Space charge limited current in a gap combined of free space and solid

The paper presents a model of the space charge limited (SCL) electron conduction in a gap with a combination of free space and dielectric solid. The SCL electron current density $J$ is solved numerically to obtain the voltage scaling of $J∼VGn$⁠, and $n$ is found to be between $n=3/2$ (a vacuum gap) and $n=2$ (a dielectric diode). The dependence of $n$ is calculated as a function of dielectric constant, electron mobility, and the relative length scale between the free space and dielectric solid. The model has been used to explain a recent experiment.