Sublattice imbalance of states in graphene nanoflakes

The energy states of π-electrons in a graphene nanoflake obtained from graphene, a well-known bipartite lattice or honeycomb lattice of carbon atoms, are studied using the tight-binding method. It is reported that the sublattice imbalance ΔN of the entire graphene nanoflake including vacancy clusters, which characterizes the electronic states, consists of those of the outer and inner edges. In nonzero-energy states, the electrons are evenly distributed between the sublattices A and B, regardless of the value of ΔN. In contrast, zero-energy states are ∣ΔN∣-fold degenerate states where the electrons are unevenly distributed on either sublattice A or sublattice B. Occasionally, large or specific graphene nanoflakes have substantial zero-energy states, which are mixed states of the nonzero-energy states and zero-energy states.


Introduction
Infinity has no end.Theoretically, a crystal is formed when a unit cell structure repeats itself infinitely; however, in reality, it has edges and is limited in size.A bipartite lattice is also broken at the edge or boundary.One such example is graphene, a honeycomb lattice of carbon atoms.Graphene and its derivatives (fullerenes such as C60 and carbon nanotubes) are very useful materials owing to their mechanical strength and suppleness, flexibility and versatility, and electronic properties such as the electron mobility and several parameters that control the energy gap.
Electrons in graphene are composed of σ-electrons in sp 2 hybrid orbitals and π-electrons, which are responsible for electrical conduction.The π-electron model of graphene is considered an efficient and suitable approach for examining electronic states in proximity to the Fermi level [1,2].The edge states of π-electrons localized near the zigzag edges of graphene nanoribbons still attract the interest of several researchers [3][4][5][6][7][8][9].
The theoretical calculation framework used to determine the electronic properties is identical to that used in our previous work on rectangular GNFs [12].We employed the simplest tight-binding Hamiltonian defined by where † c i and c i are the creation and annihilation operators for a π-electron at site i, respectively; t is the hopping integral between the nearest-neighbor sites 〈i, j〉; and ò is the site energy.Hereafter, the energies will be measured based on the value of ò in units of |t|.In this study, zero energy E = 0 corresponds to ò = 0 for the Fermi energy or the Dirac point.

Edge sites and sublattice imbalance
We consider GNFs constituted of hexagons, whose sites can be divided into inner and edge sites by the coordination number c; the former have c = 3 and the latter have c = 2.For the sake of completeness the smallest GNF is example (1) of one hexagon depicted in figure 1, which has only edge sites.The GNFs (2) and (3) of two hexagons consist of two inner sites, and 10 and 8 edge sites for (2) and (3), respectively.The number of hexagons (M), number of inner sites (N ( i) ), and other parameters for of all the GNFs illustrated in figure 1 are listed in table 1.
As presented in figure 1 and table 1, we are able to identify a relation between the number of hexagons M and the number of inner sites N ( i) in GNFs, whereas it is difficult to do the same for the total number of sites N. The relation is clearly The number of edge sites is We define the sublattice imbalance as where N A and N B are the numbers of A and B sites, respectively, in a GNF: The sublattice imbalance can be divided into those of the inner and edge sites as follows: Table 1.Specifications of the GNFs illustrated in figure 1: M denotes the number of hexagons; ) denote the numbers of inner and edge sites, respectively (number of sublattice A and B sites); and ΔN denotes sublattice imbalance.See text in section 2.2 for the dodecagonal notation.
No. where

Dodecagonal notation
N A i and ( ) N B i are the numbers of sublattices A and B of the inner sites, respectively, and ( ) N A e and ( ) N B e are those of the edge sites.
Here, we introduce another set of relations that was identified: which can be confirmed in figure 2 and table 2. Therefore, and from equation (6), naturally, , 11 The relations equations (9)-( 12) are equivalent.Further, we consider that equation (11) is useful because the sublattice imbalance of a GNF can be obtained from that of the edge sites.

Convex dodecagons
Most of the rectangular, triangular, hexagonal, parallelogram, trapezoidal, round, and other can be consolidated into a convex dodecagon with 12 edges because the smallest angle between a zigzag edge and an armchair edge is π/6.Therefore, the edges of the GNF can be characterized by 12 tangential angles and two types of zigzag and armchair edges.The tangential angle is represented by θ n for the nth edge, Examples of small GNFs.The inner sites are indicated by red circles.
where n = 1, 2, L ,12, as the convexity is specified by We start necessarily with a zigzag edge consisting of sublattice A sites with θ 1 = 0 in this study.The specifications of the dodecagonal GNF are listed in table 3. The dodecagonal GNF is described as follows: where we use two delimiters Z and A, which indicate a zigzag and an armchair type of edge, respectively, and S n following the delimiter represents the length of the edge expressed as the number of hexagons corresponding to the edge, that is, S n is the number of sublattice A or B sites constituting the zigzag edge when n is odd, and S n − 1 is the number of pairs of sublattice A and B sites constituting the armchair edge when n is even.Therefore, we introduce the following quantity:

( ) S S S S S S S S S S S S Z A Z A Z A Z A Z A Z
The number of edge sites in the dodecagonal GNF, because the pairs of sublattice A and B sites on the armchair edges never contribute toward sublattice imbalance.For example, a convex GNF with Z6A2Z2A3Z3A2Z3A3Z4A2Z2A2 in the dodecagonal notation is illustrated in figure 3, and the corresponding specifications are listed in tables 4 and 5, where equations (9)-( 11) can be confirmed.When n is odd, S n corresponds to the number of sites of the nth zigzag edge in the outer perimeter, whereas L n corresponds to that in the inner perimeter.The number of sites of all edges in the inner perimeter, which consists of the outermost sites of the inner sites, is expressed as follows: and the sublattice imbalance of the inner perimeter is expressed as follows: because the type of sublattice lattices constituting the zigzag edges in the inner perimeter is opposite to that in the outer perimeter.It is significant that the sublattice imbalance of a GNF, ΔN, can be obtained from the outer or inner perimeter using equation (11).Table 4. Specifications of the convex graphene flake Z6A2Z2A3Z3A2Z3A3Z4A2Z2A2 illustrated in figure 3 in accordance with the specifications presented in table 3. The sublattice imbalance of the outer perimeter is identical to that of the inner perimeter because L n of the zigzag edges in the outer perimeter are identical to the number of corresponding sites on the inner perimeter, whereas the types of sublattices are contrary.The number of pairs of sublattice A and B sites on the armchair edge that never contribute to the sublattice imbalance are indicated in parentheses.See the text for N ( e) , ΔN ( e) , N ( ip) , and ΔN ( ip) .Outer perimeter: Inner perimeter: The number of hexagons M in a dodecagonal GNF can be analytically obtained as follows: where M h is the height of the GNF, as follows: Applying equation (11) to the dodecagonal GNF, . 23 Because the outer perimeter, boundary, and inner perimeter of the dodecagonal GNF are closed as depicted in figure 3, where, for outer perimeters, L ae zigzag edges 1 3 3 a r m c h a i r e d g e s; 25 for boundaries, 26 and for inner perimeters, 3 a r m c h a i r e d g e s; 27 Table 5. Numbers of hexagons (M), sites (N), inner sites (N ( i) ), edge sites (N ( e) ), and sites on the inner perimeter (N ( ip) ) in the dodecagonal graphene flake (Z6A2Z2A3Z3A2Z3A3Z4A2Z2A2) illustrated in figure 3 are listed along with the values of sublattice imbalance.
Applying equations ( 23) to (29) and (30), The sublattice imbalance of the GNF can be determined from a pair of successive armchair-zigzag-armchair edges that face each other.Equation (31) becomes more simple for such GNFs, as they have edges with L n = 0 or L n = L n+6 .For example, ΔN = L 1 − L 7 in symmetric GNFs such as equilateral triangles, rectangles, and hexagons.It is also easy to obtain ΔN for the GNFs depicted in figures 1 and 2.

Vacancies and sublattice imbalance
The sublattice imbalance is also brought about by vacancies, which can be interpreted as hexagons removed from a graphene sheet as depicted in figure 4. A vacancy of sublattice A sites in figure 4(a) decreases N A in equation (4) by 1, which can be interpreted as increasing N B by 1.Thus, the vacancy contributes to the sublattice imbalance of graphene, ΔN ( h) = − 1.The contribution of a hole consisting of several vacancies to graphene is expressed as follows: which can be obtained from the sublattice imbalance of the sites on the inner perimeter of the GNF corresponding to the hole, and equations (11) and (20).The whole value of the sublattice imbalance of a GNF including holes is determined by the sublattice imbalance of the original GNF including no holes, ΔN 0 , and the contributions of the holes, ΔN ( h) : N h is the contribution of the ℓth hole and H is the number of the holes.Consequently, where, from equation (11), , 35 and equation (32) are used.The sublatice imbalance of a GNF including holes can be obtained from both the edge sites of the GNF and the sites on the inner perimeter surrounding holes inside the GNF.We will refer to them as outer edges and inner edges, respectively.The broken line represents the boundary between the outer and inner perimeters of the corresponding GNF.The contributions of holes to the sublattice imbalance of graphene can be obtained from equation (32).
However, we have already estimated the relation between the number of inner sites, N ( i) , and hexagons M (equation ( 2)).However, the relation does not hold in the case of a GNF including holes.Equation (2) indicated that an original GNF consisting of M 0 hexagons has inner sites.Eq. (3) indicates that the total number of sites of the GNF is N 0 , among which the number of edge sites is Introducing H holes to the original GNF, the number of hexagons is expressed as follows: where m ℓ is the number of hexagons constituting the ℓth hole.In the small GNF with m ℓ hexagons corresponding to the holes, the number of inner sites ) consists of vacant sites and the inner perimeter surrounding the vacancies, where equation (32) holds.The latter corresponds also to the edge sites inside the original GNF.The number of inner sites in the GNF including H holes is expressed as follows: where equations (36)-(39) are used.This relation depends not on the size and shape of voids but on the number of topological holes.Substituting equation (40) for N ( i) , it can be confirmed that the relations (9)-( 12) hold for GNFs including holes.The total number of sites is where equations (12) and (40) are used.Four possible cases obtained from equation (41) are listed in table 6.
For example, a hole corresponding to the dodecagon depicted in figure 3 in graphene contributes ΔN ( h) = − 2 to the sublattice imbalance.Furthermore, considering figure 4  (equation ( 34)).This GNF corresponds to Case III in table 6 because of N = 413.

Sublattice imbalance of states
In the π-electron model, a GNF containing N carbon atoms has N molecular orbitals (2N states for electrons).The wavefunction of a molecular orbital ψ ℓ for the ℓth energy level E ℓ (ℓ = 1, 2, L ,N) can be written as c i is the coefficient for the atomic orbital f ( i) of the ith site (i = 1, 2, L ,N) in the linear combination of atomic orbitals (LCAO) method.Let y ℓ ( ) A be the wavefunction of sublattice A sites and y ℓ ( ) B be that of sublattice B sites: Further, the intensities are expressed as We define a quantity of sublattice imbalance of states as ò ò The typical results of the numerical calculations of GNFs are presented in figure 5.The sublattice imbalance ΔN corresponds to the number of zero-energy states in any GNF.It can be found that nonzero-energy levels always have Δρ ℓ = 0, which means that the intensity is equally distributed in both sublattices A and B, irrespective of the number of sublattice sites, and zero-energy levels always have Δρ ℓ = ± 1, which means that the intensity is distributed only in either sublattice A or B. In the case of figure 5(a), the number of states is composed of the same number of states, as the number of sublattices A and B is the same.However, the energy states depicted in figures 5(b) and (c) are interesting because Δρ ℓ = 0 (r ) in the nonzero-energy ).
The nonzero-energy states are those in which the electrons are distributed evenly in both the sublattices A and B despite the value of ΔN, whereas the zero-energy states are |ΔN|-fold degenerate states in which the electrons are unevenly distributed on either sublattice A or B.

Outer edge states (edge states)
Two examples for dodecagonal GNFs are illustrated in figure 6, where the intensity distributions at zero energy and the sublattice imbalances of states are depicted.The sublattice imbalances, ΔN = 2 and ΔN = − 2, are reflected on the two zero-energy levels with Δρ ℓ = 1 and Δρ ℓ = − 1, respectively.The intensity distributions at zero energy are localized near the zigzag edges, which are completely biased on sublattice A or B sites according to Δρ ℓ = 1 or −1, as in the case of triangular GNFs [16].That is, the zero-energy levels in both (a) and (b) correspond to the result R2 defined in the preceding section, whereas the other levels correspond to R1.The characteristics can also be confirmed in other GNFs with well-defined symmetry in figures 7 and 8, of which the biases on the sublattices at the energy levels of the highest occupied molecular orbital (HOMO) or zero energy are listed in table 7. The HOMO states in the (a) rectangular, (b) hexagonal, and (c) dodecagonal GNFs with ΔN = 0 correspond to R1, whereas the GNFs (d), (e), and (f) with ΔN ≠ 0 correspond to R2.It is significant that the value of ΔN, which corresponds to the degree of degeneracy at zero energy, determines whether the πelectrons at zero energy are distributed on the sublattice A or B sites: the distribution is only on sublattice A sites for ΔN > 0, only on sublattice B sites for ΔN < 0, and on both sublattice A and B sites for ΔN = 0.A GNF with ΔN = 0 essentially has an energy gap, whereas that with ΔN ≠ 0 is gapless.This is discussed in further detail in the following sections.

Inner edge states (defect states)
Next, we consider the zero-energy states originating from the inner edges owing to vacancies in the GNFs, as described in section 2.3.Figure 9(a) and tables 8(a) present the states of HOMO in a hexagonal GNF with armchair edges, and figures 9(b)-(f) and table 8(b)-(f) present the states of HOMO or zero energy in the GNFs that holes are introduced into the original GNF (a).The GNFs (a)-(c) with ΔN = 0 have an energy gap, whereas the GNFs (d)-(f) with ΔN ≠ 0 have zero-energy states, where ΔN is the sublattice imbalance of equation (34).The so-called edge states do not appear in any GNF in figure 9 and tables 8 because the GNF has no zigzag edges in the outer perimeter.In particular, we consider that the zero-energy states in the GNFs (d)-(f), which are represented by the number of states on sublattice B sites in table 8(d)-(f), include inner edge states surrounding vacancies (defect states) because the intensities on all sublattice A sites are zero.The sublattice imbalances of states depicted in figure 10, which are affected by the sublattice imbalances owing to only inner edges, have profiles similar to the case of outer edges.However, the details cannot be revealed at present as it is not possible to apply the analysis method described in the appendix to the inner edge states.

One zigzag edge and vacancies
The zero-energy states in the GNFs containing a zigzag edge (outer edge) and voids (inner edges) are presented in figure 11 and table 9.In GNF (a) with ΔN = 4, the zero energy states are distributed only on sublattice A sites, and the edge states can be observed at the outer and inner edges.The inner edge states correspond to the defect states.In GNFs (b)-(d) with ΔN > 0, the edge states at the outer zigzag edge appear on sublattice A sites, whereas the inner edge states (defect states) cannot be observed.In GNF (e) with ΔN = 0, the zero-energy states are observed near both the outer edge on the sublattice A sites and the void on sublattice B sites.In GNF (f) with ΔN = − 1, the edge states at the outer zigzag edge on sublattice A sites never appear, whereas those at the inner  7. Rectangular GNF (a), zigzag-edged hexagonal GNF (b), and dodecagonal GNF (c) have an energy gap because the HOMO energies are nonzero, whereas triangular GNF (d), scalene zigzag-edged hexagonal GNF (e), and dodecagonal GNF (f) have zeroenergy states.The electrons at zero energy in GNFs with ΔN > 0 are biased only on sublattice A sites, whereas all sublattice B sites are nodes indicated by transparent circles.edge state (defect state) can be observed on sublattice B sites.It can be confirmed that the zero-energy states other than the case of (e) correspond to the result R2 expressed in section 3.1.The zero-energy states in the GNFs containing a zigzag edge (outer edge)and vacancies (inner edges) are presented in figure 12 and table 10.In GNFs (a), (d), and (e) with ΔN > 0, the edge states at the outer zigzag edge appear on sublattice A sites whereas the inner edge states (defect states) cannot be observed.In the GNF (b) with ΔN = 0, the zero-energy states are observed near both the outer edge on sublattice A site and the void on sublattice B sites, similar to that depicted in figure 11(e).In GNF (c) with ΔN = − 1, the edge states at the outer zigzag edge never appear, whereas those at the inner edge state (defect state) can be observed on sublattice B sites.The GNFs with the same value of ΔN in figures 11 and 12 and tables 9 and 10 have fundamentally similar profiles of the sublattice imbalance of states.That is, the profiles are determined by ΔN in accordance with the results R1 and R2.However, there are apparent exceptions: (e) in figure 11 12(c) despite the same value of ΔN = − 1.These cases will be considered in section 3.6.

Two zigzag edges and vacancies
As presented in figure 13 and table 11, we consider GNFs containing two adjacent zigzag edges: one consisting of sublattice A sites and the other consisting of sublattice B sites.GNF (a) with two Z10 edges has ΔN = 0, and GNF (d) with Z10 and Z7 edges has ΔN = 1.It can be seen that the sublattice imbalance ΔN ( h) caused by the vacancies surrounded by sublattice A or B sites, that is, the inner edges consisting of sublattice A or B sites, determines where the edge states appear.That is, it can be confirmed that the profile of the sublattice imbalance of states Δρ ℓ is determined by the value of ΔN owing to the outer and inner edges.The results R1 and R2 hold without exception.Further, figure 14 depicts rectangular GNFs consisting of a pair of holes [24], one surrounded by sublattice A sites and the other by sublattice B sites, which have ΔN = 0 because the original GNF is a rectangular GNF with ΔN 0 = 0 and a pair of holes with ΔN ( h) = 0.However, the sublattice imbalances of the states depicted in figures 14 (a) and (c) follow the result R1, whereas those depicted in (b) and (d) are not in accordance with R1 and R2.Essentially, the zero-energy levels in (b) and (d), as well as those in figures 11(e) and 12(b) should not exist because ΔN = 0.In reality, such zero-energy levels can be observed in figure 14(d).These cases will be discussed in the following sections.Table 8.Specifications of the HOMO or zero-energy states in armchair-edged hexagonal GNFs with holes depicted in figure 9. m denotes the number of hexagons constituting the hole.The sublattice imbalance ΔN consists of ΔN 0 = 0 for the original GNF and ΔN ( h) for the holes or inner edges.The intensity distributions of (a), (b), and (c) are even on both sublattice A and B sites, whereas those of (d), (e), and (f) are biased on sublattice B sites.  ).That is, the zero-energy state is unevenly distributed on either sublattice A or B, whereas the nonzero-energy state is evenly distributed on both sublattices A and B. In this section, the other energy levels with 0 < |Δρ ℓ | 1 are discussed.
In particular GNFs, originally nonzero-energy levels appear as quasi-zero-energy levels, which can be regarded as effectively zero energy within the measurement accuracy ΔE.In this case, the quasi-zero-energy levels plus the true zero-energy levels owing to sublattice imbalance ΔN are recognized as the W substantial zero-energy levels, because the quasi-zero-energy levels correspond to Q pairs of nonzero-energy levels above and below zero energy, namely, there are W energy levels E ℓ between −ΔE and ΔE.Because, in quantum mechanics, the W substantial zero-energy states are considered as mixed states, where the Iverson bracket R3: Substantial zero-energy states including quasi-zero-energy states can be represented as a mixture of nonzero-and zero-energy states.

Conclusion
The energy states of π-electrons in a GNF were studied.The sublattice imbalance ΔN of the entire GNF, which characterizes the electronic states, is reflected in the numbers of edge sites, inner sites, and sites around the vacancy clusters.It is eported that the sublattice imbalance of the GNF consists of those of the outer and inner edges through the relation between the sublattice imbalance and their sites.The nonzero-energy states are those in which the electrons are distributed evenly on both sublattices A and B despite the value of ΔN, whereas the zero-energy states are |ΔN|-fold degenerate states in which the electrons are unevenly distributed on either sublattice A or B. Particular or large GNFs occasionally have substantial zero-energy states, which are a mixture of nonzero-and zero-energy states.

Figure 3 .
Figure3.Example of dodecagonal GNFs consisting of 79 hexagons with six zigzag edges (Z) and six armchair edges (A) is represented by Z6A2Z2A3Z3A2Z3A3Z4A2Z2A2 in the dodecagonal notation.The first edge with tangential angle θ = 0 is the zigzag type, and it consists of sublattice A sites.The specifications are listed in tables 3 and 5.The outer perimeter of the GNF is formed by the edge sites whereas the inner perimeter consists of the outermost sites of the inner sites.The boundary between the outer and inner perimeters is represented by a broken line.The zigzag edges consisting of A (B) sites are represented by red (blue) lines, whereas the armchair edges are represented by yellow lines.The zigzag edges in the outer and inner perimeters are different in terms of the number of sites, S n and L n , respectively, and are contrary to each other in the type of sublattices, whereas the armchair edges are identical to each other in terms of the number of pairs of sublattice A and B sites.

Figure 4 .
Figure 4. Examples of holes in graphene or the corresponding GNFs, which consist of m hexagons represented by light blue color, contributing to sublattice imbalance ΔN ( h) : (a) a vacancy or GNF(6) depicted in figure 1 with m = 3 and ΔN ( h) = − 1, (b) a pair of vacancies or GNF(10) depicted in figure 1 with m = 4 and ΔN ( h) = 0, (c) three vacancies or GNF(21) depicted in figure 2 with m = 5 and ΔN ( h) = − 1, (d) four vacancies or GNF(25) depicted in figure 2 with m = 6 and ΔN ( h) = − 2, (e) five vacancies or GNF(22) depicted in figure 2 with m = 7 and ΔN ( h) = − 1, and (f) six vacancies or GNF(26) depicted in figure 2 with m = 8 and ΔN ( h) = − 2.The broken line represents the boundary between the outer and inner perimeters of the corresponding GNF.The contributions of holes to the sublattice imbalance of graphene can be obtained from equation (32).
as a rectangular GNF (Z15A1Z1A7Z1A1Z15A1Z1A7Z1A1 in the dodecagonal notation) with the sublattice imbalance ΔN 0 = 0 consisting of M 0 = 189 hexagons and six holes (H = 6) corresponding to å color, we find that the GNF which consists of M = 156 hexagons and N ( i) (6) = 322 inner sites from equation (40) has the sublattice imbalance D = D

Figure 6 .
Figure 6.Intensity distributions at zero energy obtained from numerical calculations of the dodecagonal GNFs: (a) Z6A2Z2A3Z3A2-Z3A3Z4A2Z2A2 and (b) Z6A2Z5A2Z1A4Z5A3Z1A3Z4A2, which contain the longest zigzag edge Z6.The sublattice imbalance of states for GNF (a), which is already presented in figure 3 and table 4, is Δρ ℓ = 1.0 at E ℓ = 0 (ℓ = 96 and 97) or otherwise Δρ ℓ = 0.0, whereas the sublattice imbalance of states for GNF (b) is Δρ ℓ = − 1.0 at E ℓ = 0 (ℓ = 128 and 129) or otherwise Δρ ℓ = 0.0.The electrons with zero energy exist only on sublattice A sites in (a) and only on sublattice B sites in (b).The intensity is normalized by the maximum value.Transparent circles indicate node sites.

Figure 7 .
Figure 7. Intensity distributions near or at zero energy in typical examples of dodecagonal GNFs whose specifications are listed in table7.Rectangular GNF (a), zigzag-edged hexagonal GNF (b), and dodecagonal GNF (c) have an energy gap because the HOMO energies are nonzero, whereas triangular GNF (d), scalene zigzag-edged hexagonal GNF (e), and dodecagonal GNF (f) have zeroenergy states.The electrons at zero energy in GNFs with ΔN > 0 are biased only on sublattice A sites, whereas all sublattice B sites are nodes indicated by transparent circles.

Figure 9 .
Figure 9. Intensity distributions at the levels of HOMO energy or zero energy in armchair-edged hexagonal GNFs: (a) original GNF: Z1A4Z1A4Z1A4Z1A4Z1A4Z1A4 (ΔN 0 = 0), (b) a vacancy pair that is surrounded by two sublattice A sites and two sublattice B sites (ΔN ( h) = 0), (c) two vacancies that are surrounded by three sublattice A sites and three sublattice B sites (ΔN ( h) = 0), (d) a vacancy that is surrounded by three sublattice B sites (ΔN ( h) = − 1), (e) a triangular hole that is surrounded by six sublattice B sites (ΔN ( h) = − 2), (f) a triangular hole that is surrounded by nine sublattice B sites (ΔN ( h) = − 3); the intensities are normalized by the maximum value.The transparent circles correspond to node sites.

Figure 10 .
Figure 10.Sublattice imbalances of states corresponding to the GNFs presented in figure9and table 8, which reflect the sublattice imbalances ΔN owing to the inner edges only.

Figure 11 .
Figure 11.Intensity distributions of the zero-energy states and the corresponding sublattice imbalances of states, which are obtained from the numerical calculations of the GNFs that contain a zigzag edge (Z10): (b) the original GNF (Z10A3Z1A1Z1A6Z1A6-Z1A1Z1A3) and the others with a vacancy surrounded by sublattice A sites (a), a vacancy surrounded by sublattice B sites (c), and triangular voids surrounded by sublattice B sites (d)-(f).The details are presented in table9.The intensities are normalized by the maximum value.The transparent circles correspond to the node sites.

Figure 13 .Table 11 .
Figure 13.Intensity distributions at zero or HOMO energy and the corresponding sublattice imbalances of states in the GNFs containing two adjacent zigzag edges and vacancies.The two adjacent edges are one with sublattice A sites and the other with sublattice B sites.Intensity distributions of the (b) original GNF (Z10A1Z10A1Z1A2Z1A9Z1A2Z1A1) for (a), (b), and (c) in the upper row; (a) GNF containing a vacancy surrounded by sublattice A sites; and (c) GNF containing a vacancy surrounded by sublattice B sites; (d) ioriginal GNF (Z10A1Z7A1Z1A3Z1A7Z1A2Z1A1) for (d), (e), and (f) in the lower row; (e) GNF containing a vacancy surrounded by sublattice B sites, and (f) GNF containing two isolated vacancies surrounded by sublattice B sites.The intensities are normalized by the maximum value, and the transparent circles correspond to the node sites.

Table 2 .
Specifications of GNFs illustrated in figure2: M denotes the number of hexagons; N ( i) ( ( ) ) denote the numbers of inner and edge sites, respectively (number of sublattice A and B sites); and ΔN denotes sublattice imbalance.See text in section 2.2 for the dodecagonal notation.

Table 3 .
Specifications of a graphene flake in the shape of convex dodecagon.θ n is the tangential angle for the nth edge.The nth edge is the zigzag type (Z) for odd numbers n or the armchair type (A) for even n.Moreover, the zigzag edges consist of sublattice A sites in the 1st, 5th, and 9th edges and sublattice B sites in the 3rd, 7th, and 11th edges.The armchair edges consist of pairs of sublattice A and B sites.The integer S n represents the length of the edge, and L n = S n − 1. See the text for details.
and table 9, (b) and (f) in figure 12 and table 10.The intensity distributions are on both sublattices A and B sites.The outer edge state on sublattice A sites and inner edge state (defect state) on sublattice B sites are observed.In particular, figure 12(f) is different from figure

Table 9 .
Specifications of the zero-energy states of the GNFs depicted in figure 11.The sublattice imbalance ΔN consists of ΔN 0 = 3 for the original GNF and ΔN ( h) for the vacancies.The number of states for the edge on sublattice A sites can be obtained from the total intensity of the rectangular part I rec of the edge states using equation (A5) with N y = 7.