  The (secondary) electron emission from multilayered Al$_2$O$_3$/TiN membranes
has been investigated with a hemispherical collector system in a scanning
electron microscope for electrons with energies between 0.3 and 10 keV. These
ultra-thin membranes are designed to function as transmission dynodes in novel
vacuum electron multipliers. Two different types, a bi-layer and a tri-layer,
have been manufactured by means of atomic-layer deposition (ALD) of aluminum
oxide and sputtering of titanium nitride. The reflection and transmission
electron yield ($\sigma_R$, $\sigma_T$) have been measured for both types of
membranes. In comparison, the tri-layer membranes outperformed the bi-layer
membranes in terms of transmission electron yield for films with the same
effective thickness. The highest transmission electron yield was measured on an
Al$_2$O$_3$/TiN/Al$_2$O$_3$ film with layer thicknesses of 5/2.5/5 nm, which
had a maximum transmission electron yield $\sigma_\text{T}^\text{max}(E_0)$ of
3.1 (1.55 keV). Furthermore, the bi-layer membranes have been investigated more
in-depth by performing an additional measurement using a positive sample bias
to separate the transmitted fraction $\eta_T$ and the transmission secondary
electron yield $\delta_T$. The transmitted fraction was used to determine the
transmission parameter $p$, which characterizes the interaction of primary
electrons (PEs) in thin films. The transmission secondary electron yield was
used to compare the energy transfer of PEs in films with different thicknesses.
