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Fano effect and bound state in continuum in electron transport through an armchair graphene nanoribbon with line defect
© Gong et al.; licensee Springer. 2013
Received: 1 May 2013
Accepted: 5 July 2013
Published: 22 July 2013
Electron transport properties in an armchair graphene nanoribbon are theoretically investigated by considering the presence of line defect. It is found that the line defect causes the abundant Fano effects and bound state in continuum (BIC) in the electron transport process, which are tightly dependent on the width of the nanoribbon. By plotting the spectra of the density of electron states of the line defect, we see that the line defect induces some localized quantum states around the Dirac point and that the different localizations of these states lead to these two kinds of transport results. Next, the Fano effect and BIC phenomenon are detailedly described via the analysis about the influence of the structure parameters. According to the numerical results, we propose such a structure to be a promising candidate for graphene nanoswitch.
PACS: 81.05.Uw, 71.55.-i, 73.23.-b, 73.25.+i
Since 2004, the monolayer graphene has been successfully realized in experiment [1, 2]. Subsequently, its intriguing properties originating from the strictly two-dimensional structure and massless Dirac fermion-like behavior of low-energy excitation have attracted intensive attention [3, 4]. Graphene can be tailored into various edge nanoribbons. Their semiconducting properties with a tunable band gap dependent on the structural size and geometry make them good candidates for the electric and spintronic devices . Due to this reason, the graphene nanoribbons (GNRs) become of particular interest. According to the edge termination types, the GNRs are generally classified into two basic groups, i.e., the armchair and zigzag GNRs [6–8]. In the tight-binding model with nearest-neighbor approximation, the zigzag GNRs are always metallic and exhibit spin-polarized edge states [6–8]. Instead, the armchair GNRs (AGNRs) show metallic characteristics when only M=3n+2 (M denotes its width with n∈i n t e g e r), whereas they are semiconducting otherwise [7–9]. Due to the advance and development of experiment, the GNRs can be successfully manufactured by different approaches, such as the high-resolution lithography and etching technique [10, 11], chemical means [12, 13], or the unzipping of carbon nanotubes [14, 15]. Besides, graphene field-effect transistors have been experimentally realized by making use of the band gap introduced in GNRs [12, 16]. These experimental progress encourage theoretical researchers to further pay attention to the electric or magnetic properties of the GNRs or GNR heterojunctions [17–23].
Because of the presence of dislocations, microcracks, grain boundaries, and phase interfaces in their growth, experimentally obtained graphene samples are not always single-crystalline materials. These abnormal mechanisms cause some significant physics properties of graphene [24–28]. Recently, a peculiar topological line defect in graphene was reported experimentally by Lahiri . This topological line defect is created by alternating the Stone-Thrower-Wales defect and divacancies, leading to a pattern of repeating paired pentagons and octagons . It was found that this line defect has metallic characteristics. Following this work, some groups proposed a valley filter based on the scattering of this line defect in graphene . Next, using a tight-binding model calculation, Bahamon et al. have observed the metallic characteristics and Fabry-P’erot oscillation phenomena in graphene line defects . After these works, researchers dedicated themselves to the discussion about the electronic and magnetic properties of graphene with a topological line defect; the line defect-based electronics has been gradually established [33–36]. Then, the influence of the line defect on the electron properties of the GNRs have become one main concern of such a field. Song et al. studied a line defect in zigzag GNR where a bulk energy gap is opened by sublattice symmetry breaking . They found that a gapless state is for a configuration which holds a mirror symmetry with respect to the line defect. Lin and Ni reported that the edge-passivated zigzag GNRs with the line defects along the edge show half-metallicity as the line defect is close to one edge . On the other hand, it has been reported that the topological line defects in the zigzag GNR can induce the tuning of antiferromagnetism to ferromagnetism. Hu et al. found that the applied strain induces the local magnetic moments on the line defect, whose coupling with those on the edges leads to a turnover of the spin polarization on one edge, making the zigzag GNR become a ferromagnetic metal at a large enough strain . In addition, Lü et al. calculated the band structure of a zigzag GNR with line defect . They observed that the lowest conduction subband of this structure connects two inequivalent Dirac points with flat dispersion, which is reminiscent of the flat-bottomed subband of a zigzag GNR. Accordingly, a valley filtering device based on a finite length line defect in graphene was proposed.
It is easy to note that the effect of the line defect in the zigzag GNRs has extensively discussed, but few works focused on the AGNRs with line defect. The main reason may be that the line defect can be extended along the zigzag GNRs. It should be certain that the line defect in the AGNRs plays a nontrivial role in the electron transport manipulation despite its terminated topology. With this idea, we, in this work, investigate the electron transport in an AGNR with line defect. We observe that the line defect induces the abundant Fano effects and BIC phenomenon in the electron transport process, which is tightly dependent on the width of the AGNR. According to the numerical results, we propose such a structure to be a promising candidate for electron manipulation in graphene-based material.
Model and Hamiltonian
Here, the index i c (m d ) is the site coordinate in the AGNR (line defect), and 〈i c ,j c 〉 (〈m d ,n d 〉) denotes the pair of nearest neighbors. t0 and t D are the hopping energies of the AGNR and line defect, respectively. ε c and ε d are the on-site energies in the AGNR and the line defect, respectively. t T denotes the coupling between the AGNR and line defect.
In general, gL/R can be numerically solved with the iteration method. In this work, we would like to analytically solve them by projecting the semi-infinite AGNR in the Green function space into a semi-infinite one-dimensional double-atom chain . By derivation, we get the coefficients of the Green function, i.e., , , and [W e ] = t0I (N) are the onsite energy, the coupling between the two atoms in each primitive cell, and the coupling between the neighboring two primitive cells of the chain, respectively. If the AGNR width M is odd, and [Ξ] j l =2δ j l + δj,l + 1 + δj,l − 1. Otherwise, and [Ξ] j l = 2δ j l − δ11 + δj,l + 1 + δj,l−1. By diagonalizing matrix [Ξ], the double-atom chain can be transformed into its molecular orbit representation, and the surface state Green function can be expressed. After this, we can obtain the surface state Green function of the semi-infinite AGNR by representation transformation.
Results and discussion
In this section, we aim to investigate the transport properties of this structure. Prior to calculation, we consider t0 to be the energy unit.
In Figure 2c,d, we present the linear conductance spectra of model C and model D. The structure parameters are considered to be the same as those in Figure 1. It can be found that here, the Fano antiresonance becomes more distinct, including that at the Dirac point. Moreover, due to the Fano effect, the first conductance plateau almost vanishes. In Figure 2c where M = 12n − 4, we find that in the case of M = 8, one clear Fano antiresonance emerges at the Dirac point, and the wide antiresonance valley causes the decrease of the conductance magnitude in the negative-energy region. In addition, the other antiresonance occurs in the vicinity of ε F = 0.03t0. When the AGNR widens to M = 20, the Fano antiresonances appear on both sides of the Dirac point respectively. It is seen, furthermore, that the Fano antiresonances in the positive-energy region are apparent, since there are two antiresonance points at the points of ε F = 0.05t0 and ε F = 0.14t0. Next, compared with the result of M = 20, new antiresonance appears around the position of ε F = − 0.08t0 in the case of M = 32. In model D, where M = 12n + 2, the antiresonance is more apparent, in comparison with that of model C. For instance, when M = 14, a new antiresonance occurs in the vicinity of ε F = 0.13t0, except the two antiresonances in the vicinity of the Dirac point. With the increase of M to M = 26, two antiresonance points emerge on either side of the Dirac point. However, in the case of M = 38, we find the different result; namely, there is only one antiresonance in the positive-energy region. This is because the widening of the AGNR will narrow the first conductance plateau. Consequently, when ε F = 0.15t0, the Fermi level enters the second conductance plateau. In such a case, the dominant nonresonant tunneling of electron inevitably covers the Fano antiresonance.
Following the above description, we next discuss the reason of the asymmetric DOS spectra of model C and model D. Note first that in the region of |ε F | → 0, [W o ] ≈ ε F I (N) + ε F [Ξ] and [W i ] = − t ε F [Ξ]. It is evident that when ε F > 0, the sign (+/−) of [W i ] j l is opposite to that of [W e ] j l , whereas the signs of them are the same in the case of ε F < 0. Such a result of electron-hole asymmetry certainly influences the surface state of the semi-infinite AGNR. Namely, when ε F > 0, the surface state of the semi-infinite AGNR will become more localized. However, the line-defect Hamiltonian is of electron-hole symmetry. Hence, in the region of ω > 0, the electron transport is weaker than that in the region of ω < 0. Due to these reasons, we see that in the four models, the effect of the line defect in the negative energy is relatively weak. Next, in the even M case, [W o ]11 ≈ 2ε F and [W i ]11 = −t ε F in the region of |ε F | → 0. This will modify the surface state properties of the semi-infinite nanoribbon. With the help of the method offered in , we have found that in the case of even M, the surface state of the semi-infinite nanoribbon can be further localized in the case of ε F > 0. Consequently, in such a case, the imaginary part of the self-energy contributed by the semi-infinite AGNR becomes small. Therefore, we can understand the reason for the asymmetric DOS states in model C and model D above and below the Dirac point.
In summary, we have investigated the electron transport through an AGNR with line defect from the theoretical aspect. As a consequence, it has been found that the line defect induces the Fano effects or the phenomenon of BIC in electron transport through this structure, which are determined by the width of the AGNR. To be specific, when M=12n−7 or M = 12n−1, the Fano effects are comparatively weak, whereas the result of BIC is abundant. However, in the configurations of M = 12n−4 or M = 12n+2, the Fano effects are dominant, and no BIC phenomenon has been observed. By paying attention to the DOS spectra, we saw that the line defect induces some localized quantum states around the Dirac point and that the different localizations of these states lead to these two transport results. Next, the influences of the changed structure parameters on the Fano effects have been presented. We believe that the numerical results are helpful for clarifying the contribution of the line defect to the electron transport in the AGNR. We propose such a structure to be a promising candidate for nanoswitch.
WJ Gong thanks Yi-Song Zheng for his helpful discussions.This work was financially supported by the National Natural Science Foundation of China (grant no. 10904010), the Fundamental Research Funds for the Central Universities (grant no. N110405010), the Natural Science Foundation of Liaoning province of China (grants no. 2013020030 and 2012020085), and the Liaoning BaiQianWan Talents Program (grant no. 2012921078).
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