Confronting theoretical results of localized and additional surface plasmon resonances in silver nanoparticles with EELS measurements

TL;DR

This paper develops a quantum model to explain large energy blueshifts and additional surface plasmon resonances in silver nanoparticles, aligning theoretical predictions with experimental EELS measurements.

Contribution

It introduces a new quantum model that accurately predicts surface plasmon resonance energies and explains large blueshifts in silver nanoparticles.

Findings

Derived size-dependent energy and line broadening of localized surface plasmons.

Proposed a generation mechanism for additional surface plasmon resonances.

Achieved excellent agreement between theoretical calculations and experimental data.

Abstract

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, 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 occurred between the center of mass and conduction electrons outside the metallic nanosphere, we firstly 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. Secondly, 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 well consistent with experimental results. Furthermore, we obtained a new 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..

Equations

$${\mathcal{H}}_C$$

$${\mathcal{H}}_r$$

$${\mathcal{H}}_c$$

$$d_{\alpha \beta }=⟨\alpha |{\xi }_z(R^3/|\vec{\xi }{|}^3-1)\Theta (|\vec{\xi }|-R)|\beta ⟩,$$

$$[-\frac{{\hslash }^2}{2m}{\nabla }^2+V_{eff}(|\vec{\xi }|)]{\psi }_{\alpha }(\vec{\xi })={ϵ}_{\alpha }\psi (\vec{\xi }),$$

$${\Omega }_q(R)={\Omega }_p(R)+\frac{2{\mathcal{A}}^2}{\hslash }\sum {}_{{}_{\{\alpha \beta \}}}{\mathcal{F}}_{\alpha }{f}_{\beta }\frac{|d_{\alpha \beta }{|}^2{ϵ}_{\beta \alpha }}{{ϵ}_{\alpha \beta }^2-(\hslash {\Omega }_p{)}^2},\text{vspace}-6pt$$

$$\hslash \gamma (R)=2\pi {\mathcal{A}}^2\sum {}_{{}_{\{\alpha \beta \}}}{\mathcal{F}}_{\alpha }{f}_{\beta }|d_{\alpha \beta }{|}^2\delta ({ϵ}_{\alpha \beta }-\hslash {\Omega }_p),$$

$$f(R)=f_0+\frac{A}{R}+\frac{B}{R^2}+\frac{C}{R^3}.$$

$$\hslash {\Omega }_a(R)=\hslash {\Omega }_p(R)+2\mathcal{A}\sum {}_{{}_{\{\alpha \beta \}}}{\mathcal{F}}_{\alpha }{f}_{\beta }|d_{\alpha \beta }|,$$

$$\mathcal{E}(ϵ,{ϵ}_{\alpha })=\frac{2}{\pi }\frac{\sqrt{{ϵ}_{{}_T}{ϵ}_{\alpha }}}{(ϵ-{ϵ}_{\alpha }{)}^2+4{ϵ}_{{}_T}{ϵ}_{\alpha }},$$

$$|\hslash {\Omega }_p-|{ϵ}_{{\alpha }_{\beta }}||\le \mathcal{A}|d_{\alpha \beta }|,$$

$$|\hslash {\Omega }_p-|{ϵ}_{{\alpha }_{\beta }}||\le (\sqrt{4{ϵ}_{{}_T}{ϵ}_{\alpha }}+\sqrt{4{ϵ}_{{}_T}{ϵ}_{\beta }}).$$

$$\hslash {\Omega }_q(R)=\hslash {\Omega }_p(R)\pm \frac{\pi {\mathcal{A}}^2}{\tau }\sum _{\{\alpha \beta \}}(f_{\beta }-f_{\alpha })|d_{\alpha \beta }{|}^2,$$

$$d_{\alpha \beta }\propto \frac{1}{V}{\int }_{R+\delta }^{\infty }{f}_{\alpha }(k_{\alpha }r)(\frac{R^3}{r^3}-1)f_{\beta }(k_{\beta }r)r^{3}dr.\text{vspace}\ast -7pt$$

$$S\approx \frac{2{\mathcal{A}}^2}{\hslash \tau }{\int }_{-Z}^0[f_{\beta }-f_{\alpha }]\frac{|d_{\alpha \beta }{|}^2{ϵ}_{\beta \alpha }}{{ϵ}_{\alpha \beta }^2-(\hslash {\Omega }_p{)}^2}d{ϵ}_{\alpha \beta },\text{vspace}-1mm$$

Full text

I. INTRODUCTION

Apart from the bulk plasmon resonance, the conduction electrons in a metallic nanoparticle (NP) support another more important self-sustaining collective oscillation, which is the well-known localized surface plasmon resonance (LSPR) and endows metallic NPs with particular abilities, such as local electromagnetic field rapid oscillation inside metallic NPs, colossal enhancement of local electric fields, extreme sensitivity to dielectric environment variations, and squeezing light beyond the diffraction limit ukr ; gra . These specialities of LSPRs render the manipulation of visible light waves at the nanoscales possible. Various applications continue to flurish in many areas, such as surface-enhanced Raman scattering pau , improvement of light confinement of photovoltaic devices haa , single-molecule detection pzi , single-photon generation and potential applications for the quantum information transfer ava , and compact laser-driven accelerators dba . It is also found applications in biochemistry and biomedical fields, such as biosensing ova , drug delivery aky , cancer phototherapy sla . Recently, real-space and real-time observation of a plasmon-induced dissociated reaction of a single dimethyl disulfide molecule has been realized emi .

Paralleling with the explorations of deep physics of LSPRs, novel findings and ideas continue to emerge. Aizpurua et al. proposed a mechanism to actively control the optical response of metallic NPs by applying an external dc bias across a narrow gap cda ; Guzzinati et al. probed the symmetry of LSPRs with phase-shaped electron beams ggi ; Govorov et al. showed that the coherent ultrafast non-dissipative energy transfer could take place between two gold NPs with an interspaced silver island rev . Such advances are boosted by the development of powerful nanoscale characterization techniques. Nowadays, electron energy-loss spectroscopy (EELS) combined with an electron monochromated and aberration-corrected scanning transmission electron mircroscopy is able to achieve energy resolution even down to $9$ meV and Ångström spatial-resolution in the studies of individual plasmonic structures wji ; aga ; jne ; mbo .

The LSPRs strongly depend on size, shape, temperature, material, dielectric environment lje ; bh ; kkl ; avi ; yoa , which provides multiple variables tailorable for various applications. Silver nanostructures are ideal for plasmon studies owing to low intrinsic losses, narrow LSPR linewidths and large optical field enhancements wji . For metal silver, the fully occupied $4d$ bands described as a polarizable medium modify the frequency of the classical Mie plasmon resonance from $\omega_{{}_{M}}=\omega_{p}/(1+2\epsilon_{m})^{1/2}$ to $\omega_{{}_{M}}=\omega_{p}/[Re{\,(\epsilon_{d})}+2\epsilon_{m}]^{1/2}$ anw ; lj , where $\omega_{p}=(4\pi\rho_{e}e^{2}/m)^{1/2}$ is the plasmon frequency of bulk metal, $\epsilon_{m}$ and $\epsilon_{d}$ respectively the dielectric constant of the environment and complex dielectric function of $4d$ bands; while $e$, $m$ and $\rho_{e}$ are the electron charge, mass and density, respectively. The localized $4d$ electrons fail to fully screen conduction electrons outside NPs, which prevails over the spill-out effect of conduction electrons and tends to blueshift the LSPR energies. Many different experiments have confirmed the LSPR energy blueshifts of silver NPs in various dielectric environments tig ; kru1 ; ckp ; mrc . It is obvious that atomic configurations in the vicinity of the surface and spill-out effect of conduction electrons become more and more prominent with the particle size decreasing, which causes the optical properties of metallic NPs to be size-dependent.

From the classical limited mean free path effect or the calculation of dielectric constant of silver particles by using quantum methods, the LSPR linewidths is usually described by $\gamma(R)=\gamma_{0}+g$ $v_{{}_{F}}/R$ kru1 ; kru2 ; kaw ; kwa ; link ; wgu1 , where $v_{{}_{F}}$ and $\gamma_{0}$ are Feimi velocity and intrinsic linewidth. The dimensionless $g$ is considered to be a constant, however, quite different values of $g$ were obtained from experimental data and theoretical models bh ; ale . The LSPR line broadening of metallic NPs is the result of exciting single particles into electron-hole states, which is the well-known Landau damping mechanism wgu1 ; jin . By generalized nonlocal optical response (GNOR) model, Mortensen et al. obtained a positive term $1/R^{2}$ for line broadenings and frequency shifts mna ; rso2 . Based on experimental data ckp , self-energy approach for particle polarizability lia , and dispersion relation corrected by the spatial spreading of induced charge app , the energy shifts of LSPRs were found to obey the same $1/R$ size-dependence. However, this $1/R$ law suffers from the divergent defect for few-nanometer metallic NPs.

It has been well established that the LSPR is the dominant response mode for metallic NPs with radii less than $20$nm to external light excitation cth ; bh ; rr ; kk . By using electron energy-loss spectroscopy (EELS), Raza et al. recently observed additional surface plasmon resonances (ASPRs) for silver NPs encapsulated in silicon nitride with radius range from $4$nm up to $20$nm rso1 , which were interpreted as the combined effect of many multipole surface plasmon resonances. While the ASPR disappearance of silver NPs with radii below $4$nm was ascribed to the decreasing of EELS signals from high-order modes. The measured ASPR energies of individual silver NPs do not have a constant value, but the multipole modes combination viewpoint of ASPRs fails to yield a feasible energy calculation method. Furthermore, except the dipole surface plasmon resonance, there are no other individual multipole modes observed in experiments.

Raza et al. also observed an extraordinarily large LSPR energy blueshift $\sim 0.9$ eV when the particle radius decreases from $4$ nm to $1$ nm. Scholl et al. observed an energy blueshift $\sim 0.5$ eV for individual ligand-free silver NPs resting on carbon films sjo , and the similar result was also observed for silver NPs dispersed on a silicon nitride substrate rso3 . While the negligible energy blueshifts $\sim 0.25$ eV appear for silver NPs embedded in solid Xe, Ar, C2H4 ckp ; kru1 . These experimental results show that the LSPR energy blueshifts of silver NPs strongly depend on their surroundings. Besides the screening effect of $4d$ electrons and the spill-out effect of conduction electrons, it is believed that the stronger quantum confinement, single-particle excitations, nonlocal response and numerous structural surface defects are the possible factors jointly determining the energy blueshifts of silver NPs cth ; lje . To understand the extraordinarily large LSPR energy blueshifts of silver NPs is an arena to test various models or theories. To the best of our knowledge, all theoretically predicted energy blueshifts of silver NPs are systematically less than experimental results rso3 ; mrc .

Nowadays, the understanding of LSPR energy shifts of metallic NPs is still poor. Quantitative predictions require the full consideration of quantum effects, which become more and more important with the particle size decreasing. The time-dependent density function theory (TD-DFT) offers the possibility to address the optical response of plasmonic systems at the quantum ab initio level rer ; vg . However, the TD-DFT becomes computationally prohibitive because their computational cost grows as fast as $O(N_{e}^{3})$ such that their reach is limited to systems with few thousands of electrons. The semi-classical hydrodynamic Drude models (HDMs) deal with differential equations of macroscopic particle density and current density rather than single electron orbitals gaining the advantage of numerical efficiency compared with TD-DFT, which manifests the HDMs as a promising tool suitable to study the optical properties of large plasmonic structures. By adding the gradient of ground electron density to the Thomas-Fermi kinetic energy, the hard-wall boundary condition of HDMs is removed and the spill-out effect can be considered, which is called quantum hydrodynamic theories (QHTs) in the literature. By assuming electrons in different states mutually collide, a viscous stress tensor yielding a dynamical correction to the kinetic energy functional is expected to play the role of the Landau damping mechanism ccr1 . It is believed that QHTs combined with suitable electron ground-state density are able to compete with the TD-DFT ccr2 . However, it is still challenging to build a QHT compatible with all experimental findings yan . It has been shown that the plasmon resonance energies of a complex nanostructure are equivalent to the electromagnetic interactions of plasmons from structures with simpler configurations pre . Therefore, a thorough understanding of basic systems can facilitate the design of highly sophisticated plasmonic nanostructures with desired optical properties. In this paper, we will use a quantum model specially constructed for metallic nanospheres to solve some unfathomed problems associated with silver NPs.

II. THE QUANTUM MODEL

For metallic NPs, the ionic cores can be treated as a uniform positively charged background according to the jellium model mat , which is extensively adopted in numerous theoretical models, such as TD-DFT ew , the matrix random-phase approximation method yc , and field theory of quantum plasmonics hva . Due to strong quantum confinement, conduction electron states are quantized into discrete levels. For a metallic nanosphere containing $N$ atoms with radius $R$ encapsulated in the medium with dielectric constant $\epsilon_{m}$, based on the jellium model one can construct a quantum model by separating the conduction electron coordinates into the coordinate of center of mass and the relative coordinates (SCRM). The total Hamiltonian $\mathcal{H}_{T}$ of SCRM can be expressed as the sum of two sub-Hamiltonians $\mathcal{H}_{C}$ and $\mathcal{H}_{r}$ respectively for describing the collective and intrinsic motions of conduction electrons, and the coupling $\mathcal{H}_{c}$ between center of mass and conduction electrons outside the nanosphere, they respectively are wg ; wgu0

[TABLE]

the sub-Hamiltonian $\mathcal{H}_{C}$ has the standard harmonic oscillator structure with the frequency $\Omega_{p}=\omega_{s}\sqrt{1-N_{out}/N}$, where $\omega_{s}$ is the unique input parameter of the SCRM and slightly varies around the classical Mie resonance frequency $\omega_{{}_{M}}$ due to numerous surface structural defects, $N_{out}$ the number of conduction electrons outside the nanosphere, and the coefficient $\mathcal{A}=\frac{e^{2}}{4\pi\varepsilon_{0}R^{3}}\sqrt{\frac{\hbar N}{2m\Omega_{p}}}$. The matrix element $d_{\alpha\beta}$ calculated between two states $|\alpha\rangle$ and $|\beta\rangle$ of $\mathcal{H}_{r}$ is given by

[TABLE]

where $\Theta(x)$ is the Heavyside step function. Within the mean-field approximation, the energy levels and corresponding wavefunctions of conduction electrons can be obtained by solving the Kohn-Sham equation

[TABLE]

where the effective potential $V_{eff}$, usually including ionic background potential, Hartree potential and exchange and correlation potnetial, can be obtained by local density approximation calculation gui2 ; mat . The quantum states of total Hamiltonian $\mathcal{H}_{T}$ can be expressed as $|I,\alpha\rangle$, where $I$ and $\alpha$ are the quantum numbers respectively characterizing the states of the center of mass and intrinsic motions of conduction electrons.

Within the SCRM framework, the LSPR frequency is given by

[TABLE]

where the sum is over all the non-degenerate state pairs $\{|0,\alpha\rangle,|1,\beta\rangle\}$ of the total Hamiltonian $\mathcal{H}_{T}$ with $0<\epsilon_{\alpha}-\epsilon_{\beta}=\epsilon_{\alpha\beta}\neq\hbar\Omega_{p}$; $f_{\beta}=1/(1+e^{(\epsilon_{\beta}-\mu)/k_{B}T})$ the Fermi-Dirac distribution; $T$, $k_{B}$ and $\mu$ the electronic temperature, the Boltzmann constant and the chemical potential respectively. We define $\mathcal{F}_{\alpha}=1-f_{\alpha}$. The line broadening of LSPRs caused by the Landau damping mechanism can be expressed as

[TABLE]

where $\delta(\epsilon_{\alpha\beta}-\hbar\Omega_{p})$ is Dirac's $\delta$ function representing the condition of energy conservation.

For a sodium nanosphere containing $1760$ atoms, the LSPR frequency and line broadening calculated by using the SCRM are perfectly consistent with the results of TD-DFT calculations wg , which shows that the SCRM is reliable to study the optical properties of metallic NPs.

III. THE LSPR ENERGY SIZE-DEPENDENCE

The LSPR energy and linewidth are of crucial significance in many applications and intrinsically limit the optimization of nanosized optical devices involving metallic NPs. Within the SCRM, the LSPR energy shift and line broadening are determined by energy states of conduction electrons and the input parameter $\omega_{s}$, which include all effects of material, shape, size, dielectric environment, and surface details, such as facets and vertices mba . The energy levels and wavefunctions of conduction electrons can be obtained in principle by solving Eq. (2). Thus, the effective potential of conduction electrons plays a central role in the optical properties of metallic NPs. However, the real effective potential of conduction electrons is so complex that to obtain accurate energy levels and numerical wavefunctions from Eq. (2) is extremely time-consuming and quickly becomes prohibitive with the particle size increasing, which is the very reason for the exploration of other calculation schemes without resorting to single energy levels and wavefunctions.

Eqs. (3) and (4) show that the size-dependences of the LSPR energy shift and line broadening determined by $|d_{\alpha\beta}|^{2}$ and the coefficient $\mathcal{A}^{2}$ have the same form. Within the SCRM framework, the general size-dependence of LSPR energies and linewidths can be worked out and the result is s1

[TABLE]

For the situation of metallic nanospheres with different sizes encapsulated in homogeneous medium, the coefficients $A$, $B$ and $C$ are approximate constants with proper dimensions. The constant $f_{0}$ is the intrinsic linewidth or the classical Mie plasmon resonance energy. Besides the first two terms, two high order terms of $1/R$ arise, which completely originate from quantum effects and are able to eliminate the divergent defect of the usual $1/R$ law for few-nanometer metallic NPs.

Fig. 1 shows the fits of Eq. (5) to measured plasmon energies of silver NPs dispersed on a silicon nitride substrate rso3 , and linewidths of individual silver NPs coated with a silica shell bh . The fitting curves are able to globally describe the experimental data, especially for the linewidths in Fig. 1 (b). The large energy spread of plasmon resonances shown in Fig. 1 (a) is related to the fact that part of experimental data are ASPR energies and the ASPR and LSPR energies do not obey the same size-dependence.

The optical properties of metallic Nps are extremely sensitive to the surface atomic configurations. The chemical control of surface layers via ligand exchange could yield abnormal optical response behaviors shen , which shows that electron density tail plays a crucial role in optical response of metallic NPs. In deducing Eq. (5), the possible chemical bonding processes occurred between surface atoms of metallic NPs and external molecules, which is able to dramatically alter the spatial distribution of conduction electron density tail, interfacial dielectric constant and the effective potential, were not considered. Therefore, Eq. (5) can not describe the LSPR size evolution behavior of metallic NPs in complex chemical environments.

IV. THE ASPRs OF METALLIC NPs

The ASPR was firstly identified as a surface mode for a semi-infinite metal by Bennett with hydrodynamic equations bal , and appeared in microscopic calculations for sodium particles lia ; ew . However, the generation mechanism of ASPRs is still unclear so far, and many quite different viewpoints of ASPRs exist. For examples, Raza et al. regard the ASPR in silver NPs as the merger of many multipole modes; Liebsch believed that the ASPR is the excitation that has dipolar angular character but with an additional node in the radial distribution of the dynamical surface screening charge compared to that of the principal Mie plasmon oscillation lia ; while Tsuei et al. deemed the ASPR the resonance in the electron-hole pair spectrum and no longer bearing purely dipolar character tku . Unlike the LSPRs, the ASPRs have neither been extensively investigated nor attracted much attention until recently.

It is surprising that there exists a simple generation mechanism of ASPRs within the SCRM framework. For a degenerate state pair $|0,\alpha\rangle$ and $|1,\beta\rangle$ with $\epsilon_{\alpha}\sim\hbar\Omega_{p}+\epsilon_{\beta}$, the perturbation energy correction to states $|0,\alpha\rangle$ and $|1,\beta\rangle$ is easy to calculate and the result is $\pm\mathcal{A}|d_{\alpha\beta}|$ for $|0,\alpha\rangle$ and $\mp\mathcal{A}|d_{\alpha\beta}|$ for $|1,\beta\rangle$. Because the sub-Hamiltonian $\mathcal{H}_{C}$ describes collective motions of all conduction electrons, the perturbation energy correction to each state of a degenerate state pair is virtually the correction to states of $\mathcal{H}_{C}$. It seems that these two sets of energy corrections to the ground state $|0\rangle$ and the first excited state $|1\rangle$ of $\mathcal{H}_{C}$ with opposite signs would offset each other and produce zero results. However, the positive energy correction $\mathcal{A}|d_{\alpha\beta}|$ to the ground state $|0\rangle$ would increase the collective oscillation energy of conduction electrons, which violates the principle that the ground state of a system would have the energy as low as possible. Therefore, the ultimate result of perturbation energy correction of the degenerate state pair $|0,\alpha\rangle$ and $|1,\beta\rangle$ is $-\mathcal{A}|d_{\alpha\beta}|$ for the ground state $|0\rangle$ and $\mathcal{A}|d_{\alpha\beta}|$ for the first excited state $|1\rangle$. Thus, all the degenerate state pairs would yield a new surface plasmon resonance with the energy given by the expression

[TABLE]

where the sum is over degenerate state pair set $\{|0,\alpha\rangle$, $|1,\beta\rangle\}$. It is natural to think of this new surface plasmon resonance as the ASPR, because in metallic NPs there only exist three kinds of plasmon resonances, namely the LSPR, ASPR and bulk plasmon resonance.

For a metallic NP with radius $R$ less than $20$ nm encapsulated in homogeneous dielectric medium, the single-particle effective potential $V_{eff}$ determining quantum states of conduction electrons is not only complex but hard to obtain. Eq. (6) shows that the ASPR energy mainly depends on matrix elements $d_{\alpha\beta}$, which are not sensitive to energy levels and wavefunctions of conduction electrons. It has been shown that most of energy levels and wavefunctions of the Schrödinger equation with a Woods-Saxon-like potential are almost the same as those of the spherical potential well of finite depth wg1 , and the observable difference between corresponding energy levels of two potentials focuses on energy levels well above the Fermi energy. However, the contribution of states with high energy levels to Eq. (6) is suppressed by the Fermi-Dirac distribution factor. On the other hand, the wavefunctions of spherical potential well of finite depth decay slightly faster well outside the metallic nanosphere than those of Woods-Saxon-like potential well. However, the deviations of matrix elements calculated by using wavefunctions of the spherical potential well of finite depth from accurate results are negligible for not very small nanospheres. Therefore, the spherical potential well of finite depth equal to the sum of Fermi energy and work function is able to substitute for the complex single-particle effective potential to calculate the ASPR energy of a metallic nanosphere.

There are no quantum states strictly satisfying the condition $\epsilon_{\alpha}=\epsilon_{\beta}+\hbar\Omega_{p}$, thus the main obstacle to calculate the ASPR energies by using Eq. (6) is how to single out all degenerate state pairs from possible quantum states $|0,\alpha\rangle$ and $|1,\beta\rangle$. The energy levels of conduction electrons can be broadened out according to the formula hni ,

[TABLE]

where $\epsilon_{{}_{T}}=(\hbar k_{0})^{2}/2m$, and $k_{0}=0.13N^{-1/3}$Å*-1*. Thus, the width of an energy level changes from zero to $2\sqrt{4\epsilon_{{}_{T}}\epsilon_{\alpha}}$. We define truly degenerate state pairs (TDSPs) and nearly degenerate state pairs (NDSPs) responsible for ASPR energies and LSPR energy shifts as state pairs with energies respectively satisfying

[TABLE]

and

[TABLE]

According to the SCRM, the ASPRs originate from the degenerate state pairs of the system. To test this viewpoint, we calculated the ASPR energies of silver NPs encapsulated in silicon nitride under the same settings of experiments done by Raza et al rso1 . The conduction electron temperature is fixed at the room temperature $T=300$K in all our calculations, and the Mie plasmon resonance frequency is calculated by using the measured frequency-dependent complex dielectric function of $4d$ band for bulk silver $\epsilon_{d}(\omega)=(59.8+55.1i)(\omega/\omega_{p})^{2}-(40.3+42.4i)(\omega/\omega_{p})+(10.05+8.06i)$ rso3 . By taking the experimental value $3.3$ of environment dielectric constant $\epsilon_{m}$ rso1 , we obtained the Mie plasmon resonance energy $\hbar\,\omega_{{}_{M}}=2.8076$ eV. The calculated and measured ASPR energies are exhibited in Fig. 2, which shows a good agreement between calculated and experimental results. Thus, we can preliminarily conclude that both the generation mechanism and calculation scheme of ASPRs based on the SCRM are reasonable. For large silver NPs, Fig. 2 shows a small discrepancy between calculated and measured ASPR energies, which is induced by using two Lorentzian or Gaussian functions with fixed full-width at half-maximum (FWHM) $0.15$ eV neglecting the variation of the LSPR line broadening with the particle size to simulate EELS spectra for determining the LSPR and ASPR energies. The first two measured ASPR energies of silver NPs with radii $4.44$nm and $5.94$nm are not consistent with calculated ASPR energies. However, these two measured surface plasmon resonances can be perfectly explained as the LSPRs.

In principle, the SCRM generation mechanism of ASPRs predicts the existence of ASPRs for all the metallic NPs. Because there are no or few TDSPs for few-nanometer metallic NPs, the ASPR energies normally shift from high ASPR energies of large metallic NPs to low energies ($\sim\hbar\Omega_{p}(R)$) according to Eq. (6). This behavior of ASPRs naturally explains the experimental observation that the ASPRs of few-nanometer silver NPs encapsulated in silicon nitride fail to be observed at high energy region containing the ASPRs of large silver NPs. Whether or not the ASPRs of few-nanometer metallic NPs to be experimentally observable strongly depends on the intensity ratio of ASPRs to background signals. The ASPR energies $\sim 2.80$ eV of silver NPs with radii $4.44$nm and $5.94$nm, and $\sim 3.11$ eV for silver NPs with radius $3.0$ nm were observed, which are shown in Fig. 3 and mistaken for the LSPR energies by Raza et al.. These experimental results are completely consistent with the SCRM predictions of ASPRs in silver NPs.

However, the SCRM generation mechanism of ASPRs does not exclude the occasional appearance of ASPRs with large resonance energies for few-nanometer metallic NPs with special sizes. Actually, for the silver NPs encapsulated in silicon nitride with radius $R\sim 2.62$ nm and the input parameter $\hbar\omega_{s}$ ranging from $2.795$ eV to $2.822$ eV, the ASPR energies vary from $3.23$eV to $3.29$eV even larger than those of large silver NPs. Small sodium clusters with special sizes are also able to support the ASPRs shown by the surface loss function and dynamical polarizability calculated respectively by using time-dependent density-functional approach and local-density approximation method lia ; ew , which indirectly shows that the SCRM generation mechanism of ASPRs in metallic NPs is reasonable.

When the radius of metallic NPs increases from few nanometers to tens of nanometers, the ASPR energy generally increases from $\sim\hbar\Omega_{p}(R)$ to high ASPR energies of large metallic NPs. When the particle radius further increases to macroscopic sizes, how does the ASPR in metallic particles evolve? Fig. 2 shows that the ASPRs in silver NPs with macroscopic sizes seem to remain the ASPR energies of large silver NPs. According to the SCRM generation mechanism, the ASPR energy is determined by matrix elements $\mathcal{A}|d_{\alpha\beta}|$ calculated between generate state pairs, which vary with the particle radius as $\sim 1/\sqrt{R}$ s1 . Therefore, the ASPR together with the LSPR in metallic NPs would evolve into the classical Mie surface plasmon resonance with the particle size increasing to macroscopic sizes. This evolving mode of surface plasmon resonances is also given by the classical electrodynamics combined with the measured dielectric function for metallic particles excited by fast moving electrons ftl ; ale2 .

The large energy spread of surface plasmon resonances for silver NPs resting on different substrates was observed in EELS experiments rso3 ; sjo . Besides shape variations, facets and the interaction between particles and the substrates, it is undoubted that the ASPRs play a significant role in the large energy spread of surface plasmon resonances.

V. THE LSPR ENERGY SHIFTS

Precise EELS experiments explicitly show that the LSPR energies of few-nanometer silver NPs shift to higher energies by remarkable deviations from the classical Mie surface plasmon resonance energy sjo ; rso3 ; rso1 . The EELS measurements also indicate that the blueshifts of LSPR energies towards higher energies are not purely monotonic but with a greater variety in peak locations when the particle size decreases to few nanometers. Both relatively small amplitude and monotonic behavior of LSPR energies predicted by known theories and models show that the LSPR energy shift of metallic NPs is poorly understood, and the extraordinarily large energy blueshifts for small silver NPs have not been satisfactorily interpreted so far mrc ; sjo ; rso3 .

If extremely powerful computation ability were possessed in the future, the TD-DFT would yield LSPR energies of metallic NPs perfectly consistent with experimental measurements. Likewise, once the sufficiently precise energy levels and wavefunctions were available, we also believe that the frequency expression Eq. (3) would produce correct results of metallic NPs. However, Eq. (3) is too sensitive to conduction electron energy levels to produce the correct results by using energy levels of the spherical potential well of finite depth. Fortunately, Eq. (3) can be transformed into an alternative form by some mathematical manipulations, which greatly lowers its sensitivity to conduction electron energy levels. The new expression of LSPR energy is s2

[TABLE]

where the signs $+$' and $-$' are respectively for the LSPR energy blue and red shifts; the sum is over NDSPs with $\epsilon_{\alpha}>\epsilon_{\beta}$. However, there are a small number of NDSPs not contributing to the LSPR energies expressed in Eq. (10), and such NDSPs should be excluded s3 . For silver NPs encapsulated in silicon nitride, we calculated the LSPR energies of silver NPs, which are measured by Raza et al. in the EELS experiments. All the calculated LSPR energies and experimental counterparts are exhibited in Fig. 3. Nevertheless, for silver NPs with radii larger than $2.58$nm, our calculated results are not consistent with measured ones. Because the line broadening of LSPRs varies with the particle radius, we think that this inconsistency is induced by the improper use of two Gaussian functions with fixed FWHM of $0.15$eV to identify the energies of LSPRs and ASPRs from EELS spectra rso1 . For silver NPs with radii smaller than $2.58$nm, only LSPRs were observed in the EELS spectra, and our calculated results are perfectly consistent with experimental measurements naturally explaining the extraordinarily large LSPR energy blueshifts measured by Raza et al. for few-nanometer silver NPs. Furthermore, we found that our calculated LSPR energies are well described by the general LSPR energy size-dependence Eq. (5), which is shown in Fig. 3.

To further test the SCRM generation mechanism of ASPRs and the LSPR calculation formula Eq. (10), we calculated the LSPR and ASPR energies of individual silver NPs resting on carbon films, which were firstly studied in EELS experiments by Scholl et al.. We noticed that the larger the particle size, the smaller the influence of carbon films on the effective dielectric constant $\epsilon_{m}$. Therefore, we can expect that the input parameter $\hbar\omega_{s}$ for silver NPs resting on carbon films would exhibit a larger variation amplitude than that of silver NPs encapsulated in homogeneous silicon nitride. According to the proposed effective dielectric constant $\epsilon_{m}=1.69$ sjo , the corresponding Mie surface plasmon resonance energy $\hbar\,\omega_{{}_{m}}=3.373$ eV is even larger than the measured LSPR energies of silver NPs with diameters $\sim 20$ nm. Considering the LSPR energy blueshifts of silver NPs, this proposed effective dielectric constant is obviously unreasonable. In our calculations, the input parameter varies in the range $3.0$ eV$<\hbar\omega_{s}<3.085$ eV. The calculated energies of LSPRs and ASPRs are shown together with measured results in Fig. 4. Most of measured plasmon resonances (blue squares) perfectly correspond to either calculated LSPRs (red squares) or ASPRs (green squares). There also exists another situation. For a silver NP with special sizes, the calculated peak locations of the LSPR and ASPR happen to be so close to each other that they virtually merge into one peak and are indistinguishable in EELS experiments. We found that three measured plasmon resonances could be perfectly explained as such merged peaks, which are indicated by arrows in Fig. 4. We found that more than half of measured surface plasmon resonances are ASPRs for silver NPs with the particle diameter ranging form $3.2$ nm to $12$ nm. Beyond this range, all measured plasmon resonances can be explained as LSPRs.

When a silver NP with diameter less than $2.0$ nm, conduction electrons have considerable possibility to stay outside the silver NP. To substitute the spherical potential well of finite depth for the real effective potential confining conduction electrons begins to deteriorate, which is the reason why the first calculated LSPR energies of Fig. 3 and Fig. 4 are explicitly smaller than experimental results. Therefore, within the SCRM framework, one has to use more precise energy levels and wavefunctions of conduction electrons to calculate LSPR energies of metallic NPs with radii below $1.5$ nm.

Much larger LSPR energy blueshifts from $0.8$ eV to $1.2$ eV appeared in our calculations for silver NPs resting on carbon films with diameters in the ranges $3.26$ nm$<D<3.32$ nm and $3.44$ nm$<D<3.50$ nm.

VI. SUMMARY

By using the SCRM, we obtained the general size-dependence of LSPR energies and linewidths for metallic NPs encapsulated in homogeneous medium. Besides the LSPR and volume plasmon resonance, the ASPR is another significant surface plasmon resonance of metallic NPs. Based on the SCRM, we proposed that the ASPR of a metallic nanosphere originates from all the TDSPs of the system composed of the center of mass and intrinsic motions of conduction electrons. Then, we implemented this SCRM generation mechanism of ASPRs in silver NPs encapsulated in silicon nitride and explained the ASPRs measured by Raza et al., which shows that the SCRM generation mechanism of ASPRs is reasonable. The ASPRs in metallic NPs are completely induced by quantum effects and would evolve into the classical Mie plasmon resonance when the particle size increases to macroscopic sizes. The SCRM generation mechanism shows that the ASPRs in metallic NPs almost always exist. For few-nanometer metallic NPs, the ASPRs generally do not disappear but shift their peak locations to low energy region $\sim\hbar\Omega_{p}$, which is supported by measured ASPR energies ($\sim 2.8$ eV) of silver NPs encapsulated in silicon nitride with radii $4.44$ nm and $5.94$ nm.

Within the SCRM framework, the LSPR of a metallic NP is determined by the transition from the first excited state $|1\rangle$ to the ground state $|0\rangle$ of the center of mass of conduction electrons with these two energy levels corrected by all non-degenerate states of the system. There are no essential difference between the origins of the LSPR and ASPR in metallic NPs. Therefore, it is somewhat strange that LSPRs can be excited by both lights and fast moving electrons, while the ASPRs can only be excited by fast moving electrons. To the best of our knowledge, there are indeed no reports of ASPRs observed in experiments by using lights to excite plasmon resonances. There are several possible reasons for the ASPRs unobserved in the light excitation experiments, such as relatively low experimental precision, the light energy range being too narrow to cover LSPR and ASPR peaks, or peak locations of the ASPR and LSPR being too close to be distinguished. It is well known that the LSPR energies of sodium NPs redshift, while the ASPR energies generally blueshift. Therefore, it seems most likely to observe the ASPRs in light excitation experiments of sodium NPs encapsulated in transparent medium.

It is usually considered that the optical properties of metallic NPs are functions of the particle size. For silver NPs resting on a silicon nitride substrate with the same size, the EELS measurements show that the amplitude and linewidth of the surface plasmon resonances can vary from particle to particle rso3 . Within the SCRM framework, it is natural to regard the LSPR and ASPR energies and line broadenings as the functions of the particle radius $R$ and input parameter $\hbar\omega_{s}$. In a general way, the input parameter $\hbar\omega_{s}$ can be expressed as $\hbar\omega_{s}=\hbar\omega_{{}_{M}}[1-\delta\epsilon_{m}/(Re(\epsilon_{d})+2\epsilon_{m})]$, where the quantity $\delta\epsilon_{m}$, the deviation from the dielectric constant $\epsilon_{m}$ of the dielectric medium encapsulating metallic NPs, can be further expressed as $\delta\epsilon_{m}=\delta\epsilon_{c}+\delta\epsilon_{d}+\delta\epsilon_{s}$, and these three terms are respectively induced by atomic configurations in the vicinity of surfaces, shape deviation from perfect spheres and effects related to the size of individual metallic NPs. The first two terms in the expression of $\delta\epsilon_{m}$ generally exist, but the third term $\delta\epsilon_{s}$ depends on experimental settings. For metallic NPs encapsulated in homogeneous medium, the first two terms $\delta\epsilon_{c}$ and $\delta\epsilon_{d}$ are main effects causing the input parameter $\hbar\omega_{s}$ to be different from the Mie surface plasmon resonance energy $\hbar\omega_{{}_{M}}$, while for metallic NPs resting on a substrate, the inhomogeneous dielectric environments and the interactions between metallic NPs and the substrate render the term $\delta\epsilon_{s}$ significant, which is the reason why the variation amplitude of the input parameter for silver NPs resting on carbon films is evidently larger than that of silver NPs encapsulated in silicon nitride. It is obvious that the input parameter $\hbar\omega_{s}$ has different values for different metallic NPs even with the same size leading to different TDSP and NDSP sets through $\hbar\Omega_{p}$ according to Eqs. (8) and (9), which would yield different energies and line broadenings of LSPRs and ASPRs. The larger the variation of the input parameter, the larger the variation of LSPR and ASPR energies and line broadenings, which naturally explains the experimental observations that the plasmon resonances of silver NPs resting on carbon films have larger energy spread than that of silver NPs encapsulated in silicon nitride. To explain the measured optical properties of metallic NPs within the SCRM framework, besides the sensitivity to particle sizes it is necessary to consider the sensitivity of optical properties to the input parameter $\hbar\omega_{s}$.

The primordial LSPR frequency formula Eq. (3) is extremely sensitive to energy levels of conduction electrons. However, the sufficiently precise energy levels of conduction electrons of a not very small metallic NP are unavailable in most cases. Fortunately, it is possible to transform Eq. (3) into an alternative form, which greatly lowers the sensitivity to the conduction electron energy levels. The calculated LSPR energies of silver NPs encapsulated in silicon nitride are not consistent with measured results except for silver NPs with radii below $2.58$ nm. We think that this inconsistency occurred for not very small silver NPs originates from the improper employment of two Gaussian functions with a fixed FWHM of $0.15$eV to identify the LSPR and ASPR energies from EELS spectra. However, our calculated results perfectly explain the large energy blueshifts of few-nanometer silver NPs encapsulated in silicon nitride measured by Raza et al..

We also calculated the LSPR and ASPR energies of silver NPs resting on carbon films, which were studied by Scholl et al.. We found that the ASPRs paly an important role in accounting for experimental observations. Almost all measured surface plasmon resonances can be explained well by the calculated LSPRs or ASPRs, while few of measured surface plasmons correspond to the merged resonance peaks composed of the LSPR and ASPR with similar energies of individual silver NPs. Besides the appearance of ASPRs, the dielectric environment of silver NPs resting on carbon films changing with particle sizes is an important reason for large energy spread of measured surface plasmon resonances. Our calculated results of silver NPs being well consistent with experimental measurements further indicate that the SCRM generation mechanism and calculation scheme of ASPRs are quite reasonable and the LSPR energy shifts of metallic NPs can be calculated by using the new formula Eq. (10). We also found much larger energy blueshifts for silver NPs resting on carbon films with particle diameters in the ranges of $3.26$ nm$<D<3.32$ nm and $3.44$ nm$<D<3.50$ nm, which is about twice as large as the LSPR energy blueshifts ($\sim 0.5$ eV) observed by Scholl et al..

For silver NPs with radii less than $1.5$nm, our calculated LSPR energies begin to be explicitly less than the measured ones, which shows that the spherical potential well of finite depth is no longer appropriate to substitute for the real effective potential $V_{eff}$. However, the optical properties of metallic NPs with radii less than $1.5$nm can be studied by using TD-DFT. For metallic NPs with radii larger than $20$ nm, the calculations based on SCRM are no longer time-saving. However, the optical properties of large metallic NPs can be studied by other models or theories, such as GNOR model and various QHTs. Therefore, the SCRM could serve as a bridge linking TD-DFT for very small metallic NPs with numerous models or theories for large metallic NPs.

Our calculated results have shown that within the SCRM framework the optical properties of metallic NPs are completely determined by the TDSPs and NDSPs of conduction electrons, which play a central role in our calculations. To the best of our knowledge, there are no precise definitions for the NDSPs and TDSPs in the literature. We defined them by Eqs. (8) and (9), which works well going with spherical potential well of finite depth to calculate the optical properties of metallic NPs with sizes in the range $1.5$ nm$<R<20$ nm. As a corollary of the SCRM, the LSPR energy shift of a few-nanometer metallic NP approximately equals to its line broadening. We hope that all theoretical predictions appeared in this paper could be tested in the future.