Fig. 4: Strong plasmon–hydron coupling.

a, Theoretical prediction for the graphene electron cooling rate in contact with different liquids, within different treatments of interactions. The cooling rate is strongly overestimated if no electron–electron interactions are taken into account (blue symbols), and underestimated if the electron–liquid interactions are considered only to first order (orange symbols). b, Graphene surface excitation spectrum \({{{\rm{Im}}}}\,[{g}_{{{{\rm{e}}}}}(q,\omega )]\), calculated at a chemical potential μ = 180 meV and temperature T_e = 623 K, renormalized by the presence of water according to equation (5). The white dashed lines are guides to the eye showing the strongly coupled plasmon–hydron mode. Inset: bare and renormalized graphene spectra at fixed wavevector q₀ = 0.15 nm⁻¹. c, Comparison between the spectrally resolved energy transfer rates obtained to first order and to arbitrary order in the solid–liquid interaction. Higher-order effects enhance the energy transfer rate at low frequencies.