We have fabricated a single quantum dot in the two-dimensional electron gas of a free-standing GaAs/AlGaAs heterostructure. Standard processing techniques are used to define the dot on a narrow mesa and selective etching undercuts the device to form a 120 nm thick free-standing beam connected to the substrate by source and drain regions. Work by Weig et al. [Phys. Rev. Lett. 92(4) (2004) 046804] on similar devices has provided evidence for phonon blockade which suppresses conduction at low temperatures and bias. Further studies are presented here that can only partially be explained by this theory. A similar gap in conductance has been seen, but evolution of the structure in a magnetic field suggests an alternative explanation of multiple dots in series, may better explain the results. (