Effective impedance of two-dimensional metal with retardation effect

Optical absorption with retardation effect is discussed for two-dimensional (2D) metal. The absorption is given by the induced Joule heat in the metal and it is proportional to Re(σ)/|ɛ|²in whichσandɛdenote conductivity and dielectric function, respectively. Here, we investigate the effective impedance in both retarded and non-retarded regions of surface plasmon by discussing the response of the current density to the electric fields. The absorption formula Re(σ)/|ɛ|²is compared with the formula Re(σ/ɛ) that is commonly used for the absorption in carbon nanotube. We show that Re(σ/ɛ) is equal to Re(σ)/|ɛ|²only in the non-retarded region. The physical reason for Re(σ/ɛ) ≠ Re(σ)/|ɛ|²in the retarded region is that the induced current density is not out-of-phase with the induced electric field, which is explained by the effective impedance for both regions. The opposite response of the current to the induced electric field distinguishes the retarded and non-retarded regions. The calculated optical absorption spectra by Re(σ)/|ɛ|²reproduce the absorption spectra by solving the Maxwell equation as a function of the angular frequency of light or incident angle relative to the 2D surface, which makes Re(σ)/|ɛ|²a general representation of absorption.