Critical currents of the quantum Hall effect in the mesoscopic regime

The critical currents associated with the breakdown of the quantum Hall effect have been measured in high-mobility, one-dimensional channels fabricated in GaAs-${\mathrm{Al}}_x{\mathrm{Ga}}_{1-x}\mathrm{As}$ heterostructures. The results differ in several respects from the breakdown in macroscopic samples. The on-set of dissipation shows little hysteresis and depends on the direction of the current with respect to the magnetic field. The value of the critical current is also independent of the filling factor across a large fraction of the Hall plateau. These observations indicate that mesoscopic considerations must be included in any theory of breakdown on this size scale.