According to the USGS, gallium arsenide use is higher than ever, but it remains niche, because it's just a minor tweak to the silicon bandgap structure, especially for bipolar circuits, where people have been fabricating high-performance CPUs in it since the 1980s. It doesn't work even adequately for CMOS, but even if it did, it doesn't solve the problems with Dennard scaling.

If you're looking for exotic materials to bail us out, it might be a little less hopeless to look to diamond or to high-temperature semiconductors.