Two-Dimensional van der Waals Epitaxy Kinetics in a Three-Dimensional Perovskite HalideYiping Wang,† Yunfeng Shi,† Guoqing Xin,‡ Jie Lian,‡ and Jian Shi*,†

†Department of Materials Science and Engineering and ‡Department of Mechanical, Aerospace, and Nuclear Engineering, RensselaerPolytechnic Institute, Troy, New York 12180, United States

*S Supporting Information

ABSTRACT: The exploration of emerging materials physicsand prospective applications of two-dimensional materialsgreatly relies on the growth control of their thickness, phases,morphologies, and film−substrate interactions. Though sub-stantial progress has been made for the development of two-dimensional films from conventional layered bulky materials,particular challenges remain for obtaining ultrathin, singlecrystalline, dislocation-free films from intrinsically non-van derWaals-type three-dimensional materials. In this report, with thesuccessful demonstration of single crystalline ultrathin large-scale perovskite halide material, we reveal and identify the favorablerole of weak van der Waals film−substrate interaction on the nucleation and growth of the two-dimensional morphology out ofnonlayered materials compared to conventional epitaxy. We also show how the bonding nature of the three-dimensional materialitself affects the kinetic energy landscape of ultrathin film growth. By studying the formation of fractal perovskites assisted withMonte Carlo simulations, we demonstrate that the competition between the van der Waals diffusion and surface free energy ofthe perovskite leads to film thickening, suggesting that extra strategies such as surface passivation may be needed for the growthof monolayer and few-layer films.

1. INTRODUCTION

Featured by the rich emerging physics1 and technologicalpromises,2 two-dimensional (2D) materials have attractedmuch attention from both the academic and industrialcommunities. Recently, endeavors have been extended fromtraditional graphene research to other van der Waals (VDW)materials exemplified by layered transition metal dichalcoge-nides3−5 and dielectric nitrides and oxides.6 Meanwhile,significant progress have been made on intrinsically nonlayeredthree-dimensional (3D) materials (i.e., non-VDW materials) bysynthesizing their 2D counterparts on selective substrates.Typical recent examples include single atomic layer FeSe onNb:SrTiO3 substrate7 boosting the superconductor Tc viapronounced interface effect; III−V compounds;8 oxides;9 andPbTe/CdTe10 2D electron gas (2DEG) systems marked bytheir quantum Hall transport phenomena.For the synthesis of the 2D form of naturally layered

materials, several sophisticated methods have been developedincluding mechanical exfoliation, vapor deposition on catalyticsubstrates, and van der Waals epitaxy.11−13 However, for non-VDW 3D materials, the conventional chemical epitaxy bymeans of molecular beam epitaxy14 or metal organic chemicalvapor deposition (MOCVD)15 is still the dominant strategy toobtain their 2D form. Such a technique places stringentrequirements16,17 on the choices of film−substrate combinationin terms of the lattice mismatch argument. This in turn narrowsdown the choices of materials for exploring innovative 2Dphysics. Fortunately, the discovery of 2D VDW epitaxy13

initially for layered materials and later18 for non-VDW 3Dmaterials may open up a new and promising window fordeveloping novel 2D materials or heterostructures (combinedwith the post transfer technique) free of substrate restriction.Progress has been made on the growth of thin films of PbSe,19

Te,20 GaAs,21 and GaN,22 as well as vertically aligned ZnOnanorods that completely overcomes the fettering of the latticemismatch.23,24 Nevertheless, most resultant film thickness (>∼20 nm) falls beyond a 2D criterion that is mandatory forenabling unique 2D physics.25 The underlying reason leading tosuch observations is believed to be the non-VDW bondingnature of the 3D nonlayered materials, which favors the growthof island morphology rather than 2D layers with higher freeenergies. Ultimately, a couple of fundamental questions arise:(1) is it theoretically possible to grow large-scale single or a fewatomic layers films out of 3D nonlayered materials by VDWepitaxy; and (2) what types of 3D materials are more feasiblefor 2D growth?In this report, we present our understanding of the 2D VDW

growth of a 3D parent material methylammonium lead chloride(MAPbCl3). MAPbCl3 is a nonlayered semiconducting materialrecently identified to be extremely efficient and promising foroptoelectronics applications. Ultrathin (sub-10 nm) and large-scale (a few tens of micrometers in lateral dimension) single

Received: July 7, 2015Revised: September 9, 2015Published: September 15, 2015

Communication

pubs.acs.org/crystal

© 2015 American Chemical Society 4741 DOI: 10.1021/acs.cgd.5b00949Cryst. Growth Des. 2015, 15, 4741−4749

crystalline 2D perovskite thin films on layered muscovite mica,a substrate very favorable for similar VDW growth for bothlayered and bulk materials,26,27 were successfully synthesized byVDW epitaxy. A classical nucleation and growth model28,29

explaining conventional epitaxy has been modified to interpretthe unique large-scale ultrathin 2D results in perovskite underthe VDW mechanism. The generalization of our VDWnucleation and growth model shows that the weak VDWinteraction plays the key role in favoring the large-scale singlecrystalline 2D growth, which is typically retarded by a strongfilm−substrate interaction in the conventional epitaxy case. Inaddition, it demonstrates that ionic and metallic crystals withdelocalized bonding characters have a greater tendency to formultrathin structures compared with covalent materials withstrongly localized bonds. Finally, with static and kinetic MonteCarlo simulations, we show that the fractal 2D morphology inperovskite materials precisely manifests the kinetic competitionbetween VDW diffusivity and thermodynamic driving force.This unique phenomenon to VDW growth suggests afundamental limit on the morphological stability of the 2Dform of a 3D material. Our findings shed light on the growthstrategies for the development of truly 2D structures out oftheir non-VDW 3D counterparts.

2. EXPERIMENTS AND RESULTS

The MAPbCl3 perovskite thin film was synthesized via achemical vapor deposition (CVD) method with dual precursorsources. A schematic drawing and further details of theexperimental setup can be found in Supporting InformationFigure S1. Figure 1a shows the optical image of the well-alignedsquare perovskite sheets with high coverage over a large area onthe mica substrate, with the typical results grown at a substratetemperature of around 220 °C and 9−10 cm away from theprecursor. All crystal sheets align along one crystallographicorientation of mica. A closer examination (Figure 1b) revealssharp perpendicular edges, consistent with the perovskite’scubic structure indicating high degree of crystallinity. Mean-

while, the uniformity in color suggests a smooth surface,confirmed by atomic force microscopy (AFM) images (Figure1c). Further observation of one of the largest square sheets(Figure 1c) shows a lateral dimension of a few tens ofmicrometers by a few tens of micrometers and thickness in thesub-10 nm range. The thickness of the ultrathin films asdetermined by AFM cross-sectional profiling (inset of Figure1c) is down to 8.7 nm. Sheets with more “whitish” color have alarger thickness ranging from 30 to 50 nm, but preserving thesame smoothness (Supporting Information Figure S2a and b).Additional SEM images confirm the square-like morphology(see Supporting Information Figure S3). The 2D sheets displaydifferent colors as the sheet thickness continues to increase(Figure 1d). Most sheets exhibit uniform color indicating gooduniformity of the sheet thickness. Inset of Figure 1d shows twotypical thicker sheets of ∼100 nm in thickness.Interestingly, morphologies other than the square are also

found popular. Figure 1e shows the typical “stair-like” zigzagedges observed at nearly every large sheet approximately 1−2cm farther away from the square ones. Such sheets featured bythe fractal morphology are always found to be larger in size andcould be grown to as large as several hundred microns in scale(see Supporting Information Figure S4a and b). Further, theclassical dendritic morphology has also been observed (Figure1f) on the substrate farthest from the precursor (∼13−14 cmaway). It should be noted that here all dendritic branches shareonly two perpendicular directions. The versatility (square,zigzag fractal, and dendritic) of the morphology and highreproducibility (∼100%) of the synthesized perovskite by theCVD VDW approach provide rich evidence on studying theirgrowth thermodynamics and kinetics.To confirm the perovskite structure and reveal its epitaxial

relation with the mica substrate, transmission electronmicroscopy (TEM) and selective-area electron diffraction(SAED) characterization were performed. Considering thevulnerable nature of MAPbCl3 to possible ion irradiationdamage during typical TEM sample preparation processing by

Figure 1.Morphology of as-grown perovskite (PVK) thin films. (a,b) Optical microscope images of well-aligned square sheets covering large areas ofmica substrate. (c) Optical image of an individual large-scale ultrathin sheet (inset showing the atomic force microscopy measurement of the filmthickness). (d) High coverage colorful square sheets with larger thickness (inset showing the magnified view of two typical examples). (e) “Stair-like”zigzag morphology of perovskite sheets. (f) Dendritic morphology expanding in two perpendicular directions. Note that none of the images are falsecolored (Scale bar: (a) 50 μm, (b) 10 μm, (c) 10 μm, (d) 50 μm (inset: 20 μm), (e) 10 μm, (f) 50 μm).

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ion milling, moisture, and organic solvents, we have developeda new TEM sample preparation method for the perovskitehalide (Figure 2a). As shown in Figure 2a, the mica substratewith perovskite sheets growing on top was first peeled off by afew mica surface layers carrying the perovskite sheets with theaid of a polydimethylsiloxane (PDMS) stamp. Perovskitesheets/mica surface layer-covered PDMS was then pressedonto LaAlO3 (LAO) substrate, upon which the peeled-off layerscan be transferred. The relatively weaker interaction betweenperovskite sheets/mica surface layers and LAO substratecompared to the VDW force within mica layers made it moremanagable and easy to transfer the perovskite sheets/micasurface layers (occasionally only the perovskite sheets weretransferred; Supporting Information Figure S5a and b) onto thecopper grid simply by pressing the grid tightly on the LAOsubstrate.Figure 2b,c shows optical images of the as-transferred large-

area “stair-like” fractal perovskite sheets (green samples inFigure 2b), as well as square ones (e.g., the blue samplehighlighted by the dash circle) on the copper grid. Figure 2dshows a low-magnification TEM image of the square sheetmarked in Figure 2c. Inset of Figure 2d shows its magnifiedoptical image. Figure 2e,f shows the SAED patterns of bothmica and perovskite sheets, respectively, with the perovskitepattern taken from Figure S5a, and the [001] zone axis wasidentified for both samples. Despite being a monoclinicstructure,30,31 mica has a quasi-hexagonal lattice composed ofpotassium ions and aluminosilicates, as shown by the SAEDpattern in Figure 2e. The perovskite sheet presents itself with aperfect set of cubic patterns, which agrees well with the natureof its cubic structure. Further electron diffraction indexingyields a lattice spacing of 5.69 Å and energy-dispersive X-ray

diffraction (EDX) spectrum (Supporting Information FigureS6) revealed the elemental signatures of N, Cl, and Pb, whichconfirmed the perovskite phase of the MAPbCl3.

32 A SAEDpattern as shown in Figure 2g taken from a perovskite sheet/mica surface layers (spot (g) in Figure 2d) illustrated theepitaxial relation between mica and the perovskite halidefeatured by a combination of two sets of diffraction patterns(square and hexagonal). Combined with our optical evidence inFigure 1 and further SAED characterizations (see SupportingInformation Figure S7a and b), we have revealed a consistentand universal VDW epitaxial relation between mica andMAPbCl3mica (001) ∥ perovskite (001), mica (200) ∥perovskite (200) (5° offset), and mica (020) ∥ perovskite (010)(5° offset)for all three perovskite morphologies (square,“stair-like” fractal, dendritic). The 5° offset along the two in-plane directions may result from the different lattice types ofthe two materials and the need for the two to maximize theVDW bonding by choosing the optimized orientation.33 It canbe concluded from the indexing of Figure 2f,g that the twoperpendicular free facets of perovskites are (100) and (010),respectively, which represent the thermodynamic preferableconfiguration of the perovskite halide. To better understandand illustrate the VDW epitaxial relation between mica and theperovskite halide, we present a tentative atomistic model,34 asshown in Figure 2h−j, where the perspective, normal, and sideviews of the perovskite/mica heterostructure are presented,respectively.The VDW growth nature of perovskites on mica is further

supported by additional evidence and arguments: (1) thelayered nature of mica barely supports chemical epitaxialgrowth; (2) if the chemical bond does exist betweenperovskites and mica, it is more natural for the perovskite to

Figure 2. Structural analysis of perovskite (PVK) thin films growing on mica. (a) Schematic drawings of the “peeling-off” TEM sample preparationtechnique of as-grown PVK thin films. (b) Optical microscope image of PVK/mica sample on the copper grid. (c) Magnified optical image of thearea marked by dashed box in (b). (d) Low magnification TEM image of a square sheet marked in (c), with the inset showing the magnified opticalimage. (e−f) Electron diffraction patterns of separate mica (hexagonal) and PVK (cubic) with zone axis both as [001]; (f) was taken from FigureS5a. (g) Electron diffraction pattern taken at the dashed circle in (d) containing both mica and PVK lattices revealing the epitaxial relation, where anapproximate 5° offset in the mica/PVK (200) direction is present. (h−j) Atomistic model of mica/PVK epitaxy from a perspective (h), normal (i),and side (j) view. Legends at the bottom mark the atom species. (Scale bar: (b) 50 μm, (c) 10 μm, (d) 1 μm (inset: 2 μm)).

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be grown on mica with a perovskite {111} surface orientedsince it is hexagonal and should fit better with the substrate; (3)additional strong evidence that may rule out chemical epitaxy isthe absence of misfit dislocations for thick films, a mostcommon way to relax the strain created by the substrate andfilm; (4) with the chemical epitaxy excluded and the epitaxialrelation still observed, based on the layered nature of mica,VDW epitaxy remains a very feasible explanation; (5) the factthat we could accidentally peel off perovskite from mica alsoindicates the existence of a weak VDW force between mica andperovskites.

3. DISCUSSIONS AND MODELINGTo understand the growth of the ultrathin non-VDWperovskite sheets on mica by VDW mechanism, a quantitativeVDW nucleation and growth model is proposed. The mostabrupt difference between conventional epitaxy and VDWepitaxy is the chemical activity of the substrate surfaceconventional epitaxy involves highly active substrate surface,while VDW substrate is almost inert. In a nucleation process,adatoms on the substrate surface bond with both the substratecharacterized by adsorption energy, Ead, and other adatomscharacterized by interatomic bonding energy, Ei, to obtainsufficient energy so as to form a nucleus that can continue togrow. Figure 3a schematically describes the atomistic nucleationprocess in a conventional and a VDW epitaxy process. In bothcases the nutrient atoms get adsorbed and diffuse on thesubstrate surface. The strong chemical bond between adatom

and substrate in the conventional case leads to a quite smallcritical nucleus (down to single atom); the deep bonding stateand the high antibonding position in this case result in a largediffusion barrier (Ed), while for the VDW case, the inertsubstrate is free from dangling bonds and only small dipolemovements exist. Thus, Ead is typically characterized by theweak VDW energy. Consequently, the weak VDW Ead leads toa much harder nucleation process since more “simultaneous”collisions of adatoms on the substrate are needed. In addition,the higher energy VDW bonding state makes the diffusionbarrier extremely low, which is indeed confirmed by recentsimulation work in the GaAs-graphene material system.21

For a quantitative understanding on the VDW process, wepresent a modified nucleation and growth model. In anucleation process, adatoms with Ead will have a lifetime onthe substrate as28

τ = −v E kTexp( / )a1

ad (1)

where v is the vibration frequency and k the Boltzmannconstant. Multiplied with the incoming atom deposition rate, R,it gives the concentration of single adatom on the substrate, N1

= Rτa. Based on Walton’s model of thin film growth inconventional epitaxy,35 at low coverage, the concentration ofthe cluster with i atoms, ni, can be related with N1 and theinteratomic bonding energy Ei by

Figure 3. van der Waals epitaxial nucleation and growth model of perovskite thin film. (a) Schematic drawing that shows different atomisticprocesses on a conventional (left) and VDW (right) substrate in terms of adsorption energy (Ead), diffusion barrier (Ed), and nuclei sizes. (b) Theplot shows how coverage of a cluster with a certain size i changes with Ead (inset shows a magnified version of Ead at 0.5−0.7 eV). Each color linereflects a different cluster size, as shown by the corresponding numbers on the right. The ball-and-stick models demonstrate the optimizedconfiguration of each cluster, with the atom legends the same as Figure 2. (c) Plot of ΔGi vs i at Ead = 0.77, 0.63 eV, showing the critical size, i*,identification process. (d) Plot of nuclei density versus Ead under capturing rate-limited (solid line) and re-evaporation rate-limited (dashed line)assumptions. Change of i* is shown by different colored lines corresponding to (b). (e,f) Optical images showing single nucleus (e) and multiplenuclei (f), with colored arrows indicating different aligning orientations. (g) Illustration that the multiple nuclei in (f) hold the same epitaxial relationobserved in Figure 2 through the crystal symmetry and 5° offset. (h) Schematic drawing of how Ead changes with thickness for three types ofmaterials. (i) Schematic drawing of the disc (green) model used to estimate the lateral growth rate and the yellow region outlining the adatom-capturing area. (j) 3D surface plots showing how deposition rate (R) and disk size (r) determine the ratio of vertical nucleation (un) and lateralgrowth (vl) with three typical Ead2 values, (k) Plot of vl and un versus R at a fixed disc size (r = 50 nm) at the same Ead2 values as in (j). (Scale bar: (e)50 μm (f) 20 μm.)

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τ= = =⎜ ⎟ ⎜ ⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝

⎞⎠

⎛⎝⎜

⎞⎠⎟

⎛⎝

⎞⎠n

NN

NN

EkT

RN

EkT

exp expii

ii a

ii

0

1

0 0 (2)

where N0 is the density of adsorptions sites available on thesubstrate. The formula contains a statistical term, (Rτa/N0)

i,that relates the probability of finding i atoms holding together,and a thermodynamics term, exp(Ei/kT), that depicts theenergy benefit of the cluster, both of which make the modelintuitively sound. On the other hand, classic nucleationtheory36 gives a similar form of ni in terms of N1 and theGibbs free energy change, ΔGi, when a cluster of i atoms isformed

τ=

−Δ⎛⎝⎜

⎞⎠⎟n

RN

GkT

expia i

0 (3)

The free energy term takes into account the energy benefitfrom vapor into solid as well as the energy penalty resultedfrom the creation of free surfaces in the shape of either a cap ora disk. Therefore, the cluster with the highest ΔGi would be themost unstable one, and thus the critical nucleus size i*, i.e.,upon capturing a new atom, the nucleation barrier ΔGi* wouldbe overcome and a stable nucleus formed. Lewis has provedearlier37 that the two modelsWalton’s and the classic oneare equivalent with the former being more suitable at a smallnumber of i since it is hard to really form a free surface withvery few atoms. Therefore, mathematically one could get ΔGi*and the critical nucleus size using Walton’s model, i.e., thecluster with the lowest coverage.In the case of the perovskite halide, from the enthalpy of

sublimation data of PbCl2 and MACl, the bond energies of Pb-Cl and MA-Cl are estimated to be 0.515 and 0.298 eV,respectively (Supporting Information Supplementary Text 1).Therefore, Ei could be acquired by maximizing the energybenefit at a given cluster size. Figure 3b plots how the coverageof different sized clusters (1−7 illustrated on the right side ofFigure 3b) differs under different adsorption energy at a typicalatomic vibration frequency 1012 s−1, a moderate deposition rateR = 1 monolayer (ML)/min and T = 500 K (the temperatureof the substrate where most perovskite sheets are found). Forsimplicity concern, we assume the same Ead for all three kindsof adatoms on the surface. The ball-and-stick models on theright of Figure 3b show the optimal configuration of everycluster. According to Figure 3b, the coverage decreasesexponentially as Ead decreases as a general trend for all i. At agiven Ead, the cluster with the lowest coverage characterizes thecritical size. When Ead > 1, corresponding to a conventionalepitaxy case, i* = 1 and nucleation proceeds easily (with acoverage value above 10−5. At low Ead value such as 0.5 eVcorresponding to a typical VDW adsorption,38 we obtain acritical nucleus size of 6 and a coverage of ∼10−25, indicatingalmost impossible nucleation event (adsorption site density is∼1019/m2). Clearly, the low VDW interaction allows a largerkinetic window (or dynamic range) for the possible growth oflarge-scale single crystalline 2D sheet due to its extremelydifficult nucleation event. Figure 3c plots the change of ΔGiwith i framed in a classic model converted from Walton’s modelin Figure 3b. Two Ead values of 0.77 and 0.63 eV areexemplified to further illustrate the identification process of i*.Further, the explicit nucleation rate, U, is determined. The

rate is proportional to the density of the critical nuclei, Ni*,which is one atom away from a stable one, and also the numberof “free” single adatoms, N1. Therefore, U is given by28

τ= = =* *UNt

DN N DR Ndd i i1 (4)

where D = (1/4)a2v exp(−Ed/kT) is the surface diffusivity ofadatoms on the substrate. The lifetime τ could be either limitedby the re-evaporation of adatoms, which makes τ follow thesame form of eq 1 when the density of stable nuclei is small, orlimited by being captured by stable nuclei before adatoms re-evaporate, which leads the lifetime to be τc = 1/DN,39 where Nis the overall density of nuclei. Integration of eq 4 yields thedensity of nuclei after a certain deposition time t

τ= ‐ ‐*N DR N t(re evaporation rate limited)a i (5)

= ‐*N RN t2 (nuclei capturing rate limited)i (6)

Nucleation rates under both scenarios are plotted against Ea inFigure 3d with t = 1 s. It is clear in both cases that the nucleidensity experiences an exponential decrease with Ead, with theadditional drop in slope (log scale) resulting from the change inthe critical nucleus size, as marked by different colors on thegraph. The quantitative results above clearly show how thechange in adsorption energy dramatically affects the nucleationprocess, which is the fundamental reason VDW epitaxy is aunique process differing from conventional epitaxy. Notably ataround Ead = 0.5 eV (the approximate value of a single bond), arough estimate of the adsorption energy in the VDW case,38

only one nucleus may be expected to exist in a reasonably largearea of 0.2 cm2. This is indeed confirmed in our experiment. Asshown in Figure 3e, despite being separate square sheets, thefact that their parallel edges do not follow the sixfold symmetryof mica is a strong indication that all of them started with onlyone nucleus; whereas by increasing the supersaturation via fastsubstrate cooling, in which case from eq 1 τa increasesexponentially and so does the single adatom concentration, weobtained square sheets with obvious different orientations,suggesting multiple nuclei, as being marked by coordinates withdifferent colors in Figure 3f. The way to determine whether it issingle nucleus or multiple nuclei is as follows: statistically, formultiple nuclei, we would not observe the growth in Figure 3esince all sixfold orientations of mica have equal probability fornucleation and the square films should align along severaldifferent orientations rather than only one orientation;similarly, epitaxial but different orientations of square films inFigure 3f rule out the possibility of a single nucleus. The samephenomenon can be found in the dendritic morphology (seeSupporting Information Figure S8a and b). Despite this, theepitaxial relation found in Figure 2 still persists in Figure 3f.Figure 3g illustrates how the sixfold symmetry of mica and thefourfold symmetry of perovskite can account for the seeminglydisordered but indeed epitaxial growth in Figure 3f, where the5° offset is taken into account. In other words, while preservingthe 5° offset (which can take place in two directions as shownin the arrows in Figure 3f), the perovskite nucleus can beoriented in 6 equiv directions corresponding to the symmetryof pseudo hexagonal mica, thus giving rise to the differentpossible orientations among different nuclei when thenucleation process is encouraged. To further confirm ouranalysis and compare the conventional epitaxy and VDW one,we conducted perovskite growth on different types of substratesincluding silicon with very active surface, graphene on silicon,graphene on mica, and non-VDW material-seeded mica (seeFigure S9a−f and the corresponding discussion in SupportingInformation). PbCl2 with stronger chemical bonds on average

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than the perovskite halide was also grown on mica (see FigureS13f and the corresponding discussion in SupportingInformation). The observed results show great agreementwith our kinetic analysis. Meanwhile, it is also noticeable thatPbCl2 is formed with a complete different morphology and at adifferent location (4−5 cm away from the furnace) comparedwith the square sheets. This observation helps to rule out anypossibility of nonuniformity in the composition of the as-grownperovskite thin film such as incorporation of PbCl2.So far, the analysis has been limited to the kinetic process of

the first layer, but what parameters dictate the evolution of 2Dfilms along the vertical direction? Assuming films grow alongvertical directions via layer-by-layer growth mechanism, thesecond layer growth would require a new round of nucleationwhich is again affected by the adsorption energy, only on adifferent “substrate” first layer. As schematically shown inFigure 3h, for intrinsically layered VDW materials, due to theirweakest VDW-type adsorption energy, the nucleation processof the second layer would be as strenuous as the first layer.Adsorption energy of 3D material, on the other hand, wouldrecover to their “normal” values as the film thickness increases.The recovery process would be very fast for localized-bondmaterials (e.g., covalent) that do not have significant long-rangeinteraction40 but slower for the delocalized-bond materials (e.g.,ionic and metallic materials) in which long-range interactionsare substantially important.41,42 The relatively smaller adsorp-tion energy of the thinner delocalized-bond materials wouldhelp suppress the nucleation rate along the vertical direction.Therefore, thinner structures of delocalized-bond materials aremore likely to promote the 2D growth.

To reveal the feasibility of atomically thin 2D film via VDWepitaxial growth, we further quantify the ratio of the nucleationrate of the second layer, un, and the lateral growth speed, vl,since monolayer film can be obtained as long as un is curbed.The mean free path of an adatom on the surface, i.e., themagnitude of the random walk it experiences before re-evaporation, is given by L = (Dτa)

1/2 = 0.5a0·exp((Ead − Ed)/2kT) (a0 is the single hop distance of adatom on substrate). Forsimplicity, we assume the first layer to be a disc with radius rand therefore the area where adatoms could potentiallyincorporate into the lattice will be the ring region around thedisc with a width of L, the dark yellow region marked in Figure3i. The nucleation event may occur on the already-formed disc.With further derivation (Supporting Information Supplemen-tary Text 2), the lateral growth rate is given by

= = +⎜ ⎟⎛⎝

⎞⎠v r t

RLN

Lr

d /d2

12

16l

0 (7)

The nucleation rate follows the same analysis as the previousone but with a different adsorption energy, Ead2, and the area ofthe disc is used to yield the exact number of nuclei instead ofthe density value. Figure 3j shows the 3D surface plots of howln(un/vl) varies with the deposition rate R and the size of thedisc r (un and vl are normalized to be the same dimension here)for three types of materials. Ead2 values of 1.5, 0.7, and 0.4 eVare used, which to some extent represent the rigid covalentbond, weak delocalized ionic bond, and VDW bond,respectively. In general, a lower Ead2 retards the nucleationprocess and thus a smaller un/vl ratio is obtained, making itmore feasible to grow a monolayer film. For all three cases, at afixed size, vl has a linear dependence on the deposition rate,

Figure 4. Fractal evolution of perovskite thin films in van der Waals epitaxy. (a−e) Morphology evolution of (a) a square sheet with {100} facetsthrough (b−d) protrusion along the ⟨110⟩ direction to (e) fractal dendrite. (f) Schematic drawing hypothesizing that the corner vertices of a squaresheet have higher adatom-capture rate than the inner region of the square edges. (g) Plot of capturing probability (P) versus edge sites from cornerto center of a square perovskite seed with Monte Carlo simulation under L = 0.1a (blue line) and L = a (red line). (h,i) 3D bar chart showing MonteCarlo simulation result for a square (h) and fractal (i) seed under L = 0.1a. Insets: (magnified) normal view of the 3D bar chart. (j,k) Morphologyevolution of perovskite films at room temperature within four months. Samples at locations 1−4 evolved from (j) 2D cone-like structure to (k)“stair-like” zigzag structure and film at location 5 disappeared; and (l,m) as-deposited perovskite square sheets (separated in (l) and semiconnectedin (m)) showing obvious lining orientations marked by the blue dashed lines. (Scale bar: (a) 10 μm, (b) 5 μm, (c) 20 μm, (d) 10 μm, (e) 20 μm, (j−m) 50 μm).

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while un is proportional to Ri*+1. Therefore, mathematically atsufficiently low R, (un/vl) ≪ 1 would always hold andmonolayer thickness could always be achieved. Such a relationis explicitly shown in Figure 3k with r = 50 nm. Therefore,according to the above analysis, among all the non-VDWmaterial systems discussed, weak delocalized-bonded 3Dmaterials would mostly favor the 2D growth via the VDWmechanism. On the other hand, it is hard, but not impossible,to obtain 2D growth for materials with higher cohesive energysince nucleation on the second layer tends to take place veryeasilybut still can be curbed as long as supersaturation iswell-controlled.Based on the above analysis, for the perovskite halide with

extremely low cohesive energy and strong ionic character, asignificant percentage of large-scale monolayer growth withproper growth conditions is expected. However, experimentally,the thinnest films we could find are always around 8 nm. Tounderstand this, further analysis assisted with static and kineticMonte Carlo simulations on the growth kinetics of fractalmorphology is conducted.Figure 4a−e presents five types of typical growth

morphologies of the perovskite halide obtained at differentlocations of substrates during several experimental attempts,with the scale bar adjusted to render a better visualization of themorphology transformation. Figure 4a shows the squaremorphology with four side facets being {100}, which followsWulff construction rule at the 2D space. For all other cases inFigure 4b−e, the film propagates along the ⟨110⟩ direction in afractal manner rather than expanding along the ⟨100⟩ directionby following the surface energy argument. Apparently, thefractal morphologies are a consequence of kinetics. The ratecompetition is proposed to stem from adatoms’ VDW-typediffusion and perovskite films’ anisotropic surface free energy.We attribute the ⟨110⟩ fractal growth direction to be aconsequence of the anisotropy in the adatom-capturingcapability of different sites at the film edges. At the initialstage of film growth, surface energy dominates and a squareshape should be expected since it is easy to achieve localequilibrium. With a square shape, the region where adatomscould potentially incorporate into the matrix is marked by boththe yellow and blue colors in Figure 4f. The outside border ofthe blue and yellow regions is formulated by the diffusionlength (L) of the adatoms. According to random walk theory,43

the probability that the adatom terminates at a certain vector rfrom the original point after random walk scales with∼exp(−| r |2/4Dt); therefore, the edge site that is closest tothe deposited atom has the highest probability to capture it.Consequently, the corner sites of the square film would havehigher probability to be incorporated by the adatom. A staticMonte Carlo simulation was carried out to prove ourhypothesis in which atoms, randomly deposited around asquare film with the side-length a, conduct random walk untilthey are captured by the matrix or a certain number of jumps isreached. Figure 4g plots the simulation results on capturingprobability (i.e., the number of atoms captured at one sitedivided by the total number of atoms captured) at differentsites under different L/a ratios. It can be seen that at small L/aratio (e.g., 0.1) a relative uniform capturing probabilitydistribution along the corner vertex to the middle of thesquare side is obtained, which is a result of the minimized arearatio of blue region over yellow one. For a larger L/a value(e.g., 1), higher capturing probability at the corner is observed.In this case, our analysis is mainly based on the assumption that

surface diffusion is much faster than crystal edge diffusion. Suchan assumption requires a considerable size of diffusion path “a”.It should be noted that when a is extremely small, i.e., at theinitial stage of the growth, the Wulff ripening process woulddominate during crystal growth as diffusions along the crystaledges become much easier. This would lead to athermodynamically stable structure rather than the kineticfractal morphology. This also explains why the fractalmorphologies are always of a larger size. A more visualizedillustration of the anisotropy in the capturing rate of differentedge sites is shown as a 3D chart in Figure 4h, with the insetbeing the normal view. Further, we simulated the adatomcapturing rate of different edge sites of a fractal morphology inFigure 4b, as shown in the 3D chart of Figure 4i. Clearly, allcorner vertices have higher capture rates than nonvertex sites;inner corner sites have slightly lower capturing rates thanoutmost corners. Higher capture rates indicate faster growth.Therefore, the corner preference serves as the morphologyinstability source that induces fractal growth. Meanwhile, localequilibrium favors the formation of {100} facets in perovskitefilm, and therefore often fractal growth morphologies with well-faceted local structures (i.e., “stair-like” zigzag contours) areobserved. The large-scale local equilibrium occurs more easilyin VDW epitaxy process as the VDW-type diffusivity is ordershigher in magnitude than the one in conventional epitaxy. Tofurther support our argument, kinetic Monte Carlo simulationis conducted that precisely predicts the well-faceted fractalmorphology by including both anisotropic capturing andequilibrating processes (see videos 1 and 2). Such fractalgrowth over a large area is different from earlier observation ofangstrom scale fractal growth of metal where diffusion isextremely limited,44,45 but often found in the formation ofsnowflakes, where the diffusivity of water molecules is high aswell.46

The high VDW-type diffusivity (surface diffusivity and edgediffusivity) and low cohesive energy of perovskites are furthersupported by more evidence. Figure 4j,k shows the morphologychange of the same sample across 4 weeks at room temperatureand ambient atmosphere. By aging, thinner films at location 5in Figure 4j sacrifice themselves (5′ in Figure 4k) andparticipate in the construction of the fractal structure of thickerfilms (1−4 in Figure 4j and 1′−4′ in (k)). Further, a few dayslater the small fractal structure at 4′ transformed to a squareone, possibly due to the ease in reaching local equilibrium(Supporting Information Figure S10). A similar change wasobserved universally where a sharp discrepancy in thicknessexisted (see Supporting Information Figure S11a−d). A livevideo was also recorded to show rapid real-time change ofmorphology overnight at room temperature (see Video 3).Accordingly, uniform and separate square films (Figure 1a,b,Figure 3e, and Figure 4i) with identical crystallographicorientation could be regarded as a consequence of the ultrahighVDW diffusivity during material growth where substratetemperature is high (220 °C). In other words, the sacrifice ofthinner films and construction of thicker films proceed muchfaster during material growth, which could precisely explainwhy many crystals share a single nucleus and these crystallinesheets line up along the ⟨110⟩ direction (blue arrows in Figure4i,m). More evidence on the manipulation of such a“relaxation” process could be found in Supporting InformationFigures S12, 13, and 14a−e). All this evidence illustrates thatthe absence of mono- or a few-layer 2D results (i.e., ≪8 nm) isvery likely caused by the high VDW-type diffusivity and low

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cohesive energy of perovskites, which lead to film thickeningduring and after material growth. Therefore, for VDWmonolayer growth of perovskite film, i.e., to prevent the filmthickening process, further strategies are required. Examples are(1) coating of second dielectric layer on top of perovskite layer,and (2) reducing growth and sample storing temperature.Indeed, the first strategy may lead to atomic artificial structurein a great analogy to layered superconducting cuperates,47

irridates,48 and nickelates.49 In fact, these two strategies havebeen found useful in preparing monolayer oxides films either inwet chemistry or in vacuum deposition growth.

4. CONCLUSIONIn summary, we show the successful growth of sub-10 nmperovskite halides on mica by VDW epitaxy mechanism. Ourproposed VDW nucleation and growth theory preciselyexplains that it is the weak VDW film−substrate interactionand low cohesive energy of the perovskite halide that lead tolarge-scale single crystalline ultrathin 2D growth. The modelalso illustrates that ionic and metallic crystals with weakdelocalized bonds tend favor 2D growth more than stronglycovalent materials. Finally, combined with Monte Carlosimulation, our model shows that the VDW-type diffusivityand low cohesive energy of perovskites determine the ultimatekinetic thickness of the perovskite films. Fractal morphology isa result of the competition between thermodynamic drivingforce and VDW diffusivity. Our findings reveal the possibility ofgrowing mono- and a few-layer 2D film via VDW epitaxy out ofnon-VDW 3D materials.9,50

■ ASSOCIATED CONTENT*S Supporting InformationThe Supporting Information is available free of charge on theACS Publications website at DOI: 10.1021/acs.cgd.5b00949.

Experimental setup, methods, materials, optical analysis,EDX and Monte Carlo simulation (PDF)Real-time morphology change video (AVI)Real-time morphology change video (AVI)Real-time morphology change video (AVI)

■ AUTHOR INFORMATIONCorresponding Author*E-mail: shij4@rpi.edu.NotesThe authors declare no competing financial interest.

■ ACKNOWLEDGMENTSJ. S. and Y. W. were supported by J.S.’s start-up fund fromRensselaer Polytechnic Institute and National ScienceFoundation under grant CMMI 1550941. J. L. acknowledgesfinancial support of the US National Science Foundation underthe award of DMR.

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