Past exam of the MPhil in micro- and nanotechnology enterprise course of the University of Cambridge / 2022 / Paper 2 / Section B / 21 / ii / d / Solution

Here the step is upward by $U=3\mathrm{eV}$, so the forbidden-region energy difference is $U-E=1\mathrm{eV}$. The bounded wavefunction isThe requested amplitude penetration depth is $\delta =1/\kappa$. Using the supplied mass and $h$,The probability density decays as $e^{-2\kappa x}$, so its $1/e$ depth is half this value. For an ideal lossless semi-infinite step below threshold, the evanescent tail does not imply net stationary particle current deep into the material; the reflected flux is unity.