The Child-Langmuir plasma sheath above a step

The plasma sheath is the space charge boundary layer that separates quasi-neutral plasma from a material surface. Child-Langmuir sheath theory assumes a plasma with cold, collisionless ions and Boltzmann electrons having a temperature $T_e$ contacts a cathode at a potential $\phi_c < 0$ (i.e., below the plasma space potential). We consider the Child-Langmuir sheath above a cathode with a step of height $d$. If $s_0$ is the planar Child-Langmuir sheath width, then far from the step we expect the sheath to be planar with a width $s_0$. However, in the neighborhood of the step the sheath must change levels, where the lateral transition length is $l$. Using a particle-in-cell code, we show that the sheath edge is well described by a hyperbolic tangent and that $l \approx s_0$.