We present a detailed analysis of the band structure of the BiAg$_2$/Ag/Si(111) trilayer system by means of high resolution Angle Resolved Photoemission Spectroscopy (ARPES). BiAg2/Ag/Si(111) exhibits a complex spin polarized electronic structure due to giant spin-orbit interactions. We show that a complete set of constant energy ARPES maps, supplemented by a modified nearly free electron calculation, provides a unique insight into the structure of the spin polarized bands and spin gaps. We also show that the complex gap structure can be continuously tuned in energy by a controlled deposition of an alkali metal.
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