Monday, April 17, 2006

Two interesting condensed matter debates

In the past couple of weeks, two interesting debates have come to my attention in condensed matter circles. The first has to do with electronic transport in graphene, and isn't really a debate - more of an interesting observation having to do with weak localization, a specific quantum correction to the classical electrical conductivity. Consider an electron propagating through a solid, scattering off of static disorder (lattice defects, grain boundaries). Feynman tells us that we have to add amplitudes for all possible paths through the material, and then square the sum of those amplitudes to get a transmission probability, assuming that all the paths add coherently. Some relevant trajectories include closed loops, that take the electron back past its starting point. For each loop like that, there is another trajectory with the loop traversed in the opposite direction. In the absence of spin-orbit scattering or magnetic fields, those loops and their time-reversed conjugates all add in phase and interfere constructively. The result is an enhanced (nonclassical) probability for the electron to back-scatter, leading to an enhanced resistance. Now, if one threads magnetic flux through those loops, electrons traversing loops in opposite directions are phase shifted relative to one another, and the constructive interference is broken. The weak localization enhancement of the resistance is suppressed at high magnetic fields, and the result is a magnetoresistance, with a field scale set by the size of the typical coherent loop. This is one of the main ways people estimate quantum coherence lengths in conductors.

What does any of this have to do with graphene? Well, here Andre Geim and coworkers look at transport in single graphene sheets, and find that weak localization is essentially absent. It turns out that the particular electronic structure of graphene implies that one can get the effect of a magnetic field if the graphene sheet isn't really flat. (For the experts: this has something to do with a pseudospin involving two equivalent sublattices on the sheet, and the breaking of that symmetry by roughness. I don't really understand this, so please let me know if there's a clear writeup about this somewhere.) Conversely, in a new paper de Heer and co-workers grow graphene epitaxially on SiC wafers, and do observe weak localization. Interesting - this seems to imply that the material grown by de Heer is in some ways intrinsically superior to that prepared by other methods. This is also roughly confirmed by the mobilities (25 m^2/Vs in de Heer's, 10 m^2/Vs in Geim's).

I'll hit the second discussion in the next post....

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