Simulation of a plasmonic sensor using kinetic theory of plasma with the Vlasov equation in MATLAB

This research proposes a mathematical model for a plasmonic sensor using kinetic theory of plasma with the Vlasov equation. A nanoantenna cavity of a plasmonic material is driven by an input electromagnetic wave, which changes the charge density and current flow in the cavity, resulting in a change in the Fermi distribution function of the charged particles. The results are achieved in terms of current density and conductivity by solving the Boltzmann transport equation, Maxwell’s equations, and Taylor series expansion in terms of perturbed electric fields with linear integro differential equations. The results are simulated using MATLAB. The changes in current density and conductivity are validated by experimental analysis of graphene plasmonic material using patch antenna with the dielectric substrates SiO2 and Al2O3. By varying the applied electric fields, current changes at the output of the plasmonic antenna are analyzed using signal-processing techniques. Wavelet transforms are used to find the space-scale behavior of the output signals, such as current density variation, voltage variation, and susceptibility change with sub-band coding techniques in terms of wavelet coefficients.


Introduction
Plasmonics is concerned with nanophotonic devices, which are optically active nanostructures. Plasmonics and photonics have also been known about for 40-50 years, but recent discoveries in nanoscience have increased in the last 10 years. Nanoscience is a field of light matter interactions, which is at the subwavelength scale in physical, chemical, and artificial nanostructures. Photonics provides solutions for nanodevices, such as nanoantenna and waveguides [1][2][3][4], Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. combining electrical and optical properties [5]. These devices are implemented for high-speed data processing and sensors [1,5]. Optical sensors based on plasmonic sensing have led to many important applications in the fields of biology, chemistry, material science, renewable energy, and information science and technology [4]. It is also being used in terahertz (THz) imaging, surface plasmon resonance (SPR) sensing, and inverse scattering using SPR phase interferometry [1,2]. These imaging devices can work in the near infrared (IR) region. The biggest advantage is that these plasmonic devices have four times fewer losses than metals [1]. An input signal is generated by a laser and the output is detected by photodiodes [2]. Today, there is great interest in plasma-filled waveguides at the nanoscale for nanolinks, which is a desirable feature because of less wiring [2,3,5]. Here, an external electromagnetic field (EMF) interacts with electromagnetic waves within a cavity, inducing coherent oscillations of electrons, which is expressed by localized surface plasmon resonance (LSPR) [1,2,4]. Some of the plasmonic materials used at the nanoscale are graphene, carbon nanotubes [2,6,7], gold [8], and silver [2], where the dimensions are tailored at the nanoscale for nanosensing devices. Cylindrical rod waveguides surrounded by a thin epsilon-mu-near zero (EMZ) shell [9] are used in optics for biochemical sensing [9]. These wires are able to overcome the issues of dispersion, attenuation and degeneration, and losses in waveguides [9]. These metastructures have the potential to bring the optical spatial domain from free space to nanosized wavelength-sized elements [10]. Epsilon negative metamaterial silver is used as the core and silica (SiO 2 ) is used as a spacer cell [10]. Absorption in the optical region by silver NP was investigated using UV-V spectroscopy [11].There is a redshift in the spectrum and SPR peak due to a change in the dielectric constant of the surrounding medium.
Graphene has the property of third-order optical nonlinearity [2,7]. Due to its nanostructure, there are dramatic reductions in energy in device operation, which enable immense applications in the field of densely packed integrated circuits with low power dissipation with increased sensitivity [8,12,13]. These sensors are device layers, which collect readings from sensors for processing. Using the principles of SPP generation [12], nanomaterial absorption and scattering characteristics are explored for sensing [1,13].
It has also been demonstrated that nanomaterials show exciting and interesting phenomena, such as subwavelength waveguiding and modulation, light tripping and filtering, and ultra-resolution imaging [4,6,12,14]. Because of this, plasmonics and nanophononics are seen in an explosive growth phase [3]. Recent development show that hybrid EM modes, and transverse electric (TE) and transverse magnetic (TM) modes are present in epsilon-near-zero (ENZ) waveguides [9]. These waveguide equations and waveguide modes are being developed in wave-based computers providing chip-scale fast integrable computing elements [10], an immense solution for metasurfaces using integrable equations with wave propagation [9]. These devices are found to have less mode degeneration, attenuation/losses, and energy dispersion [9].
Plasmonic nanostructure cross sections are higher than their physical cross sections, which have a large absorption cross section and scattering surfaces, which makes imaging easier [13,15]. Nanostructures are found to be associated with electromagnetics and optics. In recent studies, a circuit-modelingbased approach was proposed for arbitrary shaped nanostructures. By appropriately using the electromagnetic and thermal characteristics [16] of nanostructures we can apply them in several applications, such as sensing, subwavelength optical imaging and medical diagnosis, and advanced data storage systems [16].This can be further used in acoustics, fluid dynamics, and classical mechanics. A variety of plasmonic heterostructures are being used in antenna and signal processing in beam steering, communication switching, detection, transmission, transportation, and modulation in the sixth generation [17].
In a kinetic theory (KT) model of plasmonic antenna it has been observed that the current density and conductivity calculations [18][19][20], using the Vlasov equation (VE), are highly accurate compared to existing techniques [20,21]. KT in plasma was given by Lorentz force and the VE [19,20], which plays an important role in plasmonic sensing. Earlier, wave-particle interactions, particles in momentum and velocity-space, and particle distribution functions were used in plasma physics [18][19][20]. A kinetic description of plasma is essential in the thermal equilibrium state, which is defined by the VE [20]. The equation with E (electric field) and B (magnetic field) and VE represents a complete set for the derivation of plasma generation with no collision states. When there are collisions in the plasma it contributes to the current flow. The electrical biasing controls the real and imaginary part of the conductivity, which gives good dynamic control [21]. In the collision state, a new Fermi level f is defined by f 0 + δf. The Boltzmann transport equation (BTE) [22] is found to be the most accurate and widely accepted approach with a nonlinear transport equation for the simulation of the time domain [23,24] plasma transient current density.
As a nanoantenna material, graphene shows high electron mobility and excellent response in ultra-high frequency applications in high-frequency circuits and multipliers [25]. Graphene has SPP guided along a metal dielectric interface [1,3,4]. Graphene ultra-thin surfaces are composed of graphene micro-ribbons, which are suitable for terahertz generation in the far IR region and THz band [7,8,10,12]. Dynamical tuning of the Fermi energy of graphene with chemical and electrical doping [2] gives resonant frequency adjustment. Data coherency, consistency, and synchronization problems are reduced by graphene. The graphene-enabled wireless on chip is designed as nanoantennas [4,26]. These graphene nanoantennas are used in core-level communication in the THz band with improved bandwidth and area overhead [4,27]. A graphene patch over the substrate produces [2,9,20,[27][28][29] a transport channel for carriers, which gives radiation in THz. Nanoantennas made of graphene for nanosensing devices support the propagation of tightly confined SPP [7,12] waves at the metal dielectric interfaces. The graphene patch acts as a mirror, and the patch itself behaves as a resonator for SPP modes [1,4,29]. This technology also provides multitasking and broadcasting capabilities with a vast design space at the nanoscale [30,31]. The susceptibility values of graphene are found to be eight times higher than other nonlinear optical materials, hence this high nonlinearity can be used as a tool for imaging and sensing [2,7]. Recent study shows a liquid-gated graphene field effect transistor (GFET)-based biosensor model, which is analytically developed for E. coli 0157:H7 bacteria detection. The GFET was proved to possess multiparameter characteristics for disease, DNA, and clinical diagnostics [32].
In signal-processing, wavelet transform (WT) has widespread applications in the compression of images and audio. Sub-band coding schemes with quadrature mirror filter banks (QMFBs) are used to compress the data [31]. Since Fourier transforms cannot provide the time localization of a given frequency component, WTs can analyze speech and biological data and sounds [31] in both the time and frequency domains. WTs are used in chirp signal detection, gravitational wave detection, and astrophysics [31]. Time frequency analysis, which is performed in WTs, is a powerful tool in nonstationary environments. It includes partitioning, time varying filtering, denoising, and signal component extraction [31].
Here, discrete analysis of current density, charge density, and scattered signal and power spectra is performed using wavelet filters and multiresolution analysis (MRA).
Plasma-based sensing/imaging [8] also finds applications in radar technology, inverse scattering, and military application purposes, especially in enemy detection and ranging. Due to its small size and compact design it is very useful in cases where size does matter, such as in mobile communication, wireless applications, wireless data communication, remote switches, door openers, and laptops.
In this paper, the details of a mathematical analysis is proposed for a plasmonic channel (PC) using the VE and BTE in section 2. Section 3 describes the parameter estimation using WT. Section 4 presents the design and simulation of the PC using the BTE and graphene patch antenna in the THz range with a signal-processing approach for its parameter estimation and susceptibility calculations. Section 5 concludes with the results of the comparative analysis, limitations, and future scope.

Mathematical analysis of a PC
Nonrelativistic motion of an electron in an oscillating electric field is defined by the VE. Here, f is the Boltzmann function, E is the applied electric field, B is the applied magnetic field, and v is the velocity of the plasma electron in three dimensions, x, y, z. If a test particle s with charge qs is positioned in rs in plasma in stationary conditions its phase space is characterized by (r, v) (figure 1). Here, r is a measure of distance and a defines the acceleration of a particle The equation for the charged particle is written for the x, y, and z directions (figure 1). Acceleration of the particle depends on the position in the phase space defined by a x (r) , a y (r) , a z (r).
Velocity vector space is defined by v · ∇ r and the acceleration vector space a(r) ∇ v (r). ( * A1.1) Phase space is given by δΓ = δVδ 3 v, where δV is the physical volume of the plasma. The number density of an isothermal charged particle is defined by the Boltzmann relation when the thermal and the electrostatic forces acting on the plasma fluid have reached equilibrium. In equilibrium conditions, the BTE [19,33], in terms of the Fermi function f (t, r, v), is defined in time, velocity, and space. Initial electric and magnetic fields are assumed as E 0 and B 0 , respectively. The VE is a differential equation describing time evolution of the distribution function of charged particle flow in a plasma for a long-range Coulomb interaction. The interaction of charged particles with EMFs, which changes the momentum of a particle, is given by the Lorentz equation. The theory given by Vlasov starts from collisionless, and then it states the interaction with self-consistent fields generated inside the plasma with the externally applied fields.
The VE with ME define the complete charged particle dynamics for the collisionless case given by (1) For a particle with charge q and mass m, moving with velocity v, in the presence of an electric field E 0 and magnetic induction B 0 fields ( * A1.1 equation (a.4)).
If the total number of particles is N, then we have N nonlinear-coupled equations for field and particle motion trajectories.
The EMF in the medium is given by Plasma fluid dynamics are described by the moment equations. ( * A1.1) These equations are further approximated by the Vlasov-Poisson equation. The density number n and momentum nu of plasma are calculated as the zeroth-order moment and firstorder moment, respectively n =ˆfd 3 v and nu =ˆvfd 3 (6) where ρ C , J,ϵ 0 , and µ 0 denote, respectively, the total charge density, total electric current density, electric permittivity, and magnetic permeability of free space. This complete system is defined by the VE, where resonant charge exchange is known as an inelastic process for transport calculations [19]. For collisions of the charged particles, equation (1) is modified by the collision term [33] as follows where the electric field E = −∇∅ (r) , ∅ (r) is the applied electrostatic bias potential. Particle kinetic and electric potential energies remain constant in the presence of static EMFs. The electric potential Φ can be considered as the potential energy. Here, the third term q m (−∇∅ (r) , ∇ v ) is due to the chemical potential [33]. We also treat small perturba- The analytical solution of the Boltzmann equation is given by where δ is a small parameter of the first order. Here, f 0 is the Maxwell's distribution, which gives a null collision integral. If f 0 is a Boltzmann distribution function, which is defined in terms of kinetic and potential energy, (It is due to the kinetic and potential energy of the atoms [31].) * (A1.2) equation (a.7) For statistical analysis, here we take the number of collisions between charged particles, which increases when a potential field is applied across the channel. Hence, the equation given for the kth particle The electric field is E s and the magnetic field is B s , where s is varying from s = 0 to t k−1 , and t k is the collision time. The collision term accounts for the scattering process [33,34].
Here, S k is the scattering by the kth particle and summation of all k accounts for scattering by all the particles Using equations (13) and (14), with charge density and current density substituted in equation (a.6) we get The equation gives the scattered signal E 1 (figures 5 (c) and (e)) in terms of the BTE and the initial applied field E 0 .

Current density and conductivity tensor using BTE
In the presence of a background EMF, which can be resolved into a superposition of plane waves of different frequencies and wave vectors (ω, k), the conductivity terms can be computed as a function of (ω, k). Its real and imaginary parts with Maxwell's equations determine the permittivity and conductivity tensor: This equation also gives the solution for the charge carrier density in the (ω, k) domain aŝ where the whole conductivity tensor is Here, V is the velocity tensor of the PC. The real and imaginary parts of the 3 × 3 tensor determine the permittivity and conductivity tensors.

Displacement current
Using divergence theorem in the current conservation law [35] the total current density on closed surface (figure 2) in the given volume iŝ The displacement current density is The DC current on the surface S can be written as The expression of the displacement current ⟨I S (t)⟩ on the surface S at time t1 If B is the magnetic field in dl in an infinitesimal line element along the curve C and µ 0 is the magnetic constant permeability of surrounding space,¸c B.dl = µ 0 I D .
Here, I D = ϵ 0 d∅E dt dE is the electric flux, and¸B · dS = µ 0 I by Ampere's Maxwell's law [35].
Further quantum transport of N particles is given by the N single-particle pseudo-Schrodinger equation [35], and current densities are solved for quantum system Bohmian trajectories (figure 5(a)) In the polarization J D = ϵ 0 ∂E ∂t + ∂P ∂t . The first term is present in the material media and in free space.
The, I d =´´J D .dS changes in the polarization of the individual molecules of the dielectric material. Polarization is the displacement current. It is a material medium. The effect of P is in the form of change in the relative permittivity ϵ r . SPP propagates along the surface until its energy is lost either to absorption or scattering into free space. SPP waves are a shorter wavelength than a free-space wavelength. For all particle j1 to N (due to polarization in dielectric medium) The total current is the sum of the displacement and particle current in the medium.
Graphene, a boon for the scientific community, can be used in many applications. It has linear energy momentum dispersion, which provides massless Dirac fermions, which has massless extremely large electrical conductivity (10 6 m s −1 electron velocity). This property can be used for amplifiers and mixers. It has been demonstrated to be used at 100 GHz in GFETs and at 300 GHz [35]. GFETs are a young class of FETs.

Plasmonic sensing with parameter estimation techniques
For plasma, macroscopic variables are given by momentum flux, pressure, temperature, and heat flux. Figure 1 shows that by applying an electric field E to the plasma channel, the charged particles are moving with different velocities v j and traveling a distance x j , which is taken as a random function [36][37][38]. We use the KT and Klimontovich-Dupree (KD) discrete representation for the complete charge density and current density [36] calculation. The plasma charge density is given by [38] The summation is over the different charged particle species in the plasma. It is a triple integral extending over all velocity space that is over each one of the variables v x , v y, v z from −∞ to +∞

Susceptibility is taken as
If χ (r, v, t) is a component defined for momentum of type α together with unit vectorĵ then By KD discrete representation [18,19], the moment of a charged particles p is given by Using these equations, we define the plasma space into a volume dv and surface ds.

Parameter estimation and WT
The CWT of a signal x (t) with respect to the wavelet ψ(t) is defined by where k and m are real values and * denotes complex conjugation. Thus, WT ψ is a function of two variables, m, k. To find the WT of the function x(t), it should be a square integrable function. ( * A1.5) By virtue of its orthonormal properties, it is used to construct a QMFB where, in the time domain, the sequence of the basis function is defined in U = (u(k)) k∈Z , which is a low-pass filter.
Using the QMFB, we pass the signal through a low-pass and high-pass combination of filters, which gives the coarser and finer details of the signal (figures 5(d) and (f)).

Using wavelets in BTE
In the first-order perturbation equation ( * A1.6) from equation (a.25) the wavelet ψ n,k (x) defines the position vector and ψ m k (v) is the velocity vector in the BTÊ 2D wavelet The pattern of distribution of the Boltzmann distribution function after it gets perturbed by the EMF shows different spectral localization of the scale/frequency behavior in the position of velocity space and, to extract the space-scale behavior, we use the 2D WT. Figure 1 shows a plasmonic nanocavity, assuming it to be a plasmonic sensor [39]. When an external field is applied it creates perturbation in the Boltzmann distribution function f. The distribution function is summed over time t (relaxation time) and the curve is used as a good approximation for calculations of the perturbations in function f 1 as a function of f 0 . We assume an applied voltage of 5 V, a temperature of 230 K, and potential in the range 0:0.1:5 eV. For perturbed Fermi level calculations we took v = vcos(k·x − ω p * t) and e = cos(k1·x − ω k t) [38]. By increasing the temperature to 280 K, the angular frequency ω p to 2 * pi * 9 * 10 12 Hz, the angular frequency ω k to 10 8 Hz, the relaxation time τ = 0.0001:0.0001:0.01 s and integrating it with t = 1:1:100 s we obtained f 1 .

Results
Using equation (16) with a rectangular coordinate, we take approximations for an infinitesimal small volume generated because of deviation in velocity due to the movement of charged particles; we set an approximation in a spherical coordinate system d 3 v = (´´´v 2 dv sin θ dθ dφ ). A scattered EM field is calculated using the above equation shown in figure 4(a). It shows the oscillating EMF. SPP with nonlinearity in its scattering pattern is obtained using spherical coordinates.
Here, d 3 v = −µ ∈ d dv´q vf s1 (´v 2 dv sin θ dθ´dφ ) = −2π µ ϵ qv f s1´d V, where q = ne and n = 10 24 ; v is the velocity of charged particles in the plasma. Here, we take the relative permeability and permittivity as µ r = 4.4 and ϵ r = 3.1, respectively, for graphene. Figure 4(b) shows the scattered EMFs calculated with time and velocity in six-dimensional space.
The designing of the PC is carried out with a square patch and bow-tie structure of a graphene patch over SiO 2 [37] and Al 2 O 3 substrates, respectively. All simulations are performed using HFSS. The charge transport between graphene and the dielectric material, SiO 2 or Al 2 O 3 , produces scattering of EMFs in the THz range. Here, the substrate material provides a channel for charge carrier transport in graphene, which controls the carrier mobility [15]. The carrier concentration and chemical potential in the graphene-based patch can be adjusted through bias voltage, resulting in tunable characteristics. The input is given through a quarter-wave transformer to couple maximum power to the patch. The channel performance of the square patch is analyzed at different sweep frequencies, and its current density and scattering pattern are observed. Current density curves are shown on the surface of the patch (figures 4(c) and (d)). The bow-tie antenna (figure 4(e)) of graphene with the Al 2 O 3 substrate scatters the signal at 3.8 THz ( figure 4(f)). Solutions to the current density and charge density are obtained using equation (14). The current density is plotted in figure 5(a) with an applied voltage of 5 V varying charged-particle velocity. The simulated output is passed through a signal processing block where WT is used to get scattered electromagnetic field [38,39]. Figure 5 shows the WT analysis of the perturbed state of the plasma, the current generated, the scattered electric power, and susceptibility, respectively. The susceptibility of the plasmonic medium is a function of charged-particle moments. The details of the plasma flux are estimated using equations (8) and (14) and equations (30). The CWT of susceptibility gives variation in terms of wavelet coefficients of the real part and imaginary part of the signal as shown in ( figure 5(h)).

Discussion
Earlier experiments reported metasurfaces that were used to solve integral equations, which carried complex-valued EM fields and solutions via a recursive equation inside the path [10]. In this study, the Neumann series was solved as an inverse system. Chemical sensing [10] is carried out by ENZ waveguides. TE and TM modes are analyzed for such nanocylindrical structures [9]. Modes are further given by Bessel's functions of 1st kind and 2nd kind for standing waves inside the waveguides. Dispersion and abrupt bending, which cause bandwidth limitations, are reduced by ENZ and mu-near-zero medium, respectively.
These cylindrical rod waveguides are surrounded by a thin EMZ shell. Here, the structure of the classical rod (core) has a radius a and constitutive parameters ϵ d and µ d folded by a cover of radius b and with permittivity ϵ c and permeability µ c [9]. In the experiment, proof is given in the form of Bessel's function of the mth order [9]. It gives periodic waves in the azimuthal direction and traveling waves in the z direction. Outside the radius it is Hankel's function in traveling modes [9].
Similarly, an all-metallic ENZ graded-index lens was designed [40] using an array of narrow hollow rectangular waveguides for a nearly 0.7 THz frequency range using artificial metastructures. The waveguides work near the cut-off of the fundamental mode, emulating an ENZ medium.
For DNA detection using the non-equilibrium Green function [32], different variables on a GFET transfer function were tested and compared by measuring their effects on the graphene surface current density. The solution was obtained in the form of Schrodinger [32] and Poisson equations. Least squares curve fitting GFET biosensors are being used to detect hereditary infections [32]. Here, the model used I d /V gs and I d /V ds plots with a graphene 40 nm gate length: a gate oxide Si 3 N 4 of thickness 10 nm. DNA molecules can be detected by these graphene surfaces [32]. E. coli was detected via its graphene surface in the form of conductance variation: current voltage characteristics by varying the E. coli concentration. A computational model was developed for a GFET biosensor [32]. Discussing further the design and implementation of antennas, Dyadic spectral Green's function was used [41] for an integrated planar structure with a grounded dielectric slab for the spectral theory of the EMF. The equivalent circuits and the derived spectral Green's function can be used to analyze and design microstrip antennas of arbitrary shapes. A quantum-transformation-based model was also suggested [16]. In another experiment [16], an electromagnetic and thermal phenomenon Helmholtz equation and quasi static approach to the behavior of nanoparticles [16] was also observed.
Nanorods of anisotropic material arranged in triangular shapes are reported to give a plasmonic response in the form of optically induced magnetic field intensity within the triangle of the nanorod [42], which can be used for DNA detection. The size of the nanorods was 110 ± 8 nm and the diameter was 22 ± 3 nm. SPR are generated along their longitudinal and transverse axes [42]. Varying the lengths of the nanorods shifts the spectra. The nanorods are made of doublesided germanium wafers, which sense the substrate. Electron microscopy is used at 10 kV and 120 kV. Polarization p-, s-, left circulrly polarized (LCP), right circulrly polarized (RCP) is observed using COMSOL multiphysics. The scattering cross section is calculated as 1 I0¸A Re {E s × H * s }.nds [42] and I 0 = ⌈E 2 0 ⌉ (2η0) is the incident power intensity. Here, E s and H s are scattered electric and magnetic fields. It is also observed that when light falls parallel to the axis, the intensity is the highest and, using a quarter wave plate, LCP and RCP waves are investigated. This gives a chiro-optical sensor, which is right-and left-handed rotation nano ribbon (NR) plasmon mode engineering [42]. ENZ are used for nanocircuits [42].
It has also been demonstrated in various studies that graphene is an effective chemical sensor for electrolytes [43]. Graphene-based top-gate insulators are as thin as 1-5 nm in an electrolyte. Quantum scale changes in conduction can be measured with graphene [43]. A graphene-based photodetector, measuring the photon flux converting the energy of the absorbed photons to an electrical current, can be fabricated using substrates of IV and III-V semiconductors. Graphene-based strain and pressure sensors and capacitive sensors are also becoming popular because of graphene's superconductivity. Graphene oxide sheet-based SPR sensors were reported due to [44] its high covalent binding affinity for proteins. The field of plasma physics was introduced in the 1920s. The Vlasov model is more recent, and was introduced in the 1940s [45]. Using the VE, a statistical description is analyzed to investigate the collective behavior of large numbers of charged particles in the long-range plasma interaction. The theory proposed here for a plasmonic cavity also addresses the wave propagation and nonlinear properties of the electrostatic limit in the Coulomb interaction.
Compared to earlier methods, KT describes more fluid models of plasma. An important idea of this VE is that it is a self-consistent field. In the cavity, N charged particles move under the influence of their own charges and the external fields present: this drives the dynamical properties of plasma.
Plasmonic nanoantennas/channels and waveguides of micrometer to nanometer size will be promising applications in nano-wireless applications and sensing. Here, it is assumed that the plasmonic medium is close to thermal equilibrium and the collision term is linearized about a Maxwellian. The Fokker-Planck equation satisfies the important constraints of conservation of matter density, momentum, and energy. Kinetic treatment of the fluid model would be too complicated; therefore, by using a continuity equation, we obtain the first moment equation, which also contains the next higher moments [46]. Here, we get the energy equation from the moment equation. We briefly consider the various physical approximations, which are used in a fluid description. Taking cold plasma conditions, we close the equations after the zeroth-and first-order moments [46]. The derivative of momentum is simply neglected. This gives a good approximation in a variety of wave propagation problems, assuming that the phase velocities of a wave are much greater than the thermal velocities of the particles.
Our experiments prove the charge transport between graphene and dielectric materials, which was also proved in earlier experiments producing scattering of EMFs in the THz range. Here, it is shown that the substrate material used provides charge carrier transport in graphene, which is an experimental work to analyze the behavior of graphene in the presence of an external refractive index, which controls the carrier mobility [15] at the interface. It is shown in patch antenna that carrier concentration and chemical potential in a graphene-based patch, adjusted through bias voltage, results in tunable characteristics.
The VE is used for analysis of long-range interactions in electromagnetics, using Maxwell's equations. Physics applications of VEs range from magnetically confined plasmas to relativistic electromagnetism.

Applications
A variety of plasmonic hetero structures can be used for optical signals in antenna and signal processing in beam steering, communication, switching, detection, transmission, transportation, and modulation in the sixth generation [17]. The excitation properties of LSPR are used in optoelectronic conversion of optical to high-frequency GHz and THz signals in wireless communication, used in plasmonic waveguides [17], modulators, filter switches, routers, 6G ultracompact, ultrafast optical nanocircuits, and optical nanoantennas [17].
LSPR sensors are developed through variation of the size and shape of nanostructures. A change in the dielectric environment around the nanostructures changes the LSPR. SPR at the interface of a metal dielectric interface in deep nanostructure scale is used in an optical nanoplasmonic XOR logic gate [47]. SPR generated by external analyte detection techniques is demonstrated to be the best in the detection of an external refractive index, in DNA detection, and in bacteria [48,49]. Etched and tapered fiber sensors are used to detect organic pollutants and heavy metals [50]. GFET-based sensors have attracted attention because of genetic disease diagnosis. DNA sensors are implemented by optical or electrochemical transducers. Because of graphene's high surface to volume ratio, high conductance and biocompatibility properties [32], these devices are preferred for use as sensors.

Limitations
Here, the VE gives solutions by assuming perfect conductivity and neglecting the heat flow. The standard kinetic approach based on the Boltzmann equation has difficulties when applied to a description of the plasma with long-range Coulomb interactions.
The theory of pair collision is not applicable to Coulomb interaction due to divergence of the kinetic term.
Because of the difficulties associated with the long range columb interaction Boltzmann equation,VE is used for the PC.

Conclusion and future perspectives
A nonlinear model is derived for high frequencies using the KT of plasma, which gives the current density and scattered electric field from a plasma channel. This model uses the BTE and KT of plasma for the charged particle flow. For equilibrium conditions in the collisionless case, the VE with ME and moment equations completely determine the charge particle dynamics in the plasma for short duration interactions with Coulomb force. The KT describes an incompressible flow of plasmas in phase space via the distribution function f, which changes only in momentum space. By taking the perturbation at the Fermi level and defining the nonlinear wave interaction between plasmons and electromagnetic waves with the help of BTE scattering by the PC, the conductivity of the channel is simulated. Using perturbation theory and Maxwell's equations, first-order perturbation at the Fermi level is derived in terms of zeroth-order perturbation. The simulation of the PC and graphene antennas shows that plasmonic materials can confine plasmons to multiple thin metal films and wires, which interact with external fields in the emission processes. This is observed in plasmonic antennas that are tunable via a gate voltage, which provides dynamic control of the current density. The results also prove that a graphene patch over the substrate produces a plasma transport channel for carriers, which gives current and charge distribution over the patch of graphene. These PCs can work in nanodevices and circuits and as antennas, where output scattered EMF is obtained by changing the applied voltage and sweep frequencies and has wide applications in THz and optical bands. Graphene is used in many applications in nanotechnologies, which can provide viable solutions for THz technologies as transceivers. These antennas also provide wide bandwidth, which can accommodate today's demand forlarge data transfer, ranging from gigabits to terabits per second. We observe that by changing the bias voltage, we observe the change in Fermi level and plasma flux due to the fact that there is moment variation of the charged particles, which results in a change in the susceptibility of the plasma medium. Hence, by combining the KT, VE, and KD discrete representation, the change in susceptibility is measured for a PC. The WTs of the current density, scattered electric field, and susceptibility arecalculated and the changes are measured in terms of wavelet coefficients. The advantage of using WTs is that we can achieve compression to a greater extent. This model for plasmonic nanoparticles can be used in many other applications for nanonetworks and sensing. The Vlasov, defining six dimensional phase space plus time, can be used in fully nonlinear kinetic regimes, which has immense applications in fusion science and the novel field of quantum electrodynamics in high-energy density plasmas [51]. Artificial-intelligence-based devices connected through nanodevices, e.g. plasmonic devices using graphene, can provide larger memory capacity and bandwidth. These nanodevices can support the high-speed rate of next generation technologies. Plasmonics is a way toward wired and wireless ultra-high bit-rate interconnections using nanoscale chip technology.

Data availability statement
All data that support the findings of this study are included within the article (and any supplementary files).

Conflict of interest
The author declares that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
When two nearby potential differences are ∆∅, then the Boltzmann relation for the electron is given by n e (∅ 2 ) = n e (∅ 1 ) e e∆∅/KBTe . (a.5) Charge density in the channel is defined by ρ c = q s ∫ f s d 3 v, and current density in the channel is given by The charge density change with time is obtained by integrating the VE [33] over the entire velocity space, defined bŷ Taking only the first-order perturbation term f (r, v, t) = f 0 + δf 1 , solutions are obtained in terms of the perturbation equation [33]. Here, the transport parameters [19] are related to the coefficients of f 1 . The change in the energy state is defined by the change in Fermi levels, which generates conductivity [52]. We put the values of f (equation (11)) in the equation (9) ∂ (f 0 + δf 1 ) ∂t By comparing the first-order perturbation terms, we get f 1 in terms of f 0 and E 0 and B 0 , where τ v is the collision time By comparing both sides' zeroth-order perturbations (a.10) Here, ∂(f0) ∂t = 0 in equilibrium conditions. If E 1 H 1 J 1 ρ 1 is generated in the PC antenna, then the first-order perturbation at the Fermi level is calculated in terms of the zeroth-order Fermi level. Solving equations (1), (14), (a.3), and (a.4), we get ∂f1 The s particle will feel the electric field produced by all the other charges. Hence, the mean field is given by where δt is the sufficiently long time interval. Here, δt ≫ 1 τc , where τ c is the mean time between particle collisions. The mean field contributes to the acceleration in the Boltzmann equation, and E contributes to the collision integral due to random evolution [19].

A1.4
Now, the perturbed Boltzmann equation is is the Maxwell's equilibrium density performing spatiotemporal FT of the equation. Spatiotemporal FT of the perturbed equation is given by (a.14) Solving equation (a.9) ∇ · D = ρ Gauss law ρ is the free charge density in the second term related to the Gauss law. Both x (t) and ψ (t) belong to L 2 (R); these are also called energy functions. For any value of ψ k,m (t), it is a shift of ψ k,0 (t) by the m value along the time axis. Here, m represents the time shift or translation. The ψ k,0 (t) is a time-scaled version and amplitude-scaled version of ψ (t) . It tells us the amount of scaling or dilation. It is referred to as a scale or dilation variable. CWT is generated by dilates and translates of a single function. Scaled and shifted replicas of wavelet basis functions form MRA in a sequence of nested vector subspaces … ⊂ V 1 ⊂ V 0 ⊂ V −1 ⊂ . . . such that the union of these spaces is dense in the square integrable functions L 2 (R) and the intersection of these subspaces is a null vector. If x(t) ∈ W k , then x(2t)∈ W k−1 and vice versa. The output current generated from the plasma channel at the antenna terminals is connected to the SP block with WTs. Similarly, the parameters of a channel, such as the pressure dyad and moment, is calculated with approximations defined by KD discrete representation [38]. WTs are calculated for equilibrium, the perturbed Fermi level function, the scattered electric field, and power. For detail, MRA is carried out by taking a complete orthonormal set of L 2 (R) , where ψ n (x) ∈ L 2 (R). This function ψ is a function of the translates and dilates of the basis function φ . The wavelet basis function is defined as If V n is a set of basis functions, then V n ⊂ L 2 (R) and V n ⊂ V n+1 ∀n ∈ Z, the conditions for MRA. Wavelet theory is based on this MRA. For each space in V n a φ n (x) exists, such that φ n 2 n x − k 2 n , where k ∈ Z is an orthonormal basis of V n and {V i } j ∈Z is a generalized MRA. In discrete analysis, if a basis function is φ (x) = K h k φ (2x − k), where n = 1, then φ is a scale function of MRA. Here, the 2 n factor gives the graph of contraction and expansion [2,5] in the direction of x.
Thus, {{ ψ n,k } n,k∈Z is a wavelet system, and then every function f 1 is written as [53] f = If we have the first-order perturbation equation (a. 25) In one dimension, r = x and υ → v (we initially assume there is no magnetic field) To observe the real-time behavior of f 1 (t, x, v) · ψ n1k1 (x) .