Local compressibility of bilayer graphene in the quantum hall regime

In the presence of a strong magnetic field, the charge carriers in bilayer graphene (BLG) condense into a set of flat energy bands called Landau levels (LLs). Electronic compressibility measurements have historically been a powerful tool in studying the physics of partially filled LLs in two-dimensional electronic systems. In particular, electron-electron correlations arising from Coulomb interactions can introduce a negative component to the compressibility. Here we present measurements of electronic compressibility in BLG, performed locally using a scanning single electron transistor. We find that while the inverse compressibility is close to zero for $4 < |\nu| < 8$, it is markedly more negative in the lowest LL, $|\nu| < 4$. Moreover, within the lowest LL, the background inverse compressibility between integer filling also exhibits a stark even-odd asymmetry. It is more negative when starting to fill from an even filling factor than when starting to fill from an odd filling factor, exhibiting a $\nu \rightarrow \nu + 2$ symmetry and indicating the important role of the orbital degeneracy uniquely present in bilayer graphene.