Tilt Grain Boundary Topology Induced by Substrate Topography - ACS

Jul 31, 2017 - Jennifer G. DiStefanoYuan LiHee Joon JungShiqiang HaoAkshay A. MurthyXiaomi ZhangChris WolvertonVinayak P. Dravid. Chemistry of ...
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Tilt Grain Boundary Topology Induced by Substrate Topography Henry Yu,†,‡,∥ Nitant Gupta,‡,∥ Zhili Hu,‡ Kai Wang,§ Bernadeta R. Srijanto,§ Kai Xiao,§ David B. Geohegan,§ and Boris I. Yakobson*,†,‡ †

Applied Physics Program and ‡Department of Materials Science and NanoEngineering, Rice University, Houston, Texas 77005, United States § Center for Nanophase Materials Sciences, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, United States S Supporting Information *

ABSTRACT: Synthesis of two-dimensional (2D) crystals is a topic of great current interest, since their chemical makeup, electronic, mechanical, catalytic, and optical properties are so diverse. A universal challenge, however, is the generally random formation of defects caused by various growth factors on flat surfaces. Here we show through theoretical analysis and experimental demonstration that nonplanar, curved-topography substrates permit the intentional and controllable creation of topological defects within 2D materials. We augment a common phasefield method by adding a geometric phase to track the crystal misorientation on a curved surface and to detect the formation of grain boundaries, especially when a growing monocrystal “catches its own tail” on a nontrivial topographical feature. It is specifically illustrated by simulated growth of a trigonal symmetry crystal on a conical-planar substrate, to match the experimental synthesis of WS2 on silicon template, with satisfactory and in some cases remarkable agreement of theory predictions and experimental evidence. KEYWORDS: phase-field modeling, WS2, grain boundaries, topology, topography

F

topological defects (dislocations and GBs) on planar substrates occurring by chance, with probabilistic nature originating from randomness of nucleation and growth conditions. A curved substrate adds to the material’s elastic energy, possibly destabilizing the 2D crystal structure, to make defects formation more probable than on a plane. For instance, an ingenious experiment growing 2D crystal of tiny polystyrene particles-“atoms” on spherical surface exhibits branched, ribbon-like patterns,12 resembling similar setup for shrinkageinduced fractal fracturing.13 Local curvature is shown to create an effective geometrical potential, constraining the diffusion of defects,14 and even to change the critical size of nucleating grain and crystal nucleation rate.15 It can also be mentioned that an elastic strain from conforming to nonplanar substrate can additionally change the electronic band structure,16 leading, for example, to pseudo-Landau levels in graphene, as shown in theory17−20 and experiments.21,22 It is of great interest to explore how the topological defects in 2D material can be made not by chance but deterministically, by design of substrate topography.

ormation of defects is a versatile means with which to change the macroscopic material property. The broadly studied examples of it are very diverse, even for lowdimensional materials only. For instance, we have earlier shown how nanotubes form structural defects in response to tension;1 conversely, the defects’ character and concentration change the tensile strength of the tubes.2 Defects create the localized electronic states in graphene,3 while the grain boundaries (GBs) create an effective electronic conduction band gap.4 It is also shown5,6 how a two-dimensional (2D) semiconductor can surprisingly contain metallic GBs, due to polar discontinuity across the boundary. Further, carefully designed GBs of 2D metal dichalcogenides can possess substantial magnetic moment of ∼1 μB.7 These examples illustrate that, while defects can often degrade material performance, they can also impart new useful properties to it. Precise understanding and control of defect formation is highly desirable. Topological defects on flat surfaces have previously been studied. In theory, Yazyev8,9 and Liu10 have explored the energy trends among different defects as well as their energetically stable shapes for graphene. We further simulated the kinetics of defect formation in graphene, using Monte Carlo methods to reveal the growth mechanism of GBs in an atom-by-atom fashion.11 These works essentially demonstrate the formation of © 2017 American Chemical Society

Received: May 25, 2017 Accepted: July 31, 2017 Published: July 31, 2017 8612

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angle δ is for a given cone aperture angle 2a. The horizontal lines mark the orientation of the material, its intrinsic crystallographic direction. In order to wrap a piece of material seamlessly onto a cone, a wedge of angle Δ = 2π(1 - sin a) must be cut out. Then, while the two edges (red) join on the cone, the marker-lines meet at a tilt angle δ showing how a single crystal must gain a GB, in order to conform to the conical surface. The GB tilt angle is defined by the shape of the cone, namely its aperture angle, and the symmetry order n of the 2D material lattice, δ = |mod(Δ + π/n, 2π/n) − π/n|, as plotted in Figure 1b. However, how the GB is located and the kinetics of its formation require more detailed description, as follows. We describe our approach within the framework of the phase-field method (PFM). The PFM is a continuum model originating from the Landau theory25 describing phase equilibria in terms of the total free energy F of the system expanded as a functional of the order parameter, Φ(r). In the context of crystal growth, the order parameter Φ is a number which represents the local density and orientation of the material. High level atomistic methods such as DFT are not affordable, and even classical MD is impractical for the length and time scale of interest. The PFM allows for the study of phase dynamics, or material growth, not readily achievable with atomistic methods. According to Hohenberg and Halperin,26 the dynamics of phase equilibria processes, e.g., crystal growth, can be described by a Markovian equation of the form, ∂tΦ = −μδF/δΦ, where μ is defined as the mobility of the process. Depending on the system of interest, one can design different versions of the functional, F. For the study of defects in 2D crystal growth on a flat surface, a functional can be designed as in the WKC model which elegantly captures crystal orientation and GB.27,28 In the following, we first describe our approach to accounting for substrate curvature and then incorporate it into the WKC model, to proceed with direct simulation of 2D crystal growth on curved surfaces. Following the convention in the WKC model, for a 2D curved surface parametrized as S:r(u,v), the phase of a crystal can be represented by a complex order parameter Φ(r) = ϕ(r)eiθ(r). Here ϕ(r) is a normalized “density”, i.e., ϕ = 1 within a crystal grain and ϕ = 0 at a bare substrate. The complex phase θ(r) is the local orientation of the crystal, defined within −π to π; when considering crystal of Cn symmetry, the range of θ is further reduced to within −π/n to π/n. To generalize the WKC model to curved surfaces, one should use the covariant differential operators for the derivatives and surface integrals. However, this alone does not take care of the orientation field θ(r), which we describe below. The orientation field θ(r), while being well-defined in usual Cartesian coordinates on a 2D plane, needs modification in curvilinear coordinates. If we first represent the orientation of a crystal in terms of a unit vector q⃗ lying on the surface, a single crystal grain can be represented as a set of parallel vectors q⃗(r). In Cartesian coordinates, as shown in Figure 2a, the orientation field can be obtained as the angle between q⃗ and the coordinate frame, i.e., θ(r) = cos−1(q⃗·x̂), in which x̂ is the unit basis vector along the x direction. Since x̂ is constant in Cartesian coordinates, θ(r) is also constant within a single crystal grain. However, this is not the case in curvilinear coordinates, as shown in Figure 2b. If we again take the orientation as the angle between q⃗ and the coordinate frame, i.e., θ(r) = cos−1(q·⃗ û), the unit basis vector û = û(r) is now a function of position. Therefore, two parallel vectors q⃗(r1), q⃗(r2) will result in different orientations θ(r1) ≠ θ(r2), i.e., the orientation field

In fact, a prescribed GB presence in graphene on a curved conical surface has been clearly shown in our earlier atomistic study23 as well as finite embedded GB terminated by a cone(pentagon) and saddle- (heptagon) lattice disclinations. The necessity of a GB on most cones clearly complements the wellknown rule that defect-free graphene can only form “magic” pristine cones of specific aperture angles, 2a = 2 sin−1(1 − p/6), where p = 1−5 is a number of pentagons at the apex (p = 0 for flat graphene, p = 6 for nanotube).24 It is also clear from this consideration that the GB or its constituent dislocations in 2D material are caused not by the strain but by more rigorous topology requirements. In this work we analyze the growth process, to show how the topography and boundary of the nonplanar substrate, both being global properties of the surface, lead the 2D material to forming defects deterministically. To this end, to describe 2D crystal growth, the phase-field method is used, importantly augmented with a gauge modification tracking the crystallographic orientation change caused by local curvature over the conical surface. Furthermore, we present experimental results of 2D tungsten disulfide (WS2) grown over a cone-on-a-plane substrate and show how the topography of surface indeed induces GB formation of 2D crystals, with good agreement between the simulation and experiment. Before introducing any mathematical details, first Figure 1 shows schematically how 2D material on a conical substrate gains a GB. Omitting all atomistic detail, shown in ref 23 for graphene, helps to capture basic relationships, like what GB tilt

Figure 1. (a) Schematics of cutting a wedge with angle Δ out of material and wrapping the remaining piece onto a conical surface. Red thick lines show the connected edges, where GBs can form. 2a is the aperture angle of the cone and δ the tilt angle of the GB. (b) GB tilt angle as a function of cone aperture 2a, for crystals with C1, C2, C3, C4, C6, or C∞ symmetry (order n = 1, 2, 3, 4, 6, or ∞). 8613

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Gaussian curvature of the surface, an intrinsic property (shape) to the surface and invariant of the choice of parametrization.29 Therefore, Δθ sets a global condition for the self-misorientation of the 2D material, which can constrain the material to form GBs deterministically. Since θ is only well-defined within the range of −π/n to π/n, we must also evaluate Δθ within the range −π /n ≤ Δθ < π /n

Therefore, when evaluating the misorientation Δθ, accumulated in process of growth, one must always choose the principal value within the interval (eq 3). One special case is when Δθ = 2π/n ≡ 0, one of the “magic angles” of the cone crystal, a case in which no GBs will form. So far, we have presented a modification for accurately describing and tracking crystal orientation in the course of 2D material growth on curved surfaces. The closed loop misorientation Δθ is in fact the geometric phase of the surface, induced by the vector field Λ⃗ (r). We point out that the vector Λ⃗ plays the same role as the vector potential A⃗ in electromagnetics; the curl of the vector ∇ × Λ⃗ = Kn̂ is analogous to the magnetic field ∇ × A⃗ = B⃗ , with n̂ as the surface normal and K the Gaussian curvature. Although various basic shapes can be considered this way, we further specifically focus on conical surfaces, not only because it is relatively simple but also because we include basic experimental evidence for comparison. Figure 2c shows an orthogonal parametrization of the conical surface, with u ∈ (0, 2π) as the angular and v ∈ (0, R) as the radial directions. Within this parametrization, the vector field Λ⃗ ·g⃡ = (Δ/2π − 1,0). To apply theory to a conical surface, we again transform eq 2 via the Gauss−Bonnet theorem, which gives the Δθ = −∮ Λ⃗ ·dr = −∮ KdS. The Gaussian curvature K on a conical surface is zero everywhere except at the apex point, where K is singular but has a finite surface integral, ∮ KdS = Δ − 2π. These results immediately divide all closed contours on a cone into two types, as shown in Figure 2d. In the first, with contour of type C 1 , which excludes the apex, the accumulated misorientation can be calculated from eq 2 to be Δθ = 2π ≡ 0. This shows that contour C1 will cross the GB zero or an even number of times. As for the second case with contour of type C2, which includes the apex, we have Δθ = 2π − Δ. This shows that C2 will cross the GB one or an odd number of times. Therefore, one concludes that, for a single grain of 2D material, there must be 1 GB with misorientation δ as defined above; in the special case where Δ is a multiple of 2π/n, there will be no GBs according to eq 3. While we arrive at similar conclusions as depicted in Figure 1, this analysis provides a general way to predict GB formations in 2D material, given the topography of the surface. With these provisions, we are now ready to model the growth process of 2D material on a cone. While theoretical discussion above can predict whether GBs should form, it is only through the process modeling that we can learn where and how. Based on our discussion above, the substitution (eq 1) into the WKC model, gives the free energy functional we employ

Figure 2. (a) Cartesian coordinates, where q⃗0 = x̂ is the reference direction. For the same physical orientation vector q⃗(r1) and q⃗(r2), the measured orientation θ is independent of the coordinate. (b) A general curved surface, where q⃗0 = û is the reference direction. For the same physical orientation vectors q⃗(r1) and q⃗(r2), due to the curving of coordinates, the measured orientation θ1,θ2 relative to local coordinate will change. (c) 2D representation of a conical surface of central angle 2π − Δ and wedge radius R. (u,v) forms an orthogonal parametrization of the conical surface, with u along the angular direction and v the radial direction. Here q⃗0 = û is the reference direction. Due to the curving of u coordinate, the measured orientation θ1,θ2 relative to local coordinate will change, while the physical orientation remains the same. (d) 3D representation of the conical surface in (b). C1 is a closed contour excluding the apex, while C2 is a closed contour including the apex. White brushed line shows a possible position for GB.

θ(r) becomes spatially varying, while the physical orientation remains the same. Consequently, when taking the derivative of θ, simply taking the difference dθ does not take into account the change of the reference vector û. For a 2D, orthogonally parametrized surface, the correct differential should be taken as ∂E ∂G Dθ = dθ − Λ⃗ ·dr, where Λ⃗ = − v , u whose

(

2E EG

2G EG

)

components are the negative of geodesic curvatures along the u and v grid lines, respectively, and E,G are diagonal elements of E F .29 Note here that the dot the metric tensor g ⃡ = F G product Λ⃗ ·dr should be expressed as Λ⃗ ·g·⃡ (du,dv) on curved surfaces. Therefore, we finally arrive at the corrected derivative operator for θ:

(

)

∇θ → ∇θ − Λ⃗

(1)

The substitution as in eq 1 should be made wherever the derivative of θ is taken. While the quantities in eq 1 are local in space, the vector Λ⃗ keeps track of all surface topography and allows eventually reproducing correctly the topography-induced GBs, as shown in Figure 1, which we will demonstrate below. With eq 1, we can now calculate the total misorientation accumulated along a path connecting points r1,r2 on the surface, by taking the path integral δθ = ∫ rr21(∇θ − Λ⃗ )·dr. Obviously, given an arbitrary path, the value δθ will depend on the surface, its parametrization, and how the crystal was grown. However, for a closed contour C, misorientation can be calculated as Δθ =

∮ (∇θ − Λ⃗)·d r = −∮ Λ⃗·d r

(3)

(2)

From eq 2 one can clearly see that the closed loop misorientation Δθ is invariant of how the crystal was grown. Moreover, through Gauss−Bonnet theorem, we can further transform eq 2 into Δθ = −∮ Λ⃗ ·dr = −∮ KdS, where K is the

F[Φ] =

∫ [e(∇ϕ , θ)|∇ϕ|2 + f (ϕ) + s(ϕ)|(∇θ − Λ⃗)| + t(ϕ)|(∇θ − Λ⃗)|2 ]dS]

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ACS Nano Here dS = √gdudv is the differential area element on the curved surface with √g as the square root of the determinant of g;⃡ the covariant gradient operator ∇ on curved surfaces is expressed as ∇ ≡ gij∂j in the Einstein convention, where gij is the matrix element of the inverse metric tensor g−1 ⃡ . The first term (e) accounts for the energy of edges or interfaces. The second term (f), for the bulk energy of the system, is a tilted double well potential with a shallow minimum at ϕ = 0 and a lower minimum at ϕ = 1, driving the system to form crystal; the degree of tilting represents the feedstock strength or temperature effects (see Supporting Information, SI). The third and fourth terms (s and t) account for the energy inhomogeneity regarding the orientation of the crystal. Note here that a distinct feature of the WKC model is the third term (s), which has been proven necessary for a stable GB by Warren et al.27,28 The dynamics of the crystal growth process is described by the following Markovian equations, ∂ϕ δF = −μϕ , ∂t δϕ

and

∂θ δF = −μθ ∂t δθ

(5)

Figure 3. (a) A triangular WS2 crystal flake. (b) Modeled time progression of crystal growth on a flat surface (formally a cone with 2a = π). (c) Experimental SEM image of WS2 crystal grown on flat substrate. (d) Modeled progression of crystal growth on a conical surface with 2a = 19.2° (or Δ = π/3). (e) Experimental image of WS2 crystal grown on a conical substrate. (f) Simulations of crystal growth for a series of different cones, with 2a = 19.2°, 28.9°, 38.9°, 47.2°, 60°. (g) Color coded plot of the orientation field θ for (f). Blue represents ϕ = 1(grain), and white represents ϕ = 0 (substrate); in (g) cyan represents substrate.

where μϕ, μθ are mobilities for ϕ, θ respectively. For each simulation, we first initialize the calculation with a small circular grain at a point r = r0 with ϕ = 1 within the grain and ϕ = 0 elsewhere; as for the orientation, we assign a constant orientation θ = θ0 within the grain, and random values elsewhere to represent a disordered “liquid” phase. The growth process then follows eq 5. For simulations of conical surfaces in Figure 2, we ignore regions near the apex due to diminishingly small grid size. The physical justification for this is that the growth is hindered by curvature energy, if it exceeds thermal energy: K/r2 ≥ 3kBT/A where K ≈ 10 eV is the bending stiffness of WS2 or MoS2,30,31 r the curvature radius, kBT = 25 meV, times 3 atoms per unit cell area32 A = 8.5 Å2 for WS2, so the growth of WS2 stops when r ≤ 50 Å near the apex. In fact, from the experimental images (Figure 3e and Figure S2), this radius appears to be much larger, r ∼ 40−60 nm. For more simulation and parameter details see the SI.

itself, a GB forms. In parallel, Figure 3e shows SEM image of WS2 crystal grown on a conical substrate. Strikingly, the experimental crystal shape and the GB formation closely resemble the modeling results. Judging from the fact that the two edges meet nearly in parallel in Figure 3e, we deduce that the cone geometry is close to having 2a = 19.2°, as simulated in Figure 3d. We also model growth on cones of different aperture angles 2a and plot the density ϕ and orientation θ fields in Figure 3f,g, respectively. GBs can be clearly seen on the cones, except for 2a = 38.9°(Δ = 4π/3). According to eq 3, the misorientation Δθ = δ is only defined modulus 2π/3, so for 2a = 38.9°(Δ = 4π/3), one of the magic cones, no GB forms. These features are better conveyed by the color-coded orientation field, Figure 3g. As mentioned above in eqs 1 and 2, the vector Λ⃗ tracks information regarding the local curvature, leading to a continuously changing color code for orientation within the same crystal grain; however, when the crystal grain meets itself at the other side of the cone, the accumulated change in the orientation results in an abrupt jump at the interface and yields a GB. In the special case of 2a = 38.9°, the accumulated orientation misfit precisely equals 4π/3, which gives no GB for a three-fold crystal like WS2. While we have discussed relatively short-range (on the cone) features of crystal growth, we now turn to the longer-range features spreading beyond the cone and onto a flat surface, which is a more experimentally feasible substrate shape, Figure 4a. To analyze the long-range features, we take the integral in eq 2 along a closed contour C1 away from the conical surface, (Figure 4a) on the flat region, so we must have Δθ = 2π ≡ 0. This result immediately precludes the formation of a single long-range GB, since single GBs must lead to a nontrivial geometric phase Δθ ≠ 0; however, Δθ ≡ 0 does not preclude

RESULTS/DISCUSSIONS We were able to compare our simulation results with experimental data, obtained for a specific case of WS2 crystal grown on conical silicon substrate. WS2 is one of the transitionmetal dichalcogenide (TMD) 2D materials; its stable structure is of 2H phase, with a three-fold symmetry,32 Figure 3a. In order to reveal the GB, we oxidize the sample after growth is complete, and the GB, being higher in energy, are etched away in the process, leaving behind the bulk crystal grains. Photoluminescence (PL) and Raman spectroscopy were used to characterize the WS2 monolayers on the conical silicon structures, as described in the SI. Additional experimental details are described in the Methods section. Accordingly, we simulate a growing crystal of three-fold symmetry, just as WS2. Figure 3b shows the progression of crystal (the normalized density ϕ field) growing on a normal flat surface (formally, a cone with aperture angle 2a = π). Figure 3c shows the SEM image of WS2 crystal grown on a flat silicon substrate. The resemblance of the crystal shape and symmetry demonstrates the basic capability of our model to capture the main features of the growth process. More interesting results in Figure 3d show modeled progression of a single crystal grain grown on a cone with 2a = 19.2° (or Δ = π/3). One can clearly see that as the crystal grain wraps around the cone and meets 8615

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Figure 4. (a) Cone on a plane. The gray triangle represents a crystal grain growing toward the cone. The red contour C1 is taken on the plane away from the cone, with a geometric phase Δθ = 2π. (b−f) Modeled growth for cones of different apertures 2a on top of a flat surface. Data at different time steps are plotted alternatingly to show the progression. GBs are highlighted with dark green lines. Cases with only (b, c) short-range GBs and (d, f) long-range GBs formed. (g) SEM images of experimentally grown WS2 flakes (dark gray), oxidized afterward to reveal long-range GBs. White spot (pointed with red arrow) is the cone. (h, left) SEM image of WS2 flakes (dark gray) around a cone. (h, right) Modeling, with a small steep cone of aperture 2a1 = 39.3° near the center, on a shallow cone of 2a2 = 155.8°.

GBs radiating from the conical surface. Figure 4h, left, shows another SEM image, with a very interesting pattern: there is only one long-range GB on the flat surface, appearing to contradict our theoretical analysis above. However, in contrast to Figure 4g, the edges of the grain here are slightly convex, clearly indicating that the surface is not entirely flat. Measuring the bottom-left corner tilt angle, we deduce that the GB has a tilt angle of 8°; therefore the surface must be approximately a very shallow cone with an aperture angle 155.8° (or Δ = 8°). According to these assumptions, we conduct a simulation, as shown in Figure 4h, right, plotted in the same way as in Figure 4b−f. Here we simulate a triangular crystal growth on a surface with a steep cone with aperture angle 2a1 = 39.3° (measured from the side view of the experimental substrate, see SI), on top of a shallow cone with aperture angle 2a2 = 155.8°. The simulated image does indeed capture all the important features of the experiment, including the overall shape, convex edges, and the single GB. We again point out that although our simulations show variability due to the detailed kinetics of the growth process, all cases are consistent with the analysis according to theory: either zero or multiple long-range GBs form on the flat surface. In the Figure 4h, after being explained above, truly “the exception proves the rule”.

the formation of multiple GBs in the plane, since multiple GBs can yield accumulated misorientation of Δθ ≡ 0. Figure 4b−f shows a series of model results again for the three-fold crystal grain grown on different surfaces, where nucleation occurred away from the conical features. All cones have the same base radius, but different aperture angles 2a. To show the progression of the growth process, we plot results of different time steps as alternating blue-white stripes, showing the evolution of grain shape during growth. For clarity we mark the GBs with dark green lines. From our results, one can see that there is typically one GB on the cone when the crystal grain wraps around the cone, similar to what was described in Figure 3. Additionally, there is always a GB at the base of the cone, since the crystal grain on the cone will be mismatched with the grain on the flat surface, due to disrupted topography. We have recently explored the atomistic makeup of such GBs for graphene nanochimneys.33 Whether or not these features will escape the conical surface and form long-range GBs depends on the growth speeds of the different grains joining at these boundaries. Figure 4b,c shows two cases when no longrange GB forms: The huge grain in the flat valley outgrows the grain on the cone, closing around, and thus terminating all GBs at the cone. In fact, in Figure 4b, one can see that the flat grain outgrows the cone grain so much that it re-enters the cone at the far side of the cone. Figure 4d−f shows cases in which longrange GBs do form: The on-cone grain re-enters the flat valley at certain points sooner than the all-flat grain arrives, starting new grains on the flat, becoming the dominating grains at nearcone vicinity. Notably, these configurations also appear in experiments, as in Figure 4g SEM images of WS2 dark gray flakes grown over cones (white spots pointed to with red arrows). The images clearly show how the crystal flake forms multiple long-range

CONCLUSIONS In conclusion, we discussed theoretically and have shown experimental evidence of how the topography of a substrate can induce the topological changes in 2D material, to form GB deterministically, in contrast to common GB formation through random nucleation processes. The closed loop misorientation Δθ of a crystal equals the geometric phase of the substrate surface; the geometric phase is induced by a vector field Λ⃗ , a vector field analogous to the vector potential A⃗ in electro8616

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AUTHOR INFORMATION

magnetics. In addition, we incorporate our theory into the WKC phase-field method and generalize it to simulate 2D crystal grown on curved surfaces. While our theory and method can be developed for general curved surfaces, we present here growth of a three-fold symmetric crystal on conical surfaces as an example, comparable with experimental observations. Both theory and experiment discover formation of short-range (oncone) and long-range (in-valley) GBs due to the topology of the conical surface. Presented theoretical analysis is confirmed by direct simulations and further strongly supported by direct experimental evidence. Furthermore, our theory provides a guideline for the GB engineering of 2D materials via the design of substrate, leading to the control of material properties, especially electronic and magnetic for devices and circuits.

Corresponding Author

*E-mail: [email protected]. ORCID

Henry Yu: 0000-0002-1306-333X Nitant Gupta: 0000-0002-3770-5587 Kai Xiao: 0000-0002-0402-8276 David B. Geohegan: 0000-0003-0273-3139 Author Contributions ∥

These authors contributed equally.

Notes

The authors declare no competing financial interest.

ACKNOWLEDGMENTS We are grateful to Ksenia V. Bets for valuable discussions of the growth theory of 2D materials. H.Y. would also like to thank Kyle Kinneberg for helpful discussions of the mathematical details. K.W. thanks Dale K. Hensley for providing the randomly distributed conical silicon structures and Alexander A. Puretzky for the optical characterization. The synthesis of 2D materials (K.W., K.X., D.G.) was supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division and performed in part as a user project at the Center for Nanophase Materials Sciences, which is a DOE Office of Science User Facility. Work at Rice (growth theory and computer simulations) was supported by the DOE BES grant DESC0012547 and in part (relating to electro-optical properties) by the Office of Naval Research grant N00014-15-1-2372. Computer resources of the DAVinCI and NOTS clusters at Rice University are funded by NSF grant nos. OCI-0959097 and CNS-1338099. This article is a tardy yet heartfelt contribution to honor Bill Gelbart’s 70th birthday (William M. Gelbart Festschrift Special Issue J. Phys. Chem. B 2016, 120, 5787−6454).

METHODS Numerical Calculation. The modeling results are calculated by an in-house developed code based on the WKC phase field model, augmented for material growth on curved surfaces, as described in the text. The code was written in Python2.7, parallelized with the MPI4Py package.34−36 Additional modeling details and modeling parameters are described in the SI. Substrate Preparation. A combination of electron beam lithography and cryogenic reactive ion etching was used to create the conical structures. Standard 100 mm diameter silicon wafers with a ⟨100⟩ orientation were spin-coated with PMMA 495 A4 electron beam resist (Microchem Corp., Newton, MA) and exposed using an electron beam lithography system, JEOL JBX9300-FS, to define the dot arrays. The patterned wafers were then developed in 1:3 methyl isobutyl ketone (MIBK):isopropyl alcohol (IPA). Fifteen nm chromium was deposited onto the patterned wafers by electron beam evaporation, followed by a lift-off process resulting in a chromium etch mask on the areas exposed to the electron beam. The dot arrays were then etched using a cryogenic silicon etching process in an inductively coupled plasma ion etching system (Oxford Plasmalab 100). The process was carried out in a mixture of O2 and SF6 gases at −110 °C. Control of oxygen content in the SF6 flow allowed a balance between etching and passivation needed for defining the sidewall profile. Randomly distributed conical structures were achieved by adopting the same etching process without patterning the silicon wafers. Materials Growth, Structural and Optical Characterization. WS2 crystals were grown on flat SiO2/Si substrates and silicon cones by chemical vapor deposition in a home-built horizontal tubular furnace. The detailed information was described in our previous work.37 The morphologies of the silicon cones and as-grown WS2 crystals were examined with field-emission scanning electron microscopes (FE-SEM, Zeiss Merlin and FEI Novalab 600 Dual-Beam system). Spatial photoluminescence (PL) mapping, PL, and Raman spectra were measured with a 532 nm solid-state laser excitation with a spot size ∼1 μm at room temperature. The PL signals were recorded with a monochromator and a liquid-nitrogen-cooled charge-coupled device (CCD). Oxidization Process. We optimized the preferential oxidization process to reveal the GB in the WS2 crystals.38 Briefly, WS2 crystals on the substrate were loaded in the center of the horizontal tubular furnace, and the temperature was increased to 330 °C in 5 min. The whole system was then held at this temperature for 15 min. The samples were taken out for characterization after naturally cooling down to room temperature.

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