Terahertz Optoelectronic Property of Graphene: Substrate-Induced Effects on Plasmonic Characteristics

Abstract

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.

1. Introduction

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 cm²/V at room temperature [1]. This ultrahigh mobility also implies a long propagation distance of graphene surface plasmons, which are quasiparticles arising from the quantized collective oscillations of the charged carriers on the graphene surface. These quasiparticles can be excited with the assistance of a grating structure, such as graphene nanoribbons [2,3,4], a dielectric grating [5,6], or a metal grating [7,8]. Surface plasmons can also be induced near the grain boundary [9], or generated through the interaction with metal particles [10] or sound waves [11]. Besides these momentum-transfer techniques, electron-energy-loss spectroscopy [12,13] and near-field microscopy [14] are also used for the study of graphene surface plasmons. The excited surface plasmons have frequencies in the terahertz (THz) and far-infrared spectral regions. The wave number q of a graphene surface plasmon is proportional to the square of its frequency ω, i.e., q ∝ ω², when q is much smaller than the Fermi wave number (q ˂˂ k_F); correspondingly, q⁻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 [3,12], or plasmons of a metal substrate or another adjacent graphene layer [7,10,15,16,17,18,19]. In this paper, we show that the coupling strength and the deviation from q ∝ ω² 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 Section 2, we discuss the primary scattering sources and express the scattering rate as a function of the carrier energy and the Fermi energy. Once the scattering rate is determined, the optical conductivity in the THz and far-infrared spectral regions can be obtained, which is discussed in Section 3. In Section 4, we use the derived optical conductivity along with the coupling model to show the plasmon dispersion of graphene on various substrates. The characteristics of the plasmon dispersion are discussed in terms of the distance between the substrate and the graphene layer, the substrate thickness, and other physical parameters of the system. The conclusion is presented in Section 5.

2. Scattering Rate

2.1. Elastic Scattering

Many scattering mechanisms have been suggested for the explanation of the experimental observation [1,20,21,22,23,24,25]. Among all possible scattering channels, phonon scattering is the intrinsic scattering mechanism that serves as the lower bound of the scattering rate that fundamentally limits the mobility of carriers in graphene. Two types of phonon scattering mechanisms are considered in this paper for carrier transport in graphene: elastic acoustic phonon scattering and inelastic optical phonon scattering; the former is discussed in this section and the latter is discussed in Subsection 2.2. The scattering rate arising from the longitudinal acoustic phonon scattering is given by [26]

(1)

where E is the energy of charged carriers, ħ is the reduced Planck constant, k_B is the Boltzmann constant, T is the temperature, v_F (≅ 10⁶ m/s) is the Fermi velocity of carriers in graphene, D_A is the acoustic deformation potential, ρ = 7.6 × 10⁻⁷ kg/m² is the density of graphene per layer, and v_ph = 2 × 10⁴ m/s is the phonon velocity of the longitudinal acoustic mode [27]. In this paper, D_A = 18 eV is assumed, as measured in the experiments [22,25,26,28].

The impurity scattering due to the charged impurity is also considered in the literature for carrier transport in graphene [20,21,23]. For a monolayer graphene sandwiched between two media of different permittivities ε₁ and ε₂, the average dielectric permittivity of such a system is ε_avg = (ε₁ + ε₂)/ 2. The elastic scattering rate arising from the charged impurity scattering can then be written as [29]

(2)

Where n_i is the impurity density, q is the scattering wave number, = e² /2ε_avgq is the Fourier transform of the 2D potential energy, and q_s is the screening wave number. With q = 2k sin (θ/2) , where k is the wave number of carriers and θ is the scattering angle, we obtain the analytical expression for the impurity scattering rate in the limit of k ˃˃ q_s:

(3)

In fact, q_s is a function of both temperature T and wave number q. In this paper, we consider the case of a Fermi energy E_f ˃˃k_BT and a wave number q ≤ 2k_F. Within this limit, q_s is approximately the Thomas-Fermi screening wave number given by e² ρ(E_F) / 2ε_avg, where ρ(E_F) = 2E_F/ π(ħv_F)² is the density of states at the Fermi energy [29].

2.2. Inelastic Scattering

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 [30]. This SP mode has a frequency slightly higher than the associated TO mode; it can efficiently couple with electrons of graphene if the graphene layer is close enough to the polar substrate [31,32]. The SP scattering rate contributed by various SP modes v with the phonon energy ħω_v is approximately given by [33]

(4)

where the scattering wave number is , with ± standing for the phonon emission (plus sign) and absorption (minus sign) in the scattering process; e is the electronic charge; is the equilibrium phonon occupation number of the surface phonon mode v; d is the spacing between the graphene layer and the substrate; is the Fermi–Dirac function, with μ being the chemical potential; F_v is the electron-phonon coupling parameter given by [31,34]

(5)

where ε_high and ε_low are the high-frequency and low-frequency dielectric permittivities of the polar substrate, respectively, and ε₀ is the permittivity of free space. Equation (4) is a good approximation of if the phonon energy ħω_v is small. For large phonon energies, each term in the summation of Equation (4) has to be multiplied by a correction factor c_v = 1 + 0.0027ħω_v, where ħω_v is in the unit of meV [33].

Besides the aforementioned extrinsic inelastic SP scattering, there is also intrinsic inelastic scattering, i.e., optical phonons scattering, in graphene. Among various optical phonon modes, the ZO mode at the Brillouin zone center (Г − ZO) has the lowest phonon energy (110 meV) and therefore the highest phonon occupation number [35]. Because the Г − ZO mode is out-of-plane vibrations, its coupling with in-plane conduction electrons is weak. By comparison, degenerated Г − LO and Г − TO modes can efficiently couple with conduction electrons [36], but at room temperature their contributions to the scattering rate is limited because these modes are energetic (200 meV). The TO mode at the zone boundary (K − TO) has a lower energy around 160 meV; it is therefore suggested to have the highest efficiency in coupling with electrons [37]. However, as we previously showed theoretically [33], the SP modes have a higher coupling efficiency than the K − TO mode, which is also confirmed experimentally [22]. Therefore, in this paper, we only consider the surface optical phonon scattering as the dominant inelastic scattering mechanism. The K − TO phonon scattering becomes important when graphene is deposited on a substrate such as SiC or h¬-BN where the surface optical phonon scattering is inefficient.

The scattering rate as a function of the carrier energy calculated from Equations (1), (3), and (4) for graphene in the air at a distance of d = 0.34 nm above a semi-infinite SiO₂ substrate is shown in Figure 1a. The physical parameters of graphene used in the calculation are E_F ≅ μ = 100 meV, T = 300 K, and charged impurity density n_i = 4.4 × 10¹¹ cm⁻² on the same order of magnitude as the carrier density given by k_F² / π, where k_F = E_F / ħv_F. The physical parameters of SiO₂ for the calculation of can be found in Reference [32]. As can be seen from the data plotted in Figure 1a, contributes the most to the total scattering rate. For carriers of large carrier energy E, decreases with increasing E, approaching the behavior described by Equation (3). The second important scattering mechanism is , which is contributed by two surface phonon modes ħω₁ and ħω₂ of SiO₂. The overall increases with E mostly due to the fact that the density of states increases with the carrier energy. Most notably begins to increase rapidly for E > ħω₂ ≅ 156meV, which marks the onset of the intraband phonon emission of ħω₂.

The scattering rate τ⁻¹(E_F) contributed by different scattering mechanisms as a function of E_F is shown in Figure 1b. To better see the dependence of the scattering rate on E_F, the curves in Figure 1b are normalized to τ⁻¹(E_F) and are shown in Figure 1c. As can be seen, the relative weight of phonon scattering increases due to the increase of the density of states with E_F. By contrast, decreases with E_F because of the enhanced screening by the increasing carrier density .

Figure 1. Scattering rate of carriers in graphene as a function of (a) the carrier energy; and (b) the Fermi energy. Curves from top to bottom: the total scattering rate (blue curves), (red curves), and (black curves). Curves plotted in (c) are scattering rates plotted in (b) but normalized to the total scattering rate.

Figure 1. Scattering rate of carriers in graphene as a function of (a) the carrier energy; and (b) the Fermi energy. Curves from top to bottom: the total scattering rate (blue curves), (red curves), and (black curves). Curves plotted in (c) are scattering rates plotted in (b) but normalized to the total scattering rate.

3. Optical Conductivity

3.1. Drude Model

Semiclassically, the optical conductivity of monolayer graphene can be described by [29,38]

(6)

where ω is the angular frequency of the electric field. Because the behavior of τ⁻¹(E) is complicated, as we have shown in Figure 1a, the analytical solution to Equation (6) is difficult to obtain. Alternatively, we can replace τ⁻¹(E) in Equation (6) with an effective energy-independent but frequency-dependent scattering rate to obtain an analytical expression [33]:

(7)

It can be shown that if μ ˃˃ k_BT, ≅ τ⁻¹ (E_F) and Equation (7) becomes the familiar simple Drude model σ₀/(1 − iωτ_off) with σ₀ = e²E_Fτ_eff / πħ², which is commonly used in fitting the experimental data. If the condition μ ˃˃ k_BT is not valid, has to be calculated numerically. Nevertheless, with E_F ≥ 100meV and T = 300 K , it has been shown that is a weak function of frequency, and ≅ τ⁻¹ (E_F) is a fairly good approximation for Equation (7) [33].

3.2. RPA Model

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 q. To account for the nonlocality, the optical conductivity of graphene as a function of both ω and q can be derived within the random phase approximation (RPA). In the RPA, the out-of-phase response of electrons to the external field is assumed to average out to zero due to the random location of the electrons in a large quantity [39].

The polarizability of graphene within the RPA is given as [40]

(8)

where A is the area of graphene, s and s' denote the band indices having the values of 1 for the conduction band or −1 for the valance band, δ is an infinitesimal number, k' = k + q, θ is the scattering angle between k and k', and E_k = ħv_F |k|. The analytical expression for Equation (8) has previously been obtained in the limit of μ ˃˃ k_BT [40,41] and in the collisionless limit of τ⁻¹ → 0. To account for the finite scattering rate, we replace ω with ω_γ = ω + iτ⁻¹ (E_F) in Equation (8) and follow the same procedure as the one in References [40,41] assuming μ ˃˃ k_BT; the result is

(9)

where ; sgn(x) = 1 if x ≤ 0 and −1 otherwise. Equation (9) is identical to Equation (7) of Reference [42] in a slightly different form. Apparently, by setting τ⁻¹ (E_F) → 0 for ω_γ → ω, Equation (9) reduces to the collisionless results in References [40,41]. To comply with local electron conservation, Equation (9) needs to be modified to [43,44]

(10)

where ∏(q, ω) is given by Equation (9) and ∏(q, 0) is obtained by setting both τ⁻¹ (E_F) and ω to zero in Equation (9). Equation (10) is called the RPA relaxation time (RPA-RT) approximation, which properly accounts for the finite scattering rate and the conservation of local electrons [44].

The optical conductivity within the RPA-RT approximation is given by

(11)

The optical conductivity obtained from the Drude model and that from the RPA-RT approach in the long-wavelength limit (q → 0) are plotted in Figure 2. As can be seen in Figure 2a, the RPA-RT result, σ_RPA, is almost identical to the Drude result in the region ħω ˂˂ μ. By contrast, in the high-frequency region where ħω is comparable to 2μ, the Drude model fails to account for the interband transition, whereas the RPA-RT results properly account for it, as seen in Figure 2b. Due to the simplicity of the Drude model, in this paper we use Equation (7) for the conductivity of graphene. The RPA-RT conductivity, given by Equation (11), is calculated to show that the obtained results using the Drude model are good approximations in the low-energy region.

Figure 2. Optical conductivity of graphene for (a) ħω ≤ μ, and (b) ħω ≥ μ. The solid curves are derived from the Drude model (Equation (7)), and the dashed curves are obtained using the RPA-RT approach (Equation (11)). The real part and the imaginary part of the optical conductivity are plotted in red and black colors, respectively. For both figures, the parameters used are T = 300 K, μ = 100 meV, and τ⁻¹ (E_F) = 10 THz.

Figure 2. Optical conductivity of graphene for (a) ħω ≤ μ, and (b) ħω ≥ μ. The solid curves are derived from the Drude model (Equation (7)), and the dashed curves are obtained using the RPA-RT approach (Equation (11)). The real part and the imaginary part of the optical conductivity are plotted in red and black colors, respectively. For both figures, the parameters used are T = 300 K, μ = 100 meV, and τ⁻¹ (E_F) = 10 THz.

4. Plasmon Dispersion

The plasmon dispersion of graphene sandwiched between air and a substrate of a constant permittivity ε_sub is well known [40,43,45]. The physical structure is illustrated in Figure 3a. Using Maxwell’s equations with appropriate boundary conditions, the plasmon dispersion can be obtained by solving the equation:

(12)

where ε_ave = (ε_sub + ε₀)/2. Because of the finite scattering rate, q = q₁ + iq₂ has to be a complex number with q₂ ≠ 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 c in vacuum is 300 time higher than the Fermi velocity of graphene. As a result, ω/q₁ ˂˂ 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, i.e., ω/q₁ is comparable to c, is indistinguishable from the ħω axis, and thus is not considered in this paper.

Jablan et al. [43] have shown that as long as q₂ ˂˂ q₁ and ħω ˂˂ 2E_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

(13)

(14)

where τ is the inverse of τ⁻¹ (E_F). Because 2πq₁⁻¹ is the wavelength of the plasmons and q₂⁻¹ represents the distance of propagation for the plasmonic field amplitude to decrease to 1/e of its original amplitude, it is meaningful to define q₂⁻¹/q₁⁻¹ = τω as a figure of merit. Clearly, regarding surface plasmons in terms of q₂⁻¹/q₁⁻¹, a high frequency is preferred for graphene, whereas for a metal, q₂⁻¹/q₁⁻¹ decreases with increasing frequency in the optical region [46]. Note that τ is also a function of frequency [33]. For a frequency ω that is too large, above either 48 THz or ħω > 2E_F, the optical phonon scattering or the interband scattering becomes important; then, τ decreases and therefore q₂⁻¹/q₁⁻¹ 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 [3,12], and with plasmon modes of metal particles or a metal surface [7,10,15]. Coupled plasmon modes resulted from the coupling of two graphene layers have also been studied [16,17,18,19,47]. Coupled plasmon modes of these systems can be determined from the following equation when the retardation effect is ignored [15]:

(15)

where d_sub is the thickness of the substrate that has the permittivity ε_sub, d₂ is the thickness of the spacer between the graphene layer and the substrate, Г = (ε₀ + iσq/ ω)/ε₂, and ε₁, ε₂, ε₀ 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 Figure 3b,c. In the case of double-layer graphene, as shown in Figure 3d, Equation (15) is reduced by setting d₁ → 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.

Figure 3. Structures considered in this paper: (a) graphene on a semi-infinite substrate of a constant permittivity; (b) graphene on a polar substrate; (c) graphene on a metal substrate; and (d) double-layer graphene. The substrate is characterized by a permittivity of ε_sub and a thickness of d_sub. The spacer between the graphene layer and the substrate is characterized by a permittivity of ε₂ and a thickness of d₂. Below the substrate is a semi-infinite dielectric medium that has a permittivity of ε₁.

Figure 3. Structures considered in this paper: (a) graphene on a semi-infinite substrate of a constant permittivity; (b) graphene on a polar substrate; (c) graphene on a metal substrate; and (d) double-layer graphene. The substrate is characterized by a permittivity of ε_sub and a thickness of d_sub. The spacer between the graphene layer and the substrate is characterized by a permittivity of ε₂ and a thickness of d₂. Below the substrate is a semi-infinite dielectric medium that has a permittivity of ε₁.

4.1. Graphene on A Polar Substrate

Consider a graphene layer at a distance d₂ above a polar substrate, as shown in Figure 3b. The polar substrate is characterized by a permittivity given by

(16)

where ω_TO1 and ω_TO2 are the two lowest TO frequencies of the substrate with ω_TO2 > ω_TO1, and ε_int is the intermediate permittivity for frequency ω between the two TO frequencies, ω_TO2 > ω > ω_TO1. These parameters for different materials can be found in Reference [32]. Using Equations (7), (15), and (16), the coupled plasmon-phonon dispersion of graphene at a distance d₂ above a SiO₂ substrate is shown in Figure 4a, where q is normalized to k_F = E_F / ħV_F, and E_F ≅ μ = 100 meV is used; the scattering rare is chosen to be 10THz as a reasonable value given in Figure 1b. As can be seen, the dispersion of the coupled plasmon mode deviates from the decoupled mode given by Equation (12) with ε_ave = (ε₂ + ε₀)/2 (dashed curve). The deviation is especially strong around ħω = ħω₁ ≅ 59meV where the surface phonon mode is located and the coupling of plasmons and phonons is the strongest. As the distance d₂ 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 d_sub increases, the effective permittivity ε_ave increases as the space otherwise filled by air ε₁ = ε₀ is now filled with SiO₂. As a consequence, q₁ also increases for the same ħω (blue curve), as can be told from their relation in Equation (13). Therefore, we see that d_sub and d₂ can serve as tuning parameters for shaping the plasmon dispersion.

Figure 4. Plasmon dispersion of graphene on (a) SiO₂ and (b) various semi-infinite polar substrates (d_sub → ∞). The structure considered is shown in Figure 3b. In all figures, ε₁ = ε₀, ε₂ = 2ε₀, μ = 100meV, and τ⁻¹ = 10 THz are used. In (a), unless otherwise specified, d_sub = 50 nm and d₂ = 30 nm are used for the calculation. In (b), different materials for the polar substrate are represented by different colors as specified in the legend; d₂ = 30 nm is used for the solid curves, and d₂ → ∞ is assumed for the dashed curve. The black curves in (a) calculated using the Drude conductivity are plotted again in (c) to compare with the dotted curve obtained using the RPA-RT approach. The parameters used for the RPA-RT calculation are d_sub = 50 nm and d₂ = 30 nm

Figure 4. Plasmon dispersion of graphene on (a) SiO₂ and (b) various semi-infinite polar substrates (d_sub → ∞). The structure considered is shown in Figure 3b. In all figures, ε₁ = ε₀, ε₂ = 2ε₀, μ = 100meV, and τ⁻¹ = 10 THz are used. In (a), unless otherwise specified, d_sub = 50 nm and d₂ = 30 nm are used for the calculation. In (b), different materials for the polar substrate are represented by different colors as specified in the legend; d₂ = 30 nm is used for the solid curves, and d₂ → ∞ is assumed for the dashed curve. The black curves in (a) calculated using the Drude conductivity are plotted again in (c) to compare with the dotted curve obtained using the RPA-RT approach. The parameters used for the RPA-RT calculation are d_sub = 50 nm and d₂ = 30 nm

In Figure 4b, we plot the dispersion of the coupled plasmon-phonon mode for graphene at a distance d₂ 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 q increases, nonlocal effects become strong, and the results obtained in Figure 4a,b using the Drude conductivity for graphene deviate from the dispersion obtained using σ_RPA, as shown in Figure 4c for graphene on SiO₂ with d_sub = 50 nm and d₂ = 30 nm as an example.

Figure 5. Plasmon dispersion of (a) graphene on a gold substrate and (b,c) double-layer graphene. The related structures are shown in Figure 3c,d. In all figures, μ = 100 meV and τ⁻¹ = 10 THz are used for graphene. Thin solid, thick solid, and dashed curves are calculated for d₂ = 30 nm, d₂ = 50 nm, and d₂ → ∞, respectively. Black and red curves are calculated for ε₂ = 2ε₀ and ε₂ = ε₀, respectively. The dotted curves are obtained using the RPA-RT approach with parameters d₂ = 30 nm and ε₂ = 2ε₀. In (a), d_sub = 50 nm, ε₁ = 3.9 ε₀, ω_p = 8.89 eV, and γ = 17 THz are used. In (b,c), ε₁ = ε₀ is assumed. Curves for the same plasmon mode are grouped by an ellipse.

Figure 5. Plasmon dispersion of (a) graphene on a gold substrate and (b,c) double-layer graphene. The related structures are shown in Figure 3c,d. In all figures, μ = 100 meV and τ⁻¹ = 10 THz are used for graphene. Thin solid, thick solid, and dashed curves are calculated for d₂ = 30 nm, d₂ = 50 nm, and d₂ → ∞, respectively. Black and red curves are calculated for ε₂ = 2ε₀ and ε₂ = ε₀, respectively. The dotted curves are obtained using the RPA-RT approach with parameters d₂ = 30 nm and ε₂ = 2ε₀. In (a), d_sub = 50 nm, ε₁ = 3.9 ε₀, ω_p = 8.89 eV, and γ = 17 THz are used. In (b,c), ε₁ = ε₀ is assumed. Curves for the same plasmon mode are grouped by an ellipse.

4.2. Graphene on A Metal Substrate

Consider a structure shown in Figure 3c where a monolayer graphene is at a distance d₂ above a metal substrate of a thickness of d_sub deposited on a semi-infinite dielectric material. The permittivity of the metal substrate is modeled as

(17)

where ω_p is the bulk plasma frequency and γ is the scattering rate of the metal. Using Equations (7), (15) and (17), the plasmon mode of the system due to the coupling between the graphene layer and the metal slab can be numerically solved, as shown in Figure 5a, where the metal is assumed to be gold with ω_p = 8.89 eV and γ = 17 THz [48]. As can be seen, by coupling graphene with a metal substrate, the wave number q of the plasmon mode increases. This coupling effect gradually vanishes as d₂ increases. A larger ε₂ (larger ε_avg) also gives a larger q according to Equation (13). However, d_sub and ε₁ have nearly no effect on shaping the plasmon dispersion because d_sub is always much larger than the skin depth of gold, which is about c / ω_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 σ_PRA and plotted as the dotted curve in Figure 5a. At high q₁ region, the RPA-RT results start to deviate from the curves obtained using the Drude conductivity. Nevertheless, in the low-q₁ region, the Drude model is sufficiently accurate to describe the plasmon dispersion of the system.

4.3. Double-Layer Graphene

Consider the double-layer graphene shown in Figure 3d with the bottom graphene layer serving as a substrate having the conductivity σ_sub and ε₁ = ε₀. By applying the limit d_sub → 0 and substituting ε_sub in Equation (15) with

(18)

the coupled plasmon modes of double layer graphene can be solved. Note that σ_sub does not have to be equal to σ of the upper graphene layer because they do not necessarily have the same E_F or τ⁻¹. For simplicity, here we set σ_sub = σ; then with ε_sub given by Equation (18), Equation (15) becomes

(19)

which has been derived previously [47]. Equation (19) can also be derived by finding zeros of the linear-response function of the double layer graphene system [15]. The coupled plasmon modes are plotted in Figure 5b,c using Equations (7) and (19). As can be seen, the plasmon dispersion described by Equation (13) is strongly hybridized as the distance between two graphene layers d₂ decreases. We also see that similar to the case of a metal substrate, q₁ increases for the same ħω as ε₂ increases. In Figure 5b, the plasmon dispersion of the double-layer graphene is also calculated using σ_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-q₁ region.

5. Conclusions

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.