Density dependent exchange contribution to partial derivativeμ/partial derivativen and compressibility in graphene

We calculate partial derivative mu/partial derivative n (where mu=chemical potential and n=electron density), which is associated with the compressibility, in graphene as a function of n, within the Hartree-Fock approximation. The exchange-driven Dirac-point logarithmic singularity in the quasiparticle velocity of intrinsic graphene disappears in the extrinsic case. The calculated renormalized partial derivative mu/partial derivative n in extrinsic graphene on SiO(2) has the same n(-(1/2)) density dependence but is 20% larger than the inverse bare density of states, a relatively weak effect compared to the corresponding parabolic-band case. We predict that the renormalization effect can be enhanced to about 50% by changing the graphene substrate.