**Electronic Properties of Deformed Graphene Nanoribbons**

Guo-Ping Tong

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/51348

## **1. Introduction**

As early as 1947, the tight-binding electronic energy spectrum of a graphene sheet had been in‐ vestigated by Wallace (Wallace, 1947). The work of Wallace showed that the electronic proper‐ ties of a graphene sheet were metallic. A better tight-binding description of graphene was given by Saito et al. (Saito et al., 1998). To understand the different levels of approximation, Reich et al. started from the most general form of the secular equation, the tight binding Hamiltonian, and the overlap matrix to calculate the band structure (Reich et al., 2002). In 2009, a work including the non-nearest-neighbor hopping integrals was given by Jin et al. (Jin et al., 2009).

It is common knowledge that a perfect grphene sheet is a zero-gap semiconductor (semime‐ tal) that exhibits extraordinarily high electron mobility and shows considerable promise for applications in electronic and optical devices, high sensitivity gas detection, ultracapacitors and biodevices. How to open the gap of graphene has become a focus of the study. Early in 1996, Fujita et al. started to study the electronic structure of graphene ribbons (Fujita et al., 1996; Nakada et al., 1996) by the numerical method. The armchair shaped edge ribbons can be either semiconducting (*n*=3*m* and *n*=3*m*+1, where *m* is an integer) or metallic (*n*=3*m*+2) de‐ pending on their widths, i. e., on their topological properties. First-principles calculations showed that the origin of the gaps for the armchair edge nanoribbons arises from both quan‐ tum confinement and the deformation caused by edge dangling bonds (Son et al., 2006; Rozhkov et al., 2009). This result implies that the energy gap can be changed by deforma‐ tion. In 1997, Heyd et al. studied the effects of compressive and tensile, unaxial stress on the density of states and the band gap of carbon nanotubes (Heyd et al., 1997). Applying me‐ chanical force (e.g., nanoindentation) on the graphene can lead to a strain of about 10%(Lee et al., 2008). Xiong et al. found that engineering the strain on the graphene planes forming a channel can drastically change the interfacial friction of water transport through it (Xiong et

© 2013 Tong; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 Tong; licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

al., 2011). Density functional perturbation theory is a well-tested *ab initio* method for accu‐ rate phonon calculations. Liu et al. (Liu et al., 2007) studied the phonon spectra of graphene as a function of uniaxial tension by using this theory. Edge stresses and edge energies of the armchair and zigzag edges in graphene also were studied by means of the theory (Jun, 2008). Jun found that both edges are under compression along the edge and the magnitude of compressive edge stress of armchair edge is larger than that of zigzag edge. By simula‐ tions of planar graphene undergoing in-plane deformations, Chung (Chung, 2006) found that crystal structures are different from the usual hexagonal configuration. The thermody‐ namic or kinetic character of the rearrangement was found to depend on the macroscopic straining direction. Neek-Amal et al. (Neek-Amal et al., 2010) simulated the bending of rec‐ tangular graphene nanoribbons subjected to axial stress both for free boundary and support‐ ed boundary conditions. Can et al. (Can et al., 2010) applied density-functional theory to calculate the equilibrium shape of graphene sheets as a function of temperature and hydro‐ gen partial pressure. Their results showed that the edge stress for all edge orientations is compressive. Shenoy et al. (Shenoy et al., 2008) pointed out that edge stresses introduce in‐ trinsic ripples in freestanding graphene sheets even in the absence of any thermal effects. Compressive edge stresses along zigzag and armchair edges of the sheet cause out-of-plane warping to attain several degenerate mode shapes and edge stresses can lead to twisting and scrolling of nanoribbons as seen in experiments. Marianetti et al. (Marianetti et al., 2010) reveals the mechanisms of mechanical failure of pure graphene under a generic state of ten‐ sion at zero temperature. Their results indicated that finite wave vector soft modes can be the key factor in limiting the strength of monolayer materials. In the chemical activity of gra‐ phene, de Andres et al. (de Andres et al., 2008) studied how tensile stress affects *σ*and *π* bonds and pointed out that stress affects more strongly*π*bonds that can become chemically active and bind to adsorbed species more strongly. Kang et al. (Kang et al., 2010) performed a simulation study on strained armchair graphene nanoribbons. By comparison, those with strained wide archair nanoribbons can achieve better device performance. By combining continuum elasticity theory and tight-binding atomistic simulations, Cadelano et al. (Cade‐ lano et al., 2009) worked out the constitutive nonlinear stress-strain relation for graphene stretching elasticity and calculated all the corresponding nonlinear elastic moduli. Gui et al. (Gui et al., 2008) found that graphene with a symmetrical strain distribution is always a zero band-gap semiconductor and its pseudogap decreases linearly with the strain strength in the elastic regime. For asymmetrical strain distributions the band gaps were opened at the Fermi level. This is because small number of k points is chosen (Farjam et al., 2009). We also investigated the energy spectrum and gap of wider graphene ribbons under a tensile force (Wei et al., 2009) and found that the tensile force can have the gap of the ribbon opened.

**2. Graphene under uniaxial stress**

the zigzag edge, as shown in Fig. 1. Let *R* and *R* **′**

between the positions can be written in the form

which leads to the change of electronic hopping energies.

two kinds of different carbon atoms, respectively. *l* ′

atoms after deformation, respectively.

expressed as follows

and*t* ′

where*l*

tively. *l* ′

Since graphene is a monolayer structure of carbon atoms, when a force is exerted on it paral‐ lel to its plane, the positions of the atoms will change with respect to some origin in space. Let the x-axis be in the direction of the armchair edge of graphene and the y-axis in that of

fore and after deformation, respectively. According to the theory of elasticity, the relation

2

æö + æö ¢ æ ö ç ÷ <sup>=</sup> ç ÷ç ÷ ¢ è ø <sup>+</sup> è ø è ø (1)

d

where*δ*<sup>1</sup> = + *δ*(or−*δ*) is the tensile (or compression) stress along the x-direction and*δ*2is the stress in the y-direction and small compared to*δ*1, approximately equal to*δ*<sup>1</sup> / 6. When the de‐ formation of graphene occurs, the bond length between the carbon atoms changes and

**Figure 1.** Graphene sheet subjected to the tensile stress in the x-direction. Symbols A and B denote sublattices with

According to Harrison's formula (Harrison, 1980), the hopping energy after deformation is

2 0 0

0and*t*0denote the bond length and the hopping energy before deformation, respec‐

are the bond length and the hopping energy after deformation, respectively.

*l t t l*

From Fig. 1 and Eq. (1), the bond lengths between atoms A and B can be obtained

and *l* ″

1

1 0 0 1 *x x y y R R R R* d

denote the positions of a carbon atom be‐

Electronic Properties of Deformed Graphene Nanoribbons

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83

denote the bond lengths between tow adjacent carbon

æ ö ¢ <sup>=</sup> ç ÷¢ è ø (2)

**2.1. Elasticity theory**

In this Chapter, we focus on the effects of deformed graphene sheets and nanoribbons under unaxial stress on the electronic energy spectra and gaps based on the elasticity theory. Meanwhile, the energy spectrum of the curved graphene nanoribbons with the tubular warping is studied by the tight-binding approach. The energy spectrum of deformed gra‐ phene sheets subjected to unaxial stress is given in Section 2. In Section 3, we discuss the electronic properties of graphene nanoribbons under unaxial stress. The tubular warping deformation of graphene nanoribbons is presented in last Section.

#### **2. Graphene under uniaxial stress**

#### **2.1. Elasticity theory**

al., 2011). Density functional perturbation theory is a well-tested *ab initio* method for accu‐ rate phonon calculations. Liu et al. (Liu et al., 2007) studied the phonon spectra of graphene as a function of uniaxial tension by using this theory. Edge stresses and edge energies of the armchair and zigzag edges in graphene also were studied by means of the theory (Jun, 2008). Jun found that both edges are under compression along the edge and the magnitude of compressive edge stress of armchair edge is larger than that of zigzag edge. By simula‐ tions of planar graphene undergoing in-plane deformations, Chung (Chung, 2006) found that crystal structures are different from the usual hexagonal configuration. The thermody‐ namic or kinetic character of the rearrangement was found to depend on the macroscopic straining direction. Neek-Amal et al. (Neek-Amal et al., 2010) simulated the bending of rec‐ tangular graphene nanoribbons subjected to axial stress both for free boundary and support‐ ed boundary conditions. Can et al. (Can et al., 2010) applied density-functional theory to calculate the equilibrium shape of graphene sheets as a function of temperature and hydro‐ gen partial pressure. Their results showed that the edge stress for all edge orientations is compressive. Shenoy et al. (Shenoy et al., 2008) pointed out that edge stresses introduce in‐ trinsic ripples in freestanding graphene sheets even in the absence of any thermal effects. Compressive edge stresses along zigzag and armchair edges of the sheet cause out-of-plane warping to attain several degenerate mode shapes and edge stresses can lead to twisting and scrolling of nanoribbons as seen in experiments. Marianetti et al. (Marianetti et al., 2010) reveals the mechanisms of mechanical failure of pure graphene under a generic state of ten‐ sion at zero temperature. Their results indicated that finite wave vector soft modes can be the key factor in limiting the strength of monolayer materials. In the chemical activity of gra‐ phene, de Andres et al. (de Andres et al., 2008) studied how tensile stress affects *σ*and *π* bonds and pointed out that stress affects more strongly*π*bonds that can become chemically active and bind to adsorbed species more strongly. Kang et al. (Kang et al., 2010) performed a simulation study on strained armchair graphene nanoribbons. By comparison, those with strained wide archair nanoribbons can achieve better device performance. By combining continuum elasticity theory and tight-binding atomistic simulations, Cadelano et al. (Cade‐ lano et al., 2009) worked out the constitutive nonlinear stress-strain relation for graphene stretching elasticity and calculated all the corresponding nonlinear elastic moduli. Gui et al. (Gui et al., 2008) found that graphene with a symmetrical strain distribution is always a zero band-gap semiconductor and its pseudogap decreases linearly with the strain strength in the elastic regime. For asymmetrical strain distributions the band gaps were opened at the Fermi level. This is because small number of k points is chosen (Farjam et al., 2009). We also investigated the energy spectrum and gap of wider graphene ribbons under a tensile force (Wei et al., 2009) and found that the tensile force can have the gap of the ribbon opened.

82 New Progress on Graphene Research

In this Chapter, we focus on the effects of deformed graphene sheets and nanoribbons under unaxial stress on the electronic energy spectra and gaps based on the elasticity theory. Meanwhile, the energy spectrum of the curved graphene nanoribbons with the tubular warping is studied by the tight-binding approach. The energy spectrum of deformed gra‐ phene sheets subjected to unaxial stress is given in Section 2. In Section 3, we discuss the electronic properties of graphene nanoribbons under unaxial stress. The tubular warping

deformation of graphene nanoribbons is presented in last Section.

Since graphene is a monolayer structure of carbon atoms, when a force is exerted on it paral‐ lel to its plane, the positions of the atoms will change with respect to some origin in space. Let the x-axis be in the direction of the armchair edge of graphene and the y-axis in that of the zigzag edge, as shown in Fig. 1. Let *R* and *R* **′** denote the positions of a carbon atom be‐ fore and after deformation, respectively. According to the theory of elasticity, the relation between the positions can be written in the form

$$
\begin{pmatrix} R\_x' \\ R\_y' \end{pmatrix} = \begin{pmatrix} 1 + \mathcal{S}\_1 & 0 \\ 0 & 1 + \mathcal{S}\_2 \end{pmatrix} \begin{pmatrix} R\_x \\ R\_y \end{pmatrix} \tag{1}
$$

where*δ*<sup>1</sup> = + *δ*(or−*δ*) is the tensile (or compression) stress along the x-direction and*δ*2is the stress in the y-direction and small compared to*δ*1, approximately equal to*δ*<sup>1</sup> / 6. When the de‐ formation of graphene occurs, the bond length between the carbon atoms changes and which leads to the change of electronic hopping energies.

**Figure 1.** Graphene sheet subjected to the tensile stress in the x-direction. Symbols A and B denote sublattices with two kinds of different carbon atoms, respectively. *l* ′ and *l* ″ denote the bond lengths between tow adjacent carbon atoms after deformation, respectively.

According to Harrison's formula (Harrison, 1980), the hopping energy after deformation is expressed as follows

$$t' = \left(\frac{I\_0}{I'}\right)^2 t\_o \tag{2}$$

where*l* 0and*t*0denote the bond length and the hopping energy before deformation, respec‐ tively. *l* ′ and*t* ′ are the bond length and the hopping energy after deformation, respectively. From Fig. 1 and Eq. (1), the bond lengths between atoms A and B can be obtained

$$\begin{aligned} l' &= R\_\times' = l\_0(1+\delta'),\\ l'' &= \sqrt{R\_\times'^2 + R\_\times'^2} = l\_0 \sqrt{1 + \frac{1}{4}\delta + \frac{13}{48}\delta^2} \end{aligned} \tag{3}$$

The first sum is taken over *A* and all the lattice points generated from it by primitive lattice translation; the second sum is similarly over the points generated from *B*. Here*CA*and*CB* are coefficients to be determined, *RA*and*RB*are the positions of atoms *A* and *B*, respectively, and

> y

\* 0 *AA AB AB AA HEH H HE*

é ù æ ö æ ö æ ö =± + ê ú ¢ ¢ ¢¢ ç ÷¢ ¢ ¢¢ <sup>+</sup> ç ÷¢ ç ÷ ê ú è ø ë û è ø è ø

Fig. 2 shows the electronic energy spectra of deformed graphene sheets for some high sym‐ metric points*Γ*, M, and K under uniaxial stress. Because of uniaxial stress, the hexagonal lat‐ tice is distorted and the shape of the first Brillouin zone changes accordingly as the stress upon the lattice. Six "saddle" points on the boundary in the first Brillouin zone can be divid‐ ed into two groups: M and M'. At the same time, Dirac point K will drift towards the saddle point M and is accompanied by a small angle. For the convenience of comparison, we give the spectrum of undeformed graphene in Fig. 2(a). From Fig. 2 (c) and (d), we see that ten‐ sion along the armchair shape edge can reduce the band width at point *Γ*and increase the bandwidth at point M, and the result of compression is just opposite to that of tension. Fig. 2(e) tells us that tension along the zigzag shape edge can not only narrow the bandwidth at *Γ*point but decrease the bandwidth at M point as well. On the contrary, compression can si‐ multaneously increase the bandwidth at*Γ*and M. Moreover, it may be seen from Fig. 2 that whether the tensile stress or compressive stress, the result of the high symmetric point M' is always opposite to that of the point M and the energy gap cannot be opened at Dirac point (K). On the other hand, we see yet that the energy band curves between M and M' for the graphene without stress are a straight line, but for the graphene with stress the curves are not. It appears to graphene that the uniaxial stress does not open the energy gap at Dirac point. When graphene is compressed along the armchair shape edge or extended along the zigzag shape edge a small energy gap is opened at K point, which is approximately equal to 0.1eV as the stress parameter takes to be 12%. From this reason, the graphene under uniaxial

are the nearest-neighbour hopping integrals after deformation, given by Eq.(4).

<sup>2</sup> 33 3 2 2 ( , ) 4 cos cos 4 cos 22 2 *E k k t tt k R k R t k R x y y y x x y y*

= (8)

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85


1 2

(10)

*H E* y

*N* is the number of the unit cell in graphene. Substituting Eq. (6) in

For the tensile stress in the x-direction, the solution to Eq.(9) is

stress still is a semiconductor with the zero-energy gaps.

we obtain the secular equation

where *t* ′

and *t* ″

The nearest neighbour hopping integrals associated with the bond lengths are

$$t' = \frac{t\_0}{\left(1 + \delta'\right)^2}$$

$$t'' = \frac{t\_0}{\left(1 + \frac{1}{4}\delta + \frac{13}{48}\delta^2\right)}\tag{4}$$

If graphene is subject to a tensile force in the y-direction, the hopping energies are given by

$$t' = \frac{t\_0}{\left(1 - \frac{\delta}{6}\right)^2}$$

$$t'' = \frac{t\_0}{\left(1 + \frac{17}{12}\delta + \frac{109}{144}\delta^2\right)}\tag{5}$$

#### **2.2. The tight-binding energy spectrum**

Let us now consider the band structure from the viewpoint of the tight-binding approxima‐ tion. The structure of graphene is composed of two types of sublattices *A* and *B* as shown in Fig. 1. If *φ*(*r*)is the normalized orbital 2*p* z wave function for an isolated carbon atom, then the wave function of graphene has the form

$$<\left|\boldsymbol{\nu}\right> = C\_A \left|\boldsymbol{\nu}\_A\right> + C\_B \left|\boldsymbol{\nu}\_B\right>\tag{6}$$

where

$$\mid \psi\_A \rangle = \frac{1}{\sqrt{N}} \sum\_A e^{i\vec{k} \cdot \vec{R}\_A} \mid \varphi(r - R\_A) \rangle$$

and

$$\left| \left| \psi\_{\,^{B}} \right> \right| = \frac{1}{\sqrt{N}} \sum\_{\,^{B}} e^{i\mathbf{k} \cdot \mathbf{R}\_{\,^{B}}} \left| \varphi(\mathbf{r} - \mathbf{R}\_{\,^{B}}) \right> \tag{7}$$

The first sum is taken over *A* and all the lattice points generated from it by primitive lattice translation; the second sum is similarly over the points generated from *B*. Here*CA*and*CB* are coefficients to be determined, *RA*and*RB*are the positions of atoms *A* and *B*, respectively, and *N* is the number of the unit cell in graphene. Substituting Eq. (6) in

$$H\left|\psi\right> = E\left|\psi\right>\tag{8}$$

we obtain the secular equation

0

*x*

The nearest neighbour hopping integrals associated with the bond lengths are

*t*

*t*

¢ ¢ == +

¢¢ ¢ ¢ = + = ++

*lRl*

*x y*

*l RR l*

2 2 2 0

> 0 2

d

(1 )

¢ <sup>=</sup> <sup>+</sup>

*t*

*t*

0

If graphene is subject to a tensile force in the y-direction, the hopping energies are given by

0 2

d

*t*

æ ö ç ÷ - è ø

1 6

*t*

¢ =

yy

*t*

**2.2. The tight-binding energy spectrum**

84 New Progress on Graphene Research

the wave function of graphene has the form

*ik* <sup>⋅</sup>*RA* <sup>|</sup>*φ*(*<sup>r</sup>* <sup>−</sup>*RA*)

y

where

and

<sup>|</sup>*ψ<sup>A</sup>* <sup>=</sup> <sup>1</sup>

*<sup>N</sup>* ∑ *A e* 0

Let us now consider the band structure from the viewpoint of the tight-binding approxima‐ tion. The structure of graphene is composed of two types of sublattices *A* and *B* as shown in Fig. 1. If *φ*(*r*)is the normalized orbital 2*p* z wave function for an isolated carbon atom, then

> y

<sup>1</sup> ( ) *<sup>B</sup> <sup>i</sup> B B B <sup>e</sup> <sup>N</sup>*

 j

= + *C C AA BB* (6)

<sup>×</sup> <sup>=</sup> å - *k R r R* (7)

*t*

17 109 <sup>1</sup> 12 144

¢¢ <sup>=</sup> æ ö ç ÷ + + è ø

d

1 13 <sup>1</sup> 4 48

¢¢ <sup>=</sup> æ ö ç ÷ + + è ø

d

1 13 <sup>1</sup> 4 48

2

2

 d

 d

d

 d (3)

(4)

(5)

(1 ),

d

$$
\begin{vmatrix} H\_{\mathcal{A}4} - E & H\_{\mathcal{A}8} \\ H\_{\mathcal{A}8}^\* & H\_{\mathcal{A}4} - E \end{vmatrix} = 0 \tag{9}
$$

For the tensile stress in the x-direction, the solution to Eq.(9) is

$$E(k\_x, k\_y) = \pm \left[ t^{\prime 2} + 4t^{\prime}t^{\prime} \cos \left( \frac{\sqrt{3}}{2} k\_{\ne} R\_{\ne}^{\prime} \right) \cos \left( \frac{3}{2} k\_{\ge} R\_{\ne}^{\prime} \right) + 4t^{\prime 2} \cos^2 \left( \frac{\sqrt{3}}{2} k\_{\ne} R\_{\ne}^{\prime} \right) \right]^{\frac{1}{2}} \tag{10}$$

where *t* ′ and *t* ″ are the nearest-neighbour hopping integrals after deformation, given by Eq.(4).

Fig. 2 shows the electronic energy spectra of deformed graphene sheets for some high sym‐ metric points*Γ*, M, and K under uniaxial stress. Because of uniaxial stress, the hexagonal lat‐ tice is distorted and the shape of the first Brillouin zone changes accordingly as the stress upon the lattice. Six "saddle" points on the boundary in the first Brillouin zone can be divid‐ ed into two groups: M and M'. At the same time, Dirac point K will drift towards the saddle point M and is accompanied by a small angle. For the convenience of comparison, we give the spectrum of undeformed graphene in Fig. 2(a). From Fig. 2 (c) and (d), we see that ten‐ sion along the armchair shape edge can reduce the band width at point *Γ*and increase the bandwidth at point M, and the result of compression is just opposite to that of tension. Fig. 2(e) tells us that tension along the zigzag shape edge can not only narrow the bandwidth at *Γ*point but decrease the bandwidth at M point as well. On the contrary, compression can si‐ multaneously increase the bandwidth at*Γ*and M. Moreover, it may be seen from Fig. 2 that whether the tensile stress or compressive stress, the result of the high symmetric point M' is always opposite to that of the point M and the energy gap cannot be opened at Dirac point (K). On the other hand, we see yet that the energy band curves between M and M' for the graphene without stress are a straight line, but for the graphene with stress the curves are not. It appears to graphene that the uniaxial stress does not open the energy gap at Dirac point. When graphene is compressed along the armchair shape edge or extended along the zigzag shape edge a small energy gap is opened at K point, which is approximately equal to 0.1eV as the stress parameter takes to be 12%. From this reason, the graphene under uniaxial stress still is a semiconductor with the zero-energy gaps.

**Figure 3.** Structure of an armchair graphene nanoribbon with sublattices *A* and *B*. The tension is exerted on the nano‐ ribbon along the x-axis. Symbol *n* denotes the width of the nanoribbon. There are *n* sublatices *A* or *B* in a unit cell.

Since the unit cell of the nanoribbon has the translational symmetry in the *x-*direction, we can choose the plane-wave basis in the *x-*direction and take the stationary wave in the *y-*di‐ rection. For the armchair nanoribbon there are two kinds of sublattices *A* and *B* in a unit cell. Therefor, the wave functions of *A* and *B* sublattices in hard-wall conditions can be written as

<sup>1</sup> ( ,) sin ( ) , ( 1,2, , ) ( 1)

p

p

<sup>2</sup> 2 22 <sup>3</sup> ( , ) 4 cos 4 cos cos 1 12 *<sup>x</sup> x x q q Ek q t t t t k R n n*

p

é ù æ ö æ öæ ö

=± + ¢ ¢¢ <sup>+</sup> ¢ ¢¢ ¢ ê ú ç ÷ ç ÷ç ÷ ë û è ø è øè ø + +

Since the electronic energy spectrum of the perfect armchair nanoribbon depends strongly on the width of the nanoribbon, the different width has the different spectrum. For instance, the nanoribbon with widths *n*=3*m*+2 (*m* is an integer) is metallic and others are insulating. When we exert a tensile (or compressive) force on the nanoribbon along the x-axis, the metal

 p

*<sup>q</sup> k q <sup>e</sup> <sup>j</sup> q n N n*

æ ö <sup>=</sup> - = ××× ç ÷ è ø <sup>+</sup>

æ ö <sup>=</sup> - = ××× ç ÷ è ø <sup>+</sup>

where *NA*and*NB*are the normalized coefficients,*q* =1,2,...,*n* is the quantum number associat‐ ed with the wave vector *ky*, which denotes the discrete wave vector in the y-direction. When a graphene nanoribbon is subject to uniaxial stress, Eq.(11) still is available. For a nanorib‐ bon, as long as the wave vector*ky*in Eq.(10) is replaced by the discrete wave vector*ky*(*q*), we

 j

 j *j*

(11)

(12)

*r R*

*r R*

*j*

1

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87

<sup>1</sup> ( ,) sin ( ) , ( 1,2, , ) ( 1)

*<sup>q</sup> k q <sup>e</sup> <sup>j</sup> q n N n*

1

å å

*A j x*

=

y

y

*Aj*

*<sup>n</sup> ik x B x B*

*<sup>n</sup> ik x A x A*

*Bj*

*x Aj*

*x Bj*

1

å å

*B j x*

=

can obtain the energy dispersion relation of the form

nanoribbon is converted into an insulator or semiconductor.

**Figure 2.** Electronic energy spectra of graphene under uniaxial tress for some high symmetric points.

#### **3. Graphene nanoribbon under uniaxial stress**

As mentioned in Section 2, for a graphene sheet subject to uniaxial stress there are no energy gaps at Dirac point. How to open the energy gaps of graphene? Studies showed that we can realize this goal by deducing the size of graphene, i.e., changing its toplogical propertiy(Son et al., 2006). On the other hand, the band gaps of graphene nanoribbons can be mamipulated by changing the bond lengths between carbon atoms, i.e., changing the hopping integrals, by exerting a strain force (Sun et al., 2008). The nearest-neighbor energy spectrum of an arm‐ chair nanoribbon was given by the tight-binding approach and using the hard-wall aboun‐ dary condition (Zheng et al., 2007). In the non-nearest-neighbor band structure of the nanoribbon was given by Jin et al (Jin et al., 2009). In this section we use the tight-binding approach to study the energy spectrum and gap of the nanoribbon under uniaxial stress along the length direction, i.e., x-direction, of the nanoribbon, as shown in Fig. 3.

**Figure 3.** Structure of an armchair graphene nanoribbon with sublattices *A* and *B*. The tension is exerted on the nano‐ ribbon along the x-axis. Symbol *n* denotes the width of the nanoribbon. There are *n* sublatices *A* or *B* in a unit cell.

Since the unit cell of the nanoribbon has the translational symmetry in the *x-*direction, we can choose the plane-wave basis in the *x-*direction and take the stationary wave in the *y-*di‐ rection. For the armchair nanoribbon there are two kinds of sublattices *A* and *B* in a unit cell. Therefor, the wave functions of *A* and *B* sublattices in hard-wall conditions can be written as

$$\begin{aligned} \left| \left| \boldsymbol{\nu}\_{\boldsymbol{A}} (\boldsymbol{k}\_{\boldsymbol{a}}, q) \right\rangle \right| &= \frac{1}{N\_{\mathcal{A}}} \sum\_{j=1}^{s} \sum\_{x\_{k\_{j}}} e^{\boldsymbol{\mu}\_{\boldsymbol{x}} x\_{k\_{j}}} \sin \left( \frac{\pi q}{(n+1)} j \right) \left| \boldsymbol{\rho} (\boldsymbol{r} - \boldsymbol{R}\_{\boldsymbol{A}\_{j}}) \right\rangle, \quad (q = 1, 2, \cdots, n) \\\left| \left| \boldsymbol{\nu}\_{\boldsymbol{A}} (\boldsymbol{k}\_{\boldsymbol{a}}, q) \right\rangle &= \frac{1}{N\_{\mathcal{A}}} \sum\_{j=1}^{s} \sum\_{x\_{k\_{j}}} e^{\boldsymbol{\mu}\_{\boldsymbol{a}, x\_{k\_{j}}}} \sin \left( \frac{\pi q}{(n+1)} j \right) \left| \boldsymbol{\rho} (\boldsymbol{r} - \boldsymbol{R}\_{\boldsymbol{a}\_{j}}) \right\rangle, \quad (q = 1, 2, \cdots, n) \end{aligned} \tag{11}$$

where *NA*and*NB*are the normalized coefficients,*q* =1,2,...,*n* is the quantum number associat‐ ed with the wave vector *ky*, which denotes the discrete wave vector in the y-direction. When a graphene nanoribbon is subject to uniaxial stress, Eq.(11) still is available. For a nanorib‐ bon, as long as the wave vector*ky*in Eq.(10) is replaced by the discrete wave vector*ky*(*q*), we can obtain the energy dispersion relation of the form

**Figure 2.** Electronic energy spectra of graphene under uniaxial tress for some high symmetric points.

along the length direction, i.e., x-direction, of the nanoribbon, as shown in Fig. 3.

As mentioned in Section 2, for a graphene sheet subject to uniaxial stress there are no energy gaps at Dirac point. How to open the energy gaps of graphene? Studies showed that we can realize this goal by deducing the size of graphene, i.e., changing its toplogical propertiy(Son et al., 2006). On the other hand, the band gaps of graphene nanoribbons can be mamipulated by changing the bond lengths between carbon atoms, i.e., changing the hopping integrals, by exerting a strain force (Sun et al., 2008). The nearest-neighbor energy spectrum of an arm‐ chair nanoribbon was given by the tight-binding approach and using the hard-wall aboun‐ dary condition (Zheng et al., 2007). In the non-nearest-neighbor band structure of the nanoribbon was given by Jin et al (Jin et al., 2009). In this section we use the tight-binding approach to study the energy spectrum and gap of the nanoribbon under uniaxial stress

**3. Graphene nanoribbon under uniaxial stress**

86 New Progress on Graphene Research

$$E(k\_z, q) = \pm \left[ t'^2 + 4t''^2 \cos^2 \left( \frac{q\pi}{n+1} \right) + 4t't'' \cos \left( \frac{q\pi}{n+1} \right) \cos \left( \frac{3}{2} k\_z R\_z' \right) \right]^{\frac{1}{2}} \tag{12}$$

Since the electronic energy spectrum of the perfect armchair nanoribbon depends strongly on the width of the nanoribbon, the different width has the different spectrum. For instance, the nanoribbon with widths *n*=3*m*+2 (*m* is an integer) is metallic and others are insulating. When we exert a tensile (or compressive) force on the nanoribbon along the x-axis, the metal nanoribbon is converted into an insulator or semiconductor.

width *n*=6, the tensile stress can make the energy gap increase and the bandwidth decrease slightly. On the contrary, the compressive stress can decrease the gap and make the bandwidth widen. It is obvious that the energy band corresponding to quantum number *q*=*n*–1=5 plays an important role in the change of the band gap. When *n*=7, the tensile stress can make the gap narrow and the compressive stress has larger influence on the energy bandwidth, but not obvi‐ ous on the gap. It can be seen that the energy band with quantum number *q*=*n*-1=6 contributes to the gap under compressive stress and which is clearly different from the tensile situation, where *q*=*n*-2=5. As for *n*=8, whether it is tensile or compression can open the gap and the energy band contributing to the gap belongs to quantum number *q*=*n*-2=6. It fallows from this that ei‐ ther tension or compression can change the gap and the bandwidth. Therefore, the electronic

Electronic Properties of Deformed Graphene Nanoribbons

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89

**Figure 5.** Energy gaps of deformed armchair nanoribbons as a function of the width *n* (*m*). SymbolΔdenotes the ener‐

gy gap. The stress parameterδis taken to be -0.1, 0, and 0.1.

properties of armchair nanoribbons can be controlled by uniaxial stress.

**Figure 4.** Band structures of armchair graphene nanoribbons under unaxial stress with widths *n*=6, 7, 8. The stress parameterδis taken to be - 0.1, 0, and 0.1 respectively.

Fig. 4 shows the energy spectra of three kinds of the nanoribbons under uniaxial stress, and in which the next-nearest neighbor hopping integrals are taken into account. In order to facilitate comparison, the energy spectrum of the undeformed nanoribbon is given in Fig. 4. When width *n*=6, the tensile stress can make the energy gap increase and the bandwidth decrease slightly. On the contrary, the compressive stress can decrease the gap and make the bandwidth widen. It is obvious that the energy band corresponding to quantum number *q*=*n*–1=5 plays an important role in the change of the band gap. When *n*=7, the tensile stress can make the gap narrow and the compressive stress has larger influence on the energy bandwidth, but not obvi‐ ous on the gap. It can be seen that the energy band with quantum number *q*=*n*-1=6 contributes to the gap under compressive stress and which is clearly different from the tensile situation, where *q*=*n*-2=5. As for *n*=8, whether it is tensile or compression can open the gap and the energy band contributing to the gap belongs to quantum number *q*=*n*-2=6. It fallows from this that ei‐ ther tension or compression can change the gap and the bandwidth. Therefore, the electronic properties of armchair nanoribbons can be controlled by uniaxial stress.

**Figure 5.** Energy gaps of deformed armchair nanoribbons as a function of the width *n* (*m*). SymbolΔdenotes the ener‐ gy gap. The stress parameterδis taken to be -0.1, 0, and 0.1.

**Figure 4.** Band structures of armchair graphene nanoribbons under unaxial stress with widths *n*=6, 7, 8. The stress

Fig. 4 shows the energy spectra of three kinds of the nanoribbons under uniaxial stress, and in which the next-nearest neighbor hopping integrals are taken into account. In order to facilitate comparison, the energy spectrum of the undeformed nanoribbon is given in Fig. 4. When

parameterδis taken to be - 0.1, 0, and 0.1 respectively.

88 New Progress on Graphene Research

When the stress is constant, three graphs of the energy gaps with the width of the nanorib‐ bon changes are shown in Fig. 5, where (a) the width *n* is equal to 3*m*, (b) *n*=3*m*+1, and (c) *n*=3*m*+2. The results shown in Fig. 5 are inclusive of the nearest-neighbor hopping integrals. We see from Fig. 5(a) that the compressive stress can make an inflection point of the band gap minimum for the 3*m*-type nanoribbon and the width corresponding with the inflection point is about 12, and the tensile stress can not make a minimum value of the gap. For the 3*m*+1-type nanoribbon, the result shown in Fig. 5(b) tells us that the tensile stress also can produce the minimum value of the gap and the corresponding width is 19. In the case of the 3*m*+2-type nanoribbon, tensile or compression does not change the energy gap (see Fig. 5(c)). Furthermore, we found by calculations that with the inclusion of the next-nearest neighbor and the third neighbor respectively, the minimum point of the gap moves toward the direc‐ tion of the origin of coordinates (zero width), i.e., the width of the non-nearest-neighbor hopping is less than that of the nearest neighbor.

**4. The tubular warping graphene nanoribbon**

because the sp2 hybridization of a flat ribbon turns into the sp3

into account in calculating electronic energy bands.

1

1

*B j z*

=

*A j z*

=

*Aj <sup>n</sup> ik z*

*Bj <sup>n</sup> ik z*

*z Aj*

*z Bj*

In this section we choose an armchair ribbon as an example and which is bent into the tubu‐ lar shape (cylindrical shape), as shown in Fig. 7. This tubular ribbon still has the periodicity in its length direction, but its dimensionality has changed. The consequence of such a di‐ mension change is to lead to the change of the electronic energy dispersion relation. This is

bon, i.e., the curvature of graphene nanoribbons will result in a significant rehybridization of the*π*orbitals (Kleiner et al., 2001). From this reason, the *s*-orbital component must be taken

Because of the curl of the ribbon, the wavefunction of *π*electrons should be composed of the *s*- and *p*-orbital components. The wavefunctions of sublattices *A* and *B* in cylindrical coordi‐

*j j*

*j j*

<sup>=</sup> - (15)

= (16)

æ ö <sup>=</sup> ç ÷é ù + - ë û è ø å å (13)

æ ö <sup>=</sup> ç ÷é ù + - ë û è ø å å (14)

1 3 sin <sup>1</sup> 2

1 3 sin <sup>1</sup> 2

where*c*is the s-orbital component of electrons, given by (Huang et al., 2006; 2007)

*B j B j zB*

*e j ka c s c p <sup>N</sup>*

f

2 2 2sin 1 sin *<sup>c</sup>*

> 4 3 *a r*

Here*β*is a small inclined angle (Kleiner et al., 2001) between the*pz*orbital and the normal di‐ rection of the cylindrical surface, *r*is the radius of the cylindrical surface, and*a*is the distance

To clearly understand the effect of curvature, we choose the width *n*=6, 7, and 8 respectively as examples to show the characteristics of their electronic energy spectra. On the other hand,

b

b

b

*A j A j zA*

*e j ka c s c p <sup>N</sup>*

f

hybridization of a curved rib‐

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Electronic Properties of Deformed Graphene Nanoribbons

**4.1. Theoretical Model**

nates are written then

and

y

y

between two adjacent carbon atoms.

**4.2. Results and Discussion**

On the other hand, in order to make certain of the relationship between the gap and the stress, the curves of the gap versus the stress are given in Fig. 6. As shown in Fig. 6, the gap increases as the stress increases for the 3*m*- and 3*m*+1-type nanoribbons and changes in the V-shaped curve for the 3*m*+2-type nanoribbon.

**Figure 6.** Energy gaps of deformed armchair nanoribbons as a function of stressδ. SymbolΔdenotes the energy gap. Solid, dashed, and dotted lines denote widths *n*=6, *n*=7, and *n*=8, respectively.

**Figure 7.** A curved armchair graphene nanoribbon with the tubular shape. θis the central angle and *r*is the curved radius of the nanoribbon.

#### **4. The tubular warping graphene nanoribbon**

#### **4.1. Theoretical Model**

When the stress is constant, three graphs of the energy gaps with the width of the nanorib‐ bon changes are shown in Fig. 5, where (a) the width *n* is equal to 3*m*, (b) *n*=3*m*+1, and (c) *n*=3*m*+2. The results shown in Fig. 5 are inclusive of the nearest-neighbor hopping integrals. We see from Fig. 5(a) that the compressive stress can make an inflection point of the band gap minimum for the 3*m*-type nanoribbon and the width corresponding with the inflection point is about 12, and the tensile stress can not make a minimum value of the gap. For the 3*m*+1-type nanoribbon, the result shown in Fig. 5(b) tells us that the tensile stress also can produce the minimum value of the gap and the corresponding width is 19. In the case of the 3*m*+2-type nanoribbon, tensile or compression does not change the energy gap (see Fig. 5(c)). Furthermore, we found by calculations that with the inclusion of the next-nearest neighbor and the third neighbor respectively, the minimum point of the gap moves toward the direc‐ tion of the origin of coordinates (zero width), i.e., the width of the non-nearest-neighbor

On the other hand, in order to make certain of the relationship between the gap and the stress, the curves of the gap versus the stress are given in Fig. 6. As shown in Fig. 6, the gap increases as the stress increases for the 3*m*- and 3*m*+1-type nanoribbons and changes in the

**Figure 6.** Energy gaps of deformed armchair nanoribbons as a function of stressδ. SymbolΔdenotes the energy gap.

**Figure 7.** A curved armchair graphene nanoribbon with the tubular shape. θis the central angle and *r*is the curved

Solid, dashed, and dotted lines denote widths *n*=6, *n*=7, and *n*=8, respectively.

radius of the nanoribbon.

hopping is less than that of the nearest neighbor.

90 New Progress on Graphene Research

V-shaped curve for the 3*m*+2-type nanoribbon.

In this section we choose an armchair ribbon as an example and which is bent into the tubu‐ lar shape (cylindrical shape), as shown in Fig. 7. This tubular ribbon still has the periodicity in its length direction, but its dimensionality has changed. The consequence of such a di‐ mension change is to lead to the change of the electronic energy dispersion relation. This is because the sp2 hybridization of a flat ribbon turns into the sp3 hybridization of a curved rib‐ bon, i.e., the curvature of graphene nanoribbons will result in a significant rehybridization of the*π*orbitals (Kleiner et al., 2001). From this reason, the *s*-orbital component must be taken into account in calculating electronic energy bands.

Because of the curl of the ribbon, the wavefunction of *π*electrons should be composed of the *s*- and *p*-orbital components. The wavefunctions of sublattices *A* and *B* in cylindrical coordi‐ nates are written then

$$\left| \left| \boldsymbol{\nu}\_{\cdot A} \right> \right| = \frac{1}{N\_{A}} \sum\_{j=1}^{a} \sum\_{z\_{\boldsymbol{s}\_{j}}} e^{ik\_{z}z\_{\boldsymbol{s}\_{j}}} \sin \left( j \frac{\sqrt{3}}{2} k\_{\boldsymbol{\rho}} a \left| \left[ \sqrt{c\_{j}} \, \middle| \, s\_{\boldsymbol{s}\_{j}} \right] + \sqrt{1 - c\_{j}} \, \middle| \, p\_{z \, \boldsymbol{s}\_{j}} \right> \right) \tag{13}$$

$$\left| \left| \boldsymbol{\nu}\_{\boldsymbol{a}} \right> \right| = \frac{1}{N\_{B}} \sum\_{j=1}^{s} \sum\_{z\_{\boldsymbol{a}\_{j}}} e^{i \boldsymbol{k}\_{z} z\_{\boldsymbol{a}\_{j}}} \sin \left( j \frac{\sqrt{3}}{2} \boldsymbol{k}\_{\boldsymbol{\rho}} \boldsymbol{a} \right) \left[ \sqrt{c\_{j}} \left| \boldsymbol{s}\_{\boldsymbol{a}\_{j}} \right> + \sqrt{1 - c\_{j}} \left| \boldsymbol{p}\_{z \boldsymbol{a}\_{j}} \right> \right] \tag{14}$$

where*c*is the s-orbital component of electrons, given by (Huang et al., 2006; 2007)

$$c = \frac{2\sin^2\beta}{1-\sin^2\beta} \tag{15}$$

and

$$
\beta = \frac{a}{4\sqrt{3}r} \tag{16}
$$

Here*β*is a small inclined angle (Kleiner et al., 2001) between the*pz*orbital and the normal di‐ rection of the cylindrical surface, *r*is the radius of the cylindrical surface, and*a*is the distance between two adjacent carbon atoms.

#### **4.2. Results and Discussion**

To clearly understand the effect of curvature, we choose the width *n*=6, 7, and 8 respectively as examples to show the characteristics of their electronic energy spectra. On the other hand, in order to compare with the ideal flat nanoribbon, the results of the ideal ribbon along with the tubular warping ribbon are also given in Fig. 8, where black lines denote the ideal ribbon and red lines are the tubular warping ribbon.

ing ribbon becomes a carbon nanotube. Fig. 10 also shows such a fact that when a graphene

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**Figure 9.** Density of states of tubular warping armchair nanoribbons. Black and red lines are the flat and warping

nanoribbon is bent into a nanotube, its energy gap is increased.

**Figure 10.** Energy gaps as a function of the central angle (or curvature).

nanoribbons, respectively.

**Figure 8.** Band structures of the curved armchair nanoribbons with widths *n* = 6, *n* = 7, and *n* = 8, respectively. Black lines are the energy band of a perfect nanoribbon and red lines denote the band of a curved nanoribbon with the tubular shape.

By comparison, we found that the energy bandwidths become narrowed obviously for the widths *n*=6, 7, 8 and then this bending does not nearly influence on the energy gaps. This is be‐ cause the localization of electrons is enhanced from two-dimensional plane to three-dimen‐ sional curved surface. When *n*=6, the increment of the gap with respect to the flat ribbon is equal to 0.074eV. When *n*=7, the change of the gap is 0.065eV. As for *n*=8, its metallic behavior does not change as the ribbon is rolled up. Fig. 9 illustrates the density of states of the warping ribbons with widths *n*=6, 7, 8. The meaning of the black and red lines in Fig. 9 is the same as in Fig. 8. From Fig. 9, we see that the tubular warping is responsible for the energy bandwidth narrowing. The density of states of both the top of the valence band and the bottom of the con‐ duction band does not nearly change. It follows that this warping ribbon still keeps all the char‐ acteristics of the flat ribbon, especially for *n*=7. This means that the change of this dimension does not affect the electronic structure seriously. This is why we usually use a graphene sheet to study the electronic structure of a carbon nanotube. In addition, in order to show the effect of the curvature on the energy gap, a graph of the gap varying with the central angle is plotted in Fig. 10. It is apparent that for a fixed width the gap has a maximum value as the increasing of the central angle. When *n*=6, the central angle corresponding to the maximum value is between 5*π* / 4and3*π* / 2. When *n*=7, this angle approximately equals3*π* / 2. As the central angle is equal to zero, the warping ribbon becomes a flat ribbon and as the central angle goes to2*π*, the warp‐ ing ribbon becomes a carbon nanotube. Fig. 10 also shows such a fact that when a graphene nanoribbon is bent into a nanotube, its energy gap is increased.

in order to compare with the ideal flat nanoribbon, the results of the ideal ribbon along with the tubular warping ribbon are also given in Fig. 8, where black lines denote the ideal ribbon

**Figure 8.** Band structures of the curved armchair nanoribbons with widths *n* = 6, *n* = 7, and *n* = 8, respectively. Black lines are the energy band of a perfect nanoribbon and red lines denote the band of a curved nanoribbon with the

By comparison, we found that the energy bandwidths become narrowed obviously for the widths *n*=6, 7, 8 and then this bending does not nearly influence on the energy gaps. This is be‐ cause the localization of electrons is enhanced from two-dimensional plane to three-dimen‐ sional curved surface. When *n*=6, the increment of the gap with respect to the flat ribbon is equal to 0.074eV. When *n*=7, the change of the gap is 0.065eV. As for *n*=8, its metallic behavior does not change as the ribbon is rolled up. Fig. 9 illustrates the density of states of the warping ribbons with widths *n*=6, 7, 8. The meaning of the black and red lines in Fig. 9 is the same as in Fig. 8. From Fig. 9, we see that the tubular warping is responsible for the energy bandwidth narrowing. The density of states of both the top of the valence band and the bottom of the con‐ duction band does not nearly change. It follows that this warping ribbon still keeps all the char‐ acteristics of the flat ribbon, especially for *n*=7. This means that the change of this dimension does not affect the electronic structure seriously. This is why we usually use a graphene sheet to study the electronic structure of a carbon nanotube. In addition, in order to show the effect of the curvature on the energy gap, a graph of the gap varying with the central angle is plotted in Fig. 10. It is apparent that for a fixed width the gap has a maximum value as the increasing of the central angle. When *n*=6, the central angle corresponding to the maximum value is between 5*π* / 4and3*π* / 2. When *n*=7, this angle approximately equals3*π* / 2. As the central angle is equal to zero, the warping ribbon becomes a flat ribbon and as the central angle goes to2*π*, the warp‐

and red lines are the tubular warping ribbon.

92 New Progress on Graphene Research

tubular shape.

**Figure 9.** Density of states of tubular warping armchair nanoribbons. Black and red lines are the flat and warping nanoribbons, respectively.

**Figure 10.** Energy gaps as a function of the central angle (or curvature).

#### **5. Graphene nanoribbon modulated by sine regime**

A free standing graphene nanoribbon could have out-of-plane warping because of the edge stress (Shenoy et al., 2008). This warping will bring about a very small change of the elec‐ tronic energy spectrum. An ideal graphene nanoribbon only has periodicity in the direction of its length and there is no periodicity in the y-direction. To show the periodic effect in the y-direction, we modulate it with the aid of a sine periodic function

$$z = A \sin(f\mathbf{y})\tag{17}$$

By numerical calculations, we found that different modulation frequencies have different electronic band structures, i.e., the energy band structures depend strongly on the modula‐ tion frequency *f* and the modulation amplitude *A*. We take the width *n*=7 as an example to calculate the electronic energy spectrum. When the amplitude *A* is fixed, the energy band structures with different frequencies are shown in Fig. 12. It may be seen from Fig. 12 that this periodic modulation does not damage the Dirac cones, i.e., the topological property of armchair graphene nanoribbons is not destroyed. On the other hand, Fig. 12 tells us that this periodic modulation can change the energy band structure, i.e., both the bandwidth and band gap can be controlled by the modulation frequency. The density of states of electrons for an armchair graphene nanoribbon with *n*=7, modulated by using a sine function along

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**Figure 13.** Density of states of graphene nanoribbons with width *n*=7 modulated by sine regime in the direction

When the modulation amplitude, taken to be 0.1nm, is fixed, different modulation frequen‐ cies have slightly different densities of states of electrons. The main difference between the frequencies 0.0nm-1, 5.0nm-1, and 10.0nm-1 is in the conduction band and the density of states of the valence band is the same nearly. It follows that the modulation along the width direc‐ tion of the ribbon makes a notable impact for the density of states of the conduction band, especially for the high energy band corresponding to the standing wave of the smaller quan‐ tum number. In order to reveal the effect of the modulation amplitude on the electronic properties, the energy bands for the different amplitudes are calculated under certain fre‐ quency. Fig. 14 shows the band structures of the different amplitudes *A*=0.0nm, *A*=0.05nm, and *A*=0.1nm for an armchair nanoribbon with *n*=7, where the frequency *f* is taken to be

the width direction, is plotted in Fig. 13.

of the width.

where *A* is the modulation amplitude and *f* denotes the modulation frequency, i.e., modulat‐ ed number per unit length. The modulated graph of an armchair graphene nanoribbon is shown in Fig. 11.

**Figure 11.** Graphene nanoribbon modulated by sine regime in the direction of the width.

**Figure 12.** Energy spectra of graphene nanoribbons with width *n*=7 modulated by sine regime in the direction of the width. *f* is the modulation frequency.

By numerical calculations, we found that different modulation frequencies have different electronic band structures, i.e., the energy band structures depend strongly on the modula‐ tion frequency *f* and the modulation amplitude *A*. We take the width *n*=7 as an example to calculate the electronic energy spectrum. When the amplitude *A* is fixed, the energy band structures with different frequencies are shown in Fig. 12. It may be seen from Fig. 12 that this periodic modulation does not damage the Dirac cones, i.e., the topological property of armchair graphene nanoribbons is not destroyed. On the other hand, Fig. 12 tells us that this periodic modulation can change the energy band structure, i.e., both the bandwidth and band gap can be controlled by the modulation frequency. The density of states of electrons for an armchair graphene nanoribbon with *n*=7, modulated by using a sine function along the width direction, is plotted in Fig. 13.

**5. Graphene nanoribbon modulated by sine regime**

y-direction, we modulate it with the aid of a sine periodic function

**Figure 11.** Graphene nanoribbon modulated by sine regime in the direction of the width.

shown in Fig. 11.

94 New Progress on Graphene Research

width. *f* is the modulation frequency.

A free standing graphene nanoribbon could have out-of-plane warping because of the edge stress (Shenoy et al., 2008). This warping will bring about a very small change of the elec‐ tronic energy spectrum. An ideal graphene nanoribbon only has periodicity in the direction of its length and there is no periodicity in the y-direction. To show the periodic effect in the

where *A* is the modulation amplitude and *f* denotes the modulation frequency, i.e., modulat‐ ed number per unit length. The modulated graph of an armchair graphene nanoribbon is

**Figure 12.** Energy spectra of graphene nanoribbons with width *n*=7 modulated by sine regime in the direction of the

*z A fy* = sin( ) (17)

**Figure 13.** Density of states of graphene nanoribbons with width *n*=7 modulated by sine regime in the direction of the width.

When the modulation amplitude, taken to be 0.1nm, is fixed, different modulation frequen‐ cies have slightly different densities of states of electrons. The main difference between the frequencies 0.0nm-1, 5.0nm-1, and 10.0nm-1 is in the conduction band and the density of states of the valence band is the same nearly. It follows that the modulation along the width direc‐ tion of the ribbon makes a notable impact for the density of states of the conduction band, especially for the high energy band corresponding to the standing wave of the smaller quan‐ tum number. In order to reveal the effect of the modulation amplitude on the electronic properties, the energy bands for the different amplitudes are calculated under certain fre‐ quency. Fig. 14 shows the band structures of the different amplitudes *A*=0.0nm, *A*=0.05nm, and *A*=0.1nm for an armchair nanoribbon with *n*=7, where the frequency *f* is taken to be 10nm-1. When the modulation amplitude *A*=0.1nm, the band gaps corresponding to frequen‐ cies *f*=0.0nm-1, 5nm-1, and 10nm-1 are 2.580eV, 2.600eV, and 2.666eV, respectively. It seems that the band gaps linearly increase as the frequency increases. In fact, the inflection point of the smallest gap appears at *f*=6.02nm-1, where the gap is equal to 2.571eV. There are other inflection points of the gap as the frequency increases, but the gaps of these points are big compared to that of the lowest inflection point (see Fig. 15(b)).

**6. Conclusion**

**Author details**

Guo-Ping Tong\*

**References**

We investigated the electronic energy spectra of graphene and its nanoribbon subject to un‐ axial stress within the tight-binding approach. The unaxial stress can not open the energy gap of graphene at Dirac point K. But compression along the armchair shape edge or exten‐ sion along the zigzag shape edge will make a small energy gap opened at K point. From this reason, the graphene subject to uniaxial stress still is a semiconductor with the zero-energy gaps. The position of Dirac point will vary as the stress. For the armchair graphene nanorib‐ bon, the tensile or compressive stress not only can transfer the metallicity into the semicon‐ ductor, but also have the energy gap increased or decreased and the energy bandwidth widened or narrowed. Therefore, we can use the unaxial stress to control the electronic properties of armchair graphene nanoribbons. In addition, the tubular warping deformation of armchair nanoribbons does not nearly influence on the energy gap, but it is obvious to effect on the bandwidth. In addition, we also studied the periodic modulation of the shape of armchair nanoribbons by sine regime. This modulation can change its electronic proper‐

Electronic Properties of Deformed Graphene Nanoribbons

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97

The advantage of the tight-binding method is that the physical picture is clearer and the cal‐ culating process is simpler compared to the first-principles calculations. This method is suit‐ able only for narrow energy bands. Because graphene nanoribbons are the system of wider

[1] Cadelano, E., Palla, P. L., Giordano, S., & Colombo, L. (2009). Nonlinear elasticity of

[2] Can, C. K., & Srolovitz, D. J. (2010). First-principles study of graphene edge proper‐

[3] Chung, P. W. (2006). Theoretical prediction of stress-induced phase transformations

[4] de Andres, P. L., & Vergés, J. A. (2008). First-principles calculation of the effect of stress on the chemical activity of graphene. *Applied Physics Letters*, 93(17), 171915-3.

ties. For the other modulation manner, we no longer discuss it here.

monolayer graphene. *Phys. Rev. Lett.*, 102(23), 235502-4.

of the second kind in graphene. *Phys. Rev. B*, 73(7), 075433-5.

ties and flake shapes. *Phys. Rev. B*, 81(12), 125445-8.

energy bands, this method has its limitation.

Address all correspondence to: tgp6463@zjnu.cn

Zhejiang Normal University, China

**Figure 14.** Energy spectra of graphene nanoribbons with width *n*=7 modulated by sine regime in the direction of the width. The modulation frequency *f* is taken to be 10nm−<sup>1</sup> and *A* is the modulation amplitude.

**Figure 15.** Energy gaps as a function of the modulation frequency *f*.

### **6. Conclusion**

10nm-1. When the modulation amplitude *A*=0.1nm, the band gaps corresponding to frequen‐ cies *f*=0.0nm-1, 5nm-1, and 10nm-1 are 2.580eV, 2.600eV, and 2.666eV, respectively. It seems that the band gaps linearly increase as the frequency increases. In fact, the inflection point of the smallest gap appears at *f*=6.02nm-1, where the gap is equal to 2.571eV. There are other inflection points of the gap as the frequency increases, but the gaps of these points are big

**Figure 14.** Energy spectra of graphene nanoribbons with width *n*=7 modulated by sine regime in the direction of the

2.56 2.58 2.60 2.62 2.64 2.66 2.68 2.70 2.72 2.74

(c)n=9

1.86 1.88 1.90 1.92 1.94 1.96

1.46 1.48 1.50 1.52 1.54 1.56 1.58

**periodicity degree** *f*

and *A* is the modulation amplitude.

(a)n=6 (b)n=7

(e)n=12 (f)n=13

<sup>2</sup> <sup>4</sup> <sup>6</sup> <sup>8</sup> <sup>10</sup> 1.44

(d)n=10

compared to that of the lowest inflection point (see Fig. 15(b)).

96 New Progress on Graphene Research

width. The modulation frequency *f* is taken to be 10nm−<sup>1</sup>

2 4 6 8 10 1.30

**Figure 15.** Energy gaps as a function of the modulation frequency *f*.

2.60 2.65 2.70 2.75 2.80

1.60 1.65 1.70 1.75 1.80 1.85 1.90 1.95 2.00

**gap(ev)**

1.35 1.40 1.45 1.50 1.55 We investigated the electronic energy spectra of graphene and its nanoribbon subject to un‐ axial stress within the tight-binding approach. The unaxial stress can not open the energy gap of graphene at Dirac point K. But compression along the armchair shape edge or exten‐ sion along the zigzag shape edge will make a small energy gap opened at K point. From this reason, the graphene subject to uniaxial stress still is a semiconductor with the zero-energy gaps. The position of Dirac point will vary as the stress. For the armchair graphene nanorib‐ bon, the tensile or compressive stress not only can transfer the metallicity into the semicon‐ ductor, but also have the energy gap increased or decreased and the energy bandwidth widened or narrowed. Therefore, we can use the unaxial stress to control the electronic properties of armchair graphene nanoribbons. In addition, the tubular warping deformation of armchair nanoribbons does not nearly influence on the energy gap, but it is obvious to effect on the bandwidth. In addition, we also studied the periodic modulation of the shape of armchair nanoribbons by sine regime. This modulation can change its electronic proper‐ ties. For the other modulation manner, we no longer discuss it here.

The advantage of the tight-binding method is that the physical picture is clearer and the cal‐ culating process is simpler compared to the first-principles calculations. This method is suit‐ able only for narrow energy bands. Because graphene nanoribbons are the system of wider energy bands, this method has its limitation.

## **Author details**

Guo-Ping Tong\*

Address all correspondence to: tgp6463@zjnu.cn

Zhejiang Normal University, China

#### **References**


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**Chapter 4**

**Provisional chapter**

**The Cherenkov Effect in Graphene-Like Structures**

**The Cherenkov Effect in Graphene-Like Structures**

More than 70 years ago, Peierls [29] and Landau [16] performed a proof that the 2-dimensional crystal is not thermodynamically stable and cannot exist. They argued that the thermodynamical fluctuations of such crystal leads to such displacements of atoms that these displacements are of the same size as the distances between atoms at the any finite temperature. The argument was extended by Mermin [21] and it seemed that many experimental observations supported the Landau-Peierls-Mermin theory. So, the

In 2004, Andre Geim, Kostia Novoselov [13, 22, 23] and co-workers at the University of Manchester in the UK by delicately cleaving a sample of graphite with sticky tape produced a sheet of crystalline carbon just one atom thick, known as graphene. Geims group was able to isolate graphene, and was able to visualize the new crystal using a simple optical microscope. Nevertheless, Landau-Peierls-Mermin proof remained of the permanent

At present time, there are novel methods how to create graphene sheet. For instance, Dato et al. [5] used the plasma reactor, where the graphene sheets were synthesized by passing

Graphene is the benzene ring (*C*6*H*6) stripped out from their H-atoms. It is allotrope of carbon because carbon can be in the crystalline form of graphite, diamond, fullerene (*C*60),

Graphene unique properties arise from the collective behavior of so called pseudoelectrons with pseudospins, which are governed by the Dirac equation in the hexagonal lattice.

The Dirac fermions in graphene carry one unit of electric charge and so can be manipulated using electromagnetic fields. Strong interactions between the electrons and the honeycomb lattice of carbon atoms mean that the dispersion relation is linear and given by *E* = *vp*, where

The linear dispersion relation follows from the relativistic energy relation for small mass together with approximation that the Fermi velocity is approximately only about 300 times

> ©2012 Pardy, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 Pardy; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

© 2013 Pardy, licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

distribution, and reproduction in any medium, provided the original work is properly cited.

Additional information is available at the end of the chapter

"impossibility" of the existence of graphene was established.

carbon nanotube and glassy carbon (also called vitreous carbon).

*v* is so called the Fermi-Dirac velocity, *p* is momentum of a pseudoelectron.

Additional information is available at the end of the chapter

Miroslav Pardy

**1. Introduction**

Miroslav Pardy

http://dx.doi.org/10.5772/50296

historical and pedagogical meaning.

less than the speed of light.

liquid ethanol droplets into an argon plasma.

**Provisional chapter**

## **The Cherenkov Effect in Graphene-Like Structures The Cherenkov Effect in Graphene-Like Structures**

Miroslav Pardy Miroslav Pardy

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/50296

## **1. Introduction**

More than 70 years ago, Peierls [29] and Landau [16] performed a proof that the 2-dimensional crystal is not thermodynamically stable and cannot exist. They argued that the thermodynamical fluctuations of such crystal leads to such displacements of atoms that these displacements are of the same size as the distances between atoms at the any finite temperature. The argument was extended by Mermin [21] and it seemed that many experimental observations supported the Landau-Peierls-Mermin theory. So, the "impossibility" of the existence of graphene was established.

In 2004, Andre Geim, Kostia Novoselov [13, 22, 23] and co-workers at the University of Manchester in the UK by delicately cleaving a sample of graphite with sticky tape produced a sheet of crystalline carbon just one atom thick, known as graphene. Geims group was able to isolate graphene, and was able to visualize the new crystal using a simple optical microscope. Nevertheless, Landau-Peierls-Mermin proof remained of the permanent historical and pedagogical meaning.

At present time, there are novel methods how to create graphene sheet. For instance, Dato et al. [5] used the plasma reactor, where the graphene sheets were synthesized by passing liquid ethanol droplets into an argon plasma.

Graphene is the benzene ring (*C*6*H*6) stripped out from their H-atoms. It is allotrope of carbon because carbon can be in the crystalline form of graphite, diamond, fullerene (*C*60), carbon nanotube and glassy carbon (also called vitreous carbon).

Graphene unique properties arise from the collective behavior of so called pseudoelectrons with pseudospins, which are governed by the Dirac equation in the hexagonal lattice.

The Dirac fermions in graphene carry one unit of electric charge and so can be manipulated using electromagnetic fields. Strong interactions between the electrons and the honeycomb lattice of carbon atoms mean that the dispersion relation is linear and given by *E* = *vp*, where *v* is so called the Fermi-Dirac velocity, *p* is momentum of a pseudoelectron.

The linear dispersion relation follows from the relativistic energy relation for small mass together with approximation that the Fermi velocity is approximately only about 300 times less than the speed of light.

Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 Pardy; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 Pardy, licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

©2012 Pardy, licensee InTech. This is an open access chapter distributed under the terms of the Creative

The pseudospin of the pseudoelectron follows from the graphene structure. The graphene is composed of the system of hexagonal cells and it means that graphene is composed from the systems of two equilateral triangles. If the wave function of the first triangle sublattice system is *ϕ*<sup>1</sup> and the wave function of the second triangle sublattice system is *ϕ*2, then the total wave function of the electron moving in the hexagonal system is superposition *<sup>ψ</sup>* = *<sup>c</sup>*1*ϕ*<sup>1</sup> + *<sup>c</sup>*2*ϕ*2, where *c*<sup>1</sup> and are *c*<sup>2</sup> appropriate functions of coordinate **x** and functions *ϕ*1, *ϕ*<sup>2</sup> are functions of wave vector **k** and coordinate **x**. The next crucial step is the new spinor function defined as [19].

$$\chi = \begin{Bmatrix} \varrho\_1\\ \varrho\_2 \end{Bmatrix} \tag{1}$$

In other words such structures can be considered as the graphene-like structures with the appropriate index of refraction, which is necessary for the the existence of Cerenkov effect. ˇ The last but not least graphene-like structure can be represented by graphene-based

We derive in this chapter the power spectrum of photons generated by charged particle moving in parallel direction and perpendicular direction to the graphene-like structure with index of refraction *n*. The Graphene sheet is conductive contrary to some graphene-like structures, for instance graphene with implanted ions, which are dielectric media and it means that it enables the experimental realization of the Cerenkov radiation. We calculate it ˇ

To be pedagogically clear we introduce the quantum theory of the index of refraction (where the dipole polarization of matter is the necessary condition for its existence), the classical and quantum theory of Cerenkov radiation and elements of the Schwinger source theory ˇ formalism for electrodynamic effect in dielectric medium. We involve also the Cerenkov ˇ

The quantum theory of dispersion can be derived in the framework of the nonrelativistic Schrödinger equation [33] for an electron moving in dielectric medium and in the field with

and this potential energy is the perturbation energy in the Schrödinger equation

<sup>−</sup> *<sup>H</sup>*<sup>0</sup> <sup>−</sup> *<sup>V</sup>*′

*<sup>k</sup>* is the solution of the Schrödinger equation without perturbation, or,

*<sup>h</sup>*¯ *Ek t* = *ψ*0 *k e* −*iω<sup>k</sup> t*

 *ih*¯ *∂ ∂t*

*<sup>k</sup>* (*t*) and

*ψ*0

 *ih*¯ *∂ ∂t* − *H*<sup>0</sup> *ψ*0

*<sup>k</sup>* (*t*) = *<sup>ψ</sup>*<sup>0</sup> *k e* − *<sup>i</sup>*

*Fx* = −*eE*<sup>0</sup> cos *ωt*, *Fy* = *Fz* = 0. (2)

*<sup>V</sup>*′ <sup>=</sup> <sup>−</sup>*exE*<sup>0</sup> cos *<sup>ω</sup><sup>t</sup>* (3)

*<sup>ψ</sup>k*(*t*) = 0, (4)

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103

*<sup>k</sup>* (*t*) = 0. (6)

, (5)

polaritonic crystal sheet [2] which can be used to study the Cerenkov effect. ˇ

from the viewpoint of the Schwinger theory of sources [24–27, 30–32].

**2. The quantum theory of index of refraction**

Then, the corresponding potential energy is

effect with massive photons.

where for *<sup>V</sup>*′ <sup>=</sup> <sup>0</sup> *<sup>ψ</sup>k*(*t*) <sup>→</sup> *<sup>ψ</sup>*<sup>0</sup>

where *ψ*<sup>0</sup>

the periodic force

and it is possible to prove that this spinor function is solution of the Pauli equation in the nonrelativistic situation and Dirac equation of the generalized case. The corresponding mass of such effective electron is proved to be zero.

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

The graphene can be considered as the special form the 2-dimensional graphene-like structures, where for instance silicene has the analogue structure as graphene [8]. The band structure of a free silicene layer resembles the band structure of graphene. The Fermi velocity *v* of electrons in silicene is lower than that in graphene.

If we switch on an electric field, the symmetry between the *A* and *B* sublattices of silicene's honeycomb structure breaks and a gap ∆ is open in the band structure at the hexagonal Brillouin zone (BZ) points K and K'. In the framework of a simple nearest-neighbor tight-binding model, this manifests itself in the form of an energy correction to the on-site energies that is positive for sublattice *A* and negative for *B*. This difference in on-site energies ∆ = *EA* − *EB* leads to a spectrum with a gap for electrons in the vicinity of the corners of the BZ with *E*<sup>±</sup> = ± (∆/2)<sup>2</sup> + |*v***p**|2, where **p** is the electron momentum relative to the BZ corner. Opening a gap in graphene by these means would be impossible because the *A* and *B* sublattices lie in the same plane [7].

So, silicene consists of a honeycomb lattice of silicon atoms with two sublattices made of *A* sites and *B* sites. The states near the Fermi energy are orbitals residing near the K and K' points at opposite corners of the hexagonal Brillouin zone. While silicon is dielectric medium, silicene is the conductive medium with Hall effect and it is possible to study the Mach cone generated by motion of a charged particle through the silicene sheet.

On the other side, there are amorphous solids - glasses, the atomic structure of which lack any long range translational periodicity. However, due to chemical bonding characteristics, glasses do possess a high degree of short-range order with respect to local atomic polyhedra. In other words such structures can be considered as the graphene-like structures with the appropriate index of refraction, which is necessary for the the existence of Cerenkov effect. ˇ

The last but not least graphene-like structure can be represented by graphene-based polaritonic crystal sheet [2] which can be used to study the Cerenkov effect. ˇ

We derive in this chapter the power spectrum of photons generated by charged particle moving in parallel direction and perpendicular direction to the graphene-like structure with index of refraction *n*. The Graphene sheet is conductive contrary to some graphene-like structures, for instance graphene with implanted ions, which are dielectric media and it means that it enables the experimental realization of the Cerenkov radiation. We calculate it ˇ from the viewpoint of the Schwinger theory of sources [24–27, 30–32].

To be pedagogically clear we introduce the quantum theory of the index of refraction (where the dipole polarization of matter is the necessary condition for its existence), the classical and quantum theory of Cerenkov radiation and elements of the Schwinger source theory ˇ formalism for electrodynamic effect in dielectric medium. We involve also the Cerenkov ˇ effect with massive photons.

#### **2. The quantum theory of index of refraction**

The quantum theory of dispersion can be derived in the framework of the nonrelativistic Schrödinger equation [33] for an electron moving in dielectric medium and in the field with the periodic force

$$F\_X = -eE\_0 \cos \omega t, \quad F\_Y = F\_z = 0. \tag{2}$$

Then, the corresponding potential energy is

2

as [19].

The pseudospin of the pseudoelectron follows from the graphene structure. The graphene is composed of the system of hexagonal cells and it means that graphene is composed from the systems of two equilateral triangles. If the wave function of the first triangle sublattice system is *ϕ*<sup>1</sup> and the wave function of the second triangle sublattice system is *ϕ*2, then the total wave function of the electron moving in the hexagonal system is superposition *<sup>ψ</sup>* = *<sup>c</sup>*1*ϕ*<sup>1</sup> + *<sup>c</sup>*2*ϕ*2, where *c*<sup>1</sup> and are *c*<sup>2</sup> appropriate functions of coordinate **x** and functions *ϕ*1, *ϕ*<sup>2</sup> are functions of wave vector **k** and coordinate **x**. The next crucial step is the new spinor function defined

*χ* =

of such effective electron is proved to be zero.

and not by collective behavior in hexagonal sheet called graphene.

*v* of electrons in silicene is lower than that in graphene.

the BZ with *E*<sup>±</sup> = ±

*B* sublattices lie in the same plane [7].

 *ϕ*<sup>1</sup> *ϕ*2 

and it is possible to prove that this spinor function is solution of the Pauli equation in the nonrelativistic situation and Dirac equation of the generalized case. The corresponding mass

The introduction of the Dirac relativistic Hamiltonian in graphene physics is the description of the graphene physics by means of electron-hole medium. It is the analogue of the description of the electron-positron vacuum by the Dirac theory of quantum electrodynamics. The pseudoelectron and pseudospin are not an electron and the spin of quantum electrodynamics (QED), because QED is the quantum theory of the interaction of real electrons and photons where mass of an electron is the mass defined by classical mechanics

The graphene can be considered as the special form the 2-dimensional graphene-like structures, where for instance silicene has the analogue structure as graphene [8]. The band structure of a free silicene layer resembles the band structure of graphene. The Fermi velocity

If we switch on an electric field, the symmetry between the *A* and *B* sublattices of silicene's honeycomb structure breaks and a gap ∆ is open in the band structure at the hexagonal Brillouin zone (BZ) points K and K'. In the framework of a simple nearest-neighbor tight-binding model, this manifests itself in the form of an energy correction to the on-site energies that is positive for sublattice *A* and negative for *B*. This difference in on-site energies ∆ = *EA* − *EB* leads to a spectrum with a gap for electrons in the vicinity of the corners of

corner. Opening a gap in graphene by these means would be impossible because the *A* and

So, silicene consists of a honeycomb lattice of silicon atoms with two sublattices made of *A* sites and *B* sites. The states near the Fermi energy are orbitals residing near the K and K' points at opposite corners of the hexagonal Brillouin zone. While silicon is dielectric medium, silicene is the conductive medium with Hall effect and it is possible to study the

On the other side, there are amorphous solids - glasses, the atomic structure of which lack any long range translational periodicity. However, due to chemical bonding characteristics, glasses do possess a high degree of short-range order with respect to local atomic polyhedra.

Mach cone generated by motion of a charged particle through the silicene sheet.

(∆/2)<sup>2</sup> + |*v***p**|2, where **p** is the electron momentum relative to the BZ

(1)

$$V' = -e\mathbf{x}E\_0 \cos\omega t\tag{3}$$

and this potential energy is the perturbation energy in the Schrödinger equation

$$\left(i\hbar\frac{\partial}{\partial t} - H\_0 - V'\right)\psi\_k(t) = 0,\tag{4}$$

where for *<sup>V</sup>*′ <sup>=</sup> <sup>0</sup> *<sup>ψ</sup>k*(*t*) <sup>→</sup> *<sup>ψ</sup>*<sup>0</sup> *<sup>k</sup>* (*t*) and

$$
\psi\_k^0(t) = \psi\_k^0 e^{-\frac{i}{\hbar}E\_k t} = \psi\_k^0 e^{-i\omega\_k t},\tag{5}
$$

where *ψ*<sup>0</sup> *<sup>k</sup>* is the solution of the Schrödinger equation without perturbation, or,

$$\left(i\hbar\frac{\partial}{\partial t} - H\_0\right)\psi\_k^0(t) = 0.\tag{6}$$

We are looking for the solution of the Schrödinger equation involving the perturbation potential in the form

$$
\psi\_k(t) = \psi\_k^0(t) + \psi\_k^1(t), \tag{7}
$$

*<sup>ω</sup>kk*′′ <sup>=</sup> *Ek* <sup>−</sup> *Ek*′′

 *ψ*0<sup>∗</sup> *<sup>k</sup>*′ *<sup>ψ</sup>*<sup>0</sup>

*<sup>u</sup>* = ∑ *k*′

*<sup>v</sup>* = ∑ *k*′

*Ck* <sup>=</sup> <sup>−</sup>*eE*<sup>0</sup>

 −*eE*<sup>0</sup> 2¯*h* 

 −*eE*<sup>0</sup> 2¯*h* 

*xk*′*<sup>k</sup>* = *ψ*0<sup>∗</sup> *<sup>k</sup>*′ *<sup>x</sup>ψ*<sup>0</sup>

The classical polarization of a medium is given by the well known formula

define the quantum analogue form of the polarization as it follows:

The general wave function can be obtained from eqs. (7), (9), (18) and (19) in the form:

*xk*′*<sup>k</sup> ω*2

where N is the number of atom in the unite volume of dielectric medium. So we are able to

*<sup>k</sup>*′*<sup>k</sup>* <sup>−</sup> *<sup>ω</sup>*<sup>2</sup> *<sup>ψ</sup>*<sup>0</sup>

2¯*<sup>h</sup>* · *xk*′*<sup>k</sup>*

· *xk*′*<sup>k</sup> <sup>ω</sup>k*′*<sup>k</sup>* <sup>+</sup> *<sup>ω</sup> <sup>ψ</sup>*<sup>0</sup>

· *xk*′*<sup>k</sup> <sup>ω</sup>k*′*<sup>k</sup>* <sup>−</sup> *<sup>ω</sup> <sup>ψ</sup>*<sup>0</sup>

we get the following relation for *Ck* and *u* as follows:

Using the orthogonal relation

and *v* = *u*(−*ω*), or

*<sup>ψ</sup>k*(*t*) = *<sup>e</sup>*

−*iω<sup>k</sup> t ψ*0 *<sup>k</sup>* <sup>−</sup> *eE*<sup>0</sup> *<sup>h</sup>*¯ <sup>∑</sup> *k*′

and

*<sup>h</sup>*¯ . (15)

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105

*<sup>k</sup>*′′ *<sup>d</sup>*3*<sup>x</sup>* <sup>=</sup> *<sup>δ</sup>k*′*k*′′ , (16)

*<sup>ω</sup>k*′*<sup>k</sup>* <sup>+</sup> *<sup>ω</sup>* , (17)

*<sup>k</sup>*′ (18)

*<sup>k</sup>*′ (19)

. (21)

*<sup>k</sup> <sup>d</sup>*3*x*. (20)

*<sup>k</sup>*′[*ωk*′*<sup>k</sup>* cos *<sup>ω</sup><sup>t</sup>* <sup>−</sup> *<sup>i</sup><sup>ω</sup>* sin *<sup>ω</sup>t*]

*P* = *N p* = −*Nex*, (22)

where *ψ*<sup>1</sup> *<sup>k</sup>* (*t*), is the perturbation wave function correction to the non-perturbation wave function.

After insertion of formula (7) to eq. (4), we get

$$\left(i\hbar\frac{\partial}{\partial t} - H\_0\right)\psi\_k^1(t) = \frac{1}{2}exE\_0\psi\_k^0 \left(e^{-it(\omega\_k - \omega)} + e^{-it(\omega\_k + \omega)}\right). \tag{8}$$

Let us look for the solution of eq. (8) in the form:

$$
\psi\_k^1(t) = \imath \nu^{-it(\omega\_k - \omega)} + \imath \nu^{-it(\omega\_k + \omega)}.\tag{9}
$$

After insertion of (9) into (8), we get two equations for *u* and *v*:

$$\left(\hbar(\omega\_k - \omega) - H\_0\right)u = \frac{1}{2}e\mathbf{x}E\_0\psi\_{k'}^0\tag{10}$$

$$\left(\hbar(\omega\_k + \omega) - H\_0\right)v = \frac{1}{2}e\mathbf{x}E\_0\psi\_k^0. \tag{11}$$

Then, using the formal expansion

$$
\mu = \sum\_{k''} \mathbb{C}\_{k''} \psi^0\_{k''} \tag{12}
$$

we get from eq.

$$\left(\left(E\_{k''} - H\_0\right)\psi\_{k''}^0 = 0\tag{13}$$

the following equation

$$\left(\hbar \sum\_{k''} \mathbb{C}\_{k''} \left(\omega\_{kk''} - \omega\right) \psi\_{k''}^{0} = \frac{e\varepsilon E\_0}{2} \psi\_k^{0} \tag{14}$$

with

$$
\omega\_{kk''} = \frac{E\_k - E\_{k''}}{\hbar}.\tag{15}
$$

Using the orthogonal relation

$$\int \psi\_{k'}^{0\*} \psi\_{k''}^{0} d^3 \mathbf{x} = \delta\_{k'k''\prime} \tag{16}$$

we get the following relation for *Ck* and *u* as follows:

$$\mathbf{C}\_{k} = -\frac{eE\_{0}}{2\hbar} \cdot \frac{\mathbf{x}\_{k'k}}{\omega\_{k'k} + \omega'} \,, \tag{17}$$

$$
\mu = \sum\_{k'} \left( -\frac{eE\_0}{2\hbar} \right) \cdot \frac{\varkappa\_{k'k}}{\omega\_{k'k} + \omega} \psi\_{k'}^0 \tag{18}
$$

and *v* = *u*(−*ω*), or

$$
\omega = \sum\_{k'} \left( -\frac{eE\_0}{2\hbar} \right) \cdot \frac{\mathbf{x}\_{k'k}}{\omega\_{k'k} - \omega} \psi\_{k'}^0 \tag{19}
$$

and

4

potential in the form

After insertion of formula (7) to eq. (4), we get

Let us look for the solution of eq. (8) in the form:

*ψ*1

After insertion of (9) into (8), we get two equations for *u* and *v*:

*h*¯ ∑ *k*′′

 *ih*¯ *∂ ∂t* − *H*<sup>0</sup> *ψ*1 *<sup>k</sup>* (*t*) = <sup>1</sup> 2 *exE*0*ψ*<sup>0</sup> *k e*

Then, using the formal expansion

we get from eq.

with

the following equation

where *ψ*<sup>1</sup>

function.

We are looking for the solution of the Schrödinger equation involving the perturbation

*<sup>k</sup>* (*t*) + *ψ*<sup>1</sup>

*<sup>k</sup>* (*t*) = *ue*−*it*(*ωk*−*ω*) <sup>+</sup> *ve*−*it*(*ωk*+*ω*)

(*h*¯(*ω<sup>k</sup>* <sup>−</sup> *<sup>ω</sup>*) <sup>−</sup> *<sup>H</sup>*0) *<sup>u</sup>* <sup>=</sup> <sup>1</sup>

(*h*¯(*ω<sup>k</sup>* <sup>+</sup> *<sup>ω</sup>*) <sup>−</sup> *<sup>H</sup>*0) *<sup>v</sup>* <sup>=</sup> <sup>1</sup>

*u* = ∑ *k*′′

(*Ek*′′ − *H*0) *ψ*<sup>0</sup>

*Ck*′′ (*ωkk*′′ − *ω*) *ψ*<sup>0</sup>

*Ck*′′*ψ*<sup>0</sup>

*<sup>k</sup>* (*t*), is the perturbation wave function correction to the non-perturbation wave

<sup>−</sup>*it*(*ωk*−*ω*) + *e*

2 *exE*0*ψ*<sup>0</sup>

2 *exE*0*ψ*<sup>0</sup>

*<sup>k</sup>*′′ <sup>=</sup> *exE*<sup>0</sup>

<sup>2</sup> *<sup>ψ</sup>*<sup>0</sup>

*<sup>k</sup>* (*t*), (7)

. (9)

*<sup>k</sup>* , (10)

*<sup>k</sup>* . (11)

*<sup>k</sup>*′′ , (12)

*<sup>k</sup>*′′ = 0 (13)

*<sup>k</sup>* (14)

. (8)

−*it*(*ωk*+*ω*)

*ψk*(*t*) = *ψ*<sup>0</sup>

$$
\pi\_{k'k} = \int \psi\_{k'}^{0\*} \mathbf{x} \psi\_k^0 d^3 \mathbf{x}.\tag{20}
$$

The general wave function can be obtained from eqs. (7), (9), (18) and (19) in the form:

$$\psi\_k(t) = e^{-i\omega\_k t} \left\{ \psi\_k^0 - \frac{eE\_0}{\hbar} \sum\_{k'} \frac{\mathbf{x}\_{k'k}}{\omega\_{k'k}^2 - \omega^2} \psi\_{k'}^0 [\omega\_{k'k} \cos \omega t - i\omega \sin \omega t] \right\}. \tag{21}$$

The classical polarization of a medium is given by the well known formula

$$P = Np = -\text{Nex}\_{\prime} \tag{22}$$

where N is the number of atom in the unite volume of dielectric medium. So we are able to define the quantum analogue form of the polarization as it follows:

$$P = N\overline{p} = -Ne \int \psi\_k^\*(t) x \psi\_k(t) d^3 \mathbf{x}\_\prime \tag{23}$$

or, with

$$
\int \psi\_k^{0\*} \ge \psi\_k^0 d^3 \mathfrak{x} = 0,\tag{24}
$$

**3. The classical description of the Cerenkov radiation ˇ**

The equations for the potentials **A**, *ϕ* are given by equations [17, 18]

<sup>∆</sup>**<sup>A</sup>** <sup>−</sup> *<sup>ε</sup> c*2 *∂*2**A** *<sup>∂</sup>t*<sup>2</sup> <sup>=</sup> <sup>−</sup>4*<sup>π</sup>*

<sup>∆</sup>**'** <sup>−</sup> *<sup>ε</sup> c*2 *∂*2*ϕ*

with the additional Lorentz calibration condition:

After the Fourier transformation the vector potential

<sup>∆</sup>**Ak** <sup>−</sup> *<sup>ε</sup> c*2

**A***e*

*∂*<sup>2</sup>**Ak**

*<sup>∂</sup>t*<sup>2</sup> <sup>=</sup> <sup>4</sup>*π<sup>e</sup>*

1 (2*π*)3/2

electrodynamics [9].

well known:

and

we get

In electrodynamics, a fast moving charged particle in a medium when its speed is faster than the speed of light in this medium produces electromagnetic radiation which is called the Cerenkov radiation. This radiation was first observed experimentally by ˇ Cerenkov [3, 4] ˇ and theoretically interpreted by Tamm and Frank [34], in the framework of the classical

The charge and current density of electron moving with the velocity **v** and charge *e* is as it is

*c*

*e ε*

*<sup>∂</sup>t*<sup>2</sup> <sup>=</sup> <sup>−</sup>4*<sup>π</sup>*

*ε c ∂ϕ*

<sup>−</sup>*i***kx***d*3*<sup>x</sup>* <sup>=</sup> **Ak**; **<sup>A</sup>**(**x**) = <sup>1</sup>

*<sup>c</sup>*(2*π*)3/2 **<sup>v</sup>**

 *e*

div**A** +

where magnetic permeability *µ* = 1 and *ε* is dielectric constant of medium.

*̺* = *eδ*(**x** − **v***t*) (31)

**j** = *e***v***δ*(**x** − **v***t*). (32)

*e***v***δ*(**x** − **v***t*) (33)

The Cherenkov Effect in Graphene-Like Structures

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107

*δ*(**x** − **v***t*) (34)

*<sup>∂</sup><sup>t</sup>* <sup>=</sup> 0, (35)

**Ak***e*

<sup>−</sup>*i***kx***δ*(**<sup>x</sup>** <sup>−</sup> **<sup>v</sup>***t*)*d*3*x*, (37)

*<sup>i</sup>***kx***d*3*k*, (36)

(2*π*)3/2

we have

$$P = \sum\_{k'} \left( 2 \frac{Ne^2 E\_0}{\hbar} \right) \cdot \frac{\omega\_{k'k} |\mathbf{x}\_{k'k}|^2}{\omega\_{k'k}^2 - \omega^2} \cos \omega t. \tag{25}$$

Using the classical formula for polarization *P*,

$$\mathbf{P} = \frac{n^2 - 1}{4\pi} \mathbf{E},\tag{26}$$

we get for the quantum model of polarization

$$\frac{n^2 - 1}{4\pi} = \sum\_{k'} \left( 2\frac{Ne^2}{\hbar} \right) \cdot \frac{\omega\_{k'k} |\mathbf{x}\_{k'k}|^2}{\omega\_{k'k}^2 - \omega^2}. \tag{27}$$

Using the definition of the coefficients *fk*′*<sup>k</sup>* by relation

$$f\_{k'k} = \frac{2m}{\hbar} \omega\_{k'k} |\mathbf{x}\_{k'k}|^2,\tag{28}$$

we get the modified equation (27) as follows:

$$\frac{m^2 - 1}{4\pi} = \frac{Ne^2}{m} \sum\_{k'} \frac{f\_{k'k}}{\omega\_{k'k}^2 - \omega^2}. \tag{29}$$

The last formula should be compared with the classical one:

$$\frac{m^2 - 1}{4\pi} = \frac{e^2}{m} \sum\_{k} \frac{N\_k}{\omega\_k^2 - \omega^2} \tag{30}$$

where *Nk* is number of electrons moving with frequency *ω<sup>k</sup>* in the unit volume.

## **3. The classical description of the Cerenkov radiation ˇ**

In electrodynamics, a fast moving charged particle in a medium when its speed is faster than the speed of light in this medium produces electromagnetic radiation which is called the Cerenkov radiation. This radiation was first observed experimentally by ˇ Cerenkov [3, 4] ˇ and theoretically interpreted by Tamm and Frank [34], in the framework of the classical electrodynamics [9].

The charge and current density of electron moving with the velocity **v** and charge *e* is as it is well known:

$$
\boldsymbol{\varrho} = \boldsymbol{e} \boldsymbol{\delta}(\mathbf{x} - \mathbf{v}t) \tag{31}
$$

$$\mathbf{j} = e\mathbf{v}\delta(\mathbf{x} - \mathbf{v}t).\tag{32}$$

The equations for the potentials **A**, *ϕ* are given by equations [17, 18]

$$
\Delta \mathbf{A} - \frac{\varepsilon}{c^2} \frac{\partial^2 \mathbf{A}}{\partial t^2} = -\frac{4\pi}{c} \varepsilon \mathbf{v} \delta(\mathbf{x} - \mathbf{v}t) \tag{33}
$$

and

6

or, with

we have

*P* = *Np*¯ = −*Ne*

 *ψ*0<sup>∗</sup> *<sup>k</sup> <sup>x</sup>ψ*<sup>0</sup>

*<sup>P</sup>* = ∑ *k*′

> *n*<sup>2</sup> − 1 <sup>4</sup>*<sup>π</sup>* <sup>=</sup> <sup>∑</sup> *k*′

Using the classical formula for polarization *P*,

we get for the quantum model of polarization

we get the modified equation (27) as follows:

Using the definition of the coefficients *fk*′*<sup>k</sup>* by relation

 <sup>2</sup> *Ne*2*E*<sup>0</sup> *h*¯

 *ψ*∗

 ·

**<sup>P</sup>** <sup>=</sup> *<sup>n</sup>*<sup>2</sup> <sup>−</sup> <sup>1</sup> 4*π*

> <sup>2</sup> *Ne*<sup>2</sup> *h*¯ ·

*fk*′*<sup>k</sup>* <sup>=</sup> <sup>2</sup>*<sup>m</sup>*

<sup>4</sup>*<sup>π</sup>* <sup>=</sup> *Ne*<sup>2</sup>

*n*<sup>2</sup> − 1

*n*<sup>2</sup> − 1 <sup>4</sup>*<sup>π</sup>* <sup>=</sup> *<sup>e</sup>*<sup>2</sup>

where *Nk* is number of electrons moving with frequency *ω<sup>k</sup>* in the unit volume.

The last formula should be compared with the classical one:

*<sup>ω</sup>k*′*k*|*xk*′*k*|

*ω*2

2

*<sup>ω</sup>k*′*k*|*xk*′*k*|

*ω*2

*fk*′*<sup>k</sup> ω*2

*Nk ω*2

*<sup>h</sup>*¯ *<sup>ω</sup>k*′*k*|*xk*′*k*<sup>|</sup>

*<sup>m</sup>* <sup>∑</sup> *k*′

> *<sup>m</sup>* <sup>∑</sup> *k*

2

*<sup>k</sup>* (*t*)*xψk*(*t*)*d*3*x*, (23)

*<sup>k</sup> <sup>d</sup>*3*<sup>x</sup>* <sup>=</sup> 0, (24)

*<sup>k</sup>*′*<sup>k</sup>* <sup>−</sup> *<sup>ω</sup>*<sup>2</sup> cos *<sup>ω</sup>t*. (25)

**E**, (26)

*<sup>k</sup>*′*<sup>k</sup>* <sup>−</sup> *<sup>ω</sup>*<sup>2</sup> . (27)

2, (28)

*<sup>k</sup>*′*<sup>k</sup>* <sup>−</sup> *<sup>ω</sup>*<sup>2</sup> . (29)

*<sup>k</sup>* <sup>−</sup> *<sup>ω</sup>*<sup>2</sup> , (30)

$$
\Delta' - \frac{e}{c^2} \frac{\partial^2 \rho}{\partial t^2} = -4\pi \frac{e}{\varepsilon} \delta(\mathbf{x} - \mathbf{v}t) \tag{34}
$$

with the additional Lorentz calibration condition:

$$
\mathbf{div}\mathbf{A} + \frac{\varepsilon}{c} \frac{\partial \varrho}{\partial t} = \mathbf{0},
\tag{35}
$$

where magnetic permeability *µ* = 1 and *ε* is dielectric constant of medium.

After the Fourier transformation the vector potential

$$\frac{1}{(2\pi)^{3/2}} \int \mathbf{A} e^{-i\mathbf{k}\mathbf{x}} d^3 \mathbf{x} = \mathbf{A}\_{\mathbf{k}}; \quad \mathbf{A}(\mathbf{x}) = \frac{1}{(2\pi)^{3/2}} \int \mathbf{A}\_{\mathbf{k}} e^{i\mathbf{k}\mathbf{x}} d^3 k \tag{36}$$

we get

$$
\Delta \mathbf{A}\_{\mathbf{k}} - \frac{\varepsilon}{c^2} \frac{\partial^2 \mathbf{A}\_{\mathbf{k}}}{\partial t^2} = \frac{4\pi e}{c(2\pi)^{3/2}} \mathbf{v} \int e^{-i\mathbf{k}\mathbf{x}} \delta(\mathbf{x} - \mathbf{v}t) d^3 \mathbf{x},\tag{37}
$$

or,

$$
\Delta \mathbf{A}\_{\mathbf{k}} - \frac{e}{c^2} \frac{\partial^2 \mathbf{A}\_{\mathbf{k}}}{\partial t^2} = -\frac{4\pi c \mathbf{v}}{c(2\pi)^{3/2}} e^{-i\mathbf{k}\mathbf{v}t}.\tag{38}
$$

*<sup>ϕ</sup>***<sup>k</sup>** <sup>=</sup> <sup>4</sup>*π<sup>e</sup> ε*(2*π*)3/2

The intensity of the electric field has the Fourier Component as follows:

*<sup>∂</sup><sup>t</sup>* <sup>−</sup> grad*ϕ***<sup>k</sup>** <sup>=</sup> *<sup>i</sup><sup>ω</sup>*

*<sup>c</sup>*<sup>2</sup> <sup>−</sup> **<sup>k</sup>** *ε*

*<sup>c</sup>*<sup>2</sup> <sup>−</sup> **<sup>k</sup>** *ε*

The formula (48) gives **E** in the form of the moving plane wave in case that we can write

*<sup>ω</sup>* <sup>=</sup> *vkz* <sup>=</sup> *vk* cos <sup>Θ</sup> <sup>≡</sup> *kc*

cos <sup>Θ</sup> <sup>=</sup> *<sup>c</sup>*

*k*<sup>2</sup> = *k*<sup>2</sup>

*<sup>x</sup>* <sup>+</sup> *<sup>k</sup>*<sup>2</sup>

Now, let us chose the direction of the particle motion along the *z*-axis and let us introduce

*<sup>y</sup>* <sup>+</sup> *<sup>k</sup>*<sup>2</sup>

*n*(*ω*)*v*

*<sup>z</sup>* <sup>=</sup> *<sup>k</sup>*<sup>2</sup>

from which follows that the intensity of the electric field induced in the dielectric medium is:

**Ek** <sup>=</sup> <sup>−</sup><sup>1</sup> *c <sup>∂</sup>***Ak**

**<sup>E</sup>** <sup>=</sup> *ie* 2*π*<sup>2</sup>

From the last equation we have:

the cylindrical coordinates putting

At the same time

Further

4*πe* (2*π*)3/2 *<sup>ω</sup>***<sup>v</sup>**

*ω***v**

1 *<sup>k</sup>*<sup>2</sup> − *<sup>ω</sup>*2*<sup>ε</sup> c*2 *e* −*iωt*

*c*

 *i <sup>k</sup>*<sup>2</sup> − *<sup>ω</sup>*2*<sup>ε</sup> c*2 *e* −*iωt*

 *<sup>e</sup>i*(**kx**−*ωt*) *<sup>k</sup>*<sup>2</sup> − *<sup>ω</sup>*2*<sup>ε</sup> c*2

*n*(*ω*)

**Ak** <sup>−</sup> *<sup>i</sup>***k***ϕ***<sup>k</sup>** <sup>=</sup>

. (46)

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109

The Cherenkov Effect in Graphene-Like Structures

, (47)

*dkxdkydkz*. (48)

. (49)

. (50)

*<sup>z</sup>* <sup>+</sup> *<sup>q</sup>*2. (51)

*dkxdkydkz* = *q dq dϕ dkz*. (52)

On the other hand, we have:

$$
\Delta \mathbf{A} = \Delta \int \mathbf{A}\_{\mathbf{k}} e^{i\mathbf{k} \mathbf{x}} d^3 k = -\int k^2 \mathbf{A}\_{\mathbf{k}} e^{i\mathbf{k} \mathbf{r}} d^3 k \tag{39}
$$

from which we have

$$
\Delta \mathbf{A}\_{\mathbf{k}} = k^2 \mathbf{A}\_{\mathbf{k}} \tag{40}
$$

and

$$-k^2 \mathbf{A\_k} - \frac{\varepsilon}{c^2} \frac{\partial^2 \mathbf{A\_k}}{\partial t^2} = -\frac{4\pi c \mathbf{v}}{c(2\pi)^{3/2}} e^{-i\mathbf{k} \mathbf{v} t} \,. \tag{41}$$

Formula (41) shows that the dependence **Ak** on time is of the form:

$$\mathbf{A}\_{\mathbf{k}} \sim e^{-i\mathbf{k}\mathbf{v}t} = e^{-i\omega t},\tag{42}$$

where

$$
\omega = \mathbf{k} \mathbf{v}.\tag{43}
$$

At the same time

$$\frac{\partial^2 \mathbf{A\_k}}{\partial t^2} = -\omega^2 \mathbf{A\_k}.\tag{44}$$

We can transcribe eq. (41) in the following form:

$$\mathbf{A}\_{\mathbf{k}} = \frac{4\pi e}{c(2\pi)^{3/2}} \frac{\mathbf{v}}{k^2 - \frac{\omega^2 \varepsilon}{\varepsilon^2}} e^{-i\omega t}. \tag{45}$$

By analogy with the formula (45) we can derive the formula concerning the Fourier transform of *ϕ*. Or,

$$\varphi\_{\mathbf{k}} = \frac{4\pi e}{\varepsilon (2\pi)^{3/2}} \frac{1}{k^2 - \frac{\omega^2 \varepsilon}{c^2}} e^{-i\omega t}. \tag{46}$$

The intensity of the electric field has the Fourier Component as follows:

$$
\mathbf{E}\_{\mathbf{k}} = -\frac{1}{c} \frac{\partial \mathbf{A}\_{\mathbf{k}}}{\partial t} - \text{grad}\rho\_{\mathbf{k}} = \frac{i\omega}{c} \mathbf{A}\_{\mathbf{k}} - i\mathbf{k}\rho\_{\mathbf{k}} \quad =
$$

$$
\frac{4\pi e}{(2\pi)^{3/2}} \left(\frac{\omega \mathbf{v}}{c^2} - \frac{\mathbf{k}}{\varepsilon}\right) \frac{i}{k^2 - \frac{\omega^2 \varepsilon}{c^2}} e^{-i\omega t} \, \. \tag{47}
$$

*c*2

from which follows that the intensity of the electric field induced in the dielectric medium is:

$$\mathbf{E} = \frac{ie}{2\pi^2} \int \left(\frac{\omega \mathbf{v}}{c^2} - \frac{\mathbf{k}}{\varepsilon}\right) \frac{e^{i(\mathbf{k} \mathbf{x} - \omega t)}}{k^2 - \frac{\omega^2 \varepsilon}{c^2}} dk\_x dk\_y dk\_z. \tag{48}$$

The formula (48) gives **E** in the form of the moving plane wave in case that we can write

$$
\omega = \upsilon k\_z = \upsilon k \cos \Theta \equiv \frac{kc}{\mathfrak{n}(\omega)}.\tag{49}
$$

From the last equation we have:

$$\cos \Theta = \frac{\mathcal{L}}{\pi(\omega)v}. \tag{50}$$

Now, let us chose the direction of the particle motion along the *z*-axis and let us introduce the cylindrical coordinates putting

$$k^2 = k\_x^2 + k\_y^2 + k\_z^2 = k\_z^2 + q^2. \tag{51}$$

At the same time

$$
\hbar \, dk\_x dk\_y dk\_z = q \, d\mathbf{q} \, d\mathbf{q} \, dk\_z. \tag{52}
$$

Further

8

or,

and

where

of *ϕ*. Or,

At the same time

On the other hand, we have:

from which we have

<sup>∆</sup>**Ak** <sup>−</sup> *<sup>ε</sup> c*2

∆**A** = ∆

 **Ak***<sup>e</sup>*

<sup>−</sup>*k*<sup>2</sup>**Ak** <sup>−</sup> *<sup>ε</sup>*

Formula (41) shows that the dependence **Ak** on time is of the form:

*c*2

**Ak** <sup>∼</sup> *<sup>e</sup>*

*<sup>∂</sup>*<sup>2</sup>**Ak**

**Ak** <sup>=</sup> <sup>4</sup>*π<sup>e</sup>*

*c*(2*π*)3/2

By analogy with the formula (45) we can derive the formula concerning the Fourier transform

**v** *k*<sup>2</sup> − *<sup>ω</sup>*2*<sup>ε</sup> c*2 *e* −*iωt*

We can transcribe eq. (41) in the following form:

*<sup>∂</sup>*<sup>2</sup>**Ak**

*<sup>∂</sup>t*<sup>2</sup> <sup>=</sup> <sup>−</sup> <sup>4</sup>*πe***<sup>v</sup>**

<sup>−</sup>*i***kv***<sup>t</sup>* = *e*

*<sup>c</sup>*(2*π*)3/2 *<sup>e</sup>*

−*iωt*

*<sup>∂</sup>*<sup>2</sup>**Ak**

*<sup>∂</sup>t*<sup>2</sup> <sup>=</sup> <sup>−</sup> <sup>4</sup>*πe***<sup>v</sup>**

*<sup>i</sup>***kx***d*3*k* = −

*<sup>c</sup>*(2*π*)3/2 *<sup>e</sup>*

*<sup>k</sup>*<sup>2</sup>**Ak***<sup>e</sup>*

−*i***kv***t*

<sup>∆</sup>**Ak** <sup>=</sup> *<sup>k</sup>*<sup>2</sup>**Ak** (40)

−*i***kv***t*

. (38)

*<sup>i</sup>***kr***d*3*k*, (39)

. (41)

, (42)

. (45)

*ω* = **kv**. (43)

*<sup>∂</sup>t*<sup>2</sup> <sup>=</sup> <sup>−</sup>*ω*<sup>2</sup>**Ak**. (44)

$$dk\_z = \frac{d\omega}{v} \tag{53}$$

*d*2*W dzd<sup>ω</sup>* <sup>=</sup> *ie*2*<sup>ω</sup> π qdq*

write:

Then we have

 1 *<sup>v</sup>*2*ε*<sup>+</sup> <sup>−</sup> <sup>1</sup> *c*2 

determines the sign of the imaginary part of the permitivity of medium.

 1 *v*2*ε*<sup>+</sup>

*d*2*W dxd<sup>ω</sup>* <sup>=</sup> *ie*2*<sup>ω</sup>*

− <sup>1</sup> *c*2 = 1 *v*2*ε*<sup>−</sup>

from the next text. Let us introduce the new complex quantity:

2*π*

 *C*1 *du u* − *C*2 *du <sup>u</sup>* <sup>=</sup>

center in the origin of the coordinate system.

 1 *<sup>v</sup>*<sup>2</sup> <sup>−</sup> *<sup>ε</sup>*<sup>+</sup> *c*2

 <sup>−</sup> *ie*2|*ω*<sup>|</sup> *π*

We introduced in the last formula notation *ε*<sup>+</sup> = *ε*(*ω*) and *ε*<sup>−</sup> = *ε*(−|*ω*|) for *ε* at positive and negative values of *ω*. We know that in the absorptive dielectric media *ε* has the imaginary component. This imaginary component of *ε* is positive for *ω* < 0 and negative for *ω* > 0. In fact, the absorption in medium is real effect and it means that exp {−*ikx*} must correspond to the absorption for *x* > 0 for the arbitrary sign of *ω*. This experimental requirement

In formula (58) we can neglect in the numerator the imaginary part of the permitivity and

− <sup>1</sup> *c*2 = 1 *<sup>v</sup>*2*n*<sup>2</sup> <sup>−</sup> <sup>1</sup> *c*2 

On the other hand, such operation cannot be performed in the denominator which follows

 1 *<sup>v</sup>*<sup>2</sup> <sup>−</sup> *<sup>ε</sup> c*2 

In case of the absence of absorption in medium i.e. ℑ *ε* = 0, the formula (61) gives meaningless zero result. However *ε* and therefore also *u* has the nonzero imaginary part. It means that the integrals in (61) is considered in the complex plane. The contour of integration is chosen in such a way that it is going in parallel to the real axis above this axis in case of ℑ *<sup>u</sup>* > 0 (it corresponds to *<sup>ω</sup>* > 0) and it corresponds to the curve *<sup>C</sup>*1, and under the axis at ℑ *<sup>u</sup>* < 0 (i.e. *<sup>ω</sup>* < 0) which corresponds to the curve *<sup>C</sup>*2. The singular point *<sup>u</sup>* = 0 is avoided

The integration was performed as a limiting procedure along the infinitesimal circle with the

*C*1 *du u* − *C*2 *du u* 

*du*

*u* = *q*<sup>2</sup> + *ω*<sup>2</sup>

 1 *<sup>v</sup>*2*n*<sup>2</sup> <sup>−</sup> <sup>1</sup> *c*2

along the infinitesimal semi-circles above and under the axis. Thus evidently:

 *qdq*  1 *<sup>v</sup>*2*ε*<sup>−</sup> <sup>−</sup> <sup>1</sup> *c*2 

> 1 *<sup>v</sup>*<sup>2</sup> <sup>−</sup> *<sup>ε</sup>*<sup>−</sup> *c*2

The Cherenkov Effect in Graphene-Like Structures

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. (58)

111

. (59)

. (61)

. (60)

*<sup>u</sup>* <sup>=</sup> <sup>2</sup>*πi*. (62)

*q*<sup>2</sup> + *ω*<sup>2</sup>

*q*<sup>2</sup> + *ω*<sup>2</sup>

and therefore

$$
\hbar \, dk\_x dk\_y dk\_y = q \, dq \, d\varphi \, \frac{d\omega}{\upsilon} \,. \tag{54}
$$

In such a way we have for the intensity of the electrical field:

$$\mathbf{E} = \frac{ie}{\pi} \int q \, d\eta \, d\omega \left(\frac{\omega \mathbf{v}}{c^2} - \frac{\mathbf{k}}{\varepsilon}\right) \frac{e^{i(\mathbf{k} \mathbf{x} - \omega t)}}{v \left[q^2 + \omega^2 \left(\frac{1}{v^2} - \frac{\varepsilon}{c^2}\right)\right]},\tag{55}$$

where the *ϕ*-integration was already performed. The *ω*-integration involves both positive and negative frequencies.

The quantity, which is experimentally meaningful, is the energy loss of the moving particle per unit length, or, *dW*/*dz* in the prescribed frequency interval *dω*. This energy loss is in the relation with the work of force which acts on the particle by the induced electromagnetic field. The work is expressed by the formula:

$$dW = -F\_{z}dz = -e(E\_{z})\_{\mathbf{x}=\mathbf{v}t}dz,\tag{56}$$

where (*Ez*)**x**=**v***<sup>t</sup>* is the *<sup>z</sup>*-component of the electric intensity at the point where the particle is. The sign minus denotes the physical fact that the force acts against the vector of velocity, or, in the negative direction of the *z*-axis.

Thus we have:

$$\frac{d\mathcal{W}}{dz} = -e\left(\mathcal{E}\_z\right)\_{\mathbf{x} = \mathbf{v}t} = -\frac{i e^2}{\pi} \int q \, dq \, d\omega \left( e^{i(\mathbf{k}\mathbf{x} - \omega t)} \right)\_{\mathbf{x} = \mathbf{v}t} \frac{\left(\frac{\omega \mathbf{v}}{c^2} - \frac{k\_\sharp}{\varepsilon}\right)}{v \left[q^2 + \omega^2 \left(\frac{1}{\mathcal{V}^2} - \frac{\varepsilon}{c^2}\right)\right]}.$$

$$\epsilon = \frac{ie^2}{\pi} \int q \, dq \, \omega d\omega \, \frac{1}{\varepsilon} \frac{\left(\frac{1}{\overline{v}^2} - \frac{\varepsilon}{\overline{c}^2}\right)}{q^2 + \omega^2 \left(\frac{1}{\overline{v}^2} - \frac{\varepsilon}{\overline{c}^2}\right)}. \tag{57}$$

The energy loss of particle per unit length and in the frequency interval *ω*, *ω* + *dω* is obviously given as

$$\frac{d^2\mathcal{W}}{dzd\omega} = \frac{i\varepsilon^2\omega}{\pi} \int qdq \frac{\left(\frac{1}{\overline{v^2}\varepsilon\_+} - \frac{1}{\overline{c}^2}\right)}{q^2 + \omega^2\left(\frac{1}{\overline{v^2}} - \frac{\varepsilon\_+}{\overline{c}^2}\right)} - \frac{i\varepsilon^2|\omega|}{\pi} \int qdq \frac{\left(\frac{1}{\overline{v^2}\varepsilon\_-} - \frac{1}{\overline{c}^2}\right)}{q^2 + \omega^2\left(\frac{1}{\overline{v^2}} - \frac{\varepsilon\_-}{\overline{c}^2}\right)}.\tag{58}$$

We introduced in the last formula notation *ε*<sup>+</sup> = *ε*(*ω*) and *ε*<sup>−</sup> = *ε*(−|*ω*|) for *ε* at positive and negative values of *ω*. We know that in the absorptive dielectric media *ε* has the imaginary component. This imaginary component of *ε* is positive for *ω* < 0 and negative for *ω* > 0. In fact, the absorption in medium is real effect and it means that exp {−*ikx*} must correspond to the absorption for *x* > 0 for the arbitrary sign of *ω*. This experimental requirement determines the sign of the imaginary part of the permitivity of medium.

In formula (58) we can neglect in the numerator the imaginary part of the permitivity and write:

$$
\left(\frac{1}{v^2 \varepsilon\_+} - \frac{1}{c^2}\right) = \left(\frac{1}{v^2 \varepsilon\_-} - \frac{1}{c^2}\right) = \left(\frac{1}{v^2 n^2} - \frac{1}{c^2}\right). \tag{59}
$$

On the other hand, such operation cannot be performed in the denominator which follows from the next text. Let us introduce the new complex quantity:

$$
\mu = q^2 + \omega^2 \left(\frac{1}{v^2} - \frac{\varepsilon}{c^2}\right). \tag{60}
$$

Then we have

10

and therefore

*dkz* <sup>=</sup> *<sup>d</sup><sup>ω</sup>*

*dkxdkydky* = *q dq dϕ*

In such a way we have for the intensity of the electrical field:

*q dq dω*

 *ω***v** *<sup>c</sup>*<sup>2</sup> <sup>−</sup> **<sup>k</sup>** *ε*

**<sup>E</sup>** <sup>=</sup> *ie π* 

field. The work is expressed by the formula:

in the negative direction of the *z*-axis.

*dz* <sup>=</sup> <sup>−</sup>*e*(*Ez*)**x**=**v***<sup>t</sup>* <sup>=</sup> <sup>−</sup>*ie*<sup>2</sup>

*π* 

<sup>=</sup> *ie*<sup>2</sup> *π*  *q dq dω*

*q dq ωdω*

Thus we have:

*dW*

obviously given as

and negative frequencies.

*dω*

*<sup>e</sup>i*(**kx**−*ωt*)

 1 *<sup>v</sup>*<sup>2</sup> <sup>−</sup> *<sup>ε</sup> c*2

*dW* = −*Fzdz* = −*e*(*Ez*)**x**=**v***tdz*, (56)

*v q*<sup>2</sup> + *ω*<sup>2</sup>

 *ω***v** *<sup>c</sup>*<sup>2</sup> <sup>−</sup> *kz ε* 

> 1 *<sup>v</sup>*<sup>2</sup> <sup>−</sup> *<sup>ε</sup> c*2

. (57)

*v q*<sup>2</sup> + *ω*<sup>2</sup>

where the *ϕ*-integration was already performed. The *ω*-integration involves both positive

The quantity, which is experimentally meaningful, is the energy loss of the moving particle per unit length, or, *dW*/*dz* in the prescribed frequency interval *dω*. This energy loss is in the relation with the work of force which acts on the particle by the induced electromagnetic

where (*Ez*)**x**=**v***<sup>t</sup>* is the *<sup>z</sup>*-component of the electric intensity at the point where the particle is. The sign minus denotes the physical fact that the force acts against the vector of velocity, or,

> *e i*(**kx**−*ωt*) **x**=**v***t*

1 *ε*

The energy loss of particle per unit length and in the frequency interval *ω*, *ω* + *dω* is

 1 *<sup>v</sup>*<sup>2</sup> <sup>−</sup> *<sup>ε</sup> c*2 

> 1 *<sup>v</sup>*<sup>2</sup> <sup>−</sup> *<sup>ε</sup> c*2

*q*<sup>2</sup> + *ω*<sup>2</sup>

*<sup>v</sup>* (53)

*<sup>v</sup>* . (54)

, (55)

$$\frac{d^2W}{dxd\omega} = \frac{ie^2\omega}{2\pi} \left(\frac{1}{v^2n^2} - \frac{1}{c^2}\right) \left\{ \int\_{\mathbb{C}\_1} \frac{du}{u} - \int\_{\mathbb{C}\_2} \frac{du}{u} \right\}.\tag{61}$$

In case of the absence of absorption in medium i.e. ℑ *ε* = 0, the formula (61) gives meaningless zero result. However *ε* and therefore also *u* has the nonzero imaginary part. It means that the integrals in (61) is considered in the complex plane. The contour of integration is chosen in such a way that it is going in parallel to the real axis above this axis in case of ℑ *<sup>u</sup>* > 0 (it corresponds to *<sup>ω</sup>* > 0) and it corresponds to the curve *<sup>C</sup>*1, and under the axis at ℑ *<sup>u</sup>* < 0 (i.e. *<sup>ω</sup>* < 0) which corresponds to the curve *<sup>C</sup>*2. The singular point *<sup>u</sup>* = 0 is avoided along the infinitesimal semi-circles above and under the axis. Thus evidently:

$$\int\_{\mathbb{C}\_1} \frac{du}{u} - \int\_{\mathbb{C}\_2} \frac{du}{u} = \oint \frac{du}{u} = 2\pi i. \tag{62}$$

The integration was performed as a limiting procedure along the infinitesimal circle with the center in the origin of the coordinate system.

Using eq. (62) we can write the energy loss formula formula (61) in the following simple form:

$$\frac{d^2W}{dxd\omega} = \frac{c^2\omega}{c^2} \left(1 - \frac{c^2}{n^2v^2}\right)\omega. \tag{63}$$

2¯*hpkc*<sup>2</sup> cos Θ = *c*2*h*¯ <sup>2</sup>*k*<sup>2</sup> − (*h*¯ *ω*)<sup>2</sup> + 2¯*hω*

cos <sup>Θ</sup> <sup>=</sup> <sup>1</sup>

The relation between frequency and the wave number in a dielectric medium is

*<sup>ω</sup>* = *<sup>c</sup> n*

*U* = *d*3*x* 1 8*π* |**E**| <sup>2</sup> *<sup>∂</sup>*


<sup>2</sup> <sup>=</sup> *<sup>c</sup>*<sup>2</sup>

*<sup>ω</sup>*<sup>2</sup> <sup>|</sup>**<sup>k</sup>** <sup>×</sup> **<sup>E</sup>**<sup>|</sup>

we get with *β* = *v*/*c*

given by the relation

Since

where Θ is the angle between the direction of electron motion and the emission of photon. Putting *ω* = *ck*/*n* and expressing the momentum of electron in dependence of its velocity

**<sup>v</sup>** <sup>=</sup> **<sup>p</sup>***c*<sup>2</sup>

*hk*¯ 2*p* <sup>1</sup> <sup>−</sup> <sup>1</sup> *n*2 

Now, following Harris [12], we show, using the second quantization method how to derive the Cerenkov effect in a dielectric medium characterized by its dielectric constant ˇ *ε*(*ω*) and its index of refraction *<sup>n</sup>*, which is given by relation <sup>√</sup>*εµ* where *<sup>µ</sup>* is the magnetic permeability.

> *<sup>k</sup>* <sup>=</sup> *<sup>c</sup>* <sup>√</sup>*εµ*

It was shown [17] that in such dielectric medium the energy of the electromagnetic field is

rot **<sup>E</sup>** <sup>=</sup> <sup>−</sup><sup>1</sup>

*<sup>i</sup>***<sup>k</sup>** <sup>×</sup> **<sup>E</sup>** <sup>=</sup> *<sup>i</sup><sup>ω</sup>*

*c ∂***B**

*c*

<sup>2</sup> <sup>=</sup> *<sup>c</sup>*2*k*<sup>2</sup> *<sup>ω</sup>*<sup>2</sup> <sup>|</sup>**E**<sup>|</sup>

*∂ω ωε*(*ω*) + <sup>|</sup>**B**<sup>|</sup>

2 

<sup>2</sup> = *ε*|**E**|

*<sup>n</sup><sup>β</sup>* <sup>+</sup>

*p*2*c*<sup>2</sup> + *m*2*c*4. (66)

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113

The Cherenkov Effect in Graphene-Like Structures

. (68)

*k*. (69)

*<sup>∂</sup><sup>t</sup>* (71)

**B** (72)

2, (73)

. (70)

*<sup>E</sup>* , (67)

This formula was derived for the first time by Tamm and Frank in year 1937. The fundamental features of the Cerenkov radiation are as follows: ˇ


Let us remark that the energy loss of a particle caused by the Cerenkov radiation are ˇ approximately equal to 1 % of all energy losses in the condensed matter such as the bremsstrahlung and so on. The fundamental importance of the Cerenkov radiation is in its ˇ use for the modern detectors of very speed charged particles in the high energy physics. The detection of the Cerenkov radiation enables to detect not only the existence of the particle, ˇ however also the direction of motion and its velocity and according to eq. (63) also its charge.

## **4. The quantum theory of the Cerenkov effect ˇ**

Let us start with energetic consideration. So, let us suppose that the initial momentum and energy of electron is **<sup>p</sup>** and *<sup>E</sup>* and the final momentum and energy of electron is **<sup>p</sup>**′ and *<sup>E</sup>*′ . The momentum of the emitted photon let be ¯*h***k**. Then, after emission of photon the energy conservation laws are as follows:

$$
\sqrt{p^2c^2 + m^2c^4} - \hbar\omega = \sqrt{p'^2c^2 + m^2c^4} \tag{64}
$$

$$\mathbf{p} - \hbar \mathbf{k} = \mathbf{p'} \tag{65}$$

where **<sup>p</sup>**′ is the momentum of electron after emission of photon. Let us make the quadratical operation of both equations and let us eliminate *<sup>p</sup>*′ . Then we have:

$$c^2 2\hbar pk^2 \cos\Theta = c^2 \hbar^2 k^2 - (\hbar \omega)^2 + 2\hbar \omega \sqrt{p^2 c^2 + m^2 c^4}. \tag{66}$$

where Θ is the angle between the direction of electron motion and the emission of photon. Putting *ω* = *ck*/*n* and expressing the momentum of electron in dependence of its velocity

$$\mathbf{v} = \frac{\mathbf{p}c^2}{E} \,' \,\tag{67}$$

we get with *β* = *v*/*c*

12

form:

dielectric medium is.

ˇ

conservation laws are as follows:

**4. The quantum theory of the Cerenkov effect**

operation of both equations and let us eliminate *<sup>p</sup>*′

waves.

Using eq. (62) we can write the energy loss formula formula (61) in the following simple

This formula was derived for the first time by Tamm and Frank in year 1937. The

1. The radiation arises only for particle velocity greater than the velocity of light in the

3. The radiation is produced in the visible interval of the light frequencies, i. e. in the ultraviolet part of the frequency spectrum. The radiation does not exists for very short

5. The radiation generated in the given point of the trajectory spreads on the surface of the cone with the vertex in this point and with the axis identical with the direction of motion of the particle. The vertex angle of the cone is given by the relation cos Θ = *c*/*nv*.

Let us remark that the energy loss of a particle caused by the Cerenkov radiation are

approximately equal to 1 % of all energy losses in the condensed matter such as the bremsstrahlung and so on. The fundamental importance of the Cerenkov radiation is in its

use for the modern detectors of very speed charged particles in the high energy physics. The detection of the Cerenkov radiation enables to detect not only the existence of the particle,

however also the direction of motion and its velocity and according to eq. (63) also its charge.

Let us start with energetic consideration. So, let us suppose that the initial momentum and energy of electron is **<sup>p</sup>** and *<sup>E</sup>* and the final momentum and energy of electron is **<sup>p</sup>**′ and *<sup>E</sup>*′

The momentum of the emitted photon let be ¯*h***k**. Then, after emission of photon the energy

**<sup>p</sup>** <sup>−</sup> *<sup>h</sup>*¯ **<sup>k</sup>** <sup>=</sup> **<sup>p</sup>**′

where **<sup>p</sup>**′ is the momentum of electron after emission of photon. Let us make the quadratical

. Then we have:

**ˇ**

*p*2*c*<sup>2</sup> + *m*2*c*<sup>4</sup> − *h*¯ *ω* =

4. The spectral dependency on the frequency is linear for the homogeneous medium.

*ω*. (63)

ˇ

*p*′2*c*<sup>2</sup> + *m*2*c*<sup>4</sup> (64)

, (65)

.

ˇ

*c*2 <sup>1</sup> <sup>−</sup> *<sup>c</sup>*<sup>2</sup> *n*2*v*<sup>2</sup> 

*d*2*W dxd<sup>ω</sup>* <sup>=</sup> *<sup>e</sup>*2*<sup>ω</sup>*

2. It depends only on the charge and not on mass of the moving particles

fundamental features of the Cerenkov radiation are as follows: ˇ

$$\cos\Theta = \frac{1}{n\beta} + \frac{\hbar k}{2p} \left(1 - \frac{1}{n^2}\right). \tag{68}$$

Now, following Harris [12], we show, using the second quantization method how to derive the Cerenkov effect in a dielectric medium characterized by its dielectric constant ˇ *ε*(*ω*) and its index of refraction *<sup>n</sup>*, which is given by relation <sup>√</sup>*εµ* where *<sup>µ</sup>* is the magnetic permeability.

The relation between frequency and the wave number in a dielectric medium is

$$
\omega = \frac{c}{n}k = \frac{c}{\sqrt{\varepsilon\mu}}k.\tag{69}
$$

It was shown [17] that in such dielectric medium the energy of the electromagnetic field is given by the relation

$$
\delta \mathcal{U} = \int d^3 \mathbf{x} \frac{1}{8\pi} \left\{ |\mathbf{E}|^2 \frac{\partial}{\partial \omega} \omega \varepsilon(\omega) + |\mathbf{B}|^2 \right\}. \tag{70}
$$

Since

$$\text{rot } \mathbf{E} = -\frac{1}{c} \frac{\partial \mathbf{B}}{\partial t} \tag{71}$$

$$i\mathbf{k} \times \mathbf{E} = \frac{i\omega}{c} \mathbf{B} \tag{72}$$

$$|\mathbf{B}|^2 = \frac{c^2}{\omega^2} |\mathbf{k} \times \mathbf{E}|^2 = \frac{c^2 k^2}{\omega^2} |\mathbf{E}|^2 = \varepsilon |\mathbf{E}|^2. \tag{73}$$

we have:

$$\mathcal{U} = \int d^3 \mathbf{x} \frac{1}{8\pi} |\mathbf{E}|^2 \left[\frac{\partial}{\partial \omega} \omega \varepsilon + \varepsilon\right] = \int d^3 \mathbf{x} \frac{1}{4\pi} |\mathbf{E}|^2 \frac{1}{2\omega} \frac{\partial \omega^2 \varepsilon}{\partial \omega}. \tag{74}$$

We see that the energy density that a vacuum would have in a vacuum must be corrected by the factor

$$\frac{1}{2\omega} \frac{\partial}{\partial \omega} \omega^2 \varepsilon(\omega) \tag{75}$$

and it leads to renormalized **A**(**x**, *t*) as follows:

**k**,*σ*

Ω[ 1 <sup>2</sup>*<sup>ω</sup> <sup>∂</sup>*

2*πhc*¯ <sup>2</sup>

2*πhc*¯ <sup>2</sup>

*∂ω <sup>ω</sup>*2*ε*(*ω*)]*ω<sup>k</sup>*

*∂ω <sup>ω</sup>*2*ε*(*ω*)]*ω<sup>k</sup>*

1/2

The interaction Hamiltonian *<sup>H</sup>*′ is unchanged except for the change in the normalization

Now, let be the initial state |*i*� and the final state | *f*�. The transition probability per unit time

<sup>=</sup> <sup>2</sup>*<sup>π</sup>*

We use the last formula to calculate the transition probability per unit time for a free electron of momentum ¯*h***q** to emit a photon of momentum ¯*h***k** thereby changing its momentum to

1/2

**uk***<sup>σ</sup> a***k***σe*

**<sup>p</sup>** · **uk***<sup>σ</sup> a***k***σe*

*<sup>i</sup>***kx** <sup>+</sup> *<sup>a</sup>*<sup>+</sup> **<sup>k</sup>***σe* −*i***kx**

The Cherenkov Effect in Graphene-Like Structures

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*<sup>i</sup>***kx** <sup>+</sup> *<sup>a</sup>*<sup>+</sup> **<sup>k</sup>***σe* −*i***kx**

*<sup>h</sup>*¯ |�*<sup>f</sup>* <sup>|</sup>*i*�|2*δ*(*Ef* <sup>−</sup> *Ei*). (83)

×

. (84)

2*πhc*¯ <sup>2</sup>

<sup>2</sup>*<sup>m</sup>* <sup>|</sup>**<sup>q</sup>** <sup>−</sup> **<sup>k</sup>**<sup>|</sup>

<sup>4</sup>*π*2*e*2*h*¯ <sup>2</sup>|**<sup>q</sup>** · **uk***σ*|

<sup>2</sup>*<sup>ω</sup> <sup>∂</sup>*

*∂ω <sup>ω</sup>*2*ε*(*ω*)]*ω<sup>k</sup>*

2

×

. (85)

*∂ω <sup>ω</sup>*2*ε*(*ω*)]*ω<sup>k</sup>*

<sup>2</sup> − *<sup>h</sup>*¯ *<sup>ω</sup>***<sup>k</sup>**

Ω[ 1 <sup>2</sup>*<sup>ω</sup> <sup>∂</sup>* . (81)

115

. (82)

**<sup>A</sup>**(**x**, *<sup>t</sup>*) = ∑

*<sup>H</sup>*′ <sup>=</sup> <sup>−</sup> *<sup>e</sup>*

*mc* <sup>∑</sup> **k**,*σ*

is given by the first order perturbation term, or

trans prob time


and writing **v** = *h*¯ **q**/*m* be the particle velocity, we find:

trans prob time

Ω[ 1 <sup>2</sup>*<sup>ω</sup> <sup>∂</sup>*

trans prob time

**q**→**q**−**k**

<sup>=</sup> <sup>2</sup>*<sup>π</sup> h*¯ *e mc* 2 

<sup>−</sup>*i***kx**|**q**�|2*<sup>δ</sup>*

= 

cos <sup>Θ</sup> <sup>−</sup> *<sup>c</sup>*

**q**→**q**−**k**

*δ*   *h*¯ <sup>2</sup>*q*<sup>2</sup> <sup>2</sup>*<sup>m</sup>* <sup>−</sup> *<sup>h</sup>*¯ <sup>2</sup>

The matrix element in (84) is just equal to ¯*h***q***u***k***σ*. Letting <sup>Θ</sup> to be the angle between **<sup>q</sup>** and **<sup>k</sup>**

*m*2Ω*hvk* ¯ [ <sup>1</sup>

*nv* <sup>−</sup> *<sup>h</sup>*¯ *<sup>ω</sup><sup>n</sup>* <sup>2</sup>*mcv*

factor, or

*h*¯(**q** − **k**). We find:

when it moves in a medium of dielectric constant *ε*(*ω*).

Now, let us consider the Fourier transformation of the electromagnetic potential **A**:

$$\mathbf{A} = \sum\_{\mathbf{k}} \sum\_{\sigma=1,2} \left(\frac{2\pi\hbar c^2}{\Omega \omega\_k}\right)^{1/2} \mathbf{u}\_{\mathbf{k}\sigma} \left\{ a\_{\mathbf{k}\sigma}(t) e^{i\mathbf{k}\mathbf{x}} + a\_{\mathbf{k}\sigma}^+(t) e^{-i\mathbf{k}\mathbf{x}} \right\},\tag{76}$$

where the factor

$$\left(\frac{2\pi\hbar c^2}{\Omega\omega\_k}\right)^{1/2} \tag{77}$$

is a normalization factor chosen for later convenience. In other words it is chosen in order the energy of the el. magnetic field to be interpreted as the sum of energies of the free harmonic oscillators, or,

$$H = \sum\_{k\rho} \hbar \omega\_k a\_{k\sigma}^+ a\_{k\sigma\prime} \tag{78}$$

where *<sup>a</sup>*+, *<sup>a</sup>* are creation and annihilation operators fulfilling commutation relations

$$
\left[a\_{\mathbf{k}\sigma\prime}a\_{\mathbf{k}'\sigma\prime}^+\right] = \delta\_{\mathbf{k},\mathbf{k}'}\delta\_{\sigma,\sigma\prime}.\tag{79}
$$

We want the total energy rather than just the el.mag. field energy to have the form of eq. (78). And it means we are forced to replace the normalization factor by

$$\left(\frac{2\pi\hbar c^2}{\Omega[\frac{1}{2\omega}\frac{\partial}{\partial\omega}\omega^2\varepsilon(\omega)]\_{\omega\_k}}\right)^{1/2},\tag{80}$$

and it leads to renormalized **A**(**x**, *t*) as follows:

$$\mathbf{A(x,t)} = \sum\_{\mathbf{k}, \sigma} \left( \frac{2\pi\hbar c^2}{\Omega[\frac{1}{2\omega}\frac{\partial}{\partial\omega}\omega^2 \varepsilon(\omega)]\_{\omega\_\mathbf{k}}} \right)^{1/2} \mathbf{u}\_{\mathbf{k}\sigma} \left\{ a\_{\mathbf{k}\sigma} e^{i\mathbf{k}\mathbf{x}} + a\_{\mathbf{k}\sigma}^+ e^{-i\mathbf{k}\mathbf{x}} \right\}. \tag{81}$$

The interaction Hamiltonian *<sup>H</sup>*′ is unchanged except for the change in the normalization factor, or

$$H' = -\frac{e}{mc} \sum\_{\mathbf{k}\sigma} \left( \frac{2\pi\hbar c^2}{\Omega[\frac{1}{2\omega}\frac{\partial}{\partial\omega}\omega^2\varepsilon(\omega)]\_{\omega\_k}} \right)^{1/2} \mathbf{p} \cdot \mathbf{u}\_{\mathbf{k}\sigma} \left\{ a\_{\mathbf{k}\sigma} e^{i\mathbf{k}\mathbf{x}} + a\_{\mathbf{k}\sigma}^+ e^{-i\mathbf{k}\mathbf{x}} \right\}. \tag{82}$$

Now, let be the initial state |*i*� and the final state | *f*�. The transition probability per unit time is given by the first order perturbation term, or

$$
\left(\frac{\text{trans prob}}{\text{time}}\right) = \frac{2\pi}{\hbar} |\langle f|i\rangle|^2 \delta(E\_f - E\_i). \tag{83}
$$

We use the last formula to calculate the transition probability per unit time for a free electron of momentum ¯*h***q** to emit a photon of momentum ¯*h***k** thereby changing its momentum to *h*¯(**q** − **k**).

We find:

14

we have:

the factor

where the factor

oscillators, or,

*U* = *d*3*x* 1 <sup>8</sup>*<sup>π</sup>* <sup>|</sup>**E**<sup>|</sup> 2 *∂*

**<sup>A</sup>** <sup>=</sup> ∑ **k**

when it moves in a medium of dielectric constant *ε*(*ω*).

∑ *σ*=1,2 2*πhc*¯ <sup>2</sup> Ω*ω<sup>k</sup>*

*∂ω ωε* <sup>+</sup> *<sup>ε</sup>*

1 2*ω*

Now, let us consider the Fourier transformation of the electromagnetic potential **A**:

1/2

**uk***<sup>σ</sup> <sup>a</sup>***k***σ*(*t*)*<sup>e</sup>*

2*πhc*¯ <sup>2</sup> Ω*ω<sup>k</sup>*

*H* = ∑ *k*,*σ*

where *<sup>a</sup>*+, *<sup>a</sup>* are creation and annihilation operators fulfilling commutation relations

 *<sup>a</sup>***k***σ*, *<sup>a</sup>*<sup>+</sup> **k**′*σ*′ 

(78). And it means we are forced to replace the normalization factor by

Ω[ 1 <sup>2</sup>*<sup>ω</sup> <sup>∂</sup>*

is a normalization factor chosen for later convenience. In other words it is chosen in order the energy of the el. magnetic field to be interpreted as the sum of energies of the free harmonic

*<sup>h</sup>*¯ *<sup>ω</sup><sup>k</sup> <sup>a</sup>*<sup>+</sup>

We want the total energy rather than just the el.mag. field energy to have the form of eq.

2*πhc*¯ <sup>2</sup>

*∂ω <sup>ω</sup>*2*ε*(*ω*)]*ω<sup>k</sup>*

1/2

1/2

*∂*

 = *d*3*x* 1 <sup>4</sup>*<sup>π</sup>* <sup>|</sup>**E**<sup>|</sup>

We see that the energy density that a vacuum would have in a vacuum must be corrected by

<sup>2</sup> 1 2*ω*

*∂ω <sup>ω</sup>*2*ε*(*ω*) (75)

*<sup>i</sup>***kx** + *a*<sup>+</sup>

**<sup>k</sup>***σ*(*t*)*<sup>e</sup>*

−*i***kx**

*<sup>k</sup><sup>σ</sup>akσ*, (78)

<sup>=</sup> *<sup>δ</sup>***k**,**k**′ *δσ*,*σ*′ . (79)

, (80)

*∂ω*2*ε*

*∂ω* . (74)

, (76)

(77)

$$
\begin{split}
\left(\frac{\text{trans prob}}{\text{time}}\right)\_{\mathbf{q}\to\mathbf{q}-\mathbf{k}} &= \frac{2\pi}{\hbar} \left(\frac{e}{mc}\right)^{2} \left(\frac{2\pi\hbar c^{2}}{\Omega[\frac{1}{2\omega}\frac{\partial}{\partial\omega}\omega^{2}\varepsilon(\omega)]\_{\omega\_{\mathbf{k}}}}\right) \times \\\\ |\langle\mathbf{q}-\mathbf{k}|\mathbf{p}\cdot\mathbf{u}\_{\mathbf{k}\nu}e^{-i\mathbf{k}\mathbf{x}}|\mathbf{q}\rangle|^{2}\delta\left[\frac{\hbar^{2}q^{2}}{2m}-\frac{\hbar^{2}}{2m}|\mathbf{q}-\mathbf{k}|^{2}-\hbar\omega\_{\mathbf{k}}\right].
\end{split}
\tag{84}
$$

The matrix element in (84) is just equal to ¯*h***q***u***k***σ*. Letting <sup>Θ</sup> to be the angle between **<sup>q</sup>** and **<sup>k</sup>** and writing **v** = *h*¯ **q**/*m* be the particle velocity, we find:

$$
\left(\frac{\text{trans prob}}{\text{time}}\right)\_{\mathbf{q}\to\mathbf{q}-\mathbf{k}} = \left(\frac{4\pi^2 e^2 \hbar^2 |\mathbf{q}\cdot\mathbf{u}\_{\mathbf{k}\sigma}|^2}{m^2 \Omega \hbar v k [\frac{1}{2\omega} \frac{\partial}{\partial\omega} \omega^2 \varepsilon(\omega)]\_{\omega\chi}}\right) \quad \times
$$

$$
\delta\left[\cos\Theta - \frac{c}{nv} - \frac{\hbar\omega n}{2mcv}\right].\tag{85}
$$

Note that the photon is emitted at an angle to the path of the electron given by

$$\cos\Theta = \frac{c}{nv}\left[1 + \frac{\hbar\omega n^2}{2mc^2}\right].\tag{86}$$

the range of integration does not go to infinity and the integral is convergent.

cos <sup>Θ</sup> <sup>=</sup> *<sup>c</sup>*

must be the same also in case of the relativistic situation.

�

Then, the transition probability per unit time is:

Ω 1 2 *∂ ∂ω* (*ω*2*ε*)

where h.c. denotes operation of Hermite conjugation.

**p**+**k**,*λ*′→**p**,*λ*

*nv*

 <sup>1</sup> <sup>+</sup>

> �1 2 *∂ ∂ω <sup>ω</sup>*2*ε*(*ω*)

*HI* = −*e*

� � *u*+

<sup>=</sup> <sup>2</sup>*<sup>π</sup> h*¯ *e* 2 �

where *α* are the Dirac matrices, **A** is the electromagnetic potential [12].

2*πhc*¯ <sup>2</sup>

zero particles is given by the relation:

with spin 0 or 1.

**4.1. The Dirac electron**

renormalization term

potential:

*HI* = ∑ **k**,*σ* ∑ **p**,*λ* ∑ *λ*′

�trans prob time �

It is possible to show that the Cerenkov angle relation for the relativistic particles with spin ˇ

*h*¯ *ω*

This expression can be also derived using the so called Duffin-Kemmer equation for particles

We have seen that in the nonrelativistic situation the appropriate Hamiltonian involved the

Let us consider the process where an electron of momentum ¯*h*(**p** + **k**) emits a photon of momentum ¯*h***k** and polarization *σ*. The interaction Hamiltonian in case of Dirac electron is

�

Expanding *<sup>ψ</sup>*+, *<sup>ψ</sup>* and **<sup>A</sup>** by the second quantization method, we have for the interacting

**<sup>p</sup>**+**k**,*λ*(*<sup>α</sup>* · **uk***σu***p***λ*)*b*<sup>+</sup>

2*πhc*¯ <sup>2</sup>

� |*u*<sup>+</sup>

Ω 1 2 *∂ ∂ω* (*ω*2*ε*)

�

<sup>2</sup>*mc*<sup>2</sup> (*n*<sup>2</sup> <sup>−</sup> <sup>1</sup>)

� <sup>1</sup> <sup>−</sup> *<sup>v</sup>*<sup>2</sup> *c*2 

. (92)

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117

The Cherenkov Effect in Graphene-Like Structures

, (93)

*<sup>d</sup>*3*ψ*+*<sup>α</sup>* · **<sup>A</sup>***ψ*, (94)

**<sup>p</sup>**+**k**,*λ*′ *<sup>b</sup>***p***λa***k**,*<sup>λ</sup>* <sup>+</sup> *<sup>h</sup>*.*c*.

**<sup>p</sup>**+**k**,*λ*(*<sup>α</sup>* · **uk***σu***p***λ*)<sup>|</sup>

�

2 ×

, (95)

If the energy of the photon ¯*hω* is much less than the rest mass of the electron *mc*<sup>2</sup> then cos Θ ≈ *c*/*nv* which gives the classical Cerenkov angle. This can only be satisfied if the ˇ velocity of the particle is greater than *c*/*n* which is the velocity of the electromagnetic wave in medium. In vacuum where *n* = 1, *v* can never exceed *c* and so emission cannot occur.

The quantity of physical interest is the loss of energy per unit length of path of the electron. It is given by the formula:

$$\frac{d\mathcal{W}}{d\mathbf{x}} = \frac{1}{v}\frac{d\mathcal{W}}{dt} = \frac{1}{v}\sum\_{\mathbf{k},\sigma} \hbar\omega\_{\mathbf{k}} \left(\frac{\text{trans prob}}{\text{time}}\right)\_{\mathbf{q}\to\mathbf{q}-\mathbf{k}}.\tag{87}$$

Using

$$\sum\_{\sigma} |\mathbf{q} \cdot \mathbf{u}\_{\mathbf{k}} \sigma|^2 = q^2 (1 - \cos^2 \Theta) = \frac{m^2 v^2}{\hbar^2} (1 - \cos^2 \Theta) \tag{88}$$

and (for infinite Ω)

$$\lim\_{\Omega \to \infty} \sum\_{\mathbf{k}} \quad \to \quad \frac{\Omega}{(2\pi)^3} \int d^3k \tag{89}$$

and introducing spherical coordinates in *k*-space, we find:

$$\begin{split} \frac{d\mathbf{W}}{d\mathbf{x}} &= \epsilon^2 \int\_0^\infty k d\boldsymbol{k} \int\_{-1}^1 d(\cos\Theta) \frac{(1-\cos^2\Theta)\delta\left[\cos\Theta - (\frac{c}{\hbar v}) - \frac{\hbar\omega n}{2mcv}\right]}{\left[\frac{1}{2\omega}\frac{\partial}{\partial\omega}(\omega^2\varepsilon(\omega))\right]\_{n\mathbf{k}c}} = \\\\ &\frac{\epsilon^2}{\epsilon^2} \int \frac{\varepsilon(\omega)\omega^2 d\omega}{\left[\frac{1}{2}\frac{\partial}{\partial\omega}(\omega^2\varepsilon(\omega))\right]} \left[1 - \frac{c^2}{n^2v^2} \left(1 + \frac{\hbar\omega n^2}{2mc^2}\right)^2\right] .\end{split} \tag{90}$$

It is clear from this derivation that the integration over *ω* is only over those frequencies for that eq. (86) can be satisfied. Since

$$\lim\_{\omega \to \infty} n(\omega) \quad \longrightarrow \quad 1,\tag{91}$$

the range of integration does not go to infinity and the integral is convergent.

It is possible to show that the Cerenkov angle relation for the relativistic particles with spin ˇ zero particles is given by the relation:

$$\cos\Theta = \frac{c}{nv}\left[1 + \frac{\hbar\omega}{2mc^2}(n^2 - 1)\sqrt{1 - \frac{v^2}{c^2}}\right].\tag{92}$$

This expression can be also derived using the so called Duffin-Kemmer equation for particles with spin 0 or 1.

#### **4.1. The Dirac electron**

16

It is given by the formula:

Using

and (for infinite Ω)

*dW dx* <sup>=</sup> *<sup>e</sup>* 2 <sup>∞</sup> 0 *kdk* <sup>1</sup> −1

*dW dx* <sup>=</sup> <sup>1</sup> *v dW dt* <sup>=</sup> <sup>1</sup>

∑*σ*

> *e*2 *c*2

that eq. (86) can be satisfied. Since


and introducing spherical coordinates in *k*-space, we find:

lim <sup>Ω</sup>→<sup>∞</sup> ∑ **k**

*d*(cos Θ)

*∂ω* (*ω*2*ε*(*ω*))

 <sup>1</sup> <sup>−</sup> *<sup>c</sup>*<sup>2</sup> *n*2*v*<sup>2</sup>

It is clear from this derivation that the integration over *ω* is only over those frequencies for

*ε*(*ω*)*ω*2*dω*

 1 2 *∂*

Note that the photon is emitted at an angle to the path of the electron given by

cos <sup>Θ</sup> <sup>=</sup> *<sup>c</sup>*

*nv* 1 +

If the energy of the photon ¯*hω* is much less than the rest mass of the electron *mc*<sup>2</sup> then cos Θ ≈ *c*/*nv* which gives the classical Cerenkov angle. This can only be satisfied if the ˇ

velocity of the particle is greater than *c*/*n* which is the velocity of the electromagnetic wave in medium. In vacuum where *n* = 1, *v* can never exceed *c* and so emission cannot occur.

The quantity of physical interest is the loss of energy per unit length of path of the electron.

<sup>2</sup> <sup>=</sup> *<sup>q</sup>*2(<sup>1</sup> <sup>−</sup> cos2 <sup>Θ</sup>) = *<sup>m</sup>*2*v*<sup>2</sup>

→

(1 − cos2 Θ)*δ*

 1 <sup>2</sup>*<sup>ω</sup> <sup>∂</sup>*

Ω (2*π*)<sup>3</sup>

 1 +

cos Θ − ( *<sup>c</sup>*

 *nkc*

<sup>2</sup>

*h*¯ *ωn*<sup>2</sup> 2*mc*<sup>2</sup>

lim*ω*→<sup>∞</sup> *<sup>n</sup>*(*ω*) −→ 1, (91)

*∂ω* (*ω*2*ε*(*ω*))

*v* ∑ **k**,*σ h*¯ *ω***<sup>k</sup>** *h*¯ *ωn*<sup>2</sup> 2*mc*<sup>2</sup>

trans prob time

**q**→**q**−**k**

. (86)

. (87)

*<sup>h</sup>*¯ <sup>2</sup> (<sup>1</sup> <sup>−</sup> cos<sup>2</sup> <sup>Θ</sup>) (88)

*d*3*k* (89)

=

. (90)

*nv* ) <sup>−</sup> *<sup>h</sup>*¯ *<sup>ω</sup><sup>n</sup>* <sup>2</sup>*mcv*

We have seen that in the nonrelativistic situation the appropriate Hamiltonian involved the renormalization term

$$\left(\frac{1}{2}\frac{\partial}{\partial\omega}\omega^2\varepsilon(\omega)\right),\tag{93}$$

must be the same also in case of the relativistic situation.

Let us consider the process where an electron of momentum ¯*h*(**p** + **k**) emits a photon of momentum ¯*h***k** and polarization *σ*. The interaction Hamiltonian in case of Dirac electron is

$$H\_I = -e \int d^3 \psi^+ \mathbf{a} \cdot \mathbf{A} \psi,\tag{94}$$

where *α* are the Dirac matrices, **A** is the electromagnetic potential [12].

Expanding *<sup>ψ</sup>*+, *<sup>ψ</sup>* and **<sup>A</sup>** by the second quantization method, we have for the interacting potential:

$$H\_I = \sum\_{\mathbf{k}, \sigma} \sum\_{\mathbf{p}, \lambda} \sum\_{\lambda'} \left[ \frac{2\pi\hbar c^2}{\Omega\_{\frac{1}{2}} \frac{\partial}{\partial \omega'} (\omega^2 \varepsilon)} \right] \left\{ u\_{\mathbf{p}+\mathbf{k}, \lambda}^+ (\mathfrak{a} \cdot \mathbf{u}\_{\mathbf{k}\sigma} u\_{\mathbf{p}\lambda}) b\_{\mathbf{p}+\mathbf{k}, \lambda'}^+ b\_{\mathbf{p}\lambda} a\_{\mathbf{k}\lambda} + \text{h.c.} \right\}, \tag{95}$$

where h.c. denotes operation of Hermite conjugation.

Then, the transition probability per unit time is:

$$\left(\frac{\text{trans prob}}{\text{time}}\right)\_{\mathbf{p}+\mathbf{k},\lambda'\to\mathbf{p},\lambda} = \frac{2\pi}{\hbar}e^2 \left[\frac{2\pi\hbar c^2}{\Omega\_{\mathbf{\mathcal{Z}}}^{1}\frac{\partial}{\partial\omega}(\omega^2\varepsilon)}\right]|u^+\_{\mathbf{p}+\mathbf{k},\lambda}(\mathfrak{a}\cdot\mathbf{u}\_{\mathbf{k}\sigma}u\_{\mathbf{p}\lambda})|^2 \quad \times \bar{\varepsilon}$$

$$\delta\left[\sqrt{\hbar^2 c^2 |\mathbf{p} + \mathbf{k}|^2 + m^2 c^4} - \sqrt{\hbar^2 c^2 p^2 + m^2 c^4} - \hbar\omega\right] \tag{96}$$

4 ∑ *λ*=1

**<sup>p</sup>**+**k**,*λ*′*<sup>α</sup>* · **uk***σ*(*H***<sup>p</sup>** <sup>+</sup> <sup>|</sup>*E***p**|)*<sup>α</sup>* · **uk***σ*(*H***p**+**<sup>k</sup>** <sup>+</sup> <sup>|</sup>*E***p**+**k**|)*u***p**+**k***λ*′

(101) becomes:

1 2

4 ∑ *λ*′=1 *u*+

First, we note that

to show that

and

We can show also that

use the following identity:

where **a** and **b** are arbitrary vectors and

1 <sup>8</sup>|*E***p**||*E***p**+**k**|

where Trace can be evaluated with the certain difficulties.

*<sup>u</sup>***p**,*λu*<sup>+</sup>

Using the relation of completeness, the eq. (102) is just the 4 × 4 unit matrix. Therefore eq.

The trace of a product of any odd number of the matrices *αx*, *αy*, *α<sup>z</sup>* and *β* is zero. We may

**<sup>p</sup>**,*λ*. (102)

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119

1

Tr [*<sup>α</sup>* · **uk***σ*(*H***<sup>p</sup>** + |*E***p**|)*<sup>α</sup>* · **uk***σ*(*H***p**+**<sup>k</sup>** + |*E***p**+**k**|)], (103)

(*α* · **a**)(*α* · **b**) = 2(**a** · **b**)1 − (*α* · **b**)(*α* · **a**), (105)

Tr *α<sup>i</sup>* = Tr *β*. (104)

Tr *AB* = Tr *BA*, (106)

Tr (*α* · **a**)(*α* · **b**) = 4**a** · **b**. (107)

Tr (*α* · **a**)*β*(*α* · **b**)*β* = −4**a** · **b** (108)

<sup>4</sup>|*E***p**||*E***p**+**k**<sup>|</sup> <sup>=</sup>

and we may proceed to calculate the energy loss per length as we did in the nonrelativistic case.

There is one modification in this calculation. The sum over final states must include a sum over the final spin states of the electron *λ* = 1, 2. We also average over the initial spin states. Thus the general formula is of the form:

$$\frac{dW}{d\mathbf{x}} = \frac{1}{v} \frac{1}{2} \sum\_{\lambda'=1}^{2} \sum\_{\lambda=1}^{2} \sum\_{\mathbf{k}, \sigma}^{2} \hbar \omega\_k \left( \frac{\text{trans prob}}{\text{time}} \right) . \tag{97}$$

So, we must evaluate

$$\frac{1}{2} \sum\_{\lambda'=1}^{2} \sum\_{\lambda=1}^{2} (u\_{\mathbf{p}+\mathbf{k},\lambda}^{+} \mathbf{a} \cdot \mathbf{u}\_{\mathbf{k}\sigma} u\_{\mathbf{p}\lambda}) (u\_{\mathbf{p},\lambda}^{+} \mathbf{a} \cdot \mathbf{u}\_{\mathbf{k}\sigma} u\_{\mathbf{p}+\mathbf{k}\lambda'}).\tag{98}$$

Let us demonstrate the easy way of calculation of the sums. The first step is to extend the sums over *λ*′ and *λ* to include all four values. We can do this by noting that

$$\frac{|H\_{\mathbf{P}} + |E\_{\mathbf{P}}|}{2|E\_{\mathbf{P}}|} u\_{\mathbf{p},\lambda} = \begin{cases} u\_{\mathbf{p},\lambda'} & \lambda = 1,2\\ 0, & \lambda = 3,4 \end{cases} \tag{99}$$

where

$$H\_\mathbf{p} = \mathfrak{a} \cdot \mathbf{p} + \beta mc^2. \tag{100}$$

We can use the relation (99) and the similar relation with *u***p**+**k**,*λ*′ to write eq. (98) as follows:

$$
\frac{1}{2} \sum\_{\lambda'=1}^{2} \sum\_{\lambda=1}^{2} (u^+\_{\mathbf{p}+\mathbf{k},\lambda'} \mathbf{a} \cdot \mathbf{u}\_{\mathbf{k}\sigma} (H\_\mathbf{p} + |E\_\mathbf{p}|) u\_{\mathbf{p}\lambda}) \quad \times
$$

$$
\left[ \mu^+\_{\mathbf{p},\lambda} \mathbf{a} \cdot \mathbf{u}\_{\mathbf{k}\sigma} (H\_\mathbf{p+\mathbf{k}} + |E\_\mathbf{p+\mathbf{k}}|) u\_\mathbf{p+\mathbf{k}\lambda'} \right] \frac{1}{4 |E\_\mathbf{p}| |E\_\mathbf{p+\mathbf{k}}|} \,. \tag{101}
$$

Now, let us consider

$$\sum\_{\lambda=1}^{4} \mu\_{\mathbf{p},\lambda} \mu\_{\mathbf{p},\lambda}^{+}. \tag{102}$$

Using the relation of completeness, the eq. (102) is just the 4 × 4 unit matrix. Therefore eq. (101) becomes:

$$\frac{1}{2} \sum\_{\lambda'=1}^{4} \left[ u^+\_{\mathbf{p}+\mathbf{k},\lambda'} \mathfrak{a} \cdot \mathbf{u}\_{\mathbf{k}\sigma} (H\_{\mathbf{p}} + |E\_{\mathbf{p}}|) \mathfrak{a} \cdot \mathbf{u}\_{\mathbf{k}\sigma} (H\_{\mathbf{p}+\mathbf{k}} + |E\_{\mathbf{p}+\mathbf{k}}|) u\_{\mathbf{p}+\mathbf{k}\lambda'} \right] \frac{1}{4|E\_{\mathbf{p}}||E\_{\mathbf{p}+\mathbf{k}}|} = $$

$$\frac{1}{8|E\_{\mathbf{p}}||E\_{\mathbf{p}+\mathbf{k}}|} \text{Tr} \left[ \mathfrak{a} \cdot \mathbf{u}\_{\mathbf{k}\sigma} (H\_{\mathbf{p}} + |E\_{\mathbf{p}}|) \mathfrak{a} \cdot \mathbf{u}\_{\mathbf{k}\sigma} (H\_{\mathbf{p}+\mathbf{k}} + |E\_{\mathbf{p}+\mathbf{k}}|) \right] \tag{103}$$

where Trace can be evaluated with the certain difficulties.

First, we note that

18

case.

where

*δ* 

Thus the general formula is of the form:

1 2

2 ∑ *λ*′=1

1 2

 *u*+

Now, let us consider

2 ∑ *λ*′=1

2 ∑ *λ*=1 (*u*<sup>+</sup>

2 ∑ *λ*=1 (*u*<sup>+</sup>

So, we must evaluate

*dW dx* <sup>=</sup> <sup>1</sup> *v* 1 2

¯ <sup>2</sup>*c*2|**p** + **k**|<sup>2</sup> + *m*2*c*<sup>4</sup> −

2 ∑ *λ*′=1

sums over *λ*′ and *λ* to include all four values. We can do this by noting that

<sup>2</sup>|*E***p**<sup>|</sup> *<sup>u</sup>***p**,*<sup>λ</sup>* <sup>=</sup>

*H***<sup>p</sup>** + |*E***p**|

2 ∑ *λ*=1 and we may proceed to calculate the energy loss per length as we did in the nonrelativistic

There is one modification in this calculation. The sum over final states must include a sum over the final spin states of the electron *λ* = 1, 2. We also average over the initial spin states.

> 2 ∑ **k**,*σ* ¯ *ω<sup>k</sup>*

**<sup>p</sup>**+**k**,*λ<sup>α</sup>* · **uk***σu***p***λ*)(*u*<sup>+</sup>

Let us demonstrate the easy way of calculation of the sums. The first step is to extend the

We can use the relation (99) and the similar relation with *u***p**+**k**,*λ*′ to write eq. (98) as follows:

**<sup>p</sup>**,*λ<sup>α</sup>* · **uk***σ*(*H***p**+**<sup>k</sup>** <sup>+</sup> <sup>|</sup>*E***p**+**k**|)*u***p**+**k***λ*′

**<sup>p</sup>**+**k**,*λ*′*<sup>α</sup>* · **uk***σ*(*H***<sup>p</sup>** <sup>+</sup> <sup>|</sup>*E***p**|)*u***p***λ*) <sup>×</sup>

*<sup>u</sup>***p**,*λ*, *<sup>λ</sup>* = 1, 2

¯ <sup>2</sup>*c*<sup>2</sup> *p*<sup>2</sup> + *m*2*c*<sup>4</sup> − ¯ *ω*

trans prob time

. (97)

. (101)

**<sup>p</sup>**,*λ<sup>α</sup>* · **uk***σu***p**+**k***λ*′). (98)

0, *<sup>λ</sup>* <sup>=</sup> 3, 4 , (99)

*H***<sup>p</sup>** = *α* · **p** + *βmc*2. (100)

 1 <sup>4</sup>|*E***p**||*E***p**+**k**<sup>|</sup> (96)

$$\text{Tr}\,\mathfrak{a}\_{i} = \text{Tr}\,\beta.\tag{104}$$

The trace of a product of any odd number of the matrices *αx*, *αy*, *α<sup>z</sup>* and *β* is zero. We may use the following identity:

$$(\mathfrak{a}\cdot\mathbf{a})(\mathfrak{a}\cdot\mathbf{b}) = 2(\mathbf{a}\cdot\mathbf{b})1 - (\mathfrak{a}\cdot\mathbf{b})(\mathfrak{a}\cdot\mathbf{a}),\tag{105}$$

where **a** and **b** are arbitrary vectors and

$$\text{Tr } AB = \text{Tr } BA,\tag{106}$$

to show that

$$\text{Tr}\,(\mathfrak{a}\cdot\mathbf{a})(\mathfrak{a}\cdot\mathbf{b}) = 4\mathbf{a}\cdot\mathbf{b}.\tag{107}$$

We can show also that

$$\text{Tr}\,(\mathfrak{a}\cdot\mathbf{a})\beta(\mathfrak{a}\cdot\mathbf{b})\beta=-4\mathbf{a}\cdot\mathbf{b}\tag{108}$$

and

$$\operatorname{Tr}\left(\mathfrak{a}\cdot\mathbf{a}\right)(\mathfrak{a}\cdot\mathbf{b})(\mathfrak{a}\cdot\mathbf{c})(\mathfrak{a}\cdot\mathbf{d}) = $$

$$4(\mathbf{a}\cdot\mathbf{b})(\mathbf{c}\cdot\mathbf{d}) - 4(\mathbf{a}\cdot\mathbf{c})4(\mathbf{b}\cdot\mathbf{d}) + 4(\mathbf{a}\cdot\mathbf{d})4(\mathbf{b}\cdot\mathbf{c}) \tag{109}$$

for any three vectors **a**, **b**, **c**, **d**.

Using the formulas with operation Trace, we can evaluate eq. (103). We find:

$$\frac{1}{2}\left\{1-\frac{m^2c^4}{|E\_\mathbf{p}||E\_{\mathbf{p}+\mathbf{k}}|}+2\frac{(\mathbf{u}\_{\mathbf{k}\sigma}\cdot\mathbf{v}\_1)^2}{c^2}-\frac{\mathbf{v}\_1\cdot\mathbf{v}\_2}{c^2}\right\},\tag{110}$$

where we have used

$$\mathbf{v} = \frac{c^2 \mathbf{p}}{E},\tag{111}$$

and the extremal relativistic limit (**v** → *c*). We neglect this term in the remainder of the calculation. The rest of the calculation is the similar to the case with the spin 0. The only

> <sup>1</sup> <sup>+</sup>

Source theory [6, 30–32] is the theoretical construction which uses quantum-mechanical particle language. Initially it was constructed for description of the particle physics situations occurring in the high-energy physics experiments. However, it was found that the original formulation simplifies the calculations in the electrodynamics and gravity where

*h*¯ *ω*

<sup>2</sup>*mc*<sup>2</sup> (*n*<sup>2</sup> <sup>−</sup> <sup>1</sup>)

� <sup>1</sup> <sup>−</sup> *<sup>v</sup>*<sup>2</sup> *c*2  

The Cherenkov Effect in Graphene-Like Structures

, (116)

), (117)

*<sup>h</sup>*¯ *<sup>ω</sup>* }, (118)

), is the photon propagator

2 

http://dx.doi.org/10.5772/50296

. (115)

121

differences is that eq. (113) must be used instead of eq. (112). The result is

 

<sup>1</sup> <sup>−</sup> *<sup>c</sup>*<sup>2</sup> *n*2*v*2

The basic formula in the source theory is the vacuum to vacuum amplitude [30]:

The electromagnetic field is described by the amplitude (116) with the action

(*dx*)(*dx*′

)*J*

where the dimensionality of *W*(*J*) is the same as the dimensionality of the Planck constant

It may be easy to show that the probability of the persistence of vacuum is given by the

Im *<sup>W</sup>*} *<sup>d</sup>*

where we have introduced the so called power spectral function *P*(*ω*, *t*). In order to extract this spectral function from Im *W*, it is necessary to know the explicit form of the photon

*<sup>µ</sup>*(*x*)*D*+*µν*(*x* − *x*′

<sup>=</sup> exp{− �

)*J <sup>ν</sup>*(*x*′

*dtdω*

*P*(*ω*, *t*)

< 0+|0<sup>−</sup> >= *e*

where the minus and plus tags on the vacuum symbol are causal labels, referring to any time before and after space-time region where sources are manipulated. The exponential form is introduced with regard to the existence of the physically independent experimental arrangements which has a simple consequence that the associated probability amplitudes

*i <sup>h</sup>*¯ *<sup>W</sup>*(*S*)

� *ε*(*ω*)*ω*2*dω*

multiply and corresponding *W* expressions add.

and its explicit form will be determined later.


).

following formula [30]:

propagator *<sup>D</sup>*+*µν*(*<sup>x</sup>* − *<sup>x</sup>*′

*<sup>W</sup>*(*J*) = <sup>1</sup>

2*c*<sup>2</sup> �

*<sup>h</sup>*¯. *<sup>J</sup><sup>µ</sup>* is the charge and current densities. The symbol *<sup>D</sup>*+*µν*(*<sup>x</sup>* − *<sup>x</sup>*′

<sup>2</sup> <sup>=</sup> exp{−<sup>2</sup>

*h*¯

**5. The source theory of the Cerenkov effect ˇ**

*ω*2*ε*(*ω*)

the interactions are mediated by photon or graviton respectively.

1 2 � *∂ ∂ω* �

*dW dx* <sup>=</sup> *<sup>e</sup>*<sup>2</sup> *c*2

and where **v**<sup>1</sup> and **v**<sup>2</sup> are the velocities before and after emission of photon. The sum over polarizations can be carried out as was done in eq. (102). The result is that eq. (103) summed over polarization is:

$$\frac{v\_1^2}{c^2}(1-\cos^2\Theta) + \frac{1}{2}\left\{1-\sqrt{\left(1-\frac{v\_1^2}{c^2}\right)\left(1-\frac{v\_2^2}{c^2}\right)} - \frac{\mathbf{v}\_1\cdot\mathbf{v}\_2}{c^2}\right\},\tag{112}$$

where again Θ is the angle between **p** and **k** and it is given by the formula:

$$\cos\Theta = \frac{c}{n\upsilon} \left[ 1 + \frac{\hbar\omega}{2mc^2} (n^2 - 1) \sqrt{1 - \frac{v^2}{c^2}} \right]. \tag{113}$$

We have used

$$E = \frac{mc^2}{\sqrt{1 - \frac{p^2}{c^2}}}\tag{114}$$

to obtain eq. (113) from eq. (110). The second term in eq. (113) is a small correction to the result formed in the spin 0 case.

The momentum of photon is negligible in comparison with the momentum of electron. Then (**v**<sup>1</sup> ≈ **v2**) and the term in braces vanishes. This will be true in both the classical limit ¯*<sup>h</sup>* → <sup>0</sup> and the extremal relativistic limit (**v** → *c*). We neglect this term in the remainder of the calculation. The rest of the calculation is the similar to the case with the spin 0. The only differences is that eq. (113) must be used instead of eq. (112). The result is

$$\frac{d\mathcal{W}}{d\mathbf{x}} = \frac{e^2}{c^2} \int \frac{\varepsilon(\omega)\omega^2 d\omega}{\frac{1}{2}\left(\frac{\partial}{\partial\omega}\right)\omega^2 \varepsilon(\omega)} \left[1 - \frac{c^2}{n^2 v^2} \left(1 + \frac{\hbar\omega}{2mc^2}(n^2 - 1)\sqrt{1 - \frac{v^2}{c^2}}\right)^2\right].\tag{115}$$

## **5. The source theory of the Cerenkov effect ˇ**

20

for any three vectors **a**, **b**, **c**, **d**.

where we have used

over polarization is:

We have used

*v*2 1

result formed in the spin 0 case.

*<sup>c</sup>*<sup>2</sup> (<sup>1</sup> <sup>−</sup> cos2 <sup>Θ</sup>) + <sup>1</sup>

2

*nv*

cos <sup>Θ</sup> <sup>=</sup> *<sup>c</sup>*

 1 − ���� � <sup>1</sup> <sup>−</sup> *<sup>v</sup>*<sup>2</sup> 1 *c*2

where again Θ is the angle between **p** and **k** and it is given by the formula:

 <sup>1</sup> <sup>+</sup>

*h*¯ *ω*

*<sup>E</sup>* <sup>=</sup> *mc*<sup>2</sup> � 1 − *<sup>v</sup>*<sup>2</sup> *c*2

to obtain eq. (113) from eq. (110). The second term in eq. (113) is a small correction to the

The momentum of photon is negligible in comparison with the momentum of electron. Then (**v**<sup>1</sup> ≈ **v2**) and the term in braces vanishes. This will be true in both the classical limit ¯*<sup>h</sup>* → <sup>0</sup>

<sup>2</sup>*mc*<sup>2</sup> (*n*<sup>2</sup> <sup>−</sup> <sup>1</sup>)

1 2 � Tr (*α* · **a**)(*α* · **b**)(*α* · **c**)(*α* · **d**) =

Using the formulas with operation Trace, we can evaluate eq. (103). We find:

+ 2

**<sup>v</sup>** <sup>=</sup> *<sup>c</sup>*2**<sup>p</sup>**

and where **v**<sup>1</sup> and **v**<sup>2</sup> are the velocities before and after emission of photon. The sum over polarizations can be carried out as was done in eq. (102). The result is that eq. (103) summed

(**uk***<sup>σ</sup>* · **<sup>v</sup>**1)<sup>2</sup>

� �

<sup>1</sup> <sup>−</sup> *<sup>v</sup>*<sup>2</sup> 2 *c*2 �

� <sup>1</sup> <sup>−</sup> *<sup>v</sup>*<sup>2</sup> *c*2 

<sup>1</sup> <sup>−</sup> *<sup>m</sup>*2*c*<sup>4</sup> |*E***p**||*E***p**+**k**|

4(**a** · **b**)(**c** · **d**) − 4(**a** · **c**)4(**b** · **d**) + 4(**a** · **d**)4(**b** · **c**) (109)

*<sup>c</sup>*<sup>2</sup> <sup>−</sup> **<sup>v</sup>**<sup>1</sup> · **<sup>v</sup>**<sup>2</sup>

*c*2

�

*<sup>E</sup>* , (111)

<sup>−</sup> **<sup>v</sup>**<sup>1</sup> · **<sup>v</sup>**<sup>2</sup> *c*2

 

. (113)

, (112)

(114)

, (110)

Source theory [6, 30–32] is the theoretical construction which uses quantum-mechanical particle language. Initially it was constructed for description of the particle physics situations occurring in the high-energy physics experiments. However, it was found that the original formulation simplifies the calculations in the electrodynamics and gravity where the interactions are mediated by photon or graviton respectively.

The basic formula in the source theory is the vacuum to vacuum amplitude [30]:

$$<0\_+|0\_-> = e^{\frac{i}{\hbar}\mathcal{W}(S)},\tag{116}$$

where the minus and plus tags on the vacuum symbol are causal labels, referring to any time before and after space-time region where sources are manipulated. The exponential form is introduced with regard to the existence of the physically independent experimental arrangements which has a simple consequence that the associated probability amplitudes multiply and corresponding *W* expressions add.

The electromagnetic field is described by the amplitude (116) with the action

$$W(\mathbf{J}) = \frac{1}{2c^2} \int (d\mathbf{x})(d\mathbf{x}') J^\mu(\mathbf{x}) D\_{+\mu\nu}(\mathbf{x} - \mathbf{x}') J^\nu(\mathbf{x}'),\tag{117}$$

where the dimensionality of *W*(*J*) is the same as the dimensionality of the Planck constant *<sup>h</sup>*¯. *<sup>J</sup><sup>µ</sup>* is the charge and current densities. The symbol *<sup>D</sup>*+*µν*(*<sup>x</sup>* − *<sup>x</sup>*′ ), is the photon propagator and its explicit form will be determined later.

It may be easy to show that the probability of the persistence of vacuum is given by the following formula [30]:

$$|<0,+|0\_->|^2 = \exp\{-\frac{2}{\hbar}\text{Im}\,\mathcal{W}\} \stackrel{d}{=} \exp\{-\int dt d\omega \frac{P(\omega,t)}{\hbar\omega}\},\tag{118}$$

where we have introduced the so called power spectral function *P*(*ω*, *t*). In order to extract this spectral function from Im *W*, it is necessary to know the explicit form of the photon propagator *<sup>D</sup>*+*µν*(*<sup>x</sup>* − *<sup>x</sup>*′ ).

The electromagnetic field is described by the four-potentials *<sup>A</sup>µ*(*ϕ*, **<sup>A</sup>**) and it is generated by the four-current *<sup>J</sup>µ*(*c̺*,**J**) according to the differential equation [30]:

$$(\Delta - \frac{\mu \varepsilon}{c^2} \frac{\partial^2}{\partial t^2}) A^\mu = \frac{\mu}{c} (g^{\mu \nu} + \frac{n^2 - 1}{n^2} \eta^\mu \eta^\nu) I\_\nu \tag{119}$$

*<sup>P</sup>*(*ω*, *<sup>t</sup>*) = *<sup>e</sup>*<sup>2</sup>

threshold condition for the existence of the Cerenkov effect ˇ *nβ* = 1.

*D*+(*x* − *x*′

) = 1 (2*π*)<sup>3</sup>

Using exp(*ikR* cos *θ*) = cos(*kR* cos *θ*) + *i* sin(*kR* cos *θ*) and (*z* = *kR*)

cos(*<sup>z</sup>* cos *<sup>θ</sup>*) = *<sup>J</sup>*0(*z*) + <sup>2</sup>

∞ ∑ *n*=1

where *Jn*(*z*) are the Bessel functions [15], we get after integration over *θ*:

 <sup>∞</sup> 0

*kdk* <sup>∞</sup> −∞ *dω c*

*<sup>J</sup>*0(*kR*) *k*<sup>2</sup> − *<sup>n</sup>*2*ω*<sup>2</sup>

*<sup>c</sup>*<sup>2</sup> − *<sup>i</sup><sup>ε</sup>*

sin(*z* cos *θ*) =

) = 1 (2*π*)<sup>2</sup>

*dk*0*kdkdθ*

and

or, with *<sup>R</sup>* = |**<sup>x</sup>** − **<sup>x</sup>**′


*D*+(*x* − *x*′

*D*+(*x* − *x*′

**6. The Cerenkov effect in the dielectric 2D hexagonal structure ˇ**

) = (*dk*) (2*π*)<sup>3</sup>

> <sup>2</sup>*<sup>π</sup>* 0

*dθ* <sup>∞</sup> 0

<sup>4</sup>*πc*<sup>2</sup> *µωv*(<sup>1</sup> <sup>−</sup> <sup>1</sup>

where *β* = *v*/*c*. Relations (125) and (126) determine the Cerenkov spectrum and the ˇ

In case of the two dimension situation, the form of equations (119) and (120) is the same with the difference that *η<sup>µ</sup>* ≡ (1, **0**) has two space components, or *η<sup>µ</sup>* ≡ (1, 0, 0), and the Green function *D*<sup>+</sup> as the propagator must be determined by the two-dimensional procedure. I other words, the Fourier form of this propagator is with (*dk*) = *dk*0*d***k** = *dk*0*dk*1*dk*<sup>2</sup> =

> 1 **<sup>k</sup>**<sup>2</sup> <sup>−</sup> *<sup>n</sup>*2(*k*)<sup>2</sup> *<sup>e</sup>*

*kdk* <sup>∞</sup> −∞

∞ ∑ *n*=1 *dω c*

*<sup>n</sup>*2*β*<sup>2</sup> ); *<sup>n</sup><sup>β</sup>* <sup>&</sup>gt; <sup>1</sup> (125)

The Cherenkov Effect in Graphene-Like Structures

http://dx.doi.org/10.5772/50296

123

*P*(*ω*, *t*) = 0; *nβ* < 1, (126)

*ik*(*x*−*x*′ )

*eikR* cos *<sup>θ</sup>*−*iω*(*t*−*<sup>t</sup>*

(−1)*<sup>n</sup> <sup>J</sup>*2*n*(*z*) cos 2*n<sup>θ</sup>* (129)

*k*<sup>2</sup> − *<sup>n</sup>*2*ω*<sup>2</sup>

(−1)*<sup>n</sup> <sup>J</sup>*2*n*−1(*z*) cos(2*<sup>n</sup>* <sup>−</sup> <sup>1</sup>)*θ*, (130)

*e* −*iω*(*t*−*t* ′ )

′ )

, (127)

*<sup>c</sup>*<sup>2</sup> <sup>−</sup> *<sup>i</sup><sup>ε</sup>* . (128)

, (131)

with the corresponding Green function *D*+*µν*:

$$D\_{+}^{\mu\nu} = \frac{\mu}{c} (g^{\mu\nu} + \frac{n^2 - 1}{n^2} \eta^{\mu} \eta^{\nu}) D\_{+}(\mathbf{x} - \mathbf{x}'),\tag{120}$$

where *<sup>η</sup><sup>µ</sup>* ≡ (1, **<sup>0</sup>**), *<sup>µ</sup>* is the magnetic permeability of the dielectric medium with the dielectric constant *ε*, *c* is the velocity of light in vacuum, *n* is the index of refraction of this medium, and *<sup>D</sup>*+(*<sup>x</sup>* − *<sup>x</sup>*′ ) was derived by Schwinger et al. [30] in the following form:

$$D\_{+}(\mathbf{x} - \mathbf{x}') = \frac{i}{4\pi^2 c} \int\_0^\infty d\omega \frac{\sin\frac{\eta\omega}{c}|\mathbf{x} - \mathbf{x}'|}{|\mathbf{x} - \mathbf{x}'|} e^{-i\omega|t - t'|}. \tag{121}$$

Using formulas (117), (118), (120) and (121), we get for the power spectral formula the following expression [30]:

$$
\mathcal{P}(\omega, t) = -\frac{\omega}{4\pi^2} \frac{\mu}{n^2} \int d\mathbf{x} d\mathbf{x}' dt' \frac{\sin\frac{n\omega}{c} |\mathbf{x} - \mathbf{x}'|}{|\mathbf{x} - \mathbf{x}'|} \cos[\omega(t - t')] \times
$$

$$
\times \left\{ \varrho(\mathbf{x}, t) \varrho(\mathbf{x}', t') - \frac{n^2}{c^2} \mathbf{J}(\mathbf{x}, t) \cdot \mathbf{J}(\mathbf{x}', t') \right\}. \tag{122}
$$

Now, we are prepared to apply the last formula to the situations of the charge moving in the dielectric medium.

The charge and current density of electron moving with the velocity **v** and charge *e* is

$$
\boldsymbol{\varrho} = e \delta(\mathbf{x} - \mathbf{v}t) \tag{123}
$$

$$\mathbf{J} = e\mathbf{v}\delta(\mathbf{x} - \mathbf{v}t).\tag{124}$$

After insertion of eqs. (123) and (124) in equation for spectral density (122), we find:

$$P(\omega, t) = \frac{e^2}{4\pi c^2} \mu \omega v (1 - \frac{1}{n^2 \beta^2}); \quad n\beta > 1\tag{125}$$

$$P(\omega, t) = 0; \quad n\beta < 1,\tag{126}$$

where *β* = *v*/*c*. Relations (125) and (126) determine the Cerenkov spectrum and the ˇ threshold condition for the existence of the Cerenkov effect ˇ *nβ* = 1.

## **6. The Cerenkov effect in the dielectric 2D hexagonal structure ˇ**

In case of the two dimension situation, the form of equations (119) and (120) is the same with the difference that *η<sup>µ</sup>* ≡ (1, **0**) has two space components, or *η<sup>µ</sup>* ≡ (1, 0, 0), and the Green function *D*<sup>+</sup> as the propagator must be determined by the two-dimensional procedure. I other words, the Fourier form of this propagator is with (*dk*) = *dk*0*d***k** = *dk*0*dk*1*dk*<sup>2</sup> = *dk*0*kdkdθ*

$$D\_{+}(\mathbf{x} - \mathbf{x}') = \int \frac{(dk)}{(2\pi)^3} \frac{1}{\mathbf{k}^2 - n^2(k)^2} e^{i\mathbf{k}(\mathbf{x} - \mathbf{x}')},\tag{127}$$

or, with *<sup>R</sup>* = |**<sup>x</sup>** − **<sup>x</sup>**′ |

$$D\_{+}(\mathbf{x} - \mathbf{x}') = \frac{1}{(2\pi)^3} \int\_0^{2\pi} d\theta \int\_0^\infty kdk \int\_{-\infty}^\infty \frac{d\omega}{c} \frac{e^{i\mathbf{k}R\cos\theta - i\omega(t - t')}}{k^2 - \frac{\mathbf{n}^2 \omega^2}{c^2} - i\varepsilon}. \tag{128}$$

Using exp(*ikR* cos *θ*) = cos(*kR* cos *θ*) + *i* sin(*kR* cos *θ*) and (*z* = *kR*)

$$\cos(z\cos\theta) = J\_0(z) + 2\sum\_{n=1}^{\infty} (-1)^n f\_{2n}(z)\cos 2n\theta \tag{129}$$

and

22

and *<sup>D</sup>*+(*<sup>x</sup>* − *<sup>x</sup>*′

following expression [30]:

dielectric medium.

The electromagnetic field is described by the four-potentials *<sup>A</sup>µ*(*ϕ*, **<sup>A</sup>**) and it is generated by

*c*

*n*<sup>2</sup> − 1

where *<sup>η</sup><sup>µ</sup>* ≡ (1, **<sup>0</sup>**), *<sup>µ</sup>* is the magnetic permeability of the dielectric medium with the dielectric constant *ε*, *c* is the velocity of light in vacuum, *n* is the index of refraction of this medium,

) was derived by Schwinger et al. [30] in the following form:

Using formulas (117), (118), (120) and (121), we get for the power spectral formula the

*dt*′ sin *<sup>n</sup><sup>ω</sup>*

Now, we are prepared to apply the last formula to the situations of the charge moving in the

 <sup>∞</sup> 0 *dω*

*<sup>d</sup>***x***d***x**′

, *t* ′ ) <sup>−</sup> *<sup>n</sup>*<sup>2</sup>

The charge and current density of electron moving with the velocity **v** and charge *e* is

After insertion of eqs. (123) and (124) in equation for spectral density (122), we find:

(*gµν* +

*n*<sup>2</sup> − 1

*<sup>n</sup>*<sup>2</sup> *<sup>η</sup>µην*)*D*+(*<sup>x</sup>* <sup>−</sup> *<sup>x</sup>*′

sin *<sup>n</sup><sup>ω</sup>*

*<sup>c</sup>* <sup>|</sup>**<sup>x</sup>** <sup>−</sup> **<sup>x</sup>**′ |

<sup>|</sup>**<sup>x</sup>** <sup>−</sup> **<sup>x</sup>**′| *<sup>e</sup>*

*<sup>c</sup>* <sup>|</sup>**<sup>x</sup>** <sup>−</sup> **<sup>x</sup>**′ | <sup>|</sup>**<sup>x</sup>** <sup>−</sup> **<sup>x</sup>**′| cos[*ω*(*<sup>t</sup>* <sup>−</sup> *<sup>t</sup>*

*<sup>c</sup>*<sup>2</sup> **<sup>J</sup>**(**x**, *<sup>t</sup>*) · **<sup>J</sup>**(**x**′

, *t* ′ ) 

*̺* = *eδ*(**x** − **v***t*) (123)

**J** = *e***v***δ*(**x** − **v***t*). (124)

−*iω*|*t*−*t* ′ |

> ′ )]×

*<sup>n</sup>*<sup>2</sup> *<sup>η</sup>µην*)*J<sup>ν</sup>* (119)

), (120)

. (121)

. (122)

the four-current *<sup>J</sup>µ*(*c̺*,**J**) according to the differential equation [30]:

*∂*2

*<sup>∂</sup>t*<sup>2</sup> )*A<sup>µ</sup>* <sup>=</sup> *<sup>µ</sup>*

(*gµν* +

) = *i* 4*π*2*c*

(<sup>∆</sup> <sup>−</sup> *µε c*2

> *<sup>D</sup>µν* <sup>+</sup> <sup>=</sup> *<sup>µ</sup> c*

*<sup>D</sup>*+(*<sup>x</sup>* <sup>−</sup> *<sup>x</sup>*′

*<sup>P</sup>*(*ω*, *<sup>t</sup>*) = <sup>−</sup> *<sup>ω</sup>*

4*π*<sup>2</sup>

× 

*µ n*2 

*̺*(**x**, *<sup>t</sup>*)*̺*(**x**′

with the corresponding Green function *D*+*µν*:

$$\sin(z\cos\theta) = \sum\_{n=1}^{\infty} (-1)^n l\_{2n-1}(z) \cos(2n-1)\theta,\tag{130}$$

where *Jn*(*z*) are the Bessel functions [15], we get after integration over *θ*:

$$D\_{+}(\mathbf{x} - \mathbf{x}') = \frac{1}{(2\pi)^{2}} \int\_{0}^{\infty} kdk \int\_{-\infty}^{\infty} \frac{d\omega}{\mathcal{E}} \frac{J\_{0}(k\mathbf{R})}{k^{2} - \frac{\hbar^{2}\omega^{2}}{c^{2}} - i\varepsilon} e^{-i\omega(t - t')},\tag{131}$$

where the Bessel function *J*0(*z*) has the following expansion [15]:

$$J\_0(z) = \sum\_{s=0}^{\infty} \frac{(-1)^s z^{2s}}{\text{s!s!} 2^{2s}} \tag{132}$$

where the *t*

8. Or,

and

**pulse**

′

with *<sup>J</sup>*0(−*z*) = *<sup>J</sup>*0(*z*):

Cerenkov radiation. ˇ

*<sup>P</sup>*(*ω*, *<sup>t</sup>*) = *<sup>e</sup>*<sup>2</sup>

2*π*

*J* = <sup>∞</sup> 0

*<sup>P</sup>*(*ω*, *<sup>t</sup>*) = *<sup>e</sup>*<sup>2</sup>

*π µωv c*2

 <sup>1</sup> <sup>−</sup> <sup>1</sup> *n*2*β*<sup>2</sup>

current density for the charge moving along the axis is (*v* > 0)

*µωv c*2

 <sup>1</sup> <sup>−</sup> <sup>1</sup> *n*2*β*<sup>2</sup>


 <sup>∞</sup> −∞

*dxJ*<sup>0</sup> (*ax*) cos(*bx*) = <sup>1</sup>

The integral in formula (138) is involved in the tables of integrals [11] on page 745, number

In our case we have *a* = *nβω* and *b* = *ω*. So, the power spectrum of in eq. (138) is as follows

 2 *ω*

where condition *nβ* = 1 is the threshold of the existence of the two-dimensional form of the

**7. The Cerenkov radiation in two-dimensional structure generated by a ˇ**

Let us consider the electron moving perpendicularly to the 2D sheet in the pane *y* − *z* with the index of refraction *n* and the magnetic permeability *µ*. Then, the charge density and

*v*

*̺* <sup>=</sup> *<sup>e</sup>δ*(*vt*)*δ*(**x**) = *<sup>e</sup>*

*n*2*β*<sup>2</sup> − 1

√

*<sup>a</sup>*<sup>2</sup> <sup>−</sup> *<sup>b</sup>*<sup>2</sup> ; 0 <sup>&</sup>lt; *<sup>b</sup>* <sup>&</sup>lt; *<sup>a</sup>*,

*J* = ∞; *a* = *b*, *J* = 0; 0 < *a* < *b*. (139)

*P*(*ω*, *t*) = 0; *nβ* < 1, (141)

′ − *<sup>t</sup>*, we get the final formula:

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125

, *nβ* > 1, *β* = *v*/*c*. (140)

*δ*(*t*)*δ*(**x**) (142)

**J** = 0. (143)

*<sup>d</sup>τJ*<sup>0</sup> (*nβωτ*) cos(*ωτ*), *<sup>β</sup>* = *<sup>v</sup>*/*c*. (138)

The *ω*-integral in (131) can be performed using the residuum theorem after integration in the complex half *ω*-plane.

The result of such integration is the propagator *D*<sup>+</sup> in the following form:

$$D\_{+}(\mathbf{x} - \mathbf{x}') = \frac{\mathbf{i}}{2\pi\mathbf{c}} \int\_{0}^{\infty} d\omega \,\mathbf{J}\_{0}\left(\frac{n\omega}{c}|\mathbf{x} - \mathbf{x}'|\right) e^{-i\omega \left|t - t'\right|}. \tag{133}$$

The initial terms in the expansion of the Bessel function with exponent zero is as follows:

$$J\_0(z) = 1 - \frac{z^2}{2^2} + \frac{z^4}{2^2 4^2} - \frac{z^6}{2^2 4^2 6^2} + \frac{z^8}{2^2 4^2 6^2 8^2} - \dotsb \quad . \tag{134}$$

The spectral formula for the two dimensional Cerenkov radiation is of the analogue of the ˇ formula (122), or,

$$P(\omega, t) = -\frac{\omega}{2\pi} \frac{\mu}{n^2} \int d\mathbf{x} d\mathbf{x}' dt' f\_0\left(\frac{n\omega}{c} |\mathbf{x} - \mathbf{x}'|\right) \cos[\omega(t - t')] \times$$

$$\times \left\{ \varrho(\mathbf{x}, t) \varrho(\mathbf{x}', t') - \frac{n^2}{c^2} \mathbf{J}(\mathbf{x}, t) \cdot \mathbf{J}(\mathbf{x}', t') \right\},\tag{135}$$

where the charge density and current involves only two-dimensional velocities and integration is also only two-dimensional with two-dimensional *<sup>d</sup>***x**, *<sup>d</sup>***x**′ .

The difference is in the replacing mathematical formulas as follows:

$$\frac{\sin\frac{\eta\omega}{c}|\mathbf{x}-\mathbf{x}'|}{|\mathbf{x}-\mathbf{x}'|} \quad \underline{\qquad} \quad J\_0\left(\frac{\eta\omega}{c}|\mathbf{x}-\mathbf{x}'|\right). \tag{136}$$

So, After insertion the quantities (123) and (124) into (135), we get:

$$P(\omega, t) = \frac{e^2}{2\pi} \frac{\mu \omega v}{c^2} \left(1 - \frac{1}{n^2 \beta^2}\right) \int dt' f\_0\left(\frac{n \pi \omega}{c} |t - t'|\right) \cos[\omega(t - t')], \quad \beta = v/c,\tag{137}$$

where the *t* ′ -integration must be performed. Putting *τ* = *t* ′ − *<sup>t</sup>*, we get the final formula:

$$P(\omega, t) = \frac{e^2}{2\pi} \frac{\mu \omega v}{c^2} \left(1 - \frac{1}{n^2 \beta^2}\right) \int\_{-\infty}^{\infty} d\tau f\_0 \left(n \not\!\!\!/ \omega \tau\right) \cos(\omega \tau), \quad \beta = v/c. \tag{138}$$

The integral in formula (138) is involved in the tables of integrals [11] on page 745, number 8. Or,

$$J = \int\_0^\infty dx J\_0\left(ax\right) \cos(bx) = \frac{1}{\sqrt{a^2 - b^2}}; \quad 0 < b < a,$$

$$J = \infty; \quad a = b, \quad J = 0; \quad 0 < a < b. \tag{139}$$

In our case we have *a* = *nβω* and *b* = *ω*. So, the power spectrum of in eq. (138) is as follows with *<sup>J</sup>*0(−*z*) = *<sup>J</sup>*0(*z*):

$$P(\omega, t) = \frac{e^2}{\pi} \frac{\mu \omega v}{c^2} \left(1 - \frac{1}{n^2 \beta^2}\right) \frac{2}{\omega \sqrt{n^2 \beta^2 - 1}}, \quad n\beta > 1, \quad \beta = v/c. \tag{140}$$

and

24

the complex half *ω*-plane.

formula (122), or,

*<sup>P</sup>*(*ω*, *<sup>t</sup>*) = *<sup>e</sup>*<sup>2</sup>

2*π*

*µωv c*2 <sup>1</sup> <sup>−</sup> <sup>1</sup> *n*2*β*2

where the Bessel function *J*0(*z*) has the following expansion [15]:

*J*0(*z*) =

The result of such integration is the propagator *D*<sup>+</sup> in the following form:

 <sup>∞</sup> 0

*z*4 <sup>2242</sup> <sup>−</sup> *<sup>z</sup>*<sup>6</sup>

*<sup>d</sup>***x***d***x**′ *dt*′ *J*0 *<sup>n</sup><sup>ω</sup>*

> , *t* ′ ) <sup>−</sup> *<sup>n</sup>*<sup>2</sup>

where the charge density and current involves only two-dimensional velocities and

*̺*(**x**, *<sup>t</sup>*)*̺*(**x**′

integration is also only two-dimensional with two-dimensional *<sup>d</sup>***x**, *<sup>d</sup>***x**′

The difference is in the replacing mathematical formulas as follows:

*<sup>c</sup>* <sup>|</sup>**<sup>x</sup>** <sup>−</sup> **<sup>x</sup>**′ | <sup>|</sup>**<sup>x</sup>** <sup>−</sup> **<sup>x</sup>**′| −→ *<sup>J</sup>*<sup>0</sup>

So, After insertion the quantities (123) and (124) into (135), we get:

 *dt*′ *J*0 *nv<sup>ω</sup> <sup>c</sup>* <sup>|</sup>*<sup>t</sup>* <sup>−</sup> *<sup>t</sup>* ′ | 

sin *<sup>n</sup><sup>ω</sup>*

) = *<sup>i</sup>* 2*πc*

<sup>22</sup> <sup>+</sup>

*D*+(*x* − *x*′

*<sup>J</sup>*0(*z*) = <sup>1</sup> <sup>−</sup> *<sup>z</sup>*<sup>2</sup>

2*π µ n*2 

× 

*<sup>P</sup>*(*ω*, *<sup>t</sup>*) = <sup>−</sup> *<sup>ω</sup>*

∞ ∑ *s*=0

The *ω*-integral in (131) can be performed using the residuum theorem after integration in

*dωJ*<sup>0</sup>

The initial terms in the expansion of the Bessel function with exponent zero is as follows:

*<sup>n</sup><sup>ω</sup>*

<sup>224262</sup> <sup>+</sup>

*<sup>c</sup>* <sup>|</sup>**<sup>x</sup>** <sup>−</sup> **<sup>x</sup>**′

*<sup>c</sup>*<sup>2</sup> **<sup>J</sup>**(**x**, *<sup>t</sup>*) · **<sup>J</sup>**(**x**′

*<sup>n</sup><sup>ω</sup>*

*<sup>c</sup>* <sup>|</sup>**<sup>x</sup>** <sup>−</sup> **<sup>x</sup>**′


cos[*ω*(*t* − *t*

′

The spectral formula for the two dimensional Cerenkov radiation is of the analogue of the ˇ

*<sup>c</sup>* <sup>|</sup>**<sup>x</sup>** <sup>−</sup> **<sup>x</sup>**′


*z*8


> , *t* ′ )

cos[*ω*(*t* − *t*

(−1)*sz*2*<sup>s</sup>*

*<sup>s</sup>*!*s*!22*<sup>s</sup>* (132)

<sup>22426282</sup> −··· . (134)

′ )]×

.

, (135)

. (136)

)], *β* = *v*/*c*, (137)

. (133)

$$P(\omega, t) = 0; \quad n\pounds < 1,\tag{141}$$

where condition *nβ* = 1 is the threshold of the existence of the two-dimensional form of the Cerenkov radiation. ˇ

## **7. The Cerenkov radiation in two-dimensional structure generated by a ˇ pulse**

Let us consider the electron moving perpendicularly to the 2D sheet in the pane *y* − *z* with the index of refraction *n* and the magnetic permeability *µ*. Then, the charge density and current density for the charge moving along the axis is (*v* > 0)

$$
\boldsymbol{\varrho} = \boldsymbol{e} \boldsymbol{\delta}(\boldsymbol{v}t) \boldsymbol{\delta}(\mathbf{x}) = \frac{\boldsymbol{e}}{\boldsymbol{v}} \boldsymbol{\delta}(t) \boldsymbol{\delta}(\mathbf{x}) \tag{142}
$$

$$\mathbf{J} = \mathbf{0}.\tag{143}$$

After insertion of the last formulas into the spectral formula for the Cerenkov radiation (135) ˇ with regard to (136), we get

$$P(\omega, t) = \frac{e^2}{2\pi} \frac{\mu \omega}{n^2 v^2} \int dt' \delta(t) \delta(t') f\_0(0) \cos[\omega(t - t')],\tag{144}$$

After performing the t and t' integration we get

$$\int dt P(\omega, t) = \frac{e^2}{2\pi} \frac{\mu \omega}{n^2 v^2} J\_0\left(0\right). \tag{145}$$

*<sup>D</sup>*(*k*) = <sup>1</sup>

the massive photon propagator is of the form (here we introduce ¯*h* and *c*):


From eq. (149) the dispersion law for the massive photons follows:

physically meaningful and it is meaningful to study it.

The propagator for the massive photon is then derived as

 <sup>∞</sup> 0

The function (152) differs from the the original function *D*<sup>+</sup> by the factor

*dω*

sin[ *<sup>n</sup>*2*ω*<sup>2</sup>

*<sup>c</sup>*<sup>2</sup> <sup>−</sup> *<sup>m</sup>*2*c*<sup>2</sup> *h*¯ 2 ]

, *<sup>m</sup>*2) = *<sup>i</sup> c* 1 4*π*<sup>2</sup>

get the Einstein energetic equation

*D*+(*x* − *x*′

*<sup>D</sup>*(*k*, *<sup>m</sup>*2) = <sup>1</sup>

where *n* is the parameter of the medium and *m* is mass of photon in this medium.

*<sup>ω</sup>* <sup>=</sup> *<sup>c</sup> n k*2 +



where this propagator is derived from an assumption that the photon energetic equation is

<sup>2</sup> <sup>−</sup> *<sup>n</sup>*2(*k*0)<sup>2</sup> <sup>=</sup> <sup>−</sup> *<sup>m</sup>*2*c*<sup>2</sup>

Let us remark here that such dispersion law is valid not only for the massive photon but also for electromagnetic field in waveguides and electromagnetic field in ionosphere. It means that the corresponding photons are also massive and the theory of massive photons is physically meaningful. It means that also the Cerenkov radiation of massive photons is ˇ

The validity of eq. (149) can be verified using very simple idea that for *n* = 1 the Einstein equation for mass and energy has to follow. Putting **p** = *h*¯ **k**, ¯*hk*<sup>0</sup> = *h*¯(*ω*/*c*)=(*E*/*c*), we

*m*2*c*2

*<sup>h</sup>*¯ <sup>2</sup> <sup>−</sup> *<sup>i</sup><sup>ǫ</sup>*

, (147)

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127

The Cherenkov Effect in Graphene-Like Structures

, (148)

*<sup>h</sup>*¯ <sup>2</sup> , (149)

*<sup>h</sup>*¯ <sup>2</sup> . (150)

*E*<sup>2</sup> = **p**2*c*<sup>2</sup> + *m*2*c*4. (151)

1/2|**<sup>x</sup>** − **<sup>x</sup>**′

<sup>|</sup>**<sup>x</sup>** <sup>−</sup> **<sup>x</sup>**′| *<sup>e</sup>*


−*iω*|*t*−*t* ′ |

. (152)

The derived formula does not involve the Cerenkov radiation threshold. At the same time ˇ the formula does not involve the transition radiation which is generated by the charge when it is moving outside of the sheet. Nevertheless, such radiation can be easily determined by the Ginzburg method [10].

## **8. The Cerenkov effect with massive photons ˇ**

The massive electrodynamics in medium can be constructed by generalization of massless electrodynamics to the case with massive photon. In our case it means that we replace only eq. (119) by the following one:

$$\left(\Delta - \frac{\mu \epsilon}{c^2} \frac{\partial^2}{\partial t^2} + \frac{m^2 c^2}{\hbar^2} \right) A^\mu = \frac{\mu}{c} \left(g^{\mu \nu} + \frac{n^2 - 1}{n^2} \eta^\mu \eta^\nu \right) J\_{\nu \nu} \tag{146}$$

where *m* is mass of photon. The Lorentz gauge of massless photons is conserved also in the massive situation.

In superconductiviy photon is a massive spin 1 particle as a consequence of a broken symmetry of the Landau-Ginzburg Lagrangian. The Meissner effect can be used as a experimental demonstration that photon in a superconductor is a massive particle. In particle physics the situation is analogous to the situation in superconductivity. The masses of particles are also generated by the broken symmetry or in other words by the Higgs mechanism. Massive particles with spin 1 form the analogue of the massive photon.

Kirzhnits and Linde [14] proposed a qualitative analysis wherein they indicated that, as in the Ginzburg-Landau theory of superconductivity, the Meissner effect can also be realized in the Weinberg model. Later, it was shown that the Meissner effect is realizable in renormalizable gauge fields and also in the Weinberg model [35].

We will investigate how the spectrum of the Cerenkov radiation is modified if we suppose ˇ the massive photons are generated instead of massless photons. The derived results form an analogue of the situation with massless photons. According to author Pardy [25–27] and Dittrich [6] with the analogy of the massless photon propagator *D*(*k*) in the momentum representation

$$D(k) = \frac{1}{|\mathbf{k}|^2 - n^2(k^0)^2 - i\epsilon'} \tag{147}$$

the massive photon propagator is of the form (here we introduce ¯*h* and *c*):

26

with regard to (136), we get

the Ginzburg method [10].

eq. (119) by the following one:

 <sup>∆</sup> <sup>−</sup> *µǫ c*2

**ˇ**

massive situation.

representation

*<sup>P</sup>*(*ω*, *<sup>t</sup>*) = *<sup>e</sup>*<sup>2</sup>

After performing the t and t' integration we get

2*π*

**8. The Cerenkov effect with massive photons**

*∂*2 *<sup>∂</sup>t*<sup>2</sup> <sup>+</sup>

gauge fields and also in the Weinberg model [35].

*m*2*c*<sup>2</sup> *h*¯ 2

*µω n*2*v*<sup>2</sup> *dt*′ *δ*(*t*)*δ*(*t* ′

*dtP*(*ω*, *<sup>t</sup>*) = *<sup>e</sup>*<sup>2</sup>

2*π*

The derived formula does not involve the Cerenkov radiation threshold. At the same time ˇ the formula does not involve the transition radiation which is generated by the charge when it is moving outside of the sheet. Nevertheless, such radiation can be easily determined by

The massive electrodynamics in medium can be constructed by generalization of massless electrodynamics to the case with massive photon. In our case it means that we replace only

> *<sup>A</sup><sup>µ</sup>* <sup>=</sup> *<sup>µ</sup> c <sup>g</sup>µν* +

where *m* is mass of photon. The Lorentz gauge of massless photons is conserved also in the

In superconductiviy photon is a massive spin 1 particle as a consequence of a broken symmetry of the Landau-Ginzburg Lagrangian. The Meissner effect can be used as a experimental demonstration that photon in a superconductor is a massive particle. In particle physics the situation is analogous to the situation in superconductivity. The masses of particles are also generated by the broken symmetry or in other words by the Higgs

Kirzhnits and Linde [14] proposed a qualitative analysis wherein they indicated that, as in the Ginzburg-Landau theory of superconductivity, the Meissner effect can also be realized in the Weinberg model. Later, it was shown that the Meissner effect is realizable in renormalizable

We will investigate how the spectrum of the Cerenkov radiation is modified if we suppose ˇ the massive photons are generated instead of massless photons. The derived results form an analogue of the situation with massless photons. According to author Pardy [25–27] and Dittrich [6] with the analogy of the massless photon propagator *D*(*k*) in the momentum

mechanism. Massive particles with spin 1 form the analogue of the massive photon.

*µω*

After insertion of the last formulas into the spectral formula for the Cerenkov radiation (135)

ˇ

′

*<sup>n</sup>*2*v*<sup>2</sup> *<sup>J</sup>*<sup>0</sup> (0). (145)

)], (144)

)*J*0(0) cos[*ω*(*<sup>t</sup>* − *<sup>t</sup>*

*n*<sup>2</sup> − 1 *<sup>n</sup>*<sup>2</sup> *<sup>η</sup>µην*

*Jν*, (146)

$$D(k, m^2) = \frac{1}{|\mathbf{k}|^2 - n^2(k^0)^2 + \frac{m^2 c^2}{\hbar^2} - i\epsilon},\tag{148}$$

where this propagator is derived from an assumption that the photon energetic equation is

$$\left|\mathbf{k}\right|^2 - n^2(k^0)^2 = -\frac{m^2c^2}{\hbar^2},\tag{149}$$

where *n* is the parameter of the medium and *m* is mass of photon in this medium.

From eq. (149) the dispersion law for the massive photons follows:

$$
\omega = \frac{c}{n} \sqrt{k^2 + \frac{m^2 c^2}{\hbar^2}}.\tag{150}
$$

Let us remark here that such dispersion law is valid not only for the massive photon but also for electromagnetic field in waveguides and electromagnetic field in ionosphere. It means that the corresponding photons are also massive and the theory of massive photons is physically meaningful. It means that also the Cerenkov radiation of massive photons is ˇ physically meaningful and it is meaningful to study it.

The validity of eq. (149) can be verified using very simple idea that for *n* = 1 the Einstein equation for mass and energy has to follow. Putting **p** = *h*¯ **k**, ¯*hk*<sup>0</sup> = *h*¯(*ω*/*c*)=(*E*/*c*), we get the Einstein energetic equation

$$E^2 = \mathbf{p}^2 c^2 + m^2 c^4.\tag{151}$$

The propagator for the massive photon is then derived as

$$D\_{+}(\mathbf{x} - \mathbf{x}', m^2) = \frac{i}{c} \frac{1}{4\pi^2} \int\_0^\infty d\omega \, \frac{\sin[\frac{n^2 \omega^2}{c^2} - \frac{m^2 c^2}{\hbar^2}]^{1/2} |\mathbf{x} - \mathbf{x}'|}{|\mathbf{x} - \mathbf{x}'|} e^{-i\omega |t - t'|}. \tag{152}$$

The function (152) differs from the the original function *D*<sup>+</sup> by the factor

$$
\left(\frac{\omega^2 n^2}{c^2} - \frac{m^2 c^2}{\hbar^2}\right)^{1/2}.\tag{153}
$$

If we compare the potentials concerning massive and massless photons, we can deduce that also Cerenkov radiation with massive photons can be generated. So, the determination of ˇ

In case of the massive electromagnetic field in the medium, the action *W* is given by the

*<sup>µ</sup>*(*x*)*D*+*µν*(*<sup>x</sup>* <sup>−</sup> *<sup>x</sup>*′

[*gµν* + (<sup>1</sup> <sup>−</sup> *<sup>n</sup>*−2)*ηµην*]*D*+(*<sup>x</sup>* <sup>−</sup> *<sup>x</sup>*′

− <sup>2</sup>

*dtdω*

sin[ *<sup>n</sup>*2*ω*<sup>2</sup>

*P*(*ω*, *t*)

*<sup>c</sup>*<sup>2</sup> <sup>−</sup> *<sup>m</sup>*2*c*<sup>2</sup> *h*¯ 2 ]


*<sup>c</sup>*<sup>2</sup> **<sup>J</sup>**(**x**, *<sup>t</sup>*) · **<sup>J</sup>**(**x**′

where *<sup>η</sup><sup>µ</sup>* ≡ (1, **<sup>0</sup>**), *<sup>J</sup><sup>µ</sup>* ≡ (*c̺*,**J**) is the conserved current, *<sup>µ</sup>* is the magnetic permeability of the medium, *<sup>ǫ</sup>* is the dielectric constant od the medium and *<sup>n</sup>* <sup>=</sup> <sup>√</sup>*ǫµ* is the index of


where Im *W* is the basis for the definition of the spectral function *P*(*ω*, *t*) as follows:

= − �

Now, if we insert eq. (161) into eq. (160), we get after extracting *P*(*ω*, *t*) the following general

, *t* ′ ) <sup>−</sup> *<sup>n</sup>*<sup>2</sup>

Now, let us apply the formula (164) in order to get the Cerenkov distribution of massive ˇ photons. let as consider a particle of charge *Q* moving at a constant velocity **v**. In such a way

, *m*2)*J*

*<sup>ν</sup>*(*x*′

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129

), (160)

, *m*2), (161)

*<sup>h</sup>*¯ ImW, (162)

*<sup>h</sup>*¯ *<sup>ω</sup>* . (163)


 <sup>×</sup>

)]. (164)

1/2]|**<sup>x</sup>** <sup>−</sup> **<sup>x</sup>**′

, *t* ′

the Cerenkov effect with massive photons is physically meaningful. ˇ

(*dx*)(*dx*′

The probability of the persistence of vacuum is of the following form:

−<sup>2</sup> *h*¯

Im*<sup>W</sup> <sup>d</sup>*

*<sup>d</sup>***x***d***x**′ *dt*′ 

)][*̺*(**x**, *<sup>t</sup>*)*̺*(**x**′

)*J*

*<sup>W</sup>* <sup>=</sup> <sup>1</sup> 2*c*<sup>2</sup> �

> *<sup>D</sup>µν* <sup>+</sup> <sup>=</sup> *<sup>µ</sup> c*

following formula:

refraction of the medium.

expression for this spectral function:

*<sup>P</sup>*(*ω*, *<sup>t</sup>*) = <sup>−</sup> *<sup>ω</sup>*

4*π*<sup>2</sup>

cos[*ω*(*t* − *t*

*µ n*2 �

′

we can write for the charge density and for the current density:

where

From eq. (152) the potentials generated by the massless or massive photons respectively follow. In case of the massless photon, the potential is according to Schwinger defined by the formula:

$$V(\mathbf{x} - \mathbf{x}') = \int\_{-\infty}^{\infty} d\tau D\_{+}(\mathbf{x} - \mathbf{x}', \tau) = \int\_{-\infty}^{\infty} d\tau \left\{ \frac{i}{c} \frac{1}{4\pi^2} \int\_{0}^{\infty} d\omega \frac{\sin\frac{n\omega}{c}|\mathbf{x} - \mathbf{x}'|}{|\mathbf{x} - \mathbf{x}'|} e^{-i\omega|\tau|} \right\}. \tag{154}$$

The *τ*-integral can be evaluated using the mathematical formula

$$\int\_{-\infty}^{\infty} d\tau \, e^{-i\omega|\tau|} = \frac{2}{\mathrm{i}\omega} \tag{155}$$

and the *ω*-integral can be evaluated using the formula

$$\int\_0^\infty \frac{\sin a\mathbf{x}}{\mathbf{x}} d\mathbf{x} = \frac{\pi}{2}, \quad \text{for} \quad a > 0. \tag{156}$$

After using eqs. (155) and (156), we get

$$V(\mathbf{x} - \mathbf{x}') = \frac{1}{c} \frac{1}{4\pi} \frac{1}{|\mathbf{x} - \mathbf{x}'|}. \tag{157}$$

In case of the massive photon, the mathematical determination of potential is the analogical to the massless situation only with the difference we use the propagator (152) and the tables of integrals [11]:

$$\int\_0^\infty \frac{d\mathbf{x}}{\mathbf{x}} \sin \left( p\sqrt{\mathbf{x}^2 - \mu^2} \right) = \frac{\pi}{2} e^{-pu}. \tag{158}$$

Using this integral we get that the potential generated by the massive photons is

$$V(\mathbf{x} - \mathbf{x}', m^2) = \frac{1}{c} \frac{1}{4\pi} \frac{\exp\left\{-\frac{mcn}{\hbar}|\mathbf{x} - \mathbf{x}'|\right\}}{|\mathbf{x} - \mathbf{x}'|}. \tag{159}$$

If we compare the potentials concerning massive and massless photons, we can deduce that also Cerenkov radiation with massive photons can be generated. So, the determination of ˇ the Cerenkov effect with massive photons is physically meaningful. ˇ

In case of the massive electromagnetic field in the medium, the action *W* is given by the following formula:

$$W = \frac{1}{2c^2} \int (d\mathbf{x}) (d\mathbf{x}') l^\mu(\mathbf{x}) D\_{+\mu\nu}(\mathbf{x} - \mathbf{x}', m^2) l^\nu(\mathbf{x}'),\tag{160}$$

where

28

the formula:

*<sup>V</sup>*(**<sup>x</sup>** <sup>−</sup> **<sup>x</sup>**′

of integrals [11]:

) = <sup>∞</sup> −∞

*<sup>d</sup>τD*+(**<sup>x</sup>** <sup>−</sup> **<sup>x</sup>**′

and the *ω*-integral can be evaluated using the formula

After using eqs. (155) and (156), we get

, *τ*) =

 <sup>∞</sup> −∞ *dτ e*

sin *ax x*

*<sup>V</sup>*(**<sup>x</sup>** <sup>−</sup> **<sup>x</sup>**′

*dx* <sup>=</sup> *<sup>π</sup>*

) = 1 *c* 1 4*π*

In case of the massive photon, the mathematical determination of potential is the analogical to the massless situation only with the difference we use the propagator (152) and the tables

> *x*<sup>2</sup> − *u*<sup>2</sup> <sup>=</sup> *<sup>π</sup>* 2 *e*

> > exp

− *mcn*

*<sup>h</sup>*¯ <sup>|</sup>**<sup>x</sup>** <sup>−</sup> **<sup>x</sup>**′


1 |**<sup>x</sup>** − **<sup>x</sup>**′|

The *τ*-integral can be evaluated using the mathematical formula

 <sup>∞</sup> 0

 <sup>∞</sup> 0

*<sup>V</sup>*(**<sup>x</sup>** <sup>−</sup> **<sup>x</sup>**′

*dx x* sin *p* 

Using this integral we get that the potential generated by the massive photons is

, *<sup>m</sup>*2) = <sup>1</sup> *c* 1 4*π*

 <sup>∞</sup> −∞ *dτ i c* 1 4*π*<sup>2</sup>

*ω*2*n*<sup>2</sup>

*<sup>c</sup>*<sup>2</sup> <sup>−</sup> *<sup>m</sup>*2*c*<sup>2</sup> *h*¯ 2

From eq. (152) the potentials generated by the massless or massive photons respectively follow. In case of the massless photon, the potential is according to Schwinger defined by

1/2

 <sup>∞</sup> 0 *dω*

<sup>−</sup>*iω*|*τ*<sup>|</sup> = 2

sin *<sup>n</sup><sup>ω</sup>*

*<sup>c</sup>* <sup>|</sup>**<sup>x</sup>** <sup>−</sup> **<sup>x</sup>**′ |

<sup>|</sup>**<sup>x</sup>** <sup>−</sup> **<sup>x</sup>**′| *<sup>e</sup>*

*<sup>i</sup><sup>ω</sup>* (155)

. (157)

<sup>−</sup>*pu*. (158)

<sup>|</sup>**<sup>x</sup>** <sup>−</sup> **<sup>x</sup>**′| . (159)

<sup>2</sup> , for *<sup>a</sup>* <sup>&</sup>gt; 0. (156)

−*iω*|*τ*| 

. (154)

. (153)

$$D\_{+}^{\mu\nu} = \frac{\mu}{c} [\mathbf{g}^{\mu\nu} + (1 - n^{-2}) \eta^{\mu} \eta^{\nu}] D\_{+} (\mathbf{x} - \mathbf{x}', m^2), \tag{161}$$

where *<sup>η</sup><sup>µ</sup>* ≡ (1, **<sup>0</sup>**), *<sup>J</sup><sup>µ</sup>* ≡ (*c̺*,**J**) is the conserved current, *<sup>µ</sup>* is the magnetic permeability of the medium, *<sup>ǫ</sup>* is the dielectric constant od the medium and *<sup>n</sup>* <sup>=</sup> <sup>√</sup>*ǫµ* is the index of refraction of the medium.

The probability of the persistence of vacuum is of the following form:

$$|\langle 0\_+ | 0\_- \rangle|^2 = e^{-\frac{2}{\hbar} \text{Im} \mathbf{W}},\tag{162}$$

where Im *W* is the basis for the definition of the spectral function *P*(*ω*, *t*) as follows:

$$-\frac{2}{\hbar}\text{Im}\mathcal{W} \stackrel{d}{=} -\int dt d\omega \frac{P(\omega, t)}{\hbar \omega}.\tag{163}$$

Now, if we insert eq. (161) into eq. (160), we get after extracting *P*(*ω*, *t*) the following general expression for this spectral function:

$$P(\omega, t) = -\frac{\omega}{4\pi^2} \frac{\mu}{n^2} \int d\mathbf{x} d\mathbf{x}' dt' \left[ \frac{\sin[\frac{\mu^2 \omega^2}{c^2} - \frac{m^2 c^2}{\hbar^2}]^{1/2} |\mathbf{x} - \mathbf{x}'|}{|\mathbf{x} - \mathbf{x}'|} \right] \times$$

$$\cos[\omega(t - t')] [\varrho(\mathbf{x}, t) \varrho(\mathbf{x}', t') - \frac{n^2}{c^2} \mathbf{J}(\mathbf{x}, t) \cdot \mathbf{J}(\mathbf{x}', t')]. \tag{164}$$

Now, let us apply the formula (164) in order to get the Cerenkov distribution of massive ˇ photons. let as consider a particle of charge *Q* moving at a constant velocity **v**. In such a way we can write for the charge density and for the current density:

$$\boldsymbol{\varrho} = \mathbf{Q} \boldsymbol{\delta}(\mathbf{x} - \mathbf{v}t), \qquad \mathbf{J} = \mathbf{Q} \mathbf{v} \boldsymbol{\delta}(\mathbf{x} - \mathbf{v}t). \tag{165}$$

The most simple way how to get the angle Θ between vectors **k** and **p** is the use the

**<sup>p</sup>** <sup>−</sup> *<sup>h</sup>*¯ **<sup>k</sup>** <sup>=</sup> **<sup>p</sup>**′

where *<sup>E</sup>* and *<sup>E</sup>*′ are energies of a moving particle before and after act of emission of a photon with energy ¯*h<sup>ω</sup>* and momentum ¯*h***k**, and **<sup>p</sup>** and **<sup>p</sup>**′ are momenta of the particle before and

If we raise the equations (172) and (173) to the second power and take the difference of these

1/2 + *hk*¯ 2*p* <sup>1</sup> <sup>−</sup> <sup>1</sup> *n*2 

which has the correct massless limit. The massless limit also gives the sense of the parameter *n* which is introduced in the massive situation. We also observe that while in the massless situation the angle of emission depends only on *nβ*, in case of massive situation it depends also on the wave vector *k*. It means that the emission of the massive photons are emitted by

So, in experiment the Cerenkov production of massive photons can be strictly distinguished ˇ from the Cerenkov production of massless photons, or, from the hard production of spin 1 ˇ

The article is in some sense the preamble to the any conferences of ideas related to the Cerenkov effect in the graphene-like dielectric structures. At present time, the most attention ˇ is devoted in graphene physics with a goal to construct the computers with the artificial intelligence. However, we do not know, a priori, how many discoveries are involved in the

The information on the Cerenkov effect in graphene-like structures and also the elementary ˇ particle interaction with graphene-like structures is necessary not only in the solid state physics, but also in the elementary particle physics in the big laboratories where graphene can form the substantial components of the particle detectors. We hope that these possibilities

The monolithic structures can be also built into graphene-like structures by addition and re-arrangement of deposit atoms [20]. The repeating patterns can be created to form new carbon allotropes called haeckelites. The introducing such architectonic defects modifies mechanical, electrical, optical and chemical properties of graphene-like structures and it

*m*2*c*<sup>2</sup> *h*¯ <sup>2</sup>*k*<sup>2</sup>

*<sup>E</sup>* <sup>−</sup> *<sup>h</sup>*¯ *<sup>ω</sup>* <sup>=</sup> *<sup>E</sup>*′ (172)

<sup>−</sup> *<sup>m</sup>*2*c*<sup>2</sup>

<sup>2</sup>*n*<sup>2</sup> *phk*¯ , (174)

, (173)

The Cherenkov Effect in Graphene-Like Structures

http://dx.doi.org/10.5772/50296

131

conservation laws for an energy and momentum.

quadratic equations, we can extract the cos Θ in the form:

*nβ* 1 +

investigation of the Cerenkov effect in graphene-like structures. ˇ

cos <sup>Θ</sup> <sup>=</sup> <sup>1</sup>

the Cerenkov mechanism in all space directions. ˇ

will be consider in the physical laboratories.

massive particles.

**9. Perspective**

after emission of the same photon.

After insertion of eq. (165) into eq. (164), we get (*v* = |**v**|).

$$P(\omega, t) = \frac{Q^2}{4\pi^2} \frac{v\mu\omega}{c^2} \left(1 - \frac{1}{n^2\beta^2}\right) \int\_{\infty}^{\infty} \frac{d\tau}{\tau} \sin\left(\left[\frac{n^2\omega^2}{c^2} - \frac{m^2c^2}{\hbar^2}\right]^{1/2} v\tau\right) \cos\omega\tau,\tag{166}$$

where we have put *τ* = *t* ′ − *t*, *β* = *v*/*c*.

For *P*(*ω*, *t*), the situation leads to evaluation of the *τ*-integral. For this integral we have:

$$\int\_{-\infty}^{\infty} \frac{d\tau}{\tau} \sin\left(\left[\frac{n^2 \omega^2}{c^2} - \frac{c^2}{m^2}\right]^{1/2} v\tau\right) \cos\omega\tau = \begin{cases} \pi, \ 0 < m^2 < \frac{\omega^2}{c^2 \overline{v}^2} (n^2 \beta^2 - 1) \\ 0, \ m^2 > \frac{\omega^2}{c^2 \overline{v}^2} (n^2 \beta^2 - 1). \end{cases} \tag{167}$$

From eq. (167) immediately follows that *m*<sup>2</sup> > 0 implies the Cerenkov threshold ˇ *nβ* > 1. From eq. (166) and (167) we get the spectral formula of the Cerenkov radiation of massive ˇ photons in the form:

$$P(\omega, t) = \frac{Q^2}{4\pi} \frac{v\omega\mu}{c^2} \left(1 - \frac{1}{n^2 \beta^2}\right) \tag{168}$$

for

$$
\omega > \frac{mcv}{\hbar} \frac{1}{\sqrt{n^2 \beta^2 - 1}} > 0,\tag{169}
$$

and *P*(*ω*, *t*) = 0 for

$$
\omega < \frac{\mathfrak{m}cv}{\hbar} \frac{1}{\sqrt{n^2 \beta^2 - 1}}.\tag{170}
$$

Using the dispersion law (150) we can write the power spectrum *P*(*ω*) as a function dependent on *k*2. Then,

$$P(k^2) = \frac{Q^2}{4\pi} \frac{v\mu}{nc} \sqrt{k^2 + \frac{m^2c^2}{\hbar^2}} \left(1 - \frac{1}{n^2\beta^2}\right); \quad k^2 > \frac{m^2c^2}{\hbar^2} \frac{1}{n^2\beta^2 - 1} \tag{171}$$

and *P*(*ω*, *t*) = 0 for *k*<sup>2</sup> < (*m*2*c*2/¯*h*2)(*n*2*β*<sup>2</sup> − 1)<sup>−</sup>1.

The most simple way how to get the angle Θ between vectors **k** and **p** is the use the conservation laws for an energy and momentum.

$$E - \hbar \omega = E'\tag{172}$$

$$\mathbf{p} - \hbar \mathbf{k} = \mathbf{p}',\tag{173}$$

where *<sup>E</sup>* and *<sup>E</sup>*′ are energies of a moving particle before and after act of emission of a photon with energy ¯*h<sup>ω</sup>* and momentum ¯*h***k**, and **<sup>p</sup>** and **<sup>p</sup>**′ are momenta of the particle before and after emission of the same photon.

If we raise the equations (172) and (173) to the second power and take the difference of these quadratic equations, we can extract the cos Θ in the form:

$$\cos\Theta = \frac{1}{n\beta} \left( 1 + \frac{m^2 c^2}{\hbar^2 k^2} \right)^{1/2} + \frac{\hbar k}{2p} \left( 1 - \frac{1}{n^2} \right) - \frac{m^2 c^2}{2n^2 p \hbar k} \tag{174}$$

which has the correct massless limit. The massless limit also gives the sense of the parameter *n* which is introduced in the massive situation. We also observe that while in the massless situation the angle of emission depends only on *nβ*, in case of massive situation it depends also on the wave vector *k*. It means that the emission of the massive photons are emitted by the Cerenkov mechanism in all space directions. ˇ

So, in experiment the Cerenkov production of massive photons can be strictly distinguished ˇ from the Cerenkov production of massless photons, or, from the hard production of spin 1 ˇ massive particles.

## **9. Perspective**

30

*̺* = *Qδ*(**x** − **v***t*), **J** = *Q***v***δ*(**x** − **v***t*). (165)

*<sup>c</sup>*<sup>2</sup> <sup>−</sup> *<sup>m</sup>*2*c*<sup>2</sup> *h*¯ 2

*π*, 0 < *m*<sup>2</sup> < *<sup>ω</sup>*<sup>2</sup>

ˇ

0, *m*<sup>2</sup> > *<sup>ω</sup>*<sup>2</sup>

1/2 *vτ* 

*<sup>c</sup>*2*v*<sup>2</sup> (*n*2*β*<sup>2</sup> − <sup>1</sup>)

> 0, (169)

. (170)

1

*<sup>n</sup>*2*β*<sup>2</sup> <sup>−</sup> <sup>1</sup> (171)

*<sup>c</sup>*2*v*<sup>2</sup> (*n*2*β*<sup>2</sup> <sup>−</sup> <sup>1</sup>). (167)

cos *ωτ*, (166)

(168)

After insertion of eq. (165) into eq. (164), we get (*v* = |**v**|).

′ − *t*, *β* = *v*/*c*.

1/2 *vτ* 

*<sup>P</sup>*(*ω*, *<sup>t</sup>*) = *<sup>Q</sup>*<sup>2</sup>

*ω* >

*ω* <

*m*2*c*2 *h*¯ 2

 <sup>1</sup> <sup>−</sup> <sup>1</sup> *n*2*β*2 

*mcv h*¯

Using the dispersion law (150) we can write the power spectrum *P*(*ω*) as a function

4*π*

*mcv h*¯

 <sup>∞</sup> ∞

*dτ τ* sin *n*2*ω*2

For *P*(*ω*, *t*), the situation leads to evaluation of the *τ*-integral. For this integral we have:

cos *ωτ* =

From eq. (167) immediately follows that *m*<sup>2</sup> > 0 implies the Cerenkov threshold ˇ *nβ* > 1. From eq. (166) and (167) we get the spectral formula of the Cerenkov radiation of massive

> *vωµ c*2 <sup>1</sup> <sup>−</sup> <sup>1</sup> *n*2*β*2

1 *n*2*β*<sup>2</sup> − 1

> 1 *n*2*β*<sup>2</sup> − 1

> > ; *k*<sup>2</sup> >

*m*2*c*2 *h*¯ 2

*<sup>P</sup>*(*ω*, *<sup>t</sup>*) = *<sup>Q</sup>*<sup>2</sup>

where we have put *τ* = *t*

*dτ τ* sin *n*2*ω*2 *<sup>c</sup>*<sup>2</sup> <sup>−</sup> *<sup>c</sup>*<sup>2</sup> *m*2

photons in the form:

and *P*(*ω*, *t*) = 0 for

dependent on *k*2. Then,

*<sup>P</sup>*(*k*2) = *<sup>Q</sup>*<sup>2</sup>

4*π*

and *P*(*ω*, *t*) = 0 for *k*<sup>2</sup> < (*m*2*c*2/¯*h*2)(*n*2*β*<sup>2</sup> − 1)<sup>−</sup>1.

*vµ nc k*2 +

for

 <sup>∞</sup> −∞ 4*π*<sup>2</sup>

*vµω c*2 <sup>1</sup> <sup>−</sup> <sup>1</sup> *n*2*β*2

> The article is in some sense the preamble to the any conferences of ideas related to the Cerenkov effect in the graphene-like dielectric structures. At present time, the most attention ˇ is devoted in graphene physics with a goal to construct the computers with the artificial intelligence. However, we do not know, a priori, how many discoveries are involved in the investigation of the Cerenkov effect in graphene-like structures. ˇ

> The information on the Cerenkov effect in graphene-like structures and also the elementary ˇ particle interaction with graphene-like structures is necessary not only in the solid state physics, but also in the elementary particle physics in the big laboratories where graphene can form the substantial components of the particle detectors. We hope that these possibilities will be consider in the physical laboratories.

> The monolithic structures can be also built into graphene-like structures by addition and re-arrangement of deposit atoms [20]. The repeating patterns can be created to form new carbon allotropes called haeckelites. The introducing such architectonic defects modifies mechanical, electrical, optical and chemical properties of graphene-like structures and it

is not excluded that special haeckelites are superconductive at high temperatures. The unconventional graphene-like materials can be prepared by special technique in order to do revolution in the solid state physics.

[12] Harris, E. G. (1972). A pedestrian approach to quantum field theory, (John Willey and

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While the last century economy growth was based on the inventions in the Edison-Tesla electricity, the economy growth in this century will be obviously based on the graphene-like structures physics. We hope that these perspective ideas will be considered at the universities and in the physical laboratories.

## **Author details**

Miroslav Pardy

<sup>⋆</sup> Address all correspondence to: pamir@physics.muni.cz

Department of Physical Electronics, Masaryk University, Brno, Czech Republic

## **10. References**


32

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<sup>⋆</sup> Address all correspondence to: pamir@physics.muni.cz

Department of Physical Electronics, Masaryk University, Brno, Czech Republic

and in the physical laboratories.

**Author details** Miroslav Pardy

**10. References**

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is not excluded that special haeckelites are superconductive at high temperatures. The unconventional graphene-like materials can be prepared by special technique in order to

While the last century economy growth was based on the inventions in the Edison-Tesla electricity, the economy growth in this century will be obviously based on the graphene-like structures physics. We hope that these perspective ideas will be considered at the universities

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**Chapter 5**

**Electronic and Vibrational Properties of Adsorbed and**

The perennial interest in studying the physical properties of nanofilms has increased sub‐ stantially over the last few years due to the development of nanotechnologies and the syn‐ thesis of new compounds – especially those based on carbon, which are extremely

An important feature of carbon nanofilms (including those with defects) is a close relation between the electronic and phonon properties, which is exhibited, for example, in the gra‐

It is well known that graphene monolayers cannot exist as planar objects in the free state, because in flat 2D-crystals the mean-square amplitudes of the atoms in the direction normal to the layer plane diverge even at *T* =0 (see, e.g., [3]). So we can study and practically apply only such graphene, which is deposited on a certain substrate providing the stability of the plane carbon nanofilms (see, e.g., [4-6]). Only small flakes can be detached from the sub‐ strate and these flakes immediately acquire a corrugated shape [7]. When studying the elec‐ tronic properties of graphene a dielectric substrate is often used. The presence of the substrate greatly increases the occurrence of various defects in graphene and carbon nano‐ films. Our investigations make it possible to predict the general properties of phonon and

This chapter consists of three sections: first section is devoted to the calculation of local dis‐ crete levels in the electron spectra of graphene with different defects. In the second section

> © 2013 Feher et al.; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

© 2013 Feher et al.; licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

distribution, and reproduction in any medium, provided the original work is properly cited.

interesting for both fundamental research and potential applications.

electron spectra for graphene and bigraphene containing different defects.

phene-based systems with superconducting properties [1,2].

**Embedded Graphene and Bigraphene with Defects**

Alexander Feher, Eugen Syrkin, Sergey Feodosyev,

Igor Gospodarev, Elena Manzhelii,

http://dx.doi.org/10.5772/50562

**1. Introduction**

Alexander Kotlar and Kirill Kravchenko

Additional information is available at the end of the chapter

