Ex) Article Title, Author, Keywords
Current Optics
and Photonics
Ex) Article Title, Author, Keywords
Curr. Opt. Photon. 2023; 7(4): 449-456
Published online August 25, 2023 https://doi.org/10.3807/COPP.2023.7.4.449
Copyright © Optical Society of Korea.
Corresponding author: ^{*}qsli@ustb.edu.cn, ORCID 0000-0002-4908-9160
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/4.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
The localized surface-plasmon resonance has drawn great attention, due to its unique optical properties. In this work a general theoretical description of the dipole mode is proposed, using the forced damped harmonic oscillator model of free charges in an ellipsoid. The restoring force and driving force are derived in the quasistatic approximation under general conditions. In this model, metal is regarded as composed of free charges and bound charges. The bound charges form the dielectric background which has a dielectric function. Those free charges undergo a collective motion in the dielectric background under the driving force. The response of free charges will not be included in the dielectric function like the Drude model. The extinction and scattering cross sections as well as the damping coefficient from our model are verified to be consistent with those based on the Drude model. We introduce size effects and modify the restoring and driving forces by adding the dynamic depolarization factor and the radiation damping term to the depolarization factor. This model provides an intuitive physical picture as well as a simple theoretical description of the dipole mode of the localized surface-plasmon resonance based on free-charge collective motion.
Keywords: Dipole mode, Forced damped harmonic oscillator model, Free charges, Localized surface plasmon resonance (LSPR), Resonance frequencies
OCIS codes: (160.4236) Nanomaterials; (230.4910) Oscillators; (240.6680) Surface plasmons; (250.5403) Plasmonics
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
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
where
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
When the particle size, such as the radius
A dielectric ellipsoid containing free charges with relative dielectric function
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, 𝛁^{2}
where
The potential of a uniform external field
The potential inside and outside the ellipsoid can be written as
where
The expression for
where
The Laplace equation including
According to the boundary conditions (
where
where
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
where the subscript
where
Considering a point far away from the ellipsoid, the distance between them is defined as
So, the potential
where
Then, according to
The extinction and scattering cross sections based on the Drude model are expressed as follows:
where
If the scattering is small compared to the absorption,
Following Eqs. (9) and (10), we obtain the coefficients
The potential inside (
The total surface charge density
The electric field generated by free charges in an ellipsoid is
The total electric field from free charges and bound charges in the ellipsoid is
where the surface free-charge density
It can further be expressed as follows:
where the restoring force is
where
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
Because the damping coefficient
From Eq. (30), the displacement
where
Following Eqs. (17)–(19), the dipole moment of the ellipsoid is expressed as follows:
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
The extinction and scattering cross sections can be expressed as follows:
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:
where
In the same way, the scattering cross section can be written as follows:
According to the extinction and scattering cross sections obtained above, their resonance frequencies can be obtained as follows:
For a sphere (
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
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
If considering the dielectric retarded polarization and radiation damping, the dynamic depolarization factor and radiation damping term can be introduced. Further, the depolarization field
where
The results under the modified long-wave approximation can be obtained by replacing
Thus, the equation of motion is
where
The intrinsic frequency
where the intrinsic frequency or the restoring force is affected by the high-frequency dielectric function of the particle
The damping term is
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
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
It is interesting that the peak position of the extinction cross section is simply the intrinsic frequency
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
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
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).
Curr. Opt. Photon. 2023; 7(4): 449-456
Published online August 25, 2023 https://doi.org/10.3807/COPP.2023.7.4.449
Copyright © Optical Society of Korea.
Department of Applied Physics, School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China
Correspondence to:^{*}qsli@ustb.edu.cn, ORCID 0000-0002-4908-9160
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/4.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
The localized surface-plasmon resonance has drawn great attention, due to its unique optical properties. In this work a general theoretical description of the dipole mode is proposed, using the forced damped harmonic oscillator model of free charges in an ellipsoid. The restoring force and driving force are derived in the quasistatic approximation under general conditions. In this model, metal is regarded as composed of free charges and bound charges. The bound charges form the dielectric background which has a dielectric function. Those free charges undergo a collective motion in the dielectric background under the driving force. The response of free charges will not be included in the dielectric function like the Drude model. The extinction and scattering cross sections as well as the damping coefficient from our model are verified to be consistent with those based on the Drude model. We introduce size effects and modify the restoring and driving forces by adding the dynamic depolarization factor and the radiation damping term to the depolarization factor. This model provides an intuitive physical picture as well as a simple theoretical description of the dipole mode of the localized surface-plasmon resonance based on free-charge collective motion.
Keywords: Dipole mode, Forced damped harmonic oscillator model, Free charges, Localized surface plasmon resonance (LSPR), Resonance frequencies
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
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
where
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
When the particle size, such as the radius
A dielectric ellipsoid containing free charges with relative dielectric function
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, 𝛁^{2}
where
The potential of a uniform external field
The potential inside and outside the ellipsoid can be written as
where
The expression for
where
The Laplace equation including
According to the boundary conditions (
where
where
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
where the subscript
where
Considering a point far away from the ellipsoid, the distance between them is defined as
So, the potential
where
Then, according to
The extinction and scattering cross sections based on the Drude model are expressed as follows:
where
If the scattering is small compared to the absorption,
Following Eqs. (9) and (10), we obtain the coefficients
The potential inside (
The total surface charge density
The electric field generated by free charges in an ellipsoid is
The total electric field from free charges and bound charges in the ellipsoid is
where the surface free-charge density
It can further be expressed as follows:
where the restoring force is
where
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
Because the damping coefficient
From Eq. (30), the displacement
where
Following Eqs. (17)–(19), the dipole moment of the ellipsoid is expressed as follows:
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
The extinction and scattering cross sections can be expressed as follows:
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:
where
In the same way, the scattering cross section can be written as follows:
According to the extinction and scattering cross sections obtained above, their resonance frequencies can be obtained as follows:
For a sphere (
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
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
If considering the dielectric retarded polarization and radiation damping, the dynamic depolarization factor and radiation damping term can be introduced. Further, the depolarization field
where
The results under the modified long-wave approximation can be obtained by replacing
Thus, the equation of motion is
where
The intrinsic frequency
where the intrinsic frequency or the restoring force is affected by the high-frequency dielectric function of the particle
The damping term is
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
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
It is interesting that the peak position of the extinction cross section is simply the intrinsic frequency
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
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
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).