The growth and the morphology of ultrathin Fe₅₀Mn₅₀ alloy films, deposited by thermal co-evaporation of Fe and Mn on Cu(001) at room temperature are studied by scanning tunneling microscopy (STM). A high density of small islands of single and double atomic height is nucleated during the initial stages of growth of Fe₅₀Mn₅₀ on Cu(001). The film growth proceeds further in a nearly perfect layer-by-layer mode. Topographic STM images corresponding to Fe₅₀Mn₅₀ film thicknesses above 10 atomic monolayers reveal atomically flat terraces with islands or holes of monoatomic height ( ≈ 0.18 nm). In addition, a fine structure with an apparent corrugation height of about 0.05 nm is recognized. Scanning tunneling spectroscopy measurements prove the spectroscopic origin of this corrugation. It is attributed to local concentration differences of the constituents of the chemically disordered Fe₅₀Mn₅₀ alloy.