Electronic Properties of Carbon Nanostructures

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 A1,A2,A3,.... 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]

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

where X(r→)is the atomic orbital function. The overlap is zero, i.e.,

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→, we create the expressions

If we suppose that the functions Xare normalized so that ∫X*(r→−r→A(B))X(r→−r→A(B))dr→=1, then Sgives the number of the unit cells in the nanostructure. Now, the Schrödinger equation is transformed into the matrix form

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

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

where γ0=γAA=γBBis the hopping integral for the nearest neighboring atoms and a=2.46Ais 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 around zero energy is absent.

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→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].

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→. 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.

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]:

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

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) Dirac-like 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 φis the angular coordinate. Then, the resulting Dirac-like equation will have the form [13, 14]

The meaning of the particular terms is the following: eαμ, the zweibein, is connected with metric and using the corresponding tensor, it can be defined as

Here, ηαβis the metric of the plain space without curvature. Next term, Ωμ, which is the spin connection in the spinor representation, is defined as Ωμ=18ωμαβ[σα,σβ]. Here, ωμis 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

It remains to explain the sense of the gauge fields aμ,aμW,Aμ. First two of them ensure the circular periodicity. Their form is

where Nis 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

from which we get the system

Here,

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:

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]:

In this equation, vFis the Fermi velocity, s=±1denotes the value of the Kspin, η=N/6,τx,τy,τzare the Pauli matrices—these matrices have nothing to do with SOC. The points on the surface are described by the coordinates rand φ. The value of ris given by the distance from the tip (see Figure 10, right part).

SOC is incorporated using the substitutions [1]

where R=(1−η)rη(2−η),Ay=s2δp(1−η)η(2−η). Here, in the analogy to the case of the nanotube, Rrepresents the curvature [16] (the geometrical meaning is also presented in Figure 10, right part), A_yis connected with the curvature of the carbon bonds. Next, γand γ′represent the nearest neighbor and the next-nearest neighbor hopping integrals, respectively, and σx(r→)=σxcosφ−σzsinφ. Here, σx,σy,σzare the Pauli matrices, but (unlikely the τmatrices) this time they are connected with SOC. The parameters pand δare 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: δis of the order between 10⁻³ and 10−2,γ′γ~83,p~0.1. After making the transformation H^s→eiσy2φH^se−iσy2φ, which transforms the coordinate frame into the local coordinate frame, the final form of the Hamiltonian is

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

Now, the equation

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

It changes the equation into the form

Next parameters appearing in this equation are

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=−1,0,1,2,3with 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.

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.

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 −κr, where κ=1/137is the fine structure constant. This term substitutes the diagonal zeros in Eq. (30):

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.

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→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.

4.1. Electronic structure

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

Here, θis the Heaviside step function, r−,r+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 aμdepend on the chiral vector [18] of the connecting nanotube. For our purpose, this vector is (6n, 6n) and (6n, 0). In most cases, aμhas then the components

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

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

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

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

and for 0<r≤a,

In the first case, the solution is

Here, Jj(x)and Yj(x)are the Bessel functions of the integer order jand the energy ε=±vFk. 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

For the 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 ψA±of the solution. Then, from Eqs. (49) and (50) follows: for r≥a,

and for 0<r≤a,

If aφ=32and r≥a, the solution is

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

Both solutions are strictly normalizable only for j= 0. Analogous solution holds for ψB+and for ψA±if the components ψB±are chosen as zero.

There are no strictly normalizable solutions for the value aφ=12. 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 graphene 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:

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

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

which is solved by [22]

Here,

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

where Mis 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

in this case, the mass Mcorresponds 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 macroscopic effects and should be caught by the experimentalists [26].

4.4. Case of perturbed wormhole

Now we will investigate how the electronic structure changes if the number of the heptagonal 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 preservation, 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

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 negligible, so △<<1. Then, the nonzero components of the metric are

The nonzero components of the gauge fields are

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

where

Dν(ξ)being the parabolic cylinder function. The functions C△1=C△1(E),C△2=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⋅tanα, where we can suppose that the angle αdepends on the number of the defects Nlinearly, i.e., α=π2−N⋅π24. (In this case, α=π2corresponds to 0 defects and α=0corresponds to 12 defects.) Simultaneously, from Eq. (52) follows that asymptotically we have

from which follows

so

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.

In Figure 21, LDOS of zero modes is shown for a varying distance from the wormhole bridge in the units of the radius aof 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) for the zero modes of the 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 eventuality, and we only note that the corrections to LDOS would be an analogy of the corrections shown in Figure 17.