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Quasi-one-dimensional electronic states induced by an extended line defect in graphene: an analytic solution
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2014 J. Phys.: Condens. Matter 26 035302 (http://iopscience.iop.org/0953-8984/26/3/035302) View the table of contents for this issue, or go to the journal homepage for more
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Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 26 (2014) 035302 (10pp)
doi:10.1088/0953-8984/26/3/035302
Quasi-one-dimensional electronic states induced by an extended line defect in graphene: an analytic solution ¨ Liwei Jiang and Yisong Zheng Xiaoling Lu, National Laboratory of Superhard Materials, Department of Physics, Jilin University, Changchun 130012, People’s Republic of China E-mail: [email protected] Received 2 October 2013, revised 28 November 2013 Accepted for publication 28 November 2013 Published 20 December 2013 Abstract
Analytic solutions of the quasi-one-dimensional (q1D) electron states around an extended line defect in a graphene lattice are derived within the tight-binding model. Then, the electronic properties of this kind of boundary state in graphene are studied in detail. It is found that one subband composed of the even-parity boundary states emerges in the vicinity of the Dirac point. In particular, when the bulk band is gapped, such a one-dimensional subband remains in the bandgap, spanning two inequivalent valleys. In addition, this boundary state subband exhibits nontrivial dispersion, which can carry the valley polarized charge current flowing along the extended line defect. As a result, the line defect behaves like a one-dimensional channel for electronic transport. Moreover, its appearance in graphene or a hexagonal boron nitride sheet provides a promising way to print electric circuits in these two-dimensional materials. Keywords: surface states, electronic structure, line defect, graphene (Some figures may appear in colour only in the online journal)
1. Introduction
quantum devices [18–20] by manipulating the electronic spin or pseudospin. Prior to the experimental acquisition of a graphene sample, the edge state attached to the zigzag edge of a graphene lattice was theoretically studied [8–10, 21, 22]. It was found that the edge state is responsible for some unique electronic properties of graphene, such as the spontaneous edge magnetization [21] and the tunable half-metallicity [22]. Based on the chirality of the edge state, valley filtering and valley valve effects are theoretically predicted [13] to control the electronic spin in a similar way to that in the context of spintronics. Thus the concept of valleytronics paralleling spintronics was proposed [13–17]. To date, this topic has drawn much attention in the field of graphene study. More importantly, when the spin–orbital coupling is taken into account, the graphene strip becomes topologically nontrivial [23]. Accompanying the opening of the bulk bandgap, a pair of helical edge states with opposite spin polarization remain in the bandgap. As a result, the quantum spin Hall effect is expected to occur in graphene.
It is well known that in graphene [1], a one-atom thick layer of graphite, the low-energy electron excitation around the Fermi level behaves like a two-dimensional (2D) massless Dirac fermion [2, 3]. As a result, graphene presents many exotic electronic properties in contrast to conventional semiconductor materials, such as the half-integer quantum Hall effect [4–6] and reflectionless Klein tunneling [7]. Apart from the 2D bulk state, in graphene the edge state also plays an important role, and is localized at the lattice terminals [8–10] or the topological defects [11]. Due to the nature of the Dirac fermion, the graphene edge state is chiral in the sense that the pseudospin polarization depends on the electronic wavevector direction. This feature bears an analogy with the helical edge states of topological insulators [12]. It should be noted that the pseudospin in graphene refers to the sublattice [5, 6] or valley degrees of freedom [13–17] rather than the electronic physical spin. From the viewpoint of application, the edge state in graphene can be utilized to realize some functions of 0953-8984/14/035302+10$33.00
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Notwithstanding these intriguing theoretical predictions, it is still a big challenge for current experimental techniques to tailor a well-defined lattice edge to support the chiral edge state. Besides the zigzag edge, topological defects in the graphene lattice can also induce localized electronic states. In fact, many experimentally obtained graphene samples are not single-crystalline flakes. As a result, the aligned adjoining single-crystalline domains make domain boundaries intrinsic topological defects. Recently, an extended line defect of millimeter scale in an epitaxial layer of graphene was successfully fabricated [24], which was composed of a periodic repetition of one octagon plus two pentagons. Meanwhile, relevant theoretical investigations demonstrated that electronic transport through such a line defect can generate the valley filtering and valley valve effects [16]. In addition, around the line defect electronic spin polarization also occurs [25]. A zigzag-edged graphene nanoribbon embedded with an extended line defect shows a half-metallic subband structure [26]. From these theoretical investigations, one can naturally infer that the extended line defect plays a similar role to a zigzag edge. There are also numerical studies [27, 28] reporting that the extended line defect supports q1D electronic states around it. Considering that the line defect is currently an experimentally obtainable structure in a graphene sheet, in contrast to a highly qualified zigzag edge, it is of actual importance to reveal the electronic structure of the q1D states around the line defect, such as the subband profile, the localization strength of the wavefunction and a comparison between it and the edge state at a zigzag edge. In this work, we investigate analytically the q1D boundary states around a line defect in a graphene lattice. Within the tight-binding (TB) model we obtain the analytic eigen solutions of these boundary states. We find that a counterpart of the zigzag edge state is present around the line defect as an odd-parity boundary state with respect to the line defect. More interestingly, a new type of boundary state of even parity emerges. When a bandgap is opened between the bulk conduction and valence bands by a staggered potential, one subband of the even-parity boundary states remains in the bulk bandgap. In contrast to the flat dispersion of the odd-parity state, the subband profile of the even-parity boundary states exhibits notable dispersion spanning the two inequivalent valleys. Thereby the even-parity states can support valley polarized electronic transport. The boundary states near the K and K0 points possess opposite valley polarities. In these respects, the even-parity boundary states resemble the helical edge states in a topological insulator. In addition, due to the existence of the dispersive boundary state subband, an extended line defect in graphene acts as a quantum wire, which provides a feasible way to print electric circuits on graphene sheet. The rest of this paper is organized as follows. In section 2, the analytical solutions of the q1D boundary states around an extended line defect embedded in a 2D hexagonal lattice are obtained in the theoretical framework of a TB approximation. Then, in section 3, by means of the numerical calculations,
Figure 1. (a) A graphene lattice in the presence of an extended line
defect along the y direction. The shaded region defines a periodic unit with the primitive translation vector d (=2a). a is the lattice constant of the pristine graphene. The parameters t1 , t2 , τ1 and τ2 denote the nearest neighbor hopping energies in the TB model. The solid and hollow circles denote two distinct lattice points where the onsite energies of the carbon atoms are defined as +e and −e, respectively. C1 and C2 refer to two inequivalent carbon atoms at the line defect which can have distinct onsite energies denoted as e1 and e2 . The integer m measures the electronic transverse distance relative to the line defect. (b) An illustration of the TB Hamiltonian projected onto ky -space. The reduced TB parameters are labeled. To specify a lattice point, a set of indices (L, l, j) is required, as explained in (c). (c) The reduced lattice obtained by eliminating all the l = 2 and 3 sublayers of the ky -space lattice given by (b). The L interaction matrices w0o , w± o , wi and the wavefunction (cl ) are defined by the context.
we discuss the influence on the boundary states of the lattice strain, the bandgap opening and the absorption of exotic atoms around the line defect. Finally, the main results are summarized in section 4. 2. Theory
A schematic of a 2D hexagonal lattice in the presence of an extended line defect is shown in figure 1(a). The line defect consists of the periodic repetition of one octagonal plus two pentagonal rings along the y direction. Hereafter, we call such a structure line defect embedded graphene (LD-G) although it can also represent hexagonal boron nitride (h-BN) in the presence of an extended line defect. To study the electronic properties of LD-G, we adopt the π -electron TB model with the nearest neighbor approximation. In the TB Hamiltonian, the atomic onsite energies of the two sets of sublattices can be different from each other (noting that neither sublattice covers the two inequivalent atoms on the line defect, whose onsite energies can take distinct values from that of the atoms at the ordinary lattice points). This difference accounts well for the influence of the staggered potential of the substrate on a graphene sheet or two distinct atom species of an 2
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In the above equations, (cLl ) is a two-component column matrix, whose matrix element (cLl )j = Ci (ky ) is signified. To formulate concisely we will adopt the abbreviation L¯ = −L, and as the units of energy we set |t| = 1 (t ≈ −2.7 eV is the uniform hopping energy of the pristine graphene). The interaction matrices between the adjacent sublayers, as labeled in figure 1(c), are given by
h-BN sheet. For the nearest neighbor hopping energies, four different parameters are employed. As labeled in figure 1(a), the hopping energies between two ordinary lattice points are denoted as t1 and t2 . The case of t1 6= t2 implies the presence of a uniform strain. Moreover, the two hopping parameters τ1 and τ2 around the line defect may take different values from t1 and t2 , taking into account the local lattice distortion. Due to the translational symmetry of the LD-G lattice along the extended direction of the line defect, the electronic Bloch wavevector component ky is a good quantum number. Then the sub-Hilbert space spanned by the basis set 8iky (r) = P ik nd y u(r − R ) is closed for the TB Hamiltonian. √1 ni ne
[wi ] = t12 t2 (E2 − e2 − t22 )−1 [M], 2 2 2 2 −1 [w− o ] = (E − e) 1 − t1 (E − e − t2 ) [M] , 2 2 −1 2 2 [w+ o ] = (E + e) 1 − t1 (E − e − t2 ) [M] ,
Ny
where the matrix [M] takes the form " # 1, eiθ cos θ [M] = 2 −iθ , e cos θ, 1
Hereafter, such a subspace is referred to as ky -space. In the above definition of the basis function 8iky (r), u(r − Rni ) is the atomic π -orbital centered at the ith atom of the nth periodic unit. This periodic unit is actually a 1D strip, as shown in figure 1(a). d is the primitive translation vector in the y direction. Ny denotes formally the number of the periodic unit. In ky -space the electronic eigen wavefunction can be expanded in terms of the above basis set, which is given by X 9ky (r) = Ci (ky )8iky (r). (1)
which can be understood as the interaction between two inequivalent atoms at the line defect, labeled as C1 and C2 in figure 1(a). e1 and e2 stand for the onsite energies of these two atoms respectively. Away from the line defect, the onsite energies of the two sublattices, marked by solid and hollow circles as shown in figure 1(a), are designated as e and −e, respectively. Beginning with equation (2), we can work out analytical solutions of the q1D boundary states around the extended line defect if such states exist. The inversion symmetry of the LD-G lattice with respect to the line defect guarantees that the electronic eigenstates of the LD-G possess conserved parity. The wavefunction of an odd-parity state ¯ satisfies the relationship (cLl ) = −(cLl ). In particular, at the line defect one has (c00 ) = 0. Therefore, as far as the odd-parity eigenstates are concerned, the LD-G can be viewed as two pieces of independent semi-infinite hexagonal lattices with a zigzag edge. The odd-parity electronic wavefunctions at the right-hand side of the line defect follow the same Schr¨odinger equation as a zigzag-edged semi-infinite graphene lattice [8–10]. Consequently, the odd-parity boundary state at the right-hand side of the line defect has the same wavefunction and the same dispersion relation as the well-known edge state attached to the zigzag edge of graphene. In the absence of a staggered potential this kind of boundary state has zero energy and forms a dispersionless band spanning two inequivalent valleys. Many previous investigations have demonstrated that the edge state is responsible for some electronic properties unique to graphene; for example, the spontaneous magnetism [21] and valley-resolved electronic transport [13]. Now that this kind of electronic state appears as a boundary state around the line defect, one can reasonably conclude that a line defect bears some analogy with a zigzag edge. In addition to the odd-parity states, there are possibly even-parity boundary states around the line defect. By using
On the basis set 8iky (r), the TB Hamiltonian is projected into a specific ky -space which is denoted as H(ky ). It corresponds to a q1D lattice depicted by figure 1(b). The TB parameters projected into the ky -space are also labeled in this figure. Then, the Schr¨odinger equation takes the form [E − H(ky )](C) = 0 with E being the eigenenergy. The wavefunction (C) is a column matrix consisting of the electronic probability amplitudes Ci (ky ). From figure 1(b) we can see that three indices are needed to specify a lattice point. Namely, we need a notation i = (L, l, j), where L labels the layer and l = 1, . . . , 4 denotes the sublayer in a given layer. The last index, j = 1, 2, specifies the lattice point in the given layer and sublayer. It should be noted that the L = 0 layer, i.e., the line defect, has only a single sublayer (l = 0). The Hamiltonian H(ky ) and the corresponding Schr¨odinger equation can be further transformed into an analytically solvable model by deriving the indirect interaction between the l = 1 and 4 sublayers in each layer. In so doing, we need to eliminate the l = 2 and 3 sublayers in the way adopted in our previous work [29]. As a result, we obtain a further reduced lattice, as shown in figure 1(c). Corresponding to such a lattice, the electronic probability amplitude satisfies the following set of combined equations around the line defect:
L+1 L+1 t2 (cL1 ) + [w+ o ](c4 ) + [wi ](c1 ) ¯ 1¯ 0 [wi ](c14 ) + [w− o ](c1 ) + τ1 (c0 ) ¯ τ1 (c11 ) + [w0o ](c00 ) + τ1 (c11 ) 1 1 τ1 (c00 ) + [w− o ](c1 ) + [wi ](c4 ) L−1 + L [wi ](cL−1 1 ) + [wo ](c4 ) + t2 (c1 ) − L L t2 (cL−1 4 ) + [wo ](c1 ) + [wi ](c4 )
= 0,
L ≤ −2; L ≤ −2;
= 0; (2)
= 0; = 0; = 0,
L ≥ 2;
= 0,
L ≥ 2.
(4)
with θ = ky d/2 ranging from −π/2 to π/2. Besides, in equation (2) the special matrix is " # E − e1 , τ2 0 wo = , (5) τ2 , E − e2
i
L+1 L [wi ](cL4 ) + [w− o ](c1 ) + t2 (c4 ) = 0,
(3)
3
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¯
the relationship (cLl ) = (cLl ) and renormalizing the electronic probability amplitude on the line defect as (c00 )τ1 /t2 ⇒ (c00 ), from equation (2) we can obtain the Schr¨odinger equation of an even-parity state. At the right-hand side of the line defect it is given by the following set of equations:
" × 1−
t2 0 0 [w ](c ) + 2τ1 (c11 ) = 0; τ1 o 0 1 1 t2 (c00 ) + [w− o ](c1 ) + [wi ](c4 ) = 0; L−1 + L [wi ](cL−1 1 ) + [wo ](c4 ) + t2 (c1 ) − L L t2 (cL−1 4 ) + [wo ](c1 ) + [wi ](c4 )
parameters. For s = 1, 2, the eigenvalues are expressed as 2 ( E2 − e2 − t22 − λ(p)t12 1 − (−1)s νs (p) = 2λ(p)t12 t22
L ≥ 2;
= 0,
L ≥ 2.
2
2 2
E2 − e2 − t22 − λ(p)t1
The matrix [V(p)] in equation (11) can be written as " # 1, v+ (p) [V(p)] = , −1, v− (p)
(6)
= 0,
#1 )
4λ(p)t12 t22
By introducing a matrix
− 1. (12)
(13)
with the matrix elements "
#
1 −eiθ , eiθ , [ξ] = √ 1, 1 2
v± (p) =
(7)
we can diagonalize the matrix [M] as [λ] = [ξ]† [M][ξ] = diag[λ(1), λ(2)] with λ(1) = 2(1 − cos θ ) and λ(2) = 2(1 + cos θ ). Consequently, all the matrices in equation (6) can be diagonalized. They are denoted as [ξ]† [wi ][ξ] = diag[w˜ i (1), w ˜ i (2)] and [ξ]† [w± ˜± ˜± o ][ξ] = diag[w o (1), w o (2)]. Besides, we define (˜cLl ) = [ξ]† (cLl ) = [˜cLl (1), c˜ Ll (2)]T . With such a diagonalization procedure, apart from the first one, the equations in equation (6) can be rewritten in a scalar form. For p = 1 or 2, it is
±
t2 . E−e
(14)
Substituting equation (11) into (9) and defining ! ! φ1L+1 (p) c˜ L+1 1 (p) = [V(p)] , φ4L (p) c˜ L4 (p)
φ1L+1 (p) = ν1L (p)φ11 (p),
w˜ i (p)˜cL−1 ˜+ cL−1 ˜ L1 (p) = 0, o (p)˜ 1 (p) + w 4 (p) + t2 c L ≥ 2;
φ4L (p) = ν2L (p)φ00 (p).
(8)
From equation (8) we can establish a transfer matrix [T(p)] which connects the wavefunction between different layers in the way ! ! ˜ 11 (p) c˜ L+1 L c 1 (p) = [T(p)] . (9) c˜ L4 (p) c˜ 00 (p)
φ11 (p) = c˜ 11 (p) + v+ (p)˜c00 (p) = 0.
The transfer matrix is given by [T(p)] = # " + w˜ o (p)w ˜ i (p)−1 w ˜− ˜ i (p) t2−1 , w˜ + ˜ i (p)−1 o (p) − w o (p)w
(17)
To combine this condition with equation (6), we derive that the electronic probability amplitude on the line defect must conform to a constriction given as " t2 τ1−2 (E − e1 − τ2 eiθ ) − 2v+ (1) e−iθ , 0 [D](c0 ) = −t2 τ1−2 (E − e1 + τ2 eiθ ) + 2v+ (2) e−iθ , # −t2 τ1−2 (E − e2 − τ2 e−iθ ) + 2v+ (1) −t2 τ1−2 (E − e2 + τ2 e−iθ ) + 2v+ (2) c00 (1) = 0. × (18) c00 (2)
.
(10) The matrix [T(p)], a 2×2 matrix, can be readily diagonalized. It can be formally written as [T(p)] = [V(p)][ν(p)][V(p)]−1 ,
(16)
The eigenvalues of the transfer matrix given by equation (12) follow the relationship ν1 (p) · ν2 (p) = 1. If they have real solutions, we can further deduce that ν1 (p) > 1 and ν2 (p) < 1. Such a case just corresponds to an even-parity boundary state (EPBS) around the line defect. This can be seen from equation (16) which indicates that the wavefunction decays as the electron gets further away from the line defect. However, it must be noticed that the convergence of the wavefunction of the q1D eigenstate imposes a constraint condition
˜− cL1 (p) + w ˜ i (p)˜cL4 (p) = 0, t2 c˜ L−1 o (p)˜ 4 (p) + w L ≥ 2.
−t2 w˜ i (p)−1
(15)
we can obtain
t2 c˜ 00 (p) + w ˜− c11 (p) + w ˜ i (p)˜c14 (p) = 0; o (p)˜
−w ˜ i (p)−1 w˜ − o (p),
E2 − e2 − t22 − λ(p)t12 2t2 (E − e) #1 " 2 4λ(p)t12 t22 × ±1 − 1 − (E2 − e2 − t22 − λ(p)t12 )2
(11)
with [ν(p)] = diag[ν1 (p), ν2 (p)]. ν1 (p) and ν2 (p) are just the eigenvalues of the transfer matrix [T(p)]. By a simple derivation, we can obtain an explicit expression of the two eigenvalues in terms of the electronic energy E and the TB
Equation (18) implies that det[D] = 0 is essential if an EPBS is allowed around the line defect. By a simple derivation, from 4
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such a condition we obtain –
t22 2τ14
[(E − e1 )(E − e2 ) − τ22 ] +
+ 2 cos θ τ2 )v+ (1) +
t2 (2E − e1 − e2 2τ12
t2 (2E − e1 − e2 2τ12
− 2 cos θ τ2 )v+ (2) − 2v+ (1)v+ (2) = 0.
(19)
From this result we can determine the eigenenergy E of an EPBS in terms of the electronic wavevector ky and the TB parameters. With the obtained eigenenergy, from equation (18) we can work out the electronic probability amplitude (c00 ), which is expressed as ! [E2 − e2 − τ2 e−iθ − 2τ12 t2−1 v+ (1)]eiθ 0 . (20) (c0 ) = E2 − e1 − τ2 eiθ − 2τ12 t2−1 v+ (1) Then, from equations (15) and (16) we can get the wavefunction of the fourth sublayer in the Lth layer, (cL4 ) = [ς L ](c00 ),
(21)
where the matrix [ς L ] takes the form " # ν2L (1) + ν2L (2), [ν2L (2) − ν2L (1)]eiθ 1 L [ς ] = . (22) 2 [ν2L (2) − ν2L (1)]e−iθ , ν2L (1) + ν2L (2) Figure 2. Band structures of LD-G: (a) the gapless case, i.e., e = 0,
Subsequently, from equation (2) we can get the wavefunction (cL1 ) in terms of (c00 ), h i −1 (cL1 ) = −[w− t2 [ς L−1 ] + [wi ][ς L ] (c00 ). (23) o]
and (b) the gapped case with e = 0.2. The bold (cyan) lines represent the EPBS subbands which are obtained by solving the transcendental equation given in equation (19), labeled by Roman numerals from I to VIII. It should be noted that in the gapless case three discontinuous segments, named III(a), III(b) and III(c) respectively, belong to the same subband. They are separated by two small delocalized regions near two valley centers. The detail of the subband close to the K point is shown in the inset. The thin lines show the band structures obtained directly by the numerical TB calculation.
Substituting the wavefunctions (cL1 ) and (cL4 ) into the original Schr¨odinger equation [E − H(ky )](C) = 0, we can get (cL2 ) and (cL3 ). Finally, after a normalization procedure we obtain the wavefunction of an EPBS around the line defect. To describe the localization property of the EPBS, generally speaking, the inverse participation ratio [30] extensively adopted in the literature is an appropriate quantity. However, by analyzing the expression for the wavefunction given above, we can define a parameter which can intuitively describe the localization strength of the EPBS. According to equations (21)–(23), we can readily find that at any given layer, say the Lth layer, the wavefunction can always be written as a linear combination of the two terms [ν2 (1)]L and [ν2 (2)]L . Considering that |ν2 (1)| and |ν2 (2)| are both less than unity, one of the two terms is responsible for the wavefunction decay as it goes away from the line defect. Accordingly, we define a quantity η = max{|ν2 (1)|, |ν2 (2)|},
of the EPBS around the line defect. First, by solving the transcendental equation given by equation (19) we can obtain the eigenenergy E of the EPBS as a function of the electronic wavevector ky , i.e., the dispersion relation if these states form a subband. Then, from equations (20)–(24), we can calculate the wavefunction and the localization strength of these EPBSs. To begin with, we consider the simplest case of t1 = t2 = τ1 = τ2 = 1 and e = e1 = e2 = 0, which corresponds to a pristine graphene sheet in the presence of an extended line defect. The calculated result is shown in figure 2(a). It can be seen that there are indeed some subbands which are composed of the EPBSs. As labeled by the Roman numerals in figure 2(a), these subbands are located at the upper and lower boundaries of the total π -band (the conduction and valence bands of graphene), and in the region around the Dirac point, respectively. Strictly speaking, these EPBS subbands are discontinuous segments of a certain subband spanning the total Brillouin zone. For example, the segments III(a)–(c) belong to the same subband. However, they are separated by two small pieces of the bulk states near the K(π/3a) and K0 (−π/3a) points. The delocalization characteristics of
(24)
which can reasonably describe the localization strength of the EPBS around the line defect. A smaller η means a stronger localization around the line defect. 3. Numerical results
Following the analytical results presented above, we are now in a position to perform a numerical investigation 5
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the electronic states near the two valley center points can be briefly explained as follows. The variation of the lattice topology by the presence of the line defect gives rise to the Q1D boundary states around the line defect. However, the electron states near the K or K0 point are the ones of very large wavelengths. These long-wavelength states are not sensitively influenced by local lattice variation such as the line defect. Therefore, these states are delocalized or less localized than those far away from the valley centers. The global π-band structure of the LD-G shown in figure 2 is obtained by directly performing a numerical TB calculation. To do this we keep the lattice as shown in figure 1(a) infinite in the longitudinal (y) direction, but impose a periodic boundary condition in the transverse (x) direction with a large transverse size. As shown in figure 2(a), we can see that, apart from the EPBS subbands, there are also bulk bands contributed by the 2D extended states which show a Dirac cone like profile near both the K and K0 points, and the flat subband spanning the K and K0 points arising from the odd-parity boundary states. It should be noticed that the profiles of the EPBS subbands calculated from equation (19) are all consistent with the numerical TB results. More importantly, unlike the flat subband of the odd-parity boundary state, all the EPBS subbands are nontrivially dispersive. In figure 2(b) we show the band structure of LD-G for the case of e = 0.2. It is known that pristine graphene in the presence of a staggered potential can bring about a nonzero bandgap between the conduction and valence bands. For the LD-G structure, the influence of the staggered potential on the EPBS subbands is now our concern. In contrast to the case shown in figure 2(a), the two subbands at the lower boundary of the valence band remain, but in smaller regions near both terminals of the Brillouin zone. Meanwhile, a new subband (subband-VIII) at the upper side of the valence band emerges around the center of the Brillouin zone. More interestingly, subband-III persists in the bulk bandgap and the delocalized segment disappears, in contrast to the gapless case shown in figure 2(a). In order to obtain an intuitive understanding about the wavefunction of the EPBS, we need to show the electronic probability distributions of some typical EPBSs around the line defect. For such a purpose, we select two EPBSs with ky = 0.13π/a and 0.37π/a for both the gapless and gapped LD-G shown in figure 2, respectively. The former is close to the center of the Brillouin zone and the latter near the K point. We calculate the quantity |cLl1 (ky )|2 + |cLl2 (ky )|2 of these two states as a function of m, namely, the electronic transverse coordinate relative to the line defect. The calculated results are shown in figure 3. First of all, we can see that the results calculated by the theoretical approach given by equations (20) and (23) agree well with the numerical TB results. Then, from these results we infer that a nonzero bandgap can remarkably increase the localization of the EPBSs. Besides, the EPBS near the valley centers is less localized than the one close to the center of the Brillouin zone. Of course, to obtain a comprehensive understanding about the localization properties of the EPBS, we need to calculate the localization parameter η, which will be shown below. As seen above, subband-III appears in the low-energy regime with a nontrivial dispersion. In particular, it persists
Figure 3. The electronic probability distributions of two typical
boundary states in subband-III as a function of m for both the gapless and gapped LD-G. The results for the boundary states with ky = 0.13π/a and ky = 0.37π/a are shown in (a) and (b) respectively.
in the bulk bandgap of the gapped LD-G structure. These features mean that such a subband plays an important role in the electronic transport process of the LD-G. Therefore, below we will mainly focus on the electronic properties of subband-III. First, we study the influence of the hopping parameters on the EPBS in subband-III, including the subband profile and the localization strength. The numerical results for the case of changing τ1 and τ2 , which reflects the effect of the local lattice distortion around the line defect, are shown in figure 4 for the gapless LD-G. We can see from figures 4(a)–(c) that the change of these two parameters does not cause a general variation of the subband profile. However, when the ratio τ1 /τ2 deviates from unity, the delocalization regions around the K and K0 points broaden. When τ1 /τ2 < 1, the delocalization region emerges above the Dirac point (i.e., the zero point of energy). In the opposite case, namely, τ1 /τ2 > 1, the delocalization region emerges below the Dirac point. Corresponding to the cases shown in figures 4(a)–(c), the localization parameter η as a function of the electronic wavevector is plotted in figures 4(d) and (e). We can see that, besides the broadening of the delocalization regions by increasing the local lattice distortion around the line defect, the states far away from the valley centers, i.e., the K and K0 points, present stronger localization. The variation of the hopping parameters t1 and t2 , which accounts for the uniform strain field applied on the whole lattice, has a similar influence to the case of changing τ1 and τ2 . Namely, as t1 /t2 gets further away from unity, the delocalization regions increase. Such a result is shown in 6
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Figure 4. The influence of the hopping energies τ1 and τ2 on the dispersion relation (a)–(c) and the localization parameter (d)–(e) of subband-III for the gapless LD-G. The other parameters are fixed at t1 = t2 = 1 and e = e1 = e2 = 0.
Figure 5. The influence of the hopping energies t1 and t2 on the dispersion relation (a), (b) and the localization parameter (c), (d) of subband-III for the gapless LD-G. The other parameters are fixed at τ1 = τ2 = 1 and e = e1 = e2 = 0.
the valley centers, their shifts shown in figures 5(c) and (d) originate from the shifts of the valley centers by a uniform strain field which has been extensively discussed in the literature [31, 32]. For the gapped LD-G, the influence of the hopping parameters on subband-III is shown in figures 6 and 7, which resemble the gapless case. However, in contrast to the gapless case, what should be emphasized is that a nonzero
figure 5. However, as shown in figures 5(c) and (d), it is worth noting that with variation of the ratio t1 /t2 the delocalization region shows a notable shift with respect to the K (K0 ) point. When t1 /t2 < 1, it shifts towards the boundary of the Brillouin zone. On the other hand, when t1 /t2 > 1, the delocalization region shifts towards the center of the Brillouin zone. Considering that the delocalization regions adhere to 7
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Figure 6. The influence of τ1 and τ2 on the dispersion relation (a), (b) and the localization parameter (c), (d) of subband-III for the gapped
LD-G. The other parameters are fixed at t1 = t2 = 1, e1 = e2 = 0 and e = 0.2.
Figure 7. The influence of t1 and t2 on the dispersion relation (a), (b) and the localization parameter (c), (d) of subband-III for the gapped LD-G. The other parameters are fixed at τ1 = τ2 = 1, e1 = e2 = 0 and e = 0.2.
bandgap strengthens the localization of the EPBSs efficiently. As a result, the delocalization regions around the K and K0 points completely disappear. Now we turn to study the effect of the parameters e1 and e2 , i.e., the onsite energies of the two atoms C1 and C2 at the line defect. In previous work [33] it has been demonstrated that the adsorption of diatomic gas molecules such as H2 and F2 by the carbon atoms C1 and C2 can greatly alter the onsite energies of C1 and C2. Therefore, by changing
e1 and e2 in our model, we can clarify the influence of the adsorbates at the line defect on the electronic properties of the EPBSs. To describe the adsorption of the diatomic gas molecules, we need only to consider the case of e1 = e2 (denoted as e0 herein). Without loss of generality, we let t1 = t2 = τ1 = τ2 = 1 and e = 0.2. Then we change e0 from 0 to a very large value. The calculated result is shown in figure 8. From this figure we can see that in the case of e0 = 0 three EPBS subbands (III, V and VIII) emerge 8
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Figure 8. The effect of the onsite energies of the line defect atoms
Figure 9. Similar to figure 8. (a)–(d) The cases of e0 = 0, −1, −5
(e1 = e2 = e0 ) on the band structure of the LD-G. The other parameters are fixed at t1 = t2 = τ1 = τ2 = 1 and e = 0.2. (a)–(d) The cases of e0 = 0, 1, 1.5 and 1000, respectively. The bold lines (solid and dashed lines) represent the EPBS subbands, while the thin lines represent the band structures obtained directly by the numerical TB calculation.
and −1000, respectively.
defect. The case of decreasing e0 is shown in figure 9, which presents the opposite shift of the EPBS subbands. Namely, subbands-III, -V and -VIII disappear in the bulk valence band. However, a new EPBS subband occurs at the bottom of the bulk conduction band and it eventually merges into the flat subband of the odd-parity boundary states when e0 tends to negative infinity. Finally, in figure 10 we present the calculated results for the localization parameters η of other EPBS subbands, apart from subband-III. We can see that a common feature of these subbands is that the localization gets stronger when the electronic wavevector gets further away from the K and K0 points, no matter whether a bandgap is opened or not.
near the bulk bandgap. With increase of e0 , these subbands shift upward, as shown in figures 8(b)–(d). Subband-III, accompanying the total up-shift, is disconnected by a segment of the extended states, see figures 8(b) and (c). As e0 increases further, it merges into the bulk conduction band and ultimately disappears. Meanwhile, subbands-V and -VIII are separated from the bulk states and they join to form a complete subband when e0 = 1, as shown in figure 8(b). This subband gradually moves into the bulk bandgap with the continuous increase of e0 , e.g., the case of e0 = 1.5 as shown in figure 8(c). However, unlike subband-III, it does not disappear in the bulk conduction band; instead it will become completely degenerate with the odd-parity boundary state subband which shows flat dispersion at the bottom of the bulk conduction band. To show such a situation, we plot the calculated result of e0 = 1000 in figure 8(d), which corresponds actually to an infinite onsite energy on the line
4. Summary
In this work, we have obtained the analytical solutions of q1D boundary states around an extended line defect embedded in a graphene lattice. We find that the odd-parity boundary states with respect to the line defect show a flat dispersion spanning two inequivalent valleys. The wavefunctions of these states 9
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Jilin Province of China (Grant No. 20101511). Finally, we thank the High Performance Computing Center (HPCC) of Jilin University for calculation resources. References [1] Novoselov K S, Geim A K, Morozov S V, Jiang D, Zhang Y, Dubonos S V, Grigorieva I V and Firsov A A 2004 Science 306 666 [2] Semenoff G W 1984 Phys. Rev. Lett. 53 2449 [3] Ando T and Nakanishi T 1998 J. Phys. Soc. Japan 67 1704 [4] Zheng Y and Ando T 2002 Phys. Rev. B 65 245420 [5] Zhang Y, Tan Y-W, Stormer H L and Kim P 2005 Nature 438 201 [6] Novoselov K S, Geim A K, Morozov S V, Jiang D, Katsnelson M I, Grigorieva I V, Dubonos S V and Firsov A A 2005 Nature 438 197 [7] Kastnelson M I, Novoselov K S and Geim A K 2006 Nature Phys. 2 620 [8] Fujita M, Wakabayashi K, Nakada K and Kusakabe K 1996 J. Phys. Soc. Japan 65 1920 [9] Nakada K, Fujita M, Dresselhaus G and Dresselhaus M S 1996 Phys. Rev. B 54 17954 [10] Wakabayashi K, Fujita M, Ajiki H and Sigrist M 1999 Phys. Rev. B 59 8271 [11] Pereira V M, Guinea F, Lopes dos Santos J M B, Peres N M R and Castro Neto A H 2006 Phys. Rev. Lett. 96 036801 [12] Andrei Bernevig B, Hughes T L and Zhang S-C 2006 Science 314 1757 [13] Rycerz A, Tworzydło J and Beenakker C W J 2007 Nature Phys. 3 172 [14] Xiao D, Yao W and Niu Q 2007 Phys. Rev. Lett. 99 236809 [15] Zhai F, Zhao X F, Chang K and Xu H Q 2010 Phys. Rev. B 82 115442 [16] Gunlycke D and White C T 2011 Phys. Rev. Lett. 106 136806 [17] Wu Z H, Zhai F, Peeters F M, Xu H Q and Chang K 2011 Phys. Rev. Lett. 106 176802 [18] Wolf S A, Awschalom D D, Buhrman R A, Daughton J M, von Moln´ar S, Roukes M L, Chtchelkanova A Y and Treger D M 2001 Science 294 1488 ˇ c I, Fabian J and Das Sarma S 2004 Rev. Mod. Phys. [19] Zuti´ 76 323 [20] San-Jose P, Prada E, McCann E and Schomerus H 2009 Phys. Rev. Lett. 102 247204 [21] Lee H, Son Y-W, Park N, Han S and Yu J 2005 Phys. Rev. B 72 174431 [22] Son Y-W, Cohen M L and Louie S G 2006 Nature 444 347 [23] Kane C L and Mele E J 2005 Phys. Rev. Lett. 95 226801 [24] Lahiri J, Lin Y, Bozkurt P, Oleynik I I and Batzill M 2010 Nature Nanotechnol. 5 326 [25] Gunlycke D and White C T 2012 J. Vac. Sci. Technol. B 30 03D112 [26] Lin X and Ni J 2011 Phys. Rev. B 84 075461 [27] Bahamon D A, Pereira A L C and Schulz P A 2011 Phys. Rev. B 83 155436 [28] Jiang L W, Yu G D, Gao W Z, Liu Z and Zheng Y S 2012 Phys. Rev. B 86 165433 [29] Jiang L W, Zheng Y S, Yi C S, Li H D and L¨u T Q 2009 Phys. Rev. B 80 155454 [30] Wegner F 1980 Z. Phys. B 36 209 [31] Gui G, Li J and Zhong J 2008 Phys. Rev. B 78 075435 [32] Mohr M, Papagelis K, Maultzsch J and Thomsen C 2009 Phys. Rev. B 80 205410 [33] Gunlycke D, Vasudevan S and White C T 2013 Nano Lett. 13 259 [34] Song J, Liu H, Jiang H, Sun Q-F and Xie X C 2012 Phys. Rev. B 86 085437 [35] Yazyev O V and Louie S G 2010 Nature Mater. 9 806
Figure 10. The localization parameters of all the EPBS subbands
(apart from subband-III) labeled in figure 2 as a function of the electronic wavevector ky .
on one side of the line defect just copy an edge state localized at the zigzag edge of a pristine graphene sheet. With regard to these states, the line defect plays a role in separating the graphene lattice into two independent semi-infinite sheets. More interestingly, the line defect can induce EPBSs which are absent at a zigzag edge. These EPBSs form subband segments which appear around the Dirac point, as well as at the top and the bottom of the total π -band. In particular, one of the EPBS subbands with nontrivial dispersion persists in the bandgap of the bulk band if it is gapped. Such a subband provides a 1D channel to carry the current flowing along the line defect, which is robust to intravalley scattering. We have noticed that some relevant articles [34, 35] theoretically predicted highly transparent electronic transport along the line defect in graphene. This is consistent with our results on the Q1D boundary states around the line defect. A finite bandgap can efficiently enhance the localization of such a subband around the line defect. When the onsite energy of the atoms at the line defect goes far away from the Dirac point, which can be actually realized by absorbing diatomic gas molecules, the dispersion of the EPBS subband appearing in the bandgap becomes trivial. Finally, this subband tends to the flat dispersion subband of the odd-parity boundary states. Acknowledgments
This work was supported by the National Natural Science Foundation of China (Grant Nos NNSFC11074091 and NNSFC91121011) and the Natural Science Foundation of 10
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Quasi-one-dimensional electronic states induced by an extended line defect in graphene: an analytic solution
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Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 26 (2014) 035302 (10pp)
doi:10.1088/0953-8984/26/3/035302
Quasi-one-dimensional electronic states induced by an extended line defect in graphene: an analytic solution ¨ Liwei Jiang and Yisong Zheng Xiaoling Lu, National Laboratory of Superhard Materials, Department of Physics, Jilin University, Changchun 130012, People’s Republic of China E-mail: [email protected] Received 2 October 2013, revised 28 November 2013 Accepted for publication 28 November 2013 Published 20 December 2013 Abstract
Analytic solutions of the quasi-one-dimensional (q1D) electron states around an extended line defect in a graphene lattice are derived within the tight-binding model. Then, the electronic properties of this kind of boundary state in graphene are studied in detail. It is found that one subband composed of the even-parity boundary states emerges in the vicinity of the Dirac point. In particular, when the bulk band is gapped, such a one-dimensional subband remains in the bandgap, spanning two inequivalent valleys. In addition, this boundary state subband exhibits nontrivial dispersion, which can carry the valley polarized charge current flowing along the extended line defect. As a result, the line defect behaves like a one-dimensional channel for electronic transport. Moreover, its appearance in graphene or a hexagonal boron nitride sheet provides a promising way to print electric circuits in these two-dimensional materials. Keywords: surface states, electronic structure, line defect, graphene (Some figures may appear in colour only in the online journal)
1. Introduction
quantum devices [18–20] by manipulating the electronic spin or pseudospin. Prior to the experimental acquisition of a graphene sample, the edge state attached to the zigzag edge of a graphene lattice was theoretically studied [8–10, 21, 22]. It was found that the edge state is responsible for some unique electronic properties of graphene, such as the spontaneous edge magnetization [21] and the tunable half-metallicity [22]. Based on the chirality of the edge state, valley filtering and valley valve effects are theoretically predicted [13] to control the electronic spin in a similar way to that in the context of spintronics. Thus the concept of valleytronics paralleling spintronics was proposed [13–17]. To date, this topic has drawn much attention in the field of graphene study. More importantly, when the spin–orbital coupling is taken into account, the graphene strip becomes topologically nontrivial [23]. Accompanying the opening of the bulk bandgap, a pair of helical edge states with opposite spin polarization remain in the bandgap. As a result, the quantum spin Hall effect is expected to occur in graphene.
It is well known that in graphene [1], a one-atom thick layer of graphite, the low-energy electron excitation around the Fermi level behaves like a two-dimensional (2D) massless Dirac fermion [2, 3]. As a result, graphene presents many exotic electronic properties in contrast to conventional semiconductor materials, such as the half-integer quantum Hall effect [4–6] and reflectionless Klein tunneling [7]. Apart from the 2D bulk state, in graphene the edge state also plays an important role, and is localized at the lattice terminals [8–10] or the topological defects [11]. Due to the nature of the Dirac fermion, the graphene edge state is chiral in the sense that the pseudospin polarization depends on the electronic wavevector direction. This feature bears an analogy with the helical edge states of topological insulators [12]. It should be noted that the pseudospin in graphene refers to the sublattice [5, 6] or valley degrees of freedom [13–17] rather than the electronic physical spin. From the viewpoint of application, the edge state in graphene can be utilized to realize some functions of 0953-8984/14/035302+10$33.00
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Notwithstanding these intriguing theoretical predictions, it is still a big challenge for current experimental techniques to tailor a well-defined lattice edge to support the chiral edge state. Besides the zigzag edge, topological defects in the graphene lattice can also induce localized electronic states. In fact, many experimentally obtained graphene samples are not single-crystalline flakes. As a result, the aligned adjoining single-crystalline domains make domain boundaries intrinsic topological defects. Recently, an extended line defect of millimeter scale in an epitaxial layer of graphene was successfully fabricated [24], which was composed of a periodic repetition of one octagon plus two pentagons. Meanwhile, relevant theoretical investigations demonstrated that electronic transport through such a line defect can generate the valley filtering and valley valve effects [16]. In addition, around the line defect electronic spin polarization also occurs [25]. A zigzag-edged graphene nanoribbon embedded with an extended line defect shows a half-metallic subband structure [26]. From these theoretical investigations, one can naturally infer that the extended line defect plays a similar role to a zigzag edge. There are also numerical studies [27, 28] reporting that the extended line defect supports q1D electronic states around it. Considering that the line defect is currently an experimentally obtainable structure in a graphene sheet, in contrast to a highly qualified zigzag edge, it is of actual importance to reveal the electronic structure of the q1D states around the line defect, such as the subband profile, the localization strength of the wavefunction and a comparison between it and the edge state at a zigzag edge. In this work, we investigate analytically the q1D boundary states around a line defect in a graphene lattice. Within the tight-binding (TB) model we obtain the analytic eigen solutions of these boundary states. We find that a counterpart of the zigzag edge state is present around the line defect as an odd-parity boundary state with respect to the line defect. More interestingly, a new type of boundary state of even parity emerges. When a bandgap is opened between the bulk conduction and valence bands by a staggered potential, one subband of the even-parity boundary states remains in the bulk bandgap. In contrast to the flat dispersion of the odd-parity state, the subband profile of the even-parity boundary states exhibits notable dispersion spanning the two inequivalent valleys. Thereby the even-parity states can support valley polarized electronic transport. The boundary states near the K and K0 points possess opposite valley polarities. In these respects, the even-parity boundary states resemble the helical edge states in a topological insulator. In addition, due to the existence of the dispersive boundary state subband, an extended line defect in graphene acts as a quantum wire, which provides a feasible way to print electric circuits on graphene sheet. The rest of this paper is organized as follows. In section 2, the analytical solutions of the q1D boundary states around an extended line defect embedded in a 2D hexagonal lattice are obtained in the theoretical framework of a TB approximation. Then, in section 3, by means of the numerical calculations,
Figure 1. (a) A graphene lattice in the presence of an extended line
defect along the y direction. The shaded region defines a periodic unit with the primitive translation vector d (=2a). a is the lattice constant of the pristine graphene. The parameters t1 , t2 , τ1 and τ2 denote the nearest neighbor hopping energies in the TB model. The solid and hollow circles denote two distinct lattice points where the onsite energies of the carbon atoms are defined as +e and −e, respectively. C1 and C2 refer to two inequivalent carbon atoms at the line defect which can have distinct onsite energies denoted as e1 and e2 . The integer m measures the electronic transverse distance relative to the line defect. (b) An illustration of the TB Hamiltonian projected onto ky -space. The reduced TB parameters are labeled. To specify a lattice point, a set of indices (L, l, j) is required, as explained in (c). (c) The reduced lattice obtained by eliminating all the l = 2 and 3 sublayers of the ky -space lattice given by (b). The L interaction matrices w0o , w± o , wi and the wavefunction (cl ) are defined by the context.
we discuss the influence on the boundary states of the lattice strain, the bandgap opening and the absorption of exotic atoms around the line defect. Finally, the main results are summarized in section 4. 2. Theory
A schematic of a 2D hexagonal lattice in the presence of an extended line defect is shown in figure 1(a). The line defect consists of the periodic repetition of one octagonal plus two pentagonal rings along the y direction. Hereafter, we call such a structure line defect embedded graphene (LD-G) although it can also represent hexagonal boron nitride (h-BN) in the presence of an extended line defect. To study the electronic properties of LD-G, we adopt the π -electron TB model with the nearest neighbor approximation. In the TB Hamiltonian, the atomic onsite energies of the two sets of sublattices can be different from each other (noting that neither sublattice covers the two inequivalent atoms on the line defect, whose onsite energies can take distinct values from that of the atoms at the ordinary lattice points). This difference accounts well for the influence of the staggered potential of the substrate on a graphene sheet or two distinct atom species of an 2
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In the above equations, (cLl ) is a two-component column matrix, whose matrix element (cLl )j = Ci (ky ) is signified. To formulate concisely we will adopt the abbreviation L¯ = −L, and as the units of energy we set |t| = 1 (t ≈ −2.7 eV is the uniform hopping energy of the pristine graphene). The interaction matrices between the adjacent sublayers, as labeled in figure 1(c), are given by
h-BN sheet. For the nearest neighbor hopping energies, four different parameters are employed. As labeled in figure 1(a), the hopping energies between two ordinary lattice points are denoted as t1 and t2 . The case of t1 6= t2 implies the presence of a uniform strain. Moreover, the two hopping parameters τ1 and τ2 around the line defect may take different values from t1 and t2 , taking into account the local lattice distortion. Due to the translational symmetry of the LD-G lattice along the extended direction of the line defect, the electronic Bloch wavevector component ky is a good quantum number. Then the sub-Hilbert space spanned by the basis set 8iky (r) = P ik nd y u(r − R ) is closed for the TB Hamiltonian. √1 ni ne
[wi ] = t12 t2 (E2 − e2 − t22 )−1 [M], 2 2 2 2 −1 [w− o ] = (E − e) 1 − t1 (E − e − t2 ) [M] , 2 2 −1 2 2 [w+ o ] = (E + e) 1 − t1 (E − e − t2 ) [M] ,
Ny
where the matrix [M] takes the form " # 1, eiθ cos θ [M] = 2 −iθ , e cos θ, 1
Hereafter, such a subspace is referred to as ky -space. In the above definition of the basis function 8iky (r), u(r − Rni ) is the atomic π -orbital centered at the ith atom of the nth periodic unit. This periodic unit is actually a 1D strip, as shown in figure 1(a). d is the primitive translation vector in the y direction. Ny denotes formally the number of the periodic unit. In ky -space the electronic eigen wavefunction can be expanded in terms of the above basis set, which is given by X 9ky (r) = Ci (ky )8iky (r). (1)
which can be understood as the interaction between two inequivalent atoms at the line defect, labeled as C1 and C2 in figure 1(a). e1 and e2 stand for the onsite energies of these two atoms respectively. Away from the line defect, the onsite energies of the two sublattices, marked by solid and hollow circles as shown in figure 1(a), are designated as e and −e, respectively. Beginning with equation (2), we can work out analytical solutions of the q1D boundary states around the extended line defect if such states exist. The inversion symmetry of the LD-G lattice with respect to the line defect guarantees that the electronic eigenstates of the LD-G possess conserved parity. The wavefunction of an odd-parity state ¯ satisfies the relationship (cLl ) = −(cLl ). In particular, at the line defect one has (c00 ) = 0. Therefore, as far as the odd-parity eigenstates are concerned, the LD-G can be viewed as two pieces of independent semi-infinite hexagonal lattices with a zigzag edge. The odd-parity electronic wavefunctions at the right-hand side of the line defect follow the same Schr¨odinger equation as a zigzag-edged semi-infinite graphene lattice [8–10]. Consequently, the odd-parity boundary state at the right-hand side of the line defect has the same wavefunction and the same dispersion relation as the well-known edge state attached to the zigzag edge of graphene. In the absence of a staggered potential this kind of boundary state has zero energy and forms a dispersionless band spanning two inequivalent valleys. Many previous investigations have demonstrated that the edge state is responsible for some electronic properties unique to graphene; for example, the spontaneous magnetism [21] and valley-resolved electronic transport [13]. Now that this kind of electronic state appears as a boundary state around the line defect, one can reasonably conclude that a line defect bears some analogy with a zigzag edge. In addition to the odd-parity states, there are possibly even-parity boundary states around the line defect. By using
On the basis set 8iky (r), the TB Hamiltonian is projected into a specific ky -space which is denoted as H(ky ). It corresponds to a q1D lattice depicted by figure 1(b). The TB parameters projected into the ky -space are also labeled in this figure. Then, the Schr¨odinger equation takes the form [E − H(ky )](C) = 0 with E being the eigenenergy. The wavefunction (C) is a column matrix consisting of the electronic probability amplitudes Ci (ky ). From figure 1(b) we can see that three indices are needed to specify a lattice point. Namely, we need a notation i = (L, l, j), where L labels the layer and l = 1, . . . , 4 denotes the sublayer in a given layer. The last index, j = 1, 2, specifies the lattice point in the given layer and sublayer. It should be noted that the L = 0 layer, i.e., the line defect, has only a single sublayer (l = 0). The Hamiltonian H(ky ) and the corresponding Schr¨odinger equation can be further transformed into an analytically solvable model by deriving the indirect interaction between the l = 1 and 4 sublayers in each layer. In so doing, we need to eliminate the l = 2 and 3 sublayers in the way adopted in our previous work [29]. As a result, we obtain a further reduced lattice, as shown in figure 1(c). Corresponding to such a lattice, the electronic probability amplitude satisfies the following set of combined equations around the line defect:
L+1 L+1 t2 (cL1 ) + [w+ o ](c4 ) + [wi ](c1 ) ¯ 1¯ 0 [wi ](c14 ) + [w− o ](c1 ) + τ1 (c0 ) ¯ τ1 (c11 ) + [w0o ](c00 ) + τ1 (c11 ) 1 1 τ1 (c00 ) + [w− o ](c1 ) + [wi ](c4 ) L−1 + L [wi ](cL−1 1 ) + [wo ](c4 ) + t2 (c1 ) − L L t2 (cL−1 4 ) + [wo ](c1 ) + [wi ](c4 )
= 0,
L ≤ −2; L ≤ −2;
= 0; (2)
= 0; = 0; = 0,
L ≥ 2;
= 0,
L ≥ 2.
(4)
with θ = ky d/2 ranging from −π/2 to π/2. Besides, in equation (2) the special matrix is " # E − e1 , τ2 0 wo = , (5) τ2 , E − e2
i
L+1 L [wi ](cL4 ) + [w− o ](c1 ) + t2 (c4 ) = 0,
(3)
3
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¯
the relationship (cLl ) = (cLl ) and renormalizing the electronic probability amplitude on the line defect as (c00 )τ1 /t2 ⇒ (c00 ), from equation (2) we can obtain the Schr¨odinger equation of an even-parity state. At the right-hand side of the line defect it is given by the following set of equations:
" × 1−
t2 0 0 [w ](c ) + 2τ1 (c11 ) = 0; τ1 o 0 1 1 t2 (c00 ) + [w− o ](c1 ) + [wi ](c4 ) = 0; L−1 + L [wi ](cL−1 1 ) + [wo ](c4 ) + t2 (c1 ) − L L t2 (cL−1 4 ) + [wo ](c1 ) + [wi ](c4 )
parameters. For s = 1, 2, the eigenvalues are expressed as 2 ( E2 − e2 − t22 − λ(p)t12 1 − (−1)s νs (p) = 2λ(p)t12 t22
L ≥ 2;
= 0,
L ≥ 2.
2
2 2
E2 − e2 − t22 − λ(p)t1
The matrix [V(p)] in equation (11) can be written as " # 1, v+ (p) [V(p)] = , −1, v− (p)
(6)
= 0,
#1 )
4λ(p)t12 t22
By introducing a matrix
− 1. (12)
(13)
with the matrix elements "
#
1 −eiθ , eiθ , [ξ] = √ 1, 1 2
v± (p) =
(7)
we can diagonalize the matrix [M] as [λ] = [ξ]† [M][ξ] = diag[λ(1), λ(2)] with λ(1) = 2(1 − cos θ ) and λ(2) = 2(1 + cos θ ). Consequently, all the matrices in equation (6) can be diagonalized. They are denoted as [ξ]† [wi ][ξ] = diag[w˜ i (1), w ˜ i (2)] and [ξ]† [w± ˜± ˜± o ][ξ] = diag[w o (1), w o (2)]. Besides, we define (˜cLl ) = [ξ]† (cLl ) = [˜cLl (1), c˜ Ll (2)]T . With such a diagonalization procedure, apart from the first one, the equations in equation (6) can be rewritten in a scalar form. For p = 1 or 2, it is
±
t2 . E−e
(14)
Substituting equation (11) into (9) and defining ! ! φ1L+1 (p) c˜ L+1 1 (p) = [V(p)] , φ4L (p) c˜ L4 (p)
φ1L+1 (p) = ν1L (p)φ11 (p),
w˜ i (p)˜cL−1 ˜+ cL−1 ˜ L1 (p) = 0, o (p)˜ 1 (p) + w 4 (p) + t2 c L ≥ 2;
φ4L (p) = ν2L (p)φ00 (p).
(8)
From equation (8) we can establish a transfer matrix [T(p)] which connects the wavefunction between different layers in the way ! ! ˜ 11 (p) c˜ L+1 L c 1 (p) = [T(p)] . (9) c˜ L4 (p) c˜ 00 (p)
φ11 (p) = c˜ 11 (p) + v+ (p)˜c00 (p) = 0.
The transfer matrix is given by [T(p)] = # " + w˜ o (p)w ˜ i (p)−1 w ˜− ˜ i (p) t2−1 , w˜ + ˜ i (p)−1 o (p) − w o (p)w
(17)
To combine this condition with equation (6), we derive that the electronic probability amplitude on the line defect must conform to a constriction given as " t2 τ1−2 (E − e1 − τ2 eiθ ) − 2v+ (1) e−iθ , 0 [D](c0 ) = −t2 τ1−2 (E − e1 + τ2 eiθ ) + 2v+ (2) e−iθ , # −t2 τ1−2 (E − e2 − τ2 e−iθ ) + 2v+ (1) −t2 τ1−2 (E − e2 + τ2 e−iθ ) + 2v+ (2) c00 (1) = 0. × (18) c00 (2)
.
(10) The matrix [T(p)], a 2×2 matrix, can be readily diagonalized. It can be formally written as [T(p)] = [V(p)][ν(p)][V(p)]−1 ,
(16)
The eigenvalues of the transfer matrix given by equation (12) follow the relationship ν1 (p) · ν2 (p) = 1. If they have real solutions, we can further deduce that ν1 (p) > 1 and ν2 (p) < 1. Such a case just corresponds to an even-parity boundary state (EPBS) around the line defect. This can be seen from equation (16) which indicates that the wavefunction decays as the electron gets further away from the line defect. However, it must be noticed that the convergence of the wavefunction of the q1D eigenstate imposes a constraint condition
˜− cL1 (p) + w ˜ i (p)˜cL4 (p) = 0, t2 c˜ L−1 o (p)˜ 4 (p) + w L ≥ 2.
−t2 w˜ i (p)−1
(15)
we can obtain
t2 c˜ 00 (p) + w ˜− c11 (p) + w ˜ i (p)˜c14 (p) = 0; o (p)˜
−w ˜ i (p)−1 w˜ − o (p),
E2 − e2 − t22 − λ(p)t12 2t2 (E − e) #1 " 2 4λ(p)t12 t22 × ±1 − 1 − (E2 − e2 − t22 − λ(p)t12 )2
(11)
with [ν(p)] = diag[ν1 (p), ν2 (p)]. ν1 (p) and ν2 (p) are just the eigenvalues of the transfer matrix [T(p)]. By a simple derivation, we can obtain an explicit expression of the two eigenvalues in terms of the electronic energy E and the TB
Equation (18) implies that det[D] = 0 is essential if an EPBS is allowed around the line defect. By a simple derivation, from 4
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such a condition we obtain –
t22 2τ14
[(E − e1 )(E − e2 ) − τ22 ] +
+ 2 cos θ τ2 )v+ (1) +
t2 (2E − e1 − e2 2τ12
t2 (2E − e1 − e2 2τ12
− 2 cos θ τ2 )v+ (2) − 2v+ (1)v+ (2) = 0.
(19)
From this result we can determine the eigenenergy E of an EPBS in terms of the electronic wavevector ky and the TB parameters. With the obtained eigenenergy, from equation (18) we can work out the electronic probability amplitude (c00 ), which is expressed as ! [E2 − e2 − τ2 e−iθ − 2τ12 t2−1 v+ (1)]eiθ 0 . (20) (c0 ) = E2 − e1 − τ2 eiθ − 2τ12 t2−1 v+ (1) Then, from equations (15) and (16) we can get the wavefunction of the fourth sublayer in the Lth layer, (cL4 ) = [ς L ](c00 ),
(21)
where the matrix [ς L ] takes the form " # ν2L (1) + ν2L (2), [ν2L (2) − ν2L (1)]eiθ 1 L [ς ] = . (22) 2 [ν2L (2) − ν2L (1)]e−iθ , ν2L (1) + ν2L (2) Figure 2. Band structures of LD-G: (a) the gapless case, i.e., e = 0,
Subsequently, from equation (2) we can get the wavefunction (cL1 ) in terms of (c00 ), h i −1 (cL1 ) = −[w− t2 [ς L−1 ] + [wi ][ς L ] (c00 ). (23) o]
and (b) the gapped case with e = 0.2. The bold (cyan) lines represent the EPBS subbands which are obtained by solving the transcendental equation given in equation (19), labeled by Roman numerals from I to VIII. It should be noted that in the gapless case three discontinuous segments, named III(a), III(b) and III(c) respectively, belong to the same subband. They are separated by two small delocalized regions near two valley centers. The detail of the subband close to the K point is shown in the inset. The thin lines show the band structures obtained directly by the numerical TB calculation.
Substituting the wavefunctions (cL1 ) and (cL4 ) into the original Schr¨odinger equation [E − H(ky )](C) = 0, we can get (cL2 ) and (cL3 ). Finally, after a normalization procedure we obtain the wavefunction of an EPBS around the line defect. To describe the localization property of the EPBS, generally speaking, the inverse participation ratio [30] extensively adopted in the literature is an appropriate quantity. However, by analyzing the expression for the wavefunction given above, we can define a parameter which can intuitively describe the localization strength of the EPBS. According to equations (21)–(23), we can readily find that at any given layer, say the Lth layer, the wavefunction can always be written as a linear combination of the two terms [ν2 (1)]L and [ν2 (2)]L . Considering that |ν2 (1)| and |ν2 (2)| are both less than unity, one of the two terms is responsible for the wavefunction decay as it goes away from the line defect. Accordingly, we define a quantity η = max{|ν2 (1)|, |ν2 (2)|},
of the EPBS around the line defect. First, by solving the transcendental equation given by equation (19) we can obtain the eigenenergy E of the EPBS as a function of the electronic wavevector ky , i.e., the dispersion relation if these states form a subband. Then, from equations (20)–(24), we can calculate the wavefunction and the localization strength of these EPBSs. To begin with, we consider the simplest case of t1 = t2 = τ1 = τ2 = 1 and e = e1 = e2 = 0, which corresponds to a pristine graphene sheet in the presence of an extended line defect. The calculated result is shown in figure 2(a). It can be seen that there are indeed some subbands which are composed of the EPBSs. As labeled by the Roman numerals in figure 2(a), these subbands are located at the upper and lower boundaries of the total π -band (the conduction and valence bands of graphene), and in the region around the Dirac point, respectively. Strictly speaking, these EPBS subbands are discontinuous segments of a certain subband spanning the total Brillouin zone. For example, the segments III(a)–(c) belong to the same subband. However, they are separated by two small pieces of the bulk states near the K(π/3a) and K0 (−π/3a) points. The delocalization characteristics of
(24)
which can reasonably describe the localization strength of the EPBS around the line defect. A smaller η means a stronger localization around the line defect. 3. Numerical results
Following the analytical results presented above, we are now in a position to perform a numerical investigation 5
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the electronic states near the two valley center points can be briefly explained as follows. The variation of the lattice topology by the presence of the line defect gives rise to the Q1D boundary states around the line defect. However, the electron states near the K or K0 point are the ones of very large wavelengths. These long-wavelength states are not sensitively influenced by local lattice variation such as the line defect. Therefore, these states are delocalized or less localized than those far away from the valley centers. The global π-band structure of the LD-G shown in figure 2 is obtained by directly performing a numerical TB calculation. To do this we keep the lattice as shown in figure 1(a) infinite in the longitudinal (y) direction, but impose a periodic boundary condition in the transverse (x) direction with a large transverse size. As shown in figure 2(a), we can see that, apart from the EPBS subbands, there are also bulk bands contributed by the 2D extended states which show a Dirac cone like profile near both the K and K0 points, and the flat subband spanning the K and K0 points arising from the odd-parity boundary states. It should be noticed that the profiles of the EPBS subbands calculated from equation (19) are all consistent with the numerical TB results. More importantly, unlike the flat subband of the odd-parity boundary state, all the EPBS subbands are nontrivially dispersive. In figure 2(b) we show the band structure of LD-G for the case of e = 0.2. It is known that pristine graphene in the presence of a staggered potential can bring about a nonzero bandgap between the conduction and valence bands. For the LD-G structure, the influence of the staggered potential on the EPBS subbands is now our concern. In contrast to the case shown in figure 2(a), the two subbands at the lower boundary of the valence band remain, but in smaller regions near both terminals of the Brillouin zone. Meanwhile, a new subband (subband-VIII) at the upper side of the valence band emerges around the center of the Brillouin zone. More interestingly, subband-III persists in the bulk bandgap and the delocalized segment disappears, in contrast to the gapless case shown in figure 2(a). In order to obtain an intuitive understanding about the wavefunction of the EPBS, we need to show the electronic probability distributions of some typical EPBSs around the line defect. For such a purpose, we select two EPBSs with ky = 0.13π/a and 0.37π/a for both the gapless and gapped LD-G shown in figure 2, respectively. The former is close to the center of the Brillouin zone and the latter near the K point. We calculate the quantity |cLl1 (ky )|2 + |cLl2 (ky )|2 of these two states as a function of m, namely, the electronic transverse coordinate relative to the line defect. The calculated results are shown in figure 3. First of all, we can see that the results calculated by the theoretical approach given by equations (20) and (23) agree well with the numerical TB results. Then, from these results we infer that a nonzero bandgap can remarkably increase the localization of the EPBSs. Besides, the EPBS near the valley centers is less localized than the one close to the center of the Brillouin zone. Of course, to obtain a comprehensive understanding about the localization properties of the EPBS, we need to calculate the localization parameter η, which will be shown below. As seen above, subband-III appears in the low-energy regime with a nontrivial dispersion. In particular, it persists
Figure 3. The electronic probability distributions of two typical
boundary states in subband-III as a function of m for both the gapless and gapped LD-G. The results for the boundary states with ky = 0.13π/a and ky = 0.37π/a are shown in (a) and (b) respectively.
in the bulk bandgap of the gapped LD-G structure. These features mean that such a subband plays an important role in the electronic transport process of the LD-G. Therefore, below we will mainly focus on the electronic properties of subband-III. First, we study the influence of the hopping parameters on the EPBS in subband-III, including the subband profile and the localization strength. The numerical results for the case of changing τ1 and τ2 , which reflects the effect of the local lattice distortion around the line defect, are shown in figure 4 for the gapless LD-G. We can see from figures 4(a)–(c) that the change of these two parameters does not cause a general variation of the subband profile. However, when the ratio τ1 /τ2 deviates from unity, the delocalization regions around the K and K0 points broaden. When τ1 /τ2 < 1, the delocalization region emerges above the Dirac point (i.e., the zero point of energy). In the opposite case, namely, τ1 /τ2 > 1, the delocalization region emerges below the Dirac point. Corresponding to the cases shown in figures 4(a)–(c), the localization parameter η as a function of the electronic wavevector is plotted in figures 4(d) and (e). We can see that, besides the broadening of the delocalization regions by increasing the local lattice distortion around the line defect, the states far away from the valley centers, i.e., the K and K0 points, present stronger localization. The variation of the hopping parameters t1 and t2 , which accounts for the uniform strain field applied on the whole lattice, has a similar influence to the case of changing τ1 and τ2 . Namely, as t1 /t2 gets further away from unity, the delocalization regions increase. Such a result is shown in 6
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Figure 4. The influence of the hopping energies τ1 and τ2 on the dispersion relation (a)–(c) and the localization parameter (d)–(e) of subband-III for the gapless LD-G. The other parameters are fixed at t1 = t2 = 1 and e = e1 = e2 = 0.
Figure 5. The influence of the hopping energies t1 and t2 on the dispersion relation (a), (b) and the localization parameter (c), (d) of subband-III for the gapless LD-G. The other parameters are fixed at τ1 = τ2 = 1 and e = e1 = e2 = 0.
the valley centers, their shifts shown in figures 5(c) and (d) originate from the shifts of the valley centers by a uniform strain field which has been extensively discussed in the literature [31, 32]. For the gapped LD-G, the influence of the hopping parameters on subband-III is shown in figures 6 and 7, which resemble the gapless case. However, in contrast to the gapless case, what should be emphasized is that a nonzero
figure 5. However, as shown in figures 5(c) and (d), it is worth noting that with variation of the ratio t1 /t2 the delocalization region shows a notable shift with respect to the K (K0 ) point. When t1 /t2 < 1, it shifts towards the boundary of the Brillouin zone. On the other hand, when t1 /t2 > 1, the delocalization region shifts towards the center of the Brillouin zone. Considering that the delocalization regions adhere to 7
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Figure 6. The influence of τ1 and τ2 on the dispersion relation (a), (b) and the localization parameter (c), (d) of subband-III for the gapped
LD-G. The other parameters are fixed at t1 = t2 = 1, e1 = e2 = 0 and e = 0.2.
Figure 7. The influence of t1 and t2 on the dispersion relation (a), (b) and the localization parameter (c), (d) of subband-III for the gapped LD-G. The other parameters are fixed at τ1 = τ2 = 1, e1 = e2 = 0 and e = 0.2.
bandgap strengthens the localization of the EPBSs efficiently. As a result, the delocalization regions around the K and K0 points completely disappear. Now we turn to study the effect of the parameters e1 and e2 , i.e., the onsite energies of the two atoms C1 and C2 at the line defect. In previous work [33] it has been demonstrated that the adsorption of diatomic gas molecules such as H2 and F2 by the carbon atoms C1 and C2 can greatly alter the onsite energies of C1 and C2. Therefore, by changing
e1 and e2 in our model, we can clarify the influence of the adsorbates at the line defect on the electronic properties of the EPBSs. To describe the adsorption of the diatomic gas molecules, we need only to consider the case of e1 = e2 (denoted as e0 herein). Without loss of generality, we let t1 = t2 = τ1 = τ2 = 1 and e = 0.2. Then we change e0 from 0 to a very large value. The calculated result is shown in figure 8. From this figure we can see that in the case of e0 = 0 three EPBS subbands (III, V and VIII) emerge 8
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Figure 8. The effect of the onsite energies of the line defect atoms
Figure 9. Similar to figure 8. (a)–(d) The cases of e0 = 0, −1, −5
(e1 = e2 = e0 ) on the band structure of the LD-G. The other parameters are fixed at t1 = t2 = τ1 = τ2 = 1 and e = 0.2. (a)–(d) The cases of e0 = 0, 1, 1.5 and 1000, respectively. The bold lines (solid and dashed lines) represent the EPBS subbands, while the thin lines represent the band structures obtained directly by the numerical TB calculation.
and −1000, respectively.
defect. The case of decreasing e0 is shown in figure 9, which presents the opposite shift of the EPBS subbands. Namely, subbands-III, -V and -VIII disappear in the bulk valence band. However, a new EPBS subband occurs at the bottom of the bulk conduction band and it eventually merges into the flat subband of the odd-parity boundary states when e0 tends to negative infinity. Finally, in figure 10 we present the calculated results for the localization parameters η of other EPBS subbands, apart from subband-III. We can see that a common feature of these subbands is that the localization gets stronger when the electronic wavevector gets further away from the K and K0 points, no matter whether a bandgap is opened or not.
near the bulk bandgap. With increase of e0 , these subbands shift upward, as shown in figures 8(b)–(d). Subband-III, accompanying the total up-shift, is disconnected by a segment of the extended states, see figures 8(b) and (c). As e0 increases further, it merges into the bulk conduction band and ultimately disappears. Meanwhile, subbands-V and -VIII are separated from the bulk states and they join to form a complete subband when e0 = 1, as shown in figure 8(b). This subband gradually moves into the bulk bandgap with the continuous increase of e0 , e.g., the case of e0 = 1.5 as shown in figure 8(c). However, unlike subband-III, it does not disappear in the bulk conduction band; instead it will become completely degenerate with the odd-parity boundary state subband which shows flat dispersion at the bottom of the bulk conduction band. To show such a situation, we plot the calculated result of e0 = 1000 in figure 8(d), which corresponds actually to an infinite onsite energy on the line
4. Summary
In this work, we have obtained the analytical solutions of q1D boundary states around an extended line defect embedded in a graphene lattice. We find that the odd-parity boundary states with respect to the line defect show a flat dispersion spanning two inequivalent valleys. The wavefunctions of these states 9
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Jilin Province of China (Grant No. 20101511). Finally, we thank the High Performance Computing Center (HPCC) of Jilin University for calculation resources. References [1] Novoselov K S, Geim A K, Morozov S V, Jiang D, Zhang Y, Dubonos S V, Grigorieva I V and Firsov A A 2004 Science 306 666 [2] Semenoff G W 1984 Phys. Rev. Lett. 53 2449 [3] Ando T and Nakanishi T 1998 J. Phys. Soc. Japan 67 1704 [4] Zheng Y and Ando T 2002 Phys. Rev. B 65 245420 [5] Zhang Y, Tan Y-W, Stormer H L and Kim P 2005 Nature 438 201 [6] Novoselov K S, Geim A K, Morozov S V, Jiang D, Katsnelson M I, Grigorieva I V, Dubonos S V and Firsov A A 2005 Nature 438 197 [7] Kastnelson M I, Novoselov K S and Geim A K 2006 Nature Phys. 2 620 [8] Fujita M, Wakabayashi K, Nakada K and Kusakabe K 1996 J. Phys. Soc. Japan 65 1920 [9] Nakada K, Fujita M, Dresselhaus G and Dresselhaus M S 1996 Phys. Rev. B 54 17954 [10] Wakabayashi K, Fujita M, Ajiki H and Sigrist M 1999 Phys. Rev. B 59 8271 [11] Pereira V M, Guinea F, Lopes dos Santos J M B, Peres N M R and Castro Neto A H 2006 Phys. Rev. Lett. 96 036801 [12] Andrei Bernevig B, Hughes T L and Zhang S-C 2006 Science 314 1757 [13] Rycerz A, Tworzydło J and Beenakker C W J 2007 Nature Phys. 3 172 [14] Xiao D, Yao W and Niu Q 2007 Phys. Rev. Lett. 99 236809 [15] Zhai F, Zhao X F, Chang K and Xu H Q 2010 Phys. Rev. B 82 115442 [16] Gunlycke D and White C T 2011 Phys. Rev. Lett. 106 136806 [17] Wu Z H, Zhai F, Peeters F M, Xu H Q and Chang K 2011 Phys. Rev. Lett. 106 176802 [18] Wolf S A, Awschalom D D, Buhrman R A, Daughton J M, von Moln´ar S, Roukes M L, Chtchelkanova A Y and Treger D M 2001 Science 294 1488 ˇ c I, Fabian J and Das Sarma S 2004 Rev. Mod. Phys. [19] Zuti´ 76 323 [20] San-Jose P, Prada E, McCann E and Schomerus H 2009 Phys. Rev. Lett. 102 247204 [21] Lee H, Son Y-W, Park N, Han S and Yu J 2005 Phys. Rev. B 72 174431 [22] Son Y-W, Cohen M L and Louie S G 2006 Nature 444 347 [23] Kane C L and Mele E J 2005 Phys. Rev. Lett. 95 226801 [24] Lahiri J, Lin Y, Bozkurt P, Oleynik I I and Batzill M 2010 Nature Nanotechnol. 5 326 [25] Gunlycke D and White C T 2012 J. Vac. Sci. Technol. B 30 03D112 [26] Lin X and Ni J 2011 Phys. Rev. B 84 075461 [27] Bahamon D A, Pereira A L C and Schulz P A 2011 Phys. Rev. B 83 155436 [28] Jiang L W, Yu G D, Gao W Z, Liu Z and Zheng Y S 2012 Phys. Rev. B 86 165433 [29] Jiang L W, Zheng Y S, Yi C S, Li H D and L¨u T Q 2009 Phys. Rev. B 80 155454 [30] Wegner F 1980 Z. Phys. B 36 209 [31] Gui G, Li J and Zhong J 2008 Phys. Rev. B 78 075435 [32] Mohr M, Papagelis K, Maultzsch J and Thomsen C 2009 Phys. Rev. B 80 205410 [33] Gunlycke D, Vasudevan S and White C T 2013 Nano Lett. 13 259 [34] Song J, Liu H, Jiang H, Sun Q-F and Xie X C 2012 Phys. Rev. B 86 085437 [35] Yazyev O V and Louie S G 2010 Nature Mater. 9 806
Figure 10. The localization parameters of all the EPBS subbands
(apart from subband-III) labeled in figure 2 as a function of the electronic wavevector ky .
on one side of the line defect just copy an edge state localized at the zigzag edge of a pristine graphene sheet. With regard to these states, the line defect plays a role in separating the graphene lattice into two independent semi-infinite sheets. More interestingly, the line defect can induce EPBSs which are absent at a zigzag edge. These EPBSs form subband segments which appear around the Dirac point, as well as at the top and the bottom of the total π -band. In particular, one of the EPBS subbands with nontrivial dispersion persists in the bandgap of the bulk band if it is gapped. Such a subband provides a 1D channel to carry the current flowing along the line defect, which is robust to intravalley scattering. We have noticed that some relevant articles [34, 35] theoretically predicted highly transparent electronic transport along the line defect in graphene. This is consistent with our results on the Q1D boundary states around the line defect. A finite bandgap can efficiently enhance the localization of such a subband around the line defect. When the onsite energy of the atoms at the line defect goes far away from the Dirac point, which can be actually realized by absorbing diatomic gas molecules, the dispersion of the EPBS subband appearing in the bandgap becomes trivial. Finally, this subband tends to the flat dispersion subband of the odd-parity boundary states. Acknowledgments
This work was supported by the National Natural Science Foundation of China (Grant Nos NNSFC11074091 and NNSFC91121011) and the Natural Science Foundation of 10
