Study the optical and conductivity properties of two-layer graphene using mathematical modelling

Graphene is a two-dimensional allotropic form of carbon, with atoms combined into a hexagonal crystal lattice, forming a layer one atom thick. Graphene possessing exceptional optical, mechanical, physical and electrophysical properties. This article details some of the conductivityand optical in graphene. The conductivity of the two-layer graphene was measured by special mathematical models, with conductivity dependent on the voltage at the gate at different temperatures. Conductivity is faced with a jump at zero temperature and a sharp, standard rise in properties at room temperature, as shown clearly in the resulting electronic spectrum.


Introduction
Graphene is a two-dimensional allotropic form of carbon, which is formed by a layer of carbon atoms one atom thick, located in sp² hybridization and connected via σ and π bonds to a two-dimensional hexagonal crystal lattice (Fig. 1). The distance between adjacent atoms of the crystal lattice is graphene is 0.142 nm. Graphene can be represented as one atomic plane of a natural carbon speciesgraphite, separated from a bulk crystal [1].
For a long time, scientists could not get stable graphene samples. The atomic plane of graphene sought to minimize its surface energy and curled up, transformed into various allotropic forms of carbonamorphous carbon, carbon nanotubes and fullerenes [2].
Only in 2004, a group of scientists from the University of Manchester and the Institute of Microelectronics Technology Problems in Chernogolovka under the guidance of Andre Geimand Konstantin Novoselov obtained graphene using ordinary tape and tape. The graphite synthesized by them was transferred onto a 300 nm thick silicon oxide substrate. Due to light interference, sections of different plate thicknesses had different colors [3]. The thinnest sections of this plate were almost colorless. Among them, using atomic force microscopy, the structure of one atomic plane of graphene was discovered [4].
The new carbon modification has unique properties. The mechanical strength of graphene is twice that of steel. The thermal conductivity of graphene is 10 times that of copper. Graphene is characterized by a linear law of electron dispersion. Quantums of a light wave or electromagnetic radiationphotons, which are massless particles that propagate in space at the speed of light, have the same energy dependence on momentum. In graphene, electrons, like photons, have no mass. But their charge is not equal to zero and the speed of their movement is 300 times less than that of c.
Due to the linear dispersion law, graphene is a semimetal, which is a semiconductor with zero band gap and conductivity not inferior to the conductivity of copper. The mobility of electron graphene is greater than that of all known materials (100 times greater than that of silicon).
Graphene also has unique optical properties [5]. The transmission coefficient of the light wave for graphene in the region of visible light is described by the formula: where α is the fine structure constant. Thus, graphene absorbs only about 2.3% of the light waves of the visible radiation range. That is, graphene plates are almost transparent.
The transparency, good electrical conductivity and elasticity of graphene make it a promising material for the creation of solar cells and touch displays. In all respects, graphene is much better suited for such devices than the indium tin oxide currently used [6,7].
To calculate the electron current density in Graphene, we use an analog of the Landauer-Buttiker formula for the two-dimensional case, in other words, we calculate the difference in the fluxes of electrons moving from the source to the sink and back.

Calculation of the energy spectrum of grapheme
We can calculate the energy spectrum of the Craven from applying the Schrödinger equation: We represent the function  as an expansion in basis functions (not always orthogonal), numbered 1 n : Substitute equation (3) where In order for the system of homogeneous equations to have a nonzero solution, it is necessary that the determinant be equal to zero.
We reduce the solution of this equation to the usual eigenvalue problem. We write the equation in matrix form where , 1 1,0,1 nn  . We can represent the matrix H in the form   (2) is that the prohibited gap changes in the direction of the x-axis and the y-axis, that is, the properties of the material change as well.
By showing the energy gap (band gap) we can determine if the material is conductive or semiconductive or insulating and from here we can say that the graphene material has a promising future by determining the type of material conductivity and optical properties where it can be used in the manufacture of optical and electronic devices, laser devices and reagents Photovoltaic and solar cell screens.But the difference between Figure 1 and 2 is that the prohibited gap changes in the direction of the x-axis and y-axis, that is, the properties of the material change as well.

Calculation of conductivity and current density
To calculate the electron current density in Graphene, we use an analog of the Landauer-Buttiker formula for the two-dimensional case, in other words, we calculate the difference in the fluxes of electrons moving from the source to the sink and back [8]: are the electron distribution functions at the source, is the group velocity of the electron, and the transparency coefficient of the barrier D depends from the energy and momentum of a flightless particle. In our model, we neglect the tunneling effect (the calculation of this phenomenon has already been done before us), therefore, the  Summarizing the above reasoning, we can "cut off" the integration limits in the expression for the current (12), leaving only those momenta and energies for which the transmission coefficient is unity: , .  (14) it is convenient to make the change of variable ~e      , which can be interpreted as the transfer of the integration domain to the contacts [8]. The expression for the current takes the form:  (18) can be used to analytically calculate the conductivity as a function of the gate voltage at zero temperature. This problem, however, reduces to solving an algebraic equation of the fourth degree; the results of the solution are not visual. At finite temperatures, a semi-analytical calculation of conductivity is possible [9].
We substitute the expansion for the current expression: It is now interesting to trace the evolution of conductivity and linear current density as a function of the gate voltage with changing temperaturein Fig. 4 and fig. Figure 5 shows graphs for three temperatures: helium   Analyzing the obtained curves, we see that the ratio of the currents on and off on two-layer graphene is of the order of 10 at room temperature and a gate voltage variation range 0.5B V. That is why the methods for increasing the band gap in two-layer graphene attract considerable attention of researchers. we can say that the characteristic at room temperature inherits from the jump in conductivity at 0 T  a high slope. In the next subsection, we estimate this value analytically. When the Fermi energy coincides with the temperature, it can be expressed in the following formula [11]: If at zero temperature the jump in conductivity is equal to ∆G, then the derivative of conductivity at the point of the jump at a nonzero temperature can be estimated as In short form, the voltage at the gate where a and b are functions of both the spectrum of bilayer graphene and the structure parameters. For the above structural parameters (temperature is expressed in electron volts) We note that the presence of a term proportional to T in the denominator occurs again due to the presence of an isotropic shift along the momentum axis in the spectrum of bilayer graphene. Figure 6 shows a comparison of the steepness calculated by the estimation formula (solid line) with the values obtained by an exact numerical calculation. The slope value for two-layer graphene with the indicated structure and spectrum parameters is 55 · 10 3 (Ohm · m · V) −1, which is an order of magnitude greater than the similar value for the type "silicon on the insulator" (5 · 10 3 (Ohm · m · V) −1), which is considered as one of the most promising in silicon electronics [11,12].

Results
The aim of this work is to study the photon effect on graphene and carbon nanotubes.
In considering the photon effect on graphene and carbon nanotubes, the following conclusions were drawn: • The high transmission coefficient of the light wave of graphene in combination with good electrical conductivity and elasticity make it a promising material for the creation of solar cells and touch displays. • Only semiconductor modifications of graphene have optical properties. • Graphene semiconductors are the best electronic emitters. The reason for this is its high electrical conductivity. • Due to the sharp tip of graphene, high emission is possible even when low voltage is applied.
The resulting emission current has a high density and extreme stability. • Owing to their optical and semiconductor properties, subminiature size, good electrical conductivity, high emission characteristics and the ability to attach chemical graphene radicals, they have a wide range of applications in optoelectronics, measurement technology, chemical technology and biomedicine. Therefore, further detailed study of the properties of graphene is necessary. • The problem of developing economical technologies for the mass production of graphene having predetermined properties and controlled sizes is one of the most important tasks of modern science. • By calculating the energy spectrum of graphene, it was found that it varies with different axes (x, y), and also with the different energy gaps for graphene. We can determine whether the material is conductive, semi-conductive, or dielectric. Also, by finding the energy spectrum for graphene we can determine its optical properties. This means that it is possible The use of graphene in the manufacture of modern optical and electronic devices and laser devices.

Conclusion
To determine the promising capabilities of graphene-based devices, mathematical models are widely used due to technological adherence to the manufacture of real samples. The graphene conductivity model is explained by the graphene resistance depending on the voltage at the control electrode. It turns out that the maximum resistance resulting from the spread of a strong electron hole depends poorly on temperature. The characteristics of the resulting electronic spectrum show that the conductivity at zero temperature faces a jump, and this indicates the sharp and standard height of the properties at room temperature. Dual-wall graphene is used because of its rigidity and flexibility and is useful for devices that depend on the effect of photons on the graphene because the photon is emitted under the influence of the electric field by high magnetic fields.
It has been found that by calculating the energy spectrum of graphene we can determine its conductive and optical properties. This means that graphene can be used in the manufacture of modern optical and electronic devices and laser devices.