Confronting theoretical results of localized and additional surface plasmon resonances in silver nanoparticles with electron energy-loss spectroscopy measurements

Raza et al. recently observed the extraordinarily large energy blueshifts of localized surface plasmon resonances and additional surface plasmon resonances in silver nanoparticles encapsulated in silicon nitride [S. Raza, S. Kadkhodazadeh, T. Christensen, M. D. Vece, M. Wubs, N. A. Mortensen, and N. Stenger, Nat. Commun. 6, 8788 (2015)], which are not fully understood yet. By using the quantum model consisting of two subsystems, respectively, for describing the center of mass and intrinsic motions of conduction electrons of a metallic nanosphere and a coupling occurring between the center of mass and conduction electrons outside the metallic nanosphere, we first deduced the general energy and line broadening size dependence of localized surface plasmon resonances, which removes the divergent defect of usual $1/R$ size dependence. Second, we proposed that the additional surface plasmon resonance in a metallic nanosphere originates from the transition of the first excited state to the ground state of the center-of-mass subsystem with energy levels corrected by degenerate-state pairs of the system composed of the center of mass and intrinsic motions of conduction electrons. Then, we implemented this generation mechanism of additional surface plasmon resonances for silver nanoparticles encapsulated in silicon nitride, and the calculated results are consistent with experimental results. Furthermore, we obtained an energy expression of localized surface plasmon resonances, with which we successfully explained the extraordinarily large energy blueshifts of localized surface plasmon resonances in few-nanometer silver nanoparticles encapsulated in silicon nitride. Finally, we calculated the localized and additional surface plasmon resonance energies of silver nanoparticles resting on carbon films, and the calculated results perfectly explain the experimental measurements of Scholl et al. [J. A. Scholl, A. L. Koh, and J. A. Dionne, Nature (London) 483, 421 (2012)]. Within this quantum model, the optical properties of metallic nanoparticles are completely determined by degenerate-state or nearly degenerate-state pairs of the system composed of the center-of-mass and intrinsic motions of conduction electrons. Our calculations also show that additional surface plasmon resonances play almost an equal role as localized surface plasmon resonances for metallic nanoparticles excited by fast-moving electrons.

Figure 1

(a) Surface plasmon resonance energies (blue crosses) measured in the EELS experiment for silver NPs dispersed on a silicon nitride substrate [48]. The fitting function is $\hslash {\mathrm{\Omega }}_q\left(R\right)=3.25-3.46/R+51.56/R^2-117.51/R^3$, which is indicated by the green line. (b) Measured linewidths of localized surface plasmon resonances (red crosses) of individual silver NPs coated with a silica shell [22]. The fitting function is $\hslash \gamma \left(R\right)=1.14-47.39/R+729.06/R^2-3428.79/R^3$, which is indicated by the blue curve.

Figure 2

The ASPR energy of silver NPs encapsulated in homogeneous silicon nitride versus the particle radius. Blue crosses and red squares, respectively, denote measured ASPR energies displayed in Fig. 5 of Ref. [46] and calculated ASPR energies. The first two measured ASPR energies perfectly correspond to the calculated LSPR energies indicated by black empty circles. The blue squares of the lower panel denote the values of the input parameter $\hslash {\omega }_s$ for corresponding silver NPs.

Figure 3

The LSPR energy of silver NPs encapsulated in silicon nitride versus the particle radius. The blue crosses and red pentagrams, respectively, denote measured LSPR energies by Raza et al. [46] and calculated LSPR energies. The green curve indicates the fitting function $\hslash {\mathrm{\Omega }}_q\left(R\right)=2.7534+2.4960/R-1.3177/R^2-0.6452/R^3$ of Eq. (5) to calculated LSPR energies. The blue squares of the lower panel denote the values of the input parameter $\hslash {\omega }_s$ for corresponding silver NPs. In this experimental settings, the input parameter $\hslash {\omega }_s$ remains almost invariable for different silver NPs.

Figure 4

The surface plasmon resonance energy versus the particle diameter. The blue, red, and green squares, respectively, denote measured plasmon energies by Scholl et al. [47], calculated energies of LSPRs and ASPRs. The black squares denote the values of the input parameter $\hslash {\omega }_s$ for corresponding silver NPs. For clarity, all data are depicted in two plots, (a) and (b).