Absence of a Dirac gap in ferromagnetic Cr$_x$(Bi$_{0.1}$Sb$_{0.9}$)$_{2-x}$Te$_3$

Magnetism breaks the time reversal symmetry expected to open a Dirac gap in 3D topological insulators that consequently leads to quantum anomalous Hall effect. The most common approach of inducing ferromagnetic state is by doping magnetic 3$d$ elements into bulk of 3D topological insulators. In Cr$_{0.15}$(Bi$_{0.1}$Sb$_{0.9}$)$_{1.85}$Te$_3$, the material where the quantum anomalous Hall effect was initially discovered at temperatures much lower than the ferromagnetic transition, $T_C$, the scanning tunneling microscopy studies have reported a large Dirac gap $sim20-100$ meV. The discrepancy between the low temperature of quantum anomalous Hall effect ($ll T_C$) and large spectroscopic Dirac gaps ($gg T_C$) found in magnetic topological insulators remains puzzling. Here, we used angle-resolved photoemission spectroscopy to study the surface electronic structure of pristine and potassium doped surface of Cr$_{0.15}$(Bi$_{0.1}$Sb$_{0.9}$)$_{1.85}$Te$_3$. Upon potassium deposition, the $p$-type surface state of pristine sample was turned into an $n$-type, allowing spectroscopic observation of Dirac point. We find a gapless surface state, with no evidence of a large Dirac gap reported in tunneling studies.

Evidence for a Vestigial Nematic State in the Cuprate Pseudogap Phase

The CuO$_2$ antiferromagnetic insulator is transformed by hole-doping into an exotic quantum fluid usually referred to as the pseudogap (PG) phase. Its defining characteristic is a strong suppression of the electronic density-of-states D(E) for energies |E|<$Delta^*$, where $Delta^*$ is the pseudogap energy. Unanticipated broken-symmetry phases have been detected by a wide variety of techniques in the PG regime, most significantly a finite Q density-wave (DW) state and a Q=0 nematic (NE) state. Sublattice-phase-resolved imaging of electronic structure allows the doping and energy dependence of these distinct broken symmetry states to be visualized simultaneously. Using this approach, we show that, even though their reported ordering temperatures T$_{DW}$ and T$_{NE}$ are unrelated to each other, both the DW and NE states always exhibit their maximum spectral intensity at the same energy, and using independent measurements that this is the pseudogap energy $Delta^*$. Moreover, no new energy-gap opening coincides with the appearance of the DW state (which should theoretically open an energy gap on the Fermi-surface), while the observed pseudogap opening coincides with the appearance of the NE state (which should theoretically be incapable of opening a Fermi-surface gap). We demonstrate how this perplexing phenomenology of thermal transitions and energy-gap opening at the breaking of two highly distinct symmetries can be understood as the natural consequence of a vestigial nematic state , within the pseudogap phase of Bi$_2$Sr$_2$CaCu$_2$O$_8$.

Atomic-scale Electronic Structure of the Cuprate d-Symmetry Form Factor Density Wave State

Extensive research into high temperature superconducting cuprates is now focused upon identifying the relationship between the classic pseudogap phenomenon$^{1,2}$ and the more recently investigated density wave state$^{3-13}$. This state always exhibits wave vector $Q$ parallel to the planar Cu-O-Cu bonds$^{4-13}$ along with a predominantly $d$-symmetry form factor$^{14-17}$ (dFF-DW). Finding its microscopic mechanism has now become a key objective$^{18-30}$ of this field. To accomplish this, one must identify the momentum-space ($k$-space) states contributing to the dFF-DW spectral weight, determine their particle-hole phase relationship about the Fermi energy, establish whether they exhibit a characteristic energy gap, and understand the evolution of all these phenomena throughout the phase diagram. Here we use energy-resolved sublattice visualization$^{14}$ of electronic structure and show that the characteristic energy of the dFF-DW modulations is actually the pseudogap energy $Delta_{1}$. Moreover, we demonstrate that the dFF-DW modulations at $E=-Delta_{1}$ (filled states) occur with relative phase $pi$ compared to those at $E=Delta_{1}$ (empty states). Finally, we show that the dFF-DW $Q$ corresponds directly to scattering between the hot frontier regions of $k$-space beyond which Bogoliubov quasiparticles cease to exist$^{31,32,33}$. These data demonstrate that the dFF-DW state is consistent with particle-hole interactions focused at the pseudogap energy scale and between the four pairs of hot frontier regions in $k$-space where the pseudogap opens.

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$meVcdot nm^2$. 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.