1. Introduction

Plasmon oscillations in metallic nanoparticles focus growing interest because of prospective their utilization in the photovoltaics and in the photonics. The solar cells modified in nano-scale with on surface deposited metallic particles exhibit efficiency increase due to plasmon mediation in harvesting of sun light energy [1–4]. Radiation properties of plasmons in metallic nanoparticles determine also the subdiffraction plasmon-polariton low-damped propagation in plasmon wave-guides [5–8], with current and forthcoming applications in the nano-photonics.

Irradiation properties of plasmons are strongly sensitive to the nanoparticle size [9, 10]. Ultra-small metal clusters with the radius a ∈ (1,5) nm exhibit very poor irradiation [9, 11, 12] and plasmon damping in this scale is mostly due to scattering of electrons similar as for Ohmic losses including also quantum effects like Landau damping important only for ultra-small clusters [13]. With the increase of the nanoparticle size, the irradiation losses quickly rise because of strengthening of the plasmon oscillation amplitude proportional to the number of electrons. Irradiation losses of plasmon oscillations can be described by the so-called Lorentz friction [14, 15]. Accelerating charges irradiate electro-magnetic wave and the related energy loss can be accounted for as an effective electric field which hampers the movement of electrons. For the case of the oscillating dipole as of dipole-type surface plasmon in the metallic nanosphere, the Lorentz friction force is proportional to the third order time-derivative of this dipole [14,15]. The analysis of the Lorentz friction already done for surface plasmons in metallic nanospheres based on the linearization of the third order time derivative within the perturbation approach [10, 16]. This approximation results in ~a³ growth of the irradiation damping rate for plasmons with the nanosphere radius a. So strong increase of the irradiation losses means that the Lorentz friction dominates all attenuation processes in large nanospheres. The related damping quickly exceeds, however, the perturbation approach necessary condition and the approximation fails. The Lorentz friction must be in this case included beyond the perturbation approach. As it will be presented below, the related corrections are especially significant for nanospheres with radii above ca. 30 nm (for Au in vacuum). The accurate inclusion of the Lorentz friction allows afterward for the explanation of the anomalous light extinction behavior in large metallic nanospheres measured in water colloidal solutions of metallic nanoparticles with varying size [10]. The exact inclusion of the Lorentz friction significantly improved experimental data fitting especially for larger nanospheres (a > 30 nm, for Au).

For large nanoparticles various versions of the random phase approximation (RPA) turned out to be useful [11, 16, 17]. In the present report we consider metallic nanospheres described in the framework of RPA approach including damping effect within the formerly formulated model [10, 16]. In the next paragraph we analyze the corresponding dynamical equation for plasmons with damping and compare the rigorous approach to Lorentz friction with the linearized its version. The final paragraph contains the comparison of the described radiative effects of plasmon damping with the experimental data.

2. Plasmon oscillations in metallic nanosphere induced by an electric field signal and their damping

We will consider a metallic (Au) nanosphere of the radius a located in a dielectric environment with the permittivity ε in the presence of the dynamic external electric field. This corresponds to the dipole limit for interaction with light when the wave-length of incident light highly exceeds the nanosphere radius. Plasmon oscillations in the metallic nanosphere can be described by local fluctuation of the electron density similarly as in RPA approach in bulk metal by Pines and Bohm [18],

For plasmon oscillations given by Eq. (6) the dipole of the electron system, D(t), can be calculated,

The advantage of this RPA approach consists in the oscillatory form of the dynamic equation. The damping of plasmons can be thus included in a phenomenological manner, by addition of the attenuation term, $\frac{2}{{\tau }_0}\frac{\partial \mathbf{D}(\mathbf{r},t)}{\partial t}$ The damping ratio $\frac{1}{{\tau }_0}$ accounts for electron scattering losses and can be approximated by the formula [20],

The scaling of the dipole given by Eq. (7) with the nano-sphere size, ~a³, displays the fact that all electrons actually contribute to the surface plasmon oscillations. The surface modes correspond to uniform translation-type oscillations, when inside the sphere the charge of electrons is exactly compensated by jellium. We will show below that radiation losses which contribute to overall attenuation of plasmons scale also as a³ (up to ca. 30 nm), and therefore, the radiative losses quickly dominate plasmon damping at the range of 10 nm (for Au, Ag and Cu). The cross-over with respect to the size dependence of the plasmon damping occurs here [10], as it is depicted in Fig. 1.

 

Fig. 1 Comparison of various contributions to the plasmon damping (Au, vacuum) in the size-region a ∼ 10 nm, where the cross-over in the size-dependence for damping rate takes place

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The radiative losses of oscillating charges related to plasmon dipole can be expressed by the Lorentz friction [14], i.e., by the effective electric field slowing down the motion of charges,

Thus, one can rewrite Eq. (8) including the Lorentz friction term,

To apply the perturbation procedure for solution of Eq. (11) one can treat the r.h.s. of this equation as the perturbation. Thus, one obtains in zeroth step of the perturbation $\frac{{\partial }^2}{\partial t^2}\mathbf{D}(t)=-{\omega }_1^2\mathbf{D}(t)$. Therefore, within the first step of the perturbation, one can substitute the latter formula to the r.h.s. of Eq. (11), resulting in its form,

In this way the Lorentz friction was included into the total attenuation rate $\frac{1}{\tau }$, taking advantage of approximate linearization of the third order derivative in Eq. (10). Nevertheless, this approximation is justified for small perturbations only, i.e., when the second term in Eq. (13), proportional to a³, is sufficiently small to fulfill the perturbation constraints. For nanospheres with radius a < 30 nm these constraints are fulfilled and it was verified also experimentally for Au and Ag nanospheres [10]. An experimentally observed redshift of resonance surface plasmon frequency with raise of a (visible in the measured light extinction spectra [10]) in the range of nanospheres radii 5 – 30 nm agrees with the perturbation approach. Indeed, the solution of Eq. (12) is of the form D(t) = Ae^−t/τ cos(ω′t + ϕ), where ${\omega }^{\prime }={\omega }_1\sqrt{1-\frac{1}{{({\omega }_1\tau )}^2}}$, which gives the experimentally observed red shift of the plasmon resonance due to ~ a³ increase of the attenuation caused by the irradiation. The Lorentz friction term in Eq. (13) dominates plasmon damping for a ≥ 12 nm (for Au and Ag)—cf. Fig. 1. The plasmon damping increases rapidly with a and this causes the pronounced redshift of resonance frequency in coincidence with the experimental observations for 10 < a < 30 nm (Au and Ag) [10].

3. Exact inclusion of the Lorentz damping to the attenuation of dipole surface plasmons

The linearization of the third order derivative of D(t) is justified only when the attenuation term $\frac{\partial }{\partial t}\left[-\frac{2}{{\tau }_0}+\frac{2}{3{\omega }_1\sqrt{\epsilon }}{\left(\frac{a{\omega }_p}{c\sqrt{3}}\right)}^3\frac{{\partial }^2}{\partial t^2}\right]$ D(t) is sufficiently small as needed for the perturbation. In order to compare various contributions to Eq. (11) it is convenient to change to dimensionless variable t → t′ = ω₁t. The dimensionless damping ratio caused by scattering, $\frac{2}{{\tau }_0{\omega }_1}=\frac{2}{{\omega }_1}\left[\frac{v_F}{2{\lambda }_b}+\frac{v_F}{2a}\right]\simeq 0.027$ for a = 10 nm (v_F = 1.396 × 10⁶ m/s and λ_b ≃ 53 nm, ω₁ = 0.752 × 10¹⁶ 1/s) and diminishes with the increase of the radius a—cf. Fig. 1–inset. One can also estimate the dimensionless coefficient of the Lorentz friction term in Eq. (11). This coefficient, $\frac{2}{3\sqrt{\epsilon }}{\left(\frac{{\omega }_{p}a}{c\sqrt{3}}\right)}^3$, quickly increases with the radius a, as proportional to a³ (for a = 10 nm is relatively small ≃ 0.013 but for a = 20 nm increases ca. ten times to ≃ 0.104, for ε = 1).

In the case of solution of Eq. (11) by perturbation we get renormalized attenuation rate for effective damping term, $\frac{1}{{\omega }_1{\tau }_0}+\frac{1}{3\sqrt{\epsilon }}{\left(\frac{{\omega }_{p}a}{c\sqrt{3}}\right)}^3$. This term quickly achieves the value 1, for which the oscillator falls into the over-damped regime. The value 1 of this renormalized attenuation rate is attained at a ≃ 57 nm (for Au in vacuum). This is, however, an apparent artifact of the perturbation method and one can check that the exact solution of Eq. (11) sufficiently well agrees with the solution obtained by perturbation only up to a ~ 20 – 30 nm. For higher values of a the rapid increase of the approximated attenuation rate causes the discrepancy of an unacceptable level between the exact solution and that one given by the perturbation approximation. This is illustrated in Figs. 2 and 3.

 

Fig. 2 Real and imaginary parts of exact solutions for frequency and damping rate given by Eq. (14) for ε = 1; dashed lines correspond to the perturbative solution; in inset the resonance wave-length is depicted versus the radius a

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Fig. 3 Comparison of the damping rate and the resonance frequency for the oscillating solutions of Eq. (11), exact (solid lines) and approximate upon perturbation approach (dashed lines), both with respect to the nano-sphere radius a and for selected dielectric surroundings

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In Fig. 3 we have plotted the self-frequency and damping rate for dipole-type surface plasmon with respect to the nano-sphere radius a, for the approximate perturbation approach (then ${\omega }^{\prime }={\omega }_1\sqrt{1-{\left[\frac{1}{{\omega }_1\tau }\right]}^2}$)—dashed lines, and for the exact solution of Eq. (11)—solid lines. The dashed lines finish at a ≃ 57, 64, 72 nm (Au, for ε = 1, 2, 4, correspondingly), when the effective attenuation rate upon the perturbation approach reaches the limiting value 1 (then λ → ∞). At this point the perturbative oscillatory equation fall into overdamped regime with an aperiodic solution. For the accurate solution of Eq. (11) this singular behavior disappears and the oscillating solution exists for larger a as well. We see that the red-shift of the plasmon resonance is strongly overestimated in the framework of the perturbative approach to the Lorentz friction unless a < 20 − 30 nm (Au, vacuum).

The equation (11) is the third order linear differential equation and one can list the exponents of its solutions, ~e^iΩt′, in an analytical form:

The solutions given by Ω₂ and Ω₃ are of oscillating type with damping (iΩ₂ and iΩ₃ are mutually conjugated, thus Ω₂ and Ω₃ have the real parts of opposite sign and the same positive imaginary parts). The third solution, given by Ω₁, turns out to be an unstable as purely imaginary (negative). This unstable solution is a well known artifact in the Maxwell electrodynamics (cf. $ 75 in Ref. 14) and corresponds to the infinite self-acceleration of the free charge due to Lorentz friction force (i.e., corresponds to the singular solution of the equation $m\dot{\mathrm{v}}=const.\times \ddot{\mathrm{v}}$, which is associated with a formal renormalization of the field-mass of the charge—infinite for point-like charge and canceled in an artificial manner and not defined mathematically in a proper way by arbitrary assumed negative infinite non-field mass, resulting in ordinary mass of the electron). This unphysical singular particular solution should be thus discarded. The other oscillatory type solution resembles the solution of the ordinary damped harmonic oscillator, though with the distinct attenuation rate and frequency. They are expressed by analytical formulae for Ω₂ (or Ω₃) by Eq (14); for various a they are compared with the corresponding quantities found within the perturbation approach. This comparison is presented in Figs. 2 and 3. From the comparison it is clearly visible that application of the perturbation approach leads to the high overestimation of the damping rate for a > 30 nm. For 30 < a < 50 nm (Au, vacuum) the underestimation of the resonance wave-length occurs, whereas for a > 50 again the rapid and strong overestimation of the resonance wave-length takes place—cf. Fig. 2—inset. Moreover, the approximate solution terminates at a = 57 nm (Au, vacuum), beyond which the approximate oscillatory-type solution does not exist within the perturbative approach. Therefore, we can conclude that the usage of the approximate formula for the Lorentz friction damping in the form as in Eq. (13) is justified up to a ≃ 20 nm, and causes small error (in practice, negligible one for damping rate and a small underestimation of the resonance wave-length) at 20 < a < 30 nm, while for a > 30 nm these approximate values strongly differ from the exact ones. The limiting values of the radius are exemplary given for Au in vacuum and they change significantly with the relative permittivity of the dielectric surroundings, as is illustrated in Fig. 3.

In Fig. 2 one can notice that the damping rate achieves its minimum at certain value of the radius,

Table 1. Radius of the nanosphere corresponding to the minimal value of surface plasmon damping (a^*) and to its maximal value (a^**)

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3.1. On dipole approximation for plasmons and relation with the Mie theory

For sizes of nanospheres well lower than the resonant wavelength, i.e., for $\frac{2a}{500}<<1$ (for Au), one deals with the dipole approximation limit, which holds for a of order of 5 60 nm. For larger radii, e.g., a ~ 100 nm, the electrical field component of the incident resonant e-m wave depends on the position on the sphere and it excites multipole modes besides the dipole one. This problem is very well known in the Mie theory which corresponds to the decomposition of the incident planar wave into multipole fractional waves suitably to spherical boundary conditions and used next to the solution of the Maxwell equations which allows for the determination of by the geometry affected cross-sections for e-m wave scattering and extinction [9]. The material definition needed for such a calculus is taken in the Mie approach as the phenomenologically modeled dielectric function for the metallic system. To explain the crossover in the size dependence of the experimentally observed red shift of plasmon resonance with growing nanosphere radius, within the Mie theory two regimes are usually considered [21, 22]: the intrinsic size effect (for a < 20 nm, for Au) and the extrinsic one (for a > 20 nm). The former one corresponds to dipole approximation with the dielectric function of the metallic nanosphere including electron scattering (same as given by Eq. (9)) and resulting in Mie dipole response with $~\frac{1}{a}$ red shift of the resonance, whereas the latter one is addressed to larger nanospheres, a > 20 nm, when in experiment it is observed the red shift rising with a growth. The extrinsic regime resolves itself in Mie theory to inclusion of the multipole mixing in e-m response, and related lowering of resonance energy with the radius growth. The frontier between two regimes is a bit arbitrary and dictated by the experiment [22–24]. If, however, one includes the Lorentz friction losses to overall damping of plasmons, one arrives at the required cross-over in the size red-shift dependence without the need to invoke to multipole response for too small radii [10, 22]. The multipole corrections start to be unavoidable for larger radii, for a > 60 nm, as is visible in the deformation of Lorentzian shape of extinction features accompanied by large their broadening, for Au observed at a = 75 nm [10, 24]. Though the Mie theory is the fully classical solution of Maxwell equations with spherical boundary conditions and not referring to particularities of microscopic dynamics of electrons in the metallic sphere, the various effects of this dynamics can be included by appropriate modeling of the material dielectric function [21,22]. The microscopic analysis of the Lorentz friction presented in this letter upon RPA method [16] supports the modeling of the dielectric function needed to the Mie theory and can be utilized to prolong the intrinsic size effect upon the Mie approach beyond 20 nm size limit to larger radii for which the dipole approximation still holds (up to ca. 60 nm for Au). Nevertheless, since the dielectric function embraces simultaneously all energy dissipation channels in a combined manner, the inclusion of the details of size effect caused by the exact form of the Lorentz friction (out of the Mie theory as of yet [22]) to the modeled dielectric function is required. Thus, the specific size dependence of the damping time rate for plasmons caused by the Lorentz friction derived by exact solution of the related RPA equation for electron density dynamics in nanospheres (as depicted in Figs. 2 and 3) conveniently contributes to the explanation of observed size effect of plasmon resonance in the radius window 10 – 60 nm (for Au), at least.

4. Comparison with the experimental data for light extinction in colloidal solutions of metallic nano-particles with various radii

In order to compare the model of surface plasmon damping in metallic nanosphere with the experimentally observed behavior, we use our former experimental data [10] for extinction of light after trans-passing the water colloidal solution of Au nanospheres with varied radii. The extinction peaks corresponding to plasmon excitations in metal nanospheres are collected in Fig. 4, left panel—the measured features, right panel—the model fitting. A coincidence of the peak positions and their widths and the evolution with increase of the nanosphere radius is noticeable except for a region where the mixing of multipole response occurs (especially at a = 75 nm).

 

Fig. 4 Attenuation curves for nano-particles of Au obtained by measurement of extinction spectra for light passing the same small glass container filled with the water colloidal solution of metallic nano-spheres with various radii for each filling and with same volume density of metallic components—left panel (after our results [10]); the same extinction spectra calculated as Lorentzian functions with center positions and widths given by resonance frequencies and damping rates acc. to Eq.(14)—right panel

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The central frequencies of the attenuation peaks [10] with respect to the radius a of nanospheres pretty well fit to the resonant self-frequencies defined by the exact solution of Eq. (11). This fitting is presented in Fig. 5—the solid line, also an approximate perturbative curve is plotted with the dashed line. Agreement with the experimental data is excellent for the exact solution for the Lorentz friction, whereas the increasing discrepancy between the approximated (perturbative) solution and the experimental data for larger radii of nanospheres is visible.

 

Fig. 5 Resonance self-energies of surface plasmons extracted from experimental data of light extinction measurements for Au nanospheres with the radii 10, 15, 20, 25, 30, 40, 50, 75 nm in water colloidal solutions [10]; squares—experiment, dashed line—perturbative approximation, solid line—exact inclusion of the Lorentz friction

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5. Conclusion

In conclusion we can state that the Lorentz friction for dipole surface plasmons in metallic nanospheres is of great significance as it determines both the resonance frequency and the damping of plasmons. The exact inclusion of the Lorentz friction to the damping of plasmons demonstrates drawbacks of the perturbative approach to this effect. The discrepancy between the perturbative predictions and the exact solutions increases sharply with the nanosphere size and for a > 30 nm (Au, vacuum) completely dismiss the approximation. The perfect agreement of the exact solution of the dynamic equation with the experimental data verified in the wide range for nanosphere dimension (a ∈ (10,75) nm, for Au) evidences the crucial role of the Lorentz friction for plasmon oscillations in large metallic nanospheres.