Past exam of the MPhil in micro- and nanotechnology enterprise course of the University of Cambridge / 2023 / Paper 2 / Section A(ii) / 15 / iv

On an integer plateau the sheet transverse resistivity has magnitudewhere the integer $\nu$ counts occupied resolved quantum Hall edge states. Increasing $B$ at fixed sheet density reduces the Landau filling factor ${\nu }_{\mathrm{fill}}=n_{s}h/(eB)$. The observed trace is therefore a staircase toward larger $|{\rho }_{xy}|$, with transitions between plateaux rather than the classical straight line. In the plateau regions ${\rho }_{xx}\simeq 0$; dissipative peaks occur at transitions.

It is not correct to demand an exactly integer value of $n_{s}h/(eB)$ everywhere along a finite-width plateau. Localization accommodates changes in total population without changing the number of extended current-carrying channels. If spin remains unresolved, the usual spin-degenerate sequence contains even integers rather than every integer.