Odd integer quantum Hall effect in graphene:

There has been a flurry of activity in the past couple of years as a result of the discovery that graphene , a single atomic layer of a honeycomb lattice of carbon, can be produced in a simple and reliable fashion by "peeling" it off of a graphite crystal. The single graphene sheet can be contacted electrically, and forms a novel example of a two dimensional electron gas. Among the many things being studied in graphene, a highlight is the quantum Hall effect, which has recently been observed even at room temperature in graphene.

An interesting feature of the quantum Hall effect in graphene is the sequence of quantized Hall conductivities, observed at unusual multiples +-2,+-6,+-10... of e²/h, the conductance quantum. Successive quantum Hall states are separated by 4 e²/h in Hall conductivity. This can be understood as arising from the twofold spin degeneracy and a twofold valley degeneracy of the low energy electrons and holes at the two inequivalent corners of the graphene Brillouin zone. The symmetry around zero is required by time reversal invariance.

More recently, some additional quantum Hall states have been observed. Specifically +-1 and +-4 states have been seen. The +-4 states can be understood from simple spin splitting by the Zeeman effect. The +-1 state, however, requires some splitting of the valley degeneracy. This is most likely due to interactions, in close analogy to "quantum Hall ferromagnetism" in bilayer electron gases in GaAs heterostructures. This has been already suggested and studied by several authors.

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Our work:

There are some lingering questions in this picture. One is the role of disorder, which is likely quite significant. Another is the precise nature of the splitting of valley degeneracy, which gives rise to some inversion symmetry breaking. This might be of charge density wave or another type, with different spatial structures. To elucidate these issues, we carried out numerical exact diagonalization studies of the problem. We find evidence for distinct types of symmetry breaking in the lowest and first excited Landau levels. Furthermore, we find a marked sensitivity of the odd integer quantum Hall effect to disorder, specifically making the effect in the first excited Landau level (i.e. the +-3 state) much more fragile than in the lowest Landau level.

To learn more, read the paper