Density Dependent Exchange Contribution to ∂μ/∂n and Compressibility in Graphene

The band structure of graphene (a single layer of car- bon atoms), by dint of its honeycomb lattice, has linear dispersions near the K and Kpoints ("Dirac points") of the Brillouin zone. Recent developments in techniques for fabricating conducting graphene layers have thus pro- vided the physics community with a unique opportunity to study an interacting two-dimensional (2D) massless Dirac fermion system using table-top experimental equip- ment. This has led to a veritable explosion of both ex- perimental and theoretical activity in this field(1). Around the Dirac points (which we take to be the zero of energy), the kinetic energy for a "bare" electron (see below) is ǫ (0),s = sv0|k|, where k is the wavevector with respect to the Dirac point, and s = +1 and −1 for the conduction and valence bands, respectively. The elec- tron chemical potential � , which in intrinsic graphene is at zero, can be shifted up or down by doping and/or application of external gate voltages, with a concommi- tant change in the electron density. This paper reports the calculation of ∂�/∂n , which is related to the elec- tronic compressibility, in extrinsic graphene at tempera- ture T = 0 as a function of the density n. (In this pa- per, unless otherwise indicated, partial derivatives are at constant area and T = 0, and n refers exclusively to the free carrier density (i.e., the difference in electron density from that of intrinsic graphene) in the gated graphene, which we take to be substantially less (|n| ∼ 10 12 cm −2 ) than the intrinsic electron density nv (> 10 15 cm −2 ) fill- ing up the valence band. ) We obtain � (n) by evaluating the electron self-energy within the Hartree-Fock approx- imation (HFA). The HFA is a good approximation up to reasonably high values of rs (∼ the ratio of the aver- age carrier potential to kinetic energy) in parabolic-band semiconductors, and we expect it to also give reliable re- sults in graphene, where rs < 1. It is useful (and conceptually meaningful) to divide 2D graphene into three different systems depending on the band filling: bare or empty, a theoretical abstraction of just one electron in the graphene honeycomb lattice as appropriate for the single-particle band-structure cal- culation with both valence and conduction bands com- pletely (and unphysically) empty, or equivalently, the un- physical situation where the interaction between the elec- trons is turned off; intrinsic, i.e., the undoped and un- gated situation, which is a zero-gap semiconductor with a completely full (empty) valence (conduction) band and chemical potential � (= EF, since we are at T = 0) precisely at the Dirac point; extrinsic, i.e., gated/doped graphene with a tunable 2D free carrier density n of elec- trons (holes) in the conduction (valence) band, withbe- ing above (below) zero, i.e. in the conduction (valence) band. Note that only the empty system can be char- acterized by the bare, noninteracting parameters (e.g., velocity v0, density of states D0) with both intrinsic and extrinsic graphene being characterized by renormalized parameters. We emphasize that the bare graphene pa- rameters, being unphysical abstractions, cannot be ex- perimentally determined.