Voltage characteristics of hydrodynamic Dirac electron nozzles with supersonic flow

In clean Dirac electron systems such as graphene, electron-electron interactions can dominate over other relaxation mechanisms such as phonon or impurity scattering. In this limit, collective electron dynamics can be described by hydrodynamic equations. The prerequisites for electron hydrodynamics have already been fulfilled in experiments, and signatures of hydrodynamic flow have been identified in transport measurements. Here, we derive the pressure-driven hydrodynamic flow profile across a de Laval nozzle profile for Dirac electrons in the subsonic and supersonic regimes. Based on this, we resolve the local voltage characteristics, which provide clear signatures of supersonic hydrodynamic flow. In particular, we identify two distinct features in the experimentally measurable potential profile: a pronounced asymmetry of the local voltage profile on opposite sides of the nozzle, and a sharp differential resistance signature induced by an electron shock wave on the exit side of the nozzle.