Anderson Localization and general disorder of matter?

[asimov42]

Hi all,

Just did a little bit of (layperson) reading about Anderson localization and disorder in crystal structures. Here's my question: shouldn't all matter contribute to 'restricting' the wave function of e.g., an electron, whether it's in a crystal structure or not? That is, what's specific to a finite disordered lattice and Anderson localization?

That is, shouldn't things like an electron moving through water molecules (highly disordered) force localization of the electron wave function in a similar fashion?

As an extreme example, if you consider e.g., that single electron, and all the matter in, say, the observable universe, shouldn't the interference of all of the wave functions together effectively force the localization of the electron to some very small region? (since the matter distribution has reasonably large entropy) Or is it that the interference effectively 'averages' out in some way?

Thanks - sorry if the questions above are not very 'crisp'.

 

[DrDu]

Hi all,

Just did a little bit of (layperson) reading about Anderson localization and disorder in crystal structures. Here's my question: shouldn't all matter contribute to 'restricting' the wave function of e.g., an electron, whether it's in a crystal structure or not? That is, what's specific to a finite disordered lattice and Anderson localization?

Thanks - sorry if the questions above are not very 'crisp'.

Sorry if I can't give you a very crisp answer, too. The problem is a very complicated one, as there are so many different disordered systems. Andersons model is maybe one of the simplest conceivable ones, thence its importance. The Hamiltonian contains basically only two ingredients: The kinetic energy of the electrons (in the disguise of the hopping parameter t) and the disorder potential. Using a finite lattice gives you the possibility to choose the disorder potential as statistically independent at different lattice points. In a continuous model, you would have to specify some spatial correlation function leaving much more freedom.
Although the model is relatively simple, it is already hard to derive properties of it's solutions. But these properties are already quite interesting, e.g. that whether the solutions are localized or not depends on the strength of the perturbing potential and on the dimensionality of the lattice.
Finally, it was one of the first publications studying disorder in quantum systems, hence it received a lot of attention.

 

[asimov42]

Thanks DrDu, that's helpful!

I guess one of the things I'm wondering (alluded to above) is why a lattice structure like a disordered crystal is needed? Isn't it possible to, in some sense, consider a large volume of space with lots of matter as effectively being a lattice? And then shouldn't the results carry over?

 

[DrDu]

I can only provide you with a term to look for: "Random Schrödinger Operators"

 

[asimov42]

Ah, ok, I think I'm getting it a bit now. If I understand (from what I've read), the localization effect can occur even if the particle energy is higher than all of the potential barriers. However, I'm assuming there's an energy above which the effect would break down? (at some point, the electron must diffuse...)

 

[asimov42]

Hi all - I really appreciate the input from DrDu. Here's (hopefully) the last question: for effects like Anderson localization, why draw the line at, say, a finite disordered e.g. semiconductor lattice in a laboratory... this is basically 're-asking' the first question I had above. Why is it that larger scale disorder of matter (beyond some finite lattice) doesn't contribute to to the localization effect? Or is there a reason to `down weight' those effects?

 

[DrDu]

I wouldn't say that larger scale disorder does not contribute. The point is that this isn't even controversal. The interesting point in Andersons analysis is that you can trace when weak disorder wins over kinetic energy.

 

[asimov42]

Ah, ok - thanks again DrDu.

It looks like, in the Anderson model, it should in theory be possible to localize e.g., an electron, perfectly (i.e., to squash the wave function down to a delta function), if there is sufficient disorder. But this clearly can't be the case, as this would violate the uncertainty principle. Is this an issue with the model?

 

[DrDu]

Sorrz for not answering earlier.I think you missunderstand the model. The electron is never localized at a single point. Rather the nodes of the lattice correspond to some kind of valence orbital on the atoms making up the lattice. So yes, you can localize the electron to one lattice point, if you make interaction weak enough and lift all the other sites a little bit in energy.

 

[asimov42]

Hi DrDu - thanks! In another thread, I was asking - if you had, in theory, infinite randomness, would the resulting localization to a lattice point become exact (I guess this is theory what I asked before). If I do understand the Anderson model correctly, the lattice would be infinite to achieve bounded localization (with no mobility edges). In the limit, could you not push this bound to be as small as one liked (perhaps you would need infinite potentials at other lattice points?).