Topological insulators (TIs) represent a unique class of materials characterized by insulating bulk states and conducting surface states protected by time-reversal symmetry. The verification of these surface states has been a critical focus in condensed matter physics, with angle-resolved photoemission spectroscopy (ARPES) and scanning tunneling microscopy (STM) serving as two of the most powerful tools for probing their electronic structure and topological properties.

ARPES has played a pivotal role in directly visualizing the Dirac cone dispersion of TI surface states. In Bi₂Se₃, a prototypical TI, ARPES measurements revealed a single Dirac cone at the Γ point of the surface Brillouin zone, with linear dispersion extending up to approximately 0.3 eV above the Fermi level. The Dirac point, located around 0.35 eV below the Fermi level, confirmed the existence of spin-momentum-locked surface states. Spin-resolved ARPES further demonstrated the helical spin texture, where electron spins are locked perpendicular to their momentum, a hallmark of topological protection. Similar results were observed in Bi₂Te₃ and Sb₂Te₃, though variations in Dirac point position and Fermi level pinning were noted due to intrinsic doping effects.

In addition to confirming the Dirac cone, ARPES provided evidence of the absence of backscattering in TI surface states. Unlike conventional two-dimensional electron gases, where defects or impurities lead to momentum reversal and localization, topological surface states exhibit suppressed backscattering due to spin-momentum locking. This was observed in ARPES linewidth measurements, which showed minimal broadening near the Dirac point, consistent with the protection afforded by time-reversal symmetry.

STM and scanning tunneling spectroscopy (STS) complemented ARPES by offering real-space imaging and local density-of-states (LDOS) measurements. STM studies on Bi₂Se₃ surfaces revealed standing wave patterns arising from quasiparticle interference (QPI) due to scattering off defects or step edges. Fourier transforms of these patterns matched the expected scattering vectors between opposite momenta on the Dirac cone, further validating the spin-momentum locking mechanism. STS measurements near the Dirac point exhibited a V-shaped LDOS, characteristic of linearly dispersing Dirac fermions, with the minimum in conductance corresponding to the Dirac point energy.

Another key STM finding was the observation of topological surface states coexisting with bulk defects. In Bi₂Se₃, selenium vacancies often dope the bulk n-type, shifting the Fermi level into the conduction band. Despite this, STM showed that surface states remained intact, with Dirac-like features persisting even in the presence of significant disorder. This resilience underscored the robustness of topological protection against non-magnetic perturbations.

In more complex TIs, such as ternary compounds like Bi₂Te₂Se, ARPES and STM revealed additional nuances. While bulk conductivity was reduced due to compensation doping, surface states remained well-defined, with ARPES showing a Dirac point closer to the Fermi level. STM studies on these materials identified impurity resonances that interacted weakly with surface states, preserving their topological character.

Beyond binary and ternary chalcogenides, ARPES and STM were instrumental in probing the surface states of strained HgTe and quantum-confined systems. In ultrathin films of HgTe, ARPES detected a transition from a normal insulator to a TI phase as thickness exceeded a critical value (~6.3 nm), marked by the emergence of Dirac cones at the Γ point. STM measurements on these films confirmed the spatial confinement of surface states and their immunity to local defects.

The combination of ARPES and STM also shed light on magnetic perturbations in TIs. When time-reversal symmetry is broken, such as through magnetic doping or proximity coupling, a gap can open at the Dirac point. ARPES studies on Cr-doped Bi₂Se₃ observed a ~70 meV gap, while STM detected corresponding changes in LDOS near the Dirac energy. These findings aligned with theoretical predictions of a quantum anomalous Hall effect in magnetized TIs.

In summary, ARPES and STM have provided definitive experimental validation of TI surface states through direct observation of Dirac cones, spin-momentum locking, suppressed backscattering, and robustness against disorder. These techniques continue to be indispensable for exploring new topological materials and their potential applications in spintronics and quantum computing.