Static structure factor and pair correlation function of graphene

We report our theoretical investigations on the static structure factor and pair correlation function using both the density-density and spin-density response functions of a doped single graphene sheet based on the random phase approximation and on graphene's massless Dirac fermions concept. The static structure factor and pair correlation function are obtained by regularizing the dynamical polarization function, which otherwise is clearly divergent due to the interaction energy of the infinite Dirac sea of negative energy states. The local field effects have been considered in the simplistic Hubbard approximation. We find the structure factor to be dependent on the dimensionless coupling constant alpha, and for high values of coupling constant the magnetic structure factor indicates paramagnetic instability which is also corroborated from other theoretical investigations. The spin symmetric pair correlation function computed in the simplistic Hubbard approximation begins from zero at zero separation only at very high densities but the results for parallel spin and anti-parallel spin pair correlation functions expose the shortcoming of this local field approximation. This work should stimulate more investigations testing various other local field schemes and also quantum Monte Carlo based simulations.