Polarization of supported graphene by slowly moving charges

Dynamic polarization of graphene by moving external charges: Random phase approximation

“…Following Ref. 46 , we can express the induced potential Φ ind (r, z, t) in the region above graphene using the Fourier transform (r → q and t → ω) as…”

“…where Π(q, ω) is the polarization function for free graphene and ǫ bg (q) is the effective background dielectric function, which is expressed in terms of the substrate dielectric constant ǫ sub as 46 ǫ bg (q) = ǫ sub + 1 2 1 + coth(qh) ǫ sub + coth(qh) .…”

“…59 However, we note that h is defined in Ref. 46 as the position of an effective substrate surface plane where boundary conditions on the electrostatic fields are imposed. Thus, while the measured and theoretically obtained values of the graphene-substrate gap serve as a guide for the value of h, there is an uncertainty in h on the same order as the shift of the z axis discussed earlier.…”

“…46,47 This semi-classical kinetic equation (SCKE) model gave a relatively simple dielectric function for graphene that accurately described the thermal effect on plasmon dispersion 47,48 and allowed us to analyze the contributions of plasmon excitations and low-energy intra-band single-particle excitations (SPEs) to the stopping and image forces. 46 However, it remained unclear how large the density must be to validate the semi-classical model and, more importantly, what effect the inter-band SPEs that lie beyond the capability of the SCKE model have. Therefore, the first goal of this paper is to determine the conditions under which the SCKE model is applicable at zero temperature by comparing the stopping and image forces obtained using the SCKE dielectric function 46 with those obtained using the MDF-RPA dielectric function.…”

“…46 However, it remained unclear how large the density must be to validate the semi-classical model and, more importantly, what effect the inter-band SPEs that lie beyond the capability of the SCKE model have. Therefore, the first goal of this paper is to determine the conditions under which the SCKE model is applicable at zero temperature by comparing the stopping and image forces obtained using the SCKE dielectric function 46 with those obtained using the MDF-RPA dielectric function. 21,22,23 Furthermore, since we have found in Ref. 46 that a finite gap between graphene and the substrate strongly affects both forces in the SCKE model, the second goal of this paper is to examine the effect of a finite gap in the MDF-RPA model.…”