So the cool thing is the Landauer/Lego model is very general and very simple. I wanted a model that made reasonably accurate predictions, but was extremely simple. I believe I succeeded. More complex electrical and wire geometry equations do not in fact make more accurate predictions for my target variables of interest, and are vastly more complex. The number of successful predictions this model makes more or less proves its correctness, in a bayesian sense.
So for 1V wires, capacitive energy is about 7*10^-11 J/m per discharge (70 fJ/mm, a close match to the number you cite!).
Yep! and it’s even more accurate if you use the correct De Broglie electron wavelength at 1V, which is 1.23nm instead of 1nm, which then gives 81 fJ/mm. I bet there are a few other adjustments like that, but it’s already pretty close.
But look at the scaling—it’s V^2! Not controlled by Landauer.
Not really, as you point out the E = (1/2)QV = (1/2)CV^2, and Q=CV. But notice there is a minimum value of the charge Q = 1 electron charge, and a minimal value constraint on the energy per wire segment, E > Emin, thus V is constrained as well—it can not scale arbitrarily. You can use more electrons to represent the signal of course (larger wire) and lower voltage at the same Emin per segment, but there is a room temp background Landauer noise voltage around 17 mV, need a non trivial multiple of that.
The macro wire formulas are just approximations btw, for minimal nano-scale systems we are talking about single or few electrons ( I believe the spherical cow model breaks down for nano-scale wires )
The minimal element model comes from Cavin/Zhirnovdirect et al ( starting page 1725, 5th or 6th ref to ‘interconnect’, the tile model), I ref it a few times in the article. They explicitly use it for calculating minimal transistor switch energies that include minimal wires, estimate wire distances, etc, and use it to forecast end of Moore’s Law.
Communication at the nanoscale is still just a form of computation (a 1:1 function, but still erases unwanted wire states), and if it’s irreversible the Landauer Limit very much applies.
Ah, good point. But I’m still not sure the model works, because we can distribute the charge (or more generally the degrees of freedom) over the length of the wire.
Like, if the wire is only 10 nm long, adding one electron causes a way bigger voltage jump than if the wire is 500 nm long. We don’t have to add one electron per segment of wire.
I think you are correct in that you don’t actually have to have 1 electron per electron-radius (~nm) of wire—you could have a segment that is longer, but I think if you work that out it requires larger voltages to actually function correctly in terms of reliable transmission. This is all assuming we are using electron waves to transmit information, rather than ballistic electrons (but the Landauer limit will still bound the latter, just in a different way).
If you look at the spherical cow (concentric cylinder wire model), for smallish wires it reduces effectively to a constant that relates distance to capacitance, with units farads/meter.
The Landauer/Tile model predicts in advance a natural value of this parameter will be 1 electron charge per 1 volt per 1 electron radius, ie 1.602 e-19 F / 1.23 nm, or 1.3026 e-10 F/m.
So, capacitance of wires! Capacitor energy is QV/2, or CV^2/2. Let’s make a spherical cow assumption that all wires in a chip are half as capacitive as ideal coax cables, and the dielectric is the same thickness as the wires. Then the capacitance is about 1.3*10^-10 Farads/m (note: this drops as you make chips bigger, but only logarithmically).
The probability that the Landauer/Tile model predicts the same capacitance per unit distance while not also somehow representing the same fundamental truth of nature, is essentially epsilon. Somehow the spherical wire capacitance model and the spherical tile electron radius Landauer/Tile model are the same.
So the cool thing is the Landauer/Lego model is very general and very simple. I wanted a model that made reasonably accurate predictions, but was extremely simple. I believe I succeeded. More complex electrical and wire geometry equations do not in fact make more accurate predictions for my target variables of interest, and are vastly more complex. The number of successful predictions this model makes more or less proves its correctness, in a bayesian sense.
Yep! and it’s even more accurate if you use the correct De Broglie electron wavelength at 1V, which is 1.23nm instead of 1nm, which then gives 81 fJ/mm. I bet there are a few other adjustments like that, but it’s already pretty close.
Not really, as you point out the E = (1/2)QV = (1/2)CV^2, and Q=CV. But notice there is a minimum value of the charge Q = 1 electron charge, and a minimal value constraint on the energy per wire segment, E > Emin, thus V is constrained as well—it can not scale arbitrarily. You can use more electrons to represent the signal of course (larger wire) and lower voltage at the same Emin per segment, but there is a room temp background Landauer noise voltage around 17 mV, need a non trivial multiple of that.
The macro wire formulas are just approximations btw, for minimal nano-scale systems we are talking about single or few electrons ( I believe the spherical cow model breaks down for nano-scale wires )
The minimal element model comes from Cavin/Zhirnov direct et al ( starting page 1725, 5th or 6th ref to ‘interconnect’, the tile model), I ref it a few times in the article. They explicitly use it for calculating minimal transistor switch energies that include minimal wires, estimate wire distances, etc, and use it to forecast end of Moore’s Law.
Communication at the nanoscale is still just a form of computation (a 1:1 function, but still erases unwanted wire states), and if it’s irreversible the Landauer Limit very much applies.
Ah, good point. But I’m still not sure the model works, because we can distribute the charge (or more generally the degrees of freedom) over the length of the wire.
Like, if the wire is only 10 nm long, adding one electron causes a way bigger voltage jump than if the wire is 500 nm long. We don’t have to add one electron per segment of wire.
I think you are correct in that you don’t actually have to have 1 electron per electron-radius (~nm) of wire—you could have a segment that is longer, but I think if you work that out it requires larger voltages to actually function correctly in terms of reliable transmission. This is all assuming we are using electron waves to transmit information, rather than ballistic electrons (but the Landauer limit will still bound the latter, just in a different way).
If you look at the spherical cow (concentric cylinder wire model), for smallish wires it reduces effectively to a constant that relates distance to capacitance, with units farads/meter.
The Landauer/Tile model predicts in advance a natural value of this parameter will be 1 electron charge per 1 volt per 1 electron radius, ie 1.602 e-19 F / 1.23 nm, or 1.3026 e-10 F/m.
The probability that the Landauer/Tile model predicts the same capacitance per unit distance while not also somehow representing the same fundamental truth of nature, is essentially epsilon. Somehow the spherical wire capacitance model and the spherical tile electron radius Landauer/Tile model are the same.