  Thin films of the wide band gap semiconductor $\beta$-Ga$_2$O$_3$ have a high
potential for applications in transparent electronics and high power devices.
However, the role of interfaces remains to be explored. Here, we report on
fundamental limits of transport properties in thin films. The conductivities,
Hall densities and mobilities in thin homoepitaxially MOVPE grown
\mbox{(100)-orientated $\beta$-Ga$_2$O$_3$} films were measured as a function
of temperature and film thickness. At room temperature, the electron mobilities
((115$\pm$10) $\mathrm{\frac{cm^2}{Vs}}$) in thicker films (> 150 nm) are
comparable to the best of bulk. However, the mobility is strongly reduced by
more than two orders of magnitude with decreasing film thickness ((5.5$\pm$0.5)
$\mathrm{\frac{cm^2}{Vs}}$ for a 28 nm thin film). We find that the commonly
applied classical Fuchs-Sondheimer model does not explain the contribution of
electron scattering at the film surfaces sufficiently. Instead, by applying an
electron wave model by Bergmann, a contribution to the mobility suppression due
to the large de Broglie wavelength in $\beta$-Ga$_2$O$_3$ is proposed as a
limiting quantum mechanical size effect.
