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Appl. Sci. 2014, 4(1), 28-41; doi:10.3390/app4010028
Published: 20 February 2014
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.
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 . 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 , or generated through the interaction with metal particles  or sound waves . Besides these momentum-transfer techniques, electron-energy-loss spectroscopy [12,13] and near-field microscopy  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 ˂˂ kF); 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 
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 εavg = (ε1 + ε2)/ 2. The elastic scattering rate arising from the charged impurity scattering can then be written as 
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 . 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 ħωv is approximately given by 
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 . 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 , 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 . However, as we previously showed theoretically , the SP modes have a higher coupling efficiency than the K − TO mode, which is also confirmed experimentally . 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 EF ≅ μ = 100 meV, T = 300 K, and charged impurity density ni = 4.4 × 1011 cm−2 on the same order of magnitude as the carrier density given by kF2 / π, where kF = EF / ħvF. The physical parameters of SiO2 for the calculation of can be found in Reference . 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(EF) contributed by different scattering mechanisms as a function of EF is shown in Figure 1b. To better see the dependence of the scattering rate on EF, the curves in Figure 1b are normalized to τ−1(EF) 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 EF. By contrast, decreases with EF because of the enhanced screening by the increasing carrier density .
3. Optical Conductivity
3.1. Drude Model
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 .
The polarizability of graphene within the RPA is given as 
The optical conductivity within the RPA-RT approximation is given by
4. Plasmon Dispersion
The plasmon dispersion of graphene sandwiched between air and a substrate of a constant permittivity εsub 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:
where εave = (εsub + 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.  have shown that as long as q2 ˂˂ q1 and ħω ˂˂ 2EF, 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
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 :
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
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 σRPA, as shown in Figure 4c for graphene on SiO2 with dsub = 50 nm and d2 = 30 nm as an example.
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 dsub deposited on a semi-infinite dielectric material. The permittivity of the metal substrate is modeled as
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 σsub and ε1 = ε0. By applying the limit dsub → 0 and substituting εsub in Equation (15) with
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.
Conflicts of Interest
The authors declare no conflict of interest.
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