Initial Finite-Difference Time-Domain (FDTD) Modeling of Graphene Based on Intra-band Surface Conductivity

Graphene is a single two-dimensional layer of carbon atoms arranged in a hexagonal lattice, possesses interesting optical properties, and has potential for applications in optical devices. Graphene exhibits tunable surface conductivity, which arises from its electronic band structure. Graphene surface conductivity is determined by its chemical potential, which can be controlled by bias voltage and/or chemical doping. The tunability of surface conductivity allowed to tailored optical properties of graphene, making it a controllable material for optoelectronic applications. Graphene surface conductivity is applied to update the field values at each time step in the Finite-Difference Time-Domain (FDTD) method, enabling us to visualize electromagnetic (EM) wave propagation in graphene. The current article serves as a starting point for developing the FDTD approach to simulate EM wave interactions with graphene, particularly at low frequencies. In this study, we use the Kubo formula for low EM wave frequency (10-105 GHz) at ambient temperature to calculate the intra-band surface conductivity of graphene. The outcome shows that the imaginer’s intra-band surface conductivity value is relatively considerable compared to the actual value at frequencies between 102 and 104. Moreover, the chemical potential exhibits a positive linear relationship with the imaginer intra-band surface conductivity and the intra-band conductivity falls to zero as the frequency rises to NIR.


Introduction
Graphene is a two-dimensional allotrope of carbon consisting of a single layer of carbon atoms arranged in a hexagonal lattice.It is the fundamental structural element of other carbon allotropes, such as graphite or charcoal, carbon nanotubes and fullerenes [1].Due to its unique features, graphene has attracted a lot of interest in scientific research and prospective uses.It has been determined through numerous investigations that graphene exhibits high carrier mobility ranging from 8000-10000cm 2 Vs ⁄ , which is two orders of magnitude greater than the carrier mobility of metals like gold, copper, and silver, which is between 30-50 cm 2 Vs ⁄ [2].Graphene has excellent transparency to visible light, with only a negligible amount of light passing through it being absorbed [3].Graphene electrical conductivity and optical quality make it a potential material for next-generation electronic and optical components, such as transistors, flexible displays, touchscreens, and photodetectors.
Due to its distinct electrical structure and two-dimensional nature, graphene shows several unique optical features and high optical transparency.High optical conductivity (surface conductivity) of graphene is present over a broad frequency range, encompassing the visible and near-infrared (near-IR) spectrums [4].The conductivity depends on the electrons' reactivity to incident electromagnet (EM) fields and involvement in charge carrier production and transport [5,6].Additionally, there are numerous ways to dynamically modify the optical characteristics of graphene, including adding a bias voltage and/or chemical doping [7].The charge density in the graphene sheet can be changed by applying a voltage to a neighboring electrode, changing the surface conductivity.Changes in the graphene's carrier density can be actively exploited to alter its transparency, absorption, reflectivity, and how the material interacts with EM fields more generally [8,9].The special surface conductivity of graphene and its modified optical characteristics make it a promising material for applications in transformation optics.
Surface conductivity can be used to represent optical features of graphene that are closely connected to carrier density.Through various methods, including the frequency of the incident EM field and an external bias voltage, graphene surface conductivity has demonstrated transient nature and can be manipulated dynamically [7,10].Therefore, analyzing the EM field interaction with graphene and its optical properties is impossible using an analytical solution.Numerical analysis is crucial for studying and comprehending EM wave interaction with graphene and the optical characteristics of graphene.Due to the ability to incorporate the surface conductivity of graphene into algorithms, the finite-difference time-domain (FDTD) approach, among other numerical methods, can be used as an efficient numerical analysis [11].It's important to note that developing an FDTD simulation of graphene might be challenging due to the material's frequency-dependent (frequency dispersion) electrical behavior and planar nature.
In this research, we provide a preliminary design of a graphene FDTD modeling system based on the modulated surface conductivity of graphene, which depends on the frequency of EM waves and the bias voltage.Analysis begins with a quantitative examination of graphene surface conductivity as a function of frequency, particularly at low frequencies, and chemical potential related to an applied bias voltage.Using the piecewise linear recursive convolution (PLRC) method and the sub-cell perfectly matched layer (PML), we intend to create numerical methods with various approximations to address frequency dispersion and planar characteristics in the future study.

Method
Maxwell's equations are solved numerically using the finite-difference time-domain (FDTD) method to simulate the interaction of EM waves with various materials, including graphene.A time-domain technique called FDTD directly simulates the propagation and interactions of electromagnetic waves across time.The graphene sheet must be discretized into a grid of cells to use FDTD to simulate it, and each grid point's EM field values must be updated in small time steps.The electrical characteristics of graphene at each grid are required to update the EM field value at each grid.Our primary goal in this study is to characterize graphene, specifically its dispersive surface conductivity due to its twodimensional structure and the band structure that creates linear charge carrier dispersion relations (Dirac cones) [12][13][14].Graphene's surface conductivity dictates its optical and electrical properties and defines how it reacts to an electromagnetic field.
Different models can be used to express the frequency-dependent surface conductivity of graphene.The Kubo formula is one widely employed model.The electrical conductivity of materials, including graphene, can be determined using the Kubo formula, an advanced theoretical framework [15].The Kubo formula considers the interplay between charge carriers' quantum mechanical nature and the crystal lattice.The Kubo formula calculates the current-current correlation function to link the material's conductivity to its optical response.Kubo formula can be expressed as Equation (1): where   () is the Fermi-Dirac distribution expressed as: is the angular frequency of the EM field in radians,  −1 is the scattering rate in  −1 , which depends on crystal structure of graphene,   is the chemical potential in eV, which can be modulated by applying a bias voltage and/or chemical doping,  is the temperature in Kelvin, and   is Boltzmann constant.
In addition to covering a wider frequency range for EM waves, the Kubo formula gives a more thorough explanation of the conductivity of graphene [16].The first term in equation ( 1) is related to intra-band conductivity, and the second is related to inter-band conductivity.Intra-band conductivity describes conductivity resulting from charge carrier mobility within a single especially the conduction or valence band.On the other hand, beyond the mid-infrared (mid-IR) wavelengths, optical excitation is elucidated by inter-band conductivity, which quantifies the contribution of charge carriers between bands [17].Although both intra-band and inter-band contributions must be taken into account to describe the conductivity of graphene, in this study, we focus only on the interaction of low-frequency electromagnetic waves with graphene.Up to about the far-infrared (far-IR), the intra-band conductivity primarily accounts for low EM wave frequencies [17,18].The simplified expression for the intra-band conductivity,   , in graphene can be given as: Substitute Equation ( 7) and ( 8) into Equation (9).
This study determines intra-band surface conductivity at microwave to far infrared (FIR), 10-10 5 GHz, at room temperature, and chemical potential variations between 0-0.8 eV.

Results and Discussion
As Equation ( 8) shows, intra-band surface conductivity is a complex quantity.The imaginary part accounts for the graphene's capacitive response to an applied electric field, while the real part accounts for the material's conductivity.Additionally, intra-band surface conductivity is frequency-dependent and susceptible to changes in chemical potential,   , using an electrostatic bias voltage.Figure 1 shows the corresponding intra-band conductivity, where the parameters of graphene are:   = 0.5 eV,  = 0.2 ps and  = 300 K.The real and imaginary components of intra-band conductivity from equation ( 8) are shown in Figure 1.We may observe that the imaginer conductivity gradually grows at 100 GHz and exhibits a ratio increase compared to the real conductivity.At frequencies beyond 1000 GHz, however, the imaginary conductivity increases while the real component decreases to zero, which has a noticeable impact on graphene's optical and electrical responsiveness.Inter band conductivity appears to be more significant at frequencies above near-infrared (NIR) in determining how EM waves interact with graphene because real and imaginable conductivity drops to zero as frequency rises to near-infrared (NIR) levels.Designing applications like graphene-based optical transformations depends heavily on the complex conductivity of graphene, which is an essential parameter in understanding how it interacts with EM waves [19].As seen in figure 2, the tunable and considerable imaginary conductivity of graphene allows for the customization of unique electromagnetic field patterns.Figure 2 depicts Fresnel reflection at the interface of two media with various reflecting indices or surface conductivity.The Fresnel reflection occurs at the boundary between a medium with real surface conductivity, such as a perfect dielectric material, and a media with imaginary surface conductivity, which does not permit the passage of EM waves because of absorption [7].
The carrier concentration in graphene can be changed via chemical potential, \mu_c, y applying a bias voltage to an electrode, resulting in a different conductivity value at a different frequency.Figure 3 depicts the real and imaginary intra-band conductivity component as a chemical potential and frequency function, where  = 0.2 ps,  = 300 K, and chemical potential between 0 to 0.8 eV.It is clear that at low-value chemical potential,   < 0.2 eV, imaginer and real surface conductivity contributions are equal at frequencies between 100-1000 GHz.On the other hand, graphene is dominated by dielectric-like conductivity at high chemical potential l  ≥ 0.3 eV and low frequency (below 500 GHz) will enable electromagnetic propagation through graphene.This permits the application of graphene to specific EM waveguides.Compared to real conductivity, imaginary intra-band conductivity predominates at high chemical potential,   > 0.2 eV and at a frequency above 10 3 GHz.This property enables "on and off" control of EM wave propagation through graphene, which has the potential to be used as a sensor.Additionally, imaginer conductivity at frequencies of about 10 3 GHz steadily increases as chemical potential increases, allowing for control of the depth of an EM wave's penetration into graphene.All of these characteristics make graphene particularly desired for a variety of applications in electronics, optoelectronics, and sensors because they provide dynamic control over the electrical and optical properties of the material.Additionally, it is reliant on the frequency of EM waves.
The frequency-dependent conductivity values can be assigned to the graphene cells within the computational grid in order to include the surface conductivity in a finite-difference time-domain (FDTD) simulation.The conductivity terms would subsequently be included in the updated equations for the EM fields, enabling precise modeling of EM wave interactions with graphene.Equation ( 8) is divided by the thickness of the graphene to convert surface conductivity to volumetric conductivity before being inserted into numerical, ∆.The average graphene thickness is less than the EM wave's minimum wavelength.The volumetric conductivity can be expressed as: In the upcoming investigation, we will use the equivalent volumetric current,   , represented as the FDTD algorithm to achieve the dispersive property of the formulation.
Where  is the electric field at the same grid point with   .Ampère's law in the finite-difference model is used to acquire updates to the electric and magnetic fields at each grid point in small time intervals.The integral form of Ampère's law at a normal to graphene plane, e.g., xy-plane expressed as () is the Dirac Delta function implemented to perverse the finite volumetric characteristic of the current.Using the volumetric current equation ( 9) substituted with the finite-difference scheme equation (11) enables simulating time-domain behavior.This makes it possible to fully comprehend graphene's temporal reactivity, capture transient effects, and research ultrafast dynamics in graphene-based systems.Additionally, FDTD makes it possible to incorporate graphene's special material characteristics, such as its two-dimensionality and surface conductivity.FDTD simulations offer a realistic portrayal of graphene's behavior and its effect on EM wave propagation by effectively mimicking these features.

Conclusion
Intra-band conductivity is the main factor in our investigation of low frequency between 10-10 5 GHz or in the region of microwave-FIR EM wave interaction with graphene.Additionally, depending on the frequency and chemical potential, the EM wave and graphene interaction changes in this rage.Graphene exhibits a dielectric property at frequencies below 10 2 GHz and a constant chemical potential of 0.5 eV, permitting EM wave transmission through it.Imaginer intra-band surface conductivity steadily increases and becomes comparable to real conductivity at frequency between 10 2 -10 3 GHz, exhibiting the lossy dielectric property of graphene.Surface conductivity over 10 3 GHz is dominated by imaginer surface conductivity, and when the frequency approaches the near-infrared range, both real and imaginer intraband conductivity drop to zero, indicating that the interaction between EM waves and graphene at high frequencies is primarily driven by inter-band conductivity.However, our research demonstrates that by altering the chemical potential of graphene, we can dynamically change how EM waves interact with it at low frequencies.Due to increases in real conductivity at low frequencies, high chemical potential reduces graphene's lossy dielectric characteristic.At a frequency around 10 3 GHz, imaginer conductivity increases as chemical potential increases, while as the frequency increases, all chemical potential shows the same characteristic where intra-band conductivity decreases to 0. Graphene is highly desired for various electronics, optoelectronics, and sensor applications because of the frequency and chemical potential dependence of EM waves-graphene interaction.