{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,14]],"date-time":"2026-08-14T11:32:28Z","timestamp":1786707148947,"version":"build-2736575974"},"reference-count":37,"publisher":"MDPI AG","issue":"16","license":[{"start":{"date-parts":[[2026,8,14]],"date-time":"2026-08-14T00:00:00Z","timestamp":1786665600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Beijing Boson Quantum Technology Co., Ltd.","award":["2262090009"],"award-info":[{"award-number":["2262090009"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Nanomaterials"],"abstract":"<jats:p>Correlated electronic phenomena in magic-angle twisted bilayer graphene have garnered widespread research interest in two-dimensional materials science. As a powerful multiscale framework bridging atomic-scale resolution and mesoscopic structural evolution, the structural phase field crystal method has been widely adopted for graphene system studies. In this work, we develop a self-consistent XPFC model specifically for twisted bilayer graphene (tBLG) simulations. By globally optimizing the core free-energy functional parameters via a genetic algorithm, the proposed model achieves a marked improvement in consistency between the equilibrium density field and the first-principles generalized stacking fault energy surface. We further introduce a self-consistent dynamic interlayer interaction potential to replace the conventional fixed-substrate approximation, which captures the bidirectional coupling and mutual relaxation between adjacent graphene layers in a self-consistent manner. We calibrate the precise magnitude of the interlayer potential using the widths of stacking domain boundaries between distinct stacking configurations as a key metric, with the results benchmarked against atomistic simulation data. When applied to the 1.1\u00b0 magic-angle tBLG system, the model uncovers spontaneous structural relaxation driven by interlayer van der Waals interactions: low-energy AB\u2013BA stacking domains expand significantly, while high-energy AA domains shrink correspondingly.<\/jats:p>","DOI":"10.3390\/nano16161000","type":"journal-article","created":{"date-parts":[[2026,8,14]],"date-time":"2026-08-14T10:39:01Z","timestamp":1786703941000},"page":"1000","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A Self-Consistent Phase Field Crystal Method for Twisted Bilayer Graphene"],"prefix":"10.3390","volume":"16","author":[{"given":"Pingqia","family":"Wang","sequence":"first","affiliation":[{"name":"Faculty of Arts and Sciences, Beijing Normal University at Zhuhai, No. 18 Jinfeng Road, Zhuhai 519087, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Kai","family":"Liu","sequence":"additional","affiliation":[{"name":"Faculty of Arts and Sciences, Beijing Normal University at Zhuhai, No. 18 Jinfeng Road, Zhuhai 519087, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2026,8,14]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"109","DOI":"10.1103\/RevModPhys.81.109","article-title":"The electronic properties of graphene","volume":"81","author":"Guinea","year":"2009","journal-title":"Rev. Mod. Phys."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"951","DOI":"10.1126\/science.1130681","article-title":"Controlling the electronic structure of bilayer graphene","volume":"313","author":"Ohta","year":"2006","journal-title":"Science"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"151","DOI":"10.1038\/nmat2082","article-title":"Gate-induced insulating state in bilayer graphene devices","volume":"7","author":"Oostinga","year":"2008","journal-title":"Nat. Mater."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"533","DOI":"10.1038\/nature12780","article-title":"Dislocations in bilayer graphene","volume":"505","author":"Butz","year":"2014","journal-title":"Nature"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1445","DOI":"10.1038\/s41467-019-09097-x","article-title":"Stacking angle-tunable photoluminescence from interlayer exciton states in twisted bilayer graphene","volume":"10","author":"Patel","year":"2019","journal-title":"Nat. Commun."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"5309","DOI":"10.1038\/ncomms6309","article-title":"Resonant Raman spectroscopy of twisted multilayer graphene","volume":"5","author":"Zhang","year":"2014","journal-title":"Nat. Commun."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"213","DOI":"10.1038\/nphys890","article-title":"Anisotropic behaviours of massless Dirac fermions in graphene under periodic potentials","volume":"4","author":"Park","year":"2008","journal-title":"Nat. Phys."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"43","DOI":"10.1038\/nature26160","article-title":"Unconventional superconductivity in magic-angle graphene superlattices","volume":"556","author":"Cao","year":"2018","journal-title":"Nature"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1059","DOI":"10.1126\/science.aav1910","article-title":"Tuning superconductivity in twisted bilayer graphene","volume":"363","author":"Yankowitz","year":"2019","journal-title":"Science"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"240","DOI":"10.1038\/s41586-021-04121-x","article-title":"Evidence for unconventional superconductivity in twisted bilayer graphene","volume":"600","author":"Oh","year":"2021","journal-title":"Nature"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"439","DOI":"10.1038\/s41586-021-04002-3","article-title":"Fractional Chern insulators in magic-angle twisted bilayer graphene","volume":"600","author":"Xie","year":"2021","journal-title":"Nature"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"732","DOI":"10.1038\/s41567-025-02849-1","article-title":"Thermopower probes of emergent local moments in magic-angle twisted bilayer graphene","volume":"21","author":"Ghosh","year":"2025","journal-title":"Nat. Phys."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"39","DOI":"10.1038\/s41567-025-03117-y","article-title":"Optical control of orbital magnetism in magic-angle twisted bilayer graphene","volume":"22","author":"Persky","year":"2026","journal-title":"Nat. Phys."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"101525","DOI":"10.1063\/5.0267884","article-title":"Higher-order continuum models for twisted bilayer graphene","volume":"66","author":"Quinn","year":"2025","journal-title":"J. Math. Phys."},{"key":"ref_15","first-page":"7612","article-title":"Complex Strain Scapes in Reconstructed Transition-Metal Dichalcogenide Moir\u00e9 Superlattices","volume":"17","author":"Accioly","year":"2023","journal-title":"ACS Nano"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"nwaf577","DOI":"10.1093\/nsr\/nwaf577","article-title":"Strain, stress and rotation fields, and energetic features of twisted 2D materials","volume":"13","author":"Li","year":"2026","journal-title":"Natl. Sci. Rev."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"045019","DOI":"10.1088\/2632-2153\/ae1807","article-title":"An atomic cluster expansion potential for twisted multilayer graphene","volume":"6","author":"Wang","year":"2025","journal-title":"Mach. Learn. Sci. Technol."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"042004","DOI":"10.1088\/2053-1583\/ad63b6","article-title":"Multiscale computational modeling techniques in study and design of 2D materials: Recent advances, challenges, and opportunities","volume":"11","author":"Thomas","year":"2024","journal-title":"2D Mater."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"051605","DOI":"10.1103\/PhysRevE.70.051605","article-title":"Modeling elastic and plastic deformations in nonequilibrium processing using phase field crystals","volume":"70","author":"Elder","year":"2004","journal-title":"Phys. Rev. E"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"245701","DOI":"10.1103\/PhysRevLett.88.245701","article-title":"Modeling Elasticity in Crystal Growth","volume":"88","author":"Elder","year":"2002","journal-title":"Phys. Rev. Lett."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"5692","DOI":"10.1021\/nl4033928","article-title":"Epitaxial Graphene Growth and Shape Dynamics on Copper: Phase-Field Modeling and Experiments","volume":"13","author":"Meca","year":"2013","journal-title":"Nano Lett."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"13240","DOI":"10.1039\/D3CP05265F","article-title":"Phase-field crystal modeling of graphene\/hexagonal boron nitride interfaces","volume":"26","author":"Channe","year":"2024","journal-title":"Phys. Chem. Chem. Phys."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1135","DOI":"10.1140\/epjs\/s11734-022-00518-5","article-title":"A review of continuous modeling of periodic pattern formation with modified phase-field crystal models","volume":"231","author":"Starodumov","year":"2022","journal-title":"Eur. Phys. J. Spec. Top."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"031601","DOI":"10.1103\/PhysRevE.83.031601","article-title":"Phase-field-crystal methodology for modeling of structural transformations","volume":"83","author":"Greenwood","year":"2011","journal-title":"Phys. Rev. E Stat. Nonlinear Soft Matter Phys."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"035447","DOI":"10.1103\/PhysRevB.93.035447","article-title":"Structural phase field crystal approach for modeling graphene and other two-dimensional structures","volume":"93","author":"Seymour","year":"2016","journal-title":"Phys. Rev. B"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"035414","DOI":"10.1103\/PhysRevB.94.035414","article-title":"Multiscale modeling of polycrystalline graphene: A comparison of structure and defect energies of realistic samples from phase field crystal models","volume":"94","author":"Hirvonen","year":"2016","journal-title":"Phys. Rev. B"},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Qiao, H., and Liu, K. (2025). Phase-Field Crystal Method for Bilayer Graphene. Nanomaterials, 15.","DOI":"10.3390\/nano15221699"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"155438","DOI":"10.1103\/PhysRevB.92.155438","article-title":"Van der Waals bilayer energetics: Generalized stacking-fault energy of graphene, boron nitride, and graphene\/boron nitride bilayers","volume":"92","author":"Zhou","year":"2015","journal-title":"Phys. Rev. B"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"399","DOI":"10.1038\/nmat1136","article-title":"Stacking fault energies and slip in nanocrystalline metals","volume":"3","author":"Derlet","year":"2004","journal-title":"Nat. Mater."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"201","DOI":"10.1038\/s41524-021-00660-z","article-title":"Generalized stacking fault energy surface mismatch and dislocation transformation","volume":"7","author":"Feng","year":"2021","journal-title":"npj Comput. Mater."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"66","DOI":"10.1038\/scientificamerican0792-66","article-title":"Genetic Algorithms","volume":"267","author":"Holland","year":"1992","journal-title":"Sci. Am."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"1862","DOI":"10.1021\/acs.chemrev.3c00618","article-title":"Stacking Order Engineering of Two-Dimensional Materials and Device Applications","volume":"124","author":"Fox","year":"2024","journal-title":"Chem. Rev."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"804","DOI":"10.1021\/nl902948m","article-title":"Localization of Dirac Electrons in Rotated Graphene Bilayers","volume":"10","author":"Mayou","year":"2010","journal-title":"Nano Lett."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"075311","DOI":"10.1103\/PhysRevB.96.075311","article-title":"Lattice relaxation and energy band modulation in twisted bilayer graphene","volume":"96","author":"Nam","year":"2017","journal-title":"Phys. Rev. B"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"970","DOI":"10.1021\/acsaenm.2c00259","article-title":"An Atomistic Insight into Moir\u00e9 Reconstruction in Twisted Bilayer Graphene beyond the Magic Angle","volume":"1","author":"Dey","year":"2023","journal-title":"ACS Appl. Eng. Mater."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"11256","DOI":"10.1073\/pnas.1309394110","article-title":"Strain solitons and topological defects in bilayer graphene","volume":"110","author":"Alden","year":"2013","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"115501","DOI":"10.1103\/PhysRevLett.115.115501","article-title":"Quantum Monte Carlo calculation of the binding energy of bilayer graphene","volume":"115","author":"Mostaani","year":"2015","journal-title":"Phys. Rev. Lett."}],"container-title":["Nanomaterials"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2079-4991\/16\/16\/1000\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,14]],"date-time":"2026-08-14T10:54:52Z","timestamp":1786704892000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2079-4991\/16\/16\/1000"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,8,14]]},"references-count":37,"journal-issue":{"issue":"16","published-online":{"date-parts":[[2026,8]]}},"alternative-id":["nano16161000"],"URL":"https:\/\/doi.org\/10.3390\/nano16161000","relation":{},"ISSN":["2079-4991"],"issn-type":[{"value":"2079-4991","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,8,14]]}}}