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Energy-dependent apparent step heights of two-dimensional ultra-thin Pbislands grown on the Si(111)6$\times$6Au surface have been investigated by acombination of scanning tunneling microscopy, first-principles densityfunctional theory and the particle in a box model calculations. The apparentstep height shows the thickness and energy dependent oscillatory behavior,which is directly related to the spilling of electron states into the vacuumexhibiting a quantum size effect. This has been unambiguously proven byextensive first-principles scanning tunneling microscopy and spectroscopysimulations. An electronic contribution to the apparent step height is directlydetermined. At certain energies it reaches values as high as a half of theatomic contribution. The applicability of the particle in a box model to thespilling of electron states is also discussed.

Spilling of electronic states in Pb Quantum Wells