  Since the notion of topological insulator (TI) was envisioned in late 2000s,
topology has become a new paradigm in condensed matter physics. Realization of
topology as a generic property of materials has led to numerous predictions of
topological effects. Although most of the classical topological effects,
directly resulting from the presence of the spin-momentum-locked topological
surface states (TSS), were experimentally confirmed soon after the theoretical
prediction of TIs, many topological quantum effects remained elusive for a long
while. It turns out that native defects, particularly interfacial defects, have
been the main culprit behind this impasse. Even after quantum regime is
achieved for the bulk states, TSS still tends to remain in the classical regime
due to high density of interfacial defects, which frequently donate mobile
carriers due to the very nature of the topologically-protected surface states.
However, with several defect engineering schemes that suppress these effects, a
series of topological quantum effects have emerged including quantum anomalous
Hall effect, quantum Hall effect, quantized Faraday/Kerr rotations, topological
quantum phase transitions, axion insulating state, zeroth-Landau level state,
etc. Here, we review how these defect engineering schemes have allowed
topological surface states to pull out of the murky classical regime and reveal
their elusive quantum signatures, over the past decade.
