Ripples, Strain, and Misfit Dislocations: Structure of GrapheneâBoron

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Ripples, strain, and misfit dislocations: structure of graphene - boron nitride superlattice interfaces Dinkar Nandwana, and Elif Ertekin Nano Lett., Just Accepted Manuscript • DOI: 10.1021/nl505005t • Publication Date (Web): 03 Feb 2015 Downloaded from http://pubs.acs.org on February 7, 2015

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1 [λ,H] = [10,5] [λ,H] = [20,5] [λ,H] = [30,5] 2 3 λ = 20 unit cells b) 4 5 6 7 [λ,H] = [20,9] [λ,H] = [20,2] [λ,H] = [20,5] 8 9 ACS Paragon Plus Environment 10 increasing aspect ratio λ/H 11

[λ,H] = [40,5]

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Ripples, strain, and misfit dislocations: structure of graphene - boron nitride superlattice interfaces Dinkar Nandwana and Elif Ertekin∗ Department of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign, 1206 W Green Street, Urbana IL 61801, United States E-mail: [email protected]

Abstract In recently-synthesized two-dimensional superlattices of graphene and boron nitride, the atomic structure of the interface is complicated by a 2% lattice mismatch between the two materials. Using atomistic and continuum analysis, we show that the mismatch results in a competition between two strain–relieving mechanisms: misfit dislocations and rippling. For flat superlattices, beyond a critical pitch the interface is decorated by strain-relieving misfit dislocations. For superlattices that can deform out-of-plane, optimal ripple wavelengths emerge.

keywords: Graphene, hexagonal boron nitride, lattice relaxation, misfit dislocations, interface structure, heteroepitaxy

∗

To whom correspondence should be addressed

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In thin-film heteroepitaxy, two materials are integrated together with a well-defined atomic registry across a two-dimensional interface. If the materials have dissimilar lattice constants, the mismatch typically results in the formation of strain-relieving interfacial misfit dislocations. 1 These misfit dislocations can dramatically affect the optical, electronic, and thermal properties of the hybrid system, via increased scattering of electrons 2 and phonons. 3,4 In practice, the formation of misfit dislocations severely limits the materials that can be integrated together. A classic example is the Si/Ge system, in which the 4% lattice mismatch results in dislocation-decorated interfaces that largely prohibit the development of active device structures and components. 5 Despite decades of improvements to heteroepitaxial growth processes that have enabled optoelectronic devices such as light-emitting diodes, laser diodes, quantum cascade layers, and others, the formation of these dislocations continues to be a limiting factor to material integration. For a given material system, several classic critical thickness models are often invoked to predict the breakdown of the coherent interface and the formation of misfit dislocations. For instance, the classic Matthews model 6 predicts that misfit dislocations will form beyond a critical thickness when the dislocation formation energy is offset by the strain relief energy. In addition to thin-film epitaxy, misfit dislocations have also been identified in nanostructures such as Si-Ge core-shell nanowires, 7 and the notion of critical feature sizes has been extended to the growth of heteroepitaxial islands, 8,9 heterostructured nanowires, 10 and several other nanoscale geometries. Now, within the class of two–dimensional materials such as graphene and boron nitride, the formation of one-dimensional interfaces (line boundaries) separating two materials integrated within a single atomic layer (see Figure 1) has recently been demonstrated. 11–16 For example, in-situ microscopy has shown that during a sequential chemical vapor deposition process, hexagonal boron nitride grows preferentially at the edges of existing monolayer graphene domains, 11 opening routes for realizing atomically sharp interfaces on a single twodimensional sheet. In-plane heterostructures of graphene and boron nitride superlattices with controllable domain sizes have also been synthesized. 12 The use of templated graphene

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edges has been shown to enable the atomically coherent growth of two–dimensional hexagonal boron nitride. 13 More recently, the direct chemical conversion of graphene to boron nitride within a single atomic layer has also been demonstrated. 14 Excitingly, the presence of topological defects at graphene - boron nitride interfaces has even been observed. 17 These developments, and several others, 15,16 extend the framework of thin-film heteroepitaxy to a two-dimensional space of in-plane heterostructures. In this letter, we now expand the concept of critical thickness to the realm of twodimensional superlattice structures. As in-depth studies of interfacial structures and strainrelieving mechanisms 18–21 have been a key tool for improving the quality of thin-film devices, a detailed understanding of the nature of one-dimensional interfaces will also be a key enabler for next-generation optoelectronic, energy harvesting, and energy conversion devices. We find that, analogous to thin-film heteroepitaxy, in 2D a critical superlattice pitch exists beyond which the formation of strain-relieving topological defects at the interface (Figure 1) is favorable. From the nature of the topological constraints imposed by these defects, each 5|7–membered set of rings can rigorously be considered to be the two–dimensional equivalent of an edge dislocation 22–25 – associated with an extra line (rather than plane) of atoms. In Figure 1, the formation of these 2D dislocations at the interface is considered as a possible mechanism to accommodate mismatch strain (the extra line of atoms in the graphene subdomain, terminated by a dislocation at each end, is highlighted in grey). Distinct from 2D heteroepitaxy, however, we also show that strain relief avenues uniquely available in twodimensions – out-of-plane buckling and ripple formation 26–29 – delay the onset of dislocation formation, suggesting that two-dimensional heterostructures can coherently accommodate greater lattice mismatch than thin films. A similar observation of lattice-mismatch induced warping of hybrid two-dimensional materials has also recently appeared. 30 Because of its significance and widespread current interest, we consider the case of graphene/boron nitride superlattices here. However, our approach is applicable to 2D systems in general. Graphene and boron nitride are prototypical two-dimensional materials. Although isostruc-

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tural and isoelectronic, graphene is a Dirac semi-metal exhibiting a linear electronic dispersion and hexagonal boron nitride is a wide band-gap insulator (Eg > 5.0 eV). The integration of these two materials via atomically coherent interfaces can give rise to interesting functional properties. For example, ordered superlattices that integrate graphene and boron nitride are intriguing due to their tunable band gap that arises from the hybridization of pure boron nitride and pure graphene. 31 Several studies predict that controlled growth of such superlattices opens new doors to fabricate nanoscale devices for thermoelectric, 32–34 optoelectronic, 35 photovoltaic, 36 and magnetic 37 applications. Access to this diverse spectrum of applications depends critically on obtaining atomic–scale control of the interface between the graphene and boron nitride sub-domains. To reveal the stable interfacial structures of graphene - boron nitride superlattices, we analyze energies of different configurations and identify the minimum energy structure across a spectrum of geometries. The two constituent materials both exhibit a honeycomb lattice, but they possess a ∼ 2% lattice mismatch (in contrast to 2.46 ˚ A in graphene, the length of the BN unit cell is 2.51 ˚ A). A schematic illustration of the decoration of zigzag and armchair interfaces by these misfit dislocations is shown in Figure 1a and Figure 1b, respectively. Here the height H of each graphene and the boron nitride subdomain are always set equal (but our analysis can be extended to other geometries). We first consider a superlattice constrained to remain flat (e.g., by an underlying substrate, or when incorporated into a multilayer stack). Both coherent and incoherent interfaces are considered. The interface between the graphene and boron nitride subdomains is coherent (Figure 2a) when every column of atoms in graphene is in registry with a corresponding column of atoms in boron nitride. By contrast, if the strain energy in the coherent system becomes sufficiently large, it is feasible that the interface can be incoherent, decorated with atomic-scale strain-relieving misfit dislocations (Figure 2b). For the zigzag interface, an extra column of atoms is inserted into the graphene subdomain, so that every L atomic columns of boron nitride are mapped onto L + 1 atomic columns of graphene. The extra column of atoms that appears in the

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graphene subdomain is terminated by a 5|7–membered ring (misfit dislocation, Figure 2c) at both ends. For the armchair interface, by geometry the extra column of atoms must be inserted at a skew angle to the interface (Figure 1b), but otherwise serves the same purpose. These dislocations reduce the lattice misfit, but also introduce an energy penalty associated with the topological defects themselves. Whether the energetically favorable interface is coherent or incoherent reflects a competition between the lattice mismatch strain energy that is offset by the introduction of dislocations, and the energy cost to introduce them in the first place. To compare relative energies of the coherent and incoherent systems, force field based molecular simulations have been performed using energy minimization via conjugate-gradient method, as implemented within the LAMMPS molecular simulation package. 38 We use Tersoff potentials 39 with appropriate intermixing parameters to accurately model the lattice structure and elastic properties of graphene and boron nitride. The T = 0 K lattice constants, according to the potentials, are aC = 2.46 ˚ A, aBN = 2.52 ˚ A, giving a lattice mismatch of f = (aBN − aC )/aC ∼ 2.4%, in reasonable agreement with the experimental lattice mismatch of ∼ 2.0%. 12 We note that the representation of the superlattices adopted here neglects the lack of inversion symmetry in boron nitride, and therefore cannot capture effects such as the polarity of the boron nitride subdomain (terminated by B on one side and N on the other). 40–42 However, the elastic effects that we are mainly interested in are expected to be sufficiently captured. Additionally, the relaxed structures represent the T = 0 K configurations, atop which thermally induced fluctuations and ripples should also be present. 43 Figure 2d shows a comparison of the average formation energy density of the coherent and incoherent systems as a function of the dislocation spacing L for zigzag superlattices of varying half-pitch H. L and H are integers representing the number of unit cells in the direction along and perpendicular to the interface (a single unit cell with L, H = 1 is outlined in blue in Figure 2c). The formation energy Ef is the total energy of the given superlattice, relative to the energy of an equal number of atoms incorporated into isolated pure graphene

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and pure boron nitride: Ef = Esl − (NC EC + NBN EBN ) .

(1)

Here Esl is the computed total energy of the superlattice, EC is the total energy per two atoms of unstrained graphene, EBN is the total energy per BN formula unit of unstrained boron nitride, NC is the number of carbon atom pairs in the supercell, and NBN is the number of boron/nitrogen formula units in the superlattice unit cell. The formation energy density plotted in Figure 2d is equal to Ef divided by the initial (undeformed) area Asl of the unit cell. (We consider the density to facilitate the comparison of supercells of different sizes, and hence different dislocation spacing.) In Figure 2d, the dashed line labeled Ecoh is the calculated strain energy density for the coherent systems, which is found to be independent of half-pitch H and length L: Ecoh ≈ 1.96 meV/˚ A2 . In these systems, the total lattice mismatch f = 2.4% is accommodated by a constant positive strain fC in graphene (uniformly stretched) and a constant negative strain fBN in boron nitride (uniformly squished), so that

|f | = |fC | + |fBN |

.

(2)

If the elastic constants for the two materials were equivalent, the total mismatch would be evenly shared between the two subdomains: |fC | = |fBN | = |f | /2 in order to minimize the total mismatch strain energy Emm estimated by linear elasticity theory as

Emm =

Asl BN 2 C Cxxxx fBN + Cxxxx fC2 . 4

(3)

C The small difference in the elastic moduli from the empirical potentials (Cxxxx = 26.48 eV/˚ A2 , BN Cxxxx = 24.17 eV/˚ A2 ) results in a slightly unequal accommodation of lattice mismatch.

Therefore Emm is minimized by fBN = −1.25%, fC = 1.14%, giving a strain energy density in boron nitride of 1.90 meV/˚ A2 and in graphene 1.75 meV/˚ A2 . This simple elastic argument

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is in very good agreement the atomistic simulation with Ecoh ∼ 1.96 meV/˚ A2 . The slight difference between the elastic theory and the atomistic results probably comes from the excess energy due to chemical bonding between disparate materials at the interface (unrelated to elastic effects). The solid lines in Figure 2d show the formation energy density of the incoherent zigzag interface superlattices as a function of the dislocation spacing L for superlattices of given halfpitch H. When dislocation pairs are spaced closely together (small L), the average formation energy density is quite large due to the high cost of introducing a large defect concentration. As the dislocation spacing L increases, it quickly drops. However, for H = 10, according to Figure 2d, the formation energy density of the incoherent system remains above that of the coherent system for all L, and dislocation formation at the interface is never favorable. These smaller superlattices will, at equilibrium, exhibit coherent interfaces. On the other hand, for superlattices of larger half-pitch H = 20, 30, 40, the formation energy density dips below Ecoh for appropriate choices of the dislocation spacing L, suggesting that the increasing strain energy as H grows eventually induces the formation of misfit dislocations. The half-pitch H = 20 is close to the boundary between coherent and incoherent superlattices, signaling the existence of a critical for misfit dislocations in these 2D superlattices similar to the critical thicknesses of heteroepitaxial thin films. Although not shown here explicitly, we also find that the armchair interface superlattices also exhibit a critical pitch, but it is much larger with H around 45 units. Close inspection of Figure 2d shows that for sufficiently large H there is a minimum in the formation energy density around L = 40, suggesting an optimal dislocation spacing. Beyond this spacing, the formation energy density increases slowly as the dislocation spacing is increased, approaching the value Ecoh . The optimal spacing can be understood from the nature of the misfit dislocation. Misfit dislocations relieve a fraction of mismatch strain at a cost of forming defects at the interface. When an extra column of atoms is incorporated into the graphene subdomain, the dislocation pair accommodates a portion of the lattice

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mismatch f , leaving behind a residual mismatch of

f∗ =

(L)aBN − (L + 1)aC f − 1/L = . (L + 1)aC 1 + 1/L

(4)

The lattice mismatch is completely relieved at a dislocation spacing L = 1/f ∼ 42, which corresponds very well to the observed value of the minimum formation energy density occurring near L = 40. To better understand the trends in the formation energy density in Figure 2d, we use a straightforward continuum analysis motivated by several well–known models of critical feature size. 10,18,20 The total lattice misfit f is partitioned into a portion accommodated by misfit dislocations (f − f ∗ ) and another portion that remains as a residual lattice mismatch f ∗ . The total formation energy is then estimated by superposition of energy of the dislocation array and the strain energy of the residual mismatch strain field. While this approach neglects explicit interactions between the dislocation strain field and the residual mismatch strain field, it has been shown to be accurate for epitaxial thin films, 18 islands, 20 and nanowires. 10 Additionally, here it is justified a posteriori by comparison to our atomistic results. Within this approach, the formation energy density per unit cell is given by

Einc =

1 (E ∗ + 2Edis ) , Asl mm

(5)

∗ where Emm is the total strain energy arising from the residual mismatch f ∗ , and Edis is the

energy per dislocation, and the factor of two appears because each unit cell contains two ∗ dislocations (one at each interface). Emm is evaluated using Eqs. (2) and (3), with f replaced

by f ∗ . Edis is given by the 2D flat dislocation theory 44 so that Edis = (Y b2 )/(8π) ln(Lac /2rc ). Here Y is the Young’s modulus (set to the average of YC and YBN , which are quite similar), b is the magnitude of the burger’s vector (equal to the length of the unit cell aC ) and rc is the dislocation core radius, taken to be 0.94 ˚ A. 26 The formation energy density calculated in this manner, shown in Figure 2d by the open circles, reproduces the atomistic results nicely. 8


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(The largest deviations occur for small superlattices where dislocation core overlap and distortion renders the isolated dislocation theory inapplicable.) Comparing the formation energy densities of the coherent and incoherent systems via Eqs.(3) and (5), a critical halfpitch of Hcr = 19 unit cells is determined, in very good agreement with the atomistic results. If H < Hcr , the coherent interface is stable and misfit dislocation formation is unfavorable; on the other hand for larger H ≥ Hcr introduction of the dislocations sufficiently offsets the mismatch strain energy and the presence of dislocations becomes favorable. For the armchair interface superlattices, we find similar results to the zigzag case shown in Figure 2. However, the critical thickness is larger, around Hcr = 45. The larger critical thickness arises from two factors. For the armchair interface, by geometry the dislocation pair occurs at a 30◦ angle to the interface itself. The dislocations are then less effective at relieving the lattice misfit compared to the zigzag interface (where the burgers vector of the edge dislocations is parallel to the interface). Additionally, image interactions between the dislocations are less effective at reducing the formation energy of the defect array for the skew periodic arrangement. Based on these considerations, it is likely that coherence will be easier to maintain in superlattices with armchair, rather than zigzag, interfaces. It is also important to consider how the trends established above change if the superlattices are not constrained to stay flat, but can deform out of plane, as shown in Figure 3a. To assess the strain-relieving capabilities of out–of–plane deformations, 26–29 we determined the stability of the flat superlattice with respect to the formation of sinusoidal ripples of varying wavelengths. For each [L, H] superlattice, we introduced a sinusoidal ripple of wavelength λ = L in our atomistic simulations. Both the positions of the atoms, as well as the supercell lattice vectors themselves, were allowed to relax from this initial configuration to a preferred configuration. (As we will discuss later, sometimes we find that the supercell lattice vector in the interfacial direction shrinks somewhat during the relaxation). All superlattices with H ≥ 2 were found to maintain ripples for ripple wavelengths λ ≥ 5 unit cells, whereas they relaxed back to the flat geometry for λ < 5 unit cells. An example of such a relaxed, rippled

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structure is illustrated in Figure 3a for a [λ, H] = [10, 5] supercell. By contrast, the H = 1 superlattices always relax to a flat configuration for all λ, and are thus completely stable with respect to ripple formation of any wavelength. Thus, in analogy with the critical pitch rip for misfit dislocations, we observe a (remarkably small) critical pitch for rippling at Hcr =2

(for both zigzag and armchair interfaces). Moreover, ripple formation is an extremely efficient strain relief mechanism. Figure 3b shows the formation energy density as a function of ripple wavelength λ for superlattices of varying H: when ripples form the formation energy densities Ecoh are reduced by an order of magnitude relative to the flat superlattices of Figure 2b! Two additional trends are present in the atomistic results in Figure 3b. The first is that for a given H, long wavelength ripples are favored over short wavelength ripples: the λ>H (H) as ripple formation energy density quickly drops with λ towards a steady value Ecoh λ>H wavelength λ grows. Second, the limit Ecoh (H) itself depends on H: it is largest for small

H, but decreases as H increases. To give insight to these observations, Figure 4 shows the final relaxed geometry for superlattices with varying ripple wavelength λ and half-pitch H. Figure 4a shows the change to the ripple structure that occurs as the ripple wavelength λ increases from 10 to 40 unit cells at constant half-pitch H = 5 unit cells. For the smallest λ (λ/H = 2), low amplitude sinusoidal corrugations are observed to be present in boron nitride, and although they penetrate somewhat into graphene, they quickly decay and much of the graphene remains flat. For these systems, the supercell lattice vector in the interfacial direction stays at its original value L, the natural length of graphene (fs ≈ 0). On the other hand, as the ripple wavelength increases towards λ = 40 unit cells (λ/H = 8), the ripples are distributed throughout both materials. In Figure 4b, a similar change to the rippled topology is observed as λ/H increases, now at constant λ = 20 unit cells. The presence of ripples throughout both materials also corresponds with a shrinking of the lattice vector L to L(1 − fs ) along the interfacial direction (we find fs ≈ 0.1, depending on aspect ratio λ/H), so that both materials are compressed in-plane and can buckle out-of-plane. This change

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to the nature of the rippling is exhibited for all systems we considered: as λ/H increases, the structure changes from one in which the rippling is confined to the boron nitride and the graphene remains flat, to one in which the rippling is present across both materials. The profile view in Figure 4c, for large λ/H, shows that when ripples are present in both materials the amplitude in boron nitride is slightly larger than that in graphene, presumably to accommodate its longer natural length (L(1 + f ) vs. L). To explain these observations, we implemented a variational continuum model 45 based on the nonlinear Föppl-von Kármán 46 equations; the results of the continuum model are superposed with the atomistic results in Figure 3b and show very good agreement. Based on our variational model, the trends in Figure 3b and Figure 4 can be understood as a competition between two main contributions to the total elastic energy of a rippled sheet: 46 Z 1 US = Cijkl ǫij ǫkl dA 2 Z 1 κ(∇2 w)2 dA UB = 2

, (6) .

The stretching energy US describes the usual elastic strain energy (Cijkl are elastic constants and ǫij are total nonlinear strains), while the bending energy UB describes the energy associated with changes to the gaussian curvature H ≈ ∇2 w of the buckled sheet. The bending rigidity is denoted by κ, and the out-of-plane displacement field "

w(x, y) = Ac +

Abn − Ac 1 + exp Hβ y

!#

λ cos

2πx λ

.

(7)

is such that ripples of amplitude Abn λ in boron nitride decay to ripples of Ac λ in graphene with decay constant β/H. If β/H is large, the ripple quickly decays across the interface so that the two subdomain interiors have amplitude Abn λ and Ac λ respectively; whereas in the limit β/H → 0, the ripple amplitude becomes uniform (Ac + Abn )λ/2 everywhere. Thus this variational form can capture the transitional nature of the ripple geometry shown in Figure 4. The four variational parameters (all dimensionless) are the constants Ac and Abn 11


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which give the ripple amplitudes relative to the wavelength, the decay constant β, and the shrinking factor fs . The scaling properties revealed in our variational model show that while the bending energy density UB /Asl decreases with both λ and H explicitly, the stretching energy density US /Asl increases with the aspect ratio λ/H. Together, these two contributions result in an optimal set of ripple wavelengths that depend on the aspect ratio. The emergence of an optimal set of ripple wavelengths due to a competition between bending and stretching energy is reminiscent of the famous problem of rippling in a thin sheet pulled in tension, but constrained from relaxing laterally at its ends. 47 Notably, our variational model predicts that at much larger λ/H (too large to simulate atomistically) the formation energy density λ>H in Figure 3b would begin to increase again. Thus, the apparent steady value of Ecoh (H) is

an artifact of being in the intermediate regimes of λ/H where the bending energy UB has decreased, but the stretching energy US has not become too substantial yet. Of the two contributions, the bending energy density κ(∇2 w)2 /2 is more straightforward to understand: the dominant terms have scaling (1/λ2 ) and (1/H 2 ). The first term is the bending energy density of the ripples themselves, is responsible for the sharp, initial reduction in the formation energy density with λ in Figure 3b. The second comes from the modulation of the ripple amplitude across the two subdomains and gives rise to the H-dependence of λ>H Ecoh (H). In the limit of infinite half-pitch H, i.e. a single, isolated interface, our model λ>H predicts that the steady value Ecoh (H) approaches zero. We also note that it is the bending

energy UB that causes the flat system to be stable with respect to ripple wavelengths λ < 5 unit cells as described above. The second contribution, the stretching energy density Cijkl ǫij ǫkl , is more complicated, but can be largely understood from the ripple geometry transition shown in Figure 4. The dominant terms in the stretching energy density have scaling (λ/H)4 , and arise from transverse stretching. As λ/H grows, ripples of larger and larger amplitude in the boron nitride must decay within shorter and shorter regions across the interface into the graphene. This

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decay of a large amplitude ripple over a short distance introduces a large transverse strain ǫyy , the main source of UB . It is also responsible for the transition in the rippling mode in Figure 4 to a more uniformly rippled geometry: the lateral squishing makes rippling favorable in both materials, and reduces the amplitude discrepancy needed to offset the lattice mismatch. Overall, for coherent rippled structures, the competing contributions between bending energy and stretching energy give rise to optimal ripple wavelengths around 50 150 unit cells (12 - 35 nm), depending on the superlattice pitch. Given the effectiveness of ripple formation at relieving mismatch strain energy, it is interesting to consider whether misfit dislocation formation can ever be favorable in the presence of rippling (Figure 3c and 3d). Although the low formation energy densities in the rippled superlattices initially would seem to render dislocation formation unnecessary, the fact that the dislocations themselves can now also buckle out-of-plane introduces an intriguing twist. For flat dislocations in two-dimensional systems, the long-ranged 1/r decay of the strain fields gives rise to a total energy that diverges logarithmically with the system size. By contrast, in 1988 Seung and Nelson 44 predicted that the formation energy of a buckled dislocation in a two–dimensional membrane is markedly different: finite-valued and localized near the core. Localized deformations out–of–plane help relieve large strains near the dislocation core, and give rise to short-ranged strain fields. Indeed, previous studies indicate that out–of–plane relaxation 26,27,44 substantially reduces the dislocation formation energy in a two–dimensional surface. For the combination of misfit dislocations and buckling, our atomistic simulations (summarized in Figure 3c and 3d) relax to rippled systems with ripple wavelengths λ commensurate to the dislocation spacing. The ripples are no longer smooth sinusoids, but exhibit severe, sharp deformations in the vicinity of the misfit dislocation core (see also Figure 5a). In comparison to misfit dislocations in flat superlattices, the formation energy densities of buckled superlattices with dislocations are substantially lower (compare the formation energy densities in Figure 2d and 3d). Remarkably, for sufficiently large supercells we find that the

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total formation energy per supercell does not vary, but quickly converges to a constant finite value of EF ≈ 12.7 eV for all supercells considered. (A fit of the computed formation energy density of the incoherent, buckled systems to an inverse scaling relationship 12.7 eV/(Asl ) is shown in Figure 3d). To see how this energy is distributed, Figure 5b and 5c show maps of the strain energy density for both buckled and flat dislocations. For the buckled dislocations, the strain energy density is localized around the core and there is almost no residual strain away from the core. This is in contrast to the flat counterpart where the far-field exhibits a significant strain energy arising both from the slowly decaying dislocation strain fields and residual mismatch strain field (Figure 5b). Although we have not attempted to solve directly for the stress and strain fields in the rippled, dislocated superlattices, it is reasonable that the total energy can be described similarly to Eq. (5) for misfit dislocations in flat superlattices, in which the total mismatch f is partitioned into a portion accommodated by ripple formation and a portion accommodated by the misfit dislocations. Edis is the formation energy of a buckled dislocation (now finite valued), and Emm is the formation energy density associated with ripple formation to relieve the residual mismatch. The residual mismatch is again given by Eq. (4), and then Emm could again be determined variationally. If we adopt the dislocation theory proposed by Seung and Nelson 44 (derived for the case of a dislocation in a 2D membrane that is free to buckle out of plane) to estimate the buckled dislocation formation energy,

Edis = 2

Y b2 rb ln( ) 8π rc

(8)

where rb ∼ 127κ/Y b is the buckling radius, we find that 2Edis ≈ 12.6 eV. This estimate corresponds very well to our atomistic formation energy EF ≈ 12.7 eV! Thus, it appears that for these systems, EF = 2Edis + Emm ≈ 2Edis and Emm is very small. To understand this, note that even in the absence of any mismatch, the dislocation array itself will introduce its own ripple structure, with wavelength commensurate to the dislocation spacing (each

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buckled dislocation itself is a source of curvature 44 ). Thus, presumably the dislocations alone are able to accommodate the bulk of the lattice mismatch, leaving very little to zero residual mismatch to be accommodated by any rippling additional to the natural rippling induced by the dislocation array itself. To assess whether misfit dislocations are liable to form in the presence of buckling, it is useful to compare the formation energy densities in Figures 3b and 3d. With the caveat that the computed energy density differences are now quite subtle, a cross-over point between the formation energy density of rippled, coherent and rippled, incoherent systems occurs around L ≈ 120. For dislocation spacings L > 120, the formation energy density of the incoherent system appears to be lower. However, such large wavelength ripples are associated with large out-of-plane deformation and global changes to the curvature which may or may not be feasible in real systems. Therefore, these results suggest that misfit dislocations can occur in rippled systems in principle, but only when large–scale deformations (and large dislocation spacing) are possible. Thus, we conclude that interface coherency in the C-BN superlattices can be maintained for much longer es when the atoms are free to deform out–of–plane, compared to the previously observed Hcr = 19 for flat systems. Before concluding, it is important to note that the misfit dislocation models presented here (and most others including the original Matthews model 6 ) are equilibrium analyses that do not consider kinetic barriers to dislocation introduction. As is the case with thin-film heterostructures, 48,49 it may be possible that 2D superlattices can be grown dislocation–free beyond the critical dimension because of the energy barriers associated with dislocation introduction (the superlattice can remain in a metastable coherent state). In the superlattices, the effect to which kinetic effects will be important will depend on how the dislocations form in the first place (which in turn will depend on the growth mechanism). One possibility is that dislocations may nucleate at the surface of a growing superlattice and climb to the interface as in the case with thin-film epitaxial growth. Another possibility (applicable to narrower superlattices with large boundary to area ratio) is that, since the interface itself

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can be a slip plane, the dislocations may be introduced at the superlattice free boundary. The effect to which kinetic considerations delay the onset of dislocation formation remains to be seen. In summary, we predict the stable interfacial structures of lattice–mismatched grapheneboron nitride superlattice by exploring the interplay of mismatch strain and two strain– relieving mechanisms: nanoscale misfit dislocations and rippling via out–of–plane relaxation. For this, we implement atomistic total energy methods using force-field based molecular simulations. The results indicate that for flat superlattices, stability of a coherent interface is limited by a critical supercell pitch beyond which misfit dislocations, in the form of 5—7-membered rings, become energetically favorable to form. Additionally, we study the effectiveness of rippling (out-of-plane deformation) in relieving strains due to mismatch and dislocations. We observe that when out-of plane deformation is allowed, an order of magnitude reduction is observed in formation energy densities compared to the flat counterparts. Depending on superlattice , optimal ripple wavelengths are around 10 - 35 nm. Rippling also screens the long-ranged dislocation strain fields and results in finite-valued dislocation formation energy localized only to the core of the dislocation. The results demonstrate that out–of–plane deformations can be used favorably to maintain coherency in these superlattices for large geometries. Our atomistic results are well–captured by scaling arguments based on continuum theory in both flat and rippled two–dimensional membranes. The results can serve as a guide for interface engineering in graphene-boron nitride superlattice and any two–dimensional superlattice in general, since the approach presented here provides a generalized framework.

Acknowledgement We gratefully acknowledge fruitful discussions with Daryl Chrzan, Shuo Chen, and Sanjiv Sinha. We acknowledge financial support from National Science Foundation under Grant CBET-9122625. This work used the Extreme Science and Engineering Discovery Environ16


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ment (XSEDE) under Grant. No. TG-DMR120080, which is supported by National Science Foundation grant number OCI-1053575. Computational resources were also provided by the Illinois Campus Cluster program, maintained by the NCSA.

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!"#!$#%"&'()*$+(%

$),+-$")%"&'()*$+(%

./%

0%

./% Figure 1: Zigzag and armchair two-dimensional superlattices composed of ordered, alternating regions of graphene and hexagonal boron nitride. The ∼ 2% lattice mismatch between the materials results in an in-built strain in the coherent superlattice heterostructure, potentially giving rise to the formation of 5|7 topological defects that are the equivalent of misfit dislocations. The misfit dislocations on both sides form the boundary of an extra column of atoms (highlighted in gray) that help to offset the mismatch strain.

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Figure 3: Structure and energetics of graphene (black) - boron nitride (red) superlattices that are free to deform out–of–plane. (a) Coherent superlattice showing rippling in boron nitride. In this case the ripples penetrate somewhat into the graphene but decay away from the interface. (b) Formation energy density of the coherent rippled superlattice with varying ripple wavelengths λ = L. Rippling reduces the formation energy density by an order of magnitude compared to flat superlattices. The inset shows that the superlattices are unstable with respect to ripple formation for ripple wavelengths λ > 5. (c) Rippled superlattice with interface decorated by 5—7 membered rings. (d) Formation energy density of incoherent rippled superlattices with varying half-pitch H and dislocation spacing L. The atomistic formation energy densities (closed symbols) are well-described by the scaling relationship (2Edisl /Asl ) where Edisl is the finite-valued dislocation formation energy and Asl is the area of the supercell (open symbols).

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(a) coherent

incoherent (b)

(c)

0

1.2

Figure 5: (a) Topological structure of [λ, H] = [100, 10] coherent and incoherent grapheneboron nitride rippled superlattices. Sharp curvatures are exhibited near the misfit dislocation core, which relieve large localized strains. (b,c) Contour plots of atomistic formation energy density in rippled, incoherent and flat, incoherent superlattice respectively. Formation energy density in the rippled superlattice is highly localized to the core of the misfit dislocation. On the contrary, the (1/r) slowly decaying dislocation strain fields and uniform residual mismatch results in a long–ranged, extended formation energy density for the flat system.

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Ripples, Strain, and Misfit Dislocations: Structure of GrapheneâBoron

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Ripples, Strain, and Misfit Dislocations: Structure of GrapheneâBoron