Numerical study of the negative nonlocal resistance and the backflow current in a ballistic graphene system

Besides the giant peak of the nonlocal resistance $R_{NL}$, an anomalous negative value of $R_{NL}$ has been observed in graphene systems, while its formation mechanism is not quite understood yet. In this work, utilizing the non-equilibrium Green's function method, we calculate the local-current flow in an H-shaped non-interacting graphene system locating in the ballistic regime. Similar to the previous conclusions made from the viscous regime, the numerical results show that a local-current vortex appears between the nonlocal measuring terminals, which induces a backflow current and a remarkable negative voltage drop at the probe. Specifically, the stronger the vortex exhibits, the more negative $R_{NL}$ manifests. Besides, a spin-orbital coupling is added as an additional tool to study this exotic vortex, which is not a driving force for the arising vortex at all. Moreover, a breakdown of the nonlocal Wiedemann-Franz law is obtained in this ballistic system, and two experimental criteria are further provided to confirm the existence of this exotic vortex. Notably, a discussion is made that the vortex actually originates from the collision between the flowing current and the boundaries, due to the long electron mean free path and the consequent ballistic transport caused by the specific linear spectrum of graphene.