Fig. 1: SiO2 NR waveguide device and surface phonon polaritons (SPhP) properties.

a Schematic of heat conduction measurement of a SiO₂ NR waveguide, showing the heat conduction by SPhP and phonon in the waveguide between the hot and cold reservoirs, radiation loss from the waveguide, and direct background radiation between the two reservoirs. Each of these heat transfer pathways was measured and analyzed in this work to extract heat conduction mediated by SPhP in the waveguide. b SEM images of a representative SiO₂ NR device. Images of the SiO2 NR measured in this work are shown in Supplementary Fig.S2. c Dispersion relation and d Propagation length of SPhP modes supported on SiO₂ NRs. The SiO₂ NRs have rectangular cross-sections with width of 10 µm and thicknesses of 50, 500, and 1000 nm. The solid and dash curves correspond to the quasi-TM and quasi-TE modes, respectively. The insets in (c) show the |E| field distributions of two representative modes. The gray area marks the metallic region, and the light green area marks the region with refractive index less than 1 where no valid propagation mode is supported. e Propagation length of SPhP along the thermal reservoir. The thermal reservoir supports two propagation modes (Supplementary Fig. S8) and here we focus on the mode which is primarily attributed to the conversion between SPhP and heat. The thickness of SiO₂, Pt and Fe₃O₄ are set to 44 nm, 110 nm and 350 nm, as in the experiments. The refractive indices are taken from literature^28,30,31.