Physicists love simplifying idealizations, and this is especially true in the physics of materials. The simplest decent model for metals, for example, is the ideal Fermi gas, where we neglect the existence of atoms entirely and just model the electrons as noninteracting particles in some box. One step up from there, the Sommerfeld model, assumes that the electrons are in a perfectly periodic crystal lattice. In both cases, the standard semiclassical approach treats the electrons as waves but basically ignores quantum interference.
Real conductors have defects that break the lattice periodicity, like vacancies, interstitials, impurities, grain boundaries, surfaces and interfaces, etc. It's natural to wonder, are there major consequences to this "disorder"? Common sense suggests that sufficiently minor or dilute disorder can't be too important. Sure, once you break the lattice symmetry, the electronic wavefunctions can't be exactly Bloch waves anymore, but if only one atom out of 10 billion is out of place, how big a deal can it be?
In the late 1970s, a number of theorists were thinking about this problem, and they came up with some impressive insights about the role of disorder, leading to the concept of localization. The key point to consider is whether the wavefunctions in the presence of disorder are delocalized (extending "to infinity", like plane waves or Bloch waves), or whether they are localized (decaying exponentially away from some origin region where their magnitude is large). This idea can apply to wavefunctions for electrons, but it can also apply to other kinds of waves, including electromagnetic waves in inhomogeneous dielectric media (think light bouncing around in a cloud).
A major result that came out of this thinking was the scaling theory of localization. That link points to some excellent lecture notes and a couple of youtube videos by Piet Brouwer for people interested in a more technical explanation. Intuitively, if the electronic states are exponentially localized, then making a block of material bigger should lead to the conductance of that material dropping exponentially. Alternately, if the electronic states are delocalized, making a hunk of material larger should generally increase its conductance. (Think about a piece of copper wire. Now double both the length and the diameter of the wire. The conductance \(= \sigma (\pi d^2)/(4L)\) has doubled.)
Let's call \(g(L) = G(L)/(e^2/h)\) the (dimensionless) conductance of some hunk of material of size \(L\). The question is, if you increase \(L\), what happens to \(g\)? There is a scaling function \(\beta(g) \equiv d \ln g/d \ln L\) that describes this. If \(\beta(g)\) is positive, then the system is metallic. If \(\beta(g)\) is negative, then the system is insulating in the large size limit, a situation called strong localization. The technical bit is figuring out what \(\beta(g)\) looks like. (This scaling idea had many contributors, including most famously people like Anderson and Thouless)
Remarkably, in this famous paper, the conclusion is that in 2D and 1D, any disorder at all makes \(\beta(g)\) negative. Thus the surprising conclusion is that, for this model (with no interactions), in principle there are no 2D or 1D metals. (The distance scale over which the conductance decays with increasing size is the "localization length", \(\xi\), and it could be very long. That's why seeing metal-like conduction in cm-scale gated graphene or 2D electron gas samples isn't surprising or necessarily inconsistent with this. There are many subtleties here.) In 3D, the situation depends on the actual magnitude of \(g\), where if \(g\) starts too small, the system runs away toward localization as system size is increased.
This idea, that interference of scattered waves from disorder can lead to exponentially confined waves, is called Anderson Localization. This is generic to waves in disordered media, as in this famous paper where it was demonstrated for light. By the way, you can think of localization of light as an effective cavity that confines the radiation via disorder scattering, an idea which in turn led to the random laser. Just earlier this year, people successfully demonstrated 3D Anderson localization of ultrasound.
I used google gemini to code up a toy model of Anderson localization (of light) in HTML5, where the disorder is in the form of a spatially varying index of refraction. (I used periodic boundary conditions.) If the disorder is weak (5 in toy units), all the energy dumped into the middle of the space spreads out roughly equally to fill the whole region. However, if the disorder is strong (50 in toy units), the energy of the waves is localized near the origin for long simulation times. Here is the model. (No deep claims of strict accuracy here; this was quick and dirty. To really see localization in this small play area, we'd need to \(\xi\) to be small compared to the size of the region because of the periodic boundary conditions.)
The ideas here have had a very long reach, and I'll likely write more about related physics soon.
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