  Growth, electronic and magnetic properties of $\gamma'$-Fe$_{4}$N atomic
layers on Cu(001) are studied by scanning tunneling microscopy/spectroscopy and
x-ray absorption spectroscopy/magnetic circular dichroism. A continuous film of
ordered trilayer $\gamma'$-Fe$_{4}$N is obtained by Fe deposition under N$_{2}$
atmosphere onto monolayer Fe$_{2}$N/Cu(001), while the repetition of a
bombardment with 0.5 keV N$^{+}$ ions during growth cycles results in imperfect
bilayer $\gamma'$-Fe$_{4}$N. The increase in the sample thickness causes the
change of the surface electronic structure, as well as the enhancement in the
spin magnetic moment of Fe atoms reaching $\sim$ 1.4 $\mu_{\mathrm B}$/atom in
the trilayer sample. The observed thickness-dependent properties of the system
are well interpreted by layer-resolved density of states calculated using first
principles, which demonstrates the strongly layer-dependent electronic states
within each surface, subsurface, and interfacial plane of the
$\gamma'$-Fe$_{4}$N atomic layers on Cu(001).
