filling factor, landau fan, and landau index

In the paper here, there is a plot of $$R_{xx}$$ as a function of magnetic field $$B_{⊥}$$ and carrier density, $$n$$.

The filling factor jumps by 4 between adjacent lines.

The set of lines where $$R_{xx}= 0$$ seems to be defined as 'Landau fan.'

$$R_{xx}=0$$ when the fermi energy is between bands, as shown in this image below (from wikipedia).

I was also able to find in the paper here that for graphene with 4 fold degeneracy, $$\nu = \pm 4(|n|+1/2)$$.

I would like to know the general relation between the filling factor and the landau level (I wasn't able to find a clear equation showing the relation between the two. For instance, the equation I found for four fold degenerate graphene, on top of the factor of 4, it has an offset of $$1/2$$.

• The Landau level is the filling factor, both terms are employed to say the same thing. Now, for the factor 4 : in graphene, there are two types of degeneracy : there is the two equivalent sub-lattices degeneracy (usually referred to as K and K' valleys) from which comes a factor of 2. There is also an electron-hole degeneracy from which comes another factor of 2. Now you have the factor 4. For the 1/2 offset, as I am not so familiar with graphene, I forgot where it comes from. Hope it helps.
– Mary
Oct 23 '18 at 16:32