Casimir interactions in graphene systems

The non-retarded Casimir interaction (van der Waals interaction) between two free standing graphene sheets as well as between a graphene sheet and a substrate is determined. We present several different derivations of the interaction. An exact analytical expression is given for the dielectric function of graphene along the imaginary frequency axis within the random phase approximation for arbitrary frequency, wave vector, and doping. The first reference to the material graphene in the literature was made by Boehm et al. [1] in 1962. With modern technology it is now possible to produce large area graphene sheets and graphene has become one of the most advanced two-dimensional (2D) materials of today. Due to its superior transport properties it has a high potential for technological applications [2–6]. A free standing graphene sheet has a very interesting band structure. The valence and conduction bands form two sets of cones. In each set the two cones are aligned above each other with their tips coinciding at the fermi level. Thus, the fermi surface is just two points in the Brillouin zone; the value of the band gap is zero. The energy dispersion in the conduction and valence bands is linear which means that the carriers behave as relativistic particles with zero rest mass. When a graphene layer is formed on a substrate the fermi level moves up or down in energy — the sheet is doped. A graphene layer interacts with other graphene layers or with a substrate with Casimir forces, forces that were predicted [7] by Casimir in 1948. In a pioneering work Sparnaay [8] tried to experimentally verify the existence of the Casimir force between two parallel plates. However, the experimental uncertainties were of the same order of magnitude as the force itself so the experiment was non-conclusive. The interest in the Casimir force virtually exploded a decade ago. This increase in interest was triggered by a torsion pendulum experiment by Lamoreaux [9], which produced results with good enough accuracy for the comparison between theory and experiment to be feasible. This stimulated both theorists [10–15] and experimentalists [16–19] and the Casimir field has grown constantly since then. The thermal Casimir effect is not completely understood yet [20, 21]. These forces are very important in graphene systems. They are the result of many-body interactions. Other many-body effects modify the dispersion of the energy bands [22–27]. The present work is devoted to the forces. We derive the Casimir interaction between two graphene sheets, undoped and doped, and between one graphene sheet and a substrate. Numerical results are presented in the range from 1Å to 1μm for experimentally relevant doping concentrations. The derivations are performed within the non-retarded formalism. It was demonstrated in Ref. [28] that retardation effects are not important in undoped graphene. For doped graphene retardation effects are expected to show up at separations outside the range considered here. Furthermore, we present explicit expressions for the dielectric function of doped graphene on the imaginary frequency axis, in terms of real valued functions of real valued variables. We begin by calculating the interaction energy between two graphene layers. For undoped graphene retardation effects never show up [28] because of the particular band structure. For doped graphene they do for large enough separation. We limit the calculation to small enough separations for the retardation effects to be negligible. An estimate of the separation at which retardation effects become important is the separation where the non-