Temperature-Dependent Electromagnetic Surface Wave Supported by Graphene-Loaded Indium Antimonide Planar Structure

. In this study, the propagation characteristics of EM surface waves supported by the graphene-coated indium antimonide (InSb) planar waveguide have been investigated theoretically and numerically. Te modeling of graphene has been performed by use of Kubo formalism whereas the relative permittivity of indium antimonide has been calculated using Drude’s model. Te results for transverse electric (TE) and transverse magnetic (TM) polarized surface waves have been computed analytically and numerically. Te major challenge is to model the atomically thick graphene sheet over the InSb grounded slab. To get the temperature-dependent characteristic equation for the electromagnetic surface waves, the surface current boundary conditions’ approach has been employed. Te numerical results have been computed for both the TE and TM polarization states and reported that the TE does not support the propagation of surface waves. Te dispersion relation, efective mode index, phase speed, propagation length, and feld profle have been computed in Mathematica under TM polarization. Te graphene and indium antimonide have been found active for low and high Terahertz regions, respectively. As temperature increases, the plasma frequency of the InSb increases due to this reason with the increase of temperature and the resonance frequency, leading to a shift in the dispersion curve. Moreover, with the increase of temperature, the efective wave number of transverse magnetic polarized surface waves also increases. Resultantly, the confnement of such surface waves supported by graphene-loaded InSb increases. It is shown that with the variation of temperature of indium antimonide, the surface waves propagating across the interface can be tuned in the Terahertz region and can be exploited for thermo-optical sensing, near-feld communications waveguides, and graphene-based temperature sensor designing.


Introduction
In recent years, the study of surface waves has attracted signifcant attention due to their potential applications in a wide range of felds, including sensing, imaging, waveguides, flters, modulators, and surface communications [1][2][3][4][5][6][7][8].Electromagnetic surface waves (ESWs) propagate along the interface between two media, and they can be confned and manipulated at the nanoscale using plasmonic structures.Depending upon the types of partnering surfaces, diferent types of ESWs can be induced.Te ESWs which are controlled by the interface of a metal and any dielectric material having components of polaritons called surface plasmons-polaritons (SPPs) [9,10].Te EM surface waves are guided by the interfaces of two partnering materials, one with isotropic and other anisotropic dielectric in nature, that type of wave is called Dyakonov surface waves [11].Te electromagnetic surface wave propagation in which at least one of the two interacting materials is often nonhomogenous is perpendicular to the interface guiding surface waves and both interacting materials are isotropic dielectric in nature.Tat particular type of surface wave is called the Tamm wave [11].Te interfaces of two dielectric materials, where one dielectric material is jointly anisotropic and periodically nonhomogenous in a direction perpendicular to the interface, guide a Dyakonov-Tamm wave, combining the features of Dyakonov wave and Tamm wave.Te nonhomogenous electromagnetic surface wave is called the Zenneck or Sommerfeld-Zenneck wave which is incident onto a spherical or planar boundary interface with a complex Brewster angle between two homogeneous media with diferent dielectric constants [12][13][14].Electromagnetic surface waves ofer efcient and low-loss transmission of information over short distances, along with light confnement in small spaces, making them valuable for compact optical devices such as sensors, flters, and waveguides [15].Diferent types of schemes have been proposed by many researchers to control and manipulate the electromagnetic waves of diferent spectral ranges, i.e., metastructures, nearzero-index materials, and nanostructures [16][17][18].For the visible range, the diferent plasmonic platforms have been studied for strong interaction with biomolecules and nanostructures and holding potential for applications such as biosensing, drug delivery, waveguides, and nanophotonic [19][20][21].
To control the THz waves for the surface communication, biomedical sensing, DNA detection, THz antenna, flters, and absorbers, graphene has been realized as a natural metasurface [22][23][24][25].Graphene is a monoatomic, transparent sheet of carbon with its exceptional optical, thermal, electrical, and mechanical properties [26].Graphene, as proposed by Vakil and Engheta, holds immense promise as a material for actively tuning surface waves with unparalleled control.Tis potential opens up exciting opportunities for applications in surface communication, cloaking, and sensing [27].Naserpour et al. theoretically proposed the active tunability and cloaking for the graphene-coated nanodimmers and nanowires [28].Recently, the graphene-based THz metamaterials have been studied for the promising applications, i.e., biosensing, enhanced light emission, and broadband IR photodetector [29][30][31][32].Te exceptional properties of graphene make it a potential candidate for its unprecedented control and manipulation in Terahertz region.
Recently, the graphene-based temperature-assisted devices are the interest of many researchers for their potential applications in the wearable temperature sensors, thermographers, thermal current generation, thermal rectifers, and IR detectors [33][34][35].Te temperature-sensitive materials (TSM) have temperaturedependent optical, electronic, structural, and phase characteristics; i.e., with the change of external temperature, the characteristics of TSM can be tailored [36].Indium antimonide (InSb) is realized as temperature-sensitive materials, which electronically has the characteristic of semiconductors formed by the combination of indium (In) and antimony (Sb).Due to its small bandgap, indium antimonide (InSb) growth has received interest as a possible material for infrared detectors and high-speed devices in recent years [37].
InSb-based biosensors can be used to detect microorganisms and are used in temperature-dependent imaging cameras [38].Te response of the graphene towards the thermal agitation is negligible to develop the highly sensitive graphene-based thermal platforms for the communication and sensors, a graphene-loaded indium antimonide (InSb) as a temperature-sensitive material (TSM) has been proposed in the present study.As the surface waves propagate along the interface of two dissimilar media and are sensitive to the electromagnetic characteristics of patterning material, therefore, it is hypothesized that by making an interface of temperature sensitivity, the characteristics of surface may be controlled by the external temperature.Moreover, to tune the surface waves supported by InSb is the biggest challenge and need for the realization of the TSM-based devices.To meet this need, the graphene-coated InSb planar structure has been studied analytically and numerically for the active tuning and manipulation of surface waves from lower to higher THz region.Te temperature dependency and efect of graphene parameters on the dispersion curves, efective wave number, and feld profles of surface waves have been studied for both transverse electric (TE) and transverse magnetic (TM) polarizations.Te organization of the manuscript is given as the analytical modeling and formulation of the problem being provided in the second section, while the numerical results and discussion and concluding remarks are given in the sections three and four, respectively.

Analytical Formulation
In the present research work, the infuence of parametric properties, i.e., chemical potential (μ), scattering rate (τ), slab thickness (d), and temperature (T) using dispersion relation, has been analyzed.Te propagation of electromagnetic surface waves which travel along the interface of two media of diferent dielectric constants (i.e., graphene and indium antimonide) is studied.Tese waves are also known as surface plasmon polariton (SPP) which decays exponentially as the distance from the interface increases.Tere are three regions which are shown in Figure 1, upper region (z > d) which is free space, the middle region is supposed to flled with the indium antimonide (InSb), i.e., 0 < z < d and the at z � d is the interface between the graphene layer and indium antimonide has been considered, while the lower region (z < 0) whose electrical conductivity is infnite.
To physically model such planar integrated structures, the diferent approaches, i.e., Dyadic green's function approach [39], efective medium approach [40], transmission line approach [41], equivalent circuit model approach [42], and analytic absorber model approach [43], have been used in the literature.However, in the present work, the waveguide approach has been used to model the supporting electromagnetic feld interaction and propagation of the surface waves supported by the proposed structure, and the proposed approach is quite simple and easy to handle as reported in diferent works [10,44].

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TM-Polarized Surface Waves.
For transverse magnetic polarized surface waves, the electric feld (E) is directed along the interface such as in the x-axis and the magnetic feld (H) is directed along the y-axis.Te associated feld equations are as follows [11,44].
For region z > d, E (1)  x � − where are attenuation constants for their respective regions, while A and B are unknown coefcients, e − c 1 z is the exponential decay or damping, ω is the Terahertz radiation frequency, and β is known as the propagation constant.To calculate the unknown constants for the transverse magnetic surface waves, the boundary conditions are applied at z � d.Boundary conditions are given as [27] H (1)  y − H (2) E (1)  x − E (2)  x � 0.
By implementing boundary conditions on ( 1)-( 6), the characteristic equation for TM-polarized surface wave is obtained as 2.2.TE-Polarized Surface Waves.For transverse electric polarized surface waves, the feld equations are as the propagation of surface waves in which the electric feld (E) is taken along y-axis and the magnetic feld (H) is directed along x-axis, known as transverse electric TE-polarized surface waves [11,44].
For region z > d, H (1)  x � − To calculate the unknown constants (A&B) for the transverse magnetic surface waves, the boundary conditions are applied at z � d.Boundary conditions are given as H (1)  x − H (2)  x � J s � σ g E (2)  y , ( E (1)  y − E (2)  y � 0.
By implementing boundary conditions on ( 8)-( 13), the characteristic equation for the TE-polarized surface wave is obtained as

Numerical Results and Discussion
Te wave properties of surface waves on graphene-loaded indium antimonide (InSb), i.e., dispersion relation, efective mode index, penetration depth, propagation length, phase velocity, and feld profle, have been computed numerically.Te characteristics of surface waves under the temperature, thickness of the InSb slab, and chemical potential variations have been studied.Te numerical solution for each characteristic has been computed using the contour plot International Journal of Optics technique.Te scattering rate of electron-photon has been taken as τ � 0.6 ps.Te modeling of graphene monolayer has been carried out [44,45], using wolfram language-based Mathematica software.Te electromagnetic modeling of graphene has been derived using Kubo's formalism.Te optical surface conductivity of graphene is the result of the contribution of two factors, intraband and interband conduction [46].Te intraband conductivity supports graphene as a strong candidature for the tunability of surface waves in the Terahertz regime.It is caused by electron-photon processes.Te intraband conductivity [46] is given as (15) whereas the interband conductivity facilitates graphene as a potential candidate for the tunability of surface waves in the visible and infrared regime.It is caused by direct electron interband transitions.Te interband conductivity term [46] is given as Te interband conductivity dominates at THz and infrared frequencies, and graphene gives constant conductivity which is given as σ interband � e 2 /4Z.It means that a monolayer graphene sheet can absorb 2.3% of the incident light over a large frequency gap [47].Te total conductivity of graphene can be taken as the sum of intraband and interband conduction, i.e., σ g � σ intraband + σ interband .
where the termsω, μ, T, k B , Z, and τ stand for the incident frequency, chemical potential, temperature, Boltzmann constant, reduced Plank's constant, and electron-photon scattering rate, respectively.

Surface Waves on Graphene-Covered InSb.
Te efect of tuning the parameters, i.e., thickness, temperature, and chemical potentials on propagation of surface waves for graphene-covered indium antimonide slab, has been studied and calculated using Mathematica.Te amplitude of surface waves decreases as we move away from the indium antimonide slab.With respect to polarization, two types of modes can be discussed for such types of surface waves which are given as (i) TE-polarized surface wave (ii) TM-polarized surface wave Figure 2 shows the dispersion curves of TE-polarized surface waves with the variation of temperature, i.e., T � 200 K, T � 220 K, T � 240 K, T � 260 K, T � 280 K, and T � 300 K.It is predicted that the TE mode for such type of surface waves facilitated by graphene and indium antimonide structure does not exist since no variation in the straight line has been found on the tuning of temperature.All the lines obtained at various temperature variations follow a light line.So, there exists no solution for such temperature variations.Terefore, it is concluded that TE mode for electromagnetic surface waves does not exist unlike TM mode, which is consistent with the literature as given in [10,44], while in the subsequent section, the numerical results for the surface wave under TM mode have been given.

Dispersion Curve Analysis for TM Polarization.
In this section, propagation of the transverse magnetic polarized surface for indium antimonide and graphene-coated indium antimonide has been studied graphically and calculated using software pack.Te electromagnetic modeling of temperature-sensitive materials such as indium antimonide can be tunable by controlling the temperature values.For Terahertz regime, the relative permittivity of InSb is given by the Drude model [44,48].
Herein, ε ∞ is the high frequency relative permittivity, ω p is the plasma frequency, and c is the damping constant.Plasma frequency ω p can be calculated as ������������ Nq e /0.015 ε o m e  ; here, the plasma frequency (ω p ) is afected by the intrinsic carrier density "N," electronic charge "q e ," and mass of electron "m e ", i.e., q e � −1.6 × 10 − 19 C, m e � 9.1 × 10 − 31 kg, and c � π × 10 11 rad s − 1 .Te intrinsic carrier density "N" depends upon the temperature "T," bandgap "E g ," and Boltzmann constant "k B " and can be calculated as N � 5.76 × 10 20 T 3/2 exp (−E g /2k B T) where the values of band gap "E g " and Boltzmann constant "k B " are 0.26 eV and 8.62 × 10 − 5 eV/K.As intrinsic carrier density "N" depends upon the temperature "T," we can say that the plasma frequency is also dependent upon the temperature.Terefore, by tuning the temperature of indium antimonide (InSb), the plasma frequency can be adjusted.Tus, the relative permittivity of InSb can be controlled in Figure 3 by varying the temperature parameters.Te dispersion curves of TM-polarized surface waves supported by graphene-loaded InSb under the variation of temperature, chemical potential, and thickness of InSb slab have been presented graphically in Figure 4. Figure 4(a) depicts that with the increase in temperature, high resonance frequency-polarized surface waves on an InSb slab coated 4 International Journal of Optics with graphene can be achieved.Te impact of chemical potential on the dispersion curve by keeping the temperature constant has been presented in Figure 4(b).It has been found that with the increase of chemical potential, i.e., 0.2 eV, 0.4 eV, 0.6 eV, and 0.8 eV, the resonance frequency decreases.It is obvious from the fgure that the chemical potential (μ) afects the conductivity of graphene and the propagation constant starts decreasing with the increase of chemical potential, which shows that for the highly doped graphene layer, the confnement of the surface waves is low.Further, it can be deduced from this result that the skin depth of THz for graphene-coated InSb slab starts decreasing.Due to this reason, the confnement of the surface wave also decreases with the increase of chemical potential.While the efects of the thickness of InSb slab on the propagation of surface waves supported by graphenecoated InSb slab have been depicted in Figure 4(c), it is clear from the fgure that with the increase in the thickness d of the InSb slab, the resonance frequency decreases rapidly even at low THz frequency.Te ratio of the propagation constant of medium to free space is known as the efective mode index N eff .Te normalized efective wave number determines how surface waves are being confned at the interface and calculated as N eff � Re β/k o .On the graphene-coated THz-InSb slab, the confnement of the TM-polarized surface waves has been reported under temperature range, i.e., T � 200 K, 250 K, 300 K, 350 K, and 400 K in Figure 5.It is clear from the fgure that with increasing temperature, the efective wave number of transverse magnetic-polarized surface waves also increases.Resultantly, the confnement of such surface waves supported by graphene-loaded indium antimonide THz slab also increases.
To further study the confnement of the surface waves on the graphene supported THz-InSb slab, the infuence of thickness of InSb slab on the efective mode index (N eff ) under the function of frequency has been explained graphically in Figure 6.Te possible solution has been calculated using the contour plot technique.Higher values of efective mode index mean higher confnement.It is clear from the fgure that thickness of the slab plays an important role in tuning the confnement of surface waves.Te effective mode index under the diferent values of thickness, i.e., d � 20 nm, d � 40 nm, d � 60 nm, d � 80 nm, and d � 100 nm, has been analyzed.It is obvious from the fgure that with the increase of thickness of the InSb slab, the efective mode index decreases, and consequently, the confnement of such waves decreases from higher THz frequencies.Figure 7.
Propagation length (L p ) is the measurement of the distance travelled by a surface wave while travelling along the interface of two media and is given by the imaginary part of the propagation constant.Te efect of temperature on the propagation length has been presented in Figure 8. Te propagation length is normalized by wave number k o .It depicts that propagation length increases with the increase in frequency.It is also concluded that with the increase in temperature, it shifts towards a high-frequency range.
In Figure 9, the infuence of temperature on the phase velocity has been shown graphically.Phase velocity is given as V p � k o /β.To estimate the phase velocity within the temperature range of T � 200 K to 400 K, it is evident from Figure 9 that the phase velocity decreases as the frequency increases and can be adjusted by varying the temperature.
Te comparison among the feld profle of electromagnetic surface waves under diferent temperatures, i.e., T � 200 K and T � 400 K, has been presented in Figure 10.Te normalized feld profles of |E x | have been taken as a function of transverse distance (z) from the interface.It is obvious from the fgure that the feld profles are temperature-sensitive; i.e., high temperature has more intensity as compared to lower temperature.Further, it can be verifed from the fgure that as the transverse distance from the interface increases, the feld profles follow the exponential decay, which confrms the basic characteristic of surface waves.
In view of the practical implementation, the proposed structure can be easily realizable.Te synthesis of graphene can be carried out by the chemical vapor deposition on the indium antimonide substrate [44]; however, the major limitation is the excitation scheme for the surface waves in the THz regime, and to bridge this limitation, many approaches have been studied for the excitation of such evanescent modes in THz, which may be used for the realization of the work [49].Te proposed results have potential applications in the designing of the temperature-assisted THz optical devices for the communication, thermal imaging, and sensing applications.

Concluding Remarks
Te electromagnetic surface wave propagation supported by graphene-coated indium antimonide (InSb) has been studied in detail.Maxwell's equations have been solved using surface waves propagating on graphene supported by temperature-sensitive materials, such as indium antimonide, for transverse electric (TE) and transverse magnetic (TM) modes of polarization.Analytical and numerical studies have been conducted on the electromagnetic surface wave propagation on graphene coated with indium antimonide.For mathematical and graphical analysis, the Wolfram Mathematica pack has been utilized.Te subsequent conclusions from the numerical results can be derived as the electromagnetic surface waves exit for only TM polarization, while in the case of TE polarization, there is no existence of surface waves.Te propagation characteristics of surface waves are found to be temperature-sensitive.Te propagation length, penetration depth, efective mode index, and phase velocity of surface waves can be varied by tuning the temperature.It is observed that the resonance frequency of surface waves is found to increase with temperature.Te confnement of the surface waves can be tuned by the operating frequency and temperature of InSb.Te feld profle of the surface wave decreases progressively and decreases as the distance from the interface increases.Te proposed results can be used for the realization and future application of the graphene-coated TSM structure for Terahertz applications in designing thermo-optical sensing, THz communication, and near-feld thermal imaging platforms.Moreover, the work can be extended for the graphene-based curvilinear temperature-sensitive metasurfaces and bilayer asymmetric temperature-sensitive surfaces for the fexible thermo-optical sensors in THz regime as discussed in [50,51].

Figure 1 :
Figure 1: Geometry for surface waves supported by graphene-loaded InSb planar structure: (a) top view; (b) side view.

Figure 2 :
Figure 2: Infuence of temperature on the dispersion curve for the TE-polarized surface wave propagation supported by graphenecoated indium antimonide with ϵ ∞ � 15.68 and μ � 0.2 eV.

Figure 3 :
Figure 3: Dispersion curve analysis under the infuence of temperature for the TM-polarized surface wave propagation supported by indium antimonide (InSb) with ϵ ∞ � 15.68.

FigureFigure 8 :
Figure Infuence of chemical potential μ on the efective mode index N eff for the TM-polarized surface wave propagation supported by graphene-coated indium antimonide with ϵ ∞ � 15.68, d � 20 nm, and T � 400 K.