{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,31]],"date-time":"2025-12-31T03:45:45Z","timestamp":1767152745608,"version":"build-2238731810"},"reference-count":37,"publisher":"Springer Science and Business Media LLC","issue":"11","license":[{"start":{"date-parts":[[2022,11,28]],"date-time":"2022-11-28T00:00:00Z","timestamp":1669593600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2022,11,28]],"date-time":"2022-11-28T00:00:00Z","timestamp":1669593600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Nat Comput Sci"],"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    The equilibrium shape of crystals is a fundamental property of both aesthetic appeal and practical importance: the shape and its facets control the catalytic, light-emitting, sensing, magnetic and plasmonic behaviors. It is also a visible macro-manifestation of the underlying atomic-scale forces and chemical makeup, most conspicuous in two-dimensional (2D) materials of keen current interest. If the crystal surface\/edge energy is known for different directions, its shape can be obtained by the geometric Wulff construction, a tenet of crystal physics; however, if symmetry is lacking, the crystal edge energy cannot be defined or calculated and thus its shape becomes elusive, presenting an insurmountable problem for theory. Here we show how one can proceed with auxiliary edge energies towards a constructive prediction, through well-planned computations, of a unique crystal shape. We demonstrate it for challenging materials such as SnSe, which is of\n                    <jats:italic>C<\/jats:italic>\n                    <jats:sub>2v<\/jats:sub>\n                    symmetry, and even AgNO\n                    <jats:sub>2<\/jats:sub>\n                    of\n                    <jats:italic>C<\/jats:italic>\n                    <jats:sub>1<\/jats:sub>\n                    , which has no symmetry at all.\n                  <\/jats:p>","DOI":"10.1038\/s43588-022-00347-5","type":"journal-article","created":{"date-parts":[[2022,11,28]],"date-time":"2022-11-28T11:02:44Z","timestamp":1669633364000},"page":"729-735","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Defining shapes of two-dimensional crystals with undefinable edge energies"],"prefix":"10.1038","volume":"2","author":[{"given":"Luqing","family":"Wang","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Sharmila N.","family":"Shirodkar","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6406-0959","authenticated-orcid":false,"given":"Zhuhua","family":"Zhang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8369-3567","authenticated-orcid":false,"given":"Boris I.","family":"Yakobson","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2022,11,28]]},"reference":[{"key":"347_CR1","first-page":"108","volume":"3","author":"JW Gibbs","year":"1874","unstructured":"Gibbs, J. W. On the equilibrium of heterogeneous substances. Trans. Conn. Acad. Arts Sci. 3, 108\u2013248 (1874).","journal-title":"Trans. Conn. Acad. Arts Sci."},{"key":"347_CR2","unstructured":"Landau, L. D. & Lifshitz, E. M. Statistical Physics 3rd edn, 155 (Pergamon, 1980)."},{"key":"347_CR3","first-page":"449","volume":"34","author":"G Wulff","year":"1901","unstructured":"Wulff, G. Zur frage der geschwindigkeit des wachsthums und der aufl\u00f6sung der krystallfl\u00e4chen. Z. Kryst. Mineral. 34, 449\u2013530 (1901).","journal-title":"Z. Kryst. Mineral."},{"key":"347_CR4","doi-asserted-by":"publisher","first-page":"87","DOI":"10.1103\/PhysRev.82.87","volume":"82","author":"C Herring","year":"1951","unstructured":"Herring, C. Some theorems on the free energies of crystal surfaces. Phys. Rev. 82, 87\u201393 (1951).","journal-title":"Phys. Rev."},{"key":"347_CR5","doi-asserted-by":"publisher","first-page":"15859","DOI":"10.1021\/jp401566m","volume":"117","author":"E Ringe","year":"2013","unstructured":"Ringe, E., Van Duyne, R. P. & Marks, L. D. Kinetic and thermodynamic modified Wulff constructions for twinned nanoparticles. J. Phys. Chem. C. 117, 15859\u201315870 (2013).","journal-title":"J. Phys. Chem. C."},{"key":"347_CR6","doi-asserted-by":"publisher","first-page":"699","DOI":"10.1038\/nnano.2012.193","volume":"7","author":"QH Wang","year":"2012","unstructured":"Wang, Q. H., Kalantar-Zadeh, K., Kis, A., Coleman, J. N. & Strano, M. S. Electronics and optoelectronics of two-dimensional transition metal dichalcogenides. Nat. Nanotechnol. 7, 699\u2013712 (2012).","journal-title":"Nat. Nanotechnol."},{"key":"347_CR7","doi-asserted-by":"publisher","first-page":"419","DOI":"10.1038\/nature12385","volume":"499","author":"AK Geim","year":"2013","unstructured":"Geim, A. K. & Grigorieva, I. V. Van der Waals heterostructures. Nature 499, 419\u2013425 (2013).","journal-title":"Nature"},{"key":"347_CR8","doi-asserted-by":"publisher","first-page":"355","DOI":"10.1038\/s41586-018-0008-3","volume":"556","author":"J Zhou","year":"2018","unstructured":"Zhou, J. et al. A library of atomically thin metal chalcogenides. Nature 556, 355\u2013359 (2018).","journal-title":"Nature"},{"key":"347_CR9","doi-asserted-by":"publisher","first-page":"16061","DOI":"10.1038\/natrevmats.2016.61","volume":"1","author":"A Carvalho","year":"2016","unstructured":"Carvalho, A. et al. Phosphorene: from theory to applications. Nat. Rev. Mater. 1, 16061 (2016).","journal-title":"Nat. Rev. Mater."},{"key":"347_CR10","doi-asserted-by":"publisher","first-page":"7281","DOI":"10.1103\/PhysRevB.57.7281","volume":"57","author":"K Rapcewicz","year":"1998","unstructured":"Rapcewicz, K., Chen, B., Yakobson, B. I. & Bernholc, J. Consistent methodology for calculating surface and interface energies. Phys. Rev. B 57, 7281\u20137291 (1998).","journal-title":"Phys. Rev. B"},{"key":"347_CR11","doi-asserted-by":"publisher","first-page":"3113","DOI":"10.1021\/nl2011142","volume":"11","author":"Y Liu","year":"2011","unstructured":"Liu, Y., Bhowmick, S. & Yakobson, B. I. BN white graphene with \u2018colorful\u2019 edges: the energies and morphology. Nano Lett. 11, 3113\u20133116 (2011).","journal-title":"Nano Lett."},{"key":"347_CR12","doi-asserted-by":"publisher","first-page":"4294","DOI":"10.1021\/jp5097713","volume":"119","author":"D Cao","year":"2015","unstructured":"Cao, D., Shen, T., Liang, P., Chen, X. & Shu, H. Role of chemical potential in flake shape and edge properties of monolayer MoS2. J. Phys. Chem. C. 119, 4294\u20134301 (2015).","journal-title":"J. Phys. Chem. C."},{"key":"347_CR13","doi-asserted-by":"publisher","first-page":"305","DOI":"10.1016\/0039-6028(75)90254-X","volume":"51","author":"E Arbel","year":"1975","unstructured":"Arbel, E. & Cahn, J. W. On invariances in surface thermodynamic properties and their applications to low symmetry crystals. Surf. Sci. 51, 305\u2013309 (1975).","journal-title":"Surf. Sci."},{"key":"347_CR14","doi-asserted-by":"publisher","first-page":"302","DOI":"10.1016\/0039-6028(75)90253-8","volume":"51","author":"JK Lee","year":"1975","unstructured":"Lee, J. K., Aaronson, H. I. & Russell, K. C. On the equilibrium shape for a non-centro-symmetric \u03b3-plot. Surf. Sci. 51, 302\u2013304 (1975).","journal-title":"Surf. Sci."},{"key":"347_CR15","doi-asserted-by":"publisher","first-page":"83","DOI":"10.1016\/0921-5093(90)90032-X","volume":"162","author":"WJ Cahn","year":"1993","unstructured":"Cahn, W. J. & Handwerker, C. Equilibrium geometries of anisotropic surfaces and interfaces. Mater. Sci. Eng. A 162, 83\u201395 (1993).","journal-title":"Mater. Sci. Eng. A"},{"key":"347_CR16","doi-asserted-by":"publisher","first-page":"246","DOI":"10.1038\/s41565-017-0035-5","volume":"13","author":"N Mounet","year":"2018","unstructured":"Mounet, N. et al. Two-dimensional materials from high-throughput computational exfoliation of experimentally known compounds. Nat. Nanotechnol. 13, 246\u2013252 (2018).","journal-title":"Nat. Nanotechnol."},{"key":"347_CR17","unstructured":"Merck, E., Schenck, H. & Krauch, C. Chemical Reagents Their Purity and Tests (Archibald Constable, 1907)."},{"key":"347_CR18","doi-asserted-by":"publisher","first-page":"011002","DOI":"10.1063\/1.4812323","volume":"1","author":"A Jain","year":"2013","unstructured":"Jain, A. et al. Commentary: the materials project: a materials genome approach to accelerating materials innovation. APL Mater. 1, 011002 (2013).","journal-title":"APL Mater."},{"key":"347_CR19","doi-asserted-by":"publisher","first-page":"373","DOI":"10.1038\/nature13184","volume":"508","author":"L-D Zhao","year":"2014","unstructured":"Zhao, L.-D. et al. Ultralow thermal conductivity and high thermoelectric figure of merit in SnSe crystals. Nature 508, 373\u2013377 (2014).","journal-title":"Nature"},{"key":"347_CR20","doi-asserted-by":"publisher","first-page":"141","DOI":"10.1126\/science.aad3749","volume":"351","author":"L-D Zhao","year":"2015","unstructured":"Zhao, L.-D. et al. Ultrahigh power factor and thermoelectric performance in hole-doped single-crystal SnSe. Science 351, 141\u2013144 (2015).","journal-title":"Science"},{"key":"347_CR21","doi-asserted-by":"publisher","first-page":"778","DOI":"10.1126\/science.aaq1479","volume":"360","author":"C Chang","year":"2018","unstructured":"Chang, C. et al. 3D charge and 2D phonon transports leading to high out-of-plane ZT in n-type SnSe crystals. Science 360, 778\u2013783 (2018).","journal-title":"Science"},{"key":"347_CR22","doi-asserted-by":"publisher","first-page":"1418","DOI":"10.1126\/science.aax5123","volume":"365","author":"W He","year":"2019","unstructured":"He, W. et al. High thermoelectric performance in low-cost SnS0.91Se0.09 crystals. Science 365, 1418 (2019).","journal-title":"Science"},{"key":"347_CR23","doi-asserted-by":"publisher","first-page":"4410","DOI":"10.1021\/acs.nanolett.6b01541","volume":"16","author":"E Sutter","year":"2016","unstructured":"Sutter, E. et al. Electron-beam induced transformations of layered tin dichalcogenides. Nano Lett. 16, 4410\u20134416 (2016).","journal-title":"Nano Lett."},{"key":"347_CR24","doi-asserted-by":"publisher","first-page":"173104","DOI":"10.1063\/1.4934750","volume":"107","author":"R Fei","year":"2015","unstructured":"Fei, R., Li, W., Li, J. & Yang, L. Giant piezoelectricity of monolayer group IV monochalcogenides: SnSe, SnS, GeSe, and GeS. Appl. Phys. Lett. 107, 173104 (2015).","journal-title":"Appl. Phys. Lett."},{"key":"347_CR25","doi-asserted-by":"publisher","first-page":"235502","DOI":"10.1103\/PhysRevLett.105.235502","volume":"105","author":"Y Liu","year":"2010","unstructured":"Liu, Y., Dobrinsky, A. & Yakobson, B. I. Graphene edge from armchair to zigzag: the origins of nanotube chirality? Phys. Rev. Lett. 105, 235502 (2010).","journal-title":"Phys. Rev. Lett."},{"key":"347_CR26","doi-asserted-by":"publisher","first-page":"187","DOI":"10.1007\/BF00708709","volume":"1","author":"R Najafabadi","year":"1994","unstructured":"Najafabadi, R. & Srolovitz, D. J. The effect of surface relaxation and atomic vibration on the equilibrium shape of gold and copper crystallites. J. Comput. -Aided Mater. Des. 1, 187\u2013197 (1994).","journal-title":"J. Comput. -Aided Mater. Des."},{"key":"347_CR27","doi-asserted-by":"crossref","unstructured":"Einstein, T. L. in Handbook of Crystal Growth, Fundamentals 2nd edn, Vol. 1a (ed. Nishinaga, T.) Ch. 5, 215\u2013264 (Elsevier, 2015).","DOI":"10.1016\/B978-0-444-56369-9.00005-8"},{"key":"347_CR28","doi-asserted-by":"publisher","first-page":"7649","DOI":"10.1103\/PhysRevB.38.7649","volume":"38","author":"G-X Qian","year":"1988","unstructured":"Qian, G.-X., Martin, R. M. & Chadi, D. J. First-principles study of the atomic reconstructions and energies of Ga- and As-stabilized GaAs(100) surfaces. Phys. Rev. B 38, 7649\u20137663 (1988).","journal-title":"Phys. Rev. B"},{"key":"347_CR29","doi-asserted-by":"publisher","first-page":"90","DOI":"10.1016\/j.cplett.2018.05.008","volume":"702","author":"Z Wang","year":"2018","unstructured":"Wang, Z. & Pang, F. A facile way to control phase of tin selenide flakes by chemical vapor deposition. Chem. Phys. Lett. 702, 90\u201395 (2018).","journal-title":"Chem. Phys. Lett."},{"key":"347_CR30","doi-asserted-by":"publisher","first-page":"021026","DOI":"10.1088\/2053-1583\/aa6aec","volume":"4","author":"J Jiang","year":"2017","unstructured":"Jiang, J. et al. Two-step fabrication of single-layer rectangular SnSe flakes. 2D Mater. 4, 021026 (2017).","journal-title":"2D Mater."},{"key":"347_CR31","first-page":"7","volume":"74","author":"A Stroia","year":"2012","unstructured":"Stroia, A. et al. A new route for synthesis and characterization of macroporous granular silver. UPB Sci. Bull. Ser. B. 74, 7\u201318 (2012).","journal-title":"UPB Sci. Bull. Ser. B."},{"key":"347_CR32","doi-asserted-by":"publisher","first-page":"085503","DOI":"10.1103\/PhysRevLett.115.085503","volume":"115","author":"H Li","year":"2015","unstructured":"Li, H., Geelhaar, L., Riechert, H. & Draxl, C. Computing equilibrium shapes of wurtzite crystals: the example of GaN. Phys. Rev. Lett. 115, 085503 (2015).","journal-title":"Phys. Rev. Lett."},{"key":"347_CR33","doi-asserted-by":"publisher","first-page":"115502","DOI":"10.1103\/PhysRevLett.114.115502","volume":"114","author":"VI Artyukhov","year":"2015","unstructured":"Artyukhov, V. I., Hao, Y., Ruoff, R. S. & Yakobson, B. I. Breaking of symmetry in graphene growth on metal substrates. Phys. Rev. Lett. 114, 115502 (2015).","journal-title":"Phys. Rev. Lett."},{"key":"347_CR34","doi-asserted-by":"publisher","first-page":"73","DOI":"10.1038\/nature12739","volume":"505","author":"E Auyeung","year":"2013","unstructured":"Auyeung, E. et al. DNA-mediated nanoparticle crystallization into Wulff polyhedra. Nature 505, 73 (2013).","journal-title":"Nature"},{"key":"347_CR35","doi-asserted-by":"publisher","first-page":"11169","DOI":"10.1103\/PhysRevB.54.11169","volume":"54","author":"G Kresse","year":"1996","unstructured":"Kresse, G. & Furthm\u00fcller, J. Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. Phys. Rev. B 54, 11169\u201311186 (1996).","journal-title":"Phys. Rev. B"},{"key":"347_CR36","doi-asserted-by":"publisher","first-page":"3865","DOI":"10.1103\/PhysRevLett.77.3865","volume":"77","author":"JP Perdew","year":"1996","unstructured":"Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized gradient approximation made simple. Phys. Rev. Lett. 77, 3865\u20133868 (1996).","journal-title":"Phys. Rev. Lett."},{"key":"347_CR37","doi-asserted-by":"publisher","unstructured":"Wang, L. Data and Codes (Zenodo, 2022); https:\/\/doi.org\/10.5281\/zenodo.7130224","DOI":"10.5281\/zenodo.7130224"}],"updated-by":[{"DOI":"10.1038\/s43588-022-00385-z","type":"correction","label":"Correction","source":"publisher","updated":{"date-parts":[[2022,12,7]],"date-time":"2022-12-07T00:00:00Z","timestamp":1670371200000}}],"container-title":["Nature Computational Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.nature.com\/articles\/s43588-022-00347-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.nature.com\/articles\/s43588-022-00347-5","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.nature.com\/articles\/s43588-022-00347-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,12,8]],"date-time":"2022-12-08T09:41:56Z","timestamp":1670492516000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.nature.com\/articles\/s43588-022-00347-5"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,11,28]]},"references-count":37,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2022,11]]}},"alternative-id":["347"],"URL":"https:\/\/doi.org\/10.1038\/s43588-022-00347-5","relation":{},"ISSN":["2662-8457"],"issn-type":[{"value":"2662-8457","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,11,28]]},"assertion":[{"value":"23 February 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"3 October 2022","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"28 November 2022","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"7 December 2022","order":4,"name":"change_date","label":"Change Date","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"Correction","order":5,"name":"change_type","label":"Change Type","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"A Correction to this paper has been published:","order":6,"name":"change_details","label":"Change Details","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"https:\/\/doi.org\/10.1038\/s43588-022-00385-z","URL":"https:\/\/doi.org\/10.1038\/s43588-022-00385-z","order":7,"name":"change_details","label":"Change Details","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"The authors declare no competing interests.","order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interests"}}]}}