Theory of Finite-Length Grain Boundaries of Controlled Misfit Angle in

Letter pubs.acs.org/NanoLett

Theory of Finite-Length Grain Boundaries of Controlled Misfit Angle in Two-Dimensional Materials Yuanxi Wang*,†,‡ and Vincent H. Crespi*,§,∥,⊥,†,‡ †

Material Research Institute, ‡2-Dimensional Crystal Consortium, §Department of Physics, ∥Department of Chemistry, and Department of Materials Science and Engineering, Pennsylvania State University, University Park, Pennsylvania 16802, United States

⊥

S Supporting Information *

ABSTRACT: Grain boundaries in two-dimensional crystals are usually thought to separate distinct crystallites and as such they must either form closed loops or terminate at the boundary of a sample. However, when an atomically thin two-dimensional crystal grows on a substrate of nonzero Gaussian curvature, it can develop f inite-length grain boundaries that terminate abruptly within a monocrystalline domain. We show that by properly designing the substrate topography, these grain boundaries can be placed at desired locations and at specified misfit angles, as the thermodynamic ground state of a twodimensional (2D) system bound to a substrate. Compared against the hypothetical competition of growing defectless 2D materials on a Gaussian-curved substrate with consequential fold development or detachment from the substrate, the nucleation and formation of finite-length grain boundaries can be made energetically favorably given sufficient substrate adhesion on the order of tens of meV/Å2 for typical 2D materials. New properties specific to certain grain boundary geometries, including magnetism and metallicity, can thus be engineered into 2D crystals through topographic design of their substrates. KEYWORDS: 2D material, grain boundary, Gaussian curvature, conformal growth, substrate topography, nanostructure control

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deformation in the third dimension: a topological characteristic, grain boundary geometry, is controlled through topographic design of the substrate upon which the material grows. At a continuum level, the substrate topography prescribes a metric for a two-dimensional material growing on it: the growth front of the two-dimsensional (2D) layer advances along geodesics on the substrate. For a flat substrate, this is simply a uniform advance of a straight growth front.1,2,54−62 But when passing over a region of positive Gaussian curvature, that is, a conical bump on the substrate, the growth front of a single grain can self-intersect, thereby forming a semi-infinite grain boundary that projects away from the asperity, as shown in Figure 1. The fractional disclination of the resulting cone relates its Euler characteristic χ to the opening angle α through χ = 1 − sin(α/2), resulting in a grain boundary with a misfit angle of θ = 2πχ, such as shown in atomistic simulations in a previous study63 and later in the present study. (Graphene cones with 1 isolated pentagonal rings at the apex each contributing χ = 6 64−74 are a special case in that they are seamless ). More complex topographies with regions of both positive and negative

rain boundaries in two-dimensional materials such as graphene and monolayer transition metal dichalcogenides (TMD) consist of linear arrays of topological defects and cannot be annealed away by local atomic rearrangements1−3 (excepting nanoscale grain boundary loops4). They are often thought of as unfortunate consequences of nearby grains nucleating in different orientations (except in limited cases where they appear spontaneously due to stoichiometric imbalance5,6). Although grain boundaries can degrade mechanical properties,7−10 electronic transport,1,3,11,12 thermal transport,12−16 and optical response,3 they can also introduce desired properties such as magnetism,17−19 enhanced optical response,3 current filtering,11,20,21 or one-dimensional metallic character.22−25 The size, character, and composition of grain boundaries in various materials can be controlled to some degree by plastic deformation,26−30 inoculant particles,31−35 thermomechanical treatment,36−39 solute segregation,40−47 Oswald ripening,48−51 point-defect induction and agglomeration,4,5 substrate registry,22 and electron-beam irradiation.52,53 Nevertheless, direct control over the precise position, length, and misfit angle of grain boundaries in solid-state materials has remained elusive; they are usually regarded as intrinsically kinetic in origin and passively follow nucleation accidents during growth. Here, we show how control over grain boundary geometry in two-dimensional materials can be achieved through © 2017 American Chemical Society

Received: April 18, 2017 Revised: July 21, 2017 Published: August 10, 2017 5297

DOI: 10.1021/acs.nanolett.7b01641 Nano Lett. 2017, 17, 5297−5303

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Nano Letters

graphene. The layer−substrate adhesion is controlled by a separate LJ potential; Supporting Information provides further information on these potentials and the implementation of kinetic Monte Carlo. This simple model is intended to capture generic quasi-continuum behavior; detailed material-specific energetics and kinetics will be discussed later. As shown in Figure 2a, growth begins from the left with two initial fixed

Figure 1. A single grain assumes two different orientations after the growth front propagates past a conical asperity on the substrate. (a) Top and (b) side view of the resulting structure containing a semiinfinite grain boundary. The grain boundary misfit angle θ is determined by the opening angle α of the underlying cone.

curvature can define preferred sites for both the beginnings and ends of grain boundaries, as discussed below. The effect is entirely geometric in origin, so should apply to any atomically thin two-dimensional material, assuming sufficient adhesion to the substrate. The material and substrate-specific effects of atomic step edges75−77 and substrate registry are neglected here, since we focus on the universal geometrical effects of conformation to long-wavelength substrate topography; (although we note that substrate registry effects could modulate and interact with conformal growth in interesting ways, especially for faceted substrates). Related geometrical effects have been investigated previously in liquid crystals and particle packing on curved membranes.78−82 Substrate conformation af ter transfer shows a “snap-through” instability as a function of the dimensionless interlayer adhesion (normalized to bending stiffness) and substrate roughness (ratio of roughness height to wavelength);83−88 in contrast, the conformal growth considered here requires Gaussian curvature and allows for defect formation and thus cannot be characterized by this single dimensionless parameter. The present study also differentiates itself from the vast existing literature on curved 2D materials (especially sp2 carbon) by highlighting the aspect of control, that is, how the topography of a substrate can be exploited to control the nanoscale bond topology of the growing layer. Although this control does not extend to conventional grain boundaries formed by the coalescence of adjacent grains, the analysis below suggests that the achievable densities of topographically induced grain boundaries can significantly exceed that of conventional coalescence-induced grain boundaries, an advantage that will grow further as the grain size achievable in 2D materials improves; (the seamless grain coalescence that is the primary motivator for epitaxial growth on commensurate substrates is thus helpful but not necessary). We first examine general issues of adhesion during 2D conformal growth with a minimal model. Given a substrate with a certain Gaussian curvature, does a growing material of a given adhesion and stiffness conform or detach during growth over a “bump”? To investigate this question, we performed an offlattice kinetic Monte Carlo simulation of 2D growth over an asperity of Gaussian shape 7.5 Å tall with a radius (standard deviation) of 12 Å. Both the growing layer and the substrate are described by simple Lennard-Jones (LJ) interactions. Particles within the growing layer have an anisotropic interaction which depends on the angle between rij (the vector between each particle pair) and the local normal, the strength of which is tuned to match the Young’s modulus and bending stiffness of

Figure 2. Growth kinetics of a 2D layer over a Gaussian asperity, as simulated using a minimal growth model where the mechanical properties of the modeled 2D layer matches those of graphene. The growth layer (a) conforms to or (b) lifts off from the substrate curvature when the interlayer adhesion is 100 and 5 meV/Å2 respectively.

rows. If the adhesion exceeds 5 meV/Å2, then a grain boundary consisting of five- and seven-coordinate particles radiates from the apex as the growth front passes over it with an estimated grain boundary energy of ∼0.3 eV/Å (similar to the 0.27 eV/Å value for a graphene grain boundary with the same 10° misfit angle72); for weaker adhesions the growing layer detaches from the substrate, as shown in Figure 2b. For comparison, the adhesion between graphite layers is 20 meV/Å2, which is well above this value, and the adhesion of graphene to metal substrates is generally stronger,89−92 as is substrate adhesion for many other 2D materials. Thus, conformation of the growing 2D material to rough substrate topographies is anticipated to be a general phenomenon, favored when the substrate adhesion is large and the grain boundary energies and bending moduli are small. Although the local chemistries of grain boundaries in polar 2D materials differ from that of graphene (e.g., oddmember rings require homopolar bonds), the common observation of well-defined grain boundary structures without apparent complications (other than local reconstructions beyond pentagon-heptagon motifs23,24,93 with a somewhat higher grain boundary energy) suggests that their polarity should not specifically disfavor the kinetic accessibility of finite grain boundaries. Are these kinetically defined grain boundaries thermodynamically stable for real atomistic materials? To answer this, we first prepare a series of semi-infinite (cone) or finite (conesaddle) grain boundary structures for both graphene and monolayer MoS2. For graphene, we focus on grain boundaries where the constituent dislocations are pentagon−heptagon pairs.20,94,95 For MoS2, we focus on a grain boundary structure commonly observed in experiment in which two sulfur atoms bridge a homopolar Mo−Mo bond shared by the pentagonal 5298

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Figure 3. (a) Top and side views of the relaxed structures of graphene and MoS2 flakes containing semi-infinite grain boundaries of differing misfit angles. The cone opening angle is given by α and grain boundary misfit angle by θ. (b) Ground-state structure of graphene and MoS2 constrained to a surface with a local positive and negative Gaussian curvature with a finite grain boundary terminating at the two Gaussian curvature extrema.

and heptagonal rings,24,93 but the essential results should apply to any grain boundary geometry because the long-wavelength behavior is largely determined by the Euler characteristics. Trial graphene structures were relaxed in LAMMPS using the adaptive intermolecular reactive bond order (AIREBO) potential96,97 (see Supporting Information). All graphene structures were further relaxed at the tight-binding level using the DFTB+ code with Slater-Koster parameters from the “mio1-1” parameter set,98,99 although this further relaxation caused negligible structural change (not shown here). MoS2 structures were relaxed using a Stillinger-Weber potential parametrized by fitting the experimental phonon spectrum of monolayer MoS2.100 This empirical potential yields an accurate Young’s modulus due to the nature of its parametrization, although its accuracy for high strain regions near the dislocations may be limited. Figure 3a shows a sequence of semi-infinite grain boundaries such as would be produced when a growth front passes over a conical asperity of various sharpnesses. Essentially any misfit angle is possible. When the opening angle of the asperity is large, the misfit angle is small and the low linear density of dislocations produces the short-wavelength undulations visible in the side view. These undulations smooth out as the misfit angle θ increases, due to local canceling of strain fields.72 Whereas the rotational symmetry of a single isolated conical asperity means that the direction of the semi-infinite grain boundary originating at its tip is determined kinetically by the direction that the growth front passes over it (see the examples in Figures 2 and S2), the introduction of saddle-like regions paired with cone-like regions lowers the symmetry and can unambiguously define both the precise location and misfit angle of the resulting grain boundary. Such structures where positive and negative Gaussian curvatures balance out are also more compatible with realistic (i.e., asymptotically flat) substrates. Figure 3b shows the grain boundary structures of graphene and MoS2 on a surface with localized regions of positive and negative Gaussian curvature: finite-length grain boundaries

terminate with partial positive disclinations at the cone and partial negative disclinations at the saddle. Such finite grain boundaries can be experimentally identified by their large Burgers vector (as may have already been identified in ref 101). More generally, the misfit angle induced by passing over a positive curvature region is reduced when passing over a subsequent negative curvature region, that is, the Euler characteristic of a given local region gives the change in the misfit angle that occurs when a grain boundary passes over it. If multiple chains radiate from an asperity, the Euler characteristic of the asperity still determines the change of the misfit angles summed across the multiple grain boundaries, ∑iΔθi = 2πχ, so a substantial degree of control remains. Substrates where the Euler characteristics of neighboring regions cancel out are straightforward to design, an “egg carton” being one familiar example.82,102,103 More complex three-way junctions can also be engineered through suitable substrate design. We note that some related geometries of grain boundary structures have been examined as special cases during studies of the tensile strength of graphene10,63 and fracture formation in grain boundaries.104,105 Growth on three-dimensional nanoporous templates could result in Schwarzite-like structures106 but with finite grain boundaries instead of (or in addition to) isolated heptagon disclinations. The application to square lattices should not require major adjustments other than changes in grain boundary energies, while the application to quasicrystalline 2D systems such as Penrose tilings107,108 would require a reformulation of the concept of a grain boundary (in terms of perhaps of an alternating local deviation from Euler-rule planarity); this remains an intriguing topic for future work, as does application to patterns formed by physical adsorption on contoured substrates. We now investigate whether these substrate-defined finitelength grain boundaries do indeed provide the global groundstate structure, that is, that they are favored both kinetically and thermodynamically. First we simply compare the energy of a periodic structure containing finite grain boundaries with that 5299

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Figure 4. (a) Periodic structure of a graphene sheet containing two finite grain boundaries terminating at cone-like and saddle-like regions. The topography of the 2D sheet is shown in the inset, where the ± signs mark the localized positive and negative Gaussian curvature. Scale bars are omitted for clarity, because distance can be inferred from the bond lengths. (b) A structure with the positions of 16 dislocations randomized is prepared and (c) subsequently annealed into a finite grain boundary structure at 1000 K using bond rotation Monte Carlo simulations.

of pristine graphene, both in the absence of substrate. Figure 4a shows a periodic structure with the atomic positions and cell geometry fully relaxed. The defective structure suffers a energetic penalty of only 10 meV per carbon atom compared to pristine graphene, a rather small penalty that can be made arbitrarily smaller by simply reducing the areal density of the grain boundaries. The 50 meV/atom or more adhesion of graphene to a typical substrate greatly exceeds this penalty, strongly suggesting that the finite-grain-boundary sheet is energetically preferred over pristine flat graphene for contoured substrates to which it can more closely conform. A similar conclusion should apply to other 2D materials, which tend to have lower grain boundary energies, similar or stronger adhesions, and higher bending stiffness. To make this competition more rigorous, we prepared periodic graphene structures containing randomly placed dislocations on top of a contoured substrate, as shown in Figure 4b, where the substrate topography is given in the inset of Figure 4 (see Supporting Information for details on the generation of the disordered initial structures). The system was then annealed by Metropolis Monte Carlo at 1000 K with Stone−Wales bond rotation moves, similar to the implementation in a previous study.109 The Monte Carlo methodology is not intended to directly model annealing, but instead to guide a search for the global ground-state structure. Each bond rotation glides one dislocation core along its Burgers vector110 by one lattice constant and is accompanied by a full structural relaxation using the AIREBO potential. In all 30 simulation runs, the dislocations consistently migrated to form structures similar to Figure 4c where two finite-length grain boundaries produce maximal conformation to the substrate. Although 1−3 pairs of dislocations would tend to annihilate during the annealing (note that this Monte Carlo protocol allows for dislocation core migration and annihilation, but not generation), the surviving dislocations always form linear structures with pentagons (heptagons) terminating at regions of positive (negative) Gaussian curvatures, demonstrating that substrate interactions are sufficiently strong to guide the system into a ground-state configuration that hosts finite-length grain boundaries with predetermined locations and orientations. To tighten the energetic comparison between the flat/ pristine and conformal/defective sheets, we now consider alternative mechanisms by which the pristine sheet can maximize substrate adhesion. Figure 5b shows how a pristine layer could develop a radial fold extending from the apex of a

Figure 5. (a) Nucleating a grain boundary versus (b) developing a fold.

conical substrate. If the adhesion of the 2D layer with itself is similar to the layer−substrate adhesion, the energetic cost of the pristine-but-folded structure comes entirely from the curvature of the fold; (this is a conservative estimate, because layer−substrate adhesion is often stronger than layer−layer 1 adhesion). This penalty is on the order of 2 Dπr /r 2≈ 0.6 eV/Å, where, for example, D = 1.2 eV is the bending stiffness of graphene and r = 3 Å is the radius of curvature of a graphene fold.111−113 The fold energy per unit length should not vary much for different cone opening angles because the fold has similar radius irrespective of its depth. Extra deformation associated with local sheet Gaussian curvature at the initiation of the fold near the apex may add an additional constant offset to the overall fold energy. On the other hand, the grain boundary energy in graphene is only G = 0−0.4 eV/Å72 and tends to zero as the cone angle approaches 180° or 112.9° (corresponding to misfit angles 0° and 60°). Hence the conformed grain boundary structure should be more favorable than the folded pristine structure under a wide range of conditions. Conformal grain boundary formation should be even more favorable in monolayer TMDs because their grain boundary energies can be tuned to be smaller than those of graphene (by varying the metal or chalcogen chemical potential23) and their bending modulus is larger.114 We note that the actual price for grain boundary formation is paid in discrete steps, that is, whenever a dislocation is created as the grain boundary grows, whereas the energy penalty of a fold increases with its length continuously. Thus, the least favorable scenario for conformal growth is immediately after the first dislocation nucleates after passing an asperity. Even in this case, we show in the Supporting Information that grain boundary nucleation is still favored over fold formation. Note that fold formation and/or detachment from the substrate, although they 5300

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Nano Letters may be experimentally feasible,115,116 are only analyzed here as a hypothetical device to establish the boundaries of the regime within which conformal finite grain boundaries form, not to imply the final growth morphology outside of this regime is a simple fold or detached sheet (more likely, growth that favors detachment or folds would actually produce more complex morphologies such as nanoflowers117 or hierarchical hollow nanoparticles118). To show that 2D conformal growth can survive in experimental realities, we briefly discuss three common synthesis-related factors: step edges in substrates, wrinkles, and temperatures during synthesis and annealing. Step edges on substrates have different chemistries than do flat regions; although this will undoubtedly lead to material-specific kinetic effects on growth, the intrinsically geometrical origin of topographic control essentially forces grain boundary induction in regions of nonzero Gaussian curvature, so long as monolayer 2D growth is maintained. This geometrical character is apposite to general 2D materials and substrates, including amorphous substrates that do not have step edges, faceted substrates with consistent low-energy crystalline faces, and metallic substates with step edges that can be overgrown by 2D sheets without disruption.75−77 Wrinkles, commonly found in graphene on flat substrates,119 would not interrupt 2D conformal growth because they do not appear until the cooling stage of synthesis; their presence after the formation of topography-induced grain boundaries would not change the grain boundary geometry and may even generate interesting elastic interactions with the latter. Temperatures during synthesis and annealing of flat 2D sheets are generally low enough that the substrate remains solid and thus should typically retain its topography.120−122 The annealing temperature required to mobilize finite grain boundaries postsynthesis may exceed the melting point of some substrates; in those cases one will be limited to the quality achievable in as-grown systems. Even with the ground-state status assured, the conformal growth discussed here may still be interrupted by conventional grain boundaries formed by coalescence of adjacent grains. However, significant advances are being achieved in the synthesis of large single-crystal 2D monolayers by use of optimized precursors, substrates, and growth conditions that reduce seed nucleation rates and facilitate lateral growth. Such growth techniques producing large crystalline domains on flat substrates would also presumably be of sufficient quality to generate well-structured finite grain boundaries on topographic substrates. Dense networks of such grain boundaries could be grown on short-wavelength egg carton topographies (produced through, for example, anisotropic etching, nanoimprint lithography,123−125 or arrays of nanoparticles126), limited in areal density only by the strength of layer-substrate adhesion. Taking the ratio of the grain boundary energy per unit length (≤0.4 eV/Å) to a typical adhesive energy per unit area (0.02 eV/Å2), we can estimate 2 nm as the minimal separation between conformal grain boundaries. Even higher densities could be possible for stronger substrate adhesions or smaller grain boundary energies, as in TMDs (keeping in mind that the highest such densities fall outside the limits of a continuum argument). Such dense systems would embed many conformal grain boundaries within 2D flakes of sizes already achieved in single-crystal form. Grain boundary networks engineered into 2D materials at scale could potentially introduce useful properties, such as magnetism,17−19,127 one-dimensional conduction,22−25 suppression of charge-carrier scattering,128

or changes in fracture mechanics.10,102 Suitably designed substrate topography may also provide a means to facilitate the subsequent motion of grain boundaries that have formed by more traditional means.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.nanolett.7b01641. Structures (CIF) CheckCIF/PLATON report (PDF) Finite grain boundary structure preparation, minimal 2D growth model, disordered graphene structure preparation, dislocation nucleation: qualitative estimates and atomistic modeling, and additional references and figures (PDF)

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]. *E-mail: [email protected]. ORCID

Yuanxi Wang: 0000-0002-0659-1134 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors are grateful to P. Lammert and P. Varanasi for useful discussions. We acknowledge support from U.S. Army Research Office MURI Grant W911NF-11-1-0362 and the National Science Foundation Materials Innovation Platform under DMR-1539916.

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REFERENCES

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