## Abstract

We propose an effective-medium theory (EMT) for the surface charged dielectric nanoparticles (CDNPs) in the long wavelength limit, in which a homogenous CDNP is demonstrated to be equivalent to a conventional absorbing neutral particle of the same size but with different constitutive parameters (effective permittivity *ε _{e}* and effective permeability

*μ*). It is found that while the surface charge induces negligible change of magnetic permeability in particle, it gives rise to a significant change of electric permittivity. The change in permittivity depends on the charge, the particle size, and the working frequency but is independent of the constituent material. In infrared frequencies, both the real and imaginary parts of the particle permittivity may be changed considerably by surface charging. At higher frequency, the surface charge can lead to a remarkable decrease in the real part of the permittivity while keeping its imaginary part nearly unchanged. Therefore, based on EMT we can tailor the optical properties of CDNPs by optimizing their parameters, allowing for many exotic phenomena, such as vanishing scattering efficiency, great enhancement of light absorption efficiency, and surface charge induced surface plasmon resonance.

_{e}© 2013 OSA

## 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*, surrounded by free space with constitutive parameters

_{s}*ε*

_{0}and

*μ*

_{0}. 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]

*x*=

*kR*,

*y*=

*k*are the size parameters,

_{s}R*ψ*(

_{n}*x*) =

*xj*(

_{n}*x*), ${\xi}_{n}\left(x\right)=x{h}_{n}^{\left(1\right)}\left(x\right)$ are the Ricatti-Bessel functions,

*σ*is the effective surface conductivity [14, 15],

_{s}*e/m*is the charge-to-mass ratio of electron, Φ = |

_{e}*η*|

*R/ε*

_{0}is the electrostatic potential at the surface with

*η*the static surface charge density, $f=\frac{{\epsilon}_{0}{\mu}_{0}e}{{m}_{e}}=1.96\times {10}^{-6}$ is just a constant. The parameter

*γ*≈

_{s}*k*[16] at room temperature with

_{B}T/h̄*k*the Boltzmann constant,

_{B}*T*the thermodynamic temperature, and

*h̄*the Plank constant. In all our calculations,

*T*= 300 K. It should be noted that

*η*and elementary charge

*e*always have the same sign. Therefore, only the absolute value of the surface charge takes effect, independent of its sign. Equation (1) reduces to the Mie coefficients of a neutral spherical particle [13] when

*g*= 0.

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*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

_{e}*ε*and

_{e}*μ*, then we can calculate the scattering efficiency

_{e}*C*when putting the considered charged particle inside it. If

_{sca}*C*tends to be zero, we can judge that

_{sca}*ε*and

_{e}*μ*are the effective constitutive parameters for the equivalent neutral particle. Mathematically, we can just replace

_{e}*k*,

*ε*

_{0},

*μ*

_{0}, and

*x*in Eq. (1) and

*g*with ${k}_{e}=\omega \sqrt{{\epsilon}_{e}{\mu}_{e}}$,

*ε*,

_{e}*μ*, and

_{e}*k*respectively, the vanishing of the Mie coefficient indicates

_{e}x/k*C*= 0. In the long wavelength limit, only

_{sca}*a*

_{1}= 0 and

*b*

_{1}= 0 should be considered. Moreover, in the limit of

*x*→ 0, we can approximate the Ricatti-Bessel functions and their derivatives by

*ψ*

_{1}(

*x*) ≅

*x*

^{2}/3,

*ψ′*

_{1}(

*x*) ≅ 2

*x*/3,

*ξ*

_{1}(

*x*) ≅

*x*

^{2}/3 −

*i/x*, and

*ξ′*

_{1}(

*x*) ≅ 2

*x*/3 +

*i/x*

^{2}. With these approximations, one arrives at two equations for effective constitute parameters

*g*is given by Eq. (2). If

*g*= 0, the equations reduce to

*ε*=

_{e}*ε*,

_{s}*μ*=

_{e}*μ*, corresponding to the neutral particle case as expected. The differences between the effective and the original constitutive parameters are denoted as Δ

_{s}*ε*=

*ε*−

_{e}*ε*and Δ

_{s}*μ*=

*μ*−

_{e}*μ*, respectively. From Eq. (2) and Eq. (3) and considering $\left|\frac{gx{\mu}_{s}}{{\mu}_{0}}\right|=\left|\frac{f\mathrm{\Phi}{\mu}_{s}}{{\mu}_{0}}\frac{\omega}{\omega +i{\gamma}_{s}}\right|\ll 1$, we can obtain

_{s}According to Eq. (4), it can be found that though the permeability difference Δ*μ* is related to the material of the sphere and proportional to
${\mu}_{s}^{2}$, it is usually negligible due to the fact that *f*Φ ≪ 1. But, on the contrary, the permittivity difference Δ*ε* 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, Δ*ε* 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 Δ*ε* 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 Δ*ε* 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 Δ*ε* as well.

To present a clear picture of the surface charging effect, we calculate the permittivity difference Δ*ε* 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 Δ*ε* 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 2*f*Φ*ε*_{0}/*x*^{2} 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.

## 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]

*Q*=

_{abs}*Q*−

_{ext}*Q*. For a neutral scatter with the permittivity and permeability the same as those of the background medium, the scattering efficiency vanishes. While for a CDNP, even if its permittivity is different from that of the background medium, the scattering efficiency can still approach to zero when its effective permittivity is equal to the background permittivity. The reason lies in that the effective permittivity of a CDNP can be decreased by electrically charging, resulting in the same permittivity for the CDNP and the background medium. According to Eq. (4), the effective permittivity of a CDNP equals to

_{sca}*ε*

_{0}at room temperature and around the optical frequency, when the surface potential is located at Φ ≈ (

*ε*−

_{s}*ε*

_{0})

*x*

^{2}/2

*fε*

_{0}. This corresponds to the vanishing of the scattering efficiency of the CDNP. In Fig. 2, we present the scattering efficiency

*Q*and the real part of effective permittivity Re{

_{sca}*ε/ε*

_{0}} as the functions of the surface potential Φ. While the imaginary part of the effective permittivity is nearly zero as can be expected from the results shown in Fig. 1 at high frequency. It is evident that the scattering efficiency vanishes as the effective permittivity

*ε*approaches to

_{e}*ε*

_{0}as indicated by the blue dashed line. The surface potential of the CDNP for the vanishing scattering efficiency is at Φ ≈ 637 V, in accord with Φ ≈ (

*ε*−

_{s}*ε*

_{0})

*x*

^{2}/2

*fε*

_{0}.

As is well-known, for a neutral sphere in the long wave limit, the surface plasmon resonance (SPR) is located at *ε _{s}* = −2

*ε*

_{0}. 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 −2

*ε*

_{0}, 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*= −2

_{e}*ε*

_{0}, 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.

## 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.

## Acknowledgments

This work was supported by the 973 Project (No. 2011CB922004), NNSFC (Nos. 11174059 and 11274277), and the open project of SKLSP in Fudan University (No. KL2011_8).

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