Fig. 3.

Magnetohydrodynamics and the Planckian bound of dissipation. a The origin of the decrease of the electron viscosity η in a magnetic field B (schematic) perpendicular to the current flow (B = B_z) and the sample width. The viscous friction between two adjacent layers of the electron fluid moving (the arrows point along the flow direction) with different velocities (length of the arrows) is determined by the depth of the interlayer penetration of the charge carriers e⁻ (black dots). In a magnetic field, this depth is limited by the cyclotron radius. b (ρ − ρ₀)/w² as a function of B for all four WP₂ ribbons investigated at 4 K (lines), where ρ₀ is the electrical resistance in zero field. The experimental data have been fitted by the magnetohydrodynamic model in the Navier–Stokes flow limit (gray dots). c Experimentally extracted momentum-relaxing and momentum-conserving relaxation times τ_mr and τ_mc, respectively (symbols, with guide to the eye). The error bars denote the errors of the fits exemplarily displayed for 4 K in a (see Supplementary Information for details). The dashed line marks the Planckian bound on the dissipation time τ_ℏ = ℏ/(k_BT)