Moore’s Law Linear Approximation and Mathematical Analysis!

Moore’s Law Linear Approximation and Mathematical Analysis!
by Vaibbhav Taraate on 11-05-2015 at 4:00 pm

 Respected Gordon Moore has given the real computing power to the world and Respected Stephen Hawking’s work from past many years has given the reality of physics and mathematics to the universe. We can imagine the shrinking and intelligence in computing due to the real evolution of semiconductor technology. The process node has shrunk from 250 nanometer (Year 1997) to 14 nanometer (Year 2014). And continues to shrink but have limitations due to fundamental laws of Physics.

With reference to my previous article about the ”Moore’s law Limitations and Gravitational Collapse at Lower Process Nodes” I am expressing the reality due to laws of Physics and the fundamental limits of shrinking using basic mathematics.

Life is cyclic in nature and history always repeats in some forms. The 50 years era (Vacuum tube during 1904 by John Fleming to the invention of first integrated circuit by Jack Kilby during year 1958) of evolution and beginning of miniaturization and can be treated as one cycle.

Another cyclic shift in the semiconductor business happened during 1963 with the invention of CMOS. And really during year 1971 to 2012 the computational power of dense integrated circuit has doubled in approximately 2 years. It has happened due to efforts of Respected Gordon Moore during year 1971. So another cyclic shift cycle was year 1970-1971 and we have witnessed doubling of the computing power of processor in almost every 24 months.

If we perceive the nature’s laws and the reality then for almost every 50 years there is cyclic shift in the technology and the real evolution happens. The cyclic shift in the technological changes by abiding the laws of nature and physics is always welcome. But most of the time, we observe that the technological shift and the evolution which is perceived way ahead of time. From year 2015 onwards we are at the verge of crossing the limitations of Physics and even we are witnessing the real evolution and technological shift, non-linear algorithmic developments and even we are witnessing the limitations for the era of miniaturization.

In such circumstances if we try to understand the history of evolution of ITRS main process nodes (half shrink) from year 1997 (Process node 250 nanometer) to year 2014 (Process node 14 nanometer) then the processing power has doubled in almost 24 months.

Now consider simple mathematical analysis:

P1: Computing power during year 2008 at process node 45 nanometer
P2: Computing power during year 2010 at process node 32 nanometer


Then by using linear approximation theory we can establish relation as following:

P2= P1* 2 to the power of (Difference between number of years of evolution of two consecutive main ITRS process nodes/ number of years required to double the transistors)

P2=P1*2^(Δy/N)

Where Δy= Difference between number of years for process node evolution of two consecutive main ITRS process nodes and N= Number of years required to double the transistors or computing power.

So according to the present data availability from year 1997 to 2014 the computing power has doubled from one ITRS process node to another in approximately 24 months.

Therefore p2 = 2 P1
And we will get the mathematical analysis as 2P1 = P1 * 2 ^ ((2010-2008)/ N)
Therefore 2 = 2 ^ (2/N)
Where, N is equal to number of years to double the transistors in dense integrated circuits.
Therefore 1= 2/N and N=2.

But as we move forward towards the lower process nodes (With reference to ITRS main nodes) then the same mathematical linear approximation gives the result as:

P1 = Computing power for the 14 nanometer (year 2014) process node
P2 = Computing power for 10 nanometer (expected during year 2017) process node

Therefore 2P1 = P1 * 2 ^ ((2017-2014)/ N)

So to achieve double the computing power or double the transistors in dense integrated circuit it takes the duration of 3 years that is 36 months.

The real magic and understanding is the Planck scale, Planck’s constant with reference to the length of transistor.

If L1 is length of transistor for the 14 nanometer process node during year 2014 and L2 is length of transistor for the process node 10 nanometer ITRS node during year 2017 then the relationship by using the velocity of light, Planck’s constant and the Compton Wavelength can be written as:

L1 = L2 * 2 ^ ((Δy/N))so λ = λc *2 ^ ((Δy/N))

Where the Compton wavelength is the characteristics dimension of electron according to the uncertainty principle of the Heisenberg . The Compton wavelength of electron is equal to
λc=h ÷(me*c)
Where λc=Compton Wavelength
h=Plank constant
me=mass of electron
c=speed of light
So Compton Wavelength is equal to λc=(2.426*10^(-12 ))

Where the quantum wavelength is the fundamental limit to measure the spin or position of particle and even it holds the prediction of Stephen Hawking that fundamental limit for shrinking is based on the speed of light.

Therefore the relationship for the quantum computing limit at lower process node can be expressed as:

(1/(2.426*10^(-12 ) ))= (1/(Present process node))*2 ^ ((Δy)/ N)


Where, N is number of years to double the computing performance of dense integrated circuits.
Now the quantum computing limit with reference to the N=3 is as follows.

Now consider the process node of 14 nanometer then the above equation results into Δy = 37.49 years from 2014.

But if try to compute the number of years using 10 nanometer process node then the same relation gives information as

(1/(2.426*10^(-12 ) ))= (1/( process node in nanometer))*2 ^ ((Δy)/ N)

So if we consider the process node of 10 nanometer then the above equation results into Δy = 36.07 years from year 2017.

And for the process node of 5 nanometer the above equation results into Δy= 33.03 years from 2021.

The maximum limit for the quantum computing will be reached during 2030 to 2035. The reason is following, the above equation according to the linear approximation and actual data of doubling of transistors from 1997 to 2010 is considered then for the year 1997 (ITRS Process node of 250 nanometer) the Δy = 33.33 Years and for year 2012 (ITRS Process node of 22 nanometer) the Δy = 26.30.

At the lower process node, semiconductor manufacturing has real challenges due to fundamental laws of Physics. Even it can lead to the gravitational collapse as we subsequently move towards lower process nodes.

According to my observation , the cyclic technological shift and evolution in the design and process flow has started from 2014 and it will continue till year 2025. Almost during 2025, we will be able to see the high complexity stabilized algorithms using quantum computing, chemical computing. So natural cyclic technological shift will witness the quantum, chemical computing and even evolution and use of various efficient materials in the design and manufacturing of Integrated circuits. But as semiconductor manufacturing is very capital intensive, we may see the economical impact and changes in the laws for the manufacturing at the lower process node.

All the above analysis indicates that, fundamental laws have very significant impact on the Moore’s law, Rock’s law and on the overall benchmarking, predictions of the high complex products. But let us hope for the real great era of miniaturization although there are limitations and challenges due to the fundamental laws of Physics!

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