Abstract
The carbon nanostructures are perspective materials for the future applications. This has two reasons: first, the hexagonal atomic structure, which enables a high molecular variability by placing different kinds of the defects, and second, good electronic properties that can be modified for the purpose of the concrete applications with the help of the defects and of the chemical ingredients. Many kinds of the nanostructures are investigated. Here, the properties of less common forms are examined—the graphitic nanocone and graphitic wormhole.
Keywords
- graphene
- graphitic wormhole
- graphitic nanocone
- spin-orbit coupling
- zero modes
1. Introduction
The carbon nanostructures are the materials whose molecular structure is derived from graphene—the hexagonal carbon plain lattice (Figure 1). Because of their electronic structure, they are the promising materials for the construction of nanoscale devices (quantum wires, nonlinear electronic elements, transistors, molecular memory devices, or electron field emitters) and the inventions in the material science.
The planar geometry of the molecular surface is disrupted by the disclinations in the molecular structure that are most often presented by the pentagons and the heptagons in the hexagonal lattice. This change of the geometry is manifested by the positive or the negative curvature, respectively, that can be enlarged by the supply of higher number of the defects. In this way, by the supply of 1–5 pentagonal defects, we get conical structures with different values of the vortex angle (Figure 2).
One more defect can be added and a nanotube is created. This nanostructure can be considered closed as well as opened, i.e., without the cap that contains the pentagonal defects. The second case is more common (Figure 3, left part). The number of the defects can be increased up to 12, and in this way, a completely closed, spherical nanostructure arises (fullerene—Figure 3, middle part).
Analogical manipulations with the graphene lattice can be made by the supply of the heptagonal defects (Figure 4). For the case of 12 heptagonal defects, if they are placed appropriately, the wormhole structure is created (Figure 3, right part).
A lot of other variants of the graphitic nanostructures can be created using different combinations of the pentagonal and the heptagonal defects. Some of them are presented in Figure 5.
We investigate the electronic properties of several kinds of the carbon nanostructures. After the explanation of the computational methods, we demonstrate how to utilize these methods for the purpose of the investigation of graphene and some simple forms of the nanostructures—different kinds of nanoribbons and their modifications. Then, we will concentrate on the calculation of the properties of more complicated forms—the graphitic nanocone [1–3] and the graphitic wormhole [4–6]. In the first case, we consider the influence of the additional effects like the spin-orbit coupling (SOC) and the boundary effects coming from the finite size and from the extreme curvature of the surface geometry in the tip. In the second case, we investigate the effects that arise in the place of the wormhole bridge. Here, two additional effects appear: first, the SOC arising in the connecting nanotube and second, the increase of the electron mass due to relativistic effects coming from the extreme curvature of the surface geometry. As a result, the chiral massive electrons should be observed.
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
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
Here,
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
In the case of graphene, the wave function, which solves Eq. (2), can be expressed as in references [7, 8]
where the components
where
By the substitution of the solution in Eq. (5) into the Schrödinger equation (Eq. (2)), multiplying it by
If we suppose that the functions
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
where
where
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
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
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
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:
The meaning of the particular terms is the following:
Here,
It remains to explain the sense of the gauge fields
where
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
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
where
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 influenced 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]:
In this equation,
SOC is incorporated using the substitutions [1]
where
It includes the strength of SOC through the parameters
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
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
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.,
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
4. Properties of the graphitic wormhole
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 wormhole structures in the graphene bilayer during the process of the synthesis of the corresponding material.
4.1. Electronic structure
We will solve Eq. (15) in the subsection II B. In this case, the metric tensor has the form
Here,
The values of the components of
The only exception is when the chiral vector is (6
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
and for
In the first case, the solution is
Here,
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
and for
If
The second possibility for this value of
Both solutions are strictly normalizable only for
There are no strictly normalizable solutions for the value
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
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
in this case, the mass
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
The nonzero components of the gauge fields are
where (
where
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
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
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.
5. Conclusion
We performed the calculations of the electronic structure for the graphitic nanocone and the graphene wormhole. In the first case, our aim was to find the quadratically integrable solution that includes the boundary effects and considers the real geometry. This goal was partially achieved, but we need to verify the properties of the found solution close to the tip. The precision of the calculations could be improved by the better choice of the corresponding geometry, by the consideration of the discretion of the energetic spectrum coming from the finite size of the nanostructure, and by the inclusion of next effects coming from the overlap of the neighboring atomic orbitals close to the tip [2]. The localization of the electrons shown in Figures 11 and 12, especially in the case of three defects, makes the graphitic nanocone a possible candidate for the construction of the scanning probe in atomic force microscopy.
In the second case of the graphene wormhole, we presented the mathematical motivation for our prediction of the effects that should appear close to the wormhole bridge. Our predictions will be verified with the help of the geometric optimizations and ab initio calculations. On this base, the most suitable candidates for the experiments will be chosen.
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