Quantum Hall effect in vacancy-engineered $\beta$-Ag$_2$Te
Accessing surface quantum transport in topological insulators is hampered by residual bulk conduction arising from lattice defects. Here, we demonstrate a novel synthesis pathway for realizing high mobility $\beta$-Ag$_2$Te thin films where surface transport is dominant. An \textit{in-situ} vacancy engineering step as part of the molecular beam epitaxy growth process acts to modify the stoichiometry and suppress donor-type defects, enabling continuous tuning of the sheet carrier density over more than an order of magnitude through the charge-neutrality point without an external gate electrode. In the lower-carrier-density films, a fully developed $\nu=1$ quantum Hall state is observed, and Landau-level energies extracted across samples collapse onto the $E_N=v_\mathrm{F}\sqrt{2e\hbar NB}$ relation, providing evidence for the massless Dirac dispersion of the top and bottom surface states. These results establish stoichiometry-driven vacancy engineering as a versatile lithography- and gate-free approach to accessing quantum Hall transport in epitaxial topological-insulator thin films.