Abstract
We study the mobility of electrons adsorbed on thin $^{4}\mathrm{He}$ films on flat, uniform, dielectric substrates. Utilizing the time-dependent version of the Euler-Lagrange, hypernetted chain variational theory, we compute the inelastic scattering rate of an electron due to collisions with film excitations (third sound). We obtain an analytic result valid in the long-wavelength limit. In agreement with experiment, the mobility shows oscillations due to the underlying transverse film structure. The oscillations are due to the explicit appearance of the third sound speed in the scattering rate, since the third sound speed itself oscillates in conjunction with the $^{4}\mathrm{He}$ film structure. The calculated mobilities tend to be higher than reported mobilities on thin films. We attribute this difference to the contribution to the mobility from substrate structure and defects that are omitted in this model. We interpret our results as generic mobilities that are valid in the limit of perfectly smooth, structureless substrates.
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