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AA stacking, we mean that the two layers are stacked in a perfectly aligned configuration.","type":"other"},{"key":"5zt2-scbgCc26R1","doi-asserted-by":"publisher","DOI":"10.1002\/adfm.201803305","type":"journal-article"},{"key":"5zt2-scbgCc27R1","doi-asserted-by":"publisher","DOI":"10.1002\/pssb.201700033","type":"journal-article"},{"key":"5zt2-scbgCc28R1","unstructured":"See Supplemental Material at http:\/\/link.aps.org\/supplemental\/10.1103\/5zt2-scbg for more details, including details of the DFT calculations, interlayer distance, energy landscape, and interlayer and intralayer parameters landscape, as well as the Fourier components and moir\u00e9 parameters of HfS2, HfSe2, ZrS2, ZrSe2, SnS2, SnSe2. The Supplemental Material also contains Refs.\u00a0[29\u201331].","type":"other"},{"key":"5zt2-scbgCc29R1","doi-asserted-by":"publisher","DOI":"10.1088\/0953-8984\/21\/39\/395502","type":"journal-article"},{"key":"5zt2-scbgCc30R1","first-page":"446","type":"journal-article","volume":"10","author":"M. Faghihnasiri","year":"2020","journal-title":"Nanomater. Nanotechnol.","ISSN":"https:\/\/id.crossref.org\/issn\/1847-9804","issn-type":"print"},{"key":"5zt2-scbgCc31R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.96.075311","type":"journal-article"},{"key":"5zt2-scbgCc32R1","doi-asserted-by":"publisher","DOI":"10.1103\/RevModPhys.84.1419","type":"journal-article"},{"key":"5zt2-scbgCc33R1","unstructured":"DFT band structure calculations for 1T\u2212HfS2 monolayers that employ different pseudopotentials can result in more significant, albeit still small, SOI in the low-energy conduction bands (see, e.g., Fig.\u00a09 of Ref.\u00a0[34]). However, even there, near the M valley SOI has almost no impact.","type":"other"},{"key":"5zt2-scbgCc34R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.98.155204","type":"journal-article"},{"key":"5zt2-scbgCc35R1","doi-asserted-by":"publisher","DOI":"10.1063\/1.1748067","type":"journal-article"},{"key":"5zt2-scbgCc36R1","doi-asserted-by":"crossref","DOI":"10.1007\/b13586","type":"book","volume-title":"Spin-orbit Coupling Effects in Two-Dimensional Electron and Hole Systems","author":"R. Winkler","year":"2003"},{"key":"5zt2-scbgCc37R1","unstructured":"Here, we use the notation that H(k;d) is the operator in the electronic Hilbert space defined by the matrix representation H(k;d) in the basis (|\u03c8n,M\u03bd(d0)\u27e9)n\u2208{1,\u2026,6}.","type":"other"},{"key":"5zt2-scbgCc38R1","unstructured":"\u201c\u2225\u201d and \u201c\u22a5\u201d on LHS and RHS are with respect to M\u03bd and R\u0398z(M\u03bd), respectively.","type":"other"},{"key":"5zt2-scbgCc39R1","unstructured":"Position-dependent effective masses can be included in the moir\u00e9 band Hamiltonian when needed.","type":"other"},{"key":"5zt2-scbgCc40R1","doi-asserted-by":"publisher","DOI":"10.1038\/s41467-024-48511-x","type":"journal-article"},{"key":"5zt2-scbgCc41R1","unstructured":"Within the subspace of t2g states in transition metal oxide perovskites, the dominant hopping processes conserve orbital type just as the low-energy conduction states in the M-valley moir\u00e9s considered here can be described by a moir\u00e9-scale tight-binding model with hoppings that conserve valley. Hopping between particular lattice sites is strongly orbital dependent in the perovskite case and valley dependent in the moir\u00e9 case. See, e.g., Sec.\u00a05 of Ref.\u00a0[42].","type":"other"},{"key":"5zt2-scbgCc42R1","doi-asserted-by":"publisher","DOI":"10.1070\/PU1982v025n04ABEH004537","type":"journal-article"},{"key":"5zt2-scbgCc43R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.116.026803","type":"journal-article"},{"key":"5zt2-scbgCc44R1","doi-asserted-by":"publisher","DOI":"10.1038\/s41586-025-09187-5","type":"journal-article"},{"key":"5zt2-scbgCc45R1","unstructured":"For each d\u2208R2, the Bloch Hamiltonian H(x,p(x);d) of the crystal bilayer is \u039b periodic in x, where \u039b is the Bravais lattice of the (arbitrarily chosen) bottom layer. We choose \u039b and its real-space \u03a9uc (reciprocal space BZ) unit cell to be independent of d. The vertical separation between layers is fixed at each d by minimizing energy.","type":"other"},{"key":"5zt2-scbgCc46R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.98.046402","type":"journal-article"},{"key":"5zt2-scbgCc47R1","doi-asserted-by":"publisher","DOI":"10.1063\/1.4984041","type":"journal-article"},{"key":"5zt2-scbgCc48R1","doi-asserted-by":"publisher","DOI":"10.1137\/16M1088363","type":"journal-article"},{"key":"5zt2-scbgCc49R1","unstructured":"The top (bottom) layer of the quasicrystal has Bravais lattice \u039bt (\u039bb) and \u039bt=\u03bbR\u03b8z(\u039bb). For small \u03b8 and \u03bb\u22121, and for R\u2208\u039bb, \u03bbR\u03b8z(R)\u2212R\u2248(\u03bb\u22121)R+\u03b8e^z\u00d7R. Modulating this vector by \u039bt produces a position in the top layer unit cell and encodes the relative position of the nearest element of \u039bt to R\u2208\u039bb. Analogous arguments follow for the top layer. Since we choose \u039b\u2261\u039bb, even when l=t in the Hamiltonian matrix elements R\u2208\u039b is not generically in \u039bt. An additional lattice-site-dependent position shift of top layer WFs in the displaced bilayer must therefore be included when approximating quasicrystal matrix elements that involve layer t.","type":"other"},{"key":"5zt2-scbgCc50R1","doi-asserted-by":"publisher","DOI":"10.1038\/ncomms7308","type":"journal-article"},{"key":"5zt2-scbgCc51R1","volume-title":"Fundamentals of Semiconductors","author":"P.\u2009Y. Yu","year":"2007","type":"book"},{"key":"5zt2-scbgCc52R1","unstructured":"\u039bM*\u2261{\u03bbR\u2212\u03b8(G)\u2212G:G\u2208\u039b*}\u2282R2 is a Bravais lattice called the moir\u00e9 reciprocal lattice. Its dual, the moir\u00e9 lattice, is given by \u039bM={r\u2208R2:\u2009\u2009d(r)\u2208\u039b}.","type":"other"}],"container-title":["Physical Review Letters"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.aps.org\/article\/10.1103\/5zt2-scbg","content-type":"unspecified","content-version":"vor","intended-application":"syndication"},{"URL":"http:\/\/harvest.aps.org\/v2\/journals\/articles\/10.1103\/5zt2-scbg\/fulltext","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,9,29]],"date-time":"2026-09-29T22:19:49Z","timestamp":1790720389000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.aps.org\/doi\/10.1103\/5zt2-scbg"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,11,5]]},"references-count":52,"journal-issue":{"issue":"19","published-print":{"date-parts":[[2025,11]]}},"URL":"https:\/\/doi.org\/10.1103\/5zt2-scbg","relation":{},"ISSN":["0031-9007","1079-7114"],"issn-type":[{"value":"0031-9007","type":"print"},{"value":"1079-7114","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,11,5]]},"assertion":[{"value":"2024-12-06","name":"date_received","label":"Date Received","group":{"name":"publication_dates","label":"Publication dates"}},{"value":"\u00a9 2025 American Physical Society. All rights are reserved, including those for text and data mining, AI training, and similar technologies.","name":"copyright_statement","label":"Copyright statement","group":{"name":"copyright","label":"copyright"}},{"value":"2025","name":"copyright_year","label":"Copyright year","group":{"name":"copyright","label":"copyright"}},{"value":"American Physical Society","name":"copyright_holder","label":"Copyright holder","group":{"name":"copyright","label":"copyright"}}],"article-number":"196402"}}