A microprocessor is possibly the most complex machines ever built by humankind. (A supercomputer could be viewed in terms of the number of microprocessor cores used to create it).

In this post, we work out a ball-park estimate of the complexity of a microprocessor (as an example of one of the most complex machines built by humankind).

To answer this question…

A Pentium II microprocessor chip contains 7,500,000 transistors.

A Pentium IV contains 42,000,000 transistors.

See below for an evaluation of the amount of information (in bits) needed to specify or create one of these microprocessors.

Note: we consider only structure complexity, and not process complexity, in the discussion below.

**Elements of a transistor**: Substrate, gate dielectric, gate electrode, isolation structure, source, drain, contacts, interlayer dielectric, interconnect metal.

We next consider the sub-elements that constitute each of these elements mentioned above.

**Substrate**: silicon, height, width, breadth, surface clean, silicon orientation, doping level, anneal temperature, anneal time = 9 parameters.

**Gate dielectric**: oxide, height, width, breadth, growth with or without water, growth temperature, growth time (both for quality and height of oxide), x-position, y-position, z-position (wrt silicon substrate). = 10 parameters.

Gate electrode: poly-Si, height, width, breadth, doping level, anneal time, anneal temperature, x-position, y-position, z-position (wrt silicon substrate) = 10 parameters.

Gate contact metal: aluminum, height, width, breadth, x-position, y-position, z-position (wrt silicon substrate) = 7 parameters.

Gate sidewall spacer: nitride, height, width, breadth, x-position, y-position, z-position (wrt silicon substrate) = 7 parameters.

**Isolation structure**: field oxide, height, width, breadth, x-position, y-position, z-position (wrt silicon substrate), growth with or without water, growth temperature, growth time (both for quality and height of oxide) = 10 parameters.

**Source**: implant or spinon source, implant energy and dose, (leads to doping level peak, doping level depth), diffusion temperature, diffusion time = 6 parameters.

Drain: (duplicate of Soure, but on other side of gate): implant or spinon source, implant energy and dose, (leads to doping level peak, doping level depth), diffusion temperature, diffusion time = 6 parameters.

Source-contact barrier: silicide or Ti/TiN, height, width, breadth, x-position, y-position, z-position (wrt silicon substrate), temperature of anneal, time of anneal. Excess unreacted material removal = 10 parameters.

Source-contact metal: aluminum, height, width, breadth, x-position, y-position, z-position (wrt silicon substrate). = 7 parameters.

Drain-contact barrier: silicide or Ti/TiN, height, width, breadth, x-position, y-position, z-position (wrt silicon substrate), temperature of anneal, time of anneal. Excess unreacted material removal = 10 parameters.

Drain-contact metal: aluminum, height, width, breadth, x-position, y-position, z-position (wrt silicon substrate). = 7 parameters.

**Interlayer Dielectric**: oxide, height, width, breadth, x-position, y-position, z-position (wrt silicon substrate), growth with or without water, growth temperature, growth time (both for quality and height of oxide). = 9 parameters.

**Interconnect metal**:
Source interconnect: aluminum, height, width, breadth, x-position, y-position, z-position (wrt silicon substrate). = 7 parameters.

Drain interconnect: aluminum, height, width, breadth, x-position, y-position, z-position (wrt silicon substrate). = 7 parameters.

Gate interconnect: aluminum, height, width, breadth, x-position, y-position, z-position (wrt silicon substrate). = 7 parameters.

Approximation – let’s assume that each of the knobs above can take one of 128 values.

How many bits does this correspond to ? One bit = 2 states or values; two bits = 4 states; three bits = 8 states; four bits = 16 states; five bits = 32 states; six bits = 64 states; seven bits = 128 states. So the parameter-space of the knobs above, corresponds to 7 bits per parameter.

Total number of parameters from list above = 16 elements with a total of 129 parameters.

Total number of bits needed = 129 parameters x 7 bits per parameter = 903 bits.

In addition to transistors, in a microprocessor, we have resistors and capacitors and interconnect metallization lines.

**Elements of a Resistor**: Body, contact barrier for contact at positive end, contact metal for contact at positive end, contact barrier for contact at negative end, contact metal for contact at negative end, interconnect metal, interlayer dielectric.

We next consider the sub-elements that constitute each of these elements mentioned above.

**Body**: Metal or poly-si, doping, length, width, breadth, x-position, y-position, z-position = 8 parameters.

Contact at positive end: contact barrier: silicide or Ti/TiN, height, width, breadth, x-position, y-position, z-position (wrt silicon substrate), temperature of anneal, time of anneal. Excess unreacted material removal = 10 parameters.

Contact at positive end: contact metal: aluminum, height, width, breadth, x-position, y-position, z-position (wrt silicon substrate). = 7 parameters.

Contact at negative end: contact barrier: silicide or Ti/TiN, height, width, breadth, x-position, y-position, z-position (wrt silicon substrate), temperature of anneal, time of anneal. Excess unreacted material removal = 10 parameters.

Contact at negative end: contact metal: aluminum, height, width, breadth, x-position, y-position, z-position (wrt silicon substrate). = 7 parameters.

**Interconnect metal**:

Positive-contact interconnect: aluminum, height, width, breadth, x-position, y-position, z-position (wrt silicon substrate). = 7 parameters.

Negative-contact interconnect: aluminum, height, width, breadth, x-position, y-position, z-position (wrt silicon substrate). = 7 parameters.

Approximation – let’s assume that each of the knobs above can take one of 128 values = how many bits? 2,4,8,16,32,64,128; = 7 bits.

Total number of parameters from list above = 7 elements with a total of 56 parameters.

Total number of bits needed = 56 parameters x 7 bits per parameter = 392 bits.

**Elements of a Capacitor**:
Top-electrode, bottom-electrode, dielectric, contact barrier to top electrode, contact metal to top electrode, contact barrier to bottom electrode, contact metal to bottom electrode, interconnect metal to top electrode, interconnect metal to bottom electrode.

We next consider the sub-elements that constitute each of these elements mentioned above.

Bottom-electrode: Metal or poly-si, doping, length, width, breadth, x-position, y-position, z-position = 9 parameters.

Dielectric: dielectric material, length, width, breadth, x-position, y-position, z-position = 7 parameters.

Contact to top electrode: contact barrier: silicide or Ti/TiN, height, width, breadth, x-position, y-position, z-position (wrt silicon substrate), temperature of anneal, time of anneal. Excess unreacted material removal = 10 parameters.

Contact to top electrode: contact metal: aluminum, height, width, breadth, x-position, y-position, z-position (wrt silicon substrate). = 7 parameters.

Contact to bottom electrode: contact barrier: silicide or Ti/TiN, height, width, breadth, x-position, y-position, z-position (wrt silicon substrate), temperature of anneal, time of anneal. Excess unreacted material removal = 10 parameters.

Contact to bottom electrode: contact metal: aluminum, height, width, breadth, x-position, y-position, z-position (wrt silicon substrate) = 7 parameters.

**Interconnect metal**:

Top electrode -contact interconnect: aluminum, height, width, breadth, x-position, y-position, z-position (wrt silicon substrate). = 7 parameters.

Bottom electrode -contact interconnect: aluminum, height, width, breadth, x-position, y-position, z-position (wrt silicon substrate). = 7 parameters.

Approximation – let’s assume that each of the knobs above can take one of 128 values = how many bits? Counting bits -- 2,4,8,16,32,64,128; = 7 bits.

Total number of parameters from list above = 9 elements with a total of 73 parameters.

Total number of bits needed = 73 parameters x 7 bits per parameter = 511 bits.

The available literature indicates that the Pentium IV microprocessor contains 42 million transistors.

For the purposes of this calculation, let us assume an average of 1 resistor and 1 capacitor for every 10 transistors.

This gives us a total of 42 million transistors, 4 million resistors and 4 million capacitors.

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**Complexity of one transistor** = 903 bits.

42 million = 25.3 bits = ~25 bits (to capture the 42 million transistors).

So, complexity of 42 million transistors = 903 + 25 = 928 bits.

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**Complexity of one resistor** = 392 bits.

4 million = 21.9 bits = ~22 bits (to capture the 4 million resistors).

So, complexity of 4 million resistors = 392 + 22 = 414 bits.

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**Complexity of one capacitor **= 511 bits.

4 million = 21.9 bits = ~22 bits (to capture the 4 million resistors).

So, complexity of 4 million capacitors = 511 + 22 = 533 bits.

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So, not counting the interconect complexity, the total complexity of the Pentium IV microprocessor is = complexity of the transistors + complexity of the resistors + complexity of the capacitors.

**Total complexity** = 928 + 414 + 533 bits = 1875 bits.

In summary, we evaluated the structure-complexity of a Pentium IV microprocessor and came up with an estimate of 1875 bits (of information) needed to specify the microprocessor.

Since we did not consider process complexity and interconnect complexity, the number above is a lower-bound.