Size Dependence of the Plasmon Ruler Equation for Two-Dimensional Metal Nanosphere Arrays

We have utilized the discrete dipole approximation to study the localized surface plasmon resonance in infinite, periodic two-dimensional arrays of gold nanospheres with the nanospheres arranged according to the {100} face of an fcc crystal. Specifically, we have performed a systematic study of the sensitivity of both the plasmon resonance wavelength shift and extinction properties considering nanosphere diameters ranging from 20 to 100 nm, and for nanosphere gap distances ranging from 0.5 to 6 times the nanosphere diameter. In doing so, we find that the same universal decay length of the plasmon resonance wavelength shift of about 0.2 units of the nanosphere size that was previously found by Jain et al.(1) for nanoparticle dimers is also operant for two-dimensional arrays. However, we also find that the universality of the plasmon ruler is only valid for arrays with nanospheres smaller than a critical nanosphere diameter of about 70 nm, whereas for larger nanosphere diameters, a decrease in the extinction efficiency as the gap distance decreases and a reduction in the decay constant are observed. Both of these size-dependent optical responses are qualitatively interpreted using a semianalytical coupled dipole approximation that accounts for structural retardation due to the geometric arrangement of the nanospheres, as well as single sphere retardation due to both dynamic depolarization and radiative damping effects. Using the semianalytical theory, we find that the size dependence is primarily due to the effects of dynamic depolarization and structural retardation, which reduces the coupling strength, changes the extinction efficiency trend, and also reduces the decay constant of the plasmon ruler equation for larger diameter nanospheres; similar results were found for infinite, 2D arrays of nanospheres in hexagonal and simple cubic arrangements, Finally, the semianalytical theory is utilized to predict a size dependence of the plasmon ruler for dimers starting at the same critical diameter of 70 nm. However, we find that the size effect is weaker for dimers than for the array case due to the significant reduction in structural retardation for dimers as compared to the array case.