Introduction

Graphene1 has unique and highly tunable parameters, which can be exploited in novel hybrid materials and devices with numerous applications. It can be modified by doping2,3, chemical functionalization4 and geometrical restrictions, such as cutting and introduction of defects and pores5,6,7. Recently, graphene perforated with nanopores was used as a selective sieve for hydrated ions11, gases12,13 and DNA15,16.

In this work, we use large scale ab-initio calculations to describe electronic structures of nanocarbons perforated with superlattices of nanopores. We search for common principles allowing us to characterize the electronic structures of porous nanocarbons (PNC), such as porous graphene (PG), porous graphene nanoribbons (PGNR) and porous carbon nanotubes (PCNT). Although, partial results for the electronic structures of PG5,6,7 and other PNCs have been obtained17,18,19,20, general principles that would unify their electronic structures are missing.

Results

The studied PNCs are perforated with pores of different shapes, sizes and locations. Most of the results are obtained for arrays of honeycomb-shaped pores, called the “standard pore” (SP)13,14, where in each pore six C atoms (one benzene ring) are excluded from the studied nanocarbon and the dangling bonds are terminated by hydrogens. All the PNCs have their edges and pores passivated by hydrogen atoms. The details of our calculations are described in Methods.

Porous graphene nanoribbons

We begin our study by examining the band structures of porous armchair (AGNR) and zigzag (ZGNR) graphene nanoribbons. All GNRs are classified by the number of carbon dimers, N, that form the ribbon (N-GNR)20. First, we study the porous 11-AGNR and 10-ZGNR (both metallic when pristine) and elementary cells are shown in Figs. 1 (d, h). In Figs. 1 (a, b), we can see that the introduction of a periodic array of SPs in the center of 11-AGNR causes a significant band-gap opening (0.15 eV → 0.92 eV). On the other hand, both pristine and porous (same pores) 10-ZGNR have no energy band gap, as seen in Figs. 1 (e, f). When the SPs are displaced by one honeycomb cell towards the GNR-edge, the band gap in 11-AGNR shrinks ≈ 3.75 times, while 10-ZGNR remains metallic, as shown in Figs. 1 (c, g). We also checked that the PGNR band structures monotonously approach their pristine form as the separation between adjacent pores is increased. The densities of states (DOS) for the band structures presented are displayed in Figs. 1 (d, h).

Figure 1
figure 1

Band structure of: a) pristine 11-AGNR, b) 11-AGNR with centered SP, c) 11-AGNR with shifted SP, e) pristine 10-ZGNR, f) 10-ZGNR with centered SP, g) 10-ZGNR with shifted SP.

The energy scales for (b, c, f, g) cases are the same. Density of states: d) centered and shifted SP in 11-AGNR, h) centered and shifted SP in 10-ZGNR. (insets) Unit cells for 11-AGNR and 10-ZGNR with the standard pore (a is the lattice constant of the supercell).

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The energy band gaps in ZGNRs and AGNRs are known to arise from a staggered sublattice potential and a quantum confinement21,22, respectively and they depend on the ribbon width and its functionalization4,23,24. The wave functions of the HOMO and LUMO bands in AGNRs, which contribute directly to the area near Ef , are localized at the center of the ribbons, keeping their edges chemically stable4. In ZGNRs, these wavefunctions are localized at the ribbon edges. Consequently, when the SPs are positioned at the center of 11-AGNR a band gap opens in its band structure, while 10-ZGNR remains metallic. The metallicity of ZGNRs is caused by flat bands present at Ef , originating from highly localized states formed at the zigzag edges4, as discussed below. When the pore is closer to the edge of 11-AGNR, its band structure is influenced less, while no significant change is observed in 10-ZGNR.

Porous graphene

We continue to study PG that lacks explicit edges, in contrast to PGNRs. Graphene-based systems have a bipartite lattice20, with nA and nB sites per unit cell in the A and B sublattices, leading to nA − nB flat bands at the Fermi level7. The metallicity caused by these bands is highly stable to any perturbation (see Figs. 1 (f, g)). Although, nA = nB = 1 in graphene, introduction of pores or other defects may change its global sublattice balance, i.e. nA ≠ nB . The relationship between edge-localized states, zigzag-like edges of GNR and flat bands at the Fermi level was discussed in8. It turns out that even a small number of zigzag sites at the ribbon edge gives rise to highly localized states forming flat bands at Ef . These zigzag edges are characterized by a local sublattice imbalance and unbalanced π-electron spin density9. These localized states at zigzag rims were also studied in different types of pores, called “anti-molecules”, where a set of simple rules was shown to link the net number of unpaired electrons with the degeneracy of flat bands2,5,7,10,26. SPs, largely used in our study, keep the global sublattice balance of the unit cell, but their short zigzag-like rims break the local balance and may still give rise to flat bands.

In Figs. 2 (a–d), we show the band structures of (edgeless) PGs with one honeycomb and three different rectangular SP-superlattices defined by NA , NZ and displayed in Figs. 2 (e, f)25. Figure 2 (g) also shows the studied rectangular superlattice with triangular-shape pores. The honeycomb SP-superlattice has flat bands around E ≈ −1.5 eV (Fig. 2 a), which are absent in the rectangular SP-superlattices (Figs. 2 (b–d)). According to7, such quasi-flat bands at non-zero energy might be ascribed to the local sublattice imbalance (globally nA = nB ). However, the local sublattice imbalance due to SPs can not be the origin of flat bands in the band structure of the honeycomb SP-superlattice, since it has the same pores as the other structures with no flat bands. Therefore, the flat bands are most likely related with the honeycomb SP-arrangement, which might produce differently localized states. These results show that the small number of zigzag sites at the very short SP-rim can not generate flat bands at the Fermi level or at its vicinity. In contrast, even the smallest triangular-shape pores break the global sublattice balance and generate an unbalanced π-electron density9, associated with the appearance of rim-localized states and giving rise to flat bands at Ef (not shown).

Figure 2
figure 2

(top) Band structure of a) a honeycomb SP-superlattice in e) and three rectangular SP-superlattices with b) N A = 15, N Z = 4, c) N A = 7, N Z = 2 and d) N A = 7, N Z = 8.

(bottom) e) honeycomb SP-superlattice, f) rectangular SP-superlattice characterized by NA , NZ (example with NA = 15, NZ = 4 is shown), g) rectangular superlattice with triangular-shape pores.

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With these observations, we can now relate the band structures of PGNRs and PGs. Since SPs do not break the global sublattice balance in the PGNRs, their presence does not generate any new flat bands at Ef (see Fig. 1); on the contrary, we found that when AGNRs are perforated with triangular-shape pores, their band structure always contains flat bands at Ef . Even though the SPs do not create flat bands at Ef in GNRs, they still influence their band structure. While the band structures of AGNRs (Figs. 1 (b, c)) is influenced a lot, the band structures of ZGNRs with flat bands at Ef (local sublattice imbalance caused by zigzag edges) can not be significantly modified by SPs. Therefore, we conclude that all porous ZGNRs are metallic (checked by calculations).

We performed extensive ab initio calculations of electronic structures of PG-superlattices with different arrangements of the SPs. Interestingly, we found that rectangular PG-superlattices perforated with SPs and larger pores of honeycomb symmetry can be both metallic and semiconducting. It turns out that we can generalize these observations into a hypothesis that, in the first approximation, the electronic structure of these superlattices has the same type of conductivity as many parallel AGNRs or ZGNRs (of effective widths NA or NZ ), depending on the ratio ρ = NA/NZ (see Fig. 3 bottom). If ρ ≫ 1, one can see the superlattice as being “cut” into separated NA -AGNRs, while for ρ ≪ 1 the same is true for separated NZ -ZGNRs.

Figure 3
figure 3

(top) Effective replacing of porous N-AGNR by “daughter” pristine N 1 - and N 2 -AGNR ( N = 14, N 1 = 5, N 2 = 6).

(bottom) Effective replacing of rectangular SP-superlattice with NA > NZ by set of prestine AGNRs (NA = 15, NZ = 4).

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The above hypothesis was largely confirmed by our follow up calculations. For example, the conductivity in the PGs with NZ = 2 is the same as in the corresponding AGNRs: metallic for NA = 5,11,17… and semiconducting for NA = 7,9,13,15… (see also Fig. 5). If we continue with the NA = 9,15, 21,… semiconducting AGNRs and increase the initially small NZ = 2, we find that the PGs remain semiconducting for (roughly) NZ < NA , with the band gap shrinking with increasing NZ , signaling the transition to the ZGNR-dominated metallic conductivity. If we continue with the the NA = 11 metallic AGNRs, the PGs become metallic for all the NZ , since the ZGNRs that take over at NZ > NA are all metallic. Finally, when we continue from NA = 7,13…, the metallicity appears abruptly at NZ ≥ 4. In other cases, we expect that the transition between the AGNR and ZGNR-type of behavior occurs somewhere around the “diagonal”, ρ = 1. Our calculations also show that the metal-semiconductor transitions predominantly occur in two regions of the Brillouin zones, as seen in Figs. 2 (b–d) and the bands can be partially flat in the ky direction (NA = 7,13,…).

Figure 5
figure 5

Correspondence rule for GNRs and CNTs.

N - number of dimers forming GNR, p - index for the chirality vector in GNR and CNT; cond. - conductivity property (semiconducting (s) or metallic (m)); geometry defines the symmetry with respect to the mirror plane perpendicular to the ribbon and containing its axis: symmetric (s), asymmetric (a). Examples of AGNRs for N = 7, …14 and ZGNRs for N = 5, …12 are illustrated.

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Porous nanotubes

It turns out that there is also a clear correspondence between pristine CNTs and structurally analogous GNRs (GNRs can be rolled into CNTs). While ZGNRs indexed by 〈p, 0〉 are metallic for all p, only one third of AGNRs indexed by 〈p, 1〉 are metallic. Since, only one half of AGNRs can be rolled into ZCNTs, only one third ZCNTs are metallic27. We found that this mapping can be generalized to the point that every metallic/semiconducting CNT corresponds to a metallic/semiconducting GNR. The unambiguous relationship can be expressed in terms of the chirality index, p and the number of GNRs carbon dimers, N, as follows

Intuitively, we can look at this CNT-GNR correspondence as a consequence of CNT “cutting” (Figs. 4 (top, middle)), which preserves the type of conductivity. The AGNRs and ZGNRs that do not match any CNTs are all semiconducting and metallic, respectively, as summarized in Fig. 5.

Figure 4
figure 4

Cutting of: (top) (10,0)ZCNT into 〈19, 1〉AGNR and (middle) (9,9)ACNT into 〈18, 0〉ZGNR (opening of the GNRs is schematically shown).

(bottom) Removal of atoms from porous (10,10)ACNT leading to 〈17, 0〉ZGNR.

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Analogously, porous CNTs might have band structures similar to porous GNRs. Tight-binding calculations predicted28 that a line of SPs (separated by ∼ 12.8 Å) should cause band gap opening in ACNTs, whereas porous ZCNTs should be semiconducting regardless of the pore shape. These results are in contradiction with our ab-initio calculations, which show that the metallicity of pristine ACNTs (Fig. 6 a) is preserved in the porous ACNTs (Fig. 6 e), even for triangular-shape pores with clear zigzag-like rims. In metallic ZCNTs, the SP-perforation causes band gap opening, as shown in Figs. 6 (b, f), while in semiconducting ZCNTs, it causes band gap shrinking, as seen in Figs. 6 (c, g).

Figure 6
figure 6

Band structures in pristine CNTs: a) ACNT (10,10), b) ZCNT (9,0), c) ZCNT (10,10); d) DOS.

Porous CNTs: e) ACNT (10,10), f) ZCNT (9,0), g) ZCNT (10,10); h) DOS. The energy scales for (b,c) and (f, g) cases are the same.

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Discussion

The above observations allow us to build a unified model that can predict the type of conductivity in porous nanocarbons perforated with SPs and other pores that do not break the global sublattice balance. The model is based on the assumption that when NCs are perforated by a line of relatively close SPs, the type of conductivity in these PNCs is the same as in the (daughter) systems obtained from these NCs by removing all C atoms within a stripe going in the direction of the pores and having the same width as the pores (all dangling C bonds are H-terminated). We call these modified NCs the daughter systems of the original NCs (two GNRs for PGNRs, one GNR for PCNTs and a GNR-lattice for PG). This rule predicts that: (1) Porous ACNTs are metallic as the (daughter) ZGNRs; Fig. 4 (bottom) shows effective replacing of porous ACNT by pristine ZGNR. Perforating the resulting ZGNRs (and the other half of ZGNRs that can not be rolled up into CNTs) gives two metallic ZGNRs, preserving the ZGNR-metallicity. (2) Porous ZCNTs may give semiconducting or metallic AGNRs. Cutting all the AGNRs may give pairs of AGNRs with any conductivity. These results were confirmed by ab initio calculations.

We now use these rules to predict metallicity in porous AGNRs with SPs. We assume that their band gaps are EBG ≃ min (E1BG; E2BG), where E1BG, E2BG are band gaps of their two daughter AGNRs (see Fig. 3 (top)). With this inference, we can derive an analytical expression describing the dependence of the band gap on the width of the porous AGNRs. For simplicity, we consider SPs positioned in the middle of AGNRs of the width of , where a is the C-C distance and N is the number of dimers. By evaluating the widths of the pristine daughter AGNRs, we find that porous N-AGNRs are potentially metallic if the number of C-C dimers is given by at least one of these equations

i.e., if N = 5, 6, 7, 8, 9, 12, 14, 17, 18, 19, 20, ....

In Fig. 7 (top), we compare the ab-initio energy band gaps in pristine29,30 and porous AGNRs to validate the above model. In contrast to the pristine ribbons, where the metallic points emerge with the period of 3, (Nmet = 3m + 2), the band gaps of porous AGNRs have a more complex dependence. Nevertheless, the positions of the band gap minima agree with Eqns. 2.

Figure 7
figure 7

(top) Dependence of the band gap in the pristine and porous AGNR on the number of dimers (the central position of the SP).

(bottom) The same dependence in ZCNT on the chirality index.

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We can extend the assumptions used in Eqns. 2 to PNCs perforated with larger and shifted honeycomb-like pores. Their presence may still be reduced to removing from the AGNRs a layer of atoms of the width given by the pore size, where the minimum band gap of the two resulting AGNRs can determine the band gap of the porous AGNR. For example, when the SP is shifted in the 11-AGNR by one honeycomb from the ribbon center, the two daughter pristine 4-AGNRs are replaced by 2-AGNR and 6-AGNR (all semiconducting). This should lead to a band gap shrinkage, in agreement with our ab-initio calculations, presented in Figs. 1 (b, c). Alternatively, we can replace the SP by a double-size hexagonal pore with 24 C-atom excluded. If the 11-AGNR and 12-AGNR are perforated by such pores, they become semiconducting, since their cutting leads to semiconducting 2-AGNRs and 2- and 3-AGNR, respectively. These results are in agreement with ab-initio calculations, giving in 11-AGNR and 12-AGNR the band gaps of 1.1 eV and 1.18 eV, respectively. We have also tested the triple-size hexagonal pore (54 C-atoms excluded) in order to check how its long rims affect band structure of GNRs. Our calculations show that no additional features (e.g. flat bands at the Fermi level) appear when the GNR are perforated by these pores. In contrast, when these AGNRs are perforated with SPs in the ribbon center, only the 11-AGNR is semiconducting.

Finally, we discuss porous ZCNTs that can have any conductivity. In Fig. 7 (bottom), we present the energy band gap of porous ZCNTs in dependence on the chirality index, p. It exhibits similar periodicity as in the pristine ZCNTs. However, the model is not reliable in porous ZCNTs. For example, the porous ZCNT(7,0) and ZCNT(8,0) have band gaps similar to the daughter 11-AGNR and 13-AGNR, respectively. But the same is not true for the porous ZCNT(9,0) and ZCNT(10,0) paired with the daughter 15-AGNR and 17-AGNR, respectively. In principle, this failure might be caused by the fact that the AGNRs are not calculated deformed as the corresponding daughter ZCNTs31,32. However, our calculations show that the bent 17-AGNR has almost the same band gap as the pristine 17-AGNR. Therefore, a more quantitative approach needs to be used here.

It is of interest to see if other types of periodic modifications can also be used to tune the band structures of nanocarbons. To briefly examine this idea, we have replaced SPs by Stone-Wales 55–77 defects33. In Fig. 8, we show the band structures of 11-AGNR, 10-ZGNR and graphene superlattices modified in this way. The periodic array of SW 55–77 defect in 11-AGNR leads to small bang gap opening, as shown Fig. 8 (a), in analogy to 11-AGNR with SPs (Fig.1 b). The band structure of 10-ZGNR, shown in Fig. 8 (b), is not sensitive to this perturbation, as in the SP-perturbations (Fig.1 f). On the other hand, when we replace in graphene superlattices SPs with the SW 55–77 defects, we can obtain qualitatively different band structures. In particular, the band structure of graphene modified by SW 55–77 defects in the array with NA = 7 and NZ = 4 (Fig. 8 c) is similar to that of the SP-superlattice with the same NA and NZ , but here we also observe opening of a small band gap. In Fig. 8 (d), we show the band structure of the SW-graphene superlattice (NA = 9 and NZ = 4), which is semiconducting as the PG-superlattice with the same NA and NZ . These observations show that periodic defects could also be used to tune band gaps in nanocarbons, but the rules might be slightly different.

Figure 8
figure 8

Band structures of nanocarbons with periodic 55–77 Stone-Wales defects: a) 11-AGNR, b) 10-ZCNT, c) graphene superlattice with N A = 7 and N Z = 4, d) graphene superlattice with N A = 9 and N Z = 4.

Figures (e–h) show unit cells for the respective cases.

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For completeness, we have recalculated some of the above structures including spin polarization (we used a set of LDA and GGA functionals). It turns out, the band structures of NCs can be modified by the spin polarization (zigzag edges) [2], but the presence of SPs does not introduce additional magnetic features beyond the changes described already in the non-magnetic calculations. Interestingly, the presence of arrays of SW (55-77) defects in ZGNRs can alternate mutual orientation of the magnetic moments localized at the opposite edges, due to topological changes in the sublattices.

In summary, we have developed a unified picture of band structures in PNCs. Although, the proposed approach successfully describes band structures in many PNCs, it could be further refined to account for quantization, spin degrees of freedom, pore type and chirality. Similar observations were made in nanocarbons perturbed by periodic SW defects. Precise knowledge of electronic structures of these materials is essential for their applications in electronics, optics, molecular sensing and other fields.

Methods

We study the PNCs ab initio, using SIESTA 3.0-beta-1534 in supercells of (> 40 atoms) and neglect spin degrees of freedom. The length of unit cells for porous 11-AGNR and 10-ZGNR is 12.78 Å and 12.3 Å, respectively. The size of the supercells of PG used in our calculations varies between 12.3×8.52 Å (40 atoms) and 22.13 × 21.3 Å (180 atoms). We use the Perdew-Zunger LDA functional35 and pseudopotentials with the cutoff energy of 400 Ry. The calculations are done within the eigenvalue tolerance of 10−4 eV, using the DZP basis (double-zeta basis and polarization orbitals, 13 and 5 orbitals for C-atom and H-atom, respectively). The Brillouin zones of the unit cells are sampled by the Mankhorst-Pack grid36 with the spacing between the k-points of Δk < 0.01 Å−1. Geometry optimization is carried out for all the PNCs within the conjugated gradient algorithm, until all the forces are F < 0.04 eV/Å and the stress in the periodic direction is σ < 0.01 GPa. We use a.u. for k in the band structures, where only one half of the Brillouin zone is shown (symmetry).