Extended Data Fig. 1: Mode properties of metal-semiconductor nanoparticles.

a-b, Mie scattering spectra of gold nano-spheres (a) and gold nano-discs (b) in air for planar incident waves. While the excitation of higher order modes is evident for spheres, only the fundamental electrical dipole mode clearly appears for discs owing to the symmetry; the higher modes in discs are not efficiently excited by the uniform driving field. In nano-lasers, however, higher-order plasmonic modes are driven by local emitters and can be efficiently excited via near-field interactions. c, Schematic depicting mode coupling between plasmonic and semiconductor disc modes for three representative cases: (i) Non-lasing metallic luminescence when a semiconductor disc is too thin. Because of the large differences of the modes in energy, mode coupling is weak, and the lowest order modes are largely plasmonic. Because of the mode proximity, it is difficult to selectively amplify only the ED mode; (ii) Higher-order hybrid laser, where multiple dielectric-like modes are present within a gain bandwidth; (iii) A hybrid dipole laser — which may be regarded a ‘spaser’ — where the individual modes in the metal and semiconductor discs have similar energies. Strong coupling occurs between ED modes, separating the hybrid plasmon-like mode from the other hybrid modes. This mode shift may be considered as the effect of the refractive index of the semiconductor on the plasmonic mode. However, mode coupling is a more accurate explanation as the effective index experienced by the plasmonic ED mode matches the index of the ED mode in the dielectric medium. Note that the MD modes, the lowest order modes in dielectric discs, are not efficiently coupled with the plasmonic ED mode because of the field symmetry. d, FDTD simulation of metal-semiconductor discs with different diameters, as depicted in the inset. The resonance wavelength, quality factor, and mode confinement factor in the semiconductor vary as a function of the diameter ratio from 0 (III-V only) to infinity (on a gold substrate). The resonance wavelength of the hybrid ED mode increases dramatically from 600 nm to 1300 nm at size matching and then 1220 nm for oversized gold. The quality factor reaches its maximum at a diameter ratio of 0.5, partly due to optimal mode energy matching and partly due to reduced metallic absorption at 800–900 nm. The Q factor approaches slightly over 10 at infinite gold, with a quarter of electromagnetic energy residing in the semiconductor while the remaining three quarters are stored in the metal. See Supplementary Fig. 2 for more examples of Mie scattering spectra. e, Simulated Mie scattering spectra of a complex of semiconductor (n = 3.5) and gold rhombus-like particles, each with a side length of 250 nm and heights of 130 nm and 100 nm, respectively, for various gap distances. The arrow indicates the lowest-order mode at contact. f, Electric field amplitude profiles of the semiconductor-gold particles in contact (dashed outline). g, Simulated Mie scattering spectra of a semiconductor on infinite substrates for different gap distances. The arrow indicates the lowest mode at contact. h, Electric field amplitude profiles of the semiconductor-on-gold structure. i, Simulated Mie scattering spectra of a rhombus-shaped semiconductor on an infinite substrate for different rhombus interior angles.