Title: Spin-orbit proximity in MoS$_2$/bilayer graphene heterostructures

Abstract: Van der Waals heterostructures provide a versatile platform for tailoring electronic properties through the integration of two-dimensional materials. Among these combinations, the interaction between bilayer graphene and transition metal dichalcogenides (TMDs) stands out due to its potential for inducing spin-orbit coupling (SOC) in graphene. Future devices concepts require the understanding the precise nature of SOC in TMD/bilayer graphene heterostructures and its influence on electronic transport phenomena. Here, we experimentally confirm the presence of two distinct types of SOC, Ising (1.55 meV) and Rashba (2.5 meV), in bilayer graphene when interfaced with molybdenum disulphide, recognized as one of the most stable TMDs. Furthermore, we reveal a non-monotonic trend in conductivity with respect to the electric displacement field at charge neutrality. This phenomenon is ascribed to the existence of single-particle gaps induced by the Ising SOC, which can be closed by a critical displacement field. Remarkably, our findings also unveil sharp peaks in the magnetoconductivity around the critical displacement field, challenging existing theoretical models.