Sunday, November 22, 2009

Graphene, part II

One reason that graphene has comparatively remarkable conduction properties is its band structure, and in particular the idea that single-particle states carry a pseudospin.  This sounds like jargon, and until I'd heard Philip Kim talk about this, I hadn't fully appreciated how this works.  The idea is as follows.  One way to think about the graphene lattice is that it consists of two triangular lattices offset from each other by one carbon-carbon bond length.  If we had just one of those lattices, you could describe the single-particle electronic states as Bloch waves - these look like plane waves multiplied by functions that are spatially periodic with reference to that particular lattice.  Since we have two such lattices, one way to describe each electronic state is as a linear combination of Bloch states from lattice A and lattice B.  (The spatial periodicity associated with lattice A (B) is described by a set of reciprocal lattice vectors that are labeled K (K'))

Here is where things get tricky.  The particular linear combinations that are the real single-particle eigenstates can be written using the same Pauli matrices that are used to describe the spin angular momentum of spin-1/2 particles.  In fact, if you pick a single-particle eigenstate with a crystal momentum \hbar k, the correct combination of Pauli matrices to use would be the same as if you were describing a spin-1/2 particle oriented along the same direction as k.  This property of the electronic states is called pseudospin.  It does not correspond to a real spin in the sense of a real intrinsic angular momentum.  It is, however, a compact way of keeping track of the role of the two sublattices in determining the properties of particular electronic states.  

The consequences of this pseudospin description are very interesting.  For example, this is related to why back-scattering is disfavored in clean graphene.  In pseudospin language, a scattering event that flips the momentum of a particle from +k to -k would have to flip the pseudospin, too, and that's not easy.  In non-pseudospin language, that kind of scattering would have to change the phase relationship between the A and B sublattice Bloch state components of the single-particle state.  From that way of phrasing it, it's more clear (at least to me) why this is not easy - it requires rather deep changes to the whole extended wavefunction that distinguish between the different sublattices, and in a clean sample at T = 0, that shouldn't happen.

A good overview of this stuff can be found here (pdf) in this article from Physics Today, as well as this review article.  Finally, Michael Fuhrer at the University of Maryland has a nice powerpoint slide show (here) that discusses how to think about the pseudospin.  He does a much more thorough and informative job than I do here.


Kabeer said...

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Don Monroe said...

Thanks, Doug, that's really interesting, and the Fuhrer link was very nice.

I'm still confused about how pseudospin leads to low scattering , though. Your real-space description is the right direction, I think. But the scattering rate depends on a matrix element, which is determined by the scattering potential sandwiched between the initial and final states. If the scatterer is atomic-sized, then the long-range relationship between initial and final states doesn't matter, just what happens where the scattering potential is large. Similarly, a point defect can scatter one plane-wave state into another, even though they disagree in phase pretty much everywhere.

Maybe the key is that the two pseudospin states have zeroes on alternate atoms, so they're never both big on a single atom?

Doug Natelson said...

Hi Don - I see what you're saying. You're right, of course, that a point defect in a regular lattice can scatter a Bloch wave into another Bloch wave (affecting extended states even though it only "acts" at a point; it breaks the translational invariance that makes Bloch states the right solutions to the periodic potential problem). Still, I think the graphene case is different. In the graphene case, in some sense a defect would have to scatter Bloch waves from both lattices into two other particular Bloch waves from both lattices with a particular phase arrangement. I'm still finding the best way for my brain to think about this, though, I freely admit.

Anonymous said...

Backscattering due to short-range scatterers is not reduced in graphene. The reduction of backscattering is only expected for long-range scatterers, such as charged impurity disorder.

Note that the reduction of the backscattering is strongly angle dependent. The reflection is zero in exactly the backwards direction but it can be finite for other angles. See Fig. 2 in

In fact, this effect is expected to be stronger in nanotubes. There the scattering can be only in the 0 and the 180 degrees directions. The first that pointed the reduction of backscattering in carbon systems was Ando 10 years ago.

This argument to explain the high mobility in graphene and nanotubes is nice and it is often discussed in presentations. However, I believe that this has not been really demonstrated experimentally.

Don Monroe said...

Thanks, anonymous.

It makes sense, in view of my earlier comment, that the pseudospin argument would explain reduced scattering from a long-range potential (such as remote Coulomb scattering or acoustic phonons) but not a short-range potential.

I'm not familiar with the literature for graphene, but there's another implication that might be interesting to explore: Long-range potentials tend to preferentially cause small-angle scattering. This affects the coherence measured, for example, in Shubnikov-de Haas much more than it degrades the mobility (which requires significant angles). So you might expect to see much more robust SdH oscillations (and possibly quantum-Hall effects) relative to the mobility in graphene than in other 2D systems.

Measuring the "Dingle ratio" between all scattering and scattering weighted by the projected momentum change is a useful approach to quantify this.

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