How the Complementarity at Vicinal Steps Enables Growth of 2D

Letter Cite This: Nano Lett. XXXX, XXX, XXX−XXX

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How the Complementarity at Vicinal Steps Enables Growth of 2D Monocrystals Ksenia V. Bets, Nitant Gupta, and Boris I. Yakobson* Department of Materials Science and Nanoengineering, Rice University, Houston, Texas 77005, United States

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ABSTRACT: Large 2D monocrystals are highly sought after yet hard to achieve; unlike graphene, most dichalcogenides and h-BN possess low symmetry, which allows for nucleation of mutually inverted pieces, merging into polycrystals replete with grain boundaries. On vicinal substrate surfaces such growing pieces were observed to orient alike, and very recently this effect apparently enabled the growth of large single crystal h-BN. Addressing the compelling questions of how such a growth process can operate and what the key mechanisms are is crucial in guiding the substrate selection for optimal synthesis of perhaps many materials. To this end, the basic crystallography and atomistic-modeling theory presented here reveal (i) how the undulations of the ever-wandering steps do not, surprisingly, disturb the orientations of the attached 2D-nuclei, whose direction remains robust owing to complementarity between the meandering step and h-BN counterpart if their kinks have similar size of negligible misfit, δk < 0.1 Å. (ii) Stronger chemical affinity of metal to the N atoms at the zigzag edge of h-BN singles out its particular orientation, without evidence of any epitaxy, at the edge or to the surface. (iii) The monocrystal integrity requires unhindered growth spillover across the steps and the seamless healing of the residual fissures, caused by the very same steps necessary for coorientation. Molecular dynamics simulations show this happening for the steps not taller than the BN bond, s < 1.44 Å. These criteria point to [−1 1 2] steps on the Cu (110) surface, in accord with experimental results (Wang et al. Towards the growth of single-crystal boron nitride monolayer on Cu. arXiv:1811.06688 Cond. Mat. Mtrl. Sci., 2018), while other possibilities can also be predicted. KEYWORDS: h-BN, 2D material, monocrystal, vicinal surface, aligned nucleation, structural complementarity

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liftoff of inversion degeneracy was brought to fruition and enabled the production of a large monocrystal h-BN layer on a plane vicinal to Cu (110). Compelled by these experiments and to assess the possibility of a similar approach to other 2D materials, perhaps on other substrates, we wanted to understand and quantify the structural-geometrical factors as well as driving forces in play, now in the case of h-BN on Cu or possibly on Ni. To this end, we began by analyzing the geometries of the vicinal surface, when atomistic essentials are well captured in the lattice parameters and angles, and consequently the step directions and heights; accordingly, one can see to which extent the commencing h-BN layer can fit or leaves significant gaps (with high energy penalties and thus suppressed thermodynamically). We find that the role of surface epitaxy to the substrate is diminished by the dominant role of 2D-edge complementarity (geometrical fit, not an epitaxy or atomic registry) to the steps. Down to the atomic scale, the analysis reveals that there is no actual atom-to-atom registry, the definitive trait of epitaxy, and thus, the epitaxy is, strictly speaking, irrelevant.

o parallel the Czochralski growth of monocrystal ingots, which ultimately turn into silicon wafers, the ways to grow monocrystalline two-dimensional (2D) materials remain highly sought after. Significant progress in the case of graphene2,3 includes single-seed growth to an inch size4 and the approaches where multiple seeds are either co-oriented by epitaxy to the monocrystal substrate5−7 or become co-oriented even on polycrystalline support through evolutionary selection,8 all to yield a monocrystal layer of essentially unlimited size. While the latter approach seems in principle universally applicable, the reliance on epitaxy to substrate fails when the growing 2D lattice has lower symmetry. For instance, in the case of h-BN (or similarly, most of the transition-metal dichalcogenides, TMD) on high-symmetry surfaces of Ni or Cu, the emerging triangular islands9−15 are equiprobably inverted and thus cannot join to form a monocrystal; instead, the product is replete with inversion grain boundaries.10−13 This inversion degeneracy can be lifted if the substrate surface is not a low-index kind but vicinal; a small miscut angle (α) ensures the presence of atomic steps (height s), sparsely spaced (by t ∼ s/α), all roughly parallel andimportantly unidirectional, see Figure 1a. This step-prescribed directionality has been observed a number of times,10,16 see Figure 1b. Nevertheless, it came as surprise in a recent report1 that such © XXXX American Chemical Society

Received: January 11, 2019 Revised: February 13, 2019

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DOI: 10.1021/acs.nanolett.9b00136 Nano Lett. XXXX, XXX, XXX−XXX

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Figure 1. (a) Perspective view of a vicinal surface with unidirectional steps. (b) Top view of the nucleating 2D islands docked to the steps; the one shown as the dashed-line triangle is tilted, to follow the step curve. (c) Close-up view of a meandering step, composed of sparse kinks and “1D terraces” of principal direction, to which the h-BN nuclei are docked, thereby retaining their co-orientation, like houses built upright on a hill slope.

Figure 2. Complementarity of kinks at the h-BN edge (red B, blue N) to the steps of the metal surface (gray shaded). The majority of the step types result in a significant mismatch in kink height (top panel), with only a few showing differences under |δk| < 0.1 Å (middle and bottom panels). See also Table 1 for details.

The very presence of the steps facilitates the formation of a new 2D layer, which begins easier docked to a step (heterogeneous nucleation) than on a pristine terrace plane (homogeneous nucleation). Thermodynamically, attachment of the strongest binding edge to the step is most probable, and this prescribes the 2D layer orientation; if the steps on the vicinal plane were all exactly parallel, then all emerging 2D domains were also oriented alike, with a prospect to eventually fuse into a large monocrystal. However, the steps are generally known to be parallel only roughly, on average (to maintain the overall miscut direction), but can be quite meandering (Figure 1b) and even change in time and wander at high temperature.17−22 This makes an impression that the tightly docked emerging islands should also vary in orientation (Figure 1b, dashed triangle), which would preclude the overall monocrystallinity, unless some other factors realign the isles. One can see that, even if some epitaxy with the terrace takes place, it is unable to restore the order. Indeed, an estimated energy of a monolayer nucleus of size L contains two contributions, i.e., one from its stronger nearly covalent bonding to the step ∼εL (ε ≃ 1 eV/Å), and the cost of varying the weak dispersion-type “epitaxial” interaction with the terrace ∼(π/8)σL2 (σ ≃ 0.01 eV/Å2; the geometrical prefactor is chosen for a generic semicircular nucleus). Notably, these contributions scale differently with size, and the latter term can only overcome the first one at L > (8/π)ε/σ ≃ 25 nm, when the layer is already too large to detach from the step and to slide−rotate for adjusting its orientation. There is however a much simpler mechanism, originating from the very basic discreteness of atomistic makeup and filtering out this “orientation noise” among the steps. Figure 1c shows that the step-line is not really a continuous curve but

instead is discretized by the underlying lattice into a sequence of kinks of atomic size k. Consequently, the nuclei docked to the step-line need not be slanted (dashed line) but remain perfectly upright, like houses built on a slope, see Figure 1c. For this configuration to work well, the edge-step contact must remain thermodynamically favored, that is, strongly bonded and geometrically tight, bringing about an important requirement that the kinks at the metal step k must equal in size the kinks k2D at the counterpart h-BN or any other 2D layer, for that matter. In other words, the kinks must complement each other, that is, the offset δk = k − k2D must be small, to preserve the geometrical gap unchanged on the left and right of the kink, and close to its optimal magnitude for a given interface (all the kink sizes and the space gaps are defined as between atom centers, to avoid ambiguities). Noteworthy, this δk offset dictates which steps on which surfaces can support the aligned growth realization, yet it does not require cumbersome ab initio computations and can be accurately obtained from the basic lattice parameters, wellknown for most materials. Accordingly, Figure 2 shows the atomic configurations of selected interfaces of h-BN on Cu, with complementary kinks; at the h-BN edge, a single or double kink (kBN = 2.16 or 4.31 Å) is chosen, to achieve smaller |δk| for better complementarity. On the right of Figure 2, one can find that while the values of δk vary (see also Table 1), it can be very small in some cases, reaching |δk| < 0.1 Å for Cu (111) vicinal with [−1 1 0] steps, or Cu (110) vicinal with [−1 1 2] steps. Intentionally, only the zigzag N-terminated edge contact is shown, since such ZN-edge has stronger binding to the steps than others, as discussed next. Note that structural complementarity evaluated in Figure 2 and Table 1 is based on room temperature data, while thermal expansion at B

DOI: 10.1021/acs.nanolett.9b00136 Nano Lett. XXXX, XXX, XXX−XXX

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Nano Letters Table 1. Geometrical Complementarity of Kinksa Cu

Ni

surface, terrace

step

k (Å)

δk (Å)

s (Å)

k (Å)

δk (Å)

s (Å)

(0 0 1)

[0 1 0] [1 1 0] [0 0 1] [−1 1 0] [−1 1 2] [−1 1 0] [−1 −1 2]

1.80 2.54 2.54 3.60 2.08 2.20 2.54

0.36 −0.39 −0.39 0.72 0.08 −0.05 −0.39

1.80 1.80 1.27 1.27 1.27 2.08 2.08

1.75 2.47 2.47 3.50 2.02 2.14 2.47

0.41 −0.32 −0.32 0.81 0.14 0.01 −0.32

1.75 1.75 1.24 1.24 1.24 2.02 2.02

(1 1 0)

(1 1 1)

a Lattice constants are obtained from experimental reports at room temperature (300 K) as aCu = 3.60 Å, aNi = 3.50 Å, and aBN = 2.49 Å, corresponding to the h-BN kink size kBN = 2.16 Å (or 4.31 Å, if double kink is better suited to minimize the |δk|). Italicized values mark the good fit complementarity with |δk| < 0.1 Å, whereas the boldface values additionally satisfy s < 1.44 Å (covalent bond distance in h-BN).

Figure 3. Energy of contact (−ε) of h-BN with [−1 1 2] step on the vicinal (110) surface, plotted versus the h-BN edge direction angle (χ). The minima at ZN, ZB, and (local) at A are due to a simple fit of the straight metal step to a similarly straight h-BN edge. The inset shows a tossed hBN square landing on ZN, A, or ZB edges (colored respectively as blue, purple, or red). Middle panels show −ε values, computed with DFT, for ZN, A, and ZB edges for all step directions of (110) and (111) surfaces of Cu and Ni. Right panels show computed electron density distributions for the h-BN ribbon edge-docked to the [−1 1 2] step of the (110) Cu surface (top, strong bonding) and weakly van der Waals connected h-BN on the Cu (110) surface (bottom).

To further judge the preferred contact, one cannot fully rely on geometrical considerations only but needs to quantify actual chemical affinity, using the atomistic models and either classical force fields (available in the case of Ni) or more precise, yet costly, density functional theory (DFT). The data plotted in Figure 3 reveal several robust features which remain unchanged across metals, the specific atomistic construct of the contact, and use of classical or ab initio methods (see SI for details). First, the binding is always strongest for the flat, straight-line edges due to simply closer contact with the metal, while any “chiral” edges bring unbalanced kinks, a roughness of contact with strained bonds, and, therefore, weaker overall binding (this robust feature was also noted in the earlier study of nanotube nucleation on solid catalysts, see Figures 1 and 2 in ref 25. In particular, the zigzag edges bind strongest, resulting in lower energy nuclei. It is important to further discriminate between B- and N- terminated edges, and for this the reactive force field (ReaxFF) obviously is not a reliable foothold, showing preference to ZB, while all DFT results unambiguously show stronger binding for ZN; note all the blue ticks “∨” in the main plot and in the summary panels for Cu and Ni in the middle of Figure 3. The important although not surprising conclusion is that the strongest binding and lowest contact energy (by ∼0.2 eV/Å, that is ∼1−2 eV for a near

growth conditions adds only a minimal correction (Table S1). For additional examples see also extended Tables S2 and S3. The above analysis shows that the 2D islands can maintain their orientation even though the steps that they are attached to are not perfect and appear typically curved, at the microscale. Yet, the islands’ polarity further depends on what edge of the 2D nucleus is docked to the step. This, in turn, is controlled by the affinity of the h-BN to metal, the binding energy ε (per unit length) of a specific edge to the step; the larger the ε value is, the more probable is the corresponding nucleus’ docking. To support this argument one can imagine tossing a unit-sized square flake of h-BN, as illustrated in Figure 3, inset. The square is a convenient (although inessential) choice here since it exposes all three basic edges, namely, armchair A (frontier BN-units) and zigzag ZB (frontier B atoms) or ZN (frontier N atoms). The flake can land with its different side-edges to the metal, i.e., A, ZB, or ZN. Accordingly, the energy will be reduced to (const − ε). Since the “const” term is identical for the initial conditions in all three cases, the probability of each outcome is ∝ eε/kT, where ε = εZN, εZB, εAC, or ε (χ), and χ is an angle to specify the arbitrary edge direction (named chiral angle in nanotubes studies23,24). C

DOI: 10.1021/acs.nanolett.9b00136 Nano Lett. XXXX, XXX, XXX−XXX

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show that for small nuclei L such attachment (∼L) is much stronger than the interaction with the terrace plane (∼L2, the usual cause of epitaxy), and apparently, the step fully controls the 2D-island orientation. Importantly, the steps are usually not perfectly straight but wandering curved lines,17 attachment to which seemingly must disperse the nuclei orientations and undermine the formation of the monocrystal. However, we reveal that such ever-present noise among the metal step directions should not always affect the orientation of the 2D nuclei. In a simple but remarkable way, this noise is suppressed by a peculiar, figuratively speaking, “digital filter” offered by the discrete nature of atomic lattice; to detail such similarity, the analog curve describing a meandering step is sampled and digitized by the very grid of constituent atomic rows, breaking the curve into straight 1D-terrace segments, separated by the sparse kinks. Then, the nucleating 2D layer is oriented by the 1D-terraces only, while the kinks on a metal step are compensated by the complementary counter-kinks on h-BN, thus ignoring the curve line slopes (Figure 1c). We determine the measure of such complementarity as simply a size offset δk between the kink on metal and the one on h-BN. This allows derivation or prediction of specific best facets and steps, as shown in Table 1. Further, the chemical affinity of the h-BN edges to metal plays an important role in lifting the remaining degeneracy in the ways its nuclei attach to metal steps. The DFT calculations show strong preference (by 1−2 eV/nm) of N-terminated zigzag edge (ZN) docking to the steps, enough to ensure the identical orientations. Lastly, an additional requirement of the step being not too tall should permit the spillover of growth downhill or uphill, as well as the recovery of possible discontinuity cracks, as demonstrated by MD simulations. Putting these criteria together (|δk| < 0.1 Å, and s < 1.44 Å BN bond) points toward vicinal Cu (110) with steps along [−1 1 2] direction as the most likely, in accord with what was surmised from experiments.1 We note, however, that others marked in the extended Table S2 can also be worth testing, notably Ir and Pt surfaces vicinal to (110), with steps along [−1 1 2]. The described criteria can also guide experimenting with other 2D-materials syntheses.

nanometer nuclei) and therefore overwhelmingly greater probability of nucleation is for the h-BN attached to the step on metal by its zigzag ZN-side. In other words, the edge chemistry breaks the inversion degeneracy of the nucleation. Putting this together with the important finding above that the meandering curvature of the substrate steps does not disturb hBN orientation, one understands the possibility of perfect coorientation of the multiple h-BN islands, especially on vicinal surfaces highlighted in Table 1, with the negligible offsets δk for best-complementary kinks. Ensuring the co-orientation of the nuclei still cannot guarantee the formation of a monocrystal. The vicinal steps, key to aligning the individual h-BN nuclei, are on the other hand the obstacles to the desired seamless fusion of the islands. Indeed, an island formed on a terrace and arriving to its end (as a reminder, there is often a so-called Ehrlich−Schwoebel barrier17,26,27 even for simple diffusion) cannot in any obvious way connect to the next island, formed earlier one-step downward. This seems like it would leave a discontinuity fissure. If the step is very small, the upper island can spill over it, to continue growing around and join the one down continuously from its flank. This forms a short screw dislocation (of Burgers vector s, normal to the substrate plane, Figure 4). Can such crack discontinuity close, bond by

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Figure 4. Healing/closure of overhang-cracks. (a) The flakes draped over the step (s) between two terraces of Ni (1 1 0)/[1 1 2] surface form a crack-like opening in the h-BN lattice. (b, c) Molecular dynamics simulation at 500 K for 200 ps shows progressing closure of the crack.

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The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.nanolett.9b00136.

bond, in an immediate healing process, at least in the case of small steps? We verify this possibility by performing direct molecular dynamics (MD) simulations and indeed observe that the fissure readily heals, making a continuous joint island. On the other hand, if the step is tall (in excess of 1.44 Å, the would-be BN bond length), the spillover of the upper island over the step down is prohibitive, leaving the discontinuities open, with potential detriment for the layer quality. Most likely candidates are the (110) surface with [−1 1 2] steps, as highlighted in the Table 1. To summarize, motivated by recent observations of h-BN growth, we elucidate the key underlying mechanisms that enable its monocrystallinity on a vicinal metal surface.1 Contrary to earlier epitaxy-based attempts, by employing high symmetry surfaces and inevitably yielding crystallographically antiparallel domains, on a vicinal plane the layer nuclei start attached to the steps, all directed same way. We

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Effect of thermal expansion on step complementarity; step complementarity analysis of the expanded set of materials; and computational details (PDF)

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Ksenia V. Bets: 0000-0003-1070-3992 Nitant Gupta: 0000-0002-3770-5587 Author Contributions

The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript. Notes

The authors declare no competing financial interest. D

DOI: 10.1021/acs.nanolett.9b00136 Nano Lett. XXXX, XXX, XXX−XXX

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ACKNOWLEDGMENTS This work was supported by the U.S. Department of Energy (DE-SC0012547). Computer resources were provided by the National Energy Research Scientific Computing Center, supported by the DOE Office of Science (DE-AC0205CH11231), and the DAVinCI cluster at Rice University, acquired with funds from the NSF (OCI-0959097).

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DOI: 10.1021/acs.nanolett.9b00136 Nano Lett. XXXX, XXX, XXX−XXX