Magnetoconductance evolution across the topological-trivial phase transition in ${In_{x}}({Bi_{0.3}}{Sb_{0.7}})_{2-x}{Te_3}$ thin films
We investigate the evolution of electronic transport across the topological-trivial phase transition in ${\rm In}_{x}({\rm Bi}_{0.3}{\rm Sb}_{0.7})_{2-x}{\rm Te}_3$ thin films by systematically tuning the indium concentration $x$. Increasing $x$ reduces the effective spin-orbit coupling, driving a topological quantum phase transition near $x \approx 7\%$, and at higher disorder a crossover from diffusive to strongly localized transport around $x \approx 15\%$. In the diffusive regime, the magnetoconductance is well described by the Hikami-Larkin-Nagaoka formalism, with the evolution of the WAL prefactor $\alpha$ correlating with the band-inversion transition. Beyond the diffusive limit, transport crosses into variable-range hopping, accompanied by a striking reversal of magnetoconductance from negative to positive. The observed positive low-field magnetoconductance, its pronounced anisotropy, and its temperature evolution point to an orbital origin of the response. These features are naturally captured by incorporating the incoherent hopping mechanism of Raikh \textit{et al.} together with wavefunction shrinkage, rather than by conventional quantum-correction frameworks. Our results provide a unified picture of how topology, spin-orbit coupling, and disorder collectively determine the full field-temperature magnetotransport landscape in this material class, establishing a clear experimental link between the topological phase transition and the onset of incoherent hopping-dominated conduction.