Imaging Dirac-Mass Disorder from Magnetic Dopant-Atoms in the Ferromagnetic Topological Insulator Cr$_x$(Bi$_{0.1}$Sb$_{0.9}$)$_{2-x}$Te$_3$

To achieve and utilize the most exotic electronic phenomena predicted for the surface states of 3D topological insulators (TI),it is necessary to open a "Dirac-mass gap" in their spectrum by breaking time-reversal symmetry. Use of magnetic dopant atoms to generate a ferromagnetic state is the most widely used approach. But it is unknown how the spatial arrangements of the magnetic dopant atoms influence the Dirac-mass gap at the atomic scale or, conversely, whether the ferromagnetic interactions between dopant atoms are influenced by the topological surface states. Here we image the locations of the magnetic (Cr) dopant atoms in the ferromagnetic TI Cr$_{0.08}$(Bi$_{0.1}$Sb$_{0.9}$)$_{1.92}$Te$_3$. Simultaneous visualization of the Dirac-mass gap $\Delta(r)$ reveals its intense disorder, which we demonstrate directly is related to fluctuations in $n(r)$, the Cr atom areal density in the termination layer. We find the relationship of surface-state Fermi wavevectors to the anisotropic structure of $\Delta(r)$ consistent with predictions for surface ferromagnetism mediated by those states. Moreover, despite the intense Dirac-mass disorder, the anticipated relationship $\Delta(r)\propto n(r)$ is confirmed throughout, and exhibits an electron-dopant interaction energy $J^*$=145$meV\cdot nm^2$. These observations reveal how magnetic dopant atoms actually generate the TI mass gap locally and that, to achieve the novel physics expected of time-reversal-symmetry breaking TI materials, control of the resulting Dirac-mass gap disorder will be essential.