Extended Data Fig. 5: Plasmon hybridization model for thin (34 nm thick) and thick (47 nm thick) AuNCats.

a, Decomposed B2 symmetric basis mode of a LSP mode of a thin AuNCat. The mode can be expressed as a simple addition of dipole modes of each constituent Au nanoring. b, Decomposed B2 symmetric basis mode of each LSP mode of a thick AuNCat, that is, SBM and GDM. The modes can be expressed as a linear combination among two dipolar modes of each Au nanoring and a gap mode. The LSP modes of Au nanorings indicate the charge density inside the surface. The gap mode is calculated outside the surface by subtracting the two dipolar modes of Au nanorings from the SBM mode of the AuNCat. c,d, Theoretical scattering cross-section of a AuNCat (middle panel) and Au nanorings that compose the AuNCat (right and left panels) with parallel polarization of the electric field along the C₂ main axis. The inset images indicate respective surface charge distributions at each LSPR peak. Small difference in the resonance frequency of the thin AuNCat (c) means the two dipolar plasmon modes of nanorings merely couple with each other (d). e,f, Schematic illustration of the hybridization model to construct LSPRs of a AuNCat. As per plasmon hybridization between two symmetric dipole modes of each Au nanoring, simple subtraction and addition of these modes are generated. The subtracted mode has zero dipole moment and is optically dark, so that the only one LSP, the addition of ring dipolar modes, can be excited by planewaves as in c (e). In the case of the thick AuNCat, the simple subtraction and addition of these modes are intermediately formed; the modes are hybridized further with a B2 symmetric gap mode to generate the LSP modes of the AuNCat. Created SBM and GDM are optically bright with their own dipole moment, whereas the symmetric anti-bonding mode (SAM) is optically dark, which cannot be excited by planewaves (f).