**3. Volkov solution of the Dirac equation in vacuum**

Volkov solution of the Dirac equation is the mathematical solution of the Dirac equation with the plane-wave potential. The derivation of the Volkov solution of the Dirac equation in vacuum is described in the textbook of Berestetzkii et al. [12]. The four-potential in the Dirac equation

$$(\gamma(p - eA) - m)\psi = 0,\tag{5}$$

is

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

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

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),

Unique physical properties of graphene are caused by the collective behavior of the quasiparticles called pseudoelectrons having pseudospins, which move according to the

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

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

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

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

effect on graphene is to solve the Compton effect with quasielectron with zero mass.

*pµγ<sup>µ</sup>* = 0 (4)

velocity is approximately only about 300 times less than the speed of light.

the historical document.

114 Graphene - New Trends and Developments

carbon nanotube, and glassy carbon.

Dirac equation in the hexagonal lattice.

four-component spinor form as

[10] .

plasma. The authors of this method are Dato et al. [8].

$$A^{\mu} = A^{\mu}(\varphi); \quad \varphi = k\infty. \tag{6}$$

We suppose that the four-potential satisfies the Lorentz gauge condition

$$
\partial\_{\mu}A^{\mu} = k\_{\mu} \left(A^{\mu}\right)' = \left(k\_{\mu}A^{\mu}\right)' = 0,\tag{7}
$$

where the prime denotes derivation with regard to *ϕ*. From the last equation follows

$$kA = const = 0,\tag{8}$$

because we can put the constant to zero. The tensor of electromagnetic field is

$$F\_{\mu\nu} = k\_{\mu}A\_{\nu}^{\prime} - k\_{\nu}A\_{\mu}^{\prime}. \tag{9}$$

Instead of the linear Dirac equation (5), we consider the quadratic equation, which we get by multiplying the linear equation by operator (*γ*(*p* − *eA*) + *m*), [12]. We get

$$\left[ (p - eA)^2 - m^2 - \frac{i}{2} eF\_{\mu\nu} \sigma^{\mu\nu} \right] \psi = 0. \tag{10}$$

Using *∂µ*(*Aµψ*) = *<sup>A</sup>µ∂µψ*, which follows from eq. (7), and *∂µ∂<sup>µ</sup>* <sup>=</sup> *<sup>∂</sup>*<sup>2</sup> <sup>=</sup> <sup>−</sup>*p*2, with *<sup>p</sup><sup>µ</sup>* <sup>=</sup> *i*(*∂*/*∂xµ*) = *i∂µ*, we get the quadratic Dirac equation for the four-potential of the plane wave:

$$[-\partial^2 - 2i(A\partial) + e^2A^2 - m^2 - ie(\gamma k)(\gamma A')]\psi = 0. \tag{11}$$

We are looking for the solution of the last equation in the form

$$
\psi = e^{-ip\mathbf{x}} F(\varphi). \tag{12}
$$

After insertion of this relation into (11), we get with (*k*<sup>2</sup> = 0)

$$
\partial^{\mu}F = k^{\mu}F^{\prime}, \quad \partial\_{\mu}\partial^{\mu}F = k^{2}F^{\prime\prime} = 0,\tag{13}
$$

the following equation for *F*(*ϕ*):

$$2\mathbf{i}(kp)F' + [-2\mathbf{e}(pA) + \mathbf{e}^2A^2 - i\mathbf{e}(\gamma k)(\gamma A')]F = 0. \tag{14}$$

The integral of the last equation is of the form

$$F = \exp\left\{-i\int\_0^{kx} \left[\frac{e(pA)}{(kp)} - \frac{e^2}{2(kp)}A^2\right]d\varphi + \frac{e(\gamma k)(\gamma A)}{2(kp)}\right\} \frac{u}{\sqrt{2p\_0}},\tag{15}$$

where *u*/ 2*p*<sup>0</sup> is the arbitrary constant bispinor.

Al powers of (*γk*)(*γA*) above the first are equal to zero, since

$$(\gamma k)(\gamma A)(\gamma k)(\gamma A) = -(\gamma k)(\gamma k)(\gamma A)(\gamma A) + 2(kA)(\gamma k)(\gamma A) = -k^2 A^2 = 0. \tag{16}$$

Then we can write

$$\exp\left\{e\frac{(\gamma k)(\gamma A)}{2(kp)}\right\} = 1 + \frac{\varepsilon(\gamma k)(\gamma A)}{2(kp)}.\tag{17}$$

So, the solution is of the form

$$\psi\_p = R \frac{u}{\sqrt{2p\_0}} e^{iS} = \left[ 1 + \frac{e}{2kp} (\gamma k)(\gamma A) \right] \frac{u}{\sqrt{2p\_0}} e^{iS} \, \tag{18}$$

where *u* is an electron bispinor of the corresponding Dirac equation

$$(\gamma p - m)u = 0,\tag{19}$$

with the normalization condition *uu*¯ = 2*m*.

The mathematical object *S* is the classical Hamilton-Jacobi function, which was determined in the form

$$S = -p\mathbf{x} - \int\_0^{\mathbf{k}\mathbf{x}} \frac{e}{(kp)} \left[ (pA) - \frac{e}{2}(A)^2 \right] d\boldsymbol{\uprho}.\tag{20}$$

The current density is

Using *∂µ*(*Aµψ*) = *<sup>A</sup>µ∂µψ*, which follows from eq. (7), and *∂µ∂<sup>µ</sup>* <sup>=</sup> *<sup>∂</sup>*<sup>2</sup> <sup>=</sup> <sup>−</sup>*p*2, with *<sup>p</sup><sup>µ</sup>* <sup>=</sup> *i*(*∂*/*∂xµ*) = *i∂µ*, we get the quadratic Dirac equation for the four-potential of the plane wave:

*ψ* = *e*

<sup>2</sup>*A*<sup>2</sup> <sup>−</sup> *<sup>m</sup>*<sup>2</sup> <sup>−</sup> *ie*(*γk*)(*γ<sup>A</sup>*

<sup>2</sup>*A*<sup>2</sup> <sup>−</sup> *ie*(*γk*)(*γ<sup>A</sup>*

2(*kp*)

*A*2 *dϕ* +

(*γk*)(*γA*)(*γk*)(*γA*) = <sup>−</sup>(*γk*)(*γk*)(*γA*)(*γA*) + <sup>2</sup>(*kA*)(*γk*)(*γA*) = <sup>−</sup>*k*2*A*<sup>2</sup> <sup>=</sup> 0. (16)

= 1 +

<sup>2</sup>*kp* (*γk*)(*γA*)

*e*(*γk*)(*γA*)

 *u* 2*p*<sup>0</sup> *e*

)]*ψ* = 0. (11)

)]*F* = 0. (14)

, (15)

<sup>−</sup>*ipxF*(*ϕ*). (12)

, *∂µ∂µF* = *k*2*F* = 0, (13)

*e*(*γk*)(*γA*) 2(*kp*)

 *u* 2*p*<sup>0</sup>

<sup>2</sup>(*kp*) . (17)

*iS*, (18)

[−*∂*<sup>2</sup> <sup>−</sup> <sup>2</sup>*i*(*A∂*) + *<sup>e</sup>*

We are looking for the solution of the last equation in the form

After insertion of this relation into (11), we get with (*k*<sup>2</sup> = 0)

the following equation for *F*(*ϕ*):

116 Graphene - New Trends and Developments

The integral of the last equation is of the form

 −*i kx* 0

*F* = exp

where *u*/

Then we can write

So, the solution is of the form

*∂µF* = *kµF*

2*i*(*kp*)*F* + [−2*e*(*pA*) + *e*

 *e*(*pA*) (*kp*) <sup>−</sup> *<sup>e</sup>*<sup>2</sup>

2*p*<sup>0</sup> is the arbitrary constant bispinor. Al powers of (*γk*)(*γA*) above the first are equal to zero, since

> exp *e*

*<sup>ψ</sup><sup>p</sup>* <sup>=</sup> *<sup>R</sup> <sup>u</sup>*

2*p*<sup>0</sup> *e iS* = 1 + *e*

(*γk*)(*γA*) 2(*kp*)

$$j^{\mu} = \bar{\psi}\_p \gamma^{\mu} \psi\_{p'} \tag{21}$$

where *ψ*¯ *<sup>p</sup>* is defined as the transposition of (18), or

$$\bar{\psi}\_p = \frac{\bar{u}}{\sqrt{2p\_0}} \left[ 1 + \frac{e}{2kp} (\gamma A)(\gamma k) \right] e^{-iS}. \tag{22}$$

After insertion of *ψ<sup>p</sup>* and *ψ*¯ *<sup>p</sup>* into the current density, we have

$$j^{\mu} = \frac{1}{p\_0} \left\{ p^{\mu} - eA^{\mu} + k^{\mu} \left( \frac{e(pA)}{(kp)} - \frac{e^2A^2}{2(kp)} \right) \right\},\tag{23}$$

which is in agreement with the formula in the Meyer article [13].

If *Aµ*(*ϕ*) are periodic functions, and their time-average value is zero, then the mean value of the current density is

$$\overline{j^{\mu}} = \frac{1}{p\_0} \left( p^{\mu} - \frac{\varepsilon^2}{2(kp)} \overline{A^2} k^{\mu} \right) = \frac{q^{\mu}}{p\_0}. \tag{24}$$
