Determinant role of size-dependent electron relaxation processes in the nonlinear luminescence emission of resonantly excited gold nanorods

The two-photon luminescence (TPL) of gold nanoparticles (NP) was shown to result from the excitation of hot carriers, the plasmonic NP resonances playing an important role both for plasmon enhanced absorption and plasmon enhanced emission. However, the exact parameters enabling to control or optimize the NP nonlinear luminescence still need to be understood in detail. In this paper, we report the two-photon excited photoluminescence of single gold nanorods exhibiting identical aspect ratio (close to 4) and thus identical plasmonic resonances, but increasing volumes V (707 <V< 160 10$^3$ nm$^3$ i.e. rod diameters varying between 6 and 40 nm). The two-photon luminescence intensity of a high number of colloidal nanorods was investigated at the single object level, combining polarization resolved TPL and simultaneously acquired topography. Non-monotonic TPL variations are evidenced, nanorods with an intermediate size (diameter around 10 nanometers) exhibiting the highest TPL signal intensity. A model is proposed considering both the local field enhancement effects at the NP and the size-dependent electron thermalization processes. BEM (Boundary Elements Method) simulations are used to compute the fields at both the transverse and longitudinal plasmon resonance. A good fitting of the experimental data is obtained considering integration of the fields over the whole the NP volume