Dip problem of the electron mobility on a thin helium film

Electrons floating above liquid helium form an ideal two-dimensional system with an extremely high mobility. However, the mobility can change substantially when decreasing the thickness of the helium film from bulk to a thin film of a few hundred $\AA{}.$ Furthermore it is observed that for certain film thicknesses there is a pronounced dip in the mobility. We present theoretical investigations and measurements concerning this problem. Taking into account the roughness of the substrate, which supports the helium film, we find theoretically a discontinuity in the chemical potential of the electrons which results in a diplike behavior in the electron current and hence in the electron mobility. This scenario is supported by direct measurements of the electron current on substrates with different roughness and at different electron densities.