Spin–orbit interaction and negative magnetoresistance for localized electrons in InSb quantum wells

Weak-field magnetoresistance (MR) in the variable-range hopping (VRH) in the presence of spin–orbit interaction (SOI) for 2DEGs at the hetero-interface of InSb quantum wells was examined in view of the quantum interference (QI) effect. Samples with the sheet resistance, ρ>ρc=h/e2, exhibit VRH, while those with ρ<ρc exhibit weak localization (WL) at low temperatures, where h/e2 is the quantum resistance. In the WL regime, a positive magnetoresistance (MR) peak due to the weak anti-localization (WAL) with SOI is clearly observed in low magnetic field. In contrast, the low-field hopping MR remains entirely negative surviving the SOI, indicating that the hopping MR due to the QI is completely negative regardless of the SOI. This result supports the predictions based on the directed-path approach for forward-scattering paths ignoring the back-scattering return loops for the QI in the VRH.