1. Introduction

Graphene-based optical devices gained the interest of many researchers due to their exciting applications, such as chemical sensing, optical sensing spectroscopy, surface communication, and near-field imaging [1–6]. Graphene is an atomically thick allotrope of carbon that has extraordinary optical, chemical, mechanical, electrical and thermal properties [7]. Graphene plays important role in enabling the THz optics due to its unprecedented control and manipulation on the THz to IR frequency range [8–10]. Different research studies have been carried out on the graphene-based optical devices i.e., Liu et al., designed a graphene-based flexible broadband optical modulator having the bandwidth of 1.35–1.6 mm and operational speed 1.2 GHz and compared its operational working with the traditional semiconductor devices [11]. Mueller et al., fabricated the graphene-based field effect transistor (FETs) as photodetector for the fast detection and communication up to the up to data link of 10 Gb/s. To deal with the limitation regarding the mirror symmetry, the asymmetric metallization scheme was adopted [12]. Xing et al., fabricated the graphene-based optical sensor for the biochemical sensing of analyte/ cell lines/ carcinoma cells. The polarization dependent absorption under total internal reflection (TIR) condition based working scheme for the optical sensor has been adopted. The sensitivity and resolution of the designed reflective index-based optical sensor has been discussed for the two approaches i.e., surface plasmon resonance (SPR) and fiber approach, the highest sensitivity and resolution is reported as 4.3 × 107 mV/RIU and 1.7 × 10⁻⁸, respectively [13]. Pianelli et al., theoretically investigated the switchable reflection characteristics of graphene based hyperbolic metamaterial under elliptic and type-II dispersion laws and reported that the plasmonic resonance and reflection modulation can be tuned by chemical potential and number of graphene layers [14]. Dudek et al., carried out the theoretical study on the graphene based hyperbolic microcavity for the tunable transmission characteristics in the mid IR region as tunable mode resonator, intensity controller and transmission modulator via gate voltage [15]. Further, to focus the THz waves, graphene-based THz lenses, THz polarizers, and THz shifters have been studied [16–18].

Recently, the scientific community is dealing with the one of the major challenge in the graphene-based optical devices i.e., the ineffective thermal response/ sensing of the graphene [19]. Due to the single-atom thickness and low charge carrier density, the graphene is inactive against the external temperature variation [19]. To enhance the thermal response of graphene-based devices, the temperature-sensitive materials (TSM) can play an active role. The material having the temperature dependent optical or electromagnetic characteristics are called the temperature sensitive materials (TSM), vanadium dioxide (VO₂), indium antimonide (InSb), 3C-SiC and liquid crystals, are some typical examples of TSMs, which shows the temperature dependent insulator-metal transition upon the temperature variation [20–23].

Graphene loaded TSMs-based devices have potential applications in thermal rectifiers, thermo-optical devices, and thermal sensors, temperature assisted nearfield platforms and thermal imaging devices. Different studies have been carried on such thermo-optical characteristics i.e., Xu, et al., carried out the theoretical study on the theoretical near field thermal rectification among the graphene coated InSb terminal and graphene coated 3C-SiC terminal and reported that the thermal rectification efficiency is far greater for the graphene coated terminals as compared to the bare terminals. further, it is concluded that the enhancement in the rectification efficiency is due to the near field surface plasmon modes between the graphene-InSb, which enhance the heat exchange in the emitter terminal and thermal rectification efficiency increases [24]. Further, Wang et al., extended their work and carried out the radiative and heat flux characteristics of heterostructure based upon the two different terminals i.e., graphene coated InSb terminal and graphene coated 3C–SiC-nanowire terminal. It is computed that the thermal rectification efficiency increase for the graphene loaded TSMs terminal as compared to the bare terminals [25]. Moreover, it is inferred the graphene loaded InSb and 3C-SiC nanowires support the surface plasmon polaritons (SPP) modes [25]. He et al., theoretically investigated the graphene-based thermal diode composed of the three parallel slabs i.e., graphene coated hot slab-VO₂-graphene coated cold slab and reported that the graphene greatly improve the rectification efficiency of the thermal diode. The excitation of the SPP modes near to the graphene coated slabs has been presented for the forward and reverse scenario. The influence of chemical potential and slab spacing on characteristics of SPP modes and thermal rectification factor has been analyzed, and concluded that the thermal rectification factor increases 300% with the slab spacing 350 nm of the thermal diode [26]. To get the more insight interaction between the graphene loaded TSMs, the Yaqoob et al., analyzed the thermally tunable electromagnetic waves supported by the graphene loaded InSb semi-infinite slab. The numerical results are compared for the single layer graphene and the graphene loaded InSb, and reported that pristine graphene does not show any variation in the wave characteristics against the different temperatures while the dispersion curve, effective mode index, penetration depth, propagation length, and field profile are highly sensitive to the temperature and chemical potential [27].

Keeping in view of these works, the theoretical modeling of the graphene-based temperature sensitive metafilm has been presented in the present work. The temperature sensitive metafilm has been considered as graphene bilayer structure filled with the TSM (InSb). The present work has been conducted to achieve the following objectives: (i) to enhance the thermo-optical response of the graphene-based temperature sensitive metafilm, (ii) to study the temperature assisted waveguide surface modes supported by the metafilm, (iii) to enhance the thermal response of the InSb by graphene encapsulation, and (iv) to conduct the parametric study on propagation characteristics for the surface modes. The manuscript is organized as: the analytical methodology regarding the modeling of the graphene-based temperature sensitive metafilm is discussed in theoretical formulation section, while numerical results of the analytically computed results have been provided in the Section 3. The concluding remarks have been given in Section 4.

2. Theoretical formulation

The theoretical formulation for propagation of electromagnetic surface waves supported by graphene-based temperature sensitive metafilm is given in this section. The geometry of the temperature sensitive metafilm comprised of bi-layered graphene parallel structure filled with temperature-sensitive material, as depicted in Fig. 1. The structure of metafilm has been taken along z-axis i.e., the first layer of graphene with optical conductivity (${\sigma _{g1}}$) has been considered at z = 0 and second layer with optical conductivity (${\sigma _{g2}}$) has been taken at $z = d$, while the region $0 < z < d$ is supposed to filled with temperature-sensitive material i.e., indium antimonide (InSb) with tempter dependent permittivity ${\varepsilon _{Insb}}({\omega ,T} )$. However, the remaining space has been considered as free space.

 

Fig. 1. Graphene-based temperature sensitive metafilm for surface modes.

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The propagation of electromagnetic surface modes associated by the graphene-based temperature-sensitive metafilm has been considered along the x-axis, as depicted in Fig. 1. The propagation characteristics of electromagnetic surface modes are highly sensitive to the state of the polarization, like transverse electric polarization or transverse magnetic polarization. This is because the transverse electric polarization has low confinement compared to the transverse magnetic polarization [28]. Therefore, in present study, the analytical solution for the electromagnetic modes propagating along the graphene-based temperature-sensitive metafilm filled with indium antimonide has been presented for only transverse magnetic polarization. To keep the mathematical formulation simple, the optical conductivities of the both layers of graphene have been kept same i.e., ${\sigma _{g1}} = {\sigma _{g2}} = {\sigma _g}$. In all analytical calculations, the time harmonicity is taken as ${e^{ - i\omega t}}$. The field equations for transverse electromagnetic polarized surface waves modes are given below [29]:

However, the electric fields according to Eqs. (2) and (3) are computed as

To check the existence of the surface modes on the surface of graphene-based temperature-sensitive metafilm, the following boundary conditions have been employed on the fields for $z = 0$ and $z = d$, [27,29,30]

In the above equations the terms ${e^{ - {\gamma _{1,2}}z}}$ can be expanded in terms of hyperbolic functions i.e., $\cos h\textrm{}({e^{{\gamma _{1,2}}z}})$ or $\sin h\textrm{}({e^{{\gamma _{1,2}}z}})$. As the $\sin h\textrm{}({e^{{\gamma _{1,2}}z}})$ exhibits the odd modes (asymmetrical modes) while the $\cos h\textrm{}({e^{{\gamma _{1,2}}z}})$ depicts the even modes (symmetrical modes), therefore, the dispersion relations for each mode have been computed viz., the dispersion relation for odd modes:

3. Numerical results and discussion

This section deals with the numerical results for the electromagnetic surface modes under even and odd distribution scheme supported by the graphene-based temperature-sensitive metafilm. In the first part, the electromagnetic frameworks for the modeling of graphene and indium antimonide (InSb) have been discussed while in the second part, the numerical results regarding the even and odd surface modes supported by the temperature-sensitive graphene-based metafilm under the variation of different parameters have been presented.

Modeling of the graphene. The graphene has been modeled as infinitely thin and highly conductive sheet having conductivity (${\sigma _g}$) [27,31]. The optical conductivity of graphene ${\sigma _g}({\omega ,{\mu_c},T,\Gamma } )\textrm{}$ has been modeled as function of incident angular frequency ($\omega $), chemical potential (${\mu _c}$), temperature ($T$) and electron-photon scattering rate ($\Gamma $) in the frame work of Kubo formulism and realized by the following expression as [32]

Modeling of temperature-sensitive material (InSb). The InSb has been realized as temperature-sensitive material. The temperature-dependent electromagnetic modeling of InSb has been done in the frame work of hybrid Drude model as in [25,33]

In above relation, ${m^\mathrm{\ast }}$ is effective mass of free charge carriers (${m^\mathrm{\ast }} = 0.015\mathrm{\ast }{m_e}$), e is the electronic change ($e = 1.6\mathrm{\ast }{10^{ - 19}}\textrm{}C$), ${\varepsilon _o}$ is the permittivity of free space (${\varepsilon _o} = 8.85\mathrm{\ast }{10^{ - 12}}\textrm{}F/m$) and “N(T)” is the temperature-dependent charge carrier density is given as $N(T )= 5.76 \times {10^{20}}\textrm{}{T^{\frac{3}{2}}}\exp\left( { - \frac{{{E_g}}}{{{K_B}T}}} \right)\textrm{}{m^{ - 3}}$, here ${E_g}$ is the energy bandgap and ${K_B}$ is the Boltzmann constant [27,34].

To study the temperature dependence of electromagnetic characteristics of indium antimonide (${\varepsilon _{Insb}}$), the complex relative permittivity of the indium antimonide against the incident frequency is presented in Fig. 2. Figure 2(a) depicts the response of the real part, while the imaginary part is given in Fig. 2(b). It is obvious from Fig. 2(a) that with the increase of temperature $T \in [200,\textrm{}400\textrm{}K$] the real part of permittivity becomes negative, which physically depicts that with the increase of temperature, the charge carrier density (N) increases and indium antimonide is transformed from the insulator to metallic phase [25,27,34].

3.1 Characteristics of surface modes supported by temperature-sensitive metafilm

To get the physical insight of analytically computed results for the surface modes and their dependence on temperature, graphene parameters, and thickness of the TSM film, the numerical simulations were performed in the Mathematica software pack. To solve the characteristics equations for the surface modes, the contour plot analysis was implemented in the kernel. The unknown propagation constant ($\beta $) which satisfies the characteristics Eqs. (16) and (17) for even and odd distribution, respectively, was computed numerically [27,29]. To testify the accuracy of the work, the numerical results are compared with the published literature under special conditions i.e., (i) $d \to \pm \infty $, and (ii) ${\varepsilon _{insb}} = {\varepsilon _o}$ and $\mu = {\mu _o}$. Upon imposing the first condition ($d \to \pm \infty $) on Eqs. (16) and (17), the terms become i.e., $\coth ({{\gamma_2}d} )= 1$ and $\tanh ({{\gamma_2}d} )= 1$ and the Eqs. (16) and (17) become similar, which shows that the even and odd distribution of surface modes are depends upon the planar structure of the metafilm. Further solving the equations under second condition (${\varepsilon _{insb}} = {\varepsilon _o}$ & $\mu = {\mu _o}$), the terms ${\gamma _1} = {\gamma _2} = \sqrt {{\beta ^2} - k_o^2} $, lead to the dispersion relation i.e., $\beta = {k_o}\sqrt {1 - {{\left( {\frac{2}{{{\sigma_g}{\eta_o}}}} \right)}^2}} ,$ ${\eta _o} = 120\pi \mathrm{\Omega }$, for the surface modes supported by the single layer graphene computed in [9,32].

Moreover, the conditions have been applied in the kernel and the numerical results of Eqs. (16) and (17) for the dispersion relation of graphene-based temperature-sensitive metafilm converges to the published literature i.e., dispersion curve for the single layer graphene, which further confirms the accuracy of the numerical simulation of present work, as presented in Fig. 3[9,35].

 

Fig. 2. Temperature-dependent relative permittivity of InSb (${\varepsilon _{insb}}$) as function of frequency (a) real part (b) imaginary part, under values ${\varepsilon _\infty } = 15.68$, $\gamma = 0.1\textrm{}\pi \textrm{THz}$, ${E_g} = 0.26\textrm{}eV$ and ${K_B} = 8.62\mathrm{\ast }{10^{ - 5}}eV{K^{ - 1}}$.

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Fig. 3. Comparison with the published literature under special conditions (i) $d \to \pm \infty $, and (ii) ${\varepsilon _{insb}} = {\varepsilon _o}$, $\mu = {\mu _o}$, $T = 300K$, $\tau = 0.5ps$ & $\mu = 0.2\textrm{}eV$.

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In the subsequent numerical results, the propagation characteristics of surface modes supported by the graphene-based temperature-sensitive metafilm have been discussed. The dispersion curve analysis and effective mode index for even and odd modes have been computed under the variation of chemical potential (${\mu _c}$), temperature (T), and thickness of film (d). In numerical simulation of InSb, the static permittivity ${\varepsilon _\infty } = 15.68$, damping constant $\gamma = 0.1\textrm{}\pi \textrm{THz}$, energy bandgap ${E_g} = 0.26\textrm{}eV$, and Boltzmann’s constant ${K_B} = 8.62 \times {10^{ - 5}}eV{K^{ - 1}}$ were kept fixed.

3.2 Dispersion curve analysis for even and odd surface modes

In this part, the dispersion curve analysis for electromagnetic surface even and odd modes propagating on the surface of graphene-based temperature-sensitive metafilm is presented. The dispersion relation describes the response of electromagnetic waves in the medium against the frequency spectrum. To study the propagation characteristics such as dispersive and non-dispersive nature, group velocity, and phase velocity of waves in a medium, the dispersion curve analysis technique has been conducted [27,29]. The dispersion curves between the frequency and the real part of propagation constant (β) under the variation of temperature (T), chemical potential (${\mu _c}$), and films’ thickness (d) are presented in Figs. 4–6 respectively. In all the results of dispersion curves, the prorogation constant (β) is normalized by the factor 106. To verify the temperature assisted response of the surface modes, the dispersion curve analysis under different values of temperature $T \in [{200K,\textrm{}250K,300K,\textrm{}350K,\textrm{}400K} ]\textrm{}$ is presented in Fig. 4. In Fig. 4(a) the effect of temperature on the dispersion relation for the graphene based metafilm without TSM (InSb) has been presented and it can be inferred that the graphene metafilm is inactive towards the temperature without InSb. The dispersion follows the trend as reported in [36]. However, in Fig. 4(b), graphene based metafilm with TSM (InSb), It is clear that by varying the temperature, the resonance frequency, group velocity, and phase velocity of odd and even modes can be tailored (i.e., with the increase of temperature the propagation frequency starts increasing and the shifting towards the high THz frequency range). Further, it is obvious from the figure that for the low propagation constant value (β) the propagation band gap between the even and odd modes is larger as compared to the high value of propagation constant value (β). The propagation bandgaps between the even and odd modes are highly dependent upon the frequency and temperature. At the high temperature (T > 300 K), the even and odd modes are transformed into degenerate modes. Further, it is important to highlight that the with appropriate designing of the metasurfaces i.e., α-MoO3-SiC based metasurface such thermally active surface modes can be transformed into thermal radiator as reported in [37]. The group speed and phase velocity of the even and odd modes can be estimated by the dispersion curve; thus, it is clear from Fig. 4. that the even mode are slower modes as compared to the odd modes, by increasing the temperature the speed of even and odd modes can be tuned.

 

Fig. 4. Temperature-dependent dispersion curve analysis for even modes (solid lines) and odd modes (solid lines) with parameters i.e., ${\mu _c} = 0.2\textrm{}eV$, $\tau = 0.6ps$, and $d = 20nm.$ (a) graphene based metafilm without InSb with ${\varepsilon _{insb}} = {\varepsilon _o}$ (b) graphene based metafilm with InSb.

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Fig. 5. Dispersion curve analysis for even modes (solid lines) and odd modes (dashed lines) under different values of chemical potential (${\mu _c}$) of graphene metafilm (a) T = 200 K (Insulator Phase) (b) T = 300 K (Metallic Phase) with parameters i.e., $\tau = 0.6ps$ and $d = 20nm$

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In Fig. 5, the influence of chemical potential ${\mu _c} = 0.2\textrm{}eV,\textrm{}0.4\textrm{}eV,\textrm{}0.6eV,\textrm{}0.8eV\textrm{}\& \textrm{}1eV$, on the dispersion curves for even and odd modes has been analyzed. As depicted in Fig. 2, the indium antimonide InSb is temperature-dependent semiconductor, for the low temperature $T < 250K$ it behaves as insulator and for the $T > 250K\textrm{}$ it behaves as metal. Therefore, the effect of chemical on the surface even and odd modes supported by the graphene-based metafilm under both phases of indium antimonide i.e., T = 200 (Insulator) and T = 300 K (Metal) has been depicted in Fig. 5(a) and 5(b) respectively. As the chemical potential µc ${\cong} \sqrt {\mathrm{\pi \;\ }{\hbar ^2}v_f^2} {n_s}$ is associated with the charge concentration density $({n_s})$ of graphene and can be tuned by chemical doping and biasing schemes (voltage gating) [30]. It provides the additional degree of freedom to the graphene as active controller to tune the electromagnetic characteristics in THz range. It is obvious from Fig. 5(a) and 5(b) that with the increase of chemical potential the resonance frequency increases for the both phases of InSb. However, the cut-off resonance frequency has been shifted towards the higher frequency range for the case T = 300 K (metallic phase) as compared to the case T = 200 (Insulator Phase). The physical reason of the shift in the resonance frequency for the metallic phase is the reinforcement in the resonance frequency of the surface plasmons due to the coupling of the metal (InSb) plasmons with the graphene plasmons, as reported by other researchers [25,27].

Like the analysis conducted in Fig. 5, Fig. 6 deals with dispersion curve analysis for the two phases i.e., T = 200 (insulator) and T = 300 K (metallic) of the indium antimonide under different values of the TSM metafilm thickness (i.e., $d\textrm{}\epsilon \textrm{}[{20nm,\textrm{}40nm,\textrm{}60nm,\textrm{}80nm,\textrm{}100nm} ])$. It is obvious from the figure that the surface modes are highly sensitive to the thickness of the TSM metafilm—that is, for the thickness $d = 20nm$ the propagation bandgap between the even mode and odd mode is larger but as the thickness increase the bandgap between even odd modes decreases. Further, it can be inferred that the geometrical parameters of the TSM metafilm can be used for the controlling resonance conditions, excitation, and degeneracy of the even and odd modes as reported in [38].

 

Fig. 6. Dispersion curve analysis for even modes (solid lines) and odd modes (dashed lines) under different values of thickness ($d$) of TSM metafilm (a) T = 200 K (Insulator Phase) (b) T = 300 K (Metallic Phase) with parameters i.e., $\mu = 0.2eV$ and $\tau = 0.6ps$.

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3.3 Effective mode index analysis for even and odd surface modes

The effective mode index of the surface wave measures the conferment of the wave on the propagating surface and mathematically defines as ${N_{eff}} = \frac{{Re[\beta ]}}{{{k_o}}}$ [28,29]. The value of effective mode index is ratio of propagation constant in material ($Re[\beta] $) to the ratio of propagation constant in free space ${k_o}$. As the value of Neff is higher, it is considered as favorable condition for the propagation of surface waves. In this part, the confinement of the surface modes along the graphene-based temperature-sensitive metafilm structure has been analyzed under different parameters: temperature (T), chemical potential (${\mu _c}$) and TSM film thickness (d) and presented in Figs. 7–9 respectively.

 

Fig. 7. Temperature-dependent effective mode index for even modes (solid lines) and odd modes (solid lines) with parameters i.e., ${\mu _c} = 0.2\textrm{}eV$, $\tau = 0.2ps$, and $d = 20nm$

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In Fig. 7 the confinement of surface modes as function of frequency has been analyzed for each value of temperature (i.e., $T \in [{200K,\textrm{}250K,300K,\textrm{}350K,\textrm{}400K} ])$. It is clear that even surface modes have high confinement as compared to the odd surface modes. The propagation bandgap between even-odd surface modes decreases with the increase of temperature while the cut-off frequency starts shifting towards the high-frequency range.

Further, the impact of chemical potential on effective mode index of surface modes supported by the graphene-based TSM metafilm is presented in Fig. 8. The figure is analyzed in terms of two parts: T = 200 K (Insulator) and T = 300 K as Fig. 8(a) and 8(b), respectively. It can be inferred from Fig. 8 that with the increase of chemical potential, the confinement of the surface waves decreases. The major physical reason could be that the high chemical potential value has the high doping level, with losses increasing and the confinement decreasing as a result [27].

 

Fig. 8. Effective mode index analysis for even modes (solid lines) and odd modes (dashed lines) under different values of chemical potential (${\mu _c}$) of graphene metafilm (a) T = 200 K (Insulator Phase) (b) T = 300 K (Metallic Phase) with parameters (i.e., $\tau = 0.5ps$ & $d = 20nm$).

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The graphs shown in Fig. 9 explain about the impact of thickness (d) of the TSM metafilm on the confinement of electromagnetic surface modes supported the graphene-based temperature-sensitive metafilm structure for $d\textrm{}\epsilon \textrm{}[{20nm,\textrm{}40nm,\textrm{}60nm,\textrm{}80nm,\textrm{}100nm} ]$. The graphical results provided in Fig. 9(a) depict the confinement of surface modes for the insulator phase of InSb at T = 200 K. It is clear from the figure, that the even-odd modes are distinctly distributed while for the metallic phase case of InSb at T = 300 K, the even-odd modes are very closely spaced and the propagation band gap is squeezed. However, for the T = 300 K case, the resonance/cutoff frequency has been shifter to higher values as compared to the case T = 200 K. Figure 9 shows that the confinement of the surface modes decreases with the increasing the TSM metafilm thickness (d).

 

Fig. 9. Effective mode index analysis for even modes (solid lines) and odd modes (dashed lines) under different values of thickness ($d$) of TSM metafilm (a) T = 200 K (Insulator Phase) (b) T = 300 K (Metallic Phase) with parameters (i.e., $\textrm{}\mu = 0.2e$ and $\tau = 0.6ps$).

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3.4 Field distribution analysis of even and odd surface modes

To verify the nature of surface modes supported by the graphene based TSM metafilm, the normalized field distributions ($|{E_z}|$)of surface modes propagating along x-axis have been presented in Figs. 10 and 11. Figure 10(a) and 10(b) deals with the comparison between the normalized $|{E_z}|$ even and odd field distribution inside the metafilm for T = 200 K and T = 300 K respectively. In numerical computation, for the T = 200 K the propagation parameters of surface even and odd modes are taken as ($\beta = 21.77 \times {10^6}{m^{ - 1}}$ and $f = 5.02\textrm{}THz$) and ($\beta = 7.47 \times {10^6}{m^{ - 1}}$ & $f = 5.02\textrm{}THz$) respectively, while for the T = 300 K, the values ($\beta = 17.63 \times {10^6}{m^{ - 1}}$ and $f = 5.04\textrm{}THz$) and ($\beta = 5.04 \times {10^6}{m^{ - 1}}$ and $f = 5.04\textrm{}THz$) have been fixed for even and odd modes respectively. It is clear from the Fig. 10 that the field of outer region decays exponentially as the transverse distance (z) increases, which confirms that the fields are depicting the surface waves . Similarly, the filed distributions of $|{E_z}|$ for even and odd mode inside the metafilm have been presented for T = 200 K and T = 300 K in Fig. 11(a) and 11(b) respectively. It can be inferred from these figs that the amplitude of $\textrm{}|{E_z}|$ field for even and odd modes inside the metafilm decays as the wave moves away from the graphene layers, which shows that the propagating modes are the surface modes and qualify the basic criteria i.e., amplitude of the surface waves decays exponentially as the wave go away from the interface . However, it is obvious from these results that the even field distributions have the high intensity as compared to the odd field distribution.

 

Fig. 10. Normalized Field profile for even modes (solid lines) and odd modes (dashed lines) outside the graphene based TSM metafilm with parameters (i.e., $\textrm{}\mu = 0.2e$V, $\tau = 0.6ps$ and $d = 20\textrm{}nm$) (a) T = 200 K (Insulator Phase) (b) T = 300 K (Metallic phase).

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Fig. 11. Normalized Field profile for even modes (solid lines) and odd modes (dashed lines) inside the graphene based TSM metafilm with parameters (i.e., $\textrm{}\mu = 0.2e$V, $\tau = 0.6ps$ and $d = 20\textrm{}nm$) (a) T = 200 K (Insulator Phase) (b) T = 300 K (Metallic phase).

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4 Concluding remarks

A theoretical and numerical investigation on the temperature-assisted electromagnetic surface modes supported by the graphene-based TSM metafilm have been carried out. The analytical modal analysis for the even surface modes and odd surface modes has been conducted. The propagation characteristics for even/dd surface modes have been studied under different parameters: temperature (T), chemical potential (${\mu _c}$)and thickness of metafilm (d). It can be concluded that the surface modes supported by the graphene-based temperature-sensitive metafilm support the surface modes similar to the surface plasmon polariton modes. The graphene-based TSM metafilm structure highly sensitive to the external temperature and the resonance frequency, group velocity, phase velocity and confinement of the surface waves can be tailored by tuning the temperature. Moreover, the chemical potential and thickness of TSM metafilm also plays a vital role in tuning the propagation characteristics of surface modes. The accuracy of the simulated results is testified under special conditions with literature and good agreement is found. The presented results may be used in the designing the temperature assisted dual channel wave guides, THz optical switches, THz optical logic designs and flexible thermal-optical sensors.