Hydrodynamics of the viscous electron fluid in cadmium

Abstract

Thanks to electron-electron ($e$-$e$) collisions conserving momentum, metallic electron fluids are viscous. Yet, this viscosity is rarely detectable in bulk transport. Here, we report on the canonical realization of the Gurzhi effect in an elemental three-dimensional metal: cadmium. Using focused ion beam microstructuring to tune the effective thickness, we detected a low-temperature size-dependent resistivity upturn in a finite window sandwiched between ballistic and diffusive regimes. Within this window, the electrical conductivity displays a simultaneous quadratic dependence on both sample size and temperature -- fingerprint of a hydrodynamic flow. This leads us to quantify the amplitude and the temperature dependence of kinematic and dynamic viscosity of the electron fluid. In cadmium, in contrast with graphene and $^3$He, the rate of momentum-conserving $e$-$e$ collisions is not set by the main Fermi energy, but by Lilliputian energy scales and inter-valley bottlenecks.