Realization of ideal flat band by rotated d-orbitals in Kagome metals

Recently there has been intense interest in Kagome metals, which are expected to host flat bands (FBs). However, the observed FBs are non-ideal as they are not flat over the whole 2D Brillouin zone and overlap strongly with other bands. Most critically, the theoretical conditions for the existence of ideal FB in Kagome metals, beyond the simplest Kagome lattice model, are unknown. Here, based on tight-binding model analyses of the interplay between d-orbital symmetry and underlying Kagome lattice symmetry, we establish such conditions of minimal FB model. We show that for a pure TM Kagome lattice, only <!--[if gte msEquation 12]> style='font-size:12.0pt;mso-ansi-font-size:12.0pt;mso-bidi-font-size:12.0pt;
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normal'>y ^{style='mso-bidi-font-style:normal'>2}} , , and orbitals do not. The condition for the latter four orbitals to produce a FB is to rotate them so that they will conform the C₃ symmetry of the Kagome lattice. For intercalated TM Kagome lattices, the Kagome-hexagonal intercalation exhibits always a FB, while the case for the Kagome-trigonal intercalation is conditional. Another condition to isolate the FB arising from individual d-orbitals is to increase the crystal field splitting. These are a clear explanation for the existence of ideal FB in 2D MOF/COF instead of 3D or exfoliated 2D kagome metals.