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

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.


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 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.

OPEN ACCESS
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 , i.e., , when q is much smaller than the Fermi wave number (q ˂˂ k F ); correspondingly, 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 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.

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 is the energy of charged carriers, is the reduced Planck constant, is the Boltzmann constant, T is the temperature, ( ) is the Fermi velocity of carriers in graphene, is the acoustic deformation potential, is the density of graphene per layer, and is the phonon velocity of the longitudinal acoustic mode [27].In this paper, 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 and , the average dielectric permittivity of such a system is .The elastic scattering rate arising from the charged impurity scattering can then be written as [29] ( Where is the impurity density, q is the scattering wave number, is the Fourier transform of the 2D potential energy, and is the screening wave number.With , 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, 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 B T and a wave number .Within this limit, is approximately the Thomas-Fermi screening wave number given by , where is the density of states at the Fermi energy [29].

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 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 being the chemical potential; is the electron-phonon coupling parameter given by [31,34] (5 2 where and are the high-frequency and low-frequency dielectric permittivities of the polar substrate, respectively, and is the permittivity of free space.Equation ( 4) is a good approximation of if the phonon energy is small.For large phonon energies, each term in the summation of Equation ( 4) has to be multiplied by a correction factor , where 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 ( ) has the lowest phonon energy (110 meV) and therefore the highest phonon occupation number [35].Because the mode is out-of-plane vibrations, its coupling with in-plane conduction electrons is weak.By comparison, degenerated and 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 ( ) 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 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 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 above a semi-infinite SiO 2 substrate is shown in Figure 1a.The physical parameters of graphene used in the calculation are , , and charged impurity density on the same order of magnitude as the carrier density given by , where .The physical parameters of SiO 2 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 and of SiO 2 .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 , which marks the onset of the intraband phonon emission of .

The scattering rate contributed by different scattering mechanisms as a function of is
shown in Figure 1b.To better see the dependence of the scattering rate on , the curves in Figure 1b are normalized to 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 .By contrast, decreases with because of the enhanced screening by the increasing carrier density . high

Drude Model
Semiclassically, the optical conductivity of monolayer graphene can be described by [29,38] ( where ω is the angular frequency of the electric field.Because the behavior of is complicated, as we have shown in Figure 1a, the analytical solution to Equation ( 6) is difficult to obtain.Alternatively, we can replace in Equation ( 6) with an effective energy-independent but frequency-dependent scattering rate to obtain an analytical expression [33]: It can be shown that if μ ˃˃ k B T, and Equation ( 7) becomes the familiar simple Drude model with , which is commonly used in fitting the experimental data.If the condition μ ˃˃ k B T is not valid, has to be calculated numerically.
Nevertheless, with and , it has been shown that is a weak function of frequency, and is a fairly good approximation for Equation ( 7) [33].

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, and denote the band indices having the values of for the conduction band or for the valance band, is an infinitesimal number, , is the scattering angle between and , and .The analytical expression for Equation ( 8) has previously been obtained in the limit of μ ˃˃ k B T [40,41] and in the collisionless limit of .To account for the finite scattering rate, we replace with in Equation ( 8) and follow the same procedure as the one in References [40,41] assuming μ ˃˃ k B T; the result is (9) where ; if and otherwise.Equation ( 9) is identical to Equation (7) of Reference [42] in a slightly different form.Apparently, by setting for , 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] ( where is given by Equation ( 9) and is obtained by setting both and 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 ( ) are plotted in Figure 2. As can be seen in Figure 2a, the RPA-RT result, , is almost identical to the Drude result in the region ħω ˂˂ μ.By contrast, in the high-frequency region where is comparable to , 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.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 , , and .

Plasmon Dispersion
The plasmon dispersion of graphene sandwiched between air and a substrate of a constant permittivity 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 .Because of the finite scattering rate, has to be a complex number with 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/q 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, i.e., is comparable to c, is indistinguishable from the axis, and thus is not considered in this paper.Jablan et al. [43] have shown that as long as q 2 ˂˂ q 1 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 is the inverse of .Because is the wavelength of the plasmons and represents the distance of propagation for the plasmonic field amplitude to decrease to of its 1/ e original amplitude, it is meaningful to define as a figure of merit.Clearly, regarding surface plasmons in terms of , a high frequency is preferred for graphene, whereas for a metal, decreases with increasing frequency in the optical region [46].Note that is also a function of frequency [33].For a frequency that is too large, above either 48 THz or , the optical phonon scattering or the interband scattering becomes important; then, decreases and therefore 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 is the thickness of the substrate that has the permittivity , is the thickness of the spacer between the graphene layer and the substrate, , 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 .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.semi-infinite dielectric medium that has a permittivity of .

Graphene on a Polar Substrate
Consider a graphene layer at a distance above a polar substrate, as shown in Figure 3b.The polar substrate is characterized by a permittivity given by (16) where and are the two lowest TO frequencies of the substrate with , and is the intermediate permittivity for frequency between the two TO frequencies, .
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 above a SiO 2 substrate is shown in Figure 4a, where q is normalized to , and is used; the scattering rare is chosen to be 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 (dashed curve).The deviation is especially strong around where the surface phonon mode is located and the coupling of plasmons and phonons is the strongest.As the distance 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 increases, the effective permittivity increases as the space otherwise filled by air is now filled with SiO 2 .As a consequence, also increases for the same (blue curve), as can be told from their relation in Equation (13).Therefore, we see that and can serve as tuning parameters for shaping the plasmon dispersion.
In Figure 4b, we plot the dispersion of the coupled plasmon-phonon mode for graphene at a distance 2 d 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 RPA  , as shown in Figure 4c for graphene on SiO 2 with sub 50 nm d  and 2 30 nm d  as an example.

Graphene on A Metal Substrate
Consider a structure shown in Figure 3c where a monolayer graphene is at a distance above a metal substrate of a thickness of deposited on a semi-infinite dielectric material.The permittivity of the metal substrate is modeled as (17) where is the bulk plasma frequency and 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 and [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 increases.A larger (larger ) also gives a larger q according to Equation (13).However, 100 meV and have nearly no effect on shaping the plasmon dispersion because is always much larger than the skin depth of gold, which is about 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 and plotted as the dotted curve in Figure 5a.At high region, the RPA-RT results start to deviate from the curves obtained using the Drude conductivity.Nevertheless, in the low-region, the Drude model is sufficiently accurate to describe the plasmon dispersion of the system.

Double-Layer Graphene
Consider the double-layer graphene shown in Figure 3d with the bottom graphene layer serving as a substrate having the conductivity and .By applying the limit and substituting in Equation ( 15) with (18) the coupled plasmon modes of double layer graphene can be solved.Note that does not have to be equal to of the upper graphene layer because they do not necessarily have the same or .For simplicity, here we set ; then with 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 decreases.We also see that similar to the case of a metal substrate, increases for the same as increases.In Figure 5b, the plasmon dispersion of the double-layer graphene is also calculated using 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-region.

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.

Figure 1 .
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 2 .
Figure 2. Optical conductivity of graphene for (a) , and (b) .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 , , and .

Figure 3 .
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 and a thickness of .The spacer between the graphene layer and the substrate is characterized by a permittivity of and a thickness of .Below the substrate is a

Figure 4 .
Figure 4. Plasmon dispersion of graphene on (a) SiO 2 and (b) various semi-infinite polar substrates ( ).The structure considered is shown in Figure 3b.In all figures, , , , and are used.In (a), unless otherwise specified, and are used for the calculation.In (b), different materials for the polar substrate are represented by different colors as specified in the legend; is used for the solid curves, and 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 and .

Figure 5 .
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, and are used for graphene.Thin solid, thick solid, and dashed curves are calculated for , , and , respectively.Black and red curves are calculated for and , respectively.The dotted curves are obtained using the RPA-RT approach with parameters and .In (a), , , , and are used.In (b,c), is assumed.Curves for the same plasmon mode are grouped by an ellipse.