**4. Properties of the graphitic wormhole**

The graphs of LDOS based on the found solution are sketched in **Figure 12** for the same modes and numbers of the defects as in **Figure 11**, i.e., −1≤ *j* ≤3. In spite of our expectations, this time the behavior of the found result is the same for arbitrary number of the defects, i.e., the

**Figure 12.** Graphs of LDOS of the graphitic nanocone influenced by the Coulomb interaction (including the influence

Now, we would like to verify the possible quadratic integrability of the solution found for the case of the additional effect coming from the charge simulation. In **Figure 13**, we see the

only and of the simultaneous influence of SOC and the Coulomb interaction. We see here that in comparison with the first case, in the second case, the decrease of LDOS close to *r* = 0 is much faster and one could suppose that the quadratic integrability of the acquired solution is achieved here. To gain confidence with our conclusion, we have to do the integration of LDOS in the investigated interval close to *r* = 0 in all the outlined cases. This task is still in progress.

**Figure 13.** Behavior of LDOS for zero energy close to *r* = 0 for different numbers of defects in the conical tip: influence of

SOC only (left) and the simultaneous influence of SOC and the Coulomb interaction (right).

→ from the tip, −1 ≤ *j* ≤ 3 and for different numbers of the defects.

<sup>→</sup> variable close to zero energy for the case of the influence of SOC

of SOC) for different distances *r*

44 Recent Advances in Graphene Research

dependence of LDOS on *r*

**3.3. Comparison of the results**

appearance and the uniqueness of the peak in the case of three defects are distorted.

The wormhole is understood as a form that arises when two graphene sheets are connected together with the help of the connecting nanotube. This can be achieved by the supply of the heptagonal defects on both sides of the given nanotube. The number of the defects can vary from 1 to 12. The composition of the graphitic wormhole is depicted in **Figure 14**: it consists of the connecting nanotube and two (perturbed or unperturbed) graphene sheets. The places of the connections are called the wormhole bridges. Because of the physical limitations, the radius of the nanotube must be much larger than its length (this fact is ignored in **Figure 14** for the better illustration of the composition). The limit case of 12 defects is described in references [4, 17], in the other cases, we speak about the so-called "perturbed wormhole" Here, using the formalism of the subsection II B, we derive the electronic structure for both cases, and we will find out the form of the zero modes on the wormhole bridge. Furthermore, we investigate the influence of the additional effects that could appear here due to extreme curvature in the place of the wormhole bridge—the relativistic mass acquisition of the present electrons. This effect together with the effect of SOC that appears in the carbon nanotubes [16] could lead to the appearance of the zero modes of the chiral massive electrons in the place of the wormhole bridge. This could serve as a useful instrument for the detection of the worm‐ hole structures in the graphene bilayer during the process of the synthesis of the correspond‐ ing material.

**Figure 14.** The composition of the graphitic wormhole.

#### **4.1. Electronic structure**

We will solve Eq. (15) in the subsection II B. In this case, the metric tensor has the form

$$\mathbf{g}\_{\mu\nu} = \Lambda^2(r\_{\pm}) \begin{pmatrix} 1 & 0 \\ 0 & r\_{\pm}^2 \end{pmatrix}, \qquad \Lambda(r\_{\pm}) = \left( a \wedge r\_{\pm} \right)^2 \theta(a - r\_{\pm}) + \theta(r\_{\pm} - a). \tag{33}$$

Here, *θ* is the Heaviside step function, *r*−, *r*<sup>+</sup> are the polar coordinates corresponding to the lower and the upper graphene sheet, respectively and *a* = *r*−*r*+ is the radius of the wormhole.

The values of the components of *aμ* depend on the chiral vector [18] of the connecting nanotube. For our purpose, this vector is (6*n*, 6*n*) and (6*n*, 0). In most cases, *aμ* has then the components

$$a\_{\phi} = \frac{3}{2}, \qquad a\_r = 0. \tag{34}$$

The only exception is when the chiral vector is (6*n*, 0), where *n* is not divisible by 3. Then,

$$a\_{\phi} = \frac{1}{2}, \qquad a\_r = 0. \tag{35}$$

Knowledge of the spin connection is also needed—the values of the components have the form

$$
\Omega\_{\phi} = -\frac{\mathrm{i}}{2} \sigma\_3 \left( r \frac{\Lambda'(r)}{\Lambda(r)} + \mathrm{l} \right), \qquad \Omega\_r = 0. \tag{36}
$$

All these expressions we substitute into Eq. (15). The resulting equation is

$$\text{div}\_{\boldsymbol{F}} \boldsymbol{\sigma}^{\mu} (\boldsymbol{\hat{\boldsymbol{\sigma}}}\_{\mu} + \boldsymbol{\Omega}\_{\mu} \mp \text{i}\boldsymbol{a}\_{\mu}) \boldsymbol{\upmu}^{\pm} = \boldsymbol{\varepsilon} \boldsymbol{\upmu}^{\pm}. \tag{37}$$

**Figure 15.** Local density of states on the bridge of the graphitic wormhole for different values of *aφ*.

Here, each sign ± corresponds to a different Dirac point (the corner ofthe reciprocal unitlattice). We get these four possibilities: for *r* ≥*a*,

Electronic Properties of Carbon Nanostructures http://dx.doi.org/10.5772/63633 47

$$-\mathrm{i}\nu\_{F}\left(\left.\partial\_{r}+\frac{1}{r}\mathrm{i}\partial\_{\boldsymbol{\theta}}\mp\frac{a\_{\boldsymbol{\theta}}}{r}+\frac{1}{2r}\right|\boldsymbol{\nu}\_{\boldsymbol{\theta}}^{\pm}=\varepsilon\boldsymbol{\nu}\_{\boldsymbol{\varrho}}^{\pm},\qquad-\mathrm{i}\nu\_{F}\left(\left.\partial\_{r}-\frac{1}{r}\mathrm{i}\partial\_{\boldsymbol{\theta}}\pm\frac{a\_{\boldsymbol{\theta}}}{r}+\frac{1}{2r}\right|\boldsymbol{\nu}\_{\boldsymbol{\varrho}}^{\pm}=\varepsilon\boldsymbol{\nu}\_{\boldsymbol{\varrho}}^{\pm}\right.\tag{38}$$

and for 0<*r* ≤*a*,

( )

Here, *θ* is the Heaviside step function, *r*−, *r*<sup>+</sup> are the polar coordinates corresponding to the lower and the upper graphene sheet, respectively and *a* = *r*−*r*+ is the radius of the wormhole.

The values of the components of *aμ* depend on the chiral vector [18] of the connecting nanotube. For our purpose, this vector is (6*n*, 6*n*) and (6*n*, 0). In most cases, *aμ* has then the components

<sup>3</sup> , 0.

The only exception is when the chiral vector is (6*n*, 0), where *n* is not divisible by 3. Then,

<sup>1</sup> , 0. <sup>2</sup> *<sup>r</sup> a a*

Knowledge of the spin connection is also needed—the values of the components have the form

i () 1 , 0. 2 () *<sup>r</sup>*

 y ey

<sup>2</sup> *<sup>r</sup> a a*

j

j

3

All these expressions we substitute into Eq. (15). The resulting equation is

s

m s

j *r r r*

æ ö L¢ W =- + W = ç ÷

i ( i) . *<sup>F</sup> v a*

mmm

**Figure 15.** Local density of states on the bridge of the graphitic wormhole for different values of *aφ*.

We get these four possibilities: for *r* ≥*a*,

Here, each sign ± corresponds to a different Dirac point (the corner ofthe reciprocal unitlattice).

q

 q

= = (34)

= = (35)

è ø <sup>L</sup> (36)

± ± ¶ +W = m (37)

è ø (33)

( ) , ( ) / ( ) ( ). <sup>0</sup> *g r r ar a r r a <sup>r</sup>*

æ ö = L ç ÷ L = -+ -

± ± ± ±±

2 2

2 1 0

±

mn

46 Recent Advances in Graphene Research

$$\mathrm{i}\mathbf{v}\_{F}\left(\frac{r}{a}\right)^{2}\left(\boldsymbol{\mathcal{O}}\_{r}-\frac{1}{r}\mathrm{i}\boldsymbol{\mathcal{O}}\_{\boldsymbol{\theta}}\pm\frac{a\_{\boldsymbol{\theta}}}{r}-\frac{1}{2r}\right)\boldsymbol{\nu}\_{\boldsymbol{\theta}}^{\pm}=\mathrm{e}\boldsymbol{\nu}\_{\boldsymbol{\varrho}}^{\pm},\qquad\mathrm{i}\mathbf{v}\_{F}\left(\frac{r}{a}\right)^{2}\left(\boldsymbol{\mathcal{O}}\_{r}+\frac{1}{r}\mathrm{i}\boldsymbol{\mathcal{O}}\_{\boldsymbol{\theta}}\mp\frac{a\_{\boldsymbol{\theta}}}{r}-\frac{1}{2r}\right)\boldsymbol{\nu}\_{\boldsymbol{\varrho}}^{\pm}=\mathrm{e}\boldsymbol{\nu}\_{\boldsymbol{\uprho}}^{\pm}.\tag{39}$$

In the first case, the solution is

$$\begin{split} \boldsymbol{\Psi}^{\pm} = \begin{pmatrix} \boldsymbol{\Psi}\_{\boldsymbol{A}}^{\pm}(\boldsymbol{r}, \boldsymbol{\varrho}) \\ \boldsymbol{\Psi}\_{\boldsymbol{B}}^{\pm}(\boldsymbol{r}, \boldsymbol{\varrho}) \end{pmatrix} = \boldsymbol{c}\_{1} \begin{pmatrix} \boldsymbol{J}\_{/\mp \boldsymbol{a}\_{\boldsymbol{\varrho}} - 1/2}(kr) \\ -\text{sign}\boldsymbol{\varepsilon} \, \boldsymbol{J}\_{/\mp \boldsymbol{a}\_{\boldsymbol{\varrho}} + 1/2}(kr) \end{pmatrix} \\ + \boldsymbol{c}\_{2} \begin{pmatrix} \boldsymbol{Y}\_{/\mp \boldsymbol{a}\_{\boldsymbol{\varrho}} - 1/2}(kr) \\ -\text{sign}\boldsymbol{\varepsilon} \, \boldsymbol{Y}\_{/\mp \boldsymbol{a}\_{\boldsymbol{\varrho}} + 1/2}(kr) \end{pmatrix}. \end{split} \tag{40}$$

Here, *Jj* (*x*) and *Yj* (*x*) are the Bessel functions of the integer order *j* and the energy *ε* = ± *vF k*. To calculate LDOS, similarly as in the previous section, Eq. (22) is used. In **Figure 15**, different behavior of LDOS, depending on the gauge field *aφ*, is manifested.

#### **4.2. Zero modes**

Forthe presented possibilities, we investigate the zero modes—solutions of the Dirac equation for the zero energy. For this purpose, we consider zero values of the component *ψ<sup>A</sup>* <sup>±</sup> of the solution. Then, from Eqs. (49) and (50) follows: for *r* ≥*a*,

**Figure 16.** Comparison of the properties of the wormhole and the plain graphene: (a) local density of states, (b) zero modes.

$$\left(\stackrel{\circ}{\mathcal{O}}\_r - \frac{1}{r} \mathrm{i}\mathcal{O}\_\theta \mp \frac{a\_\phi}{r} + \frac{1}{2r}\right) \nu\_\text{\tiny.}^\pm = 0\tag{41}$$

and for 0<*r* ≤*a*,

$$\left(\partial\_r - \frac{1}{r} \mathrm{i}\mathcal{O}\_\theta \pm \frac{a\_\phi}{r} - \frac{1}{2r}\right) \nu\_\text{B}^\pm = 0. \tag{42}$$

If *a<sup>φ</sup>* <sup>=</sup> <sup>3</sup> <sup>2</sup> and *r* ≥*a*, the solution is

$$
\psi\_{\mathcal{B}}^{-}(r,\phi) \sim r^{-j-2}e^{\mathrm{i}\cdot\phi}.\tag{43}
$$

The second possibility for this value of *aφ* is 0<*r* ≤*a*, the corresponding solution is then

$$
\psi\_B^{-}(r,\varphi) \sim r^{-j+2}e^{\iota\_j\varphi}.\tag{44}
$$

Both solutions are strictly normalizable only for *j* = 0. Analogous solution holds for *ψ<sup>B</sup>* + and for *ψA* ± if the components *ψ<sup>B</sup>* ± are chosen as zero.

There are no strictly normalizable solutions for the value *a<sup>φ</sup>* <sup>=</sup> <sup>1</sup> <sup>2</sup> . It means that in this case, the zero modes do not exist.

On the base of these results, one could expect a strong localization of LDOS near Fermi energy on the wormhole bridge. It is demonstrated in **Figure 16a**, where LDOS of the plain gra‐ phene is supplied for the comparison. It could be experimentally observed. In **Figure 16b**, we see the comparison of the zero modes of these two structures at different distances from the wormhole bridge.

#### **4.3. Case of massive fermions**

In the continuum gauge field theory, zero mass of the fermions in the Dirac equation is considered (in other words, it is very small in comparison with energy). On the other hand, the extreme curvature of the investigated structure leads to such values of the Fermi velocity that cause the appearance of the relativistic effects. The changes of the Fermi velocity due to curvature and other effects were demonstrated in references [13, 19]. As a result, the mass of the fermions becomes considerable, similarly as in the bilayer graphene [20, 21]. This effect is strengthened by the effective mass acquisition during the motion along the tube axis that happens due to the extreme size difference between the graphene sheets and the wormhole radius. This change of the space topology of graphene from 2D to 1D is similar to the string theory compactification. It means that we can image the wormhole connecting nanotube as the 1D object.

So we need to incorporate a mass term into the Dirac equation (Eq. (15)). To solve this problem, we go through the system of the corresponding equations (Eq. (20)) and transform it into the following differential equation of the second order:

$$
\left(\partial\_{\xi\xi} - \frac{1}{2\operatorname{g}\_{\xi\xi}}\partial\_{\xi}\operatorname{g}\_{\xi\xi} + \frac{\tilde{j}}{2}\sqrt{\frac{\operatorname{g}\_{\xi\xi}}{\operatorname{g}\_{\phi\phi}^{3}}}\partial\_{\xi}\operatorname{g}\_{\phi\phi} - \tilde{j}^{2}\frac{\operatorname{g}\_{\xi\xi}}{\operatorname{g}\_{\phi\phi}} + E^{2}\operatorname{g}\_{\xi\xi}\right)u\_{j} = 0. \tag{45}$$

To simplify the calculations, the cylindrical geometry is supposed: the radius vector of the point at the surface changes as

Electronic Properties of Carbon Nanostructures http://dx.doi.org/10.5772/63633 49

$$\vec{R} = (R\cos\varphi, R\sin\varphi, \xi),\tag{46}$$

where *R* is the radius of the cylinder. In this case, Eq. (45) is considerably simplified:

$$
\left(\partial\_{\lesssim \mp} + E^2 - \frac{\tilde{j}^2}{R^2}\right)\mu\_j = 0,\tag{47}
$$

which is solved by [22]

$$
\mu\_{/}(\xi) = A e^{k\xi} + B e^{-k\xi}.\tag{48}
$$

Here,

2 i ( , )~ . *j j <sup>B</sup> r re*

The second possibility for this value of *aφ* is 0<*r* ≤*a*, the corresponding solution is then

Both solutions are strictly normalizable only for *j* = 0. Analogous solution holds for *ψ<sup>B</sup>*

On the base of these results, one could expect a strong localization of LDOS near Fermi energy on the wormhole bridge. It is demonstrated in **Figure 16a**, where LDOS of the plain gra‐ phene is supplied for the comparison. It could be experimentally observed. In **Figure 16b**, we see the comparison of the zero modes of these two structures at different distances from the

In the continuum gauge field theory, zero mass of the fermions in the Dirac equation is considered (in other words, it is very small in comparison with energy). On the other hand, the extreme curvature of the investigated structure leads to such values of the Fermi velocity that cause the appearance of the relativistic effects. The changes of the Fermi velocity due to curvature and other effects were demonstrated in references [13, 19]. As a result, the mass of the fermions becomes considerable, similarly as in the bilayer graphene [20, 21]. This effect is strengthened by the effective mass acquisition during the motion along the tube axis that happens due to the extreme size difference between the graphene sheets and the wormhole radius. This change of the space topology of graphene from 2D to 1D is similar to the string theory compactification. It means that we can image the wormhole connecting nanotube as

So we need to incorporate a mass term into the Dirac equation (Eq. (15)). To solve this problem, we go through the system of the corresponding equations (Eq. (20)) and transform it into the

> <sup>1</sup> 0. 2 2 *<sup>j</sup> j g g g g j Eg u*

To simplify the calculations, the cylindrical geometry is supposed: the radius vector of the

3

æ ö ç ÷ ¶- ¶ + ¶ - + = è ø

jj x jj

*g gg* xx

2 2

xx

% % (45)

xx

jj

2 i ( , )~ . *j j <sup>B</sup> r re*

y j- - -

y j- - +

are chosen as zero.

There are no strictly normalizable solutions for the value *a<sup>φ</sup>* <sup>=</sup> <sup>1</sup>

*ψA* ±

if the components *ψ<sup>B</sup>*

48 Recent Advances in Graphene Research

zero modes do not exist.

wormhole bridge.

the 1D object.

**4.3. Case of massive fermions**

±

following differential equation of the second order:

xx x xx

xx

point at the surface changes as

j

j (43)

(44)

+ and for

<sup>2</sup> . It means that in this case, the

$$k = \sqrt{\frac{\tilde{j}^2}{R^2} - E^2}.\tag{49}$$

In references [23, 24], in a very similar form the dispersion relation is given for the massive 1D Dirac equation:

$$k = \sqrt{M^2 - E^2},\tag{50}$$

where *M* is the mass of the corresponding fermion. From reference [22], an analogy indeed follows between the 2D massless and 1D massive case. On this base, we rewrite Eq. (45) into the form

$$
\left(\partial\_{\ddagger\xi} - \frac{1}{2\mathbf{g}\_{\ddagger\xi}}\partial\_{\xi}\mathbf{g}\_{\ddagger\xi} + \frac{\tilde{j}}{2}\sqrt{\frac{\mathbf{g}\_{\ddagger\xi}}{\mathbf{g}\_{\ddagger\wp}^{3}}}\partial\_{\xi}\mathbf{g}\_{\phi\wp} - \tilde{j}^{2}\frac{\mathbf{g}\_{\ddagger\xi}}{\mathbf{g}\_{\phi\wp}} + (E^{2} - M^{2})\mathbf{g}\_{\ddagger\xi}\right)\boldsymbol{\mu}\_{\/} = \mathbf{0},\tag{51}
$$

in this case, the mass *M* corresponds to the fermion in the altered conditions. Now we find the corrections of LDOS of the graphitic wormhole for different values of *M*. It is shown in **Figure 17**. Our prediction is that these massive particles arising in the wormhole nanotubes could create energy bulks on the wormhole bridge and in the close area that should be experimentally measured by the STM or by the Raman spectroscopy [25]. Moreover, this effect could be strengthened by the effect of SOC present in the connecting nanotube that was described in reference [16] for the nanotubes and in Section 3 for the nanocone. This effect causes next energy splitting and as the result, the aforementioned chiral massive electrons could appear.

Another possibility to identify the wormhole structure comes from the fact that the massive particles could create strain solitons and topological defects on the bridge of the bilayer graphene that should propagate throughout the graphene sheet. These are almost macroscop‐ ic effects and should be caught by the experimentalists [26].

**Figure 17.** Comparison of LDOS for different masses of fermions at different distances from the wormhole bridge.

#### **4.4. Case of perturbed wormhole**

Now we will investigate how the electronic structure changes if the number of the heptago‐ nal defects on the wormhole bridge is lowered—in this way, the perturbed wormhole is created.

In **Figure 18**, the possible forms of this structure are depicted. Due to symmetry preserva‐ tion, only the even numbers of the defects, i.e., 2, 4, 6, 8, or 10, are considered.

The metric of the sheets can be draught by the radius vector

$$\vec{R}(z,\varphi) = \left(a\sqrt{1+\omega z^2}\cos\varphi, a\sqrt{1+\omega z^2}\sin\varphi, z\right),\tag{52}$$

where △ is a positive real parameter; its value is derived from the number of the defects of the wormhole. In the case of *N* = 2 defects, we can say that the value of this parameter is negligi‐ ble, so △ < <1. Then, the nonzero components of the metric are

$$\mathbf{g}\_{zz} = \mathbf{l} + \frac{a^2 \omega^2 z^2}{\mathbf{l} + \omega z^2} \sim \mathbf{l} + a^2 \omega^2 z^2, \qquad \mathbf{g}\_{\varphi\varphi} = a^2 (\mathbf{l} + \omega z^2). \tag{53}$$

The nonzero components of the gauge fields are

Another possibility to identify the wormhole structure comes from the fact that the massive particles could create strain solitons and topological defects on the bridge of the bilayer graphene that should propagate throughout the graphene sheet. These are almost macroscop‐

**Figure 17.** Comparison of LDOS for different masses of fermions at different distances from the wormhole bridge.

Now we will investigate how the electronic structure changes if the number of the heptago‐ nal defects on the wormhole bridge is lowered—in this way, the perturbed wormhole is

In **Figure 18**, the possible forms of this structure are depicted. Due to symmetry preserva‐

tion, only the even numbers of the defects, i.e., 2, 4, 6, 8, or 10, are considered.

( ) 2 2 *Rz a z a z z* ( , ) 1 cos , 1 sin , ,

jj

=+ + <sup>r</sup> V V (52)

The metric of the sheets can be draught by the radius vector

j

**4.4. Case of perturbed wormhole**

created.

ic effects and should be caught by the experimentalists [26].

50 Recent Advances in Graphene Research

$$a\_{\boldsymbol{\phi}} = N \mid \mathbf{4}, \qquad a\_{\boldsymbol{\phi}}^{\boldsymbol{\eta}} = -(2m+n) \nmid \mathbf{3}, \tag{54}$$

**Figure 18.** Different forms of the perturbed wormhole: (a) Two defects, (b) Four defects, (c) Six defects, (d) Eight de‐ fects, (e) Ten defects.

where (*n, m*) is the chiral vector of the connecting nanostructure. Then, regarding the form of the spin connection and by the substitution into Eq. (15), we get the solution

$$\Psi\_{\ \perp}(z) = C\_{\perp} D\_{\nu\_{\parallel}}(\xi(z))e^{\psi \rho} + C\_{\perp 2} D\_{\nu\_{\perp}}(i\xi(z)))e^{\psi \rho},\tag{55}$$

$$\begin{split} \boldsymbol{\Psi}\_{s}(\boldsymbol{z}) &= \frac{\boldsymbol{C}\_{\boldsymbol{\omega}}}{E} \Big( \hat{\boldsymbol{\partial}}\_{\boldsymbol{z}} \boldsymbol{D}\_{\boldsymbol{\nu}\_{1}}(\boldsymbol{\xi}(\boldsymbol{z})) - \frac{\tilde{\boldsymbol{J}} \boldsymbol{D}\_{\boldsymbol{\nu}\_{1}}(\boldsymbol{\xi}(\boldsymbol{z}))}{a} (\boldsymbol{1} - \frac{1}{2} \boldsymbol{\omega}^{2} \boldsymbol{z}^{2}) \Big) \boldsymbol{e}^{-\boldsymbol{\psi}\boldsymbol{\rho}} \\ &+ \frac{\boldsymbol{C}\_{\boldsymbol{\omega}}}{E} \Big( \hat{\boldsymbol{\partial}}\_{\boldsymbol{z}} \boldsymbol{D}\_{\boldsymbol{\nu}\_{1}}(\boldsymbol{i}\_{\boldsymbol{\nu}}^{\boldsymbol{\varepsilon}}(\boldsymbol{z})) - \frac{\tilde{\boldsymbol{J}} \boldsymbol{D}\_{\boldsymbol{\nu}\_{1}}(\boldsymbol{i}\_{\boldsymbol{\nu}}^{\boldsymbol{\varepsilon}}(\boldsymbol{z}))}{a} (\boldsymbol{1} - \frac{1}{2} \boldsymbol{\omega}^{2} \boldsymbol{z}^{2}) \Big) \boldsymbol{e}^{-\boldsymbol{\psi}\boldsymbol{\rho}}, \end{split} \tag{56}$$

where

$$\nu\_1 = i \frac{a^2 \Delta - 4a^2 E^2 + 4ia \sqrt{\Delta} \tilde{j} + 4 \tilde{j}^2}{8a \sqrt{\Delta} \tilde{j}}, \qquad \nu\_2 = -i \frac{a^2 \Delta - 4a^2 E^2 - 4ia \sqrt{\Delta} \tilde{j} + 4 \tilde{j}^2}{8a \sqrt{\Delta} \tilde{j}},\tag{57}$$

$$
\zeta(z) = (-\omega)^{1/4} \left( \sqrt{\frac{a}{2\tilde{j}}} + \sqrt{\frac{2\tilde{j}}{a}} z \right),
\tag{58}
$$

*Dν*(*ξ*) being the parabolic cylinder function. The functions *C*△<sup>1</sup> =*C*△1(*E*), *C*△<sup>2</sup> =*C*△2(*E*) serve as the normalization constants. We see the graph of the local density of states in **Figure 19** (left part).

In the case of more than two defects, the value of Δ is not negligible, and we can get only the numerical approximation of LDOS. The derivation of the value of the parameter Δ follows from **Figure 19** (right part). From this figure it follows that in the middle part, the upper branch of the graphene sheet converges to the line *z* = *x* ⋅ tan*α*, where we can suppose that the angle *α* depends on the number of the defects *N* linearly, i.e., *<sup>α</sup>* <sup>=</sup> *<sup>π</sup>* <sup>2</sup> <sup>−</sup> *<sup>N</sup>* <sup>⋅</sup> *<sup>π</sup>* <sup>24</sup> . (In this case, *<sup>α</sup>* <sup>=</sup> *<sup>π</sup>* <sup>2</sup> corre‐ sponds to 0 defects and *α* =0 corresponds to 12 defects.) Simultaneously, from Eq. (52) follows that asymptotically we have

$$\vec{R}(z \to \infty, \varphi) \to \left(a\sqrt{\omega z \cos \varphi}, a\sqrt{\omega z \sin \varphi}, z\right), \tag{59}$$

**Figure 19.** Left: LDOS on the bridge of the graphitic perturbed wormhole. Right: Derivation of the Δ parameter.

from which follows

$$z = x \cdot \tan \alpha = \left( a \sqrt{\Delta} \right)^{-1} x,\tag{60}$$

so

#### Electronic Properties of Carbon Nanostructures http://dx.doi.org/10.5772/63633 53

$$\Delta = \frac{1}{a^2 \tan^2 \alpha} = \frac{1}{a^2 \tan^2 \left(\frac{\pi}{2} - N \cdot \frac{\pi}{24}\right)}.\tag{61}$$

In **Figure 20**, we see the comparison of LDOS for different kinds of the perturbed wormhole. From the plots it follows that the intensity is rising with the increasing number of the defects, and it is closer and closer approaching the results in **Figure 15**, where the case of 12 defects is shown.

where

part).

n

52 Recent Advances in Graphene Research

that asymptotically we have

from which follows

so

2 22 2 2 22 2

 n

1/4 <sup>2</sup> () ( ) , <sup>2</sup> *a j z z j a*


44 4 44 4 , , 8 8 *a a E ia j j a a E ia j j i i a j a j*

> æ ö =- + ç ÷

> > è ø

*Dν*(*ξ*) being the parabolic cylinder function. The functions *C*△<sup>1</sup> =*C*△1(*E*), *C*△<sup>2</sup> =*C*△2(*E*) serve as the normalization constants. We see the graph of the local density of states in **Figure 19** (left

In the case of more than two defects, the value of Δ is not negligible, and we can get only the numerical approximation of LDOS. The derivation of the value of the parameter Δ follows from **Figure 19** (right part). From this figure it follows that in the middle part, the upper branch of the graphene sheet converges to the line *z* = *x* ⋅ tan*α*, where we can suppose that the angle *α*

sponds to 0 defects and *α* =0 corresponds to 12 defects.) Simultaneously, from Eq. (52) follows

jj

( ) <sup>r</sup> V V (59)

*Rz a z a z z* ( , ) cos , sin , , ®¥ ®

**Figure 19.** Left: LDOS on the bridge of the graphitic perturbed wormhole. Right: Derivation of the Δ parameter.

*zx a x* tana

( ) <sup>1</sup>

, -

V V % % (57)

% <sup>V</sup> % (58)

<sup>2</sup> <sup>−</sup> *<sup>N</sup>* <sup>⋅</sup> *<sup>π</sup>*

=× = V (60)

<sup>24</sup> . (In this case, *<sup>α</sup>* <sup>=</sup> *<sup>π</sup>*

<sup>2</sup> corre‐

1 2

x

depends on the number of the defects *N* linearly, i.e., *<sup>α</sup>* <sup>=</sup> *<sup>π</sup>*

j

**Figure 20.** Comparison of LDOS for different numbers of the defects in the perturbed wormhole at different distances *d* from the wormhole bridge.

**Figure 21.** Zero modes of the perturbed wormhole for different numbers of the defects.

In **Figure 21**, LDOS of zero modes is shown for a varying distance from the wormhole bridge in the units of the radius *a* of the wormhole center. It was also acquired in the numerical way. For the unperturbed case (0 defects), the resulting plot resembles a line. In reference [27], the exponential solution is found for this case but with a very slow increase, so this could be that case. It is also seen from the plot that for the increasing number of the defects, the solution is approaching expressions in Eqs. (54) and (55)forthe zero modes ofthe unperturbed wormhole.

Of course, the massive fermions could also appear in the case of the perturbed wormhole. We will not perform a detailed derivation of the electronic structure for the case of this eventual‐ ity, and we only note that the corrections to LDOS would be an analogy of the corrections shown in **Figure 17**.
