Past exam of the quantum condensed matter course of the University of Cambridge / 2019 / Section B / 5 / d

For interface area $\mathcal{A}$, each resolved spin branch has one state per flux increment $h/e$, so its degeneracy of a Landau level isIf $\nu$ such branches are occupied, including fractional occupation of the highest, the filling factor givesThe filling counts spin and any other internal branches separately; it is an integer when a whole number of resolved levels are full.

From the visible PDF axes, approximately ten unsplit periods lie between $1/B=1.0$ and $2.5{\mathrm{T}}^{-1}$, giving $\Delta (1/B)\simeq 0.15{\mathrm{T}}^{-1}$. Consecutive unsplit orbital peaks differ by two occupied spin branches, so carrier density from spin-degenerate quantum oscillations givesThe precision is limited by reading the plotted trace, roughly a few percent. The factor two assumes only spin remains unresolved in that part of the graph; counting individual spin-split peaks would instead use a one-branch filling increment.