Despite numerous experimental and theoretical studies on the highly miscible Sn=Cuð001Þ system, a conclusive detailed atomic picture of this rich alloy surface phase diagram is still warranted. Depending of the surface coverage of Sn, a rich variety of Sn=Cuð001Þ surface structures may be expected, ranging from the so-called phase I to phase IV. An extreme lack of atomic details about them hinders the overall comprehension of this alloy system. We focus on examining the surface energetics and thermodynamics of the Sn=Cuð001Þ structures to identify the low-energy structures for the experimentally observed phases I-IV. We also discuss the surface electronic structure of these low-energy Sn=Cuð001Þ structures in terms of their surface work functions and surface dipole moments. Finally, we compare the simulated scanningtunneling-microscopy (STM) images of these Sn=Cuð001Þ phases with available experimental STM measurements. We believe this work sets a good theoretical platform for an accurate further investigation of the Sn=Cu bimetallic surface-alloy system for surface-sensitive applications in, e.g., heterogeneous nanocatalysis.