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C 2011 American Chemical Society

Observational Geology of Graphene, atthe NanoscaleBoris I. Yakobson†,* and Feng Ding‡

†Department of Mechanical Engineering and Materials Science, Department of Chemistry, and the Richard E. Smalley Institute for Nanoscale Science and Technology,Rice University, Houston, Texas 77005, United States, and ‡Institute of Textiles and Clothing, Hong Kong Polytechnic University, Hong Kong, China

Vista after vista . . . range beyond range.Thornton Wilder

What one sees and recognizes de-pends on the scope and the per-mitted detail of our worlds. Walk

around an office, and the world appearscomfortably flat and unadventurous. Yeta not-so-distant journey—or simply abinocular—may reveal a mountain ridge oran ocean coast, that is, if you are not in Texas.Embark on an expedition of Magellanianscale, and you may prove the Earth's round-ness and perhaps discover continents. Re-cent advances in microscopy instrumenta-tion reveal both the new, finer atomisticdetails of a material's makeup and, at thesame time, the organization on larger scales.One monoelemental material is of particu-larly keen and sustained interest.A monatomic layer of sp2-carbon, once

obtained by reduction of graphite oxide1

and called graphene, remained largely over-looked until the physical properties of thismaterial, cleanly peeled off mechanically,2

attracted great attention from researchersacross a range of fields. Exfoliated this way,pieces are usually small and the atomicstructure appears as clean hexagonal tiling,with minor elastic undulations and ratherrare point defects—a mono- or divacancy,an interstitial carbon atom, or a Stone-Walesdefect 5/7|7/5. Importantly, these defectsare not only rare, but they also easily vanishupon annealing: a local transformation andaddition or deletion of an atom restoresa fully perfect lattice. This contrasts withnon-annealable, topological defects. A strik-ing example is a humble lonely pentagon,which imposes a conical shape on thewhole lattice around it3 and obviously can-not be eliminated without rearranging anentire macroscopic area. Another exampleis an extended series of pentagons “5” andheptagons “7”, or rather a series of their

pairs 5/7. If such series has a balancedpolarity (as in 5/7/5/7|7/5/7/5), then it canbe annealed locally. If the polarity along thelinear series is maintained, it has globalconsequences and together these 5/7s en-sure that the domains/grains separated bysuch grain boundaries (GBs) are also mis-matched in their orientations.

The article by Kim et al. in this issue of ACSNano,4 which is essentially concurrent witha Nature paper by Huang et al.,5 presents avivid picture of polycrystalline graphene, afirst careful mapping of the grains-domainsreoriented in-plane, tilted with respect toeach other at different angles (Figure 1a).

The article by Kim et al. in

this issue of ACS Nano

presents a vivid picture of

polycrystalline graphene, a

careful mapping of the

grains/domains reoriented

in-plane, tilted with

respect to each other at

different angles.

* Address correspondence tobiy@rice.edu.

10.1021/nn200832y

ABSTRACT As the available length ranges expand, graphene begins to show its anticipated

polycrystallinity. Its texture, revealed with modern comprehensive microscopy in recent work by Kim

et al., includes coherent domains/grains oriented randomly yet with an intriguing degree of

regularity. The domains are stitched together by pentagons and heptagons aligned into the grain

boundaries. The challenge is now to deduce the mechanisms of formation based on observations and

then to find ways to control the morphology toward useful properties and applications.

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In a tour de force of modern micro-scopy, the studies use complemen-tary methods and careful analysis toyield otherwise unavailable infor-mation. While with a greater scopethe dark-field imaging and localdiffraction techniques enable theresearchers to measure the domainmisorientations, on a smaller scale,the aberration-corrected high-reso-lution transmission electron micro-scopy (HRTEM) further shows theintimate atomistic structure of theseam lines, the borders between thedomains. The relative tilt angle var-ies from 0 to 30�, due to the hex-agonal symmetry of real graphene,instead of a 180� range in the sche-matic depiction of Figure 1a. Datacollected from sufficiently largeareas show a non-obvious distribu-tion that peaks at smaller and great-er tilt angles. To enlarge the sampleset, a composite distribution fromthree sources4,5,6 is plotted inFigure 1b,c. In spite of subtle differ-ences in the growth conditions, allsamples have been produced oncopper foils by chemical vapor de-position (CVD),7 which providesgood reason for considering suchcombined statistics. Since the indi-vidual reports suggest peaks nearcertain angles, it is instructive tosee what features remain robustacross experiments by differentgroups. Furthermore, one can eitherdirectly add the counts (assigningequal weights to each GB found,Figure 1b) or first normalize the dataand thus assign the same weight toeach report (Figure 1c). Eventhough the peaks become less con-spicuous, a visible dominance of thelow angle <7� and especially near28�GBmust be telling us something.This angle distribution is sustainedacross the experiments, althoughthe different growth conditions dochange the size (as large as 3-10 μm in ref 4 or only ∼0.2 μm inone growth method of ref 5) andshapes of the grains. Moreover, onenotices not only the generic threeGBs per node in ref 4 but also oftenmultiple GBs radiating from the ap-parent growth centers in ref 5. It is

reasonable to speculate that a cleanersubstrate and consequently slowhomogeneous nucleation would re-sult in larger domains, whereas thepresence of centers for heteroge-neous nucleation should speed itup, producing smaller grains, likelyemanating from the shared hubs. Inspite of the apparent grain mor-phology sensitivity to the growthconditions, the GB angle distributionappears reproducibly non-uniform.Just months before Kim et al.'s4

and Huang et al.'s5 reported obser-vations, three theoretical studiesappeared,8-10 in which the struc-tures and energies of tilt GBs weresystematically compared as a func-tion of the tilt angle. All three viewedthe GBs as 5/7 chains, similar ofcourse to the old views of graphite11

and of nanotube junctions,12,13

where the change of chirality isequivalent to the tilt angle betweenthe grains. As a natural step fromgeometry to physics, all groups8-10,14

computed the GB energies, shownin the composite plot of Figure 2a.The GB degenerates into a perfectzigzag line at 0� and similarly de-generates into a perfect armchairline at 60�, both with zero energy.As the tilt and the number of 5s and7s both increase, the energy γ goesup, as well, yet less than in simpleproportion to the defect occur-rence. For higher 5/7 defect density,there is a mutual cancellation, a“destructive interference” of thestress fields, similar to optics. In-deed, a 5/7 dislocation is not astress-free void nor a crack, but im-portantly, it carries a known com-pression on the 5 side (extra plane-row, blue in Figure 2b,c) and anopposite sign tension on the 7 side(missing plane-row, red in Figure 2b,c). Obviously, these fields are mu-tually canceled when brought inproximity, thus higher defect den-sity reduces the bond prestrain(which makes the GB seeminglystronger) and the total energy. Con-sequently, the high-density GB hasa slightly lower energy even thoughit contains more defects. Figure 2cillustrates this with the computed

continuum stress field (namely, aTr σ = σxx þ σyy) around an edgedislocation, visibly greater for a sin-gle dislocation (^) than for a con-densed array in a large angle GB.One could even construct a Heisen-berg form of a Hamiltonian, as H ∼Σi|bi|

2 - ΣijJijbi 3bi (note that in thehexagonal lattice the Burgers vectorb accepts six discrete values), andhave a basic model of GB handy.Positive “exchange integral” Jij ori-ginates from the elastic fields can-cellation for the collinear bs andsuggests that such defects want toalign and condensate. Direct ato-mistic calculations do show suchan energy dip at 32�, especially pro-nounced in theaccuratedensity func-tional theory (DFT) computations.10

It is tempting to relate this lower

Figure 1. Even in a solidly assembledpuzzle, the mismatch in textureorientations creates a pattern of tiltboundaries (a). Composite distribu-tions of the tilt angles from experi-ments, (b) with equal weight pereach grain boundary occurrence,in magenta,4 pink,6 and red,5 or (c)with equal weight per each report,in purple,4 light blue,6 and blue,5

all consistently non-uniform.

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energy with the higher presenceof such GB angles in experiments(the equivalent angle would be28�, within the observed peaks,Figure 1b,c), yet one can find noreason for such an association.Complementary to the global do-

main mapping, the intrinsic atomic-scale makeup of the GB lines is ofgreat interest. One thing is to pro-pose a logical bond topology ofGB11 or to compute its precise geo-metries through energy minimiza-tion,8-10,14 yet another thing is tosee it—and believe it. This is whathigh-resolution microscopy offersto an inquisitive observer.4 HRTEMreveals only 5s and 7s, predomi-nantly next to each other—that is,the disclinations paired up into dis-locations of smallest Burgers vec-tors ((1,0), and with neithervacancies nor interstitials. This sug-gests good local equilibration atgrowth conditions: any low-energyvoid site is filledwith a carbon atom,every high-energy interstitial C isexpelled, and every unnecessaryStone-Wales defect is switchedoff. Even though it remains unclearhow statistically representative animage or two might be, the re-ported pattern4 is quite remarkable,especially in its striking visual simi-larity with the related pattern, ob-tained completely independently.5

One is left to wonder to what extentthis is indeed a most typical GBstructure, or a coincidental similar-ity. While resolving the planar atomicpositions, the HRTEM may concealthe three-dimensional landscape,and theoretical calculations canhelp restoring it. To this end, onecan extract all atomic x-y positionsand then perform full energy mini-mization in x-y-z space, as in ref 8.The resulting structure in Figure 3shows significant off-planewarping.(It is possible then that the visibly“distorted” hexagons in the ADF-STEM image5 are simply not parallelto the plane of view.) This warpingof a free-suspended GB area is ofcourse suppressed if the structure islaid on a substrate, with accordinglychanged energies. The meandering

of GB is another striking feature thatmay or may not be typical. Both thismeandering and the aperiodic posi-tions of the 5/7s imply that the GB isunlikely the lowest energy structurefor a given tilt angle.Energy is an omnipotent and

loved measure in physics, and itsminimization guides various pro-cesses, in particular, the structuralstability or the fluctuations off equi-librium, whose probabilities corre-late with ∼exp(-E/kbT). However,both dislocations and GB are non-thermodynamic objects; they donot emerge as results of fluctuationsbut are kinetically “frozen” and donot need to obey the laws of statis-tical physics in this sense. In thisrespect, a GB is similar to a moun-tain ridge—a strikingly improbableobject from the energy minimiza-tion viewpoint, yet not uncommon.To try to understand the GB mor-phology and the tilt angle distribu-tions one should track back alongtheir past origin pathways and notsimply compare their present ener-gies. Similarly, a geologist is forcedto do retro-engineering because theobjects of his study were shaped byslow processes long ago. When thecontinental activity almost seized inPrecambrian times or the Paleo-cene, there was not a soul there towatch and to record what was goingon and how. The geologist does hisfieldwork now, observeswhat is avail-able, and tries to deduce how itcame about. For the nanostructures,

the formation processes may be toofast for an in situ record, sowe resortto a similar approach: record thefinal state morphology and thenguess what factors could havecaused its formation. “Observa-tional geology” of graphene, madepossible by the advances in micro-scopy, suggests that its monocrys-talline domains and the GBs betweenthem must originate during growthand the tilt angle distribution maybe determined by the degree ofepitaxy to the substrate at the ear-liest stages of nucleation.

The possibility that the GBs are“induced” by the underlying textureof polycrystalline Cu is ruled outsince the size of graphene domainsis at least an order of magnitude

Figure 2. Grain boundary energy varies with the tilt angle and is sensitive to thelocal organization of the 5 and 7 pairs. (a) Data shown from different computa-tions: hollow circles9 (TBA), blue triangles14 (classical force field), red circles(AIREBOpotential as in ref 8), and green squares10 (DFT). (b) Computed continuumstress field from an isolated edge dislocation, and (c) from a series in the grainboundaries, with apparent mutual cancellation, explaining a dip in energy near∼30�; blue is for compression and red is for tension.

A grain boundary is

similar to a mountain

ridge—a strikingly

improbable object

from the energy

minimization

viewpoint, yet not

uncommon.

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smaller than the granularity of cop-per foils in use.4,5 If graphenegrowth was not epitaxial at all, onewould expect random grain orien-tations and a uniformdistribution ofthe GB tilt. On the other hand,strong epitaxial match to a mono-crystal substrate would permit onlyvery few GB angles and sharp peaksin the tilt distribution for the CVDproduced graphene. Small sp2-nu-clei of graphene are likely to startnear the substrate steps15 and arealso likely to have preferred orienta-tions, not necessarily the same asfor an extended graphene sheet. Asthe nucleus island grows, its mobi-lity on a substrate rapidly decreasesand soon it adapts an orientationthat remains frozen, unchangedafterward. Investigations of the low-est energy clusters can be rathertedious15 but offer a possible wayto understand GB tilts. For example,while a coronene-like cluster has itslattice properly packed with hexa-gons embracing the metal atoms(Figure 4a), interestingly, a largercluster of circumcoronene type C54,according to one calculation,16 istilted at an angle of ∼14�. If anisland preserves this orientationand eventually encounters anothergrain frozen in a mirror-twin posi-tion as in Figure 4b, the emerg-ing grain boundary has a tilt of(bingo!) ∼28�. The emerging contactboundary is illustrated in Figure 4c.Although this may well be a coin-cidence, the possibility of graphenedomains being trapped in the localminima at earlier stages in theirformation is a probable cause forthe complexity of the grain orienta-tions. More frequent grain orienta-tions may correspond to theirepitaxial match to Cu(111) at earlystages of nucleation and small is-land growth. If nucleation is favorednear the steps on ametal surface, asboth calculations15 and some experi-ments17 suggest, then the orienta-tions of such steps and terraces canplay a definitive role in themorphol-ogy of finished graphene “quilts”(Figure 5). Who knew that the dull-gray material graphite, when seen

through the rigorous technologicaleyes of modern microscopy, wouldturn into such a post-impressionistplay of color and form? Yet here it is,the colored map of the united do-

mains of graphene, inviting and un-explored in many corners.

PRACTICAL CONSEQUENCESAND LOOKING TO THE FUTURE

What are the practical conse-quences beyond the fascinatingpatterns, maps, and colored quiltsrevealed by microscopy? Generally,

polycrystals can outperform mono-crystals, sometimes impressively. Inmagnetic materials, GBs suppressthe Bloch walls mobility and helpto tune material hardness (magneticcoercivity); this seems irrelevant togenerally nonmagnetic graphenes,except for the ferromagnetism ofthe defects themselves. In me-chanics, well-known grain bound-ary strengthening culminates inthe Hall-Petch relationship, wherethe smaller domains suppress thedislocation mobility, so the yield

Figure 3. Pentagons (blue) and heptagons (red) in a grain boundary should resultin noticeable off-plane 3D landscapes,8 obtained here by computational relaxa-tion, for an example based on aberration-corrected annular dark-field scanningtransmission electron microscopy (ADF-STEM) image,5 bottom. Bottom partadapted with permission from ref 5. Copyright 2011 Nature Publishing Group.

Figure 4. “Multi-epitaxial” growth process may result from different ground stateorientations relative to the metal substrate, when a smaller cluster (a) is thenrotated by ca. 14�. If trapped in these orientations (b), two such islands can form ahigh-angle boundary. Preferential nucleation at a step15 can further determine asubset of specific tilts, which requires further detailed study.

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stress increases inversely to the do-main size, Δσyield ∼ 1/

√d. Gra-

phene, however, is nonductile; its5/7 dislocations are sessile at prac-tical temperatures.11 Grain bound-aries work as stress concentratorsfor easier brittle failure, as recentexperimental tests show, as well.5

Electrical resistance does not ap-pear to be significantly increasedby the apparently irregular grainboundaries,5 while theoretical anal-ysis suggests that for the chargecarriers almost complete transpar-ency or reflection can both be ex-pected from the periodic arrays ofdislocations.18 On the basis of re-cent observations and the basic factthat theGBstructure is notwell equili-brated but is rather frozen in disorder,it may be challenging to engineer aperiodic array accurately. An interest-ing exception is a special case of non-tilt GB, a translational variety wherestrong order in a defect sequence

558558558558

is well-maintained by its rigid at-tachment to the zigzag rows in thenative lattice, whose halves arestacked slightly off (hcp and fcc)on the Ni(111) surface.19

Beyond the captivating patternsunearthed by “nano-geology” fieldwork, there lie various tantalizingpossibilities. There is a striking dif-ference between geology and na-noscience: In the former, one canfigure out the origin of the observedstructures but cannot repeat theexperiment and cannot direct itsconditions toward a desired out-come. In a nanotechnology lab, thisis possible. The emerging under-standing relates the specific scaleswith the periods of nanoformations.Locally, in the smallest atomisticscale (∼0.5 nm), the GBs appearwellequilibrated and at the lowest avail-able energy, that is why one sees nounnecessary disorder, but only in-evitable 5/7s (or 4|8s in the sheartype GB). On a somewhat largerscale (∼1-10 nm), the 5/7s becomeunruly and the GB lines meander,with the lowest energy no longerbeing a reliable foothold. One canspeculate that these shapes are fro-zen, inherited from the instabilitiesof the growing fronts of individualdomains. Indeed, the front of grow-ing graphene either can remainstraight (probably zigzag, as a slow-est-growing facet20) or at certainconditions can develop undula-

tions through the known Mullins-Sekerka instability in diffusion-con-trolled aggregation. At this inter-mediate time, shapes at the frontmay freeze as a tortuous GB whenthe two fronts finallymeet and closethe gap. This can possibly be avoidedby changing growth conditions.On the even larger scales of theentire domains (50-1000 nm), theirorientations can in fact be estab-lished even earlier, at the beginningnucleation stage. A set of selectedlow-energy small islands can ulti-mately develop into the non-uni-form distribution of the tilt GB inthe complete polycrystal. Better un-derstanding each stage of growthcan enable morphological controlat the corresponding length scalesand ultimately allow us to engineerthe networks of built-in one-dimen-sional nanostructures with desiredmechanical, chemical, magnetic,and electronic (both longitudinaland transverse) properties.

Acknowledgment. Work at Rice wassupported by the Office of Naval Re-search, the Robert Welch Foundation,and the National Science Foundation.We are grateful to Y. Liu, V. I. Artyukhov,andM. Liu for help and useful discussions.

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Figure 5. A 25 μmview, with color-coded orientations of graphene domains, eachsmaller than the copper grains, yet some visibly grown across the copper grainboundaries, marked out by white lines. (Courtesy of P. Huang and D. A. Muller,Cornell University.)

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