There are many researchers going on to overcome the limits of silicon and let the Moore's law live into the next decade (although somebody has pointed out that strictly speaking the Moore's law is no longer valid today). Graphene as I mentioned several times in these posts or other materials like molybdenum disulphide, are possible candidates.
Now I read a news from the University of Cambridge that a joint team of researchers from Cambridge, the University of Singapore and the A*STAR Data Storage Institute in Singapore have demonstrated that it is possibile to use certain materials, like chalcogenide glass, that have the property of changing phase under a tiny spark of voltage can be used to perform at the same time processing and storage, something that in today's chip takes place in two separate structures.
Phase changing materials (PCM) are not new, they were first studied in 1960ies. Water is a phase changing material that upon freezing re-arrange its atomic structure, and have been studied for quite a while. When a phase change occurs their atomic structure changes and this change can be detected.
The change, in some materials, can be extremely rapid, in the order of half a nanosecond, and that by itself would support high switching speed that in turns translate into high processing speed if such a transition is used as a result of a logic gate.
Additionally, since the operation performed would be at the same time a processing and storage operation (whilst in today's chip that would imply a sequence of reading-transporting- processing- transporting- storing) the actual computational speed can be 500 to 1,000 times faster than in today's chips.
Add to this the fact the creating a phase transition requires much less energy than operating a silicon logic gate and you get an amazing new chip
So far researchers have been able to get speed in the order of 10 ns, something that would make PCM chips competitive with flash memory but not good to replace DRAM and microprocessors. Additionally, there is the huge gap separating lab experiments from industrial production processes so it is premature to say if these approaches will eventually replace silicon chips. Still it is nice to see that researchers are looking into radically new computational paradigms and it is interesting to notice that these look like the ones that Nature has selected to operate our brains...