Past exam of the physics of Nanoelectronic Systems course of the University of Cambridge / 2013 / Paper 1 / 1 / a / Solution

Use $e>0$, $B>0$ along $+z$, and $U(y)$ for the electron's confinement potential energy. The spin-resolved Landau level branches in the smooth-edge approximation areThe spin labels denote upper and lower energies; the sign of a material's effective $g$ factor is not inferred from the stated positive splitting. In units of $\hslash {\omega }_c$, the four occupied bulk energies are $1/3,2/3,4/3,5/3$. A chemical potential in $(5/3,7/3)$ therefore gives filling factor four. The branches rise at both confining edges and cross the common Fermi energy once at each edge. Each crossing is one spin-resolved quantum Hall edge state; there are four on either side. The two spin branches within an orbital level are separated by $\hslash {\omega }_c/3$, while the intervening gap to the next orbital pair is $2\hslash {\omega }_c/3$.