Disorder plays a critical role in modifying the electronic properties of topological insulators, particularly in how it affects topological protection. The robustness of topological surface states against backscattering is a defining feature, but disorder can introduce mechanisms that degrade this protection. Key phenomena include Anderson localization and the formation of mobility gaps, which alter transport properties. Doped topological insulators like Bi₂Se₃ provide a clear platform for studying these effects, while weak antilocalization serves as a contrasting phenomenon that highlights the interplay between disorder and quantum coherence.

Topological protection in ideal systems arises from time-reversal symmetry and strong spin-orbit coupling, which prevent elastic backscattering of surface states. However, disorder breaks translational invariance and can introduce scattering mechanisms that compromise this protection. Anderson localization occurs when strong disorder causes electron waves to interfere destructively, localizing states and suppressing conductivity. In three-dimensional topological insulators, bulk disorder can hybridize with surface states, leading to localization effects even in the topological surface band. For example, doping Bi₂Se₃ with magnetic or non-magnetic impurities introduces scattering centers that can localize carriers, depending on the impurity concentration and strength of disorder.

Mobility gaps emerge when disorder broadens energy levels, creating regions where states are localized rather than extended. Unlike a true bandgap, a mobility gap does not imply an absence of states but rather a lack of conducting pathways. In Bi₂Se₃, high levels of disorder can cause the Dirac point to become obscured by a mobility gap, reducing surface state conductivity. Experimental studies have shown that increasing disorder shifts the Fermi level into the bulk conduction or valence band, masking the topological surface states. This effect is particularly pronounced in systems with high defect densities or inhomogeneous doping.

Weak antilocalization contrasts with Anderson localization by enhancing conductivity at low temperatures due to quantum interference. In topological insulators, spin-momentum locking suppresses backscattering, leading to constructive interference of time-reversed electron paths. This results in a positive magnetoresistance, a hallmark of weak antilocalization. However, as disorder increases, the phase coherence length decreases, weakening the antilocalization signature. In doped Bi₂Se₃, the competition between weak antilocalization and Anderson localization can be observed by tuning the disorder strength. At low disorder, weak antilocalization dominates, while at higher disorder, localization effects prevail.

The distinction between these effects lies in the nature of disorder. Non-magnetic disorder primarily induces elastic scattering, which can lead to Anderson localization if phase coherence is preserved. Magnetic disorder, on the other hand, breaks time-reversal symmetry explicitly, potentially opening a gap in the surface states and destroying topological protection. In Bi₂Se₃ doped with magnetic ions like Cr or Fe, the suppression of weak antilocalization and the emergence of a gap have been observed, demonstrating the sensitivity of topological states to magnetic perturbations.

Transport measurements provide direct evidence of these phenomena. In weakly disordered Bi₂Se₃, the surface states exhibit a metallic temperature dependence with a pronounced weak antilocalization correction. As disorder increases, the temperature dependence becomes insulating, signaling the onset of localization. Magnetotransport studies reveal crossover regimes where both weak antilocalization and localization contribute, depending on the Fermi level position and disorder strength. These observations align with theoretical predictions that topological protection is robust against moderate disorder but breaks down when disorder exceeds a critical threshold.

The role of disorder in topological insulators extends beyond transport properties. Angle-resolved photoemission spectroscopy (ARPES) studies of disordered Bi₂Se₃ show broadening of the surface state dispersion, indicative of increased scattering. However, the Dirac cone remains intact up to moderate disorder levels, confirming the resilience of topological states. At higher disorder, the ARPES spectra become diffuse, suggesting hybridization between surface and bulk states due to impurity potentials. This underscores the importance of controlling disorder to maintain topological protection in practical devices.

Comparing different types of disorder reveals further nuances. Point defects, such as Se vacancies in Bi₂Se₃, act as Coulomb scatterers, affecting carrier mobility but not necessarily destroying topological protection. Extended defects, like grain boundaries, introduce stronger scattering potentials that can localize carriers more effectively. The interplay between defect types and their distribution determines whether weak antilocalization or Anderson localization dominates the low-temperature transport behavior.

In summary, disorder significantly impacts topological protection through mechanisms like Anderson localization and mobility gaps. Doped Bi₂Se₃ serves as a model system for studying these effects, where the balance between weak antilocalization and localization depends on disorder strength and type. While moderate disorder preserves topological surface states, excessive disorder degrades their coherence and conductivity. Understanding these limits is crucial for designing robust topological materials for applications in spintronics and quantum computing. The competition between quantum coherence and localization remains a central theme in the study of disordered topological insulators.