In a recent article in Nature, you can find the details of a RISC-V CPU built using carbon nanotubes. Of course, Nature is a pricey proposition, but you can probably find the paper by its DOI number if you bother to look for it. The researchers point out that silicon transistors are rapidly reaching a point of diminishing returns. However, Carbon Nanotube Field Effect Transistors (CNFETs) overcome many of these disadvantages.
The disadvantage is that the fabrication of CNFETs has been somewhat elusive. The tubes tend to clump and yields are low. The paper describes a method that allowed the fabrication of a CPU with over 14,000 transistors. A wafer gets nanotubes grown all over it and then some of them are removed. In addition, some design rules mitigate other problems.
In particular, a small percentage of the CNFETs will become metallic and have little to no bandgap. However, the DREAM design rules can increase the tolerance of the design to metallic CNFETs with no process changes.
Before you get too excited, limitations in channel length and contact size keep the processor running at a blazing 10 kHz. To paraphrase Weird Al, your operating system boots in a day and a half. The density isn’t great either since working around stray and metallic CNFETs means each transistor has multiple nanotubes in use.
On the other hand, it works. New technology doesn’t always match old technology at first, but you have to crawl before you walk, and walk before you run.