  We have synthesized Fe$_{1+y}$Te thin films by means of molecular beam
epitaxy (MBE) under Te-limited growth conditions. We found that epitaxial
layer-by-layer growth is possible for a wide range of excess Fe values, wider
than expected from what is known on the bulk material. Using x-ray magnetic
circular dichroism spectroscopy at the Fe L$_{2,3}$ and Te M$_{4,5}$ edges, we
observed that films with high excess Fe contain ferromagnetic clusters while
films with lower excess Fe remain nonmagnetic. Moreover, x-ray absorption
spectroscopy showed that it is possible to obtain films with very similar
electronic structure as that of a high quality bulk single crystal
Fe$_{1.14}$Te. Our results suggest that MBE with Te-limited growth may provide
an opportunity to synthesize FeTe films with smaller amounts of excess Fe as to
come closer to a possible superconducting phase.
