Transverse magnetic focusing ($B_j=2j\hslash k_F/(eL)$)

A magnetic field curves ballistic carrier trajectories between an injector and a collector. For openings separated along a specular edge by $L$, peaks occur near $L=2j{r}_c$ with $r_c=\hslash k_F/(e|B|)$, giving $|B_j|=2j\hslash k_F/(eL)$. Field polarity and carrier sign select the connection direction. Electron phase coherence can produce fine structure but is not required for the basic semiclassical focusing geometry.

In a parabolic band, transport mobility and mean free path obey $\ell =\mu \hslash k_F/e$. The rough condition $\ell \gtrsim L$ gives $\mu \gtrsim eL/(\hslash k_F)$, whereas surviving a first semicircle uses path length $\pi L/2$. Clear peaks usually need a larger margin and a specular boundary, so this is an observability scale rather than a universal sharp threshold.