For example, when a constant voltage (the gate voltage) is applied between graphene and a metallic layer, separated by a very thin insulator, the resulting electrical field will modify the quantity of conduction carriers and thus graphene's electrical conductivity. Restricting ourselves to linear, spatially local materials, the constitutive relations take the form: In equations (1.14) and (1.15), spatial and temporal nonlocality are displayed as convolutions between constitutive tensors and fields. In this case, the 2D analogue of the constitutive equation (1.25) is written as. 1. This duality between the zero-gap-semiconductor picture and the metal-with-an-empty-valence-band picture makes graphene particularly interesting for applications in many photonic devices that require conducting but transparent thin films. Thus, pure graphene can be regarded as a zero-gap semiconductor. The real part of this term (zero for {\gamma }_{c}=0) contributes to energy absorption or dissipation due to the intraband electrons. The volume current density {{\boldsymbol{J}}}_{\omega }^{{\rm{(cond)}}} is then {{\boldsymbol{J}}}_{\omega }^{{\rm{(cond)}}}={{\boldsymbol{K}}}_{\omega }/\delta, and from equation (1.25) it follows that, Therefore, according to equation (1.29), the effective dielectric constant of this homogeneous film is, Recalling the constitutive equation (1.30) and taking into account the fact that the time-averaged rate of work done per unit time per unit volume by the electromagnetic fields on a current volume distribution is \mathrm{Re}\;\{{{\boldsymbol{J}}}_{\omega }\cdot {{\boldsymbol{E}}}_{\omega }^{*}\}/2 [1], it follows that the time-averaged rate of work done per unit time per unit surface by the electromagnetic fields on the graphene charge carriers is. The behavior of {\boldsymbol{P}} and {\boldsymbol{M}} under the influence of the fields results from the microscopic structure of the material. The conductivity plots shown in Fig. 3. Although piecewise homogeneous materials are inhomogeneous, the inhomogeneities are completely confined to the boundaries. For spatially local materials, only the angular frequency ω is relevant. Most relevantly for our present purposes, {\rho }^{{\rm{(2D)}}} and {\boldsymbol{K}} do not generally vanish at a graphene boundary when graphene is treated as an infinitely thin layer characterized by the constitutive equation (1.30). This expression shows that for {\gamma }_{c}=0, \mathrm{Re}\;{\sigma }^{{\rm{inter}}}=0 and \mathrm{Im}\;{\sigma }^{{\rm{inter}}}\lt 0, that is, the interband contribution to the conductivity is purely imaginary and negative. Macroscopically averaged bound charges and currents are described in terms of the polarization density {\boldsymbol{P}}({\boldsymbol{r}},t), representing the electric dipole moment per unit volume, and the magnetization density {\boldsymbol{M}}({\boldsymbol{r}},t), representing the magnetic dipole moment per unit volume. To find out more, see our, Browse more than 100 science journal titles, Read the very best research published in IOP journals, Read open access proceedings from science conferences worldwide, Copyright © 2016 Morgan & Claypool Publishers, Graphene Optics: Electromagnetic Solution of Canonical Problems, Optimization of Immobilization of Nanodiamonds on Graphene, Doping graphene with a monovacancy: bonding and magnetism, Radiative Properties of Semiconductors: Graphene. In the high-frequency limit the total conductivity becomes mostly real and tends to the universal value {e}^{2}/4\hslash. where \theta (\hslash \omega -2| \mu | ) is a step function. In this case, {\boldsymbol{P}}=0, {\boldsymbol{M}}=0 and {{\boldsymbol{J}}}^{{\rm{(cond)}}}=0, and from equations (1.8) and (1.9) we obtain. Phys. For undoped (no chemical additions) and ungated (zero gate voltage) graphene at T=0\ {\rm{K}}, the charge carrier density n0 is very low, but it can be tuned by chemical additions (doping) or with the help of a constant electric field (electric field effect, gate voltage). 5 Howick Place | London | SW1P 1WG. Therefore, {\sigma }^{{\rm{intra}}} corresponds to a surface impedance Z (the inverse of the surface conductivity), with \mathrm{Im}\;Z\lt 0, that is, an inductive surface impedance. Find out more. The vector \mathrm{Re}\;{\boldsymbol{k}} points in the direction at which the phase of the wave propagates in space (the direction of the phase velocity). The inverse of \sigma (\omega ), the coefficient of proportionality linking the tangential component of the electric field and the surface current along the plane of the sheet, is usually called surface impedance ([1], pp 354–6, [14]). To turn the surface conductivity of a graphene layer into a uniaxial anisotropic permittivity, two additional parameters must be therefore introduced: Using these two parameters, the two parallel (or in-plane) components and the perpendicular (or out-of-plane) component of the permittivity tensor are given by, $$\varepsilon_{||}(\omega, \Gamma, \mu_c, T) = \varepsilon_r + i\frac{\sigma(\omega, \Gamma, \mu_c, T)}{\varepsilon_0 \omega \Delta} \mathrm{\ and\ } \varepsilon_\bot = \varepsilon_r$$, The above uniaxial anisotropic material description can be introduced into a simulation in FDTD and MODE using the analytic material type. A short calculation shows that plane time harmonic fields in the form, satisfy equation (1.45) and are solutions of the homogeneous Maxwell equations provided {\boldsymbol{k}}\cdot {\boldsymbol{a}}=0 (transversality condition, wave vector perpendicular to fields) and {\boldsymbol{k}}\cdot {\boldsymbol{k}}={\omega }^{2}\varepsilon (\omega )\mu (\omega )/{c}^{2} (dispersion relation).

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