Past exam of the MPhil in micro- and nanotechnology enterprise course of the University of Cambridge / 2021 / Paper 2 / Section B / 20 / iv / c / Solution

The transport mean free path is $\ell =v_F{\tau }_{\mathrm{tr}}$, while $\mu =e{\tau }_{\mathrm{tr}}/m^{\ast }$. Eliminating the scattering time gives $\ell =\mu \hslash k_F/e$. The usual ballistic focusing mobility criterion $\ell \gtrsim L$ therefore givesA more geometrically faithful first-peak survival criterion uses the semicircular path length $s=\pi L/2$, giving $\mu \gtrsim 17.4{\mathrm{m}}^2{\mathrm{V}}^{-1}{\mathrm{s}}^{-1}$. There is no sharp universal visibility threshold: clearly resolved peaks favour $\ell \gg s$, small aperture broadening and specular reflection. The mobility inferred in part (ii) corresponds to $\ell \simeq 314\mu \mathrm{m}$, comfortably above both estimates. Higher-order peaks can still be reduced by boundary roughness even when bulk mobility is high.