† Corresponding author. E-mail:
In this paper we propose a type of new analytical method to investigate the localized states in the armchair graphene-like nanoribbons. The method is based on the tight-binding model and with a standing wave assumption. The system of armchair graphene-like nanoribbons includes the armchair supercells with arbitrary elongation-type line defects and the semi-infinite nanoribbons. With this method, we analyze many interesting localized states near the line defects in the graphene and boron-nitride nanoribbons. We also derive the analytical expressions and the criteria for the localized states in the semi-infinite nanoribbons.
Graphene, as a novel two-dimensional material, has attracted a lot of research interest in this decade since it was discovered in 2003.[1] It has many special electronic, optic and magnetic properties due to its energy band structure.[2,3] For example, graphene has the Dirac-like electron with the Klein tunneling effect.[4,5] It has the half integer quantum Hall effect.[6]
With the restriction of one dimension, the bulk graphene forms the graphene nanoribbons (GNR). Armchair GNR (aGNR) and zigzag GNR (zGNR) are the two basic types.[7] There are edge states on the edges of the zGNR.[7] The early tight binding (TB) model shows that these edge states appear near the Fermi level with a large density of states.[7,8] With the Hubbard model and the first-principles calculations, it is found that these edge states actually have an energy gap near the Fermi level with opposite spin polarizations on two edges.[9,10] Some half-metal structures can be prepared by the atom modulations in zGNR systems.[11–13] Furthermore, the properties of some complex structures, such as the transport properties of the double-layer graphene with line defects, the graphene superlattice and doubly-stacked zGNR have been studied by many methods.[14–16] In the thermal physics aspect, there are also a lot of researches. For example, the ballistic thermoelectric properties for the bended GNRs and the graphene with impurities have been investigated by the first-principles calculations and the quantum transport theory.[17,18]
For aGNR, the TB calculation results show that they may be conductors or semiconductors.[7,19] The first-principles results show that for the metal-type aGNR in the TB model, in fact, there is still a very small gap.[20,21] However, the TB method is still a brief and useful model. It employs the pz orbital near the Fermi level to describe the electrical properties of GNR. The results basically agree with the first-principles calculations. In the GNR with a uniform stress, the TB model also gives roughly the same results as the first-principles calculation.[22,23]
For the uniform aGNR, an analytical solution was proposed.[24] In this solution, the transverse wave function of the ribbon was assumed to have a standing-wave form. Under this assumption, the electron states in a semi-infinite aGNR or in a uniform aGNR block were also studied.[25–27]
However, these analytical studies are only suitable for uniform nanoribbons. For inhomogeneous aGNR systems, to our knowledge, there has been no simple analytical expression and the energy band has to be obtained numerically. In this paper we develop a new one-dimensional (1D) analytical method which is suitable for the inhomogeneous aGNR systems. Within this 1D method, we calculate the energy band for aGNRs in the presence of various line defects or uniaxial strains. We also extend this method to the non-equilibrium Green’s function (NEGF) theory for the transmission calculation. In the case of very large aGNR systems, our method can obtain accurate results without heavy computations because it only needs to deal with small matrices.
In the semi-infinite or block aGNR systems, the zigzag edges support some localized electron states, which are similar to the edge states in zGNR.[25–27] These states were also verified from the surface Green’s function calculation in our previous work.[28] On the zigzag edges of bilayer graphene, the solutions of localized states were also proposed.[29] These localized states are due to the zigzag edges or defects in graphene. In other materials such as silicon quantum dots and nanocrystals, people also found such local states. They are related to the dangling bonds, impurities or defects on curved surfaces with the energy in band gaps.[30–32]
In this paper we employ this 1D method to investigate these localized states in the aGNR or the boron-nitride nanoribbons with line defect. We obtain analytic solutions and the critical condition for these localized states in semi-infinite aGNRs. We also find some asymmetrical localized states in the boron-nitride nanoribbon with a line defect.
The rest of this paper is organized as follows. In Section
For the simple aGNR as shown in Fig.
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![](cpb_26_12_127310/cpb_26_12_127310_ieqn1.gif)
![](cpb_26_12_127310/cpb_26_12_127310_ieqn2.gif)
![](cpb_26_12_127310/cpb_26_12_127310_ieqn3.gif)
![](cpb_26_12_127310/cpb_26_12_127310_ieqn4.gif)
![](cpb_26_12_127310/cpb_26_12_127310_ieqn5.gif)
![](cpb_26_12_127310/cpb_26_12_127310_ieqn6.gif)
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Then the following standing-wave-formed wavefunction is obtained:
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With this form of wavefunction and the hard wall boundary condition, we have
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![](cpb_26_12_127310/cpb_26_12_127310_ieqn7.gif)
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We substitute this form of the wavefunction into the Schrödinger equation with the tight-binding approximation and obtain the eigenvalue equation for the band calculation.[24] This standing-wave assumption of the wavefunction can reduce any uniform (in y direction) aGNR system into a 1D system. Figure
Now we generalize this idea to the supercell system, including the supercell with some line defects. Usually the line defect in graphene consists of the periodically repeated one octagonal and two pentagonal rings in the y direction.[33,34] Here in this paper we only tune down some hopping parameters along the y direction, just as the bonds in the GNR are elongated by stress,[22,23] thereby presenting a simple ‘line defect’ (see Fig.
We assume that the supercell system has N columns in the y direction and 2M sub-unit cells in the x direction. In the TB approximation, the wavefunction can be written as
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![](cpb_26_12_127310/cpb_26_12_127310_ieqn10.gif)
![](cpb_26_12_127310/cpb_26_12_127310_ieqn11.gif)
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![](cpb_26_12_127310/cpb_26_12_127310_ieqn12.gif)
![](cpb_26_12_127310/cpb_26_12_127310_ieqn13.gif)
![](cpb_26_12_127310/cpb_26_12_127310_ieqn14.gif)
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Similarly, we have the following equation for the B-type atoms in an aGNR supercell
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Substituting Eq. (
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We see that the two equations above are j-independent. This means that the system is transformed into the 1D form. Then we employ the Bloch boundary condition
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Substituting this Bloch boundary condition into Eq. (
The equations above are for the uniform (in the x direction) aGNR supercell. For the supercell with a line defect, we assume that the defect only changes the hopping integral t as in the strained GNR,[22,23] equations (
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![](cpb_26_12_127310/cpb_26_12_127310_ieqn16.gif)
![](cpb_26_12_127310/cpb_26_12_127310_ieqn17.gif)
![](cpb_26_12_127310/cpb_26_12_127310_ieqn18.gif)
![](cpb_26_12_127310/cpb_26_12_127310_ieqn19.gif)
![](cpb_26_12_127310/cpb_26_12_127310_ieqn20.gif)
![](cpb_26_12_127310/cpb_26_12_127310_ieqn21.gif)
![](cpb_26_12_127310/cpb_26_12_127310_ieqn22.gif)
![](cpb_26_12_127310/cpb_26_12_127310_ieqn23.gif)
With the approach above, we set up a new one-dimensional model formulism for the band structure calculation of an aGNR supercell system.
Here we consider a simple semi-infinite aGNR as shown in Fig.
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From Eqs. (
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![](cpb_26_12_127310/cpb_26_12_127310_ieqn24.gif)
![](cpb_26_12_127310/cpb_26_12_127310_ieqn25.gif)
![](cpb_26_12_127310/cpb_26_12_127310_ieqn26.gif)
![](cpb_26_12_127310/cpb_26_12_127310_ieqn27.gif)
![](cpb_26_12_127310/cpb_26_12_127310_ieqn28.gif)
![](cpb_26_12_127310/cpb_26_12_127310_ieqn29.gif)
![](cpb_26_12_127310/cpb_26_12_127310_ieqn30.gif)
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Iterating the formula above, we obtain the following result:
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Since we assume
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Combining the boundary equation (Eq. (
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We see that
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For a convergent solution, it is apparent that p satisfies the condition
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We find that this local state also exists in finite-sized graphene nanoribbons (or graphene quantum dots).[35] In Ref. [35],
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![](cpb_26_12_127310/cpb_26_12_127310_ieqn33.gif)
![](cpb_26_12_127310/cpb_26_12_127310_ieqn34.gif)
![](cpb_26_12_127310/cpb_26_12_127310_ieqn35.gif)
Finally, if A-type atom and B-type atom are different as shown in Fig.
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![](cpb_26_12_127310/cpb_26_12_127310_ieqn36.gif)
We also use this 1D model to calculate the transmission spectrum or local density of states (LDOS) for armchair graphene-like nanoribbons with the NEGF theory.[37] The system has to be uniform in the y direction for the feasibility of our 1D method. From Eqs. (
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![](cpb_26_12_127310/cpb_26_12_127310_ieqn37.gif)
![](cpb_26_12_127310/cpb_26_12_127310_ieqn38.gif)
![](cpb_26_12_127310/cpb_26_12_127310_ieqn39.gif)
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![](cpb_26_12_127310/cpb_26_12_127310_ieqn40.gif)
![](cpb_26_12_127310/cpb_26_12_127310_ieqn41.gif)
![](cpb_26_12_127310/cpb_26_12_127310_ieqn42.gif)
![](cpb_26_12_127310/cpb_26_12_127310_ieqn43.gif)
![](cpb_26_12_127310/cpb_26_12_127310_ieqn44.gif)
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And the transmission coefficient of mode p can be calculated from the NEGF theory[29]
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![](cpb_26_12_127310/cpb_26_12_127310_ieqn45.gif)
![](cpb_26_12_127310/cpb_26_12_127310_ieqn46.gif)
Firstly, we use Eq. (
Our method can also calculate the band in aGNR with uniaxial strain.[23] As the method developed in Refs. [22] and [23], the three bond vectors in the aGNR are changed with the uniaxial strain σ in the x direction as
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We choose the supercell sizes with 2M = 2 and N = 23, 24, 25, which correspond to the cases of N + 1 = 3q, 3q + 1, and 3q + 2.[20] We calculate the band gaps in aGNR with different strains as shown in Fig.
With this supercell method, we now calculate the band structure of the aGNR supercell with a line defect as shown in Fig.
Figure
Now we study another case where the supercell has the size N = 7, 2M = 4. In this case, N is an odd number. We find that if we chose the value of (N + 1)/2 or (N − 1)/2p, the total number of bands is 2(N + 1)M or 2(N − 1)M, which is not equal to the number of bands in the 2D-TB model (2MN).
Here we give a detailed analysis of this discrepancy. As we stated previously, each p value corresponds to a type of standing wave in the y direction in the 1D model. In a simple aGNR, A-type and B-type atoms are equivalent to each other, so p value can be chosen from 1 to N.[24] Here in our supercell (with line defects) case, A-type and B-type atoms are not equivalent to each other. There are
In Fig.
We notice that this 1D-supercell method gives the same band result as the 2D-TB method, only if the aGNR supercell is uniform in the y direction. For a very large system our 1D-supercell method has a much small computation load, since it only needs to solve
Now we begin to analyze the localized states in these aGNRs. The edge states are well known in the zigzag GNR. Here we find that in the armchair GNR, if there is some line defect, there also exist such edge states, which are localized near the line defect. These localized states result from the zigzag edge structures, and they have been discussed in the literature before.[27,37]
We calculate an aGNR supercell (N = 23, 2M = 16) with a line defect on the supercell boundaries. The hopping integrals in the defect are set to be
Then we tune the hopping integral t1 to a lower energy (−1.8 eV), the energy band with p = 8 becomes lower than the band with p = 9. And the electron density distribution with p = 9 is plotted as well (see Figs.
From Subsection
In Figs.
Now we consider the localized state near the edge of a semi-infinite aGNR. As discussed in Subsection
![]() | Fig. 6. (color online) Electron density distribution of the localized state in a semi-infinite aGNR (N = 23, p = 9). Panel (a) shows the overhead view and panel (b) the side view of the 3D plot. |
From Fig.
Boron-nitride (BN) nanoribbon is very similar to the graphene nanoribbon, except that in the boron–nitride nanoribbon, A-type and B-type atoms are different (Fig.
![]() | Table 1.
Tight binding parameters for the boron-nitride nanoribbon. . |
We first calculate the energy bands of two single-column armchair BN nanoribbons with the geometric sizes: (2M = 2, N = 6) and (2M = 2, N = 8) as shown in Figs.
Then we choose an armchair BN nanoribbon supercell with a line defect and the size parameters N = 23, 2M = 16 (see Fig.
In Figs.
In this paper we develop a new 1D method for the armchair GNR-like systems, including the supercells with line defects and uniaxial strains. Our method gives the same results as the 2D-TB method. However, our method may save a lot of computation load for very large systems. By this method we investigate the properties of many localized defect edge states in these systems. A condition for these localized states to exist in the semi-infinite aGNR is derived as
We also study the BN nanoribbon supercell. We find that there exists some asymmetric localized state in these BN nanoribbons. For all of these defect states, the localization degree depends on two factors: the defect intensity (the hopping integral) and the lateral standing-wave index (p value).
This method is suitable for the band calculation of the armchair-like 2D materials. We will apply it to these materials in the future study.
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