The picture from Eli Yablonovitch described here is basically right as far as I can tell, and Jacob Cannell’s comment here seems to straightforwardly state why his method gets a different answer [edit: that is, it is unphysical]:
But in that sense I should reassert that my model applies most directly only to any device which conveys bits relayed through electrons exchanging orbitals, as that is the generalized electronic cellular automata model, and wires should not be able to beat that bound. But if there is some way to make the interaction distance much much larger—for example via electrons moving ballistically OOM greater than the ~1 nm atomic scale before interacting, then the model will break down.
[The rest of this comment has been edited for clarity; the comment by Steven Byrnes below is a reply to the original version that could be read as describing this as a quantitative problem with this model.] As bhauth points out in a reply, the atomic scale is a fraction of a nanometer and the mean free path in a normal metal is tens of nanometers. This is enough to tell us that in a metal, information is not “relayed through electrons exchanging orbitals”.
Valence electrons are not localized at the atomic scale in a conductor, which is part of why the free electron model is a good model while ignoring orbitals. The next step towards a quantum mechanical model (the nearly-free modification) considers the ionic lattice only in reciprocal space, since the electrons are delocalized across the entire metal. The de Broglie wavelength of an electron describes its wavefunction’s periodicity, not its extent. The mean free path is a semiclassical construct, and in any case does not provide a “cell” dimension across which information is exchanged.
The “tile”/cellular-automaton model comes from Cavin et al., “Science and Engineering Beyond Moore’s Law” (2012) and its references, particularly those by Cavin and Zhirnov, including Shankar et al. (2009) for a “detailed treatment”. As @spxtr says in a comment somewhere in the long thread, these papers are fine, but don’t mean what Jacob Cannell takes them to mean.
That detailed treatment does not describe energy demands of interconnects (the authors assume “no interconnections between devices” and say they plan to extend the model to include interconnect in the future). They propose the tiling framework for an end-of-scalingprocessor, in which the individual binary switches are as small and as densely packed as possible, such that both the switches and interconnects are tile-scale.
The argument they make in other references is that at this limit, the energy per tile is approximately the same for device and interconnect tiles. This is a simplifying assumption based on a separate calculation, which is based on the idea that the output of each switch fans out: the output bit needs to be copied to each of around 4 new inputs, requiring a minimum length of interconnect. They calculate how many electrons you need along the length of the fan-out interconnect to get >50% probability of finding an electron at each input. Then they calculate how much energy that requires, finding that it’s around the minimal switching energy times the number of interconnect tiles (e.g. Table 28.2 here).
For long/”communication” interconnects, they use the same “easy way” interconnect formula that Steven Byrnes uses above (next page after that table).
The confusion seems to be that Jacob Cannell interprets the energy per tile as a model of signal propagation, when it is a simplifying approximation that reproduces the results of a calculation in a model of signal fan-out in a maximally dense device.
I understand the second part of this comment to be saying that Jacob & I can reconcile based on the fact that the electron mean free path in metal wires is actually much larger than 1 nm. If that’s what you’re saying, then I disagree.
If the lowest possible interconnect loss is a small multiple of kT/(electron mean free path in the wire), then I claim it’s a coincidence. (I don’t think that premise is true anyway; I think they are off by like 4 OOM or something. I think there is like 6 OOM room for improvement in interconnect loss compared to Jacob’s model, so replacing 1 nm with copper mean free path = 40 nm in Jacob’s model is insufficient to get reconciliation.)
I think that, if there were two metal wires A & B, and wire A had 10× higher density of mobile electrons than B, each with 10× lower effective mass than B, but the electrons in A have 100× lower mean free path than B, then the resistivities of A & B would be the same, and in fact we would not be able to tell them apart at all, and in particular, their energy dissipation upon transmitting information would be the same.
One point of evidence, I claim, is that if I give you a metal wire, and don’t tell you what it’s made of, you will not be able to use normal electrical equipment to measure the electron mean free path for that wire. Whereas if the electron mean free path was intimately connected to electronic noise or binary data transmission or whatever, one might expect that such a measurement would be straightforward.
Oh, no. I just meant to highlight that it was a physically incorrect picture. Metallic conduction doesn’t remotely resemble the “electronic cellular automata” picture, any version of which would get the right answer only accidentally, I agree. A calculation based on information theory would only care about the length scale of signal attenuation.
Even for the purposes of the cellular model, the mean free path is about as unrelated to the positional extent of an electron wavefunction as is the de Broglie wavelength.
The picture from Eli Yablonovitch described here is basically right as far as I can tell, and Jacob Cannell’s comment here seems to straightforwardly state why his method gets a different answer [edit: that is, it is unphysical]:
[The rest of this comment has been edited for clarity; the comment by Steven Byrnes below is a reply to the original version that could be read as describing this as a quantitative problem with this model.] As bhauth points out in a reply, the atomic scale is a fraction of a nanometer and the mean free path in a normal metal is tens of nanometers. This is enough to tell us that in a metal, information is not “relayed through electrons exchanging orbitals”.
Valence electrons are not localized at the atomic scale in a conductor, which is part of why the free electron model is a good model while ignoring orbitals. The next step towards a quantum mechanical model (the nearly-free modification) considers the ionic lattice only in reciprocal space, since the electrons are delocalized across the entire metal. The de Broglie wavelength of an electron describes its wavefunction’s periodicity, not its extent. The mean free path is a semiclassical construct, and in any case does not provide a “cell” dimension across which information is exchanged.
The “tile”/cellular-automaton model comes from Cavin et al., “Science and Engineering Beyond Moore’s Law” (2012) and its references, particularly those by Cavin and Zhirnov, including Shankar et al. (2009) for a “detailed treatment”. As @spxtr says in a comment somewhere in the long thread, these papers are fine, but don’t mean what Jacob Cannell takes them to mean.
That detailed treatment does not describe energy demands of interconnects (the authors assume “no interconnections between devices” and say they plan to extend the model to include interconnect in the future). They propose the tiling framework for an end-of-scaling processor, in which the individual binary switches are as small and as densely packed as possible, such that both the switches and interconnects are tile-scale.
The argument they make in other references is that at this limit, the energy per tile is approximately the same for device and interconnect tiles. This is a simplifying assumption based on a separate calculation, which is based on the idea that the output of each switch fans out: the output bit needs to be copied to each of around 4 new inputs, requiring a minimum length of interconnect. They calculate how many electrons you need along the length of the fan-out interconnect to get >50% probability of finding an electron at each input. Then they calculate how much energy that requires, finding that it’s around the minimal switching energy times the number of interconnect tiles (e.g. Table 28.2 here).
For long/”communication” interconnects, they use the same “easy way” interconnect formula that Steven Byrnes uses above (next page after that table).
The confusion seems to be that Jacob Cannell interprets the energy per tile as a model of signal propagation, when it is a simplifying approximation that reproduces the results of a calculation in a model of signal fan-out in a maximally dense device.
I understand the second part of this comment to be saying that Jacob & I can reconcile based on the fact that the electron mean free path in metal wires is actually much larger than 1 nm. If that’s what you’re saying, then I disagree.
If the lowest possible interconnect loss is a small multiple of kT/(electron mean free path in the wire), then I claim it’s a coincidence. (I don’t think that premise is true anyway; I think they are off by like 4 OOM or something. I think there is like 6 OOM room for improvement in interconnect loss compared to Jacob’s model, so replacing 1 nm with copper mean free path = 40 nm in Jacob’s model is insufficient to get reconciliation.)
I think that, if there were two metal wires A & B, and wire A had 10× higher density of mobile electrons than B, each with 10× lower effective mass than B, but the electrons in A have 100× lower mean free path than B, then the resistivities of A & B would be the same, and in fact we would not be able to tell them apart at all, and in particular, their energy dissipation upon transmitting information would be the same.
One point of evidence, I claim, is that if I give you a metal wire, and don’t tell you what it’s made of, you will not be able to use normal electrical equipment to measure the electron mean free path for that wire. Whereas if the electron mean free path was intimately connected to electronic noise or binary data transmission or whatever, one might expect that such a measurement would be straightforward.
Oh, no. I just meant to highlight that it was a physically incorrect picture. Metallic conduction doesn’t remotely resemble the “electronic cellular automata” picture, any version of which would get the right answer only accidentally, I agree. A calculation based on information theory would only care about the length scale of signal attenuation.
Even for the purposes of the cellular model, the mean free path is about as unrelated to the positional extent of an electron wavefunction as is the de Broglie wavelength.