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Multiple Exciton Generation in Graphene Nanostructures James McClain and Joshua Schrier* Department of Chemistry, HaVerford College, HaVerford, PennsylVania 19041 ReceiVed: February 9, 2010; ReVised Manuscript ReceiVed: July 20, 2010
Multiple exciton generation has the promise of improving photovoltaic device efficiency. However, occurrence of this phenomenon in semiconductor nanocrystals is controversial. Using the Pariser-Parr-Pople Hamiltonian and an impact ionization mechanism, we examined over 800 finite graphene nanostructures and have identified cove-periphery hexagons, armchair-periphery hexagons, zigzag parallelograms, rectangular, and cove-periphery rectangular structures with efficient low-threshold biexciton generation. Using the experimental exciton cooling rate, we estimate that absorbed photons can generate between 1.3 and 1.8 biexcitons per absorbed photon in optimal structures. Introduction In conventional photovoltaic devices, an absorbed photon generates a single electron-hole pair, and any energy greater than the band gap, Eg, is dissipated as heat when the excited carriers relax. If instead the single photon were to generate multiple electron-hole pairs, a phenomenon referred to as multiple exciton generation (MEG), the photovoltage (roughly equal to Eg) would remain unchanged, but the current (proportional to the number of excitons generated) would be increased, raising the efficiency of solar energy utilization.1 The many reports of efficient MEG in semiconductor nanocrystals have been reviewed by Nozik1 and McGuire et al.2 However, experiments by Nair and co-workers have cast doubt on these claims.3,4 The seemingly contradictory results may be due to a sensitive dependence on nanocrystal surface chemistry5 or occurrence of photoionization yielding MEG-like signals in early experiments.2 In device applications, optical analysis of PbSe nanocrystal photovoltaics failed to find external quantum efficiencies greater than unity,6 although photoconductive detectors using nanocrystals show signatures of MEG.7 From the theoretical perspective, initial explanations of efficient MEG based on an impact ionization mechanism and the favorable ratio of the density of biexciton states to the density of exciton states in nanocrystals determined by Zunger and co-workers were quite optimistic.8,9 However, more detailed calculations of this mechanism examining the value of the matrix elements entering into the evaluation of the rate of impact ionization, such as the work by Allan and Delerue10,11 and Rabani and Baer,12 do not suggest efficient MEG in semiconductor nanocrystals. (Alternative mechanisms, e.g., direct photoexcitation of multiple exciton states,13 have also been proposed.) The present work considers the impact ionization mechanism for MEG in graphene nanostructures. A related process, singlet fission, where a highly excited singlet exciton state decays into two triplet exciton states has been observed in covalently linked tetracene molecules.14 In contrast, the impact ionization process involves the decay of a highly excited singlet exciton state into two singlet exciton states, occurring by the following process: If one (or both) of the carriers in the initial excited state has a large kinetic energy, * To whom correspondence should be addressed. E-mail: jschrier@ haverford.edu.
this can excite an additional electron-hole pair, generating a three-particle “trion” state. For example, a highly excited electron loses energy to form a negative trion (the original electron, but now in a lower energy state, plus the newly excited electron-hole pair). The initial hole is still present, so the system now has a total of two electron-hole pairs (i.e., a biexciton). A similar process can occur for the holes, proceeding through the formation of a positive trion. While Allan and Delerue10,11 and Rabani and Baer12 have shown this to be inefficient in all but the smallest semiconductor nanocrystals, there are several reasons to suspect that twodimensional (2D) carbon-based structures may be more favorable: (1) The Coulombic interactions are larger in graphene than in semiconductors. Comparison of the Ohno-Klopman or Mataga-Nishimoto forms of these integrals used in the present work and in previous tightbinding calculations of nanocrystals, for example, ref 15, shows that while the onsite Coulomb matrix elements for Si and Se are comparable to those of π-electrons in planar hydrocarbons, those for Cd are about half of this value. (2) The Coulomb interaction in semiconductors is screened by the dielectric constant of the material, reducing its effective strength. In nanocrystals, the dielectric constant is reduced as the size of the nanocrystal decreases,16 but it is still several times larger than what would be experienced by a polyaromatic hydrocarbon dissolved in an organic solvent or polymer matrix. Consequently, one would expect the strength of the Coulombic matrix elements to be larger, potentially increasing the MEG rate. (3) The 2D quantum particle-in-a-box model suggests that the density of trion states (DOTS) can be tuned by creating one narrow direction and one long direction, resulting in a set of narrowly spaced levels with a finite band gap. In principle, this could also be done with semiconductor nanorods. However, the symmetrical band structure of graphene will cause this optimization to be equally effective for both the positive and negative trion density of states. (4) Localization of band-edge states near zigzag edges in finite graphene nanoribbons has been predicted from density functional calculations,17 and could be used to create both localized and delocalized states near the band-edge. These considerations motivated us to examine the trion-mediated MEG process in a variety of finite graphene nanostructures.
10.1021/jp101259m 2010 American Chemical Society Published on Web 08/09/2010
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Methods
A d3r d3r′ ψr(r) ψs(r) ψu(r′) ψt(r′)/ε|r - r′| ) ∑ ∑ ∑ ∑ CrµCsνCuFCtσ
Vrsut )
Model Hamiltonian. The electronic structure of the graphene nanostructures was modeled using the self-consistent field (SCF) Pariser-Parr-Pople (PPP) π-electron theory, which has been widely used in quantum chemistry.18 The π-electron Hamiltonian is solved within a basis of pz atomic orbitals, φµ, located on site µ, and assumed to have zero-differential-overlap (ZDO). Indices r, s, t, u denote general (either electron or hole) molecular orbital (MO) states (obtained from the SCF calculation), and µ, ν, F, σ denote atomic orbital (AO) indices. Variations in bond strength due to charge redistribution in the molecule are accounted for using nearest-neighbor resonance integrals given by19
µ
×
[A
F
ν
σ
d3r d3r′
)
[
(
2 (eV) 3
)]
(1)
thus incorporating dependence on the charge-density matrix elements,
Pµν ) 2
∑ crµcrν
(2)
r∈occ
calculated from the previous SCF iteration. Within the ZDO approximation, the only Coulombic integrals are of the form
γµν )
φµ(r) φµ(r) φν(r′) φν(r′) |r - r′| ≡ (µµ|νν)
A d3r d3r′
(3)
]
(8)
∑ ∑ Crµ′Csµ′CuF′CtF′(µ′µ′|F′F′)/ε
(9)
µ
PPP
≈
F
ν
σ
F′
Equation 7 results from substituting in the definition of the eigenstates as linear combinations of atomic orbitals, and a uniform dielectric constant, ε. (Within PPP theory of aromatic molecules, such as the graphene nanoribbons considered here, ε ) 1.) Equation 8 is a restatement using the simplified notation for the Coulombic integrals in the AO basis used in eq 3. Equation 9 follows from the zero-differential-overlap approximation inherent in PPP theory.18 The two-center terms may be evaluated using either the MN (eq 4) or OK (eq 5) forms discussed above. Calculation of Trion Formation Rates. Following the discussion of Rabani and Baer,12 MEG occurs via the formation of positive (negative) trions resulting from the decay of high energy holes (electrons). Indices i, j, k are used to denote hole states, and a, b, c to denote electron states. A particular exciton (ia) state decays with a rate Γia that is the sum of the decay rates of the hole, Γ+ i , and electron, Γa , trion formation rates,
Γia ) Γi+ + Γa The one-center Coulomb integral is obtained from the difference between the experimental ionization potential and electron affinity of a carbon atom,19 γµµ ) 10.84 eV. Two-center integrals are expressed in terms of the one-center integral and a function of distance, Rµν, between atoms µ and ν, using an empirical functional form. We have investigated both the MatagaNishimoto (MN),
γµν ) e2 / [Rµν + (e2 /γµµ)]
(4)
and Ohno-Klopman (OK), 2 γµν ) e2 / √Rµν + (e/γµµ)2
(5)
forms in this work. Following SCF convergence, one obtains single-particle MO states that are linear combinations of AOs, for example,
ψr )
∑ crµφµ
(6)
µ
with corresponding eigenvalues εr. A general Coulombic integral, Vrsut, can be expressed in terms of AO integrals specified by PPP theory as
(7)
∑ ∑ ∑ ∑ CrµCsνCuFCtσ(µν|Fσ)/ε µ′
βµν ) -2.318exp 0.335 Pµν -
φµ(r) φν(r) φF(r′) φσ(r′) ε|r - r′|
(10)
The trion formation rates are computed from Fermi’s golden rule by
∑
Γi+ )
4π |(2Vjikb - Vkijb)|2δ(εi - {εk + εj - εb}) p j,k,b (11)
Γa )
4π |(2Vacjb - Vabjc)|2δ(εa - {εb + εc - εj}) p c,b,j (12)
∑
using the integrals, Vrsut, defined in eq 9. As in the treatments of Rabani and Baer12 and Allan and Delerue,11 we have used the single-particle states for these evaluations. We used a 25 meV window for the delta-function in eq 11 and eq 12. The number of biexcitons produced is a competition between the trion formation processes and the exciton cooling rate of the initial exciton. This results in a kinetic model for the average number of excitons produced as a function of energy,
Nex(E) )
∑ pia(E) [2Γia + γd]/(Γia + γd)
(13)
ia
where pia(E) is the probability of generating a particular exciton consisting of electron-state a and hole-state i, and γd is the cooling rate of a single exciton to its lowest state.12 Experimental
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studies of epitaxial graphene indicate that γd is in the range of 1-6.7 ps-1,20 that are used as the optimistic and pessimistic limits, respectively, in the calculations below. An optimal structure for solar energy utilization will have a dense spectrum of single-exciton absorption lines in the 2-3Eg range and efficient biexciton generation for each of those absorptions. To quantify this property, we have defined a biexciton efficiency metric as follows: (1) Compute the number of excess excitons produced as a function of energy using eq 13, with γd ) 1 ps-1. (Use of other values will give the same qualitative results.) (2) Assign these to 100 evenly spaced bins in the range from 2Eg to 3Eg and compute the average number of biexcitons generated in that interval. (3) The MEG metric is the area contained under this histogram. A value of one indicates perfect biexciton generation for every incident photon in the 2Eg-3Eg range, and a value of zero indicates a complete absence of biexciton generation over the same range. The latter case results from either an absence of single exciton absorption in that range or an absence of biexciton generation. Validity of the Single-Particle State Model. It is known that the excited states of molecular systems can depend strongly on the proper treatment of electron-hole interactions,21 which are absent from the single-particle based methodology used by Rabani and Baer,12 Allan and Delerue,11 and the present work. The single-particle eigenfunctions are correct to first-order in perturbation theory, but the eigenenergies of the virtual states correspond to the energy of an additional electron moving in the self-consistent field of the molecule, rather than a quasiparticle excitation energy.22 Consequently, the interpretation of the difference between the “hole” (HOMO and below) and “electron” (LUMO and above) Hartree-Fock eigenergies overestimates the true value of the excitation energy. It is possible to correct each excited state by constructing a modified Fock operator eliminating this additional spurious interaction, and then diagonalizing it to obtain a new set of virtual orbitals,23–25 but this is too computationally demanding for the many excited states needed here. For this initial study, we justify neglecting these corrections for the following reasons: (1) The first-order perturbative correction to the energy is an additional two-electron integral between the final and initial states.22 We have computed these for several structures and found that the values of these matrix elements to be roughly constant, with variations on the order of 50 meV, and thus comparable to the window that we have chosen for the delta-function above. Therefore, while the reported HOMO-LUMO gaps are overestimated throughout the paper, the relative energies of different excited states will be correct. This is further supported by the qualitative similarity of the MN and OK results, despite their differences in HOMO-LUMO gap. (2) The single-particle wave functions are correct to first order,22 as described above. Therefore, the resulting integrals that we evaluate for the rates will be approximately correct. (3) The trion formation process involves many simultaneous possible channels. We have examined a few structures and found that these individual trion formations all have comparable rates. Thus, moving one or two states outside the delta-function window will have a minor effect on the total trion decay rate, since most of the contributions remain unchanged. Results and Discussion Figures 2-6 shown in the main text show the results using the Mataga-Nishimoto (MN) integral form. Each of these has a corresponding figure using the Ohno-Klopman (OK) integral form in the Supporting Information.
McClain and Schrier
Figure 1. Graphene nanostructure types considered in this work. For clarity, the double bonds are not shown. (a) Cove-periphery hexagons (CPH(zz)), labeled by number of zigzag edges, zz; (b) armchairperiphery hexagons (APH(ac)), labeled by number of armchair edges, ac; (c) zig-zag parallelograms (P(w,l)), labeled by number of zigzag edges in width (w) and length (l); (d) rectangular (R(zz,ac)) structures, labeled by number of zigzag (zz) and armchair (ac) edges; (e) coveperiphery rectangular (CPR(zz,ac)) structures, labeled by number of zigzag (zz) and armchair (ac) edges.
Cove-Periphery Hexagons. The cove-periphery hexagonal (CPH) structures are labeled by summing the two outwardpointing zigzag vertices at the ends and the inward pointing zigzag vertices inside the cove, as illustrated for the CPH(4) structure shown in Figure 1a, which has been synthesized by Do¨tz et al.26,27 Figure 2a shows the HOMO-LUMO gap and biexciton generation metric for a series of structures with this type of geometry. As expected from the particle-in-a-2D-disk model, the HOMO-LUMO gap decreases monotonically as a function of edge length (which is proportional to radius). The OK calculations shown in the Supporting Information are qualitatively the same. The CPH(8) structure has the highest MEG metric according to both MN and OK calculations, and Figure 3a shows the number of excitons produced as a function of the excitation energy. The red and black lines show the average number of excitons produced using the slow (γd ) 1 ps-1) and fast (γd ) 6.7 ps-1) bounds of the exciton cooling rate, respectively. Efficient biexciton generation occurs at low onset, but the widely spaced lines indicate that many incident photons in the range 2-3E/Egap do not create biexcitons. This is also the case according to the OK calculations, shown in the Supporting Information. Armchair-Periphery Hexagons. These structures are labeled by summing the number of outward-facing armchair bonds along the edge of the hexagon, as illustrated for the APH(3) structure shown in Figure 1b. Both APH(2) (hexa-peri-hexabenzocoronene)28 and APH(4)29 have been synthesized.27 Figure 2b shows the HOMO-LUMO gap and biexciton generation metric as a function of this length. As with the CPH results described above, the HOMO-LUMO gap decreases monotonically as a function
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Figure 2. HOMO-LUMO gap (left axis, blue dotted curve) and biexciton generation metric (right axis, red solid curve) calculated using the Mataga-Nishimoto integral form, as a function of size for (a) coveperiphery hexagon (CPH) and (b) armchair-periphery hexagon (APH) structures. Corresponding figures using the Ohno-Klopmann integral form are qualitatively similar and are shown in the Supporting Information.
of edge length. However, the maximum value of the biexciton generation metric, reached for APH(3), is approximately a factor of 5 smaller than the CPH structures. The OK results are qualitiatively the same (see the Supporting Information). Figure 3b examines the case of APH(3). Although there are several excitations that have efficient biexciton generation, most of the transitions are slow. In the corresponding OK calculations, the rates are all uniformly low. Parallelograms. The parallelogram (P) structures are be labeled by the number of zigzag edges in the (width,length) directions, for example, P(2,4) for the example shown in Figure 1c. Pyrene corresponds to P(2,2) in this notation. The length and width are symmetric upon rotation of the structure, so P(n,m) is identical to P(m,n). Similar structures were studied in the theoretical paper of Ezawa.30 Parallelograms with width and length each ranging from 1 to 30 zigzag edges are shown in Figure 4 for the MN calculation and in the Supporting Information for the corresponding OK calculations. As the box length increases, the HOMO-LUMO gap decreases, as shown in Figure 4a, consistent with a particle-in-a-box model. Very small gaps prevented SCF convergence for the larger structures, so those points are not shown on the graph. The biexciton generation metric values shown in Figure 4b are predominantly zero, but there are sharp peaks at several points, consistent with the considerations of the DOTS in a 2D particle-in-a-box discussed in the introduction. Among these, P(8,15) has the highest metric for the parallelogram graphene nanostructures in the MN calculation, and P(8,14) has the highest metric in the OK calculation. Efficient structures are clustered, for example, P(9,13) and P(9,14),
Figure 3. Average number of excitons produced as a function of excitation energy in units of the HOMO-LUMO gap (Egap), calculated using eq 13. Calculations using the optimistic exciton cooling rate (γd ) 1 ps-1) and pessimistic (γd ) 6.7 ps-1) values are shown in red and black, respectively. (a) CPH(8); (b) APH(3); (c) P(8,15); (d) R(33,6); (e) CPR(30,9).
P(8,15) and P(8,17), P(7,22) and P(7,23) all have high MEG generation metrics. The OK calculations generally predict a higher biexciton generation metric with less abrupt changes as a function of size than the MN predictions. Figure 3c shows the average number of excitons produced as a function of excitation energy for P(8,15), which has a biexciton generation metric of 0.24. The predicted HOMO-LUMO gap of this structure is 1.2 (MN) or 1.8 eV (OK). The absorptions leading to biexciton formation are densest above 2.5Eg, and in general the average number of excitons produced is about 1.3 or 1.8
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Figure 5. (a) HOMO-LUMO gap and (b) biexeciton generation metric for rectangular (R) graphene nanostructures, calculated using MatagaNishimoto integral form.
Figure 4. (a) HOMO-LUMO gap and (b) biexciton generation metric for parallelogram (P) graphene nanostructures, calculated using the Mataga-Nishimoto integral form.
excitons/photon for the pessimistic and optimistic exciton decay rates, respectively. The corresponding OK calculation for P(8,15) (see the Supporting Information) predicts slightly lower rates, but a denser spectrum of absorptions at high energies and a lower absorption onset compared to the MN predictions. Rectangles. Calculations on rectangular (R) structures of the form shown in Figure 1d were carried out for the range of 3-19 armchair edges (1.5-7.5 nm) and 3-33 zigzag edges (1-8 nm). Labeling these as R(zz,ac), Figure 1d shows a R(5,3) structure. R(2,2) (perylene), R(2,3) (terrylene),31 and R(2,4) (benzo[1,2,3cd:4,5,6-c′d′]diperylene)32 have all been synthesized. Similar structures were considered in the theoretical paper of Hod et al.17 Figure 5b shows that the efficient structures have between 6 and 8 armchair edges and more than 15 zigzag edges (The OK results, shown in the Supporting Information, predict slightly larger metrics for shorter zigzag edge length structures, but are otherwise qualitatively the same.) This is due to localization of the band edge states to the zigzag edges previously observed in density functional theory calculations of both periodic33 and finite17 graphene nanoribbons. In contrast, the highly excited states are delocalized over the structure. A large spatial overlap between the slowly varying parent exciton state and the edgelocalized daughter trion state maximizes eq 9 giving a fast rate. When the structure is very wide, the parent state is spread over the whole structure, and the overlap is small, giving a small rate of trion formation. When the structure is too small, the edge states are delocalized over both edges, and the relative phase between the band edge state and parent state can cause eq 9 to go to zero. In the region where the number of armchair edges is between 6 and 8, these two competing effects are
balanced. Figure 3d shows the number of exciton states generated as a function of energy for the R(33,6) structure, which was the best according the MN calculations. The predicted HOMO-LUMO gap (Figure 5a) of this molecule is 1.2 eV (MN) or 1.6 eV (OK), which is similar to that predicted for P(8,15) above. The number of absorbances leading to biexciton formation is both denser and extends to lower energy than the P(8,15) structure discussed above, according to the MN predictions. In contrast, the OK calculations predict R(33,6) to have slightly lower biexciton generation than the P(8,15) structure. Cove-Periphery Rectangles. Figure 1e shows the labeling scheme for cove-periphery rectangles (CPR) by zigzag and armchair lengths of (zz,ac), with the example of CPR(5,3). Do¨tz et al.26 have synthesized CPR(3,4). Using the MN approximation, the two best structures are CPR(30,9) and CPR (33,8). Using the OK approximation, the two best structures are CPR(27,7) and CPR(33,7). The most efficient structures according to MN are in a band where the number of armchair edges is betweeen 8 and 10 and the number of zigzag edges is greater than 19. The OK predictions are qualitatively the same, but they predict a wider range of optimal number of armchair edges (between 7 and 10) and a sharp increase in biexciton generation when the number of zigzag edges is greater than 24. This is similar to the rectangular structures discussed above. Figure 3e shows that CPR(30,9) extends the onset of biexciton generation to lower energies than the R(33,6) structure discussed above. The OK calculations predict a higher onset of biexciton generation, but a more closely packed set of absorptions in the 2.5-3 Egap range. The HOMO-LUMO gap for this molecule is predicted to be between 1 eV (MN, Figure 6b) and 1.5 eV (OK, Figure 5b), which is lower than the previously discussed structures, and hence better suited for solar energy utilization.1 Other Structures. We also computed molecules 35-38, 40-43, and 46-53 in Mu¨llen and co-workers’ review of synthetic graphene nanostructures.27 All of these were found to
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J. Phys. Chem. C, Vol. 114, No. 34, 2010 14337 cated earlier studies on semiconductor nanocrystals.5 Since our calculations identify a broad range of widths and lengths over which MEG generation will occur, subtractive methodologies may also be feasible for the larger rectangular structures. Narrow graphene nanoribbons may be produced either by unzipping carbon nanotubes36,37 or by nickel-nanoparticle etching38 (the latter favors production of zigzag edges), followed by reactive ion etching lithography39 or masked conversion to graphane40 to produce finite lengths. Functionalizing the edges of these structures with short branched alkanes (to increase their solubility in organic solvents41) will have no effect on the present results (aside from distortions of the molecule from planarity), since the optical properties described above depend only on the π-electrons. Finally, these structures may be incorporated into organic photovoltaics following the procedures used to construct existing P3HT/graphene organic photovoltaics,42 or used as sensitizers for Gra¨tzel cells.43 Conclusion
Figure 6. (a) HOMO-LUMO gap and (b) biexeciton generation metric for cove-periphery rectangle (CPR) graphene nanostructures, calculated using Mataga-Nishimoto integral form.
have small metric values on the order of 10-2 or less in both the MN and OK calculations (see the Supporting Information). These structures are small and do not have the density of trion states necessary to have a large MEG efficiency metric, although some individual transitions were found to have fast trion generation rates. Features Giving Rise to Efficient MEG. Considering eqs 11 and 12, is the biexciton generation rate more dependent on the Vrsut integrals or the density of trion states (DOTS)? Figure S6 in the Supporting Information compares the biexciton generation metric versus a “DOTS-only” metric, computed by setting all of the Vrsut ) 0.1. A perfect linear correlation would indicate that high MEG efficiency results from the DOTS terms; a lack of correlation would indicate that while the DOTS is necessary, the variation of the integrals ultimately determines which structures have efficient biexciton generation. We found a strong qualitative correspondence (Spearman rank correlation 0.80) between the DOTS and the final rate. Therefore, tuning the DOTS is a good first step in designing these structures, though in many cases small integral values reduce the biexciton generation rate. However, localized states on the zigzag edges tend to keep these integrals high (vide supra). Moreover, the integrals decrease more slowly when the band edge states are localized on the zigzag edge rather than delocalized over the whole structure. In contrast, nanocrystal wave functions are essentially particle-in-a-box states delocalized over the whole structure, so the values of the integrals decrease rapidly with increasing diameter.12 In principle, localized states similar to those discussed here could be created using heterostructured nanocrystals. Experimental Feasibility. The structures described above are generalizations of molecules that have previously been synthesized using organic chemistry techniques.27,29,34,35 Synthesizing well-defined molecules eliminates the uncertainties involved in the surface structure effects that may have compli-
In conclusion, we have examined over 800 graphene nanostructures in search of multiple exciton generation. We have demonstrated that the particular functional form used for the semiempirical Coulombic interactions does not matter, as both give qualitiatively similar predictions. The most promising are rectangular and cove-periphery rectangular structures, where a broad range of sizes are capable of efficient biexciton generation at low excitation energies and are relatively insensitive to size variations. These structures may be constructed using either bottom-up organic synthesis or top-down subtractive approaches. Moderate HOMO-LUMO gaps (which are overestimated in the present calculations) allow for efficient utilization of solar photons. In the best structures, absorbed photons in the range of 2-3Eg generate an average of 1.3-1.8 excitons, depending on whether the experimental lower or upper bound of the carrier cooling rate in graphene is used. The main conclusion of this work is that MEG is efficient in a variety of carbon-based nanostructures, beyond the carbon nanotube photodiodes recently demonstrated by McEuen and co-workers.44 Several future theoretical studies are suggested by the current work. (1) Semiconductor nanoparticles have a decreased carrier cooling rate compared to the bulk due to phonon bottlenecks, so γd in eq 13 is a function of size and could be potentially smaller (better) for graphene nanostructures as compared to the bulk. This could be calculated using the procedure of Kilina et al.45 (2) We have considered only planar structures. However, edge stresses introduce nonplanar rippling,46–48 and the resulting local strains can induce create potential barriers and localized states.49 Alternatively, nonarmchair/zigzag reconstructions can occur to minimize the edge stresses.48 The resulting nonplanarity can be modeled in PPP theory by scaling the nearest-neighbor resonance integral by the cosine of the twist angle,18 or else by using semiempirical or ab initio models that incorporate all of the valence electrons to account for the resulting σ-π hybridizations. (3) Extensions to ab initio Hamiltonians and treatment of electron correlation would also resolve the differences between the different semiempirical models described above. Acknowledgment. This work was supported by the Haverford College Startup Fund. J.S. acknowledges the Donors of the American Chemical Society Petroleum Research Fund for partial support. J.M. was supported by the Haverford College Provost’s Summer Research Fund. Supporting Information Available: Figures showing Ohno-Klopman integral results and DOTS-only metric values
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