Exact dispersion relations for the hybrid plasmon-phonon modes in graphene on dielectric substrates with polar optical phonons

Sharare Jalalvandi; Sara Darbari; Mohammad Kazem Moravvej-Farshi

doi:10.1364/OE.434274

1. Introduction

Carrier transport is strongly affected by electron-phonon interactions [1–4], as inevitable sources of scattering, leading to sharp changes in the behavior of electronic systems, such as plasmons — i.e., collective oscillations of the conduction electrons. Over the years, graphene has attracted ample theoretical and experimental attention due to its unique physical and thermal properties, making it suitable for various applications [5–7]. In particular, graphene plasmons possess noticeable advantages [8–11], such as long-lived excitations confining enhanced electric fields in tiny volumes and the ability to control the plasmon peak’s frequency through an electrostatic gating mechanism.

Some experimental [12–17] and theoretical [18–22] works have focused on the hybridization of plasmon and phonon modes in the past decade. In general, two different types of phonon can affect plasmons (intrinsic and extrinsic). Lattice vibrations and defects in graphene are two inherent scattering sources in the absence of extrinsic phonons, dominating carrier transport and mobility. Intervalley [23] and intravalley acoustic [17,18] and optical [20–22] phonon scatterings are the three types of intrinsic phonon scattering mechanisms. Intervalley scattering, induced by high momentum and high energy phonons, becomes significant at high temperatures. The so-called D Raman peak from Raman scattering data for graphene, which is highly sensitive to the fabrication process and experimental measurements setup, shows intervalley scattering is an impurity- and defect-assisted process [23–26]. Under particular conditions, however, it is possible to reduce the deteriorating effects of intervalley scattering on plasmons [26]. Low-energy acoustic phonons induce quasielastic intravalley phonon scattering. This mechanism becomes significant at low temperatures where high-energy intravalley optical phonons are absent. The effects of the longitudinal acoustic phonons are more pronounced than the other modes due to their stronger coupling with plasmons [17,19]. For working frequencies higher than those of the acoustic phonons, their deteriorating effect on plasmons is negligible. Although the energy of the intravalley optical phonons is not low enough to be considered a scattering channel, they bring plasmons of higher frequencies into the intraband single-particle excitation (SPEintra) continuum via optical phonon emission (PE) [12,16].

Besides the intrinsic optical phonons, polar optical phonons (POPs) on the surface of substrates, such as SiC [12,13], hBN [14,15], and SiO₂ [16], have shown undeniable effects on the graphene surface plasmons. From the first studies on the impact of surface polar optical phonons (S-POPs) on graphene transport [18], compared to the remote phonon scattering in two-dimensional (2D) electron gas in channels of Si metal-oxide-semiconductor field-effect transistors (MOSFETs) [27,28], theoretical and experimental efforts have been made with enthusiasm. When the surface plasmons of graphene, placed on a polar substrate, are excited at frequencies about the S-POPs’, they are strongly affected by the electric dipoles induced via vibrations of the electrically positive and negative atoms on the surface of the substrate. Investigations show that S-POPs can change plasmon dispersion dramatically, especially near the phonon frequencies [15–16].

Although the measured results on variations of dispersion in the presence of intrinsic and extrinsic phonons for graphene on nonpolar and polar substrates are available experimentally, the lack of precise theoretical dispersion relations for the plasmonic modes and hybrid plasmon-phonon modes are apparent. So, we seek the exact dispersion relations of the coupled modes in the structured graphene placed on a polar substrate, with one, two, or three vibrational frequencies. Moreover, the existing theoretical predictions [12,29] show graphene surface plasmons fall within the interband single-particle excitation continuum and start to damp (i.e., decay into electron-hole pairs) at a critical wavevector (q_crit), where the plasmon radian frequency exceeds the cutting radian frequency of ω_crit = −v_Fq_crit + 2μ_C/ħ. Here, ℏ represents the reduced Planck’s constant, and v_F≈10⁶ m·s⁻¹ and μ_C are graphene Fermi velocity and chemical potential. Nonetheless, based on the experimental data [16], the damping process starts at a smaller critical wavenumber for frequencies higher than the frequency of intrinsic optical phonon. This work aims to address this discrepancy by refining the 2D polarizability, Π₀. The new Π₀ overcomes the two existing limitations: (i) inability to indication of the starting point for the mode damping and (ii) the complexity of the previous relations for graphene dispersion at high wavenumbers, q [12,29]. Our simple dispersion models bring in two imperative results. That is a method of calculating the plasmonic content in a coupled mode and the corresponding mode lifetime. In most graphene plasmonic-based optoelectronic devices, there is an essential need to measure or obtain the output current [30], experimentally or theoretically. When phonons vibrations are absent, the device output current depends on the entire plasmons content [31]. Otherwise, the scenario for calculating the device output differs [30]. In other words, in calculating the output device current in the presence of plasmons-phonons interactions, knowledge of pure plasmonic content surviving the interactions is vital. We will show later, in the next section, by knowing an exact dispersion relation, one can obtain the pure plasmonic content. To the best of our knowledge, such a dispersion relation is not available to date.

In this paper, we model a precise dispersion relation for the plasmon-phonon modes in graphene placed on nonpolar and polar dielectric substrates having one, two, or three phonons vibrational frequencies, based on the existing absorption experimental data [12,16,30], enabling us to predict the precise output currents of devices like plasmon-enhanced photodetectors, solar cells, and modulators. Moreover, another vital parameter is the coupled-mode lifetime, also captured in this study.

Different research groups have employed various methods like first-principle calculations [2,3], Boltzmann equation approach [20], and effective-mass approximation [32] to investigate the impact of the interaction between graphene intrinsic optical phonons and electrons on its surface plasmons dispersion. The same methods are also applicable to the interactions of the extrinsic phonons (S-POPs) with graphene electrons. Here, we use random phase approximation (RPA) instead.

Through the electromagnetic theory, we know surface plasmons propagate in a medium with a near-zero dielectric constant. Hence, the first step to reach our goal is to obtain the graphene dielectric function in the presence of phonons. In doing so, we need to determine the effective electron-electron potentials mediated by both the intrinsic optical phonons and S-POPs, first. Then, employ these potential values in the RPA expansion of the dielectric function and, finally, equate it to zero. With some algebraic manipulations, we achieve the exact and complete relation for the coupled modes dispersion, for the first time, to the best of our knowledge. In this investigation, we ignore the impact of possible defects in graphene. Nevertheless, we model the edge scattering effect by a broadening factor.

The rest of the paper is organized as follows. In the second section, we review the scheme for describing the electron-electron potentials mediated by intrinsic optical phonons and the extrinsic S-POPs. Then, after a brief introduction of the electron interactions in a many-body system, including phonons and the RPA dielectric expansion, we calculate the desired dispersion function. Besides, we obtain the plasmons lifetime in the single-particle excitation and phonon emission regions while introducing the phonons lifetime. Then, we discuss the results obtained, including the modified dispersion relation for graphene in the absence of extrinsic phonons, beyond 2D classical one, the exact hybrid dispersion relations for graphene on polar substrates with up to three phonons, plasmon content of the coupled modes, and their lifetime. At last, we close the paper with conclusive remarks in the final section.

2. Theory

2.1 Intrinsic optical phonons

The lattice vibrations, responsible for optical phonons in graphene, change the separation between neighboring carbon atoms, modifying the overlap integral between them. Using perturbation theory, let us write the electron-electron potential mediated by optical phonons in the form of [32]

(1)$$V_{\textrm{el - el}}^{\textrm{op}} = \frac{1}{{2A}}\sum\limits_{{{\mathbf k}_1}{\sigma _1}} {\sum\limits_{{{\mathbf k}_2}{\sigma _2}} {\sum\limits_{{\mathbf q}\lambda } {\frac{1}{A}{{|{g_{{\mathbf q}\lambda }^{\textrm{op}}} |}^2}{{\cal G}}_\lambda ^0(\omega )\hat{c}_{{{\mathbf k}_1} + {\mathbf q},{\sigma _1}}^\dagger \hat{c}_{{{\mathbf k}_2} + {\mathbf q},{\sigma _2}}^\dagger {{\hat{c}}_{{{\mathbf k}_2}{\sigma _2}}}{{\hat{c}}_{{{\mathbf k}_1}{\sigma _1}}}} } } \equiv \frac{1}{{2{A^2}}}\sum\limits_{{\mathbf k},\sigma ,{\mathbf q},\lambda } {v_\lambda ^{\textrm{op}}\hat{c}_{{\mathbf k} + {\mathbf q},\sigma }^\dagger {{\hat{c}}_{{\mathbf k}\sigma }}} ,$$

where A is the area of graphene, k and q are the wave vectors of the electrons and phonons,σ represents the spin of the electron, λ denotes phonon identifier, ${g_{{\mathbf q}\lambda }^{\textrm{op}}}$ is the transition matrix elements related to electron-optical phonon interaction, ${{\cal G}}_\lambda ^0$ introduces the phonon Green’s function, and $\hat{c}_{{\mathbf k} + {\mathbf q},\sigma }^\dagger$ and ${{\hat{c}}_{{\mathbf k},\sigma }}$ are the creation and annihilation operators, respectively. For intrinsic optical phonons, the identifier λ is omitted from ${{|{g_{{\mathbf q}\lambda }^{\textrm{op}}} |}^2}$ and ${{\cal G}}_\lambda ^0$ as in [20,32]

(2a)$${|{g_{\mathbf q}^{\textrm{op}}} |^2} = \frac{{\hbar D_{\textrm{op}}^2}}{{2{\rho _m}{\omega _{\textrm{op}}}}},$$

and

(2b)$${{\cal G}}_{}^0 = \frac{{2{{{\omega _{\textrm{op}}}} / \hbar }}}{{{{({\omega + {i / {{\tau_{\textrm{op}}}}}} )}^2} - \omega _{\textrm{op}}^2}}.$$

D_op ≈ 11 eV·Å⁻¹ is the optical phonon deformation potential, ρ_m ≈ 7.6×10⁻⁷ kg·m⁻² is graphene mass density [20], ħω_op= 198 meV and τ_op ≈ 70 fs [16] are the intrinsic optical phonon energy and lifetime, and ω represents the input optical signal frequency. From Eq. (1), one can interpret similarity to the simple electron-electron Coulomb interaction [33] with coupling strength of $(1/A){|{g_{\mathbf q}^{\textrm{op}}} |^2}{{{\cal G}}^0}$. Moreover, we have

(3)$${v^{\textrm{op}}} = {|{g_{\mathbf q}^{\textrm{op}}} |^2}{{{\cal G}}^0}(\omega ),$$

which is the potential between two electrons mediated by an intrinsic optical phonon in graphene, with a definite wave vector.

2.2 Extrinsic surface optical phonons

Due to the optical nature of the surface phonons, an equation similar to Eq. (1) can be used for S-POPs, except for different transition matrix elements and green’s function, as in [20]

(4)$$v_\lambda ^{\textrm{POP}} = \sum\limits_\lambda {{{|{g_{{\mathbf q}\lambda }^{\textrm{POP}}} |}^2}{{\cal G}}_\lambda ^0(\omega ).}$$

in which

(5)$${|{g_{{\mathbf q}\lambda }^{\textrm{POP}}} |^2} = \frac{{\pi {e^2}}}{{{\varepsilon _0}}}{{\cal F}}_\lambda ^2\frac{{\textrm{exp}({ - 2q{z_0}} )}}{q} \approx \frac{{\pi {e^2}}}{{q{\varepsilon _0}}}{{\cal F}}_\lambda ^2,$$

and ${{\cal G}}_\lambda ^0$ is similar to Eq. (2b), in which ω_op and τ_op are replaced by the extrinsic phonons frequency, ω_λ, and lifetime, τ_POP. In Eq. (5), e is the elementary charge, ε₀ is the free-space permittivity, and z₀ ≈ 35 Å is the graphene-substrate van der Waals separation [20]. Notice, since qz₀≪1, the term exp(−2qz₀) in Eq. (5) becomes almost one. Fröhlich coupling strength, ${{\cal F}}_\lambda ^2$, describes the magnitude of the polarization field of the phonon labeled λ [33]. For substrates with only one dominant phonon (λ = 1), such as SiC, we have

(6)$${{\cal F}}_1^2 = \frac{{\hbar {\omega _1}}}{{2\pi }}\left( {\frac{1}{{\varepsilon_\infty^{} + 1}} - \frac{1}{{\varepsilon_0^{} + 1}}} \right).$$

For those with more than one dominant phonon mode, each phonon mode is associated with one Fröhlich coupling strength — e.g., for AlN, Al₂O₃, h-BN, and HfO₂ with λ = 1 and 2, we have,

(7a)$${{\cal F}}_1^2 = \frac{{\hbar {\omega _1}}}{{2\pi }}\left( {\frac{1}{{{\varepsilon_{\textrm{i,1}}} + 1}} - \frac{1}{{{\varepsilon_0} + 1}}} \right)$$

and

(7b)$${{\cal F}}_2^2 = \frac{{\hbar {\omega _2}}}{{2\pi }}\left( {\frac{1}{{\varepsilon_\infty^{} + 1}} - \frac{1}{{\varepsilon_{\textrm{i,1}}^{} + 1}}} \right),$$

while for substrates like SiO₂ with (λ = 1, 2, and 3), we have

(8a)$${{\cal F}}_1^2 = \frac{{\hbar {\omega _1}}}{{2\pi }}\left( {\frac{1}{{\varepsilon_{\textrm{i,1}}^{} + 1}} - \frac{1}{{\varepsilon_0^{} + 1}}} \right),$$

(8b)$${{\cal F}}_2^2 = \frac{{\hbar {\omega _2}}}{{2\pi }}\left( {\frac{1}{{\varepsilon_{\textrm{i,2}}^{} + 1}} - \frac{1}{{\varepsilon_{\textrm{i,}1}^{} + 1}}} \right),$$

and

(8c)$${{\cal F}}_3^2 = \frac{{\hbar {\omega _3}}}{{2\pi }}\left( {\frac{1}{{\varepsilon_\infty^{} + 1}} - \frac{1}{{\varepsilon_{\textrm{i,2}}^{} + 1}}} \right).$$

In Eqs. (7) and (8) ω₁ < ω_λ≥2, ε_∞, ε_i,1(2), and ε₀ are the optical, intermediate, and static permittivity of the substrate, ε_i,1 represents the permittivity for the frequency range of ω₁ < ω < ω₂, and ε_i,2 is related to the range ω₂ < ω < ω₃. The values of the above parameters for different substrates are given in Table 1.

Table 1. Various permittivities, the dominant phonon energies, and related Fröhlich coupling constants for dielectrics substrates SiC, Al₂O₃, HfO₂, AlN, ZrO₂, ZrSiO₄, hexagonal BN (hBN), and SiO₂

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2.3 Dielectric function

The RPA is a powerful tool for finding plasmons dispersion relation, starting with the graphene dielectric function. In this approach, a single electron is exposed to a self-consistent field, including the externally applied electric field plus the one induced by other electrons. For RPA to result in an excellent approximation, we must pay attention to the Wigner-Seitz parameter (rs) [29], which describes the density of an interacting system. For a small value of rs (e.g., ≈ 0.5 for graphene, which is independent of electron concentration [29]), the system interacts weakly, and RPA becomes an excellent approach. Furthermore, changing electron concentration in graphene does not affect the accuracy of the RPA method.

Consider an electron, in the absence of an external field, moves through a weakly interacting system, going from the state k₁ to state k₂, influenced by two dominant potentials: (i) a Coulomb potential induced by other electrons (V_C = e²/2qε₀) [16], and (ii) the potential between electrons mediated by phonons (V_ph = v^op + v_λ^POP). Feynman diagrams [35] (Table 2) show all possible routes through which the state of an electron can change from k₁ to k₂. As observed from the table, an electron may propagate freely going from k₁ to k₂, provided k₂ = k₁. Moreover, the freely moving electron may undergo a scattering process with a phonon of polarizability Π₀ and then continue to move freely again before the second scattering occurs. The latter process may occur many times. Hence, the total possibility is the sum of all possible routes, which may be summarized in an infinite series

(9)$$\Uparrow = \uparrow + \begin{array}{c} \uparrow \\ {\boxed{{{\Pi _0}}}}\\ \uparrow \end{array} + \begin{array}{c} \uparrow \\ {\boxed{{{\Pi _0}}}}\\ \uparrow \\ {\boxed{{{\Pi _0}}}}\\ \uparrow \end{array} + \begin{array}{c} \uparrow \\ {\boxed{{{\Pi _0}}}}\\ \uparrow \\ {\boxed{{{\Pi _0}}}}\\ \uparrow \\ {\boxed{{{\Pi _0}}}}\\ \uparrow \end{array} + \cdots = \uparrow + \left\{ { \uparrow + \begin{array}{c} \uparrow \\ {\boxed{{{\Pi _0}}}}\\ \uparrow \end{array} + \begin{array}{c} \uparrow \\ {\boxed{{{\Pi _0}}}}\\ \uparrow \\ {\boxed{{{\Pi _0}}}}\\ \uparrow \end{array} + \cdots } \right\} \times \begin{array}{c} \uparrow \\ {\boxed{{{\Pi _0}}}}\\ {} \end{array} = \uparrow + \begin{array}{c} \uparrow \\ {\boxed{{{\Pi _0}}}}\\ \Uparrow \end{array}$$

For simplicity, we omitted the indices of the diagrams shown in Table 2 from Eq. (9). Here we consider optical phonons as the dominant scatterer neglecting other scatterers, like acoustic phonons, defects, and ionized impurities. In a weakly interacting system, the probability of successive scattering incidences equals the possibilities of all individual scattering incidences multiplied by each other. So, we can simplify Eq. (9) into [22]

(10)$${V_{\textrm{eff}}} = ({{V_\textrm{C}} + {V_{\textrm{ph}}}} )+ {V_{\textrm{eff}}} \times {\Pi _0} \times ({{V_\textrm{C}} + {V_{\textrm{ph}}}} ),$$

where V_eff represents the upward double arrow and (V_C + V_ph) signifies the upward single arrow. Hence, we can write the effective dynamical potential as

(11)$${V_{\textrm{eff}}} = {{({{V_C} + {V_{\textrm{ph}}}} )} / {[{1 - ({{V_C} + {V_{\textrm{ph}}}} ){\Pi _0}} ]}}.$$

Moreover, the dielectric function from RPA, ε(q, ω), can be expressed by the effective potential [22,35],

(12)$${V_{\textrm{eff}}} = {{({{V_\textrm{C}} + {V_{\textrm{ph}}}} )} / {\varepsilon ({q,\omega } )}}.$$

Table 2. Feynman diagrams and their descriptions

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By comparing Eq. (11) and Eq. (12), and using V_C, Eq. (3), and Eq. (4) we get

(13)$$\varepsilon ({q,\omega } )= {\varepsilon _{\textrm{env}}} - \left( {\frac{{{e^2}}}{{2q{\varepsilon_0}}} + {{|{g_{\mathbf q}^{\textrm{op}}} |}^2}{{{\cal G}}^0}(\omega )} \right.\left. { + \sum\limits_\lambda {{{|{g_{{\mathbf q}\lambda }^{\textrm{POP}}} |}^2}{{\cal G}}_\lambda^0(\omega )} } \right){\Pi _0},$$

where ε _env is the average of the dielectric constants of the upper and lower media surrounding the graphene. In our model, we consider air as that the upper-medium.

As mentioned in the Introduction, we need to modify the existing formula for Π₀ used in Ref. [8] to remove the limitations in predicting a precise starting point for the decaying continuum and correct behavior for surface plasmons at high q values to resemble that obtained experimentally [12,15,16].

Comparing the experimental data in Refs. [12,16,24] with classical 2D plasmon dispersion given [8], we see for ℏω > 198 meV, the measured plasmons peak’s frequency is blue-shifted compared to that of the 2D classical model. A further observation from this comparison is the discrepancy in the variations of dispersion for high q values —- i.e., ω $\propto$ q^1/2 for classical dispersion and ω $\propto$ q^3/2 for the experimental dispersion [12]. To eliminate these deviations, we have added a q-dependent term, γ, in the denominator of the classical Π₀, as in

(14a)$${\Pi _0} \approx \frac{{{\mu _\textrm{C}}{q^2}}}{{\pi {\hbar ^2}({{{({\omega + \,{i / {{\tau_{\textrm{pl}}}}}} )}^2} - \gamma_{}^2} )}},$$

wherein. μ_C= ±ℏv_F|k_F| with k_F representing the carriers’ Fermi velocity and wavenumber, and τ_pl is the plasmons lifetime, which we will introduce in the following subsection. As an appropriate choice for the dependencies of γ on the plasmons wavenumber to eliminate the deviation in the model from the experimental data, we suggest

(14b)$${\gamma ^2} = \frac{1}{8}\nu _\textrm{F}^2{q^2}{[{\exp ({{{q^{\prime}} / q}} )- 1} ]^{ - 1}},$$

where

(14c)$$q^{\prime} = \frac{{2\pi {\varepsilon _0}{\varepsilon _{\textrm{env}}}}}{{{e^2}{\mu _\textrm{C}}}}{({\hbar {\omega_{\textrm{op}}}} )^2}.$$

is the plasmon’s wavenumber at the frequency ω = ω_op obtained from classical 2D graphene dispersion [8]. For q≪q’, [exp(q'/q)−1]⁻¹→0 and γ→0, and hence Π₀ approaches the classical one. As q increases, Π₀ and dispersion relation start deviating from those obtained by the classical model. Nonetheless, for large wavenumbers, where ℏω ≫ ℏω_op, a Taylor series expansion of the exponential term in Eq. (14b) results in γ $\propto$ q^3/2. Using this and substituting Eq. (14) back into Eq. (13) and setting ε(q, ω) = 0 and ignoring the effects of the optical phonons, we see that ω $\propto$ q^3/2 at high qs which is consistent with the experimental data [12,15,16]. Hence Eq. (14) is valid for all ranges of plasmons wavenumber and energy.

2.4 Lifetime calculations

2.4.1 Plasmon lifetimes

Plasmons are composed of an ensemble of free electrons. Hence, their lifetimes can be the same, τ_pl, helping us evaluate Π₀ as in Eq. (14) and the plasmon-phonon coupled-mode lifetime, which we will calculate in the next section. Plasmons damp via different processes like scattering from background ionized impurities and the structured graphene edges, optical phonon emission [16], and electron-hole generation within the single-particle excitation region [29]. Hence, we can express τ_pl as a function of these processes characteristic times,

(15)$$\tau _{\textrm{pl}}^{ - 1} = \tau _0^{ - 1} + \tau _{\textrm{PE}}^{ - 1} + \tau _{\textrm{SPE}}^{ - 1} + \tau _{\textrm{ed}}^{ - 1},$$

where τ₀≈ 85 fs comes from ionized impurity scattering in graphene [16]. The accuracy of graphene fabrication controls this characteristic time. The more well-constructed the sample is, the less scattering by defects and disorders will be. τ_PE is the lifetime of the plasmon due to the phonon emission. When plasmon energy exceeds that of the inherent optical phonon, ħω_op = 198 meV, they damp through emitting optical phonons and entering the PE continuum [12,16]. This kind of scattering is an unavoidable process and is dominant for frequencies higher than ≈ 48 THz. It is highly dependent on temperature and becomes insignificant at low temperatures, where the number of inherent optical phonons is low, making the rate of the plasmons-phonons scattering negligible. As the temperature increases, the number of phonons increases, and the scattering rate becomes significant [17]. τ_SPEis the lifetime in the single-particle excitation continuum. It is due to interband transitions [29], which disrupt the balance of electron density with chemical potential μ_C. As long as the electron density at this level is low, this characteristic time is insignificant. Besides, τ_ed represents the lifetime of electrons due to the edge effects. Scattering by the edges of the structure changes the momentum dramatically and causes high-momentum intervalley scatterings in graphene [23–26]. Same as τ₀, τ_ed is also highly sensitive to the fabrication process and experimental measurements setup.

The existing theoretical model [29] shows plasmons start decaying via electron-hole generation within the intraband energy range of ħv_Fq – 2μ_C < ħω < ħv_Fq or within the interband transitions continuum. Nonetheless, the experimental results [12,16] suggest the plasmons decay for ħω ≥ ħω_op ― i.e., damping by phonon emission and decaying into the intraband continuum before the interband transitions start. The lifetime of the plasmon in the PE continuum is [16]

(16a)$$\tau _{\textrm{PE}}^{ - 1} = \frac{{{g_{\textrm{PE}}}}}{\hbar }|{\hbar ({\omega + {\omega_\alpha }} )+ {\mu_\textrm{C}}} |\cdot [{{\mathbb{F}}({\omega ,{\omega_\alpha }} )+ 2} ],$$

where g_PE ≈ 0.018,

(16b)$${\mathbb{F}}({\omega ,{\omega_\alpha }} )= \textrm{erf}\left( {\frac{{\hbar \omega - \hbar {\omega_\alpha }}}{{2{k_B}T}}} \right) + \textrm{erf}\left( {\frac{{ - \hbar \omega - \hbar {\omega_\alpha }}}{{2{k_B}T}}} \right),$$

represents the thermal broadening effect, with k_B and T as the Boltzman constant environment temperature, and ω_α=ω_op. Moreover, τ_SPEconsists of two components, one related to the intraband, τ_SPE,intra, and the other with the interband transitions, τ_SPE,inter

(17a)$$\tau _{\textrm{SPE}}^{ - 1} = \tau _{\textrm{SPE,intra}}^{ - 1} + \tau _{\textrm{SPE,inter}}^{ - 1}.$$

in which

(17b)$$\tau _{\textrm{SPE,intra}}^{ - 1} ={-} \frac{{{g_{\textrm{SPE,intra}}}}}{\hbar }|{\hbar ({\omega + {\omega_\alpha }} )- {\mu_C} - 3} |\cdot {\mathbb{F}}({\omega ,{\omega_\alpha }} ),$$

with g_SPE,intra ≈ 0.009, ω_α=v_Fq (i.e., the critical frequency at which the intraband transition starts), and

(17c)$$\tau _{\textrm{SPE,inter}}^{ - 1} = \frac{{{g_{\textrm{SPE,inter}}}}}{\hbar }|{\hbar ({\omega + {\omega_\alpha }} )+ {\mu_C}} |\cdot [{{\mathbb{F}}({\omega ,{\omega_\alpha }} )+ 2} ],$$

with g_SPE,inter ≈ 0.02 and ω_α = −v_Fq+2μ_C/ℏ (i.e., the critical frequency at which the interband transition starts).

In a periodically structured graphene with the periodicity d and filling factor F, as the physical width of graphene (i.e., w = Fd) shrinks edge scattering becomes significant, changing the momentum of the electrons notably. Hence, the scattered electrons decay via the intervalley phonon emission. The plasmons wavenumber in this periodic structure satisfies a formula like q_pl ≈ π / (w – w₀), where w₀ ≈ 28 nm is the electrically inactive width of graphene [16]. We model τ_ed, representing the lifetime of electrons in the presence of the edge effect, [16]

(18)$${\tau _{\textrm{ed}}} \approx \frac{{2\pi F \times {{10}^{ - 6}}}}{q}\textrm{.}$$

2.4.2 Phonon lifetime

The edge scatterings, resulting in the emission of high-energy intervalley phonon scattering, can have a significant impact on the S-POPs’ lifetime, provided w < 90 nm (i.e., equivalent to q ≈ 4×10⁷ m⁻¹) [16]. Hence, the S-POPs’ lifetime is

(19)$${\tau _{\textrm{POP}}} \approx {\tau _{\textrm{POP,0}}}\textrm{exp}\left( { - \frac{q}{{4 \times {{10}^7}}}} \right),$$

where τ_POP,0≈ 200 fs is the lifetime of surface optical phonons in the absence of edge scattering [16,30]. We use Eq. (19) for calculating Green’s function (${{{\cal G}}^0}$) for the extrinsic phonons.

3. Results and discussion

3.1 Plasmon dispersion

To determine the plasmon dispersion, experimentally, the extinction spectrum of a superlattice of graphene is measured at different periodicities (i.e., various wavenumbers). Then, by locating the extinction peaks in the q-ω plane, one can find the dispersion graph [16]. Theoretically, one may obtain the extinction spectrum (i.e., the loss function) from the imaginary part of the inverse dielectric function,

(20)$$L \approx{-} {\mathop{\rm Im}\nolimits} {\{{\varepsilon ({q,\omega } )} \}^{ - 1}}.$$

Although the loss function does not express an exact relation to obtaining the plasmons peaks’ frequencies, it offers valuable information through the graphical description of the q-ω plane, indicating where the incident light is lost by plasmon excitations.

3.1.1 Nonpolar substrates

Concerning the structured graphene placed on a nonpolar substrate (i.e., with no S-POPs), we obtain the plasmon dispersion relation by considering the new Π₀ given in Eq. (14),

(21)$$\omega = \sqrt {\frac{{{e^2}{\mu _\textrm{C}}}}{{2\pi {\hbar ^2}{\varepsilon _0}{\varepsilon _{\textrm{env}}}}}q + \gamma _{}^2} .$$

As mentioned in Section II, for high q values where ω ≫ ω_op the dependence of ω and γ on q are similar. Using Eq. (20), we obtained the plasmonic mode loss spectrum in the q-ω plane as illustrated in Fig. 1, together with a color bar indicator. The solid white curve in this figure shows how the plasmons peaks’ frequency varies with the wavenumber (Eq. (21)). In these calculations, we considered a graphene layer with a chemical potential of μ_C = 0.3 eV [16] transferred on a diamond-like-carbon (DLC) substrate with a dielectric constant of ε_DLC = 5 (i.e., ε_env = 3). Moreover, the green dots represent the classical 2D dispersion (i.e.,γ = 0), and the dashed lines indicate the boundaries of regions with allowed damping processes. The data obtained from Eq. (21) is in excellent agreement with those obtained experimentally shown in [16], within the entire q-ω plane, unlike the classical 2D dispersion. Further key features observed from this figure are: (i) As plasmons frequency reaches the optical phonons’ frequency (i.e., the horizontal white dashes), the plasmons start to damp into the PE continuum. (ii) While the plasmonic mode decaying by the phonon emission at high frequencies, the excitation of an interband single-particle enhances the damping process (i.e., within the SPE_inter region). (iii) The plasmons’ peak frequency moves away from the SPE_itnra region as q increases.

Fig. 1. Plasmons modes loss spectra (Eq. (20)) in graphene with μ_C = 0.3 eV on a DLC substrate. The white solid curve represents the corresponding peak frequency plasmons wavenumber (Eq. (21)). The dashed lines show the allowed damping boundaries. The green dots represent the classical 2D dispersion (i.e., γ = 0).

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3.1.2 Polar substrates

The S-POPs of a polar substrate with λ ( = 1, 2, or 3) dominant vibrating modes split the graphene surface plasmons into λ+1 branches of coupled plasmonic modes. To find the exact dispersion relation for every allowed coupled mode, we set Eq. (13) zero, and after some algebraic manipulations, obtain the intended dispersion relations for graphene on a given polar substrate. For a substrate with a single S-POP, Eq. (21) reduces to

(22a)$$\begin{aligned} \omega _{1,2}^{\textrm{pl - ph}}({q,{\mu_\textrm{C}}} )&= \Omega \left\{ {1 + \frac{1}{{2{\Omega ^2}}}({\omega_1^2 + {\gamma^2}} )} \right. \mp \left[ {1 + \frac{1}{{4{\Omega ^4}}}{{({\omega_1^2 - {\gamma^2}} )}^2}} \right.\\ &{\left. {{{\left. { + \frac{1}{{{\Omega ^2}}}\left( {\frac{{8\pi {{\cal F}}_1^2}}{\hbar }{\omega_1} - ({\omega_1^2 - {\gamma^2}} )} \right)} \right]}^{1/2}}} \right\}^{1/2}}\end{aligned}$$

where ω₁^pl-ph and ω₂^pl-ph(>ω₁^pl-ph) represent the peak frequencies of the two coupled plasmon-phonon modes and

(22b)$${\Omega ^2} = \frac{{q{\mu _\textrm{C}}{e^2}}}{{4\pi {\varepsilon _0}{\varepsilon _{\textrm{env}}}{\hbar ^2}}},$$

which is a function of q and μ_C.

We substituted Eq. (13) in Eq. (20) and solved it to obtain the loss spectra for the hybrid plasmon-phonon modes on the surface of graphene with μ_C = 0.57 eV [12] placed on the surface of a SiC substrate with only one dominant S-POPs of energy ħω₁ = 116 meV and the dielectric constant of ε_SiC= 6.5 in the q-ω plane, as illustrated in Fig. 2.

Fig. 2. Hybrid modes loss spectra (Eq. (20)) in graphene with μ_C = 0.57 eV on a SiC substrate with a single POP-vibrating mode. The two white solid curves represent the corresponding peak frequencies versus the plasmons wavenumber (Eq. (22)). Inset depicts a zoomed-in view of the plots about S-POPs energy (ħω₁/μ_C ≈ 0.2). The dashed lines show the allowed damping boundaries.

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The two solid white curves shown in this figure depict the dependencies of the peak frequencies of the hybrid modes on the plasmons wave number (Eq. (22)). The white dashes here represent the boundaries of regions with allowed damping processes. Comparison of these results with similar experimental results presented in Ref. [12] confirms excellent agreements between the two. The inset shows a zoomed-in view of the plots around the splitting energy (ℏω₁ / μ_C ≈ 0.2), assisting visualization. We can easily observe that far from this energy, where S-POPs have negligible effects on the plasmonic modes, and hence the coupled modes peak frequencies (Eq. (22)) and plasmonic modes (Eq. (21)) coincide, showing solely electrons participate in the excitations. As $\hbar \omega _{1,2}^{\textrm{pl - ph}} \to \hbar {\omega _1}$ the plasmon-phonon hybridization prevails. Another distinct key feature observed in this figure is around the S-POPs excitation energy. The lower mode (i.e., the vibrational mode) starts saturating within the SPE_intra continuum, and the higher one (i.e., the electronic mode) starts moving away from the SPE_intra region. Further observations indicate that as q increases, the weight of the mode intensity transfers from the lower mode to the higher one and enhances until it reaches the onset of the PE continuum, beyond which it damps similar to what has been observed experimentally [12,16]. Moreover, as the plasmon's energy increases beyond that of the graphene intrinsic phonon (i.e., the horizontal white dashes), in addition to damping, the higher mode broadens. When the plasmons wavenumber and energy enter the SPE_inter continuum, the corresponding mode damps more rapidly because of single-particle excitations in addition to the phonon emission.

When graphene is on any polar substrate with two fundamental S-POPs modes, we can obtain the peak frequencies for all three hybrid coupled plasmon-phonon modes by equating Eq. (13) to zero for λ = 1, 2,

(23a)$$\omega _{1,2}^{\textrm{pl - ph}} = {\textrm{Re}} \sqrt {\alpha - \frac{1}{2}({\beta + {\chi / \beta }} )\pm i\frac{{\sqrt 3 }}{2}({\beta - {\chi / \beta }} )} ,$$

and

(23b)$$\omega _3^{\textrm{pl - ph}} = {\textrm{Re}} \sqrt {\alpha + ({\beta + {\chi / \beta }} )} ,$$

where ω₁^pl-ph < ω₂^pl-ph < ω₃^pl-ph and α, β, and χ are as given in Appendix A.

Solving Eq. (20) we obtain the loss spectra for the three coupled plasmon-phonon modes on graphene with μ_C = 0.3 eV, as illustrated in Fig. 3. As can be seen from this figure, the behavior of the three hybrid coupled plasmon-phonon modes in the allowed damping continua is similar to the behavior of the hybrid modes shown in Fig. 2. For ħωs far away from ħω_{1, 2}, where S-POPs become ineffective, the modes become purely plasmonics. Nonetheless, as ħω → ħω_{1, 2}, the dispersions change dramatically, as observed in the inset. Moreover, as q increases gradually from zero, the weight of the mode intensity moves from the first mode to the second and then from the second to the third mode, and enhances until it reaches the onset of the PE continuum, beyond which it broadens and damps into SPE_intra faster than before.

Fig. 3. Hybrid modes loss spectra (Eq. (20)) in graphene with μ_C = 0.3 eV on an AlN substrate with two S-POPs vibrating modes. The three solid white curves represent the traces of the corresponding peak frequencies (Eq. (23)). Inset depicts a zoomed-in view of the plots around the two S-POPs excitation energies (ħω_1,2/μ_C ≈ 0.28 and 0.35). The dashed lines show the allowed damping boundaries.

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Next, we obtain the peak frequencies for hybrid coupled plasmon-phonon modes induced on the surface of graphene placed on a polar substrate with three S-POPs. In doing so, we equate Eq. (13) to zero for λ = 1, 2, and 3, and after some algebraic manipulations get the formula for the peak frequencies,

(24a)$$\omega _{1,2}^{\textrm{pl - ph}} = {\textrm{Re}} \sqrt {\Delta - {{{\Phi ^{{1 / 2}}}({1 \pm {{{\Gamma ^{1/2}}} / {{\Phi ^{3/4}}}}} )} / {6{\Lambda ^{{1 / 6}}}}}} ,$$

(24b)$$\omega _{3,4}^{\textrm{pl - ph}} = {\textrm{Re}} \sqrt {\Delta - \Phi {{^{{1 / 2}}({ - 1 \pm {{{{\rm K}^{1/2}}} / {{\Phi ^{3/4}}}}} )} / {6{\Lambda ^{{1 / 6}}}}}} .$$

Parameters Δ, Φ, Γ, Κ, and Λ in Eq. (24) are introduced in Appendix B.

Now, by substituting Eq. (13) into Eq. (20), presuming the polar substrate is SiO₂ with three dominant phonons at the energies ħω_{1, 2, and 3} = 100, 134, and 148.8 meV and the dielectric constant of ε_SiO2 = 2.4, we obtain the loss spectra for the hybrid coupled plasmon-phonon modes on the graphene with μ_C = 0.5 eV [16]. Figure 4 shows the four dispersive hybrid modes in the q-ω plane. The four solid white curves represent the corresponding modes’ peak frequencies. The results plotted here are in excellent agreement with the experimental data presented in [16]. Moreover, this figure displays all key features mentioned for polar substrates with one and two surface vibrational excitations.

Fig. 4. Hybrid modes loss spectra (Eq. (20)) in graphene with μ_C = 0.43 eV on a SiO2 substrate with tree S-POPs vibrating modes. The solid white curves represent the traces of the corresponding peak frequencies (Eq. (24)). Inset depicts a zoomed-in view of the plots around the three POPs excitation energies (ħω_1,2,3 /μ_C ≈ 0.233, 0.312, and 0.346). The dashed lines show the allowed damping boundaries.

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All data shown thus far, except for those shown in Fig. 3, are obtained from the dispersion relation, Eq. (13), we have modeled using the existing experimental data [12,16,30], via modifying the electron polarizability, Π₀, as given in Eq. (14). Only for the substrate with two dominant vibrational frequencies, the experimental data are not available. Nonetheless, we presumed our accurate model for the cases with none, one, and three vibrational modes can also be precise for the substrate with two vibrational modes.

The scattering rate of the electrons by phonons is proportional to ω_λ. Hence, electrons track slower vibrations. That means the scattering rate decrease as the speed of motion increases. In substrates with four or more phonons ω_λ≥4 is high enough for neglecting their scattering effect on the graphene surface plasmons.

3.2 Plasmonic content

For calculating the output current of graphene-based optoelectronic devices, such as plasmonic photodetectors [30], placed on a polar substrate, we need to know the pure plasmonic content of the hybrid coupled plasmon-phonon modes, we have discussed in the earlier subsection. Recall that for a polar substrate with n vibrating modes on the surface, the number of hybrid coupled modes is n + 1 (Eq. (22)-Eq. (24)). The plasmonic content for either of the given dispersion mode is given by [28,30],

(25)$${\Theta _i} = {{\prod\limits_{\lambda = 1}^n {({\omega_{\textrm{pl - ph,}i}^2 - \omega_\lambda^2} )} } / {\prod\limits_{j \ne i = 1}^{n + 1} {({\omega_{\textrm{pl - ph,}i}^2 - \omega_{\textrm{pl - ph,j}}^2} )} }}.$$

Now, we substitute Eqs. (22)-(24) into (25) to obtain the pure plasmonic content related to the dispersion modes in graphene placed on any given polar substrate. The dots, dashes, dotted-dashes, and solid curves in Fig. 5(a), (5(b)), and 5(c) illustrate the numerical data for the plasmonic content for the three substrates (i.e., SiC, AlN, and SiO₂), using the same chemical potential for graphene as before. The solid curve and dashes, in each part, represent the behavior of the plasmonic contents for the first and second dispersive modes on the respective polar substrate. The dots-dashes in part (b) or (c) represent the plasmonic content of the third mode on the AlN or SiO₂ substrate, whereas the dots only shown in part (c) display the content of the fourth mode on graphene placed on SiO₂ substrate. The solid circles, diamonds, and squares in Fig. 5(c) represent the existing experimental data for the plasmonic content of modes ω₁^pl-ph, ω₂^pl-ph, and ω₄^pl-ph when graphene is on SiO₂ substrate (extracted from [30]), demonstrating excellent agreements with our developed model. The inset in each part depicts the corresponding dispersion modes zoomed in about the S-POPs excitation energies ħω_λ / μ_C.

Fig. 5. Plasmonic content of the hybrid modes in graphene on (a) SiC (b), AlN, and (c) SiO₂ substrates. All conditions are the same as those used for Figs. 2, 3, and 4. Blue circles, magenta diamonds, and green squares in (c) represent the existing experimental data (extracted from [30]) for the plasmonic contents of the first, second, and fourth modes when graphene is on SiO₂ substrate. Insets display the corresponding coupled-mode dispersions.

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As can be observed from this figure, for small qs, (i.e., q/k_F ≪ 0.05), where ℏω₁^pl-ph≪ ℏω₁, the corresponding plasmon content for each of the three cases shown in this figure is mainly plasmonic. As q/k_F increases and ℏω₁^pl-ph→ℏω_1, the plasmonic content of the first mode starts transferring to the second mode (see Fig. 2), and its content becomes predominantly vibrational for q/k_F > 0.1. Contrary to the first mode, the second hybrid mode content, at q/k_F ≪ 0.05 for all three given substrates, is dominantly vibrational, and as q increases, the plasmonic content increases. Although this increase for SiC continues until it becomes predominantly plasmonic for large qs, for the other two substrates, the content maximizes as ω₂^pl-ph→ ω₂ and then starts to drop fast and hence becomes predominantly vibrational again. The dots-dashes show that the plasmonic content of the third mode for the polar substrate with two vibrational modes behaves like that of the second mode on the substrate with only one S-POPs mode (Fig. 5(b)), whereas the behavior of the one for the polar substrate with three vibrational excitations is somewhat similar to the content of the second hybrid mode for the substrate with two S-POPs modes (5(c)). Similar to those for the second mode on SiC and the third mode on AlN substrates, the plasmonic content of the fourth mode on SiO₂ substrate grows as q increases until it becomes predominantly plasmonic (q / k_F > 0.2). Moreover, as can be observed from Fig. 5(c), the peak content for the third mode is about an order of magnitude smaller than that of the second mode. That is because of the (ω₃ − ω₂) << (ω₄ − ω₃).

3.3 Hybrid mode lifetime

Here, we model the coupled plasmon-phonon mode lifetime, τ_c, which depends on both electrons’ and phonons’ relaxation times (τ_pl and τ_POP). Hence, to obtain τ_c, we need to know the plasmonic and phononic contents. Knowing the plasmonic content for each of the three exemplar substrates, Θ_i(%) from Eq. (25) and the corresponding τ_pl and τ_POP from (15) and (19), we can deduce the coupled-mode lifetime as

(26)$${\tau _{\textrm{c,i}}} = {\Theta _\textrm{i}}{\tau _{\textrm{pl}}} + ({1 - {\Theta _\textrm{i}}} ){\tau _{\textrm{POP}}}.$$

The dots, dashes, dotted-dashes, and solid curves in Fig. 6 illustrates the results of our calculations for the lifetimes of the hybrid coupled modes versus wavenumber q on graphene placed on either of the three exemplar substrates — i.e., SiC (Fig. 6(a)), AlN (Fig. 6(b)), and SiO₂ (Figs. 6(c)). Consistent with data illustrated in Fig. 5, for small wavenumbers (i.e., q / k_F ≪ 0.05), where the first mode (ℏω₁^pl-ph) is predominantly plasmonic, the corresponding hybrid coupled-mode lifetime for each of the three examples is τ_c1 ≈ τ₀ = 85 fs, indicating the dominance of the impurity scattering in that region. As q and ℏω₁^pl-ph increase, τ_c1 decreases due to the edge scatterings until the mode’s content becomes almost vibrational when ℏω₁^pl-ph→ ℏω₁. Then, any further increase in q, while ℏω₁^pl-ph is saturating just below ℏω₁, the phononic content gradually increases (Fig. 5), and τ_c1 starts to rise until the dispersive mode damps into the SPE_intra continuum (Figs. 2–4), beyond which τ_c1 decreases rapidly. Concerning the plasmonic content of the higher hybrid modes in either of the three examplar cases, they all start at their corresponding phonon excitation frequencies for very small wavenumbers — i.e., they are predominantly vibrational (τ_ci(i≥2) ≈ τ_POP). As q and energy increase, the hybrid mode lifetimes approach the plasmons lifetimes (τ_{pl, i(≥2)}). Further increase in q increases the edge scattering lifetime (τ_ed), τ_PE, and τ_SPE, reducing the total hybrid mode lifetime. The solid circles in Fig. 6(c) represent the existing experimental data for the lifetime of the fourth mode (ω₄^pl-ph) when graphene is on SiO₂ substrate (extracted from [16]), demonstrating an excellent agreement with our developed model.

Fig. 6. The lifetimes of the hybrid modes in graphene on (a) SiC (b), AlN, and (c) SiO₂. All conditions are the same as those used for Figs. 2, 3, and 4. Green squares in (c) represent the existing experimental data (extracted from [16]) for the lifetime of the fourth mode when graphene is on SiO₂ substrates. Insets display the corresponding coupled-mode dispersions.

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As depicted by dashes in Fig. 6(b) and 6(c) and dots-dashes in Fig. 6(c), at intermediate frequencies, the behavior of the corresponding lifetimes is readily comprehensible by considering the vibrational contents. At wavenumbers for which the hybrid modes are out of the SPE_intra continuum, an increase in the phononic content (1−Θ) increases the total lifetime. As the hybrid modes enter the intraband damping region, the lifetime decreases even faster to a few fs.

4. Conclusion

Using the random phase approximation method, we have derived an exact dispersion relation for the plasmonic mode on the graphene surface placed on a nonpolar substrate. Our numerical results show that the graphene surface plasmons on a DLC substrate for large wavenumbers vary as ω (q) $\propto$ q^3/2, dissipating in the SPE_inter continuum and away from the SPEintra continuum consistent with the experimental measurements published by others. On the contrary, the classical 2D dispersion formula, derived by others, shows large wavenumbers ω $\propto$q^1/2, entering the SPE_intra continuum. Along the same line, using the frequency of each S-POPs mode on the surface of the polar dielectric substrates with λ (=1, 2, or 3) vibrational excitations, we have derived exact dispersion relations for λ+1 hybrid coupled plasmon-phonon modes. Considering graphene monolayers placed on polar substrates like SiC with one, AlN with two, and SiO₂ with three S-POPs vibrational frequencies, we have calculated the loss spectra and dispersion relations for all allowed hybrid plasmon-phonon modes in graphene. These results are in excellent agreement with the existing experimental data obtained by others. Moreover, we have found the plasmonic content of each hybrid mode and the related lifetime versus q. Employing these results, one can readily predict the output currents of optoelectronic devices based on graphene plasmonics made on polar and nonpolar dielectric substrates.

Appendix A: parameters of Eq. (23)

Parameter α, β, and χ given in Eq. (23) are

(27)$$\alpha = \frac{2}{3}{\Omega ^2} + \frac{1}{3}\left( {{\gamma^2} + \sum\limits_{\lambda = 1}^2 {\omega_\lambda^2} } \right),$$

(28)$$\beta = \sqrt[3]{{\delta + \sqrt {{\delta ^2} - {\chi ^3}} }},$$

and

(29)$$\chi = {\alpha ^2} - \eta ,$$

wherein

(30)$$\delta = \alpha \left( {\chi - \frac{1}{2}\eta } \right) + \phi ,$$

(31)$$\eta = \frac{2}{3}{\Omega ^2}\left( {\frac{{ - 4\pi }}{\hbar }\sum\limits_{\lambda = 1}^2 {{{\cal F}}_\lambda^2{\omega_\lambda }} + \sum\limits_{\lambda = 1}^2 {\omega_\lambda^2} } \right) + \frac{1}{3}\left( {{\gamma^2}\sum\limits_{\lambda = 1}^2 {\omega_\lambda^2} + \prod\limits_{\lambda = 1}^2 {\omega_\lambda^2} } \right),$$

and

(32)$$\phi = \left[ {{\Omega ^2}\left( {\frac{{ - 4\pi }}{\hbar }\sum\limits_{\scriptstyle\lambda ,\mu = 1\atop \scriptstyle\lambda \ne \mu}^2 {{{\cal F}}_\lambda^2{\omega_\mu }} + \prod\limits_{\lambda = 1}^2 {\omega_\lambda^2} } \right) + \frac{1}{2}{\gamma^2}\prod\limits_{\lambda = 1}^2 {\omega_\lambda^2} } \right]\prod\limits_{\lambda = 1}^2 {\omega _\lambda ^2} .$$

Appendix B: parameters of Eq. (24)

We show the details of parameter A to F given in Eq. (24) here,

(33)$$\Delta = {{\left( {({2{\Omega ^2} + {\gamma^2}} )+ \sum\limits_{\lambda = 1}^3 {\omega_\lambda^2} } \right)} / 4},$$

(34)$$\Gamma ={-} 54{\kappa _2}{\Lambda ^{1/2}} - 18{\kappa _1}{\Lambda ^{{1 / 3}}}{\Phi ^{{1 / 2}}} - {\Phi ^{3/2}},$$

(35)$$\Phi = {\kappa _1}({{\kappa_1} - 6{\Lambda ^{1/3}}} )- 12{\kappa _3} + 9{\Lambda ^{{2 / 3}}},$$

(36)$${\rm K} ={+} 54{\kappa _2}{\Lambda ^{1/2}} - 18{\kappa _1}{\Lambda ^{{1 / 3}}}{\Phi ^{{1 / 2}}} - {\Phi ^{3/2}}.$$

(37)$$\Lambda = {{{{\kappa _1^3} / {27}} + \kappa _2^2} / 2} + {{4{\kappa _1}{\kappa _3}} / 3} + {{{{({3{\kappa_4}} )}^{{1 / 2}}}} / {18}},$$

The parameters κ_j (j = 1, 2, 3, 4) in (B1)-(B5) are,

(38)$${\kappa _1} ={-} 6{\Delta ^2} + {\xi _1},$$

(39)$${\kappa _2} = 8{\Delta ^3} - 2\Delta {\xi _1} + {\xi _2} + {\xi _3},$$

(40)$${\kappa _3} = 3{\Delta ^4} - {\Delta ^2}{\xi _1} + \Delta ({{\xi_2} + {\xi_3}} )+ {\xi _4},$$

(41)$${\kappa _4} = \kappa _2^2({27\kappa_2^2 + {\kappa_1}({4\kappa_1^2 + 144{\kappa_3}} )} )+ \kappa _1^2{\kappa _3}({16\kappa_1^2 + 128{\kappa_3}} )+ 256\kappa _3^3.$$

Parameters ξ_j (j = 1, 2, 3, and 4) in (B6)-(B9) are

(42)$${\xi _1} = \sum\limits_{\lambda = 1}^3 {\left[ { - \Omega _\lambda^4 + ({2{\Omega ^2} + {\gamma^2}} )\omega_\lambda^2 + \frac{1}{2}\sum\limits_{\mu \ne \lambda = 1}^3 {\omega_\lambda^2} \omega_\mu^2} \right]} ,$$

(43)$${\xi _2} ={-} \frac{1}{2}\sum\limits_{\scriptstyle\lambda ,\mu ,\nu = 1\atop \scriptstyle\lambda \ne \mu \ne \nu }^3 {\Omega _\lambda ^4({\omega_\mu^2 + \omega_\nu^2} )} ,$$

(44)$${\xi _3} = \frac{1}{2}({2{\Omega ^2} + {\gamma^2}} )\sum\limits_{\scriptstyle\lambda ,\mu = 1\atop \scriptstyle\lambda \ne \mu }^3 {\omega _\lambda ^2} \omega _\mu ^2 + \prod\limits_{\lambda = 1}^3 {\omega _\lambda ^2} ,$$

(45)$${\xi _4} = \frac{1}{2}\sum\limits_{\lambda \ne \mu \ne \nu = 1}^3 {\Omega _\lambda ^4\omega _\mu ^2\omega _\nu ^2} - ({2{\Omega ^2} + {\gamma^2}} )\prod\limits_{\lambda = 1}^3 {\omega _\lambda ^2} ,$$

where

(46)$$\Omega _\lambda ^4 = \frac{{8\pi {\Omega ^2}}}{\hbar }\,{{\cal F}}_\lambda ^2{\omega _\lambda },\quad \textrm{with }\lambda = 1,2,3$$

Funding

Tarbiat Modares University (IG-39703).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Material	Parameters
Material	ε ₀	ε _i,1	ε _i,2	ε _∞	ħω₁ (meV)	ħω₂ (meV)	ħω₃ (meV)	Φ²₁ (meV)	Φ²²₂ (meV)	Φ²₃ (meV)	Ref.
SiC	9.7	−	−	6.5	116.0	−	−	0.735	−	−	[12,34]
Al₂O₃	12.53	7.27	−	3.20	56.1	110.1	−	0.420	2.053	−	[28]
HfO₂	22.0	6.58	−	5.03	21.6	54.2	−	0.304	0.293	−	[28]
AlN	9.14	7.35	−	4.80	83.60	104.96	−	0.282	0.88	−	[28]
ZrO₂	24.0	7.75	−	4.00	25.02	70.80	−	0.296	0.966	−	[28]
ZrSiO₄	11.75	9.73	−	4.20	38.62	116.00	−	0.908	1.830	−	[28]
hBN	5.09	4.575	−	4.10	101.7	195.7	−	0.258	0.520	−	[20,34]
SiO₂	3.90	3.65	3.55	2.40	100	134	144.8	0.107	0.101	1.7	[16,28,30]

Diagram	Description
$⇑_{k_{1}}^{k_{2}}$	The electron propagates through the system under potential and its state changes from k₁ to k₂.
$↑_{k_{1}}^{k_{2}}$	The electron propagates freely (k₁ = k₂).
${Π_{0}}_{↑ k_{1}}^{↑ k_{2}}$	The electron propagates under potential in the system and is scattered by a phonon, the electron polarizability is Π_0, and its state changes from k₁ to k₂.

Material	Parameters
Material	ε ₀	ε _i,1	ε _i,2	ε _∞	ħω₁ (meV)	ħω₂ (meV)	ħω₃ (meV)	Φ²₁ (meV)	Φ²²₂ (meV)	Φ²₃ (meV)	Ref.
SiC	9.7	−	−	6.5	116.0	−	−	0.735	−	−	[12,34]
Al₂O₃	12.53	7.27	−	3.20	56.1	110.1	−	0.420	2.053	−	[28]
HfO₂	22.0	6.58	−	5.03	21.6	54.2	−	0.304	0.293	−	[28]
AlN	9.14	7.35	−	4.80	83.60	104.96	−	0.282	0.88	−	[28]
ZrO₂	24.0	7.75	−	4.00	25.02	70.80	−	0.296	0.966	−	[28]
ZrSiO₄	11.75	9.73	−	4.20	38.62	116.00	−	0.908	1.830	−	[28]
hBN	5.09	4.575	−	4.10	101.7	195.7	−	0.258	0.520	−	[20,34]
SiO₂	3.90	3.65	3.55	2.40	100	134	144.8	0.107	0.101	1.7	[16,28,30]

Diagram	Description
$⇑_{k_{1}}^{k_{2}}$	The electron propagates through the system under potential and its state changes from k₁ to k₂.
$↑_{k_{1}}^{k_{2}}$	The electron propagates freely (k₁ = k₂).
${Π_{0}}_{↑ k_{1}}^{↑ k_{2}}$	The electron propagates under potential in the system and is scattered by a phonon, the electron polarizability is Π_0, and its state changes from k₁ to k₂.

Exact dispersion relations for the hybrid plasmon-phonon modes in graphene on dielectric substrates with polar optical phonons

Abstract

1. Introduction

2. Theory

2.1 Intrinsic optical phonons

2.2 Extrinsic surface optical phonons

2.3 Dielectric function

2.4 Lifetime calculations

2.4.1 Plasmon lifetimes

2.4.2 Phonon lifetime

3. Results and discussion

3.1 Plasmon dispersion

3.1.1 Nonpolar substrates

3.1.2 Polar substrates

3.2 Plasmonic content

3.3 Hybrid mode lifetime

4. Conclusion

Appendix A: parameters of Eq. (23)

Appendix B: parameters of Eq. (24)

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (6)

Tables (2)

Equations (58)

Optics Express

Sara Darbari	https://orcid.org/0000-0002-3769-0215
Mohammad Kazem Moravvej-Farshi	https://orcid.org/0000-0002-8954-6418