I've been thinking more about explaining what we mean by "insulators", in light of some of the insightful comments. As I'd said, we can think about three major classes of insulators: band insulators (a large gap due to single-particle effects (more below) exists in the ladder of electronic states above the highest occupied state); Anderson insulators (the highest occupied electronic states are localized in space, rather than extending over large distances; localization happens because of disorder and quantum interference); and Mott insulators (hitherto neglected electron-electron interactions make the energetic cost of moving electrons prohibitively high).
The idea of an energy gap (a big interval in the ladder of states, with the states below the gap filled and the states above the gap empty) turns out to be a unifying concept that can tie all three of these categories together. In the band insulator case, the states are pretty much single-particle states (that is, the energy of each state is dominated by the kinetic energies of single electrons and their interactions with the ions that supply the electrons). In the Anderson insulator case, the gap is really the difference in energy between the highest occupied state and the nearest extended state (called the mobility edge). In the Mott case, the states in question are many-body states that have a major contribution due to electron-electron interactions. The electron-electron interaction cost associated with moving electrons around is again an energy gap (a Mott gap), in the ladder of many-body (rather than single-particle) states.
I could also turn this around and talk in terms of the local vs. extended character of the highest occupied states (as Peter points out). In the ideal (infinite periodic solid) band insulator case, all (single-particle) electronic states are extended, and it's the particular lattice arrangement and electronic population that determines whether the highest occupied state is far from the nearest unoccupied state. In the Anderson case, quantum interference + disorder leads to the highest occupied states looking like standing waves - localized in space. In the Mott case, it's tricky to try to think about many-body states in terms of projections onto single-particle states, but you can do so, and you again find that the highest relevant states are localized (due, it turns out, to interactions). Like Peter, I also have been meaning to spend more time thinking hard about insulators.
Coming soon: a discussion of "metals".
Nice posts. I'm a high energy guy trying to learn some condensed matter theory. I've found your posts helpful and clear.
ReplyDeleteDoug wrote:
ReplyDelete>In the ideal (infinite periodic
>solid) band insulator case, all
>(single-particle) electronic >states are extended,
But this goes back to what I wrote earlier.... Although this language is appropriate if one describes such systems in terms of Bloch functions, Kohn showed that this is not the essentially point. He showed that if one has a completely filled band (e.g. a "band" insulator) then one can write the many-electron wavefunction as a Slater determinant of EITHER delocalized Bloch waver OR localized Wannier functions. This is not possible for a partially filled band. There one must write it as Bloch waves.
Since the constituent wavefunction of a band insulator can be written using a complete set of local functions, localization is a ground state property of both Anderson and band insulators. In this sense band insulators are just as localized as Anderson insulators.
This point of view is not discussed in a textbook that I am aware of, but is more or less standard theory in the ferroelectric community. For instance the modern theory of polarization rests on the instrisic localization of wavefunctions... even in band insulators. Frankly, I was shocked the first time I learned about all this! :)
There are some nice pedagogical introductions by Raffaele Resta on this stuff.
This paper provides a unifying view of Band and Mott insulators:
ReplyDeletehttp://arxiv.org/abs/cond-mat/0301338
Ofcourse one needs to understand the more conventional way of looking too (which you provided) for getting a better understanding.
Nice posts Doug. I'm trying to get up to speed here in my new academic career so I've missed most of this discussion. However, on the topic of insulators it might be worth mentioning the whole new idea of the "topological insulator". ( See for example http://www.nature.com/nature/journal/v452/n7190/abs/nature06843.html ). It turns out that band insulators can be very nontrivial -- and more recent work extends many of these ideas to interacting insulators as well. I don't know if this is unified with Anderson insulators though.
ReplyDelete