Supersonic flow and hydraulic jump in an electronic de Laval nozzle

Electronic systems with global momentum conservation can be described as a hydrodynamic fluid. Clean material systems with strong carrier-carrier interactions, such as graphene [1] or WTe₂ [2], can reach this regime, leading to novel electronic phenomena such as whirlpools, superballistic conductance and Poiseuille flow, all analogues of incompressible flow.

Compressible flow, where the drift velocity of the carriers is comparable to the sound velocity of the fluid and the fluid density is no longer constant, has been unexplored in electronic systems. In this work, we use the the de Laval nozzle geometry [3] to accelerate the carriers to supersonic speeds, which then relax abruptly to subsonic velocities at a shock [3]. The tunable, low electronic speed of sound in bilayer graphene makes this realizable in experiment.

Our work investigates discontinuities in electronic transport consistent with supersonic flow [4]. Kelvin probe force measurements identify regions with supersonic flow and observe a hydraulic jump in the local potential, the equivalent of a shock wave for liquids. This is the first demonstration of compressible electron flow, and we will discuss possible flow instabilities that can now be realized.

[1] A. Lucas, K.C. Fong, J. Phys. Condens. Matter 30, 053001 (2018)

[2] A. Aharon-Steinberg et al. Nature 607, 74-80 (2022)

[3] K. Moors, O. Kashuba, T. L. Schmidt. arxiv:1905.01247 (2019)

[4] J. Geurs, T. Webb et al. arxiv:2509.16321 (2025)