1. Introduction

The charged particles, which are not uncommon naturally, i. e., the dusty plasmas in the solar system [1–3] and colloidal particles at liquid interfaces [4–6] are somehow electrically charged, have drawn much attention in electromagnetic wave scattering research recently, due to their special properties on the scattering [7, 8] and absorption [9, 10]. It has been found that the Rayleigh approximation is not suitable for solving the scattering of a charged spherical particle even in the long wavelength limit [11]. However, the Mie theory [12, 13], which is ever a powerful method for solving the scattering problem by an electrically neutral particle, could be applied for studying the scattering effect by a charged sphere with modified Mie coefficients [14–17]. The surface charge on the sphere behaves just like a coating. A natural question is if this “coating” can be equivalent to the change of material parameters of the particle.

In this paper, we present the formulation of an effective-medium theory (EMT) [18–21] for the surface charged spheres in the long wavelength limit. By doing so, a surface charged sphere is, in the long wavelength limit, found to be equivalent to a neutral particle with the same radius but different effective permittivity and permeability. As a result, the scattering problem is reduced to the one that we are familiar with, and both the Rayleigh approximation and the traditional Mie theory apply. Analytical expressions for the effective permittivity and permeability are derived, from which the novel scattering effect by the surface charged sphere could be well explained and predicted. The difference between the effective and original permeability is negligible, while the difference on the permittivity could be significant. The imaginary part of the effective permittivity of the surface charged sphere can be considerably increased by surface charging, explaining why the absorption efficiency can be remarkably enhanced [9]. More interestingly, around the optical frequency and at room temperature, the real part of the effective permittivity can be substantially decreased along with a negligible increment on the imaginary part when the dielectric particle with very small radius is surface charged. A particle with zero or even negative permittivity can be designed by sufficiently surface charging, without resorting to specific micro-structures design [21–25]. Some exotic physical consequences accompanied at nanoscale are discussed. Our numerical results show that the vanishing scattering efficiency and surface plamon resonance (SPR) could be achieved by adjusting the surface charge of the dielectric nanoparticles (NPs). Since the SPR frequency depends on surface charge on the particle, it is expected that charging dielectric NPs may provide an alternative way for the achievement of a controllable plasmonic resonant frequency of NPs, or in turn, this special feature may serve as a handle for determining the surface charge that are ubiquitously present on NPs. Most importantly, we expect our theory could provide an approach to study the interactions of the charged colloidal particles and characterize its optical properties.

2. Theoretical analysis

Let us consider a spherical particle of radius R, composed of a homogeneous material with permittivity ε_s and permeability μ_s, surrounded by free space with constitutive parameters ε₀ and μ₀. The circular frequency of the incident wave is ω, and the wavenumbers outside and inside the scatter are $k=\omega \sqrt{{\epsilon }_0{\mu }_0}$ and $k_s=\omega \sqrt{{\epsilon }_s{\mu }_s}$, respectively. If the sphere is electrically neutral, the Mie coefficients are just given by Eq. (4.53) in Ref. [13]. But once the sphere is surface charged, the formulae are no longer valid. The electrons on the particle surface are driven by the electromagnetic field, forming the surface current and converting the boundary conditions. Considering the discontinuity of the tangential component of magnetic field on the surface, the Mie coefficients for the charged sphere should be transformed to [14–16]

In the long wavelength limit, i. e., x ≪ 1 and y ≪ 1, the high order Mie coefficients are expected to be negligible and the scattering effect of the charged sphere is dominated by n = 1 terms. Within the coherent-potential approximation [18–21], we can retrieve the effective permittivity ε_e and permeability μ_e of the charged sphere when treated it as a neutral one. The idea of the EMT is as follows. Suppose we have a homogenous background medium with constitutive parameters ε_e and μ_e, then we can calculate the scattering efficiency C_sca when putting the considered charged particle inside it. If C_sca tends to be zero, we can judge that ε_e and μ_e are the effective constitutive parameters for the equivalent neutral particle. Mathematically, we can just replace k, ε₀, μ₀, and x in Eq. (1) and g with $k_e=\omega \sqrt{{\epsilon }_e{\mu }_e}$, ε_e, μ_e, and k_ex/k respectively, the vanishing of the Mie coefficient indicates C_sca = 0. In the long wavelength limit, only a₁ = 0 and b₁ = 0 should be considered. Moreover, in the limit of x → 0, we can approximate the Ricatti-Bessel functions and their derivatives by ψ₁(x) ≅ x²/3, ψ′₁(x) ≅ 2x/3, ξ₁(x) ≅ x²/3 − i/x, and ξ′₁(x) ≅ 2x/3 + i/x². With these approximations, one arrives at two equations for effective constitute parameters

According to Eq. (4), it can be found that though the permeability difference Δμ is related to the material of the sphere and proportional to ${\mu }_s^2$, it is usually negligible due to the fact that fΦ ≪ 1. But, on the contrary, the permittivity difference Δε is independent of the material, it is proportional to the surface potential and inversely proportional to the square of the size parameter x. As a consequence, Δε could be significant when the size of the sphere is very small compared with the wavelength and the surface potential Φ is large enough. Therefore, in our work the charged dielectric NPs (CDNPs) are considered to obtain a strong effect. Since the imaginary part of Δε is always positive, the absorption efficiency by a charged sphere is usually greater than that by a neutral sphere [9]. While the real part of Δε is always negative, resulting in the decrement of the effective permittivity to some extent for a surface charged sphere. We can also find from Eq. (4) that the circular frequency ω of the incident beam has influence on permittivity difference Δε as well.

To present a clear picture of the surface charging effect, we calculate the permittivity difference Δε for a CDNP of the size parameter x = 0.05 as function of circular frequency. The results are given in Fig. 1, where the negative real part and the positive imaginary part of Δε are indicated by red solid line and blue dashed lines, respectively. It can be observed that with the increase of the circular frequency, the decrement of the real part increases fast in the first stage and then comes to a saturation value 2fΦε₀/x² quickly, while the increment of the imaginary part exhibits an obvious increase at a lower frequency and then vanishes at higher frequency. With these intriguing properties, one can significantly decrease the effective permittivity of a CDNP by adjusting its surface potential, especially at high frequency, meanwhile keep the imaginary part of the effective permittivity nearly unchanged. In particular, we can tailor the optical properties of a CDNP by designing its effective relative permittivity to be 1, zero, or even negative through controlling its surface potential. With this special handle, many exotic scattering effects can be realized. In the following, we will present some typical examples.

 

Fig. 1 The permittivity difference Δε is plotted as a function of circular frequency ω for the fixed surface potential Φ = 600 V and the fixed size parameter x = 0.05.

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3. Numerical results and discussion

The scattering properties of a particle is characterized by the scattering, extinction, and absorption efficiency, which can be measured according to the Mie coefficients with [9, 15]

 

Fig. 2 The scattering efficiency Q_sca (red solid line) and effective permittivity ε_e (blue dashed line) versus surface potential Φ at a fixed size parameter x = 0.05. Other parameters are ω = 1.5 × 10¹⁵, ε_s = 2ε₀, and μ_s = μ₀.

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As is well-known, for a neutral sphere in the long wave limit, the surface plasmon resonance (SPR) is located at ε_s = −2ε₀. Rosenkrantz and Arnon [9] proposed that the SPR could be realized when a spherical scatter with positive permittivity was surface charged. With our EMT, this novel phenomenon can be well explained. As we have already shown above, the effective permittivity of a dielectric sphere can be significantly decreased by surface charging. If the parameters are properly selected for a CDNP, the real part of the effective permittivity will arrive at −2ε₀, resulting in the appearance of the SPR. According to Eq. (3), the condition for the occurrence of SPR is $\frac{f\mathrm{\Phi }}{x^2}\approx 1+\frac{\text{Re}\left\{{\epsilon }_s\right\}}{2{\epsilon }_0}$. In Fig. 3(a), we present the absorption efficiency and the effective permittivity as the function of size parameter x for a CDNP with the radius 5 nm. The peak of the absorption efficiency indicates the occurrence of the SPR as can be observed from the red solid line, which appears exactly at ε_e = −2ε₀, as marked by the black dash-dot line. In addition, under different surface potentials the SPR can still be observed but at different circular frequencies, which is clearly demonstrated in Fig. 3(b). As a result, charging a neutral particle can provide an alternative way to tune the SPR frequency of NPs. Besides measuring the frequency of the SPR, our theory may in turn serve to determine the surface potential or surface charge of the CDNP.

 

Fig. 3 (a) The absorption efficiency Q_abs (red solid line) and the real part of effective permittivity ε_e (blue dashed line) versus size parameter x at a fixed surface potential Φ = 500 V. (b) The absorption efficiency Q_abs versus size parameter x under different surface potentials. Other parameters are R = 5 nm, ε_s = 2ε₀, and μ_s = μ₀.

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4. Conclusion

In conclusion, based on the coherent-potential approximation, we have proposed an effective medium theory for the surface charged particles. Analytical expressions for the effective permittivity and permeability are presented. It is found that the real part of the effective permittivity will be reduced while the imaginary part will be increased when the sphere is surface charged. As a result, on the one hand, the absorption efficiency of nanoparticles can be remarkably enhanced by surface charging. On the other hand, when the sphere is very small, a significant decrement of the effective permittivity can be realized, leading to many exotic phenomena such as vanishing scattering and surface plasmon resonance. As the surface plasmon resonance frequency depends on the surface charge on a particle, it is expected that surface charging may provide an alternative way for tuning the surface plasmon resonant frequency of a dielectric nanoparticle, or in turn, the latter may serve as a handle for determining the surface charge that are ubiquitously present on nanoparticles.