## 1. Introduction

In modern age Engineers have paved the way for a new generation of faster, more powerful cell phones, computers and other electronics by developing a practical technique to replace silicon with carbon on large surface. The capability of silicon, the material at the heart of computer chips has been harnessed beyond its limits by engineers and carbon has come up as an integrating replacement for the same. The material called“Graphene” which is a single layer of atoms arranged in honeycomb lattice could let electronics to process information and produce radio transmission 10 times better than silicon based devices.

For theorists, such a system is also of great interest because it provides a physical realization of two-dimensional field theories with quantum anomalies. Indeed, the continuum limit of the effective theory describing the electronic transport in graphene is that of two-dimensional massless Dirac fermions. The reported and predicted phenomena include the Klein paradox (the perfect transmission of relativistic particles through high and wide potential barriers), the anomalous quantum Hall effect induced by Berry phases and its corresponding modified Landau levels and the experimental observation of a minimal conductivity.

From the point of view of its electronic properties, graphene is a two-dimensional zero-gap semiconductor with the cone energy spectrum, and its low-energy quasiparticles are formally described by the Dirac-like Hamiltonian [1, 2].

Where *k* (for example, photons) but the role of the speed of light is played here by the Fermi velocity

From a crystallographic point of view, the graphene is a triangular Bravais lattice with a diamond-shaped unit tile consisting of two sites so one gets the honeycomb structure. The very unique feature of the graphene band structure is that the two lowest-energy bands, known as the valence and the conduction bands, touch at two isolated points located at the corners of the Brillouin zone. In the immediate vicinity of these degeneracy points, known as the Dirac points, the band structure is a cone. In natural graphene samples, there is exactly one electron per site, and thus, at zero temperature, all levels in the valence band are filled (a situation known as half-filling). As a result, the energy of the last occupied level precisely slices the band structure at the Dirac points. The low-energy excitations of this system are then described by the massless two-dimensional Weyl-Dirac equation and their energy dispersion relation

With this approach we can review different phenomena such: The Dirac point with a double-cone structure for optical fields, an optical analogy with Dirac fermions in graphene, can be realized in optically homogenous metamaterials. The condition for the realization of Dirac point in optical systems is the varying of refractive index from negative to zero and then to positive.

Also we give a support to the similitude of the band structure of a macroscopic photonic crystal with the electronic band structure of grapheme, which is experimentally much more difficult to access, allows the experimental study of various relativistic phenomena. With our analytical and numerical analysis we hope to verify that, similar behaviors exist to electrons in graphene treated as mass-less particle, ie, electron wave propagation.

This chapter presents a short review on the chiral propagation of electron waves in monolayer graphene and optical simulation with optical field in the negative-zero-positive index metamaterial NZPIM and its close connection. Section II presents an enhanced vector diagram of Maxwell’s equations for chiral media with quasi parallel electromagnetic fields,*Zitterbewegung* of optical pulses near the Dirac point inside a negative-zero-positive index metamaterial, and shows that the chiral field near the Dirac point becomes a diffusive wave The last sections describe the theoretical description of minimal conductivity in graphene under chiral approach and absorption of light by quasi 2D Dirac fermions.

## 2. A vector diagram of Maxwell’s equations for chiral media with E ∥ H

The idea of representing Maxwell’s time-harmonic equations in homogeneous isotropic media by vector diagram as put forward by Wilton [3] and by S. Uckun [4] deserves consideration. All the common relations between field and potential quantities implied by Maxwell’s equations can be represented by a diagram. It is started that the diagram not only illustrates Maxwell’s equations, but also many of the methods for constructing diagram are based on the formal similarity between many theorems of vector calculus and those of vector algebra.

An isotropic chiral medium is a macroscopically continuous medium composed of equivalent chiral objects that are uniformly distributed and randomly oriented. A chiral object is a three-dimensional body that cannot be brought into agreement with its mirror image by translation and rotation. An object of this sort has the property of handedness and must be either left-handed or right-handed. An object that is not chiral is said to be achiral, and thus all objects are either chiral o achiral. Due to their novel properties and wide applications in microwave and radar engineering, chiral media has been undergoing extensive research during the last years. That is why this study aims to cover chiral medium for the representation of Maxwell’s equations in vector diagram form. In a chiral media a cross coupling between electric and magnetic filed exists. Thus, the vector diagram has vectors along all three coordinate axes whereas the vector diagram presented by Wilton [3] for achiral media has vectors only in one plane with H vector normal to it.

### 2.1. Vector diagram construction

Assuming

Chirality is introduced into the theory by defining the following constitutive relations to describe the isotropic chiral medium [5]

Where the chirality admittance

For a graphical representation of the above relationships, following Wilton’s procedure [3], let us assume vector-differential operator,

As shown in Figure 1, three transverse coordinate axes are chosen as

Since

That is shown in Figure 1.

Following the Uckun’s approach [5], we substitute Equation (7) into Equation (6) having

Substituting Equation (9) and Equation (5) into Equation (2) gives

Here

And eliminate the term in parentheses in Equation (10). The choice in Equation (11) can be known as a chiral Lorentz gauge. Then Equation (10) will be simplified to

Divide both sides of Equation (12) by

( Figure 1, shows this vectorial equation).

The difference between our approach and the Uckun’s procedure [4], is that we take the chiral media characterized by

Placing the value of

by rearranging this equation we obtain

will be obtained as shown in Figure 1. In this figure we put

Taking divergence of Equation (5) and using Equations (3) and (4) in it

will be derived. To find the projection of

Similarly, getting gradient of both sides of Equation(11), using the vector identify

So it will be obtained as parallel component of A to

By using Equations (8), (13), (14), (16), and (17) the vector diagram of Lorenz gauge can be completed as shown in Figure 1, where all Maxwell’s relations and potential quantities appear.

Now let us examine derivation of some relations from the diagram. For example, it is seen that the component of

Taking the gradient in both sides of Equation (18) and dividing it by scalar value

This is the known continuity equation. Since the divergence of the curl of any vector is identically zero, the divergence of Equation (2) yields.

as expected. By using the vector calculus a few possible equations from the vector diagram can be written as follows

For example, adding Equations (13) and (14) side by side and using Equation (8) will give Equation (24) which shows the correctness of the equation derived from the diagram 1. Instead of Lorenz gauge we can choose Coulomb’s gauge.

In Equation (10) so that it will take the form

where the subscript

will be obtained. Placing the values of Equations (5) and (6) into Equation (2) will give

and value of

Combining these equations with Equation (26) and using Equation (7) we have

By using the same coordinates axes

As seen in Figure 1, Lorenz gauge are the best choice because these make

This Beltrami condition is useful to numerical calculations in graphene. We apply this approach to a two dimensional chiral graphene slab. This result cannot be obtained with the Uckun’s approach [4]. In terms of chiral magnetic potential

The dispersion relation of the transversal wave is

That is

So we have

The novel result here is that in our chiral theory we do not make

In the next section we study the situation when the refractive index is negative.

## 3. Chiral waves in graphene acting as metamaterial media

Metamaterials are composite materials in which both permittivity and permeability possess negative values at some frequencies has recently gained considerable attention [see e.g., [8-12]. This idea was originally initiated by Veselago in 1967, who theoretically studied plane wave propagation in a material whose permittivity and permeability were assumed to be simultaneously negative [11]. Recently Shelby, Smith, and Schultz constructed such a composite medium for the microwave regime, and experimentally showed the presence of anomalous refraction in this medium [10]. Previous theoretical study of electromagnetic wave interaction with omega media using the circuit-model approach had also revealed the possibility of having negative permittivity and permeability in omega media for certain range of frequencies [9].

The anomalous refraction at the boundary of such a medium with a conventional medium, and the fact that for a time-harmonic monochromatic plane wave the direction of the Poynting vector is antiparallel with the direction of phase velocity, can lead to exciting features that can be advantageous in design of novel devices and components. For instance, as a potential application of this material, compact cavity resonators in which a combination of a slab of conventional material and a slab of metamaterial with negative permittivity and permeability are possible. The problems of radiation, scattering, and guidance of electromagnetic waves in metamaterials with negative permittivity and permeability, and in media in which the combined paired layers of such media together with the conventional media are present, can possess very interesting features leading to various ideas for future potential applications such as phase conjugators, unconventional guided-wave structures, compact thin cavities, thin absorbing layers, high-impedance surfaces, to name a few. In this section, we will first present a brief overview of electromagnetic properties of the media with negative permittivity and permeability, and we will then discuss some ideas for potential applications of these materials.

Such a medium is therefore termed left-handed medium [12]. In addition to this ‘‘left-handed’’ characteristic, there are a number of other dramatically different propagation characteristics stemming from a simultaneous change of the signs of

The pseudoscalar

According to Maxwell’s equations, electromagnetic waves propagating in a homogeneous dielectric magnetic material are either positive or negative transverse circularly polarized waves, and can be expressed as

where

If the phase velocity and energy flow are in the same directions, and from Maxwell’s equation, one can see that the electric

To advance in our propose we considerer other more popular representation to describe a chiral medium, [12] as

in which electromagnetic coupling terms are added to the basic terms. Bi-isotropy or bianisotropy is used for calling such constitutive equations, according to the parameters to be scalars or tensors. If

There is a long dispute on strong chiral medium since it was introduced theoretically. Traditional electromagnetic conclusions have limited us to understand strong chirality, i.e.

In Ref. [14], the reason for traditional restriction of chirality parameters was concluded

as: 1) The wave vector of one eigenwave will be negative; 2) The requirement of a positive definite matrix to keep positive energy:

With the exploration of backward-wave medium, we know that negative wave vector, or opposite phase and group velocities, are actually realizable. And there is an unfortunate mathematical error in the second reason: in linear algebra, only if it is real and symmetric, positive definite matrix is equivalent to that all eigenvalues should be positive. The matrix (14) is a complex one, making the analysis on restriction of positive energy meaningless.

Actually, in a strong bi-isotropic medium with constitutive relations as Eqs. (1) and (2), the energy can be drawn as

Mathematically, the amount of energy density propagated is proportional to the magnitude of the Poynting vector

The concept of parallel fields is important in the theoretical formulation of: Space electromagnetism and vacuum, the classical and quantum gravitational fields, the study of elementary particles, operator and Dirac matrices, fields and chiral electrodynamics [15, 16]. If we put

In this case the total density energy is may be

In this special case where the energy propagated in one direction is equal to that propagated in the opposite direction, there is no net energy flow in the medium and the sum of the two TEM waves form what is generally known as a standing wave. The condition for a standing wave is that the time average of

In terms of Eqs. (41-42), if

First, with the assumption that

Second, it is the effectiveness of linear models. Similar to the case that linear optical and electromagnetic models can no longer deal with very strong optical intensity and electromagnetic field, we introduce nonlinear optics to take into account the higher order terms of polarization.

If the spatial dispersion is strong enough, the higher order coupling terms cannot be neglected as before. People used to mistake strong chirality with strong spatial dispersion, hence adding a limitation to chirality parameter,

However, the strong spatial dispersion is embodied in the BF model, e.g. the value of T, while the strong chirality is represented by the Pasteur model, e.g. the ratio of

Based on Eqs. (41)-(43), we have computed T and

From Figure 4. we see that near of

Solving Beltrami's equation with boundary conditions, we can demonstrate the optical *Zitterbewegung* effect by means of electromagnetic pulses propagating through a negative-zero-positive index metamaterial (NZPIM) if we make

Thus in optics, the Beltrami’s equation for electromagnetic waves can be reduced to the Helmholtz equation.

In later sections we discuss some effects such as *Zitterbewegung* of optical pulses, diffusion phenomenon and tunneling rate of dirac electron in graphene.

## 4. Two component equations and tunneling rate of dirac electron in graphene

The usual choice of an orthogonal set of four plane-wave solutions of the free-particle Dirac equation does not lend itself readily to direct and complete physical interpretation except in low energy approximation. A different choice of solutions can be made which yields a direct physical interpretation at all energies. Besides the separation of positive and negative energy states there is a further separation of states for which the spin is respectively parallel or antiparallel to the direction of the momentum vector. This can be obtained from the Maxwell’s equation without charges and current in the

where:

and *I* is the two-by-two identity matrix and the Fermi velocity

The Hamiltonian commutes with the momentum vector*H* and

Thus for a simultaneous eigenstate of *p* or –*p*, corresponding to states for which the spin is parallel or antiparallel, respectively, to the momentum vector like a graphene system.

A simultaneous eigenfunction of *H* and *p* will have the form of a plane wave

where the *E* can have either of the two values.

We now demand that

The eigenvalue equation is

Since

The physical interpretation of the solutions is now clear. Each solution represents a homogeneous beam of particles of definite momentum p, of definite energy, either

In this section, we study the tunneling rate of Dirac electrons in graphene through a barrier with an intense electromagnetic field. A one transport phenomenon in graphene is the chiral tunneling [1, 2, 17, 18]. In mono layer graphene a perfect transmission through a potential barrier in the normal direction is expected. This tunneling effect is due to the chirality of the Dirac electrons, which prevents backscattering in general. This kind of reflectionless transmission is independent of the strength of the potential, which limits the development of graphene-based field-effect transistors (FET). The perfect transmission can be suppressed effectively when the chiral symmetry of the Dirac electrons is broken by a laser field, when the n-p junctions in graphene are irradiated by an electromagnetic field in the resonant condition [19,20].We consider a rectangular potential barrier with height

where

where *e* is the electron charge and the chiral potential vector is

Since the tunneling time is of order of sub-picosecond and the potential

In order to study such a strongly time-dependent scattering process, we employ the finite- difference time-domain (FDTD) method to solve Eq. (47) and Eq. (48) numerically in the time-domain [21]. In the traditional FDTD method, the Maxwell’s equations are discretized by using central-difference approximations of the space and time partial derivatives. As a time-domain technique, the FDTD method can demonstrate the propagation of electromagnetic fields through a model in real time. Similar to the discretization of Maxwell’s equations in FDTD, we denote a grid point of the space and time as

For any function of space and time

These eqs. can be replaced by a finite set of finite differential equations like:

For computational stability, the space increment and the time increment need to satisfy the relation

At the boundary, one-dimensional Mur absorbing boundary conditions are used [21-22]. To compare our results with [25] which use linear polarization for the vector potential

Numerical simulations are shown in Fig. 7. The following parameters are used in our calculation: the peak position

When there is no pump beams, a perfect chiral tunneling can be found [see Fig. 7 (a)]. This result is consistent with that of Ling et al. [25]. But when the sample is irradiated by an intense non resonant laser beam, a reflected wave packet appears [see Fig. 7 (d)]. The perfect transmission is suppressed. By analyzing the transmitted wave packet and the reflected wave packet, we can obtain the tunneling rate.

To explain the suppression of chiral tunneling, we first investigate the chiral potential wave in the barrier within a rotating wave approximation [23, 24]. Figure 7 (a) shows the renormalized band as a function of momentum k with intensity

Here, the important point is that we make

Under intense light beams, the dressed states are strongly mixed with valence states and conduction states. Therefore, the chiral symmetry of Dirac electrons in graphene can be broken and perfect chiral tunneling is strongly suppressed. Numerical results are shown in Fig. 8 (left) with pump intensity

Figure 8 (left), show that the reflectance decreases, and the transmittance increases as

## 5. *Zitterbewegung* of optical pulses near the Dirac point inside a negative-zero-positive index metamaterial

This optical analog of the *Zitterbewegung* effect is similar to two-dimensional (2DPC) photonic crystal.

The *Zitterbewegung* effect can appear in solids, because the electron at the touching point between two interacting bands in a solid also obeys the massless Dirac equation [26]. In recent years, a growing attention has been devoted to the simulations or demonstrations of the *Zitterbewegung* effects in controllable physical systems, such as, ultra cold atoms, superconductors, semiconductor nanostructures with spin-orbit coupling, a single trapped ion, and graphene.

For a homogenous medium, when a light field is polarized in the *z* direction, the Beltrami’s equation

with a wave number

In the case of grapheme near of

where*z*-direction) corresponding to the same frequency

Note that the positive and negative branches of the band structure coexist. Equation (58) is the massless Dirac equation of the light fields in homogenous materials, which is the same as that of electrons in graphene [27]. Therefore, for the NZPI metamaterial [satisfying Eq. (58)], we will have the Dirac point with a double-cone structure for the light field at frequency

which are the same as that of the massless Dirac equation at zero energy. [29]. Therefore, it can be predicted that the behavior of light fields at *L* have the total energy transmittance

which tells us that the propagation of light field at (or near) *L* / 1 scaling, a main characteristic of the diffusion phenomenon. This effect can be obtained if

Here, the Dirac point with the double-cone structure for the light field can also be realized in a homogenous negative-zero-positive index (NZPI) medium [26], in which the two-dimensional Helmholtz equation could be written as the two-dimensional massless Dirac equation. The condition for the realization of the Dirac point in the homogenous optical medium is the index varying from negative to zero and then to positive with frequency, and the light field also obeys eq. (45) near the Dirac point.. Thus, in this work, we show that the *Zitterbewegung* effect with optical pulses appears near the Dirac point in NZPIM slabs. With the realization of the NZPIMs in experiments, we believe that it already has a great possibility to observe *Zitterbewegung* effect with optical pulses in the GHz region.

## 6. Theoretical description of minimal conductivity in graphene under chiral approach

The first results on the theoretical combination of graphene electron transport properties and a characteristic property of Dirac chiral fermions were obtained by Katsnelson et al, [2]. The intrinsic nature of Dirac fermions gives rise to minimal conductivity even for an ideal crystal, that is, without any scattering processes. The simplest way for the theoretical consideration is the Landauer approach [29]. Assuming that the sample is a ring of length *y* direction; author used the Landauer formula to calculate the conductance in the *x* direction (Fig. 9).

The convenient boundary conditions are not physical, but to get finite transparency one should choose*.*

In the coordinate representation the Dirac equation (59) at zero energy takes the form (

The solutions of these equations are

Due to periodicity in *y* direction both wave functions should be proportional to*, n* = 0*,±*1*,±*2*,...* This means that the dependence on the *x* is also fixed:

the wave functions are proportional to*x* = 0 and*.* The wave functions in the sample are supposed to have the same *y*-dependence, that is,*.*

Requiring continuity of the each wave function at the edges of sample, one can find the transmission coefficient:

Where

Further, one should assume that

The conductance then equals

Zitterbewegung − circular motion of elementary particles caused by an interference between positive and negative energy states − leads to the fluctuation of the position of an electron. This relativistic “jittering” of an electron in graphene could be interpreted in terms of classical physics as an interaction of electron with some potential caused by the presence of disorder in crystal. Therefore, the Zitterbewegung plays a role of “intrinsic” disorder in the system which appears in the presence of minimal conductivity of the ideal crystal (without scattering) even at zero temperatures.

Under this chiral approach, we show that the optical transparency of suspended graphene can be defined by the fine structure constant,

There is a small group of phenomena in condensed matter physics, which are defined only by the fundamental constants and do not depend on material parameters. Examples are the resistivity quantum *h* is the Planck constant and *e* the elementary charge). By and large, it requires sophisticated facilities and special measurement conditions to observe any of these phenomena. Here, we show that such a simple observable as the visible transparency of graphene [1] is defined by the fine structure constant,

Optical properties of thin films are commonly described in terms of dynamic or optical conductivity *G*. For a 2D Dirac spectrum with a conical dispersion relation *G* was theoretically predicted [32, 33] to exhibit a universal value*E* is much larger than both temperature and Fermi energy*G* also implies that all optical properties of graphene (its transmittance *T*, absorption *R*) can be expressed through fundamental constants only (*T*, *R* are unequivocally related to *G* in the 2D case). In particular, it was noted by Kuzmenko *et al* [34] that *G* – both *T* and *R* are observable quantities that can be measured directly by using graphene membranes.

## 7. Absorption of light by quasi 2D Dirac fermions

Here, following [35], we show how the universal value of graphene’s opacity can be understood qualitatively, without calculating its dynamic conductivity. Let a light wave with electric field

Taking into account the momentum conservation*D* is the density of states at

The interaction between light and Dirac fermions is generally described by the Hamiltonian

where the first term is the standard Hamiltonian for 2D Dirac quasiparticles in graphene [1] and

Here

This results in *R*<<1 as discussed above), its opacity (1 – *T*) is dominated by the derived expression for

In the case of a zero-gap semiconductor with a parabolic spectrum (e.g., bilayer graphene at low ε), the same analysis based on Fermi’s golden rule yields

On a more general note, graphene’s Hamiltonian *H* has the same structure as for relativistic electrons (except for coefficient *v*F instead of the speed of light *c*). The interaction of light with relativistic particles is described by a coupling constant, a.k.a. the fine structure constant. The Fermi velocity is only a prefactor for both Hamiltonians

Thus, we have found that the visible opacity of suspended graphene is given by π α within a few percent accuracy and increases proportionally to the number of layers *N* for few-layer graphene. Its dynamic conductivity at visible frequencies is remarkably close to the universal value of

Electrons from the valance band (bottom) are excited into empty states in the conduction band with conserving their momentum and gaining the energy *E*= h ω.

The approximation of 2D Dirac fermions is valid for graphene only close to the Dirac point and, for higher energies ε, one has to take into account such effects in graphene’s band structure as triangular warping and nonlinearity.

Most theories suggest

## 8. Conclusions

In this chapter we presented a short review on the chiral propagation of electron waves in monolayer graphene and optical simulation with optical field in the negative-zero-positive index metamaterial NZPIM and its close connection. Section II presented an enhanced vector diagram of Maxwell’s equations for chiral media with quasi parallel electromagnetic fields,*Zitterbewegung* of optical pulses near the Dirac point inside a negative-zero-positive index metamaterial, showed that the chiral field near the Dirac point becomes a diffusive wave. The last sections described the theoretical description of minimal conductivity in graphene under chiral approach and absorption of light by quasi 2D Dirac fermions.

We reviewed the formulation of graphene’s massless Dirac Hamiltonian, under the chiral electromagnetism approach, like a metamaterial medium, hopefully demystifying the material’s unusual chiral, relativistic, effective theory. The novel result here was that in our theory we did not make

There is a small group of phenomena in condensed matter physics, which are defined only by the fundamental constants and do not depend on material parameters. Examples are the resistivity quantum

Also we give a support to the similitude of the band structure of a macroscopic photonic crystal with the electronic band structure of graphene, which is experimentally much more difficult to access, allows the experimental study of various relativistic phenomena. With our analytical and numerical analysis we hope to verify that, similar behaviors exist to electrons in graphene treated as mass-less particle, ie, electron wave discovered only 8 years ago graphene is already one of the most studied carbon allotropes. But this material still poses a lot of theoretical and experimental questions, which have to be answered.

## Acknowledgments

I wish to thank to colleague Jorge Benavides Silva of the EIEE for many useful discussions on particles and photons