This invention minimizes the thermal resistance and maximizes the power density of a power transistor by mounting the transistor in flip-chip fashion on a heat sink/heat spreader and conducting the heat from the active semiconductor layer through the heat sink/heat spreader (as opposed to through the low conductivity substrate). Illustratively, the semiconductor device package comprises: a high electron mobility transistor (HEMT) formed in a layer of Gallium Nitride (GaN) having a first major surface; at least one metal contact pad making thermal contact with the layer of GaN on its first major surface; a heat sink/heat spreader in electrical and thermal contact with the contact pad(s) on the first surface; and a substrate on which the heat sink is mounted.