For the prototypical Ge/Si(001) system, we show that at high growth temperature a new type of Stranski-Krastanow islands is formed with side facets steeper than {111} and high aspect ratio. Nano-goniometric analysis of the island shapes reveals the presence of six new facet groups in addition to those previously found for dome or barn-shaped islands. Due to the highly multi-faceted island shape and high aspect ratio, the new island types are named "cupola" islands and their steepest {12 5 3} side facet is inclined by 68°to the substrate surface.
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