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From what I have read, the problem with modern semiconductors/electronics seems to be quantum tunnelling and heat. The root of these problems is the size of the devices. The electrons are leaking out, and currents are causing active materials to melt.

How far have we become in this regard? Can we make our devices even smaller? What is being done to maintain advancements in computing power? What is the main research, particularly in quantum mechanics and in solid state physics, being done to compute faster using less energy and space?

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The main solution is to give up Silicon altogether since it is very inefficient, compared to some of the compound semiconductors(such as GaAs). However, GaAs will eventually reach the same limitations. In any case, you can't really make a transistor from half an atom, therefore there are some physical limits as to how small you can make a transistor and we're very close to that. Therefore, current focus is on cheap and widespread use of compound semiconductors.

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Well, seems that Intel can often find some material to get closer to the physical limit. But the limit can't be reached, your transistor needs at least 1 atom.

Another limit is on the clock frequency, which is essentially due to material's intrinsic property (mobility, or speed of electrons). Graphene may have a good chance for its ultrahigh mobility.

One practical solution should be parallel computing, as the CPU of our PCs has more and more physical cores. GPU computing is another way out.

As for quantum computers, it's very hard to say as there are plenty of theoretical and technical obstacles.

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