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).
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