**2. Computational formalism**

The electronic structure can be characterized by the density of states (DOS)—the number of the electronic states per the unit interval of energies. This quantity can be used as the measure of the density of the electrons and generally we can say that the higher value of DOS, the higher conductivity. With the help of DOS, the electric field can be calculated as well. Besides DOS, one more quantity is defined—the local density of states (LDOS). It is DOS related to the unit area of the molecular surface or to the unit area of the surface in the space of the wave vector *k* <sup>→</sup> . Then, the quantities depend on the variables as follows:

$$LDOS = DOS(E), \quad LDOS = LDOS(E, \vec{r}), \quad \text{resp.} \quad LDOS = LDOS(E, \vec{k}). \tag{1}$$

Two methods are used for the calculation of LDOS. The first one is used for the periodical structures with the planar geometry (plain graphene, nanotubes, nanoribbons, etc.), the second one is used for the structures that are aperiodical or that have the curved geometry (fullerene, nanocone, wormhole, nanotoroid, etc.). We outline here the base of these methods. Both the methods start on solving the Schrödinger equation for the electron bounded on the molecular surface

$$
\hat{H}\psi = E\psi.\tag{2}
$$

Here,

$$
\hat{H} = \frac{\hat{k}^2}{2m} + \hat{V}(\vec{r}) + \hat{U}(\vec{r}),
\tag{3}
$$

*V* ^ representing the potential of the periodic crystal, *<sup>U</sup>* ^ representing the external potential that is responsible for the curvature.

#### **2.1. Periodical structures with planar geometry**

For the periodical structures, the external potential in Eq. (3) is zero, and the carbon lattice can be divided into several sublattices, each containing equivalent atomic sites. We can denote the sites corresponding to the different sublattices as *A*, *B*, *C*, ... or *A*1, *A*2, *A*3, .... We can find a unit cell—the smallest possible cell in the structure that contains all possible inequivalent atomic sites (**Figure 6**).

In the case of graphene, the wave function, which solves Eq. (2), can be expressed as in references [7, 8]

$$
\bar{\nu} = C\_A \bar{\nu}\_A + C\_B \bar{\nu}\_B,\tag{4}
$$

where the components *ψA*, *ψB* correspond to the particular sublattices. In the tight-binding approximation, we postulate the solution in the form

$$\|\boldsymbol{\nu}\|\_{\mathcal{A}(\mathbb{B})} = \sum\_{\boldsymbol{A}(\mathbb{B})} \exp[\mathbf{i}\vec{k}\cdot\vec{r}\_{\boldsymbol{A}(\mathbb{B})}] X(\vec{r}\cdot\vec{r}\_{\boldsymbol{A}(\mathbb{B})}),\tag{5}$$

where *X* (*r* <sup>→</sup> ) is the atomic orbital function. The overlap is zero, i.e.,

$$\int X^\*(\vec{r} \cdot \vec{r}\_A) X(\vec{r} \cdot \vec{r}\_B) d\vec{r} = 0. \tag{6}$$

By the substitution of the solution in Eq. (5) into the Schrödinger equation (Eq. (2)), multiplying it by *ψ*† and making the integration over *r* <sup>→</sup> , we create the expressions

of the density of the electrons and generally we can say that the higher value of DOS, the higher conductivity. With the help of DOS, the electric field can be calculated as well. Besides DOS, one more quantity is defined—the local density of states (LDOS). It is DOS related to the unit area of the molecular surface or to the unit area of the surface in the space of the wave vector

Two methods are used for the calculation of LDOS. The first one is used for the periodical structures with the planar geometry (plain graphene, nanotubes, nanoribbons, etc.), the second one is used for the structures that are aperiodical or that have the curved geometry (fullerene, nanocone, wormhole, nanotoroid, etc.). We outline here the base of these methods. Both the methods start on solving the Schrödinger equation for the electron bounded on the molecular

For the periodical structures, the external potential in Eq. (3) is zero, and the carbon lattice can be divided into several sublattices, each containing equivalent atomic sites. We can denote the sites corresponding to the different sublattices as *A*, *B*, *C*, ... or *A*1, *A*2, *A*3, .... We can find a unit cell—the smallest possible cell in the structure that contains all possible inequivalent atomic

In the case of graphene, the wave function, which solves Eq. (2), can be expressed as in

 y

where the components *ψA*, *ψB* correspond to the particular sublattices. In the tight-binding

r r rr

exp[i ] *A(B) A(B) A(B)*

= × å *k r X(r - r ),*

yy

*A(B)*

<sup>→</sup> ) is the atomic orbital function. The overlap is zero, i.e.,

,

ˆ*H E* y y

ˆ2 ˆ ˆˆ 2

*DOS DOS(E), LDOS = LDOS(E,r), LDOS = LDOS(E,k).* <sup>=</sup> resp. <sup>r</sup> <sup>r</sup> (1)

<sup>=</sup> . (2)

^ representing the external potential that

*<sup>k</sup> H V(r)+U(r), <sup>m</sup>* = + r r (3)

= + *C C AA BB* (4)

(5)

<sup>→</sup> . Then, the quantities depend on the variables as follows:

^ representing the potential of the periodic crystal, *<sup>U</sup>*

approximation, we postulate the solution in the form

y

**2.1. Periodical structures with planar geometry**

is responsible for the curvature.

*k*

34 Recent Advances in Graphene Research

surface

Here,

*V*

sites (**Figure 6**).

references [7, 8]

where *X* (*r*

**Figure 6.** Unit cells for different periodical structures: graphene (left), different kinds of nanoribbons (right). The gra‐ phene structure is considered to be infinite in 2D, the nanoribbons are considered to have final width and the second size is infinite as well.

$$H\_{ab} = \int \boldsymbol{\upmu}\_a^\* H \boldsymbol{\upmu}\_b \, \mathrm{d}\vec{r}, \quad \mathrm{S} = \int \boldsymbol{\upmu}\_A^\* \boldsymbol{\upmu}\_A \, \mathrm{d}\vec{r} = \int \boldsymbol{\upmu}\_B^\* \boldsymbol{\upmu}\_B \, \mathrm{d}\vec{r}, \quad a, b \equiv A, B. \tag{7}$$

If we suppose that the functions *X* are normalized so that *∫X* \* (*r* <sup>→</sup> −*r* → *A*(*B*) )*X* (*r* <sup>→</sup> −*r* → *A*(*B*) )d*r* <sup>→</sup> =1, then *S* gives the number of the unit cells in the nanostructure. Now, the Schrödinger equation is transformed into the matrix form

$$
\begin{pmatrix} H\_{\mathcal{A}4} & H\_{\mathcal{A}8} \\ H\_{\mathcal{A}4} & H\_{\mathcal{B}8} \end{pmatrix} \begin{pmatrix} C\_{\mathcal{A}} \\ C\_{\mathcal{B}} \end{pmatrix} = ES \begin{pmatrix} C\_{\mathcal{A}} \\ C\_{\mathcal{B}} \end{pmatrix}. \tag{8}
$$

The eigenvalues of the matrix in this equation are the energy eigenvalues, and they create the electronic spectrum. For this purpose, first, we need to determine the values of the matrix elements *Hab*. From their definition follows

$$\begin{split} \mathbf{H}\_{ab} &= \left[ \boldsymbol{\upmu}^{\*}\_{a} H \boldsymbol{\upmu}\_{b} \mathbf{d} \vec{r} = \sum\_{a,b} \exp\left[-\mathbf{i} \,\vec{\mathbf{k}} \cdot \left( \vec{r}\_{\vec{a}} \cdot \vec{r}\_{\vec{a}} \right) \right] \right] X^{\*} \left( \vec{r} \cdot \vec{r}\_{a} \right) H X \left( \vec{r} \cdot \vec{r}\_{b} \right) \mathbf{d} \vec{r} \\ &= \sum\_{a,b} \exp\left[-\mathbf{i} \,\vec{\mathbf{k}} \cdot \left( \vec{r}\_{\vec{a}} \cdot \vec{r}\_{\vec{a}} \right) \right] \boldsymbol{\upgamma}\_{a}, \end{split} \tag{9}$$

where *γab* ab denotes the corresponding hopping integral. The eigenvalues are labeled by *k* → and in the nearest-neighbor approximation, they can be expressed as

$$E(\vec{k}) = \pm \gamma\_o \sqrt{I + 4\cos^2\frac{k\_\flat a}{2} + 4\cos\frac{k\_\flat a}{2}\cos\frac{k\_\flat a\sqrt{3}}{2}},\tag{10}$$

where *γ*<sup>0</sup> =*γAA* =*γBB* is the hopping integral for the nearest neighboring atoms and *a* =2.46*A* is the distance between the next-nearest atomic neighbors. DOS and LDOS are then defined as

$$DOS(E) = \frac{I}{\pi} \lim\_{a \to 0} \text{Im} \int \frac{d\vec{k}}{E \cdot E(\vec{k}) + \text{i.o.}}, \qquad LDOS(E, \vec{k}) = \mathcal{S}(E \cdot E(\vec{k})).\tag{11}$$

The corresponding graphs of electronic spectrum and DOS are given in **Figure 7**. We see that for zero energy, the density of states has zero value. This property is typical for the semimetallic nanostructures. For the metallic nanostructures, a peak appears for zero energy. In the first case, a gap is present around zero in the electronic spectrum. Its width can be influenced by the additional defects in the hexagonal structure or by the chemical admixtures and in this way, a material with the predefined properties can be synthesized. In the second case, the gap around zero energy is absent.

**Figure 7.** The electronic spectrum for *ky* = 0 (left) and *k*x = 0 (middle) and the density of states (right) of graphene. The solid and dashed lines correspond to the positive and the negative energy values, respectively. In the bottom, we see the form of the electronic spectrum for an arbitrary value of the wave vector.

In a similar way, but with a more complicated structure of the wave function in Eq. (4) and for a larger size of the matrix in Eq. (8), the electronic spectrum and DOS can be found for other nanostructures like the nanoribbons in the right side of **Figure 6** [9].

The results we see in **Figure 8**. In the left part, the shape of the segment of the concrete nanoribbon is present. The plot of the electronic spectrum and DOS are given in the middle and in the right part, respectively. The direction of the wave vector *k* <sup>→</sup> is considered longitudinal. The upper part corresponds to the nanoribbon with zigzag edges [10] and with 12 atomic sites in the unit cell (see **Figure 6**). That is why the size of the appropriate matrix in Eq. (8) would be 12 × 12 and its spectrum contains 12 eigenvalues. This corresponds to 12 lines in the graph of the electronic spectrum. The graph of DOS shows a zero energy peak that signalizes the metallicity of this kind of nanostructure. It is a typical property for the zigzag nanoribbons unlike the armchair nanoribbons [10].

where *γ*<sup>0</sup> =*γAA* =*γBB* is the hopping integral for the nearest neighboring atoms and *a* =2.46*A* is the distance between the next-nearest atomic neighbors. DOS and LDOS are then defined as

The corresponding graphs of electronic spectrum and DOS are given in **Figure 7**. We see that for zero energy, the density of states has zero value. This property is typical for the semimetallic nanostructures. For the metallic nanostructures, a peak appears for zero energy. In the first case, a gap is present around zero in the electronic spectrum. Its width can be influenced by the additional defects in the hexagonal structure or by the chemical admixtures and in this way, a material with the predefined properties can be synthesized. In the second case, the gap

**Figure 7.** The electronic spectrum for *ky* = 0 (left) and *k*x = 0 (middle) and the density of states (right) of graphene. The solid and dashed lines correspond to the positive and the negative energy values, respectively. In the bottom, we see

In a similar way, but with a more complicated structure of the wave function in Eq. (4) and for a larger size of the matrix in Eq. (8), the electronic spectrum and DOS can be found for other

The results we see in **Figure 8**. In the left part, the shape of the segment of the concrete nanoribbon is present. The plot of the electronic spectrum and DOS are given in the middle

<sup>→</sup> is considered longitudinal.

the form of the electronic spectrum for an arbitrary value of the wave vector.

nanostructures like the nanoribbons in the right side of **Figure 6** [9].

and in the right part, respectively. The direction of the wave vector *k*

d

<sup>r</sup> r r <sup>r</sup> (11)

*1 k DOS(E)= - LDOS(E,k)= (E - E(k)).*

w

0

w

p®

around zero energy is absent.

36 Recent Advances in Graphene Research

<sup>d</sup> lim Im , <sup>i</sup>

ò

*E - E(k)+*

**Figure 8.** Electronic spectrum and DOS of different kinds of nanoribbons.

The middle and the bottom part correspond to some modifications of the previous form—the nanoribbon with the reconstructed edges. This causes the enlargement of the unit cell (**Figure 6**) and, consequently, a more complicated structure of the electronic spectrum. The metallic properties depend on the concrete kind of the modification: for the nanostructure in the middle part, the zero energy peak in DOS is preserved, whereas it disappears for the nanostructure in the bottom part. Furthermore, in the first case, the electronic spectrum gets a more complicated structure—the number of the Dirac points, where the lines are crossing, is doubled. This feature remains and strengthens for the larger width: in **Figure 9**, the form of the electronic spectrum is depicted for the same kind of the nanostructure and its width is three times larger.

#### **2.2. Structures with curved geometry**

For the curved structures, the nontrivial geometry is described by the external potential *U* (*r* → ) in Eq. (3). Because of the aperiodicity, the eigenvalues cannot be labeled by the wave vector *k* <sup>→</sup> . Nevertheless, the solution of the Schrödinger equation (Eq. (2)) can be expressed with the help of the solutions for the previous case as so labeling by the wave vector will still play a key role in the following procedure.

$$
\Psi(\vec{r}) = \int \mathrm{d}\vec{k} \,\nu\_{\vec{k}}(\vec{r}),\tag{12}
$$

For the purpose of the calculations, we express the wave function that solves Eq. (2) in the case of zero external potential in the form of the so-called "Luttinger–Kohn base" [11]:

$$
\hat{\mathbf{r}}\,\boldsymbol{\Psi}\_{\vec{k}}(\vec{r}) = f\_{\vec{A}}(\vec{\kappa}\,)e^{i\vec{k}\cdot\vec{r}}\,\boldsymbol{\Psi}\_{\vec{A}}(\vec{\mathcal{K}},\vec{r}) + f\_{\vec{B}}(\vec{\kappa}\,)e^{i\vec{k}\cdot\vec{r}}\,\boldsymbol{\Psi}\_{\vec{B}}(\vec{\mathcal{K}},\vec{r}),\tag{13}
$$

where *κ* <sup>→</sup> =*k* <sup>→</sup> − *K* <sup>→</sup>, *<sup>K</sup>* <sup>→</sup> being the Dirac point and *E*(*<sup>k</sup>* <sup>→</sup> ) is the appropriate eigenvalue for the zero external potential. After the substitution of this expression into Eq. (12), we get

$$\Psi(\vec{r}) = \int d\vec{\kappa} \left( f\_{\mathcal{A}}(\vec{\kappa} \,) e^{|\vec{\kappa} \cdot \vec{r}|} \, \mathcal{V}\_{\mathcal{A}}(\vec{\mathcal{K}}, \vec{r}) + f\_{\mathcal{B}}(\vec{\kappa} \,) e^{|\vec{\kappa} \cdot \vec{r}|} \, \mathcal{V}\_{\mathcal{B}}(\vec{\mathcal{K}}, \vec{r}) \right) . \tag{14}$$

**Figure 9.** Electronic spectrum for extra wide nanoribbon with reconstructed edges.

This wave function will be substituted into Eq. (3). In reference [12], a sequence of steps is described whose result is the transformation of this equation to a two-dimensional (2D) Diraclike equation for the massless fermion. In the practical calculations, a suitable choice of the coordinates is useful. In our case, we will suppose the rotational symmetry. Then, we perform the transformation of the coordinates: (*x*, *y*, *z*)→(*ξ*, *φ*), where *φ* is the angular coordinate. Then, the resulting Dirac-like equation will have the form [13, 14]

$$\mathbf{i}\,\sigma^{\alpha}\mathbf{e}\_{\alpha}^{\mu}[\hat{\boldsymbol{\ominus}}\_{\mu} + \boldsymbol{\Omega}\_{\mu} - \mathbf{i}\boldsymbol{a}\_{\mu} - \mathbf{i}\boldsymbol{a}\_{\mu}^{W} - \mathbf{i}\boldsymbol{A}\_{\mu}]\Psi = E\Psi. \tag{15}$$

The meaning of the particular terms is the following: *e<sup>α</sup> <sup>μ</sup>*, the zweibein, is connected with metric and using the corresponding tensor, it can be defined as

$$\mathbf{g}\_{\mu\nu}(\mathbf{x}) = \mathbf{e}\_{\mu}^{\alpha}(\mathbf{x})\mathbf{e}\_{\nu}^{\beta}(\mathbf{x})\boldsymbol{\eta}\_{\alpha\beta}. \tag{16}$$

Here, *ηαβ* is the metric of the plain space without curvature. Next term, *Ωμ*, which is the spin connection in the spinor representation, is defined as *Ω<sup>μ</sup>* <sup>=</sup> <sup>1</sup> <sup>8</sup> *ω<sup>μ</sup> αβ σα*, *σβ* . Here, *ω<sup>μ</sup>* is a more usual form of the spin connection. For its definition, we have to stress that the rotational symmetry is supposed. Then, it has the form

$$
\alpha\_{\boldsymbol{\phi}}^{12} = -\alpha\_{\boldsymbol{\phi}}^{21} = \mathbf{l} - \frac{\overline{\mathcal{O}\_{\boldsymbol{\xi}}} \sqrt{\mathbf{g}\_{\boldsymbol{\phi}\boldsymbol{\phi}}}}{\sqrt{\mathbf{g}\_{\boldsymbol{\xi}\boldsymbol{\xi}}}} = 2o\boldsymbol{\alpha}, \qquad \alpha\_{\boldsymbol{\xi}}^{12} = \alpha\_{\boldsymbol{\xi}}^{21} = \mathbf{0}. \tag{17}
$$

It remains to explain the sense of the gauge fields *aμ*, *aμ <sup>W</sup>* , *Aμ*. First two of them ensure the circular periodicity. Their form is

$$a\_{\phi} = N \,/\, 4, \qquad a\_{\phi}^{\;\,\,\,} = -\frac{1}{3}(2m+n), \tag{18}$$

where *N* is the number of defects and (*n,m*) is the chiral vector. The last term, *Aμ*, represents one possible additional effect—the magnetic field.

The rotational symmetry enables to find the solution of Eq. (15) with the help of the substitution

$$\Psi^T = \begin{pmatrix} F\_1 \\ F\_2 \end{pmatrix} = \frac{1}{\sqrt[4]{\mathbf{g}\_{\rho\rho}}} \begin{pmatrix} \boldsymbol{\mu}\_f(\boldsymbol{\xi}) \mathbf{e}^{i\rho \boldsymbol{\cdot}} \\ \boldsymbol{\nu}\_f(\boldsymbol{\xi}) \mathbf{e}^{i\rho(\boldsymbol{\cdot}+1)} \end{pmatrix}, \qquad \boldsymbol{j} = \mathbf{0}, \pm \mathbf{1}, \ldots, \tag{19}$$

from which we get the system

$$\frac{\partial\_{\boldsymbol{\xi}}\boldsymbol{\mu}\_{j}}{\sqrt{\mathbf{g}\_{\boldsymbol{\xi}\boldsymbol{\xi}}}} - \frac{\tilde{j}}{\sqrt{\mathbf{g}\_{\boldsymbol{\varphi}\boldsymbol{\varphi}}}}\boldsymbol{\mu}\_{j} = E\boldsymbol{\nu}, \qquad -\frac{\partial\_{\boldsymbol{\xi}}\boldsymbol{\nu}\_{j}}{\sqrt{\mathbf{g}\_{\boldsymbol{\varphi}\boldsymbol{\xi}}}} - \frac{\tilde{j}}{\sqrt{\mathbf{g}\_{\boldsymbol{\varphi}\boldsymbol{\varphi}}}}\boldsymbol{\nu}\_{j} = E\boldsymbol{\nu}. \tag{20}$$

Here,

**2.2. Structures with curved geometry**

38 Recent Advances in Graphene Research

role in the following procedure.

y

*k*

where *κ*

<sup>→</sup> =*k* <sup>→</sup> − *K* <sup>→</sup>, *<sup>K</sup>*

For the curved structures, the nontrivial geometry is described by the external potential *U* (*r*

in Eq. (3). Because of the aperiodicity, the eigenvalues cannot be labeled by the wave vector

<sup>→</sup> . Nevertheless, the solution of the Schrödinger equation (Eq. (2)) can be expressed with the help of the solutions for the previous case as so labeling by the wave vector will still play a key

For the purpose of the calculations, we express the wave function that solves Eq. (2) in the case

i i *r r <sup>k</sup> A AB B (r)= f ( )e (K, r)+ f ( )e (K, r),*

 × × r r r r <sup>r</sup> r r r r rrr

 k

> k

 y× × <sup>Y</sup> <sup>ò</sup> r rr r r r r r r r r <sup>r</sup> (14)

 k

<sup>→</sup> ) is the appropriate eigenvalue for the zero

 ky

d *<sup>k</sup>* Y*(r)= k (r),* y ò <sup>r</sup> r r r

of zero external potential in the form of the so-called "Luttinger–Kohn base" [11]:

external potential. After the substitution of this expression into Eq. (12), we get

k

 y

i i d *r r A AB B (r)= f ( )e (K, r)+ f ( )e (K, r) ( ).*

k

 ky

<sup>→</sup> being the Dirac point and *E*(*<sup>k</sup>*

kk

**Figure 9.** Electronic spectrum for extra wide nanoribbon with reconstructed edges.

→ )

(12)

(13)

$$
\tilde{j} = j + 1/2 - a\_{\phi} - a\_{\phi}^{\prime\prime} - A\_{\psi}. \tag{21}
$$

From the solution, LDOS is defined as the square of the absolute value of the wave function. In this case, we get it as the sum of squares of the absolute values of its *ξ*-components:

$$LDOS(E, \xi) = |\mu(E, \xi)|^2 + |\nu(E, \xi)|^2. \tag{22}$$

#### **3. Properties of the graphitic nanocone**

The graphitic nanocone is a nanostructure that can be created from the plain graphene by the insertion of the pentagonal defects into the hexagonal structure. The number of these defects can vary from one to five. In this way, the conical tip arises and its smoothness and the vortex angle are given by the number of the defects and their placement. Then, the real geometry of the graphitic nanocone and the pure conical geometry are different (**Figure 10**, left part). The value of the vortex angle *φ* for the purely conical geometry can be calculated as

$$\sin\frac{\varphi}{2} = 1 - \frac{N}{6},$$

where *N* is the number of the pentagonal defects in the conical tip.

The electronic structure of the graphitic nanocone with purely conical geometry and without any additional effects was investigated in reference [15]. There, the solution of Eq. (15) for this case is derived. Here, using the gauge field theory approach, we introduce the results of the calculations in different approximations: the nanocone with purely conical geometry influ‐ enced by the SOC [1] and the same case with the additional effect of the Coulomb interaction coming from the charge placed into the conical tip [3]. The reason is following: it is one of the possibilities how to simulate the real geometry in the conical tip.

#### **3.1. Electronic structure influenced by the spin-orbit coupling**

In the case of the purely conical structure, this form is assigned to the Hamiltonian in the Schrödinger equation (Eq. (2)) [15]:

$$
\hat{H}\_0 = \begin{pmatrix} H\_1 & 0 \\ 0 & H\_{-1} \end{pmatrix} \qquad \hat{H}\_{0s} = \mathrm{i}\hbar \mathbf{v}\_F \left\{ \mathbf{r}^\prime \partial\_r - \mathbf{r}^\times \boldsymbol{r}^{-1} \left[ (\mathbf{I} - \boldsymbol{\eta})^{-1} \left( \mathbf{s} \partial\_\boldsymbol{\varphi} - \frac{\mathbf{3}}{2} \mathbf{i} \boldsymbol{\eta} \right) - \frac{1}{2} \boldsymbol{\tau}^\times \right] \right\}. \tag{24}
$$

In this equation, *vF* is the Fermi velocity, *s* = ± 1 denotes the value of the *K* spin, *η* = *N* / 6, *τ <sup>x</sup>* , *τ <sup>y</sup>* , *τ <sup>z</sup>* are the Pauli matrices—these matrices have nothing to do with SOC. The points on the surface are described by the coordinates *r* and *φ*. The value of *r* is given by the distance from the tip (see **Figure 10**, right part).

SOC is incorporated using the substitutions [1]

1/2 . *<sup>W</sup> jj aa A* =+ - - jj

2 2 *LDOS E u E v E* ( , ) | ( , )| | ( , )| . xx

**3. Properties of the graphitic nanocone**

40 Recent Advances in Graphene Research

From the solution, LDOS is defined as the square of the absolute value of the wave function. In this case, we get it as the sum of squares of the absolute values of its *ξ*-components:

The graphitic nanocone is a nanostructure that can be created from the plain graphene by the insertion of the pentagonal defects into the hexagonal structure. The number of these defects can vary from one to five. In this way, the conical tip arises and its smoothness and the vortex angle are given by the number of the defects and their placement. Then, the real geometry of the graphitic nanocone and the pure conical geometry are different (**Figure 10**, left part). The

value of the vortex angle *φ* for the purely conical geometry can be calculated as

where *N* is the number of the pentagonal defects in the conical tip.

possibilities how to simulate the real geometry in the conical tip.

**3.1. Electronic structure influenced by the spin-orbit coupling**

1 1 1

*H Hv r s*

*s Fr*

t

Schrödinger equation (Eq. (2)) [15]:

*H*

, *τ <sup>y</sup>*

*η* = *N* / 6, *τ <sup>x</sup>*

0 0 1


distance from the tip (see **Figure 10**, right part).

*H*

sin 1 , 2 6 j

The electronic structure of the graphitic nanocone with purely conical geometry and without any additional effects was investigated in reference [15]. There, the solution of Eq. (15) for this case is derived. Here, using the gauge field theory approach, we introduce the results of the calculations in different approximations: the nanocone with purely conical geometry influ‐ enced by the SOC [1] and the same case with the additional effect of the Coulomb interaction coming from the charge placed into the conical tip [3]. The reason is following: it is one of the

In the case of the purely conical structure, this form is assigned to the Hamiltonian in the

*y x z*

 h

, *τ <sup>z</sup>* are the Pauli matrices—these matrices have nothing to do with SOC. The

æ ö ì ü ï ï é ù æ ö <sup>=</sup> ç ÷ = ¶- - ¶- - í ý ê ú ç ÷ è ø ï ï î þ ë û è ø <sup>h</sup> (24)

j


 ht

<sup>0</sup> 3 1 ˆ ˆ i (1 ) i . <sup>0</sup> 2 2

 t

In this equation, *vF* is the Fermi velocity, *s* = ± 1 denotes the value of the *K* spin,

points on the surface are described by the coordinates *r* and *φ*. The value of *r* is given by the

 j

> x

% (21)

= + (22)

*<sup>N</sup>* = - (23)

$$
\hat{\mathcal{O}}\_r \rightarrow \hat{\mathcal{O}}\_r - \frac{\delta \eta'}{4 \gamma R} \sigma\_x(\vec{r}), \qquad \mathrm{i}\hat{\mathcal{O}}\_\phi \rightarrow \mathrm{i}\hat{\mathcal{O}}\_\phi + \mathrm{s}(\mathbb{I} - \eta) A\_\gamma \sigma\_\gamma,\tag{25}
$$

**Figure 10.** Left: the deviation of the geometry of the graphitic nanocone from the geometry of the real nanocone. Right: The notation of the variables in the nanocone.

where *<sup>R</sup>* <sup>=</sup> (1 <sup>−</sup> *<sup>η</sup>*)*<sup>r</sup> η*(2 − *η*) , *Ay* =*s* 2*δp* (1 <sup>−</sup> *<sup>η</sup>*) *η*(2−*η*). Here, in the analogy to the case of the nanotube, *R* represents the curvature [16] (the geometrical meaning is also presented in **Figure 10**, right part), *Ay* is connected with the curvature of the carbon bonds. Next, *γ* and *γ* ′ represent the nearest neighbor and the next-nearest neighbor hopping integrals, respectively, and *σx*(*r* <sup>→</sup> )=*σ <sup>x</sup>* cos*φ* −*σ <sup>z</sup>* sin*φ*. Here, *σ <sup>x</sup>* , *σ <sup>y</sup>* , *σ <sup>z</sup>* are the Pauli matrices, but (unlikely the *τ* matrices) this time they are connected with SOC. The parameters *p* and *δ* are connected with the hopping integrals and the atomic potential, respectively. Their closer explanation can be also found in reference [16]. The following choice of their values was used: *δ* is of the order between 10−3 and 10−<sup>2</sup> , *γ* ′ *<sup>γ</sup>* <sup>~</sup> <sup>8</sup> <sup>3</sup> , *p* ~0.1. After making the transformation *H* ^ *<sup>s</sup>* <sup>→</sup>*e*<sup>i</sup> *σy* <sup>2</sup> *<sup>φ</sup>H* ^ *se* −i *σy* <sup>2</sup> *<sup>φ</sup>*, which transforms the coordinate frame into the local coordinate frame, the final form of the Hamiltonian is

$$\begin{split} \hat{H}\_s &= \hbar \mathbf{v}\_F \begin{pmatrix} 0 \\ -\hat{\boldsymbol{\sigma}}\_r + \mathbf{i} \frac{1}{r} \frac{\boldsymbol{\varepsilon}}{r} \boldsymbol{\sigma}\_x(\vec{r}) - \frac{\mathbf{i} \mathbf{s} \hat{\boldsymbol{\sigma}}\_\boldsymbol{\sigma}}{(1-\eta)r} - \frac{A\_\circ}{r} \boldsymbol{\sigma}\_\circ - \frac{3\eta}{2(1-\eta)r} - \frac{1}{2r} \\ \hat{\boldsymbol{\sigma}}\_r - \mathbf{i} \frac{1}{r} \boldsymbol{\xi}\_x \boldsymbol{\sigma}\_x(\vec{r}) - \frac{\mathbf{i} \mathbf{s} \hat{\boldsymbol{\sigma}}\_\boldsymbol{\sigma}}{(1-\eta)r} - \frac{A\_\circ}{r} \boldsymbol{\sigma}\_\circ - \frac{3\eta}{2(1-\eta)r} + \frac{1}{2r} \\ 0 \end{pmatrix} \tag{26}$$

It includes the strength of SOC through the parameters *ξx*, *ξy*:

$$
\xi\_x = \frac{\delta \gamma' \sqrt{\eta (2 - \eta)}}{4(1 - \eta)\gamma}, \qquad \xi\_y = A\_y + \frac{1}{2(1 - \eta)}.\tag{27}
$$

Now, the equation

$$
\hat{H}\_s \mathcal{V}(r, \varphi) = E \psi(r, \varphi) \tag{28}
$$

will be solved for the calculation of LDOS. Similarly as in Eq. (19), we can do the following factorization due to the rotational symmetry:

$$\psi(r,\varphi) = e^{\psi\circ} \begin{pmatrix} f\_{/\uparrow}(r) \\ f\_{/\downarrow}(r) \\ g\_{/\uparrow}(r) \\ g\_{/\downarrow}(r) \end{pmatrix} \tag{29}$$

It changes the equation into the form

$$
\begin{pmatrix}
0 & 0 & \begin{matrix} \mathcal{O}\_r + \frac{F}{r} & -\frac{1}{r}C \\\\ 0 & 0 & -\frac{1}{r}D \\\\ -\mathcal{O}\_r + \frac{F-1}{r} & \frac{1}{r}D & 0 \\\\ \frac{1}{r}C & -\mathcal{O}\_r + \frac{F-1}{r} & 0 & 0
\end{matrix}
\end{pmatrix}
\begin{pmatrix}
f\_{\uparrow\uparrow}(r) \\\\ f\_{\downarrow\downarrow}(r) \\\\ \mathbf{g}\_{\uparrow\uparrow}(r) \\\\ \mathbf{g}\_{\downarrow\uparrow}(r)
\end{pmatrix} = E \begin{pmatrix}
f\_{\uparrow\uparrow}(r) \\\\ f\_{\downarrow\downarrow}(r) \\\\ \mathbf{g}\_{\uparrow\uparrow}(r) \\\\ \mathbf{g}\_{\downarrow\downarrow}(r)
\end{pmatrix}.
\tag{30}
$$

Next parameters appearing in this equation are

$$F = \frac{sj}{1-\eta} - \frac{3}{2} \frac{\eta}{1-\eta} + \frac{1}{2}, \qquad C = \xi\_\times - \xi\_\times, \qquad D = \xi\_\times + \xi\_\times. \tag{31}$$

In reference [1], a numerical method is introduced in detail that helps to find the solution of this system. Using a modified version of Eq. (22) (we sum up the squares of absolute values of four components instead of two components), we calculate LDOS from this solution. For different numbers of the defects in the conical tip, we see the resulting LDOS in **Figure 11**. It involves the modes *j* = −1, 0, 1, 2, 3 with the same weight. While for the case of one and two defects in the tip, arbitrary energy and *r* = 0, LDOS grows to infinity, in the case of three defects in the tip, this effect appears close to zero energy only. Using a more thorough analysis, one could find out that the peak for the case of three defects corresponds to the case of the mode *j* = −1 and for other modes and the same number of defects, the behavior is the same as in the case of one and two defects.

(2 ) <sup>1</sup> , . 4(1 ) 2(1 ) *<sup>x</sup> y y <sup>A</sup>*

<sup>ˆ</sup> (, ) (, ) *H r Er <sup>s</sup>* yj

i

(, ) ( )

*f r r e g r*

j  x

> yj

will be solved for the calculation of LDOS. Similarly as in Eq. (19), we can do the following

( ) ( )

*j j j j j*

*f r*

æ ö ç ÷ ç ÷ <sup>=</sup> ç ÷ ç ÷ ç ÷ è ø

 ¯ ¯

( )

*g r*

() () <sup>i</sup> 0 0 () () . 1 i () () 0 0

*<sup>F</sup> gr gr <sup>D</sup> r r gr gr*

3 1 , , . 1 21 2 *x y x y*

In reference [1], a numerical method is introduced in detail that helps to find the solution of this system. Using a modified version of Eq. (22) (we sum up the squares of absolute values of four components instead of two components), we calculate LDOS from this solution. For different numbers of the defects in the conical tip, we see the resulting LDOS in **Figure 11**. It involves the modes *j* = −1, 0, 1, 2, 3 with the same weight. While for the case of one and two defects in the tip, arbitrary energy and *r* = 0, LDOS grows to infinity, in the case of three defects in the tip, this effect appears close to zero energy only. Using a more thorough analysis, one

xx

= - + =- =+ - - (31)

*j j <sup>r</sup>*

è øè ø - -¶ + è ø

h

<sup>=</sup> (28)

() ()

*j j*

 ¯ ¯ ¯ ¯

æ öæ ö

*j j*

 xx

*<sup>r</sup> j j*

*<sup>F</sup> <sup>C</sup> r r fr fr <sup>F</sup> <sup>D</sup> fr fr r r <sup>E</sup>*

ç ÷ç ÷ - ¶+

(29)

(30)

¢ - <sup>=</sup> = + - - (27)

 h

h g

y j

<sup>i</sup> 0 0

æ ö ¶+ - ç ÷

*r*

<sup>=</sup> - -¶ +

*sj F CD*

i 1 0 0

h

> h

*r*

*<sup>F</sup> <sup>C</sup> r r*

Next parameters appearing in this equation are

h

dg h

x

factorization due to the rotational symmetry:

It changes the equation into the form

Now, the equation

42 Recent Advances in Graphene Research

From these results follows that there could be a strong localization of the electrons in the tip, especially near zero energy in the case of three defects. Now, we will be interested, if this behavior remains the same after the inclusion of some boundary effects that should simulate the real geometry of the nanostructure. Furthermore, we would like to ensure in this way the quadratical integrability of the solution.

**Figure 11.** Three-dimensional graphs of LDOS of the graphitic nanocone influenced by SOC. Here, LDOS corresponds to the sum of the solutions corresponding to *j* = 1,0,1,2,3. The number of the defects in the tip in the particular cases: *N* = 1 (left), *N* = 2 (middle) and *N* = 3 (right).

#### **3.2. Incorporation of the boundary effects by a charge simulation**

The influence of the charge considered in the conical tip is expressed in the Hamiltonian by the presence of the diagonal term <sup>−</sup> *<sup>κ</sup> <sup>r</sup>* , where *κ* =1 / 137 is the fine structure constant. This term substitutes the diagonal zeros in Eq. (30):

$$
\begin{pmatrix}
0 & -\frac{\kappa}{r} & -\frac{\mathbf{\dot{\mathcal{E}}}\_r}{r}D & \mathbf{\hat{\mathcal{O}}}\_r + \frac{F}{r} \\
\frac{\mathbf{\dot{\mathcal{E}}}\_rC & -\boldsymbol{\mathcal{O}}\_r + \frac{F-1}{r} & 0 & -\frac{\kappa}{r}
\end{pmatrix}
\begin{pmatrix}
f\_{\boldsymbol{\mathcal{I}}\circ\boldsymbol{c}}(r) \\
f\_{\boldsymbol{\mathcal{I}}\circ\boldsymbol{c}}(r) \\
g\_{\boldsymbol{\mathcal{I}}\circ\boldsymbol{c}}(r) \\
g\_{\boldsymbol{\mathcal{I}}\circ\boldsymbol{c}}(r)
\end{pmatrix} = E \begin{pmatrix}
f\_{\boldsymbol{\mathcal{I}}\circ\boldsymbol{c}}(r) \\
f\_{\boldsymbol{\mathcal{I}}\circ\boldsymbol{c}}(r) \\
g\_{\boldsymbol{\mathcal{I}}\circ\boldsymbol{c}}(r) \\
g\_{\boldsymbol{\mathcal{I}}\circ\boldsymbol{c}}(r)
\end{pmatrix}.
\tag{32}
$$

In this way, the parallel influence of both the Coulomb interaction and SOC is considered [3]. To solve the resulting equation, we use the analogy of the numerical method used in [1]—this analogy is presented in reference [3]. From the calculated results, LDOS is calculated using Eq. (22) again.

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

**Figure 12.** Graphs of LDOS of the graphitic nanocone influenced by the Coulomb interaction (including the influence of SOC) for different distances *r* → from the tip, −1 ≤ *j* ≤ 3 and for different numbers of the defects.

#### **3.3. Comparison of the results**

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 dependence of LDOS on *r* <sup>→</sup> variable close to zero energy for the case of the influence of SOC 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).
