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How does doping go with the nanometer scale?
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Nominal Animal:

--- Quote from: VinzC on June 07, 2021, 10:58:11 am --- I'm still interested in knowing (the orders of magnitude) the doping concentrations in high scale integrated circuits, for my own curiosity but also because it's a ratio I tell my students. It is too unfortunate that the sources I viewed do not themselves source their numbers.
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That is, I believe, because this number (of dopant atoms per bulk atom) is so variable, and actual the dopant density (typically in atoms per cubic centimeter) is easier to refer to in practice, just like T3sl4co1l mentioned above.

If you look at the math (density of states, or even just number of charge carriers in different temperatures), depending on the bulk atoms and the acceptor/donor atoms, the change in properties varies wildly.  Instead of referring to the fraction of atoms, materials physicists at least are actually referring to how strongly the doping changes the bulk properties; and depending on the elements in question, that may change the "limit ratios" by several orders of magnitude.  The actual dopant density (number per cubic centimeter) is "just a technical detail".

If you look at e.g. the Wikipedia Doping article, it says "When on the order of one dopant atom is added per 100 million atoms, the doping is said to be low or light. When many more dopant atoms are added, on the order of one per ten thousand atoms, the doping is referred to as high or heavy."  However, it has no references to that claim; it just shows that we're talking about BIG differences, and even at very high or heavy doping, the dopants are still "rare".

The Wikipedia semiconductor article mentions that doping pure germanium with arsenic at 1:100,000 increases the electrical conductivity by a factor of 10,000.  I'd call that heavy doping, even though it is one tenth of what the other article calls "heavy doping".

If you were to ask for my opinion, I'd say 1:100,000 or more is heavy doping, and 1:10,000,000 or less is light doping, and amounts in between can be either depending on the atom elements involved, because doping is such a sensitive matter.  (Oh, darn, that's a physicist dad joke. :palm: Sorry.)


--- Quote from: VinzC on June 07, 2021, 10:58:11 am ---just not applicable to a planar model (is that the appropriate term?) at those ridiculously small scales but to a 3D model, which makes sense to me.
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Yes to both.

Put simply, in continuous volume, the effects of a single dopant reaches tens to hundreds of lattice cells in each direction (depending on the lattice atoms and dopant atoms). Both Si and Ge have FCC lattices, with lattice sizes (lattice constant, "edge length") 0.543nm and 0.566nm, respectively; so we're talking about nanometers to tens of nanometers in silicon and germanium semiconductors.  Any kind of barrier, be it discontinuity in the lattice structure (grain boundary in polycrystalline silicon, or other defect) will "block" it, however; and the exact electron densities in any particular structure needs quantum mechanics to be modelled properly (and such software exists, e.g VASP, Dalton, Siesta, Turbomole) because the results are not intuitive or similar to physical objects in macro scale; and those simulations are slow.

So, we can do very small features in 2D, if we keep the size in the third dimension large enough to have the necessary continuous volume.  FinFETs are, for just this reason, called non-planar or 3D transistor.

(A student might point out that wouldn't the fin in a FinFET act like it was less doped than the bulk it was etched on – because obviously the effect must wane as distances increase –, to which the answer is "yes, of course; but the range of semiconductor properties where such features work is wide enough".)

To get much smaller – that is, closer to the single-atom transistor limit, which we already know is at least theoretically possible with spintronics but basically merges classical logic gates and quantum computing –, we need to switch to using non-doped semiconductors for the small-volume details.  Because surface or grain boundary has completely different properties than bulk material, we can use polycrystalline material in similar ways to doping: the grain size dictates the properties.  Other possibilities currently being researched involves nanowires, nanotubes and layered structures; the layered structures being the cheapest, but nanowires and nanotubes being hyped the most.
VinzC:

--- Quote from: Nominal Animal on June 07, 2021, 05:09:24 pm ---[...] doping is such a sensitive matter.  (Oh, darn, that's a physicist dad joke. :palm: Sorry.)
--- End quote ---
:-DD :-+
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