Metallic nanostructures have attracted widespread attention for their fantastic optical properties, due to the abundant free electrons in metals. When these electrons are constrained within a limited volume, their collective motion (which is driven by the external light field and modulated by bound charges at the interface of the metal and the medium) generates the localized surface-plasmon resonance (LSPR) [1]. The LSPR of metallic nanoparticles exhibits intense absorption and scattering cross-sections in the far field. It can generate large local electric field enhancement in some regions [2, 3], and a local heat source from the plasmonic nanoparticles [3, 4]. The resonance wavelengths can be tuned by size and shape of the plasmonic nanoparticles, while recently doped semiconductors have attracted extra attention for tunable resonances due to their variable carrier density [3, 5–9]. The LSPR has applications in a wide range of fields, including biosensors [7, 10–12], solar cells [8, 12, 13], nanoantennae [14], and therapeutic applications [3, 4, 6, 15–17].

The LSPR is the interaction of the incident light with metal nanoparticles, which can be described by Maxwell’s equations and the dielectric function of the metal (in the Drude model). For quantitative research, numerical simulations are performed using many algorithms, such as the finite-difference time-domain (FDTD) and discrete-dipole approximation (DDA), for arbitrary shapes of particles. Mie theory [18] provides a rigorous solution for spherical nanoparticles. A typical way in textbooks to analyze the LSPR is by considering a dielectric sphere with the dielectric function of the metal under a uniform external field, in a quasistatic approach [18]. For the electric dipole mode, the resonance condition (Frӧhlich condition) can be derived from one coefficient of Mie theory, or the quasistatic approach with the dielectric function of the Drude model [18]. As we all know, the Drude model arises from the collective motion of free charges in metals, so the LSPR in Mie theory and the quasistatic approximation is associated with the collective motion of free charges through the dielectric function of the metal. Additionally, the hydrodynamic model [19] and the hybridization model [20] are also used to describe the LSPR based on the collective motion of free electrons, but those models are not as intuitive as the forced damped harmonic oscillator model. A more intuitive physical picture is that free charges oscillate within the domain of a nanoparticle under the electric field of the incident light. The harmonic oscillator model is reasonable for this collective motion. This model can explain many phenomena, including damping [21], coupling [22], and resonance frequencies in the near field and far field [23, 24]. There is no thorough theoretical derivation for the LSPR based on this model, except for our previous work [1]. In that work, the metal is divided into free charges and the dielectric background, which has a dielectric function. Free charges generate the collective motion under the electric field, and the influence of free charges does not appear in the dielectric function. We have derived the theoretical description for the dipole mode of an ellipsoidal nanoparticle under the special condition that there is a π/2 phase difference between the displacement and the external field at the resonance frequency. Nevertheless, this is not a general theoretical description, for some important information is not contained in it due to this special condition. Moreover, size effects are also not included in the previous results. According to the modified long-wavelength approximation, the dynamic depolarization factor and radiation damping term can be introduced into the forced damped harmonic oscillator model [25, 26].

In this paper, we first derive a theoretical description based on a subwavelength ellipsoid under general conditions. The restoring force, driving force, and extinction and scattering cross sections of the LSPR and its resonance frequencies, are obtained using the forced damped harmonic oscillator model. We then compare the results, including the extinction and scattering cross sections, from our model to those from the dielectric function of the Drude model. To account for the size effects, we introduce the dynamic depolarization factor and radiation damping term into the harmonic oscillator model. Finally, we discuss some conclusions from the harmonic oscillator model.

The relative dielectric function of the metal ε(ω) always includes a contribution from free charges, which is described by the Drude model. ε(ω) is usually described as follows [27]:

(1)ε(ω)=ε∞−ωp2ω2+γ2+iωp2γω3+γ2ω

where ω_p is the plasma frequency and γ is the damping coefficient. The first term ε_∞ is the high-frequency dielectric function arising from the bound charges of the lattice, and the second and third terms arise from the Drude model based on free charges.

In our model, the metal is divided into two parts: The dielectric background, and free charges. Free charges include free electrons (negative) and the related lattice background (positive), with equal charge densities. In other words, the metal is regarded as a dielectric that contains equal negative and positive free charges. The bound charges of the dielectric background generate the relative dielectric function ε_∞. The response of free charges to light is considered as a collective motion driven by an external field, rather than included in the dielectric function of the metal. The approach in which the metal is divided into two parts is similar to that of the Huang equations, in which the ionic crystal is divided into two parts: the positive and negative ions, and the bound charges of those ions [28].

When the particle size, such as the radius a, is much smaller than the wavelength λ of light, for example a < λ/20 [29], the particle is regarded to be in a uniform external field, which is the quasistatic approximation (the electrostatic approach) [18]. To obtain analytical results, the particle is usually assumed to be a sphere to derive the fundamental conclusions. Sometimes it is assumed to be an ellipsoid, to include the shape effects. On the other hand, many practical nanoparticles can be approximated as ellipsoids.

A dielectric ellipsoid containing free charges with relative dielectric function ε_∞ is located in a medium with relative dielectric function ε_m under a uniform external electric field E₀, as shown in Fig. 1. The magnitude of the charge density of the positive and negative free charges in this ellipsoid is set as ρ. Two ellipsoids formed by the positive and negative free charges separate from each other under the external field, and the distance between the centers of the two ellipsoids is set as d, which can be regarded as the displacement of the oscillator formed by free charges. a and b are the principal semiaxes of the ellipsoid.

Figure 1.Schematic diagram of the forced damped harmonic oscillator model. In this model, free charges in a dielectric ellipsoid with relative dielectric function ε_∞ are divided into two ellipsoids of equal and opposite charge density under the external field E0. The distance between the centers of the two ellipsoids is d. The relative dielectric function of the medium is εm. a and b are the principal semiaxes of the ellipsoid.

When the light irradiates this ellipsoid, free charges accumulate on its surface and generate an additional electric field. The electric field in the ellipsoid includes the external electric field, the additional electric field from the accumulated surface free charges, and that from the surface bound charges from the lattice and the surrounding medium. The collective motion of free charges driven by the external field in a nanoparticle should be a forced motion with damping effects, so it is reasonable to use the forced damped harmonic oscillator model to describe the LSPR.

In the electrostatic approach, the potential in space satisfies the Laplace equation, &#x1d6c1;²φ₁ = 0 and &#x1d6c1;²φ₂ = 0, where φ₁ and φ₂ are the potential inside and outside the ellipsoid respectively. The potential is solved in ellipsoidal coordinates [30, 31]. The relationship between ellipsoidal coordinates and Cartesian coordinates is

(2)x2a2+u+y2b2+u+z2c2+u = 1

where a, b, and c are the principal semiaxes of the ellipsoid. This equation has three different real roots: ξ, η, and ζ.

The potential of a uniform external field E0 along the x axis is written as

(3)φ0=−E0x=∓E0(ξ+a2)(η+a2)(ς+a2)(b2−a2)(c2−a2)

The potential inside and outside the ellipsoid can be written as

(4)φ1=φ1'=C1φ0

(5)φ2=φ0+φ2'=1 + C2∫ξ∞ ds Rs (s  + a 2)φ0

where φ'₂ is from the surface charges of the ellipsoid. φ'₂ is set as

(6)φ2'  = φ0F(ξ)

The expression for F(ξ) is

(7)F(ξ)=C2∫ξ∞ ds Rs (s  + a 2)

where Rs=(s+a2)(s+b2)(s+c2).

The Laplace equation including F(ξ) is

(8)d2Fdξ2+dFdξddξln[Rξ(ξ+a2)]=0

According to the boundary conditions (ξ = 0),

(9)φ1,H=φ2,H

(10)−εmε0∂φ2,H∂n+ε∞ε0∂φ1,H∂n = σf

where σf is the surface free-charge density function that we set. φ_1,H and φ_2,H are the potential inside and outside the ellipsoid in our harmonic oscillator model respectively. The subscript H indicates the harmonic model. The normal derivative of the potential in ellipsoidal coordinates is

(11)∂φ∂n=∂φh1∂ξ

where h1=(ξ−η)(ξ−ζ)/2Rξ.

If there are no free charges, or the response of free charges is attributed to the dielectric function as in the Drude model, the boundary conditions are

(12)φ1,D=φ2,D

(13)−εmε0∂φ2,D∂n+εε0∂φ1,D∂n = 0

where the subscript D indicates the Drude model. Those are the usual boundary situations for an ellipsoid with the relative dielectric function ε(ω). Here we give the well-known results in textbooks [18], to compare to our results later. The potential inside and outside the ellipsoids with the Drude model are obtained as follows:

(14)φ1,D=εmεm+ (ε−εm)n(x)φ0

(15)φ2,D=1−12abc(ε−εm )εm +(ε−εm )n(x)∫ξ∞ ds Rs(s + a 2 ) φ0

where n^(x) is the depolarization factor along the x direction, which is closely associated with the shape. The depolarization factor is

(16)n(x)=abc2∫0∞ ds Rs(s+a 2 )

Considering a point far away from the ellipsoid, the distance between them is defined as r. When r → ∞, ξ → ∞ in the ellipsoidal coordinates, and then ξ ≈ r² [31].

(17)∫ξ∞ ds Rs(s + a2)=∫r2 ∞ ds s 5/2=23r3

So, the potential φ₂'(r) is

(18)φ2'(r)=E0xr3V4πε−εm(ε−εm)nx+εm

where V = 4πabc/3 is the volume of the ellipsoid. The potential is expressed in terms of the dipole moment as φ₂'(r) = p/4πε₀ε_mr². Thus, the dipole moment p_D of the ellipsoid with the Drude model can be obtained:

(19)pD=43πabcε0εmε−εmεm+ (ε−εm)nxE0

Then, according to p = ε₀ ε_mαE₀, the polarizability α_D is

(20)αD=43πabcε−εmεm+ (ε−εm)nx

The extinction and scattering cross sections based on the Drude model are expressed as follows:

(21)Cext,D=kImα=43πabckImε−εmεm+(ε−εm)nx

(22)Csca,D=k46πα2=827πk4a2b2c2 ε−εm εm +(ε−εm )n x 2

where k is the wave vector.

If the scattering is small compared to the absorption, C_ext,D can be replaced by C_abs,D in Eq. (21) [18].

3.1. General Theoretical Results for the Harmonic Oscillator Model

Following Eqs. (9) and (10), we obtain the coefficients C₁ and C₂ as follows:

(23)C1=εm+2a2h1n(x)σf/ε0φ0(ε∞−εm)n(x)+εm

(24)C2=a3bch1σf/ε0φ0−abc(ε∞−εm)/2(ε∞−εm)n(x)+εm

The potential inside (φ_1,H) and outside (φ_2,H) the ellipsoids can be written as follows:

(25)φ1,H=εm(ε∞−εm)n(x)+εmφ0+2a2h1n(x)σf/ε0(ε∞−εm)n(x)+εm

(26)φ2,H=φ0+a3bch1σf/ε0φ0−abc(ε∞−εm)/2(ε∞−εm)n(x)+εm⋅φ0∫ξ∞ ds Rs(s +  a2 ) .

The total surface charge density σ_total, including free charges and bound charges, is

(27)σtotal=−ε0（∂φ2∂n−∂φ1∂n）=1(ε∞−εm)n(x)+εmσf−φ02a2h1(ε∞−εm)ε0(ε∞−εm)n(x)+εm

The electric field generated by free charges in an ellipsoid is E_f = ρdn^(x)/ε₀ [1], where d is displacement between the centers of the two ellipsoids formed by the positive and negative free charges.

The total electric field from free charges and bound charges in the ellipsoid is

(28)Etotal=σtotalσfEf=ε∞−εmεm+(ε∞−εm)n(x)E0n(x)+ρd/ε0εm+(ε∞−εm)n(x)n(x)

where the surface free-charge density σ_f is expressed as σ_f = ρxd/²a2h₁ [1]. The first term is related to the driving force, for it is associated with the external field. The second term is related to the restoring force, for it is proportionate to the displacement d. Then the equation of motion is

(29)md¨+mγd˙=eE0−Etotal

It can further be expressed as follows:

(30)md¨+mγd˙+mω02d=eE0εmε∞−εmn(x)+εm

where the restoring force is

(31)F=−mω02d=−m1ε∞−εm+εm/n(x)ωp2d

where ω²_p = ρe/ε₀m; The constants e, m, and ε₀ are the elementary charge, the free electron’s mass, and the vacuum permittivity respectively.

The actual driving force in Eq. (30) does not equal the force of the incident light field; it is weakened by the dielectric functions of the medium and dielectric background.

The resonance frequency is

(32)ωR=ω02−γ22

Because the damping coefficient γ is practically much smaller than ω₀, the above equation can be approximated as follows:

(33)ωR≈ω0=1 ε∞ −εm +εm /n(x)ωp

From Eq. (30), the displacement d can be obtained as follows:

(34)d=eE0mεm(ε−εm)n(x)+εm1ω02−ω2−iωγexp(−iωt)=AE0exp(iφ)exp(−iωt),

where

(35)A=emεm(ε∞−εm)n(x)+εm1(ω02−ω2)2+(ωγ)2

(36)exp(iφ)=ω02−ω2+iωγ(ω02−ω2)2+(ωγ)2

Following Eqs. (17)–(19), the dipole moment of the ellipsoid is expressed as follows:

(37)pH=43πabcε0εm1(ε∞−εm)n(x)+εmE0⋅[ρAeiφ/ε0+(ε−εm)].

The polarization of the metal arises from the dipole moment formed by the positive and negative free charges and the polarization of the bound charges in the dielectric background.

The polarizability is

(38)αH=43πabc1(ε∞−εm)n(x)+εm[ρAeiφ/ε0+(ε∞−εm)]

The extinction and scattering cross sections can be expressed as follows:

(39)Cext,H=kImα=43πkabc ⋅Imε∞−εm(ε∞−εm)n(x)+εm+ρAeiφ/ε0(ε∞−εm)n(x)+εm

(40)Csca,H=k46πα2=827πk4a2b2c2⋅ ε∞ −εm (ε∞ −εm )n (x) +εm + ρAe iφ /ε0 (ε∞ −εm )n (x) +εm 2

The extinction and scattering cross sections obtained from the two models (harmonic oscillator model and Drude model) are equivalent. The extinction cross section can be written as follows:

(41)Cext,D=Cext,H=43πkabcεm[(ε∞−εm)n(x)+εm]2ωp2⋅ωγ(ω02−ω2)2+(ωγ)2,

where ρ is converted to ω_p.

In the same way, the scattering cross section can be written as follows:

(42)Csca,D=Csca,H=827πk4a2b2c21ε∞ −εm nx+εm 2⋅ε∞ −εm 2+2εm ε∞ −εm ε∞ −εm nx +εm ωp2ω02−ω2 ω02−ω2 2 +ωγ 2 +εm ε∞−εmn x+εm 2ωp4 ω02−ω2 2 +ωγ 2

According to the extinction and scattering cross sections obtained above, their resonance frequencies can be obtained as follows:

(43)ωext2=ω02

(44)ωsca2≈ω02−[2(ε∞−εm)n(x)−εm]γ22εm−(ε∞−εm+9ε∞n(x))n(x)γ43εmω02+(εm2+ε∞2+ε∞εm)n(x)2γ4εm2ω02.

For a sphere (n^(x) = 1/3),

(45)ωsca2≈ω02−(2ε∞−5εm)γ26εm−ε∞−εm(3εm−2ε∞)γ418εm2ω02−（ε∞−εm） ​γ418εmω02

For the scattering cross section, the resonance frequency obtained by Eq. (42) is analytical but cumbersome, so we give the approximate result as shown in Eq. (44), under the condition ω²₀ >> γ². Under this condition the third and fourth terms in Eq. (44) are relatively small, so ω_sca is close to ω0 but less than it for the usual high-frequency dielectric functions of materials, the effect of which can be found in many studies [32, 33].

3.2. Dynamic Depolarization Factor and Radiation Term

In the quasistatic approximation, size effects are not included. If considering size effects, the modified long-wavelength approximation [25, 26] can be used. The depolarization field E_p is

(46)Ep=−n (x)P/ ε0

If considering the dielectric retarded polarization and radiation damping, the dynamic depolarization factor and radiation damping term can be introduced. Further, the depolarization field E_p can be written as follows:

(47)Ep=−neff(x)Pε0 =−(n(x)−b1k2−ib2k3)Pε0

where n_eff^(x) is the effective depolarization factor with the inclusion of the finite-wavelength correction [26], k is the wave vector, and b₁ and b₂ are coefficients. b₁k² is associated with the dielectric retarded polarization, and b₂k³ is associated with the radiation damping [25]. In addition, for oblate and prolate ellipsoids, the depolarization factors and the dynamic depolarization factors and radiation terms can be found in the [26, 34, 35].

The results under the modified long-wave approximation can be obtained by replacing n^(x) with n_eff^(x) [26]. The electric field in the ellipsoid is

(48)Etotal=σtotalσfEf=ε∞−εmεm+(ε∞−εm)neff(x)E0neff(x)+ρd/ε0εm+(ε∞−εm)neff(x)neff(x).

Thus, the equation of motion is

(49)md¨+mγ'd˙+mω02d=[εm(ε∞−εm)(n(x)−b1k2)+εm2]2+[b2k3εm(ε∞−εm)]2[(ε∞−εm)(n(x)−b1k2)+εm]2+[b2k3(ε∞−εm)]2 ⋅eE0exp(iϕ),

where

(50)exp(iϕ)=εm(ε∞−εm)(n(x)−b1k2)+εm2+ib2k3εm(ε∞−εm)[εm(ε∞−εm)(n(x)−b1k2)+εm2]2+[b2k3εm(ε∞−εm)]2.

The intrinsic frequency ω0 is

(51)ω02=(ε∞−εm)(n(x) −b1 k2 )2+εm(n(x)−b1k2)+b23k6(ε∞−εm)[εm+(ε∞−εm)(n(x) −b1 k2 )]2+[(ε∞−εm)b2 k3 ]2ωp2,

where the intrinsic frequency or the restoring force is affected by the high-frequency dielectric function of the particle ε_∞, the dielectric function of the medium ε_m, the plasma frequency ω_p, and the dynamic depolarization factor and the radiation damping term.

The damping term is

(52)γ'=γ+1ωb2k3εmωp2[εm+(ε∞−εm)(nx−b1k2)]2+[(ε∞−εm)b2k3]2,

where the first term arises from the damping of the collective motion of free charges, and the second term arises from the radiation damping of the electric dipole formed by the free and bound charges. The driving force has a phase delay compared to the force of the external electric field E₀, as shown in Eq. (50), which arises from the radiation damping of the dipoles.

3.3. Discussion

The displacement of the harmonic oscillator is that between the positive and negative free charges, and its magnitude is influenced by the high-frequency dielectric function of the particle, the dielectric function of the medium, and the dynamic depolarization factor and the radiation damping term. In our model the displacement is related to free charges, as shown in Fig. 1. The maximum displacement corresponds to the maximum accumulation of surface free charges.

The actual driving force is not equal to the force of the incident light field. It is affected by the dielectric function of the surrounding medium, the dielectric function of the dielectric background, and the shape factor n^(x). Considering also the dynamic depolarization factor and radiation term, the actual driving force is also affected by them. Furthermore, when the radiation term is considered, a phase delay exists between the actual driving force and the external field.

It is interesting that the peak position of the extinction cross section is simply the intrinsic frequency ω0 of the material. The resonance frequency of the scattering cross section is slightly smaller than that of the extinction cross section. The resonance frequency of the harmonic oscillator equation [see Eq. (32)] is lower than ω0, which is attributed to the peak position for the near field [23]. In the oscillator model, this resonant frequency generates the maximum displacement of the oscillator.

As we know, when the frequency tends to zero, the imaginary part of the dielectric function of the Drude model as shown in Eq. (1) will tend to infinity, which is the deficiency of the Drude model. The Drude model is not used in our harmonic oscillator model. In our model there is no conflict, because free charges always accumulate on the surfaces and generate the dipole moment, even when the frequency tends to zero. Our model avoids the limitation of the dielectric function of the metal. The two models are based on the same physical picture, so they are equivalent, and the same extinction and scattering cross section results can be obtained. In addition, the damping γ for both models is the same, due to the same collective motion of free charges.

The forced damped harmonic oscillator model, which is a very intuitive physical picture for the LSPR, can be used to understand the physical source of the LSPR caused by the collective motion of electrons in metal. Based on this harmonic oscillator model, the LSPR is no longer limited to metals. Semiconductors with free charges can also exhibit the LSPR, and the resonance can be modulated by the density of the doping [3, 5–9]. In recent years a few carriers in a small nanoparticle, even four carriers in a ZnO nanoparticle [36], have been found to sustain the surface plasmon from the collective motion of those carriers [9]. In understanding these phenomena, the picture of carriers oscillating under the driving field is more intuitive than that of the collective motion of the carriers, which is attributed to the dielectric function of the nanoparticle. In the harmonic oscillator model, the density of the carriers determines the magnitude of the scattering and extinction, and the resonance frequencies. Our model provides a basic theoretical understanding of plasmonic phenomena.

We have derived analytical expressions for the LSPR, based on the collective motion of free charges by the forced damped oscillator model. The extinction cross section and scattering cross section and their resonance frequencies were derived under the harmonic oscillator model, the results of which are consistent with those of the Drude model. This means those two models are equivalent, due to the same collective motion picture, and their parameters are the same, such as γ. After we further introduce size effects, the expression can be extended by including the effective depolarization factor in the harmonic oscillator model. The restoring force and the driving force will be modified by the high-frequency dielectric function of the dielectric background, the dielectric function of the medium, the depolarization factor of the ellipsoid, the dynamic depolarization term, and the radiation term. The driving force has a phase shift from the external field, due to the radiation term. According to our model, the dipole plasmon mode arises from the collective motion of free charges, modulated by the bound charges and radiation of the dipole moment formed by free charges and the bound charges. This theoretical model describes an intuitive picture of the LSPR that can be applied to particles containing free charges in metals or semiconductors.

The authors declare no conflict of interest.

No data were generated or analyzed in the current study.

Central Universities (FRF-BR-19-002B); Scientific Research Foundation for the Returned Overseas Chinese Scholars (48th); Beijing Higher Education Young Elite Teacher Project (No YETP0391).