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The terahertz plasmon dispersion of a multilayer system consisting of graphene on dielectric and/or plasma thin layers is systematically investigated. We show that graphene plasmons can couple with other quasiparticles such as phonons and plasmons of the substrate; the characteristics of the plasmon dispersion of graphene are dramatically modified by the presence of the coupling effect. The resultant plasmon dispersion of the multilayer system is a strong function of the physical parameters of the spacer and the substrate, signifying the importance of the substrate selection in constructing graphene-based plasmonic devices.

Graphene is a two-dimensional layer of carbon atoms in a honeycomb lattice. This unique atomic arrangement results in a linear energy-momentum dispersion of carriers in graphene and an ultrahigh carrier mobility exceeding 200,000 cm^{2}/V at room temperature [^{2}, when _{F}); correspondingly, ^{−1} is much smaller than the physical dimension of the graphene structure. This simple relation can be altered by the presence of other quasiparticles, such as surface phonons of a polar substrate [^{2} the relation can be described by a simple analytical model, regardless of the type of quasiparticles being coupled.

This paper is organized as follows. In

Many scattering mechanisms have been suggested for the explanation of the experimental observation [_{B} is the Boltzmann constant, _{F} (≅ 10^{6} m/s) is the Fermi velocity of carriers in graphene, _{A} is the acoustic deformation potential, ^{−7} kg/m^{2} is the density of graphene per layer, and _{ph} = 2 × 10^{4} m/s is the phonon velocity of the longitudinal acoustic mode [_{A} = 18 eV is assumed, as measured in the experiments [

The impurity scattering due to the charged impurity is also considered in the literature for carrier transport in graphene [_{1} and _{2}, the average dielectric permittivity of such a system is _{avg} = (_{1} + _{2})/ 2. The elastic scattering rate arising from the charged impurity scattering can then be written as [_{i} is the impurity density, ^{2} /2_{avg}_{s} is the screening wave number. With _{s}:
_{s} is a function of both temperature _{f} ˃˃_{B}_{F}. Within this limit, _{s} is approximately the Thomas-Fermi screening wave number given by ^{2} _{F}) / 2_{avg}, where _{F}) = 2_{F}/ _{F})^{2} is the density of states at the Fermi energy [

Using Maxwell’s equations and appropriate boundary conditions, a surface optical phonon (SP) mode can be solved for a planar interface between two semi-infinite dielectrics with one dielectric characterized by a transverse optical (TO) frequency [_{v}_{v}_{high} and _{low} are the high-frequency and low-frequency dielectric permittivities of the polar substrate, respectively, and _{0} is the permittivity of free space. Equation (4) is a good approximation of _{v}_{v}_{v}_{v}

Besides the aforementioned extrinsic inelastic SP scattering, there is also intrinsic inelastic scattering,

The scattering rate _{2} substrate is shown in _{F} ≅ _{i} = 4.4 × 10^{11} cm^{−2} on the same order of magnitude as the carrier density given by _{F}^{2} / _{F} = _{F} / _{F}. The physical parameters of SiO_{2} for the calculation of _{1} and _{2} of SiO_{2}. The overall _{2} ≅ 156meV, which marks the onset of the intraband phonon emission of _{2}.

The scattering rate ^{−1}(_{F}) contributed by different scattering mechanisms as a function of _{F} is shown in _{F}, the curves in ^{−1}(_{F}) and are shown in _{F}. By contrast, _{F} because of the enhanced screening by the increasing carrier density

Scattering rate of carriers in graphene as a function of (

Semiclassically, the optical conductivity of monolayer graphene can be described by [^{−1}(^{−1}(_{B}^{−1} (_{F}) and Equation (7) becomes the familiar simple Drude model _{0}/(1 − i_{off}) with _{0} = ^{2}_{F}_{eff} / ^{2}, which is commonly used in fitting the experimental data. If the condition _{B}_{F} ≥ 100meV and ^{−1} (_{F}) is a fairly good approximation for Equation (7) [

In the Drude model, the nonlocal effect is ignored. To describe the organized oscillation of electrons due to the long-range nature of Coulomb force among them, the Drude model is insufficient when the oscillation is characterized by a large wave number

The polarizability of graphene within the RPA is given as [_{k} = _{F} |_{B}^{−1} → 0. To account for the finite scattering rate, we replace _{γ}^{−1} (_{F}) in Equation (8) and follow the same procedure as the one in References [_{B}^{−1} (_{F}) → 0 for _{γ}^{−1} (_{F}) and

The optical conductivity within the RPA-RT approximation is given by
_{RPA}, is almost identical to the Drude result in the region

Optical conductivity of graphene for (a) ^{−1} (_{F}) = 10 THz.

The plasmon dispersion of graphene sandwiched between air and a substrate of a constant permittivity _{sub} is well known [

where _{ave} = (_{sub} + _{0})/2. Because of the finite scattering rate, _{1} + i_{2} has to be a complex number with _{2} ≠ 0 for Equation (12) to be valid. In Equation (12) and below, we have ignored the retardation effect in view of the fact that the speed of light _{1} ˂˂ c is generally true as we shall see in the following figures for the plasmon dispersion of graphene on various substrates. The region where the retardation effect is important, _{1} is comparable to

Jablan _{2} ˂˂ q_{1} and _{F}, the Drude model is sufficient to determine the plasmon dispersion described by Equation (12) without the need of the RPA-RT approach. By plugging Equation (7) in Equation (12), we obtain
^{−1} (_{F}). Because 2_{1}^{−1} is the wavelength of the plasmons and _{2}^{−1} represents the distance of propagation for the plasmonic field amplitude to decrease to 1/_{2}^{−1}/_{1}^{−1} = _{2}^{−1}/_{1}^{−1}, a high frequency is preferred for graphene, whereas for a metal, _{2}^{−1}/_{1}^{−1} decreases with increasing frequency in the optical region [_{F}, the optical phonon scattering or the interband scattering becomes important; then, _{2}^{−1}/_{1}^{−1} might decrease with increasing frequency.

The plasmon dispersion of graphene can be greatly altered by coupling graphene plasmons with other quasiparticles. Both theoretically and experimentally, it has been shown that graphene plasmons can couple with surface phonons of polar substrates [_{sub} is the thickness of the substrate that has the permittivity _{sub}, _{2} is the thickness of the spacer between the graphene layer and the substrate, Г = (_{0} + i_{2}, and _{1}, _{2}, _{0} are respectively the permittivity of the medium below the substrate, the permittivity of the spacer, and the permittivity of free space above the graphene layer, as shown in _{1} → 0. In the following, we thoroughly study the coupled plasmon dispersion of these systems while considering the existence of a finite scattering rate of carriers in graphene.

Structures considered in this paper: (_{sub} and a thickness of _{sub}. The spacer between the graphene layer and the substrate is characterized by a permittivity of _{2} and a thickness of _{2}. Below the substrate is a semi-infinite dielectric medium that has a permittivity of _{1}.

Consider a graphene layer at a distance _{2} above a polar substrate, as shown in _{TO1} and _{TO2} are the two lowest TO frequencies of the substrate with _{TO2} > _{TO1}, and _{int} is the intermediate permittivity for frequency _{TO2} > _{TO1}. These parameters for different materials can be found in Reference [_{2} above a SiO_{2} substrate is shown in _{F} = _{F} / _{F}, and _{F} ≅ _{ave} = (_{2} + _{0})/2 (dashed curve). The deviation is especially strong around _{1} ≅ 59meV where the surface phonon mode is located and the coupling of plasmons and phonons is the strongest. As the distance _{2} between graphene and the substrate increases, the dispersion of the coupled plasmon mode (red curve) approaches that of the decoupled mode. As the thickness of the substrate _{sub} increases, the effective permittivity _{ave} increases as the space otherwise filled by air _{1} = _{0} is now filled with SiO_{2}. As a consequence, _{1} also increases for the same _{sub} and _{2} can serve as tuning parameters for shaping the plasmon dispersion.

Plasmon dispersion of graphene on (_{2} and (_{sub} → ∞). The structure considered is shown in _{1} = _{0}, _{2} = 2_{0}, ^{−1} = 10 THz are used. In (_{sub} = 50 nm and _{2} = 30 nm are used for the calculation. In (_{2} = 30 nm is used for the solid curves, and _{2} → ∞ is assumed for the dashed curve. The black curves in (_{sub} = 50 nm and _{2} = 30 nm

In _{2} above a semi-infinite substrate of various materials. As can be seen, some dispersion relations are characterized by two peaks because they are associated with substrates that have two surface phonon modes of energies within the plotted range. Note that as _{RPA}, as shown in _{2} with _{sub} = 50 nm and _{2} = 30 nm as an example.

Plasmon dispersion of (^{−1} = 10 THz are used for graphene. Thin solid, thick solid, and dashed curves are calculated for _{2} = 30 nm, _{2} = 50 nm, and _{2} → ∞, respectively. Black and red curves are calculated for _{2} = 2_{0} and _{2} = _{0}, respectively. The dotted curves are obtained using the RPA-RT approach with parameters _{2} = 30 nm and _{2} = 2_{0}. In (_{sub} = 50 nm, _{1} = 3.9 _{0}, _{p} = 8.89 eV, and _{1} = _{0} is assumed. Curves for the same plasmon mode are grouped by an ellipse.

Consider a structure shown in _{2} above a metal substrate of a thickness of _{sub} deposited on a semi-infinite dielectric material. The permittivity of the metal substrate is modeled as
_{p} is the bulk plasma frequency and _{p} = 8.89 eV and _{2} increases. A larger _{2} (larger _{avg}) also gives a larger _{sub} and _{1} have nearly no effect on shaping the plasmon dispersion because _{sub} is always much larger than the skin depth of gold, which is about _{p} = 0.22 nm in the THz frequency region. We also calculated the plasmon dispersion for graphene on different metal substrates; the dispersion curves overlapped and can hardly be distinguished. Therefore, the choice of the metal substrate is not important regarding the plasmon dispersion within the THz frequency region. The plasmon dispersion of the system is also calculated using _{PRA} and plotted as the dotted curve in _{1} region, the RPA-RT results start to deviate from the curves obtained using the Drude conductivity. Nevertheless, in the low-_{1} region, the Drude model is sufficiently accurate to describe the plasmon dispersion of the system.

Consider the double-layer graphene shown in _{sub} and _{1} = _{0}. By applying the limit _{sub} → 0 and substituting _{sub} in Equation (15) with
_{sub} does not have to be equal to _{F} or ^{−1}. For simplicity, here we set _{sub} = _{sub} given by Equation (18), Equation (15) becomes
_{2} decreases. We also see that similar to the case of a metal substrate, _{1} increases for the same _{2} increases. In _{PRA} and plotted as the dotted curve. As in the case of graphene on a metal substrate, the Drude model is a good approximation for the plasmon dispersion in the low-_{1} region.

In this paper, the plasmon dispersion of graphene on various substrates is systematically investigated. We start from the calculation of the scattering rate and the optical conductivity. Once these physical parameters are known, the plasmon dispersion of a multilayer system consisting of graphene on dielectric and/or plasma thin layers can be determined. We show that the characteristics of the plasmon dispersion are a strong function of the distance between the graphene layer and the substrate, the permittivity of the spacer, the surrounding permittivity, and various physical parameters of the substrate. Our studies show the importance of the substrate selection as well as the system configuration in designing graphene-based plasmonic optoelectronic devices in the THz frequency region.

This work was supported by U.S. Air Force AOARD under Grant Award No. FA2386-13-1-4022.

The authors declare no conflict of interest.

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