We have fabricated high quality In0.5Al0.5As/In0.5Ga0.5As high electron mobility transistors (HEMTs) with thin buffer layers lattice-mismatched on GaAs substrates by molecular beam epitaxy (MBE). Our step-graded InyAl1-yAs buffer layers efficiently terminate lattice-misfit dislocations at each step interface according to cross-sectional transmission electron microscopy (TEM) and plan-view TEM observations, The buffer layers facilitate high mobility (approximate values of 10,000 cm(2)/V . s at room temperatures). even though the total buffer thickness is smaller than 600 nm. The mobility is comparable to that for HEMTs grown on InP substrates and better than that reported for HEMTs grown on GaAs with much thicker buffers. For a device with a 0.5 mu m long, 20 mu m wide gate and a 600 nm buffer layer, the peak extrinsic G(m) is 750 mS/mm, which is higher than that for previously reported HEMTs with the same gate length, including devices on InP substrates. In0.5Al0.5As/In0.5Ga0.5As single quantum wells (SQWs) grown on the step-graded InyAl1-yAs buffer layers show intense, sharp photoluminescence spectra, comparable to those of SQWs grown lattice-matched on InP. These results show that the tin step-graded buffer enabled growth of high-quality In0.5Al0.5As/In0.5Ga0.5As heterostructures on GaAs substrates.