Strongly localized states and giant optical absorption induced by multiple flat-bands in AA-stacked multilayer armchair graphene nanoribbons

We propose an AA-stacked multilayer graphene nanoribbon with two symmetrical armchair edges as a multiple flat-band (FB) material. Using the tight-binding Hamiltonian and Green’s function method, we find that the FBs are complete and merged into many dispersive bands. The FBs cause multiple strongly localized states (SLSs) at the sites of the odd lines in every sublayer and a giant optical absorption (GOA) at energy point 2t, where t is the electronic intralayer hopping energy between two nearest-neighbor sites. By driving an electric field perpendicular to the ribbon plane, the bandgaps of the FBs are tunable. Accordingly, the positions of the SLSs in the energy regime can be shifted. However, the position of the GOA is robust against such field, but its strength exhibits a collapse behavior with a fixed quantization step. On the contrary, by driving an electric field parallel to the ribbon plane, the completeness of FBs is destroyed. Resultantly, the SLSs and GOA are suppressed and even quenched. Therefore, such ribbons may be excellent candidates for the design of the controllable information-transmission and optical-electric nanodevices.


Introduction
The flat band (FB) system [1][2][3], possessing dispersionless subband in its band structure, has attracted an increasing interest recently. One is partly because this system is a possible platform to understand the fundamental physics of the topological insulator [4,5], the fractional quantum Hall effect [6][7][8], and the superconducting transition [9][10][11]. Other is partly because this system has some significant applications in the nanotechnology, for example, the manipulation of the FB can induce a metal-insulator transition [12] and an insulator-metal transition [13]. Therefore, there are many efforts to obtain the FB systems by translationally constructing the invariant lattice systems such as Lieb lattice [14][15][16], Kagome lattice [17,18], hexagonal structure [19,20], and their composite structures [21,22], etc.
It is well known that the monolayer graphene (GR) is a natural hexagonal structure [23]. And its one-dimensional counterparts, e.g. the armchair-edged ribbons (AGNRs) and the zigzag-edged ones (ZGNRs), are the basic elements in the graphene-based nanodevice [24][25][26]. All pristine ZGNRs are the FB systems which have been extensively studied [27][28][29][30]. In fact, the pristine AGNRs with two symmetrical edges in atomic configuration, i.e. containing an odd number of the carbon dimer atom lines crossing the ribbon width, are also the FB systems. And two complete FBs at energies E = ±t (t is the hopping energy between two nearest-neighbor sites) always exist in such AGNRs [28,31]. However, to our knowledge, the attentions toward them have been insufficient except the investigation on the ferromagnetism induced by the FBs [31]. It is partly owing to that it is difficult to manipulate an FB and remain its completeness by introducing an external electric field based on the structure of a single plane. Moreover, the properties of the electronic transports are not changed obviously even if the FBs are destroyed because they are merged into the dispersive bands.
Recently, the AA-stacked graphite [32], i.e. the bilayer and the multilayer graphene samples are fabricated experimentally [33,34]. In such samples, only atoms at the equivalent sites of two nearest-neighbor sublayers generate adjacent to each other with a weak coupling. Therefore, it is easy to cut them into one-dimensional armchair-and zigzag-edged nanoribbons. The electronic properties of the bilayer nanoribbons have some investigations [35][36][37][38]. It was found that some good properties in the monolayer nanoribbons still partly remain in the bilayer ones such as three categories of the semiconducting and metallic conduction according to the widths of the armchair ribbons [36,37]. Motivated on this evidence, here, we propose an AA-stacked multilayer armchair-edged graphene nanoribbon (AAMLAGNR) with two symmetrical edges as a multiple FB material.
In this paper, we theoretically investigate the band structure of the pristine AAMLAGNRs, by directly solving the tight-binding Hamiltonian. We show that the AAMLAGNRs with two symmetrical edges are the multiple FB systems, in which the complete FBs are merged into the dispersive bands. The FBs cause multiple strongly localized states (SLSs) only at the sites of the lines with odd order numbers, counting from one edge to the other edge. And these states can be obtained by calculating the local density of states (LDOS) at the corresponding sites based on the Green's function approach. Additionally, the FBs also produce a giant optical absorption (GOA) at energy ω = 2t ( = 1), which is displayed by a prominent sharp peak in the optical absorption spectrum A(ω). Moreover, the SLSs and the GOA are tunable by applying an electric field perpendicular or parallel to its sublayers, which is easily achievable in nowadays nanotechnological experiments.

Hamiltonian and method
The sketch of a pristine (M, N)-AAMLAGNR is depicted in figure 1, where it contains M sublayers in thickness (figure 1(a)) and each sublayer has N carbon dimer lines in width ( figure 1(b)). When N is odd, its two armchair edges are symmetrical in atomic configuration, otherwise they are antisymmetrical. In layer m, an unit cell α (the rectangle in figure 1(b)) contains 2N carbon atoms. We describe the atom sites in the cell as (m, n, α, β), where β = A, B denote the sites of atom A and B in dimer line n, respectively. Using the tight-binding model in the molecular orbital representation [39,40], the Hamiltonian can be written as with , Here, the state vector |m, n, α, A(B) denotes the molecular orbital of carbon atom (m, n, α, A(B)). H 1 and H 2 describe the nearest intralayer hopping with strength t and the interlayer hopping with strength γ, respectively. H 3 describes the field-staggered on-site potentials U m and U n , which are induced by the perpendicular electric field (E z in figure 1(a)) and the parallel electric field (E y in figure 1(b)) to its sublayers, respectively. In our paper, the intrinsic interactions between electrons in the GRs, such as the weak spin-orbit interaction, are neglected just like most studies. Of course, the interactions between electrons can cause interesting properties [41,42]. For example, a small local Coulomb interaction can generate a large intervalley gap at the assistance of the FB in the dice model [42]. We believe that some manipulations of the FB in the monolayer GRs, such as local Coulomb interactions induced by dopants [31], also are good ways to adjust our results which we will consider in the future investigations. In order to derive the band structure of the pristine systems, we employ a Fourier transform to H for a (M, N)-AAMLAGNR along the translational-invariant x direction (see figure 1(b)). The total wavefunction can be expressed as where parameters L x and k x denote the total number of unit cells of the ribbon and the electron momentum along the x direction, respectively. R α,x is the x-position of a site in unit cell α, and ψ β (k x , m, n) is the coefficient function of the waverfunction at site (m, n, α, β). Substituting equations (1) and (2) into the Schrödinger equation H|Ψ = E|Ψ , we obtain and where a is the intralayer lattice constant between two nearest carbon atoms. Applying the hard-wall boundary conditions to the thickness of ribbon in the z direction (see figure 1(a)) as and to the width of ribbon in the y direction (see figure 1(b)) as Taking equations (5) and (6) into equations (3) and (4), we can numerically obtain the band structure E(k x ) for the (M, N)-AAMLAGNR with definite M and N.
In this work, we also need to obtain the LDOS at arbitrary site in the (M, N)-AAMLAGNR, in which contains the information of the SLSs originating from the FBs. In the tight-binding model, the Green's function at site (m, n, α, β) in cell α can be written as where δ is a positive infinitesimal real quantity. Thus the LDOS at site (m, n, α, β) in the ribbon can be expressed as Although the electronic transport is insensitive to the FBs, the electronic hopping can realize between the states of two FBs. When an appropriate electric field is applied to a (M, N)-AAMLAGNR, electrons are excited from the occupied valence band E v (k x ) to the unoccupied conduction band E c (k x ). The vertical optical transitions with the same wavevector (Δk x = 0) occur, which can be described by the optical absorption spectrum with the x-polarized photons. According to the Kubo formula, the spectrum is expressed by [43,44] where C is a constant determined by the system parameters, p x is the x-component of the momentum operator, f(E c/v ) is the Fermi-Dirac distribution function in conduction/valence band. At zero temperature, f(E v ) = 1 and f(E c ) = 0 are taken when the on-site potential U m(n) is chosen symmetrically around the Fermi level (E F = 0). By drawing A(ω), one can see a GOA peak at a fixed energy point 2t in all AAMLAGNRs with two symmetrical edges.

Results and discussion
In this section, we present the numerical examples of the band structure (E(k x )), the SLSs, and the optical absorption spectrum (A(ω)) for a few-layer (M, N)-AAMLAGNR without and with external electric field, respectively. In our calculations, the nearest-neighbor intralayer hopping t = 2.57 eV and the interlayer hopping γ = 0.36 eV are taken according to the first-principles calculation [45]. The total units of ribbon L x = 5001 and the positive infinitesimal real quantity δ = 0.001 are fixed in the LDOS(ω) in equation (8) and the A(ω) in equation (9).

SLSs and optical absorption without external electric field
We firstly give the bandgap ΔE of AAMLAGNRs as the function of width (∼ N) and thickness (∼ M) in figure 2. We find that the '3p' conducting rule in the AGNRs [27,28] only applies to the AAMLAGNRs with p = 1. As shown in figure 2(a), the (M, 3)-AAMLAGNR with N = 3p and the (M, 4)-AAMLAGNR with N = 3p + 1 are semiconducting. Naturally, the (M, 5)-AAMMLAGNR with N = 3p + 2 is metallic. However, when p = 2 is taken, the (4, 6)-AAMMLAGNR with N = 3p and the (4, 7)-AAMLGNR with N = 3p + 1 are semiconducting while the (6, 6)-and (6, 7)-AAMMLAGNRs are metallic. Here, the '3p' rule is destroyed. More clearly, when p 3 is satisfied, all the (M, N)-AAMLAGNRs are metallic with gaps ΔE = 0. This is different from the AA-stacked bilayer nanoribbons [36,37], and is also very different from the AB-and ABC-stacked multilayer nanoribbons [46].  (M, 4)-AAMLAGNRs because the pristine 3-and 4-AGNRs have very large bandgaps ΔE > 0.5t [28]. Therefore, the (M, 3)-and (M, 4)-AAMLAGNRs are stable semiconducting channels which have potential applications as current switch in the molecular devices [47,48]. Moreover, they can also be produced experimentally in nowadays nanotechnology since a rigid monatomic linear chain of carbon atoms has been successfully carved out from graphene by employing energetic electron irradiation inside a transmission electron microscope [49]. Figure 3 shows the band structure E(k x ) for a set of the pristine AAMLAGNRs without external electric field. It is easily seen that all ribbons are metallic since p = 3 is taken. However, obviously, only the (5, 11)and (6, 11)-AAMLAGNRs have additional 10-and 12-FBs [the (red) solid lines in figures 3(c) and (d)], respectively, which are merged into many dispersive bands. This property originates from the fact that a (N, M)-AAMAGNR is composed of M-layered N-AGNRs. When N is odd, the two FBs with energies E = t and −t always exist in the band structures of such AGNRs [28,31]. Accordingly, it is natural to having 2M-FBs in the (M, N)-AAMLAGNRs with N = odd number. Moreover, the existence of interlayer coupling with strength γ causes the conduction (valence) FBs to discrete symmetrically around E = t(−t). These FBs can induce multiple SLSs at the sites of the odd lines in the AAMLAGNRs, which will be marked by the sharp peaks in LDOS(ω) discussed as follow.
In figure 4, we display the calculated LDOS(ω) at different carbon sites in the (5, 11)-and (6, 11)-AAMLAGNRs, respectively. Similar to the monolayer AGNRs [31], a site-selective localization rule exists in these systems. For the (5, 11)-AAMLAGNR, we can see that the sharp and high peaks of LDOS appear at the sites of the odd carbon dimer lines (see figure 4(a) for layer 1, (b) for layer 2, and (c) for layer 3). Such peaks are the signs of the SLSs induced by the FBs. Note that the SLSs can change into the compact localized states (CLSs) [50,51] by gapping the FBs away from the dispersive energy spectrum [52]. The CLSs are ideal candidates for the transmission of information [53]. In contrast, the SLSs vanish at the sites of the even lines (see figure 4(d) for layer 1, (e) for layer 2, and (f) for layer 3). In this case, there are only lower oscillation LDOS peaks due to the Van Hove singularity in quasi-1D systems. This above rule is also applicable to the (6, 11)-AAMLAGNR (see figures 4(g)-(i)). Moreover, there also is an FB-selective occupancy rule in the AAMLAGNRs with M = odd number. The numbers of the SLSs are unequal at the sites of layer 1 ( figure 4(a)), layer 2 ( figure 4(b)), and layer 3 (figure 4(c)) in the (5, 11)-AAMLAGNR while they are equal at the sites of any sublayer in the (6, 11)-AAMLAGNR (figures 4(g) and (i)). In order to understand the two rules, we use Onipko's approach [54] to obtain the analytical expression of the electronic band structure in the pristine AAMLAGNR without external electric field. Hence we have can be written as and those at the sites of the even lines (n = 2, 4, 6, . . .) can be written as where the parameters C A(B) and D A(B) are structure functions determined by the position of the site and the normalization condition over an unit cell in the AAMLAGNR. For the (5, 11)-AAMLAGNR, when the integer j 1 = 6 is taken, the FB energies E FB = ±t − 2γ cos(πj 2 /6) with j 2 = 1, 2, 3, 4, 5 can be derived from equation (10), thus 10-FBs exist in the figure 3(d). Moreover, their corresponding 10-SLSs all appear at sites of the odd lines in layer 1 (see figure 4(a)) since the wavefunction components ψ A(B) (k x , 1, n) = 0 are satisfied according to equation (11). However, when j 2 = 3 and m = 2 are taken, E FB = ±t and ψ A(B) (k x , 2, n) = 0 are obtained. Accordingly, the SLSs at E = ±t disappear in layer 2 ( figure 4(b)). Equivalently, the SLSs at E = ±0.86t and ±1.14t disappear in layer 3 (figure 4(c)), which can be deduced from equations (10) and (11) when j 2 = 2 and m = 3, j 2 = 4 and m = 3 are taken, respectively. According to equation (12), it is easily to be found the wavefunction components ψ A(B) (k x , m, n) = 0 at E FB = ±t − 2γ cos(πj 2 /6) are robust against all values m for the even numbers n. Therefore, the SLSs do not exist at the sites of the even lines (see figures 4(d)-(f)). It is worth mentioning that for the (6, 11)-AAMLAGNR, the part absence of SLSs does not occur (see figures 4(g) and (h)) since all wavefunction components ψ A(B) (k x , m, n) = 0 are right in equation (11) at the FB energies E FB = ±t − 2γ cos(πj 2 /7) in equation (10).   Figure 5 shows the optical absorption spectrum A(ω) and FB structure E(k x ) in the (M, N)-AAMLAGNRs without external electric field. In figure 5(a), a GOA peak abruptly appears at ω = 2t ( = 1) (blue solid line), comparing to other peaks (black dash line in figure 5(a1)). This property stems from the electronic excitation from the occupied valence band to the unoccupied conduction band in  an FB-pair with gap ΔE = 2t. As shown in figure 5(b), there are 5-paired FBs with gap ΔE = 2t in the (5, 11)-AAMLAGNRs. Such FB-pair is mainly induced by electrons in a same sublayer of the AAMLGNRs, according to equation (10). Therefore, the superposition effects of the electronic excitations, induced by the FB-pair of the different sublayers, produce the giant absorption peak at ω = 2t. It is also acceptable that the giant peak increases linearly as the layers M in the (M, 11)-AAMLAGNRs (see figure 5(c)) while it is insensitive to the width N in the (5, N)-AAMLAGNRs with N = odd number (see figure 5(d)). Accordingly, it is possible to obtain the layer-number of the AAMLAGNR with two symmetrical armchair edges by probing the GOA at ω = 2t.

Perpendicular electric-field manipulation on SLSs and optical absorption
Nextly, we turn to a perpendicular electric field manipulation on the above discussed FBs, SLSs, and optical absorption spectrum A(ω) in the pristine AAMLAGNRs. When an uniform electric field with strength E z is perpendicularly applied to the AAMLAGNRs as shown in figure 1(a), all sites in layer m obtain a same potential U m = U 1 + (m − 1)(U M − U 1 )/(M − 1). Here, potentials U M and U 1 are for the sites of layers M and 1, respectively. To fix the Fermi level E F = 0, we take U 1 = −U M . We choose the total potential U = 2U M to describe the electric field since the field strength E z = U/[(M − 1)de] is defined, where d and e are the interlayer distance and electron charge, respectively. Figure 6 shows the band structure E(k x ) and the LDOS(ω) under a perpendicular electric field in the (5, 11)-AAMLAGNR. Comparing to figure 6(a) with U = 1t and figure 6(b) with U = 2t, an enhancing electric field can widen the bandgaps ΔE between two nearest conduction (valence) FBs. Therefore, the lowest conduction (highest valence) FB very closes to the Fermi level when U = 2t is taken ( figure 6(b)). At this case, the completeness of FBs remain unchanged. Accordingly, a perpendicular electric field can shift the positions of the SLSs (see the sharp peaks of the LDOS in figures 6(c) and (d)). However, the field manipulation on the strength of the SLSs is complicated. As the total potential U increases, comparing to figures 6(c) and (d), some SLSs become strong, others become weak. Therefore, such electric field can destroy the FB-selective occupancy rule in figure 4. But, the site-selective localization rule in figure 4 is robust against the perpendicular electric field (see from figures 6(c)-(f)). Figure 7 displays the FB structure E(k x ) and the optical absorption spectrum A(ω) under a perpendicular electric field in the (5, 11)-AAMLAGNR. Despite the fact that the electric field can shift the FBs, there is still 5-paired FBs with bandgap E = 2t (figure 7(a)) when the total potential U = 1t is taken. Therefore, the strength of giant absorption A(ω) at ω = 2t (figure 7(b)) almost remains unchanged, comparing to the zero-field case in figure 5(a). However, when the potential U = 2t is taken, the FB-pairs with bandgap E = 2t reduces to 3-pairs (figure 5(c)). As a result, the peak of A(ω) at ω = 2t decreases in figure 5(d) by a comparison with figure 5(c). This is reasonable since the other 2-paired FBs with bandgap E 1 = 2t are located above and down the Fermi level E F = 0, respectively. Electronic excitation in two occupied valence FBs is forbidden. Apparently, such excitation also does not occur in two unoccupied conduction FBs. Motivated on this, we can produce a quantized optical absorption spectrum by driving a perpendicular electric field as shown in figure 8. Figure 8 displays the optical absorption spectrum A(ω) at ω = 2t as the function of the total potential U in the (M, 11)-AAMLAGNRs. One can see that a conspicuous set of quantized steps exists in this system. For instance, in the (10, 11)-AAMLAGNR, the potential U varies away 0, the spectrum A(ω) 10.4 is relatively stable (figure 8(a)). When the potential U 1.84t is adjusted, the value A(ω) abruptly jumps down from 10.4 to 8.4, where A(ω) 8.4 is the absorption peak of the (8,11)-AAMLAGNR at the zero case ( figure 8(a)). Continually, steps A(ω) 6.3 of the (6, 11)-AAMLAGNR and A(ω) 4.2 of the (4, 11)-AAMLAGNR ( figure 8(b)) appear one by one. Similarly, the peak A(ω) of the (9, 11)-AAMLAGNR at the zero electric field can degenerate into that of the (7, 11)-AAMLAGNR, of the (5, 11)-AAMLAGNR, and of the (3, 11)-AAMLAGNR. Accordingly, it is possible that the giant absorption peaks of the (10, 11)and (9, 11)-AAMLAGNRs collapse to the cases of bilayer and monolayer AGNRs, respectively.

Parallel electric-field manipulation on SLSs and optical absorption
Lastly, we discuss a parallel electric field manipulation on the above discussed FBs, SLSs, and optical absorption spectrum A(ω) in the pristine AAMLAGNRs. When an uniform electric field with strength E y is applied and parallel to the layer-planes ( figure 1(b)), different potentials U n = U 1 + (n − 1)(U N − U 1 )/(N − 1) are given to the sites of different lines n in every sublayer, where potentials U N and U 1 are for lines N and 1, respectively. To fix the Fermi level E F = 0, we take U 1 = −U N . Additionally, we choose the total potential U = 2U N to describe the electric field since the field strength E y = 2 √ 3U/[3(N − 1)ae] is defined. Figure 9 exhibits the results of above manipulations in the (5, 11)-AAMLAGNR. The parallel electric field gives the different potentials U n to the sites of the different line n in the same sublayer m, but the same on-site potential distribution exists in the different sublayers. Therefore, we can see that the completeness of FBs are destroyed more and more severely with the increase of the total potential U (see figures 9(a) and (b)), but the FB positions remain unchanged by a comparison with the case of the zero electric field ( figure 3(c)). Correspondingly, the positions of SLSs are also very stable. Their peaks become lower as the potential U increases (see figures 9(c) and (d)). Specifically, when the potential U = 1t is taken, the SLSs are indistinguishable because they have hidden into other electron states ( figure 9(d)). Therefore, the parallel electric field can suppress the SLSs and hide them, which may be useful in the secure transmission of information by hiding data [55]. Here, the sharp peak of the giant optical absorption A(ω) at ω = 2t also is strongly suppressed by the parallel electric field, comparing figure 9(e) with U = 0.5t to figure 5(a) with U = 0. As expected, when a strong electric field with U = 1t is taken, such peak disappears nearly (figure 9(f)). Therefore, the strength of the giant optical absorption is tunable by driving a parallel electric field, but its position is robust against such field.

Summary
In this paper, we have theoretically studied the band structure, the local density of states and the optical-absorption spectrum in the pristine (M, N)-AAMLAGNRs by using the tight-binding model. The '3p' conducting rule of the AGNRs only applies to the AAMLAGNRs with p = 1. It is said that the (M, 3)and (M, 4)-AAMLAGNRs are semiconducting, and the (M, 5)-AAMLAGNR is metallic. Moreover, their bandgaps are very large and insensitive to the thickness M, which should be useful as a narrow semiconducting channel in the molecular devices. When the integer p > 2 is taken, all AAMLAGNRs are metallic. Interestingly, 2M-FBs exist in the (M, N)-AAMLAGNRs with two symmetrical edges, i.e. N is an odd number. Moreover, they are complete. A site-selective localization rule exists in this system. Namely, these complete FBs induce the SLSs at the sites of the odd lines in every sublayer, which can be exhibited by the sharp peaks of the LDOSs at the corresponding sites. An FB-selective occupancy rule also exists in this system. Namely, in the (M, N)-AAMLAGNRs with M = odd number, electrons in different sublayers occupy different FBs, which has been revealed by the different number of the sharp peaks in the SLSs. A giant optical absorption abruptly appears and holds at the energy point ω = 2t ( = 1), which's strength is determined mainly by the number of the layers of AAMLAGNRs.
Additionally, the completeness of the FBs are robust against the perpendicular electric field but their positions can be shifted. The FB-selective occupancy rule in the zero field is destroyed. Some SLSs become increasingly strong while others become decreasingly weak as the total potential U increases. The giant optical absorption at ω = 2t is relatively stable under a suitable electric field. However, by adjusting the potential U, the giant optical absorption of the (M, N)-AAMLAGNR can collapse to the zero-field cases of the (M − 2, N)-AAMLAGNR, (M − 4, N)-AAMLAGNR, . . . , in turn. Resultantly, a quantized steps of absorption spectrum is observed in this system. On the contrary, by applying a parallel electric field, the positions of the FBs remain unchanged while their completeness is destroyed. The SLSs and the giant optical absorption become increasingly weak. The position of the giant absorption at ω = 2t is also robust against such electric fields. The SLSs can change into the CLSs by gapping the FBs away from the dispersive energy spectrum. Therefore, such AAMLAGNRs may be ideal candidates for the transmission of information. These systems may also be excellent candidates for the design of the controllable optical-electric FB-devices since the giant optical absorption exists and is tunable. It needs to be pointed out that the SLSs and GOA do not appear in the antisymmetrical AAMLAGNRs with even dimer lines in width since there are no FB in their band structures.
It is worth mentioning that to enhance the stability of the graphene nanoribbons, the carbon atoms at the two edges are often passivated by other atoms such as the hydrogen-and oxygen-passivations [56]. For the H-passivated case, an ab initio calculation shows that the bond lengths at the edges, parallel to dimer lines, are shortened by 3.3%-3.5% for the 12-, 13-, and 14-AGNRs as compared to those in the middle of the ribbon [57]. A 3.5% decrease would cause the nearest-neighbor hopping energy at the edges t 1 = 1.12t [58]. Our calculations find that the multiple FBs, SLSs, and GOA also exist in the H-passivated AAMLAGNRs with two symmetrical edges (no show here). The edge passivations change their positions and values, but not inherently.