Silicon "reached its limit" a long time ago if you consider the speed of individual transistors. This is why III-V semiconductors like GaAs and GaN and not silicon are used in microwave circuits.
However, if you are talking about the speed of computers the answer is more complicated. The reason is the speed of a computer depends on more than just the speed at which you can clock your processor. An obvious example would be that modern computers tend to contain several CPUs ("cores") which speeds of the execution of programs that are written to take advantage of this. Hence, it is possible to make faster computers without using faster transistors.
The reason why this is relevant is that "which technology will replace silicon" depends on more than just how quickly you can turn on and off a transistor: you also have to be able to scale up the technology in such a way that you can reliably fabricate at least a few tens of millions of transistors on a wafer (for a CPU much more) AND and the process needs to be cost-efficient.
The problem is that whereas there are many "fast" technologies are around nothing can compete with silicon when it comes to scalability and cost. Hence, silicon will be around for a VERY long time; certainly longer than 2020.
That said, some "novel" technologies can potentially co-exist with siliicon. One example would be optical busses for inter-chip interconnects. It is also possible that some new materials will be introduced in the silicon process for making vias etc (maybe even something as "exotic" as carbon nanotubes).
There are some indications that the "limit" will not actually be the material but the cost of fabrication. A good example would be that the price of the lithographer (the machine that that "draws" the circuit on the chip) that will be needed to make CPUs in a couple of CPU generations is about the same as the price of a whole factory just a decade ago (they already exist a prototypes, as far as I understand there are two in the whole world, Intel has one); and in a typical factory you need a few of them.
There are probably only 3-4 companies in the world that can afford to build the required factories and they will only invest the money if they think they can sell a LOT of CPUs at a reasonable price. Hence, once we start to see major deviations from Moore's law it is likely to be due to economical and not technological factors.