Sub-Nanometer Width Armchair Graphene Nanoribbon Energy Gap

Letter pubs.acs.org/JPCL

Sub-Nanometer Width Armchair Graphene Nanoribbon Energy Gap Atlas Carlos-Andres Palma,*,†,¶ Manohar Awasthi,‡,¶ Yenny Hernandez,§,∥ Xinliang Feng,§,⊥ Klaus Müllen,*,§ Thomas A. Niehaus,*,‡ and Johannes V. Barth† †

Physik-Department, Technische Universität München, James-Franck-Straße 1, 85748 Garching, Germany Institute I - Theoretical Physics, University of Regensburg, D-93040 Regensburg, Germany § Max Planck Institute for Polymer Research, Ackermannweg 10, 55128 Mainz, Germany ‡

S Supporting Information *

ABSTRACT: Extended all-sp2-carbon macromolecules have the potential to replace silicon in integrated nanometer-scale devices. Up to now, studies on the electronic properties of such structures, for example, graphene nanoribbons, have been focused mostly on the infinitely long limit, which is inadequate when approaching future devices with sub-10 nm control. Moreover, their electronic variation has not been systematically assessed as a function of chemically diverse edge termini. Such knowledge is central when prototyping potential allcarbon circuits. Here, we present a graphene nanoribbon energy gap atlas based on density functional tight-binding spin-polarized calculations of nearly ten thousand randomly generated nanoribbons with a maximal nominal width of 1 nm and an armchair long edge. We classify ribbon families and show that their energy levels are strongly dependant on their termini edge states. We notably reveal modulation of the bulk energy gap by 0.3 eV through minimal edge modifications and put forward simple rules for inducing antiferromagnetic edge states.

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he advent of graphene1 has resurged interest in the optical,2 electronic,3,4 and spin5−7 properties of extended polyaromatic hydrocarbons (PAHs). Of the PAHs graphene families, sub-nanometer graphene nanoribbons (GNRs) are of special interest, not only because they are the only bottom-up atomically precise fabricated, extended graphene nanostructures to date8,9 but also because they represent potential components in the future of all-carbon organic (spin-)electronics10 including ballistic transport.11 The potential integration of finite AGNRs in device elements motivates a comprehensive description of the energy gap modulation as a function of length. Moreover, a level of nanofabrication uncertainty at the termini edges of nanoribbons remains for both bottom-up on-surface,8 solution,9,12 and specially top-down13,14 fabrication protocols of armchair graphene nanoribbons15,16 (AGNRs). For instance, a general solution synthesis procedure of a cyclodehydrogenated polyphenylene has been shown to yield possibly different termini12 (and most likely structure GNR 1a, Figure 1). For the on-surface synthesis approach,8,17,18 at least three different edges have been identified with markedly different electronic properties in 7-AGNR,19 that is, in AGNRs with a width of seven carbon atoms in their backbone. These examples additionally motivate the systematic study of different termini edges of finite AGNRs, as these can cause AGNR’s electronic and magnetic properties to transition from benzenoid (no biradical character) to (semi)quinoid radicaloids.20 This versatility of AGNRs contrasts GNRs featuring zigzag edges in their repeating backbone (ZGNRs),5,7 whose proper© 2015 American Chemical Society

ties are largely independent of defects in their short edge termini.21,22 The simplest model short edge terminus in AGNRs is usually approximated to be a zigzag edge.15,16,23 With increasing width of the AGNRs these termini zigzag edges follow an acene series, from a simple naphthalene edge to anthracene and tetracenes, cf. Figure 1. For oligoanthrenes (or simply, anthenes members of the 7-AGNR family), a biradical structure dominates the ground state.24,25 This ground state was found to be of antiferromagnetic character,5 giving rise to the frontier orbital electronic properties of some AGNRs.23 However, only for model nanographene systems can one employ simple considerations to predict antiferromagnetism. For example, Lieb’s theorem in the Hubbard model predicts Heisenberg antiferromagnets26 only with an adequate partition of the graphene sublattices,27 which maximizes the number of nonadjacent sites. This equivalently corresponds to the partitioning of a molecular structure in non-Kekulé subunits, that is, units where alternating single and double bonds of the valence electrons cannot be drawn without leaving unpaired electrons. Therefore, the need for models which can estimate radical character7,20,23 of AGNRs and their electronic and magnetic properties is essential. Stability of antiferromagnets for instance, is critical for tailoring the barrier between alphaReceived: June 2, 2015 Accepted: July 20, 2015 Published: August 7, 2015 3228

DOI: 10.1021/acs.jpclett.5b01154 J. Phys. Chem. Lett. 2015, 6, 3228−3235

Letter

The Journal of Physical Chemistry Letters

Figure 1. Schematics of armchair graphene nanoribbons (AGNR) and density functional tight-binding (DFTB) benchmarks to optical spectra and density functional theory. (a) Schematic representation of two 9-AGNRs showing two different short edge termini: armchair and zigzag. (b) Experimental optical gaps (open squares) of PAHs comparison to length dependence of GNR 1 by DFTB (solid circles). The absorption spectra of GNR 1a is shown in the inset. (c) Optical absorption of a variety of PAHs synthesized in our group starting from hexabenzocorone (HBC, C42) to a large PAH C114H30 from different preparations: Subli., Sublimated film; THF, tetrahydrofuran solution; Sm., Smeared film; TCB, 1,2,4trichlorobenzene solutions; DMF, Dimethyl sulfoxide solutions. Benzene (C6H6) and triphenylene (C18H12) absorption spectra not shown. The optical gap of each compound is obtained as the first maxima in the absorption times the energy squared spectrum (“inverse Tauc’s plot”), in order to remove dispersive background due to solid-state preparation. (d) Comparison of the energy gaps for distinct AGNRs widths between the present study and the density functional theory (DFT) in ref 23. The energy gaps in blue are compared between the reported for periodic structures in ref 23 (blue line open circles) and with the “bulk” gap (see main text) of long AGNRs (207 Å, blue solid circles).

model benzenoid AGNR (Figure 1b inset, GNR 1a with n ∼ 25, ∼207 Å in length) was obtained from solid-state absorption spectra as 2.12 ± 0.05 eV (Figure 1b red stars). This value correlates remarkably well (2.32 eV) with the one calculated for 610 carbons (207 Å in length) by DFTB. The correlation indicates that with increasing length, the exciton binding energy is on the order of the density functional underestimation of the energy gap.31 Such phenomenological optical and energy gap correlation is further highlighted taking into account that the optical gaps of benzenoid PAHs (Figure 1c) follow almost perfectly the energy gap of GNR 1 as a function of number of carbons (Figure 1b squares). Not surprisingly, benzenoid PAHs are characterized by a strong electron localization32 and their electronic properties do not deviate from sub-nanometer-wide benzenoid AGNRs. With this in mind, the study of variations in the (spin-)electronic properties of GNRs due to chemically diverse short termini edges reduces to nonbenzenoid GNRs. First, we set to characterize model AGNRs15,16 as a function of length with zigzag edge termini. Figure 2a,b shows polar plots of 5-, 7-, and 9-AGNR along with a benzenoid GNR. Unlike previous studies, we vary the edge termini for each AGNR as depicted. In order to understand the influence of the edge termini on the energy gap, we employ two classifications: we first identify each GNR as benzenoid (not featuring any zigzag termini edges) or by the zigzag edge termini exterior to repeating backbone width (edge/width) ratio. Then, we identify AGNRs as unpolarized (UnP) or antiferromagnetic (AF). The latter classifications were attributed by finding the

beta spin symmetry (for, e.g. spintronic and memory devices28,29). Thus, models providing a comprehensive characterization of even-number-of-atoms randomly generated all-sp2-carbon AGNR structures are a compulsory departure point for engineering all-carbon electronics. In the present work, we analyze length and short termini edge effects on the energy gap of a plethora of AGNRs with nominal width of less than 1 nm, by means of spin-polarized density functional tight-binding (DFTB) calculations. First, we show how energy gaps closely correlate with polyaromatic and ribbon optical gaps, suggesting that exciton-binding energies compensate self-energy corrections not included in the DFTB model. This serves as a starting point for phenomenological energy gap engineering. Second, we unravel short termini edge effects in the structures of ten thousand randomly generated graphene ribbons. These ribbons are generated with maximum width, length and number of atoms of 9.67 Å 20.7 nm, and 896 carbon atoms, respectively. Particularly, we extend previous studies23,30 and show that the termini edge structures determine magnetoelectric modulation in AGNRs. Our systematic study evaluating different termini features of finite-length graphene nanoribbons is critical to the modulation of the electronic and magnetic properties of AGNRs en route to molecular (spin-)electronics. The first studies of optical gap measurements in atomically precise nanoribbons9,12 motivate benchmarks with state-of-theart theories to prepare the phenomenological energy gap engineering of graphene nanoribbons. The optical gap of the 3229


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Figure 2. DFTB energy gaps of AGNRs (a,b) Polar plot of the energy gaps of model AGNRs. The radii gives the length of the GNR (from ∼30 Å to 370 Å), the angle the energy gap and the color code the width. For the sake of generality, width and lengths are plotted before structure optimization, employing C−C bond lengths of 1.40 Å throughout. The GNRs are classified as having a simple gap (a) or as GNRE having an edge gap and a bulk gap (b) depending on the criterion described in the main text. The lower half of the polar plot classifies the GNRs according to the zigzag edge/GNR width ratio below 1 or above 1. A tag “b” refers to structures without zigzag edges, that is, fully benzenoid. The hashtag (#) indicates how many families (i.e., a GNR with the same thickness and edge termini but different length) belong to the indicated edge ratio. (c,d) Polar plots of the energy gaps of more than five thousand randomly generated GNRs following the same classification as in (a,b).

that the energy gaps of periodic 7-AGNR in Figure 1d have a similar origin to “bulk”, long 3c structures.30 Figure 2a shows that the LUMO of 3c is delocalized, like the “bulk” state LUMO+1 of 3a, which comes into play when calculating periodic structures of 7-AGNR where edge termini are ignored (all structures with MOs shown have been geometrically optimized). When different edge termini are considered, the energy gap remains constant as a function of length for 3a and 3b23 (Figure 2b). Note that geometrical relaxation of all structures was performed using unpolarized ground states and, therefore, does not play a role in the emergence of the edge states (Supporting Information Figure S1 shows nearly identical geometries for 3a and 3c). The energies arising from delocalized MOs with larger cross sections, can be expected to govern optical absorption,34,35 whereas localized states might exhibit tunneling36 and possibly nanomagnetism.28,29 Thus, though zigzag edges in graphene nanostructures cause the

ground state from unpolarized and spin-polarized collinear DFTB calculations,33 respectively (see Experimental Methods). When the ground state is found to be polarized, the AGNR is identified as an AGNRE. When an AGNRE is encountered, its energy gap is referred to as “edge” gap when frontier orbital renderings indicate that this gap is due to states localized at the edges. The energy gap due to delocalized orbitals spawning the repeating unit of the AGNR is referred to as “bulk” gap, as mentioned previously. Figure 2b (left and right panel) shows AGNRs that fulfill this edge gap (E) criteria. The necessity of defining a bulk and edge gap is evident when comparing Figure 1d with Figure 2a,b. At the 370 Å length limit, molecules 3a, 3b, and 3c (belonging to 7-AGNR) show energy gaps of 0.15, 0.18, and 1.4 eV, respectively (Figure 2a,b, yellow lines). The “bulk” energy gap for long AGNRs 3c (1.4 eV) is close to that reported by DFT23 for a periodic 7-AGNR structure calculated without edge termini (∼1.5 eV) (Figure 1d). Thus, it is clear 3230


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Figure 3. Random termini edge sampling reveals AGNR family subtypes. (a) Sampled exceptions to inferred structure−property trends: 6a−c, and other 5-AGNRs and 4-AGNRs with termini edge/width ratio equal or higher than one, do not feature polarized states. 7a,b, Termini edge structures showing corresponding spin difference (HOMOup − HOMOdown) plots for AGNRs with edge/width ratios below one, conserve the number of Clar’s sextets typical for antiferromagnetic AGNRE. The isosurface cutoff for spin-polarization is chosen as 2% of the maximum spin-polarization for optimal image rendering. 7c, a 5-AGNR can also feature edge states when non-Kekulé edges are introduced. 7d, asymmetric, non-Kekulé edges give rise to triplet states. The DFTB and DFT PBE with a triple-ζ polarized basis set isosurface rendering of the HOMO−1 are compared. (b) Energy diagram (eV) in the Kohn−Sham orbital picture for the four subtypes of AGNR structures found: unpolarized, antiferromagnetic, non-Kekulé antiferromagnetic, and triplet. The HOMO of the selected archetypes is also shown.

addition, some structures belong to 5-AGNR, similar to those in the model structures 2a and 2b. These AGNRs do not adhere to edge/width ratio rules because of their (3n − 1)AGNR character, where n is an integer (Figure 3, 6a,b). This behavior has previously23 been denoted 5- and 8-AGNR as quasimetallic AGNRs due to their apparent lack of termini edge states and low energy gaps. Thus, it becomes only relevant to analyze any exceptions to the edge/width ratio rule for the edge-state-forming AGNRs in Figure 2d. Here, more than 50% of the structures seem not to follow an edge/width ratio preference. A remarkable example is structure 4c, which becomes antiferromagnetic upon changing the geometry of the edge while keeping a constant edge/width ratio relatively to the parent structure 4b. It is clear that in this case, Clar’s sextet configuration is decreased from three sextets in the repeating tetracene unit of parent structure 4b, to two sextets. It suffices to remove one atom on each edge (red circle) to recover the nonantiferromagnetic (unpolarized) high sextet configuration 4d, simultaneously lowering the bulk gap for the 50 Å long molecule by ∼0.1 eV. This exception to the edge/width ratio intuitively follows Clar’s rule, evocating that acene structures with minimal Clar’s sextets retain a multiradical character.42 However, Clar’s rule cannot be used for predicting (lack of) antiferromagnetism, specially in large molecules where combinations of armchair and zigzag termini edges are involved (cf. 7a in Figure 3). Another apparent rule exception occurs when narrow zigzag edges hide a larger zigzag edge structure like in structure 7b. These large tetracene edges confer an antiferromagnetic stability of up to 0.08 eV with respect to the unpolarized state, for 7b AGNR lengths of 28 Å. Energy differences much larger than the thermal energy at 300 K

expression of localized ferromagnetic or antiferromagnetic states,5,6,37−40 the previous example suggests that termini zigzag edges are not a sufficient condition for inducing magnetic properties in AGNRs. Having benchmarked the model and shown that zigzag termini are not a sufficient condition for magnetic properties, we turn to the main question addressed in this study: How do chemically diverse short edge termini modulate the energy gap and magnetic properties of finite AGNRs? In order to systematically determine edge vs electronic property relationships41 in AGNRs, we randomly generate nearly ten thousand AGNRs, which include 361 different combinations of edge termini with widths ranging from 3-AGNR (i.e., 1.47 Å wide oligoparaphenylenes) to 9-AGNR (9.67 Å wide). The generator provides AGNRs with either no symmetry between left and right edges, mirror symmetry or inversion symmetry (see Supporting Information for molecular orbital plots and energies of chosen structures). From the randomly generated AGNRs, we find that 72% do not feature edge states (Figure 2c), the rest fulfilling the GNRE criterion (Figure 2d). Up to 95% of edges that lack antiferromagnetic states are classified as benzenoid or have an edge/width ratio below one. Again, the edge/width ratio is defined as the longest zigzag morphology belonging to the edge termini divided by the width of the AGNR backbone. A particular example is AGNR 4b, which in analogy to model 3c loses its antiferromagnetic edge state by lowering the edge/ width ratio. In fact, only 5% of the structures not possessing an edge gap have an edge/width ratio exceeding one. However, close inspection reveals that most of these structures are asymmetric, lacking mirror or edge inversion symmetry. In 3231


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Figure 4. DFTB energy levels of AGNRs and outlook. (a) Combined HOMO and LUMO energies polar plots of the randomly generated ribbons in Figure 2c,d and b−e). A concept half-adder all-carbon sp2 blueprint employing 8-AGNR diodes (orange, in panel c) chosen from the random generated ribbons. A and B are voltage inputs, depending on their respective inverses A′ and B′. The grounded Sum and C are the read outs. The positive sign (+) indicates the positions where a load is applied. Out-of-plane electrodes contacting all lettered symbols are shown in black circles.

remarkable properties of such chosen structures is that removal of few carbon atoms affects their energy gaps by up to 0.33 eV. For the 28 Å long 8-AGNRs, the gaps are increased by only removing two atoms, from a 0.53 eV HOMO−LUMO gap (5a) to 0.63 and 0.86 eV (5b, 5c, respectively). Finally, it is of interest to explore the properties of the GNRs toward the design of electronic device elements. Besides the energy gap modulation, AGNRs may be plotted to depict energy differences between their frontier orbitals (Figure 4a). Although the energy of the unoccupied frontier orbitals is known to be underestimated by density functional methods, the occupied orbital energies from spin-polarized calculations33 are a valid point of departure for the design of hole-transport molecular diodes.47 From the polar plot in Figure 4a, it can be observed that long AGNRs cover a large HOMO energy range (in absolute units), from ∼ −5.1 eV (7-AGNRs) to ∼ −4.5 eV (8-AGNRs). Moreover, within GNRs of equal thicknesses, GNR 5a and 5b can be regarded as archetypes for the conceptual design of a hole-transport molecular diode, that is, where holes can be either transported through the (depopulated) HOMO (black arrow, Figure 4c) or blocked by the 0.2 eV gap between 5a and 5b. Figure 4e illustrates how an allcarbon molecular half-adder, depicted in Figure 4b, could be envisioned with such diodes. Here, A and B are the voltage inputs with their respective inverses A′ and B′, whereas the grounded Sum and C are the read outs. Such a three-branched hexagonal diode grid is understood with ease: The Sum is positive (returns a value of one) when at least one out of the two left and top branches is positively biased. In order to leave one branch positively biased and return a Sum read out of one, only A or B (but not both) must be positively biased. When both A and B are biased positively, that is, both inputs are one, their inverses A′ B′ are both neutral, and the left and top branches voltages drop to zero; the Sum read out is zero. (Figure 4b−d). Similar trigonal architectures using antiferromagnetic AGNRs may be imagined for the fabrication of molecularly frustrated magnetic devices.48 Besides such intriguing function, the construction of highly symmetric

suggest increasing room temperature stability of the magnetic state.43 Altogether, edge/width ratios >1 in long sub-nanometer wide AGNRs, constitute a powerful molecular design rule for antiferromagnetic edge states with increasing antiferromagnetic stability, in analogy to the proposed stability of singlet biradicals (∼two unpaired electrons calculated from natural orbital occupations20) in zigzag graphene nanoribbons (ZGNRs).7 Interestingly, our study includes sampling and identification of “Clar’s goblet”44 AGNRs, comprising non-Kekulé edges (specifically triangulene) attached to 5-AGNR. These nonKekulé edges introduce a biradical character by-design, which also can be explained by Lieb’s theorem.27 Moreover, we find that asymmetric non-Kekulé edges can be attached to AGNRs (7d in Figure 3a). The resulting biradical structures exhibit distinct energies for both radical subunits, opening up design rules for triplet states. For instance, the triplet state 7d in Figure 3a features large stabilization energies, of up to 1.45 eV (1.55 eV using DFT with the B3LYP functional, see Experimental Methods and Supporting Information) with respect to the singlet state. Although the strategy of combining non-Kelulé edges for by-design spin chemistry is well known,45,46 armchair graphene nanoribbons offer the possibility of fine-tuning the energy of the triplet state by increasing the backbone’s ribbon length with respect to the second, shorter non-Kekulé edge. Figure 3b summarizes all four different electronic subtypes of AGNRs arising from different termini edges sampled in our study: unpolarized, antiferromagnetic, non-Kekulé antiferromagnetic, and triplet. Of all the widths investigated, 8-AGNRs (orange lines) are the only ones not present in Figure 2d and, thus, do not feature edge states within the limited sampling of the edge structural space employed. We expect finite 3n − 1 family 8-AGNR to reveal edge states when introducing non-Kekulé zigzag termini as in the case of 5-AGNR. 8-AGNR are also potentially important building blocks of macromolecular electronic circuitry because their family covers a large energy gap region in the 30−50 Å length range. Three different 8-AGNRs are depicted in Figure 2, AGNR 5a, 5b, and 5c. One of the 3232



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hexagonal device grids can be envisioned in conjunction with supramolecular self-assembly protocols: each branch featuring molecular recognition exclusively between A−A ends, B−B ends, and SUM−SUM ends (Figure 4e, the “AB” electrode input refers to a Y-door of inputs A′ and B′). Evidently, the employed GNR generation algorithm, and hence the proposed macromolecular branches for the concept half-adder, do not include any synthetic chemistry design rules. As such, it is clear that the hypothesized molecular control presented in Figure 4 is unreachable by currently available solution or on-surface chemistry methods. Notwithstanding, the employed structure generator can be straightforwardly modified to generate complex structures (and subproducts) from established chemical synthesis and self-assembly protocols. In summary, our systematic study and polar plots of randomly generated sub-nanometer ribbons reveal important structure−property relationships. (1) Most AGNRs with zigzag edge termini are neither antiferromagnetic, nor feature localized edge states (2) (3n − 1)-AGNR families can be tailored to provide antiferromagnetic states when introducing non-Kekulé edge structures (3) a ratio between the width of the termini edge with respect to the width of the AGNR backbone higher than one, identifies edge states in 32 out of 46 sampled AGNR families (4) the ground states of all even-number-of-atoms subnanometer width AGNRs have the potential to be antiferromagnetic or triplet at high temperatures. Ultimately, our screening of compounds following in-silico chemical synthesis generation protocols instead of random generation could simultaneously allow prediction of function and synthesis/ assembly of (two-dimensional) macromolecular (spin)electronics device blueprints from their small molecule precursors. In this regard, our work is a step forward, elucidating fabrication-structure−property relationships affording molecular engineering protocols for all-organic (spin-) electronics.

Letter

ASSOCIATED CONTENT

S Supporting Information *

Additional computational methods and DFTB+ validation benchmarks to DFT. Spin difference plots of 73 structures, complete list of molecular orbital energies for 9214 structures and DFTB+ scripts are provided separately as additional data. This material is available free of charge via the Internet .The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpclett.5b01154.

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]. *E-mail: [email protected]. *E-mail: [email protected]. Present Addresses ∥

(Y.H.) Departamento de Fı ́sica, Universidad de los Andes, Carrera 1 18A-10, Bloque Ip. Bogotá, Colombia. ⊥ (X.F.) Chair of Molecular Functional Materials, Technische Universität Dresden, 01062 Dresden, Germany. Author Contributions ¶

(C.-A.P., M.A.) Equally contributing authors.

Notes

The authors declare no competing financial interests.

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ACKNOWLEDGMENTS Work financially supported by the European Research Council via Advanced Grant MolArt (Grant 247299) and the German Science Foundation (DFG, GRK 1570). The authors thank Phillip Shemella, Ari Seitsonen, and Hartmut Schlichting for helpful discussions. We thank Lukas Dössel, Moussa Saleh, and Lileta Gherghel for providing the UV−vis data used in this work. C.-A.P. thanks Shengda Wang and Francesco Rao for help with the python networkx plotting code. Angel Rubio and Duncan Mowbray are acknowledged for preliminary GWcorrected tight-binding studies.

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EXPERIMENTAL METHODS UV−vis spectra were recorded in quartz cuvettes or substrates. DFTB is an approximate valence-only DFT method that employs localized atomic orbitals as basis functions.49 Calculations were performed with the DFTB+ package50,51 using the carbon and hydrogen mio-0−151 Slater−Koster parameters. For calculating benzenoid and antiferromagnetic AGNRs under a general framework, collinear self-consistent charge26 spin-polarized calculations33 were carried out for all structures in the spin-restricted singlet (unpolarized), biradical singlet (antiferromagnetic), and triplet configuration. The electronic configuration with the lowest energy was taken as the ground state. DFTB incurs in additional stabilization of delocalized states,52,53 potentially underestimating emergence of edge states, therefore serving as a higher bound for predicting magnetic states. Although spin contamination is usually a minor issue in density functional methods, it may affect the length dependent onset of antiferromagnetism with respect to multideterminant methods.54 Additional firstprinciples DFT calculations were performed with the ADF computational package for selected compounds in order to validate the DFTB results. This comparison is presented in the Supporting Information together with more details on the computational protocol.

REFERENCES

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Sub-Nanometer Width Armchair Graphene Nanoribbon Energy Gap

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