**2. Graphene**

**1. Introduction**

112 Graphene - New Trends and Developments

equation:

the form

enough energy left, the process may be repeated.

process with the symbolic equation

*dσ <sup>d</sup>*<sup>Ω</sup> <sup>=</sup> <sup>1</sup> 2

*e*4 *m*2*c*<sup>4</sup>  *ω ω*

The Compton effect is the light-particle interaction where the wavelength of scattered photon is changed. The difference between the Compton effect and the Einstein photoeffect consists in the fact that during the photoeffect, the energy of photon is transmitted to electron totally. Compton used in his original experiment [1] the energy of the X-ray photon (∼20 keV) which was very much larger than the binding energy of the atomic electron, so the electrons could be treated as being free. Compton scattering usually refers to the interaction involving only the electrons of an atom. However, the nuclear Compton effect was confirmed too. The effect is important because it demonstrates that light cannot be explained purely as a wave phenomenon. The Compton experiment proved that light is composed of particle-like objects with energy *E* = *h*¯ *ω*. The interaction between electrons and high-energy photons is such that the electron takes part of the photon initial energy, and a photon containing the remaining energy is emitted in a different direction from the original. If the scattered photon still has

Compton, in his paper [1], derived a simple formula relating the shift of wavelength to the scattering angle of the X-rays by postulating that each scattered X-ray photon interacted with only one electron. His paper involves the information on experiments for verification of the

where *λ* is wavelength of the scattered X-ray and *θ* is the angle between the incident and scattered X-ray. The scattering was considered in the laboratory frame where electron was at rest. Let us remark immediately that eq. (1) has also a limit for *m* → 0, if and only if *θ* → 0. It corresponds to the situation of graphene sheet, where the mass of the so-called pseudoelectron can be considered zero. This limit can be verified immediately by the visible light and not by the X-rays. In case of using the X-rays in graphene, we get still the original Compton process with the real electrons (ionization process in graphene) and not the process with the so-called pseudoelectrons. The considered process was the so-called one-photon

The differential cross section corresponding to eq. (2) was derived by Klein and Nishina in

<sup>2</sup> *ω ω* + *ω <sup>ω</sup>* <sup>−</sup> sin2 *<sup>θ</sup>*

It is the ratio of the number of scattered photons into the unit solid angle Ω over the number of incident photons. At the present time with the high-power lasers, there is a possibility

*mc* (<sup>1</sup> <sup>−</sup> cos *<sup>θ</sup>*), (1)

*γ* + *e* → *γ* + *e*. (2)

. (3)

*<sup>λ</sup>* <sup>−</sup> *<sup>λ</sup>* <sup>=</sup> *<sup>h</sup>*

Graphene is a two-dimensional carbon sheet which is very consistent (100 times stronger than steel). It is a very good conductor of heat and electricity. Graphene was investigated by theorists for decades; however, it was first generated in the laboratory in 2003. Being two-dimensional, it interacts in a special way with light and with other materials. Researchers have discovered the bipolar transistor effect, the charge ballistic transport, and the large quantum oscillations.

Technically, graphene is a crystalline allotrope of carbon with two-dimensional geometrical properties. More than 70 years ago, Peierls [3] and Landau [4] proved that the two-dimensional crystal is not stable from the viewpoint of thermodynamics and cannot exist. The thermodynamic displacements of atoms in such a crystal are of the same size as the distances between atoms at any finite temperature. Mermin [5] accepted the theoretical arguments in 1968, and it seemed that many experimental observations were in accord with the Landau-Peierls-Mermin theory.

However, in 2004, Geim and Novoselov [6], [7] with co-workers at the University of Manchester in England produced a crystalline sheet of carbon just one-atom thick. Then, the Geim group was able to isolate graphene and was able to visualize the new crystal medium using a simple optical microscope. The Landau-Peierls-Mermin proof remained as the historical document.

After some time, the new sophisticated methods generating graphene sheets were invented. The graphene sheets were, e.g., synthesized by passing liquid ethanol droplets into an argon plasma. The authors of this method are Dato et al. [8].

Graphene is composed of the benzene rings (*C*6*H*6) without their H-atoms. Graphene is only one of the crystalline forms of carbon which crystallize as diamond, graphite, fullerene (*C*60), carbon nanotube, and glassy carbon.

Unique physical properties of graphene are caused by the collective behavior of the quasiparticles called pseudoelectrons having pseudospins, which move according to the Dirac equation in the hexagonal lattice.

The Dirac fermions in graphene carry unit electric charge. Strong interactions between the electrons and the carbon atoms result in linear dispersion relation *E* = *vF*|**p**|, where *vF* is the so-called Fermi-Dirac velocity, **p** being the momentum of a pseudoelectron. The Fermi velocity is approximately only about 300 times less than the speed of light.

The pseudospin of the pseudoelectron follows from the hexagonal form of graphene. Every hexagonal cell system is composed of the systems of two equilateral triangles. The fermions in the triangle sub-lattice systems can be described by the wave functions *ϕ*<sup>1</sup> and *ϕ*2. Then the adequate wave function of the fermion moving in the hexagonal structure is their superposition, or *ψ* = *c*1*ϕ*<sup>1</sup> + *c*2*ϕ*2, where *c*<sup>1</sup> and *c*<sup>2</sup> are functions of coordinate **x** and functions *ϕ*1, *ϕ*<sup>2</sup> are functions of the wave vector **k** and coordinate **x**. The crucial step in graphene theory is the definition of the bispinor function with components *ϕ*1, *ϕ*<sup>2</sup> [9].

The relativistic generalization of nonrelativistic equation *E* = *vF*|**p**| is evidently the Dirac-Weyl equation for the description of neutrino which can be transcribed in four-component spinor form as

$$p\_{\mu}\gamma^{\mu} = 0\tag{4}$$

and it is possible to prove that this spinor function is solution of the Pauli equation in the nonrelativistic situation. The corresponding mass of such effective electron is proved approximately to be zero. So, it follows from this formalism that to describe the Compton effect on graphene is to solve the Compton effect with quasielectron with zero mass.

The introduction of the Dirac relativistic Hamiltonian in graphene physics has the physical meaning that we describe the graphene physics by means of electron-hole medium. It is the analogue of the Dirac theory of the electron-positron vacuum in quantum electrodynamics. However, the pseudoelectron and pseudospin in graphene physics are not an electron and the spin of quantum electrodynamics (QED), because QED is the relativistic quantum theory of the interaction of real electrons and photons where mass of an electron is defined by classical mechanics and not by collective behavior in the hexagonal sheet called graphene.

The graphene sheet can be considered as the special form of the more general 2D-graphene-like sheets, where, for instance, silicene has the similar structure as graphene [10] .

On the other hand, amorphous solids – glasses – lack long-range translational periodicity in the atomic structure. However, due to chemical bonds, glasses do possess a high degree of short-range order with respect to local atomic polyhedra. It means that such structures can be considered as the graphene-like structures with the appropriate index of refraction, being necessary for the existence of the Cherenkov effect and the Compton effect in dielectric ˇ medium.

The last but not least graphene-like structure can be represented by graphene-based polaritonic crystal sheet [11], which can be used for the Cherenkov effect and the Compton ˇ effect in the graphene-like dielectric medium with the index of refraction *n*.
