I taught we already had gallium-arsenide transistors. The problem is cost as it is reserved for application where power enveloppe is very thin (earing aids) and switching speed is critical (telecom equipment).
Another problem with GaAs and other III-V semiconductors is that they do not scale well, and so you can not pack as many transistors on a chip, and so they just can not compete with silicon in logic. They are quite useful for other applications, but not in your computer. Besides the low temperature hurdle, it's not clear if these new materials will face the same cost and scalability problems as III-Vs.
Maybe Computer Science should be in the College of Theology. -- R. S. Barton