What does negative filling mean for Landau levels?
Answers
HEY SAKNA HERE
First, let me say that I find the discussion in Hierarchy of fillings for the FQHE in monolayer graphene particularly clear on this topic, and Goerbig's RMP is also a good resource. The paper you mention focuses on the n=0 n=0 Landau level in graphene, so I will as well.
The basic idea is that there is a difference in how we label filling ratios in conventional semiconductor 2DEGs and in materials with Dirac-like spectra, such as graphene. In the 2DEG case ν semi νsemi is measured relative to the empty lowest Landau level (n=0 n=0). Hence ν semi ≥0 νsemi≥0 as you're probably familiar with.
In graphene it is natural to instead measure filling ratios relative to the neutrality point (the Dirac node). Due to particle-hole symmetry the n=0 n=0 Landau level is half-filled at zero energy. Hence ν=0 ν=0 in the graphene case corresponds to a half-filled Landau level, rather than an empty one as would be implied by ν semi =0 νsemi=0, and negative filling ratios become possible.
The final ingredient to note is that the Landau levels in graphene have a four-fold degeneracy due to spin and valley symmetries. Applying a magnetic field breaks this degeneracy and you get four sublevels. In the case of the n=0 n=0 Landau level, two of the sublevels end up below the Dirac point, and two above it. The n=0 n=0 Landau level is then said to be empty at ν=−2 ν=−2 and completely filled at ν=2 ν=2, and the graphene filling ratio can be related to the one in a 2DEG counting by ν=ν semi −2 ν=νsemi−2.
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Explanation:
Further, when ⌫ Landau levels are filled, there is a gap in the energy spectrum: to ... The current is I = e˙x, which means that, in ... we've set up our computation we get a negative Hall resistivity rather than positive;.