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The electrical and structural properties of AlN/GaN heterostructures grown by molecular beam epitaxy on sapphire are compared with those of AlGaN/GaN heterostructures. The structural characteristics as assessed by x-ray diffraction show little difference but the electron density in the two-dimensional electron gas is about twice higher for AlN/GaN structures with only slightly lower mobility than in AlGaN/GaN. By proper choice of the Fe doping in GaN(Fe) and the thickness of unintentionally doped GaN layers, the composite buffer of the structure can be made semi-insulating. The current through the AlN/GaN structures is determined by tunneling through the AlN barrier and is much higher than that for AlGaN/GaN films due to the lower thickness of AlN compared to AlGaN. Increasing the thickness of AlN from 3 to 4 nm decreases the leakage current by about an order of magnitude.