Tannersfriend: And for the FET you will always need P and N channels, right?
Well, there are two types of FETs: the n-type and the p-type. For CMOS you need both, and this is what has been done since decades with silicon.
However, with gallium arsenide only n-types could be build, and that's why people couldn't create something like GaAs CMOS. Then Geoff Taylor stepped in.