We take the constant rate of increase in computing power for granted. One of the founders of Intel, Moore, estimated once the rate of the number of transistors $$n$$ in an integrated chip to be exponential, doubling every year. The smaller size of these transistors allows manufactures to pack more and more in each chip every year, making processors more powerful, memory capacities larger, and overall, computers faster.

Moore’s law can be stated as: $$n(t)=2^{frac{t-t_0}{2}}$$ where $$n(t)$$ is the number of transistors on a chip at some time in the future $$t$$, $$n(t_0)$$ is the known number of transistors for a time $$t_0$$.

This exponential trend cannot be sustained forever, at some point the transistor will be so small that quantum effects will dominate and the laws of physics that make the transistor work will change dramatically, effectively making them stop working. J. Powel looked at this in The Quantum Limit to Moore’s Law (subscrition required). Here, I reproduce a similar calculation with compatible results.

**The question I am interested in is: given all things equal, what is a good estimate for the year that we will reach the quantum regime where transistors won’t be able to get any smaller? **

First, we need one data point of the number of transistors in an integrated circuit at a certain year. For this, I used Intel’s own data that says that in 2007 they could manufacture transistor of lenght $$45nm=45times10^{-9}m$$. By assuming that the number of transistors is inversely proportional to the size of the transistor, we can estimate $$n(t_0)$$.

Now, we need to estimate the number of transistors in an integrated circuit just before they transistor reaches the quantum limit. For this, I chose the transistor to be of the order of the Compton wavelength of the electron, or 10^{-12} m . The Compton wavelength $$lambda=frac{h}{m_e c}$$ is the dimension of an electron from Heisenberg’s uncertainty principle. If you are in this regime, the electron behaves mostly as a wave, not as a particle and the electronic properties of the transistor will change, making it unusable. It is a fundamental physical limitation of the size of a transistor.

With this numbers at hand, I solved for $$t$$, and found out the year where Moore’s law will break:

Year $$2038$$ is when Moore’s law will not hold anymore.

The purpose of this excersize is NOT to predict the end of the computer growth as we know it, as this is bound to fail, but to stress the need for understanding electron transport in the quantum regime. The quantum regime is something we will reach in my lifetime, and a new theory of electronics in the fully quantum regime is needed before we get there.

But, also, this number misteriously agrees with XKCD’s end-of-the-world prediction in a manner different from what XKCD intended.

Creepy.

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mobilis in mobili