_{1}

^{*}

We study an interface between two media separated by a strictly 2D sheet. We show how the amplitude reflection coef- ficient can be modeled by that for an interface where the 2D sheet has been replaced by a film of small but finite thick- ness. We give the relationship between the 3D dielectric function of the thin film and the 2D dielectric function of the sheet. We apply this to graphene and show how the van der Waals interaction between two graphene sheets is modified when going from the 2D sheet description to the thin film description. We also show the wrong result from keeping the 2D dielectric function to represent the film medium.

There are no strictly 2D (two-dimensional) systems in the real world. However, if the carriers are strongly confined in one direction they have quantized energy levels for one spatial dimension but are free to move in two spacial dimensions [

In Section 2 we give the amplitude reflection coefficients for an interface between two media, in Section 2.1 we show how these are modified when a 2D sheet is inserted at the interface, in Section 2.2 we show how these are modified when instead a film of finite thickness is inserted, and in Section 2.3 we show how a 2D sheet can be modeled by a homogeneous film of finite thickness. In Section 3 we illustrate this modeling for graphene. Before we end with a brief summary and conclusion Section, 5, we discuss in Section 4 how the results for 2D sheets and thin films can be used in non-planar structures.

For the present task we need a geometry consisting of two regions and one interface,. For planar structures there are two types of mode [

and

respectively. Note that r_{ji} = −r_{ij} holds for both mode types. If retardation is neglected there is only one mode type and the amplitude reflection coefficient is

In the above equations, is the dielectric function of medium i, c is the speed of light in vacuum, and k is the length of a wave vector in the plane of the interface. We have suppressed the arguments of the functions in Equations (1)-(3). The amplitude reflection coefficients and the γ-functions are functions of k and ω. The dielectric functions are functions of ω, only, i.e. spatial dispersion is neglected. Inclusion of spatial dispersion in the bulk dielectric functions is possible [11,12] but would lead to much higher complexity and negligible effects for the present problem.

There are different formulations of electromagnetism in the literature. The difference lies in how the conduction carriers are treated. In one formulation these carriers are lumped together with the external charges to form the group of free charges. Then only the bound charges contribute to the screening. We want to be able to treat geometries with metallic regions. Then this formulation is not suitable. In the formulation that we use the conduction carriers are treated on the same footing as the bound charges. Thus, both bound and conduction charges contribute to the dielectric function. In the two formalisms the E and B fields, the true fields, of course are the same. However, the auxiliary fields the D and H fields are different. To indicate that we use this alternative formulation we put a tilde above the D and H fields and also above the dielectric functions. See [

The amplitude reflection coefficient gets modified if there is a 2D layer at the interface. We treat [

where the polarizability, , of the 2D sheet is obtained from the dynamical conductivity, ,

and the dielectric function is

For TM modes the tangential component of the electric field, which will induce the external current, is parallel to k, so the longitudinal 2D dielectric function of the sheet enters. The bound charges in the 2D sheet also contribute to the dynamical conductivity and the polarizability.

The modified amplitude reflection coefficient for a TE mode is [

where the polarizability, , of the 2D sheet is obtained from the dynamical conductivity, ,

and the dielectric function is

For a TE wave the electric field is perpendicular to k, so the transverse 2D dielectric function of the sheet enters.

If retardation is neglected there is only one mode type and the amplitude reflection coefficient is

Now we have in Equations (4), (7), and (10) the amplitude reflection coefficients for an interface between two media with a 2D sheet sandwiched in between. To be noted is that spatial dispersion of the 2D sheet can be included without any complications. This spatial dispersion can have important effects [13-16]. In next section we will show the corresponding results when instead of a 2D sheet we have a thin film sandwiched between the two media.

For the present task we need a geometry consisting of three regions and two interfaces, i|j|k.

For this composite interface the amplitude reflection coefficient for a wave impinging from the i side is [10,17]

where d_{j} is the thickness of the film j. This expression is valid for TMand TE-modes when retardation is included and also for the modes when retardation is neglected. The appropriate amplitude reflection coefficient from Equations (1)-(3) should be used in the expression on the right hand side. In next section we show how the effect of a strictly 2D sheet can be modeled by a film of finite thickness with the proper choice of 3D dielectric function.

Let us study a 2D-sheet placed in a time varying electric field, E, in the plane of the film. There will be a surface current density,. Since we want to treat this 2D sheet as a thin film of finite thickness, δ, we let this current be spread evenly through the thickness of the film. The volume current density, j, is then and since it follows that

Now, since

and

we find that

Thus, in problems with 2D sheets one can treat the sheets as thin 3D films where the 3D polarizability above is used. To check if this is a reasonable approach we insert the expressions in Equations (1)-(3) into Equation (11) where now d_{j} is δ and. If we now let the film thickness, δ, go towards zero we reproduce the results in Equations (4), (7), and (10). Thus in the limit the model is exact.

We will now calculate the Casimir interaction energy between two free standing undoped graphene sheets in vacuum. To make it as simple as possible we neglect retardation and perform the calculations for zero temperature. Retardation effects are actually very small in graphene [6,18].

In a general point, z, in the complex frequency plane, away from the real axis the polarizability is [

where v is the carrier velocity which is a constant in graphene (), and g represents the degeneracy parameter with the value of 4 (a factor of 2 for spin and a factor of 2 for the cone degeneracy). In the numerical calculations we use the value [^{5} m/s for v. The polarizability in Equation (16) is valid for T = 0 when only the contributions from the Dirac cones are included and there is no gap at the Fermi level. Effects of finite temperature and modifications due to a gap, that in some situations open up, can be dealt with [

If we now treat the graphene sheet as a thin film of thickness δ the polarizability of the film material should be chosen as

and on the imaginary frequency axis it is

The van der Waals (vdW) and Casimir interactions can be derived in many different ways.

One way is to derive the interaction in terms of the electromagnetic normal modes [

where is the condition for electromagnetic normal modes. For the present geometry the mode condition function is

where the index 1 stands for vacuum and the index 2 for the film material.

Using Equation (11) with the proper functions for our problem inserted we get

where

For strictly 2D sheets the corresponding mode condition function is

The result of Equation (19) with the mode condition function from Equation (23) is shown as the solid curve in ^{−3} in front of the integral. The result is a straight line in ^{−3}. The dashed curve in

When we treat the graphene sheets as thin films of thickness δ the momentum, k, enters in more places in the integrand with the effect that the result has a more complicated dependence on d. However in those additional places k always enters in the combination kδ. This means that there will be a universal correction factor, from treating the sheet as a film, that depends on d/δ. This correction factor is shown as the solid curve in

In this section we discuss how one may proceed in nonplanar structures, One may, e.g. have a cylinder or a sphere coated by a graphene or graphene-like film. The spatial dispersion complicates things. In these structures the momentum, k, is no longer a good quantum number for the normal modes. However, the problems are often dominated by long wavelengths. The 3D polarizability for a graphene film in the long wavelength limit is

Fortunately the wave number is now absent from the expression and nothing hinders the use of this expression in non-planar structures.

We have shown a way to modify the dielectric function when a homogeneous film of finite thickness is used to simulate a strictly 2D sheet or vice versa. As an illustration we calculated the van der Waals interaction between two undoped graphene sheets. The particular screening in graphene leads to a very simple van der Waals interacttion which follows a pure power law, the same power law as that for the Casimir interaction between two ideal metal half spaces. From this follows that the retardation effects are virtually absent. The general results presented here are valid for any 2D sheet, e.g. a 2D electron gas. In that case the spatial dispersion effects are more complicated [13-16].

We found a universal correction factor, as function of d/δ, to the interaction between the graphene sheets when they were modeled by films of thickness δ. There was a maximum of 19 % over shoot at separations around 8δ.

We further pointed out how to proceed in the case of non-planar structures as to avoid problems caused by spatial dispersion.

We are grateful for financial support from the Swedish Research Council, VR Contract No. 70529001.