Can we make electron drift velocity faster than light by reducing area of resistor?

Question

We know that $I= nqAV_d$.

Can we send high current ($I$) through a "fat wire" (more $A$) then reduce $A$ at the resistor so much that $V_d$ becomes faster than light in order to maintain $I$?

Answer 1

A metal would melt before the drift velocity reaches anywhere near the speed of light (besides all the other mechanisms preventing the drift velocity from getting that high).

In semiconductors, and likely in metals at very high current densities as well, the drift velocity eventually stops increasing linearly with the electric field. This is known as velocity saturation and is mainly due to carriers scattering by emitting optical phonons. The saturation velocity is on the order of $10^7\ \text{cm/s}$ for most semiconductors, about 3 orders of magnitude lower than the speed of light.

Answer 2

tldr: not by reducing area, but by slowing light, maybe.

If we restrict our discussion to the speed of light in a medium, it is possible to slow down the phase or group velocity of light while in a material.

However, I am not sure in practice whether you can reduce the speed of light in a semiconductor or dielectric below the drift velocity of electrons simultaneously moving through that material in a circuit.

Note that at fixed applied voltage, reducing cross-sectional area $A$ does not help as drift velocity is $v_d=\mu E$, where electron mobility $\mu$ depends on the material and electric field $E=\Delta V / \Delta x$ depends on the length $\Delta x$ of the element for fixed voltage $\Delta V$.

Rather, it is more of a materials and photonics problem whether you can pull this off. On the materials side, you simultaneously want high electron mobility $\mu$ and slow light (e.g. high dielectric constant = relative permittivity: remember $v=c/\sqrt{\epsilon_r\mu_r}$). It is a fun exercise to plug in numbers and see how far off you are for a specific material.

In my opinion, the best bet is to forget about the permittivity, optimize mobility and use clever photonics. In Hao et al. (2015) they slow the group velocity of (a narrow band of infrared) light in graphene by a factor of 163: equaling $1.839\cdot10^6\ \text{m/s}$.

If you can maintain freestanding graphene's amazing mobility of $200,000\ \text{ cm}^2/(\text{V}\cdot\text{s})=2\cdot10^3\ \text{m/s}\cdot\text{cm/V}$, you only need to apply $1\ \text{kV}$ across a centimeter of graphene to reach Hao et al.'s speed of slowed light in graphene.

Interestingly, in a chapter on breakdown of a graphene nanoribbon field-effect transistor by Amiri and Ghadiry (2017) they simulate electric fields above $1\ \text{MV/cm}$. (There are probably better, experimental references for this.)

So it seems to me if you create a photonic design that simply maintains high mobility, graphene might be able to handle the electric field necessary to go into an interesting $v_g < v_d\, (< c)$ regime, where $v_g$ is the group velocity of some bandwidth of light in graphene and $v_d$ is the drift velocity of electrons in graphene.

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Other thoughts:

In principle, if the phase velocity is reduced below the speed of electrons in the material, you should be able to get Cherenkov radiation, which is a cool effect. See this paper on Cherenkov radiation in integrated nanophotonic structures for a flavor of this: they model a single moving charge with thoughts on an application like a particle detector. (There are multiple papers on Cherenkov generation of phonons by increasing the drift velocity past the speed of sound in a material, but of course that is sound, not light. This is related to the velocity saturation in semiconductors that Puk mentioned.)