# Equations of the Model

We investigate plasmons in an ungated graphene structure with a heavily-doped p^{+}-Si substrate, where the graphene layer is exposed on the air, as well as a gated graphene structure with the substrate and a metallic top gate, which are schematically shown in Figs. 11.1a and b, respectively. The thickness of the substrate is assumed to be sufficiently larger than the skin depth of the substrate surface plasmons. The top gate can be considered as perfectly conducting metal, whereas the heavily doped Si substrate is characterized by its complex dielectric constant.

**Figure 11.1 **Schematic views of (a) an ungated graphene structure with a heavily doped Si substrate where the top surface is exposed on the air and (b) a gated graphene structure with a heavily doped Si substrate and a metallic top gate.

Here, we use the hydrodynamic equations to describe the electron motion in graphene [26], while using the simple Drude model for the hole motion in the substrate (due to virtual independence of the effective mass in the substrate on the electron density, in contrast to graphene). In addition, these are accompanied by the self-consistent

2D Poisson equation (the formulation used here almost follows that for compound semiconductor high-electron-mobility transistors, see Ref. [25]). Differences are the hydrodynamic equations accounting for the linear dispersion of graphene and material parameters of the substrate and dielectric layers. In general, the existence of both electrons and holes in graphene results in various modes such as electrically passive electron-hole sound waves in intrinsic graphene as well as in huge damping of electrically active modes due to the electron-hole friction, as discussed in Ref. [26]. Here, we focus on the case where the electron concentration is much higher than the hole concentration and therefore the damping associated with the friction can be negligibly small. Besides, for the generalization purpose, we formulate the plasmon dispersion equation for the gated structure; that for the ungated structure can be readily found by taking the limit *W _{t}*

*—>*°° (see Fig. 11.1).

Then, assuming the solutions of the form exp(/7cx - *icot),* where *к* = 2я/Я and *со* are the plasmon wavenumber and frequency (Я denotes the wavelength), the 2D Poisson equation coupled with the linearized hydrodynamic equations can be expressed as follows:

where *tp _{0}}* is the ac (signal) component of the potential, E

_{e},

*m*and v

_{e},_{e}are the steady-state electron concentration, the hydrodynamic "fictitious mass,” and the collision frequency in graphene, respectively, and e is the dielectric constant which is different in different layers. The electron concentration and fictitious mass are related to each other through the electron Fermi level,

*fi*and electron temperature,

_{e},*T ■*

* ^{1}* e-

In the following we fix *T _{e}* and treat the fictitious mass as a function of Eg. The dielectric constant can be represented as

where *e _{t},* e

_{b}, and e

_{s}are the

*static*dielectric constants of the top and bottom dielectric layers and the substrate, respectively, £2

_{S}= ^4

*ne*/

^{2}N_{s}*m*is the bulk plasma frequency in the substrate with

_{h}G_{s}*N*and

_{s}*m*being the doping concentration and hole effective mass, and v

_{h}_{s}is the collision frequency in the substrate, which depends on the doping concentration. The dielectric constant in the substrate is a sum of the static dielectric constant of Si,

*e*= 11.7 and the contribution from the Drude conductivity. The dependence of the collision frequency, v

_{s}_{s}, on the doping concentration,

*N*is calculated from the experimental data for the hole mobility at room temperature in Ref. [27].

_{s},We use the following boundary conditions: vanishing potential at the gate and far below the substrate, *(pj _{z=} щ* = 0 and

*(pj*= 0; continuity conditions of the potential at interfaces between different layers,

_{z=} and

*a continuity condition of the electric flux density at the interface between the bottom dielectric layer and the substrate in the z-direction,*z=.b->o',

_{Wb+0}=

*e*+o = e

_{b}d(pjdz_{z}=-w_{b}_{s}cW<3z|

_{z=}

*-w*and a jump of the electric flux density at the graphene layer, which can be derived from Eq. (11.1). Equation (11.1) together with these boundary conditions yield the following dispersion equation

_{b}-o',

where

and *H _{bx}* = cothfcWbt. In Eq. (11.5), the term A

_{c}on the right-hand side represents the coupling between graphene plasmons and substrate surface plasmons. If

*A*were zero, the equations F

_{c}_{gl}.(co) = 0 and F

_{sub}(co) = 0 would give independent dispersion relations for the former and latter, respectively. Qualitatively, Eq. (11.8) indicates that the coupling occurs unless

*kW*» lor

_{b}*kW*<к 1, i.e., unless the separation of the graphene channel and the substrate is sufficiently large or the gate screening of graphene plasmons is effective. Note that the non-constant frequency dispersion of the substrate surface plasmon in Eq. (11.10) is due to the gate screening, which is similar to that in the structure with two parallel metal electrodes [28]. Equation (11.5) yields two modes which have dominant potential distributions near the graphene channel and inside the substrate, respectively. Hereafter, we focus on the oscillating mode primarily in the graphene channel; we call it "channel mode,” whereas we call the other mode "substrate mode.”

_{t}