Engineering phase-frustration-induced flat bands in an aza-triangulene covalent kagome lattice

Nature Materials (2026)Cite this article π-Conjugated covalent organic frameworks provide a versatile molecular scaffold for the realization of designer quantum nanomaterials. Strong electron–electron correlation within these artificial lattices can give rise to exotic phases of matter. Their experimental realization, however, requires precise control over orbital symmetry, charge localization and band dispersion, all arising from the effective hybridization between molecular linkers and nodes. Here we present a modular strategy for constructing diatomic kagome lattices from aza-[3]triangulene nodes, in which a D3h-symmetric ground state is stabilized through resonance contributions from a cumulenic linker. First-principles density functional theory and scanning tunnelling spectroscopy reveal that the hybridization of a sixfold-degenerate set of edge-localized Wannier functions in the unit cell gives rise to orbital-phase-frustration-induced non-trivial flat bands. These results establish a general design principle for engineering orbital interactions in organic lattices and open a pathway towards programmable covalent-organic-framework-based quantum materials with correlated electronic ground states.This is a preview of subscription content, access via your institution Access Nature and 54 other Nature Portfolio journals Get Nature+, our best-value online-access subscription $32.99 / 30 days cancel any timeSubscribe to this journal Receive 12 print issues and online access $259.00 per yearonly $21.58 per issueBuy this articleUSD 39.95Prices may be subject to local taxes which are calculated during checkoutThe X-ray crystallographic coordinates for structure 1 reported in this study have been deposited at the Cambridge Crystallographic Data Centre (CCDC)75 under deposition number CCDC 2416619. These data can be obtained free of charge from the CCDC via www.ccdc.cam.ac.uk/data_request/cif.

The Supplementary Information contains detailed synthetic procedures and characterization of precursors 1 and 2; the electronic structure of linear A[3]T chains and cyclic (A[3]T)6; the bonding analysis for linear A[3]T chains and cyclic (A[3]T)6; Supplementary Figs. 1–13; the X-ray crystal structure data of 1; and Supplementary Tables 1–6. Source data are provided with this paper.The DFT code and pseudopotentials can be downloaded from the Quantum Espresso website70,71. For this study, we used version 6.7 for the DFT–PBE calculations. The Wannier90 code can be downloaded from the Wannier90 website72. The TB simulation code can be downloaded from the PythTB website74. For this study, we used version 1.8.0 for the TB fitting. X-ray crystal structure data were refined using the CrysAlisPro software package76.Kang, M. G. et al. Topological flat bands in frustrated kagome lattice CoSn. Nat. Commun. 11, 4004 (2020).Article PubMed PubMed Central CAS Google Scholar Calugaru, D. et al. 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The Cambridge Structural Database. Acta Crystallogr. B 72, 171–179 (2016).Article CAS Google Scholar CrysAlisPro v.1.172.43.143a (Oxford Diffraction/Agilent Technologies UK, 2024).Download referencesThis work was primarily supported by the US Department of Energy (DOE), Office of Science, Basic Energy Sciences (BES), Materials Sciences and Engineering Division under contract DE-AC02-05CH11231 (F.R.F., S.G.L.; Nanomachine programme KC1203; molecular design, on-surface growth, TB analyses) and contract DE-SC0023105 (F.R.F.; molecular precursor synthesis) and by the National Energy Research Scientific Computing Center, a US DOE Office of Science User Facility operated under contract DE-AC02-05CH11231, awards BES-ERCAP0037553 (S.G.L.) and DDR-ERCAP0032308 (B.Q.); the Office of Naval Research under contracts N00014-24-1-2134 (F.R.F.; STM imaging and STS characterization) and N00014-19-1-2503 (F.R.F.; STM instrumentation); the National Science Foundation under contract DMR-2325410 (S.G.L.; DFT simulations), TACC Frontera (S.G.L.) and ACCESS resources at the Stampede3 (S.G.L.); and the Heising-Simons Faculty Fellows Program at the University of California (UC), Berkeley (F.R.F.). The UC Berkeley Molecular Graphics and Computation Facility in the College of Chemistry (CoC-MGCF) is supported in part by National Institutes of Health (NIH) contract S10OD034382; the UC Berkeley NMR facility in the College of Chemistry (CoC-NMR) is supported in part by NIH contract S10OD024998; and the CoC X-ray facility is supported in part by NIH contract S10-RR027172. We thank H. Çelik and the CoC-NMR for assistance with spectroscopic characterization. We thank K. Durkin and D. Small of the CoC-MGCF for support with computational resources. We thank Z. Zhou and QB3 for assistance with electrospray ionization mass spectrometry measurement. We thank N. Settineri for assistance with X-ray crystal structure analysis. H.X.W. acknowledges support from the Agency for Science, Technology, and Research (A*STAR) Singapore National Science Scholarship programme; B.Q. acknowledges support from the Kavli ENSI Graduate Student Fellowship programme and the Bakar Institute of Digital Materials for the Planet Postdoc Fellowship programme; and Y.Y. thanks Q. Sun and B. Yuan for valuable discussions and training in nc-AFM imaging.These authors contributed equally: Yuyi Yan, Fujia Liu, Weichen Tang.Department of Chemistry, University of California, Berkeley, CA, USAYuyi Yan, Fujia Liu, Han Xuan Wong, Boyu Qie & Felix R. FischerDepartment of Physics, University of California, Berkeley, CA, USAWeichen Tang & Steven G. LouieMaterials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA, USAWeichen Tang, Steven G. Louie & Felix R. FischerKavli Energy NanoSciences Institute, University of California Berkeley and Lawrence Berkeley National Laboratory, Berkeley, CA, USABoyu Qie & Felix R. FischerBakar Institute of Digital Materials for the Planet, College of Computing, Data Science, and Society, University of California Berkeley, Berkeley, CA, USABoyu Qie & Felix R. FischerSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarY.Y., F.L., B.Q. and F.R.F. initiated and conceived the research; F.L. and B.Q. designed, synthesized and characterized the molecular precursors; Y.Y. performed on-surface synthesis and STM characterization and analysis; W.T., B.Q., S.G.L. and F.R.F. performed DFT calculations as well as theoretical analyses; H.X.W. assisted with data interpretation; S.G.L. and F.R.F. secured funding and supervised all aspects of the project; and B.Q., Y.Y., F.L., W.T., S.G.L. and F.R.F. wrote the manuscript. All authors contributed to the scientific discussion.Correspondence to Boyu Qie, Steven G. Louie or Felix R. Fischer.The authors declare no competing interests.Nature Materials thanks the anonymous reviewers for their contribution to the peer review of this work.Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.a, Molecular structure and DFT calculated orbitals for high-spin D3h and low-spin C2v symmetric A[3]T (M06-2X/def2-TZVP) (red arrow, α-spin; blue arrow, β-spin). b, Selected natural orbitals in CAS(5,5) for low-spin C2v symmetric A[3]T with different occupations. c, Selected natural orbitals in CAS(5,6) for high-spin D3h symmetric A[3]T with different occupation. d, Optimized energy for high-spin D3h and low-spin C2v symmetric A[3]T at different levels of theory.Source dataSynthesis of 4,8,12-tris(dichloromethylene)-8,12-dihydro-4H-benzo[9,1]quinolizino[3,4,5,6,7-defg]acridine (1) and 8,12-bis(dichloromethylene)-8,12-dihydro-4H-benzo[9,1]quinolizino [3,4,5,6,7-defg]acridin-4-one (2) from 4H-benzo[9,1]quinolizino[3,4,5,6,7-defg]acridine-4,8,12-trione.a–c, STM topographic image of densely packed self-assembly of 1 after sublimation on Au(111). (a, Vs = –600 mV, It = 20 pA; b, Vs = –200 mV, It = 30 pA; c, Vs = –100 mV, It = 20 pA). d–e, STM topographic images of 2D-A[3]TCOF grown from 1 at 523 K. (d, Vs = –1000 mV, It = 30 pA; e, Vs = –400 mV, It = 30 pA). f, Constant-height BRSTM image (Vs = –10 mV, CO-functionalized tip) 2D-A[3]TCOF. g–i, STM topographic image of densely packed self-assembly of 2 after sublimation on Au(111). (g, Vs = –600 mV, It = 20 pA; h, Vs = –200 mV, It = 30 pA; i, Vs = –100 mV, It = 20 pA). j–n, STM topographic images of linear A[3]T chains and cyclic (A[3]T)6 grown from 2 at 523 K. (j, Vs = –1200 mV, It = 30 pA; k, Vs = –400 mV, It = 30 pA; l, Vs = –200 mV, It = 50 pA; m, Vs = –200 mV, It = 50 pA; n, Vs = –1200 mV, It = 20 pA). Linear A[3]T chains range in length between 10–40 monomer units. All STM experiments were performed at T = 4.5 K.Source dataa, Constant-height BRSTM image of a representative segment of the p6mm lattice of 2D-A[3]TCOF (Vs = –10 mV, Vac = 10 mV, f = 455 Hz, CO-functionalized tip). b, nc-AFM image of a representative segment of the p6mm lattice of 2D-A[3]TCOF (Vs = –10 mV, f = 27.64 kHz, CO-functionalized tip). c, Constant-height BRSTM image of a representative segment of the p6mm lattice of 2D-A[3]TCOF (Vs = –10 mV, Vac = 10 mV, f = 455 Hz, CO-functionalized tip). Inset shows the Fourier transform (FT) of the electronic structure of the Kagome lattice (note: clear signal intensity is observed in the first BZ). d, nc-AFM image of a representative segment of the p6mm lattice of 2D-A[3]TCOF (Vs = –10 mV, f = 27.64 kHz, CO-functionalized tip). Inset shows the Fourier transform (FT) of the atomic structure of the Kagome lattice (note: no signal intensity is observed in the first BZ suggesting the QPI patterns are purely electronic in nature). e–f, Two representative nc-AFM images showing the edge termination of a locally ordered 2D-A[3]TCOF island (Vs = –10 mV, f = 27.64 kHz, CO-functionalized tip). g, Constant-height BRSTM image of a representative linear A[3]T chain featuring segments of s-cis and s-trans linked A[3]T cores (Vs = –10 mV, Vac = 10 mV, f = 455 Hz, CO-functionalized tip). h, nc-AFM image of a A[3]T chain end featuring the C = O and C = CH2 end groups.Source dataa, Constant-height BRSTM image of a self-assembled lattice of cyclic (A[3]T)6 (Vs = –10 mV, Vac = 10 mV, f = 455 Hz, CO-functionalized tip). Unit cell is highlighted by a white dashed line. b, Schematic representation of the weak C–H•••O = C hydrogen bonding pattern that directs the assembly of (A[3]T)6 on the Au(111) surface.Source dataa–aa, Constant-height dI/dV maps recorded at the indicated sample voltage biases on the same 2D-A[3]TCOF depicted in Fig. 3 using different STM tips (Vac = 10–20 mV, f = 455 Hz, CO-functionalized tip). All STM experiments were performed at T = 4.5 K.Source dataa–ak, (top) FT of constant-height dI/dV maps recorded at the indicated sample voltage biases (Vac = 10 mV, f = 455 Hz, CO-functionalized tip). (bottom) Normalized FT signal intensity (I/I0) as a function of the magnitude of the momentum vector |k | (with respect to the Γ point) within the first BZ. Median normalized intensity (μ½) indicated as dashed line.Source dataThe unrestricted and restricted singlet simulations show identical ground states with = 0 and are 0.80 eV lower in energy than the open-shell triplet state.Source dataa, Phase diagram for the TB Kagome band structure as a function of t2 and t3 (in units of t1). Colour gradient indicates the size of the band gap in |Eg (eV)/t1 | ; Type-I (pxy;D), Type-II (K±;K±), Type-III (K−;K+), Type-IV (K+;K−), and mixed metallic Kagome band structures are separated dotted lines. Filled circles mark the position of examples in (b–f) that fall within the depicted window. b, Typical TB band structure of a mixed metallic phase. Dispersive Dirac bands cross the EF. c,Typical TB band structure of a Type-I phase. The top two Dirac bands (D) arise typically from s-orbital hopping in a hexagonal lattice while the four band VB complex (pxy) is usually derived from pxy-orbital hopping. d, Typical TB band structure of a Type-II phase. Pairs of two Dirac bands bordered at lower energy by a flat band (K+, the sign in the exponent indicates the sign of the lattice hopping; positive the flat band is at found at lower energy; negative the flat band is at found at higher energy). e, Typical TB band structure of a Type-III phase (K−; K+). (F) Typical TB band structure of a Type-IV phase (K+; K−).Source dataa, Molecular model of a 2D-C4-A[3]TCOFs assembled from aza-[3]triangulene core and buta-1,3-diyne linkers. b, DFT-PBE band structure for a freestanding 2D-C4-A[3]TCOF and corresponding truncated NNN TB Hamiltonian in the Wannier basis. c, Molecular model of a 2D-C6-A[3]TCOFs assembled from aza-[3]triangulene core and hexa-1,3,5-triyne linkers. d, DFT-PBE band structure for a freestanding 2D-C6-A[3]TCOF and corresponding truncated NNN TB Hamiltonian in the Wannier basis. e, Molecular model of a 2D-C8-A[3]TCOFs assembled from aza-[3]triangulene core and octa-1,3,5,7-tetrayne linkers. f, DFT-PBE band structure for a freestanding 2D-C8-A[3]TCOF and corresponding truncated NNN TB Hamiltonian in the Wannier basis. g, Molecular model of a 2D-C2PhC2-A[3]TCOFs assembled from aza-[3]triangulene core and 1,4-diethynylbenzene linkers. h, DFT-PBE band structure for a freestanding 2D-C2PhC2-A[3]TCOF and corresponding truncated NNN TB Hamiltonian in the Wannier basis.Source dataSupplementary Figs. 1–13, Tables 1–6, Materials and General Methods and Discussion of one-dimensional A[3]T chains and (A[3]T)6.Source data for Supplementary Fig. 1.Source data for Supplementary Fig. 2.Source data for Supplementary Fig. 3.Source data for Supplementary Fig. 4.Source data for Supplementary Fig. 5.Source data for Supplementary Fig. 6.Source data for Supplementary Fig. 7.Source data for Supplementary Fig. 8.Source data for Supplementary Fig. 9.Source data for Supplementary Fig. 10.Source data for Supplementary Fig. 11.Source data for Supplementary Fig. 12.Source data for Supplementary Fig. 13.Statistical source data.Unprocessed image data.Statistical source data and unprocessed image data.Statistical source data.Statistical source data.Unprocessed image data.Unprocessed image data.Unprocessed image data.Unprocessed image data.Statistical source data.Statistical source data.Statistical source data.Statistical source data.Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.Reprints and permissionsYan, Y., Liu, F., Tang, W. et al. Engineering phase-frustration-induced flat bands in an aza-triangulene covalent kagome lattice. Nat. Mater. (2026). https://doi.org/10.1038/s41563-026-02528-3Download citationReceived: 17 September 2025Accepted: 02 February 2026Published: 27 February 2026Version of record: 27 February 2026DOI: https://doi.org/10.1038/s41563-026-02528-3Anyone you share the following link with will be able to read this content:Sorry, a shareable link is not currently available for this article. Provided by the Springer Nature SharedIt content-sharing initiative