogeneity increases with decreasing $x$, showing magnetic clusters separated by gapless regions and an in-plane magnetic correlation length $\xi_{\vert \vert }$ $\sim$ 8-QL. We also find spatially localized double and single resonance peaks in the gapless regions, and their areal densities peak near $T_{c}$. We attribute the resonance sites to isolated Cr impurities, which couple with the spins of surrounding Dirac electrons and form localized topological spin textures of a long lifetime. With increasing interlayer magnetic field, the resonance sites diminish and the gap distribution becomes more homogeneous.