Methods of forming a layer of silicon germanium include forming an epitaxial layer of Si1-xGex on a silicon substrate, wherein the epitaxial layer of Si1-xGex has a thickness that is less than a critical thickness, hc, at which threading dislocations form in Si1-xGex on silicon; etching the epitaxial layer of Si1-xGex to form Si1-xGex pillars that define a trench in the epitaxial layer of Si1-xGex, wherein the trench has a height and a width, wherein the trench has an aspect ratio of height to width of at least 1.5; and epitaxially growing a suspended layer of Si1-xGex from upper portions of the Si1-xGex pillars, wherein the suspended layer defines an air gap in the trench beneath the suspended layer of Si1-xGex.