Soft Crystals in Flatland: Unraveling Epitaxial Growth Michael D. Ward* Department of Chemistry and the Molecular Design Institute, New York University, 100 Washington Square East, New York, New York 10003-6688, United States ABSTRACT: Thin film epitaxy typically invokes a superposition of a pair of rigid two-dimensional lattices with a well-defined orientation governed by some form of commensurism. A report by Meissner et al. in this issue of ACS Nano demonstrates that the organization of organic molecules on substrates may not be that simple, as static distortion waves involving miniscule shifts of atomic positions from substrate lattice points can lead to orientations of a molecular film that cannot be described by often used models. Herein, we provide some highlights of epitaxy, with a focus on configurations that reflect the delicate balance between intermolecular interactions within a molecular film and molecule−substrate interactions. Although geometric models for explaining and predicting epitaxial configurations can be used to guide synthesis of materials, their use must recognize energetic factors and the possibility of more complex, and possibly less predictable, interface structures. pitaxya word that derives from “epi” (upon) + “taxy” (order), likely originating from ordered formation of Greek soldiers marching on a battlefieldhas been a staple of materials synthesis for decades, as witnessed by the fabrication of epitaxial thin film devices ranging from diode lasers1 to spintronics.2 Stimulated by the versatility of organic synthesis and the promise of devices with properties that can be tuned through molecular design, the past few decades have witnessed the emergence of organic semiconductors, including thin films that promise new devices ranging from light-emitting diodes to field-effect transistors. This has provoked considerable interest in the structure and properties of molecular films and the role of epitaxy in their formation and organization on substrates. Molecular films, however, introduce significantly greater complexity. Molecules are large, oddly shaped, and their plane group symmetries tend to be low and unlike the substrates upon which they form. Consequently, simple descriptions of epitaxy, wherein a pair of primitive lattice vectors in the overlayer are aligned with a pair of primitive lattice vectors in the substrate and the misfit is calculated from the difference in the magnitude of the lattice vectors, usually do not apply. Interpreting the epitaxial configuration, let alone predicting it, can be thorny, made somewhat impenetrable by the use of (sometimes) interchangeable terms like quasiepitaxy,3,4 rotational epitaxy,5 van der Waals epitaxy, linearly commensurate structures,6 axially commensurate structures,7 coincident epitaxy, and higher order commensurism. Nonetheless, models and computationally efficient geometric matching algorithms have been devised to assist in the design and characterization of epitaxy,8−10 based on the presumption of correspondence between lattice positions and potential energy
surfaces11 that enable the interface to be described by a superposition of plane waves. It seemed reasonable to assume that these models would aptly describe molecular films in perpetuity. The report by Meissner et al. in this issue of ACS Nano12 has undermined that presumption and made the design of molecular films more interesting although probably more complicated. The presumptive models of epitaxy thought to describe molecular films can be illustrated using simple models, described nearly 15 years ago13 in an attempt to create a grammar of epitaxy in real space rather than reciprocal space, so it would be digestible to a broader audience. The epitaxial interface can be described fully by seven parameters, the lattice parameters of the substrate (a1, a2, and α), the lattice parameters of the overlayer (b1, b2, and β), and the azimuthal angle, θ, between the lattice vectors a1 and b1. The substrate and overlayer lattice vectors for a given azimuthal orientation θ are related through the four-element transformation matrix [C], in which the matrix coefficients describe the relationship between the lattice vectors of the substrate and overlayer.
E
© 2016 American Chemical Society
⎡ b1 ⎤ ⎡ a1 ⎤ ⎡ p q ⎤⎡ a1 ⎤ ⎢ ⎥ = [C]⎢ ⎥ = ⎢ ⎢ ⎥ ⎣ a 2 ⎦ ⎣ r s ⎥⎦⎣ a 2 ⎦ ⎢⎣b2 ⎥⎦ b1 sin(α − θ ) b sin(θ ) ; q= 1 ; a 2 sin(α) a 2 sin(α) b sin(α − θ − β) b sin(θ + β) r= 2 ; s = 12 a1 sin(α) a 2 sin(α)
p=
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Figure 1. (Top) Schematic representation of three possible modes of epitaxy that illustrate (left) a commensurate overlayer; (middle) a pointon-line (POL) coincident overlayer, illustrating two possible unit cell choices, and dashed lines representing real-space supercells whose vertices coincide with substrate lattice points; (right) an POL coincident overlayer in which every other row of overlayer lattice points sits between primitive substrate directions. The matrix [C] is denoted at the top of each panel. (Bottom, left) Comparison of the potential wells describing the intermolecular interactions between molecules within a molecular film (Eintra) and the molecule−substrate interactions for the case where the former are dominant. The elastic constants define the curvature of the potential wells. Coincidence is more likely if the potential well for Eintra is steep, corresponding to a large elastic constant, and the interface potential is shallow, corresponding to a smaller elastic constant for lateral repositioning of the overlayer molecules. Under this condition, a coincident configuration is preferable to expanding the molecular layer to achieve a commensurism. (Bottom, right) Simplified description of the dependence of the mode of epitaxy on the relative magnitude of the intralayer and interlayer elastic constants. Adapted with permission from ref 13. Copyright 2001 Wiley.
The matrix [C] can then be used to sort epitaxial configurations. The values of the matrix elements depend on the choice of the primitive lattice vectors describing the primitive unit cell. All primitive unit cells, however, must have equal area and therefore identical matrix determinants. Consequently, an overlayer unit cell can always be constructed with a reciprocal basis vector b1* that coincides with a substrate reciprocal basis vector a1* for the case of commensurism and certain kinds of coincidence. These vectors are related by b1* = ma1*, where m is an integer. This can be described straightforwardly in real space, however, as illustrated in Figure 1. When all the matrix elements are integers, the overlayer will be commensurate with the underlying substrate. If the matrix elements are all rational numbers but with only two integers, confined to one column, every lattice point of the overlayer will rest on at least one primitive lattice line of the substrate, a condition that can be described generally as point-online (POL) coincidence. In real space, this affords a supercell defined by the phase-coherent registry with the substrate, evident from the point-on-point coincidence at the corners of the supercell. If the supercell positions coinciding with substrate lattice points are considered energetically preferred, this condition implies that overlayer lattice points on the perimeter are less favorable though not necessarily repulsive. The choice for the overlayer is dictated by whether it is energetically preferable for the overlayer to remain intact and to rotate to an azimuthal angle to achieve coincidence or whether molecule− substrate interactions dominate, straining the overlayer to
achieve commensurism. Other modes of epitaxy can be defined; for example, a matrix [C] containing four rational numbers but with integers in different columns corresponds to a kind of POL coincidence but with every other row of overlayer lattice points residing between rows of substrate lattice points. Pointon-line coincidence has been implicated for organic overlayers too numerous to list here; for example, perylene tetracarboxylic dianhydride (PTCDA) on HOPG (Figure 2),14,15 thioindigo on HOPG and MoS2,16 and perylene on KCl.17 Epitaxy need not be limited to organic molecules on conventional highsymmetry substrates, although the superposition of two low-
Figure 2. (a) Scanning tunneling microscope image of a monolayer of PTCDA on HOPG revealing a 3 × 1 coincident supercell. The coincident relationship is depicted schematically in (b). The overlayer has lattice vectors b1 = 16/3a1 ± 1/3a2 and b2 = ±4a1 + 9a2, equivalent to an azimuthal rotation of θ = ± 3.2°. Adapted with permission from ref 14. Copyright 1992 Springer. 6425
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of triangles in which displacements of only 0.52 Å, on average, point away from the corners of the triangles. This distortion is a consequence of a gain in molecule−substrate energy accompanying strain in the HBC overlayer, which is permitted by some elasticity in the HBC overlayer. This translates to noninteger elements of the matrices for both the room temperature (RT) and low temperature (LT) phases, which with less precision could be mistaken for commensurate.
symmetry complex lattices can be expected to be more restrictive. Nonetheless, line-on-line (LOL) coincidence was reported for the organic−organic heterolayer system PTCDA on HBC on graphite.18 Line-on-line is similar to POL in that the overlayer molecules reside on parallel, equally spaced lines, but these lines are not constrained to primitive substrate lattice lines. Like many POL examples, the observed epitaxial ordering corresponded to a minimum in potential energy.
⎡ 5.147(5) −0.969(4)⎤ ⎥ [C]RT = ⎢ ⎢⎣ 0.969(4) 6.116(5) ⎥⎦
The report by Meissner et al. in this issue of ACS Nano has made the design of molecular films more interesting although probably more complicated
⎡ 5.113(5) −0.986(4)⎤ ⎥ [C]LT = ⎢ ⎢⎣ 0.985(1) 6.092(6) ⎥⎦ ⎡ 5 −1⎤ [C]commensurate = ⎢ ⎥ ⎣1 6 ⎦
So what did Meissner et al. do to perturb this quaint understanding? They built on classic literature by Novaco and McTague, who posited static distortion waves to describe incommensurate overlayers of argon atoms on a graphite surface.19 Novaco and McTague were prescient, stating “More subtle effects occur when adsorbate−adsorbate interactions dominate lateral variations in adsorbate− substrate forces. Then the spacing of the adsorbed species will be only perturbed to a greater or lesser degree, forming an incommensurate interfacial structure with periodic strains caused by the substrate field. We examine here some consequences of this distortion of the adsorbate structure and show that, although for infinite systems it is independent of relative translation of the two lattices, it does depend on their relative orientation. As a result, the lowest energy state can have a definite nonsymmetry relative angle of orientation.” Novaco and McTague were referring here to configurations that stabilize the overlayer−substrate system overall, recognizing that this could occur even for incommensurate arrangements, wherein “each adatom is statically displaced from its ideal lattice site, and these displacements vary sinusoidally across the monolayer.” Fast forward to Meissner et al., who used a combination of ultraprecise low-energy electron diffraction, scanning tunneling microscopy (STM), and density functional theory to reveal the existence of static distortion waves (SDWs) for overlayers of the organic molecule hexa-perihexabenzocoronene (HBC), a kind of “nanographene”. Though other overlayers have been suggested to conform to the SDW model because local distortions were observed, such as rubrene on Bi(001),20 a sinusoidal incommensurism across the entire substrate can be difficult to ascertain. In the case of HBC on graphite, the SDW structure was deduced from a Moiré pattern
OUTLOOK AND CHALLENGES The findings of Meissner et al. are fundamentally interesting from the perspective of understanding intermolecular forces and substrate-directed organization. Regulating the structure of organic thin films is essential for new technologies, although most applications will rely on polycrystalline films devoid of epitaxial registry with a substrate. There is a resurgence of interest in the use of epitaxy for materials synthesis, however. For example, although a “hard material”, Bi2Se3 films grow on epitaxial graphene/SiC(0001) by “van der Waals epitaxy”, initiated through two-dimensional (2D) nucleation (Figure 3).21 Winding of the resulting growth fronts around the pinning centersstep edges of the substrate or by domain boundariesduring the coalescence of the 2D island created spirals that mimicked classical dislocation spirals. The origin of dislocations is not well understood generally, particularly for organic films and crystals, and the use of epitaxial substrates with nanomorphological features seems a reasonable strategy for spawning dislocations as well as synthesizing films with unique electronic properties. In the case of Bi2Se3, a strong topological insulator, STM measurements confirmed that the one-dimensional helical mode of a line defect was not supported. In a separate report, current enhancement at step edges of Bi2Se3 compared with terraces measured by conductive probe atomic force microscopy was attributed to spin−orbit coupling and topological effects.22 Principles of epitaxy are increasingly being explored for the synthesis of even a broader range of materials, including bulk
Figure 3. (a) Scanning tunneling microscope image of a Bi2Se3 film grown on epitaxial graphene/SiC acquired at room temperature, showing the formation of spirals. (b) View of a spiral and line profile across AB. (c) Atomic resolution image of the spiral core acquired at 78 K, showing a step originated from the spiral core in the center of the image. Adapted with permission from ref 21. Copyright 2012 American Physical Society. 6426
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crystals. The structure of the first epitaxial layer can be deterministic in the formation of bulk organic crystals grown on a substrate. In 1998, our laboratory demonstrated this result for the organic superconductor bis(ethylenedithiolo)tetrathiafulvalene triiodide, wherein the orientation and structure of a coincident precursor film on graphite dictated the orientation and polymorph of bulk crystals grown from the first layer.23 In 2001, we reported a geometric lattice analysis program dubbed GRACE (geometric real-space analysis of crystal epitaxy) that generates a ranking of lattice matches between a selected range of crystal planes for crystal polymorphs, with user input of a substrate.24,25 We used GRACE to select a substrate for nucleation of a particular polymorph of 5-methyl-2-[(2-nitrophenyl)amino]-3-thiophenecarbonitrile, otherwise known at “ROY” for the red−orange− yellow colors of its many polymorphs. Recently, GRACE was used to deduce that the (001) face of crystalline δ-pyrazinamide served as a heteroepitaxial template for the nucleation of new metastable polymorphs of a sulfonamide.26 Heteroepitaxy on organic crystal substrates, with surfaces that are inherently more complex than conventional van der Waals substrates, will require analysis of the contributions from intermolecular interactions with surface functional groups and surface topography, as well as crystal lattice matching.27 Unraveling these factors, coupled with efficient database search algorithms for selecting the proper combinations of substrate and epitaxial alignment, could lead to more rigorous control of thin film properties and crystal polymorphism, impacting electronic materials and pharmaceuticals.
polymorphs through epitaxy. The incorporation of computed strain energy, though built into previous geometric models by identifying coincident lattice matches within designated strain constants,8−10 is a valuable feature, particularly given the observation by Meissner et al. that overlayer structures may depart from conventional coincident models. Needless to say, the use of epitaxy as a synthetic tool is likely to grow, and efforts should be made to create accessible epitaxy programs and a common database of epitaxial systems.
AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
ACKNOWLEDGMENTS The author gratefully acknowledges support from the MRSEC Program of the National Science Foundation under Award Number DMR-1420073. REFERENCES (1) Nakamura, S. The Roles of Structural Imperfections in InGaNBased Blue Light−Emitting Diodes and Laser Diodes. Science 1998, 281, 956−961. (2) Hirohata, A.; Sagar, J.; Fleet, L. R.; Parkin, S. S. P. Heusler Alloy Films for Spintronic Devices. Springer Ser. Mater. Sci. 2016, 222, 219− 248. (3) So, F. F.; Forrest, S. R.; Shi, Y. Q.; Steier, W. H. Quasi-Epitaxial Growth of Organic Multiple Quantum Well Structures by Organic Molecular Beam Deposition. Appl. Phys. Lett. 1990, 56, 674. (4) Forrest, S. R. Ultrathin Organic Films Grown by Organic Molecular Beam Deposition and Related Techniques. Chem. Rev. 1997, 97, 1793−1896. (5) Ashida, M. The Orientation Overgrowth of Metal-phthalocyanines on the Surface of Single Crystals. II. Vacuum-condensed Films of Copper-phthalocyanine on Alkali Halides. Bull. Chem. Soc. Jpn. 1966, 39, 2632−2638. (6) Grey, F.; Bohr, J. An Alternative Explanation for Epitaxial Growth: The Case of fcc(111) on bcc(110). Appl. Surf. Sci. 1993, 65, 35−40. (7) Paik, S. M.; Schuller, I. K. New Calculational Method for Epitaxial Energy: Application to an Axial Commensurate Interface. Phys. Rev. Lett. 1990, 64, 1923−1926. (8) Zur, A.; McGill, T. Lattice Match: An Application to Heteroepitaxy. J. Appl. Phys. 1984, 55, 378−386. (9) Hillier, A. C.; Ward, M. D. Epitaxial Interactions between Organic Overlayers and Ordered Substrates. Phys. Rev. B: Condens. Matter Mater. Phys. 1996, 54, 14037−14051. (10) Epicalc geometric lattice match program available for download at http://wardresearchgroup.com/ResearchSoftware. (11) Last, J. A.; Hooks, D. E.; Hillier, A. C.; Ward, M. D. The Physicochemical Origins of Coincident Epitaxy in Molecular Overlayers: Lattice Modeling vs. Potential Energy Calculations. J. Phys. Chem. B 1999, 103, 6723. (12) Meissner, M.; Sojka, F.; Matthes, L.; Bechstedt, F.; Feng, X.; Müllen, K.; Mannsfeld, S. C. B.; Forker, R.; Fritz, T. Flexible 2D Crystals of Polycyclic Aromatics Stabilized by Static Distortion Waves. ACS Nano 2016, DOI: 10.1021/acsnano.6b00935. (13) Hooks, D. E.; Fritz, T.; Ward, M. D. Epitaxy and Molecular Organization on Solid Substrates. Adv. Mater. 2001, 13, 227−241. (14) Ludwig, C.; Gompf, B.; Glatz, W.; Petersen, J.; Eisenmenger, W.; Möbus, M.; Zimmermann, U.; Karl, N. Video-STM, LEED and XRay Diffraction Investigations of PTCDA on Graphite. Z. Phys. B: Condens. Matter 1992, 86, 397−404.
Principles of epitaxy are increasingly being explored for the synthesis of even a broader range of materials, including bulk crystals. Recently, a new computational framework for the selection of optimal substrates for epitaxial growth using first-principles calculations of formation energies, elastic strain energy, and topological information was introduced (Figure 4).28 Using
Figure 4. Conceptual framework for an epitaxy database search routine. Reproduced from ref 28. Copyright 2016 American Chemical Society.
epitaxial growth of metastable VO2 compounds on TiO2 substrates, the model incorporated a geometric unit cell area matching between the substrate and the target compound, as well as strain energy density. Qualitative agreement with experimental observations for the epitaxial growth of known VO2 polymorphs was realized, promising the use of this approach for guidance on experimental efforts aimed at forming computationally predicted, but elusive, new materials or 6427
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(15) Ludwig, C.; Gompf, B.; Petersen, J.; Strohmaier, R.; Eisenmenger, W. Z. STM Investigations of PTCDA and PTCDI on Graphite and MoS2. A Systematic Study of Epitaxy and STM Image Contrast. Z. Phys. B: Condens. Matter 1994, 93, 365−373. (16) Petersen, J.; Strohmaier, R.; Gompf, B.; Eisenmenger, W. Monolayers of Tetrachloro-Thioindigo and Thioindigo in the STM: Orientational Disorder and the Absence of Photochromism. Surf. Sci. 1997, 389, 329−337. (17) Fryer, J. R.; Ewins, C. Epitaxial Growth of Thin Films of Perylene. Philos. Mag. A 1992, 66, 889−898. (18) Mannsfeld, S. C. B.; Leo, K.; Fritz, T. Line-on-Line Coincidence: A New Type of Epitaxy Found in Organic-Organic Heterolayers. Phys. Rev. Lett. 2005, 94, 056104. (19) McTague, J. P.; Novaco, A. D. Substrate-Induced Strain and Orientational Ordering in Adsorbed Monolayers. Phys. Rev. B: Condens. Matter Mater. Phys. 1979, 19, 5299−5306. (20) Wang, J.-Z.; Lan, M.; Shao, T.-N.; Li, G.-Q.; Zhang, Y.; Huang, C.-Z.; Xiong, Z.-H.; Ma, X.-C.; Jia, J.-F.; Xue, Q.-K. STM Study of a Rubrene Monolayer on Bi(001): Structural Modulations. Phys. Rev. B: Condens. Matter Mater. Phys. 2011, 83, 235433. (21) Liu, Y.; Weinert, M.; Li, L. Spiral Growth without Dislocations: Molecular Beam Epitaxy of the Topological Insulator Bi2Se3 on Epitaxial Graphene/SiC(0001). Phys. Rev. Lett. 2012, 108, 115501. (22) Macedo, R. J.; Harrison, S. E.; Dorofeeva, T. S.; Harris, J. S.; Kiehl, R. A. Nanoscale Probing of Local Electrical Characteristics on MBE-Grown Bi2Te3 Surfaces under Ambient Conditions. Nano Lett. 2015, 15, 4241−4247. (23) Last, J. A.; Hillier, A. C.; Hooks, D. E.; Maxson, J. B.; Ward, M. D. Epitaxially-Driven Assembly of Crystalline Molecular Films on Ordered Substrates. Chem. Mater. 1998, 10, 422−437. (24) Mitchell, C. A.; Yu, L.; Ward, M. D. Selective Nucleation and Discovery of Organic Polymorphs through Epitaxy with Single Crystal Substrates. J. Am. Chem. Soc. 2001, 123, 10830−10839. (25) GRACE geometric lattice match program available for download at http://wardresearchgroup.com/ResearchSoftware. (26) Gawade, R. L.; Chakravarty, D. K.; Kotmale, A.; Sangtani, E.; Joshi, P. V.; Ahmed, A.; Mane, M. V.; Das, S.; Vanka, K.; Rajamohanan, P. R.; Puranik, V. G.; Gonnade, R. G. Additive Mediated Syn-Anti Conformational Tuning at Nucleation to Capture Elusive Polymorphs: Remarkable Role of Extended π-Stacking Interactions in Driving the Self-Assembly. Cryst. Growth Des. 2016, 16, 2416−2428. (27) Chadwick, K.; Chen, J.; Myerson, A. S.; Trout, B. L. Toward the Rational Design of Crystalline Surfaces for Heteroepitaxy: Role of Molecular Functionality. Cryst. Growth Des. 2012, 12, 1159−1166. (28) Ding, H.; Dwaraknath, S. S.; Garten, L.; Ndione, P.; Ginley, D.; Persson, K. A. Computational Approach for Epitaxial Polymorph Stabilization through Substrate Selection. ACS Appl. Mater. Interfaces 2016, 8, 13086−13093.
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