The aim of this chapter is to investigate the effect of size, shape and environment on the optical properties of metallic nanoparticles in a large spectral range (λ = 300–900 nm) using quasi-static approximation for nanoparticles of sizes (D = 10–40 nm) and Mie theory for nanoparticles of sizes (D = 40–100 nm). Extinction (scattering and absorption) cross-sectional spectrum of nanoparticles is obtained for different diameters embedded in different matrices. Collective oscillation of electrons in conduction band in metallic nanoparticles is known as surface plasmon resonance (SPR) phenomena. SPR of metallic nanoparticles has significant applications in optics, communications and biosensors. We present in this chapter the effects of the interparticle distance on the optical response of gold dimer nanoparticles of 100 nm diameter. The extinction spectra of dimer nanoparticles are calculated by using generalized Mie theory.
- metallic nanoparticles
- noble metals
- optical properties
- surface plasmon resonance
- extinction cross section
- dimer of nanoparticles
Nanomaterials are macroscopic systems but constructed and organized from elementary bricks of nanometric dimensions known as nanoparticles. They have many fields of application, in the fields of optics [1, 2], magnetism [3, 4], electronics [5, 6], telecommunications [7, 8], superconductors [9, 10], chemical catalysis  or biological marking .
The phenomena of absorption, scattering and reemission of an electromagnetic wave by particles of micrometric or even nanometric size are numerous and varied. For example, sunlight incident on the Earth’s atmosphere is dispersed by gas molecules and suspended particles (aerosols), giving rise to a blue sky, white clouds and various optical phenomena such as the rainbows or the halos. Another example that concerns us more closely is that of the variety of colors of stained-glass windows whose origin is based on the very specific scattering properties of light by very fine metallic grains trapped in transparent glasses .
The metallic clusters of nanometric size, well below the optical wavelength, have been known and exploited for centuries for their spectacular optical properties (stained glass, ceramics) . Take, for example, the case of gold nanoparticles, which can be obtained in colloidal form by chemical synthesis. When diluted in solution, they exhibit varying colors, ranging from red to violet as their size decreases. Other colors like green and blue can be obtained by also playing on their shape. These optical properties are a consequence of the dielectric confinement in these objects whose size is less than the wavelength of the excitatory light and which is at the origin of the well-known phenomenon of surface plasmon resonance (SPR), which dominates the extinction spectrum in the visible domain. This oscillation is analogous to that of an electron gas in a massive system (plasmon mode) but modified by the presence of interfaces.
Among the nanoscale systems that possess very interesting optical properties, metals and especially noble metals (gold, silver and copper) have been the most widely studied in this way [15, 16, 17]. Noble metals and gold in particular lend themselves well to the synthesis of such materials, thanks to their resistance to aging (oxidation), even in a divided form. Great progress has been made in understanding the optical properties of composite materials based on nanoparticle well as in their methods of synthesis. These range from techniques of precipitation of metal salts in glasses or gels, to the assembly of preformed particles, chemically (colloid) or physically (sources of clusters).
In this chapter, we recall the theoretical results concerning the optical properties of bulk and confined metal systems. Particular emphasis is placed on the remarkable origin and properties of surface plasmon resonance (SPR) and its dependence on the size, shape and dielectric environment of nanoparticles. We study the basic formalisms for the calculation of the different cross sections of nanoparticle interaction with light (extinction, scattering and absorption) in the framework of:
Quasi-static approximation for spherical and spheroidal particles
Mie theory for homogeneous spherical particles.
Finally, we briefly discuss the effects of the interparticle distance on the optical response of dimer nanoparticles based on the generalized Mie theory.
2. Drude model and plasma frequency
The first situation envisaged, since it is the simplest, is that of a metal whose optical properties are essentially due to the behavior of the conduction electrons (as for the alkali metals). The Drude model  is then quite appropriate to describe the behavior of these quasi-free electrons. In this model, the conduction electrons of solid metal, considered as independent, move almost freely. These electrons undergo random collisions with other particles (other electrons, phonons, defects, etc.), with a probability per unit of time given by the electronic collision (or relaxation) rate . The corresponding time
In this equation, is the displacement of the electron around an average position, and the second term is a damping term related to the different collision processes cited above. An oscillating solution of this equation is as follows:
The displacement of Drude’s electrons induces a dipole moment , and the electron gas polarization, defined as the dipolar moment per unit volume, is thus written as:
where is the number of free electrons per unit volume and is called plasma frequency of bulk metal. The Drude model allows us to introduce simply the notion of dielectric constant, a characteristic magnitude of the optical response of a solid.
3. Dielectric constant and dielectric susceptibility
In a material, the two macroscopic vector quantities, which are the electric field and the electric displacement , are connected together by the relation
where is the vacuum permittivity and is the induced electrical polarization. Since the intensity of the applied field is not too high, the polarization is connected linearly with through the dielectric susceptibility χ of the medium according to the relation:
which allows us to establish a relation between the electric displacement and the electric field by defining the dielectric constant such that:
The dielectric constant is generally written in complex form:
The real and imaginary parts of are written as:
In the optical domain where the frequencies are such that , these expressions are simplified in the form:
We then see, in a classical approach, how the core electrons are introduced in the study of the optical properties of noble metals.
4. Interband absorption: contribution of d electrons
The optical response of bulk noble metals cannot be completely described by the Drude model. To take into account the contribution of core electrons (d band electrons) in the optical response of noble metals, a term must be added to the dielectric constant as calculated from the Drude model (Eq. 9). The dielectric constant can be written in the following way:
and are complex ( and ).
The first term represents the intraband component of the dielectric constant. It is connected to the optical transitions of a free electron of the conduction band to a higher energy level of the same band. This term is well described by the Drude model, hence the index
The role of the core electrons d is manifested by the deviation from the Drude function. It is mainly observed in the imaginary parts of . The strong growth of the imaginary component , observed around 620
5. Extinction, scattering and absorption of wave by a sphere in the quasi-static approximation
We study the optical response of a metal sphere in the quasi-static approximation, that is to say when the wavelength λ of the incident radiation (plane wave) is much greater than the diameter D of the sphere. In this case, it is possible to consider that the applied electromagnetic field is uniform at each instant and at any point of the object volume (no delay effect). The sphere of radius R is characterized by a complex dielectric constant. The dielectric constant of the environment is real since this dielectric medium is transparent in our frequency domain (). In the quasi-static approximation and if the temporal variation of the applied field is set aside, the problem returns to that of a sphere subjected to a uniform field . The electric potentials take the form :
Given the symmetry of the problem, the azimuthal angular dependence does not intervene, and then the system is reduced to a two-dimensional problem:
where is written as
The application of the electric field induces a polarization of the sphere (Figure 3) and the appearance of a charge surface distribution:
One can write:
α is defined as the polarizability of the sphere, and V = 4/3πR3 is the volume of the particle. The extinction and scattering cross sections in general form are
The essential point here is that the polarization and consequently the different cross sections can become very important if the common term to their denominator vanishes or takes very low values. Knowing that the dielectric constant of the metal (Eq. 5) depends on the frequency of the excitation wave, a resonance phenomenon is observed when
will be minimum.
If the imaginary component or its variation is minimal, then the expression (Eq. 20) can cancel out for a pulsation such as . By writing the dielectric constant of the determined metal in the case of noble metals (Eq. 8) and using the Drude model to explain the contribution of free electrons (Eq. 7), we deduce the expression of resonance pulsation :
In the dipolar approximation, the pulsation is conventionally interpreted as the oscillation pulsation of the electronic cloud with respect to the ions of the nanoparticle. Such a phenomenon is associated with a collective excitation of electrons within the metallic nanoparticles and known as surface plasmon resonance due to the fact that the restoring force which intervenes in the oscillatory motion is mainly due to the charges located at the surface as this is illustrated in Figure 3 . By explicating the real and imaginary parts of the dielectric constant of the metal and the wave vector k, the extinction and scattering cross sections (Eqs. 17 and 18) can be written as:
Here, and we note
Thus, for nanoparticles such as
6. Dielectric constant of a confined system
In the previous description, the dielectric constant of the particles was taken as that of solid bulks. This explains, for example, the absence of size effects on the value of the SPR in the quasi-static approximation. Strictly speaking, it is expected that the effective dielectric constant of the particles will be different from that of the solid bulk, essentially due to the fact that the confinement and the presence of an interface with the external environment must introduce significant modifications. The confinement can be taken into account in the Drude model by introducing phenomenologically a collision effective term for free electrons with surfaces. When the particle size is smaller than the mean free path of electrons, the collision frequency with the surfaces of a sphere of radius R is proportional to , where
The total collision rate is written as 
where is the rate of intrinsic electronic collisions (electron/electron and electron/phonon) of the infinite solid, and the parameter
The dielectric constant of the bulk solid is written as:
and the dielectric constant of a confined metal:
The interband transitions are little modified for sizes up to 3 nm  and then ; it is therefore possible to obtain the expression of the dielectric constant of a confined metal from Eq. (28) and a knowledge of . can be estimated by subtracting from the experimental dielectric constant (table) [21, 22] the Drude bulk part (Eq. 9):
In the optical domain, the frequencies are such that and ; the expression of the dielectric constant of confined metal is reduced to:
It should be noted that the profile of the spectra is given by Eq. (22) in which the modified value of the dielectric constant of the particle (Eq. 31) is introduced. In the case where the surface plasmon resonance is far from the interband threshold (like silver), it is possible to show that the spectrum adopts a quasi-Lorentzian profile. The full width at half maximum (FWHM) can be expressed approximately in the form [27, 28]:
The width of the resonance spectrum depends not only on the modified electronic collision rate but also on the imaginary part of the interband dielectric constant in the vicinity of the plasmon resonance. For silver with
7. Environment effect on the surface plasmon resonance
From the expression (Eq. 22), it is clear that the extinction cross-sectional spectrum is very sensitive to the value of . For a given metal, the SPR profile can be modified by its environment by means of its dielectric constant . The spectral position will move to red by increasing (Eq. 21). The extinction cross section of silver and gold nanoparticles in different environments was calculated (Figures 6 and 7). The dielectric constant used for the metallic nanoparticle was deduced from bulk metal by adding a surface contribution with
8. Shape effect
So far, we have considered spherical nanoparticles. Experimentally, we are often led to study deformed particles which adopt ellipsoidal forms. On the basis of the notion of polarizability of particles, we establish here the general formulas of extinction cross sections for ellipsoids (semiaxes: a, b and c) in the quasi-static approximation (a, b and c < <λ) and discuss the shape effect on the optical response.
By analogy with the case of the sphere, it is possible to show that the polarizability of an ellipsoid when an electric field is applied to it in one of the
To obtain this result, the Maxwell equations are solved in a system of ellipsoidal coordinates. The establishment of the boundary conditions at the interface between the particle and the external environment makes it possible to express the electrical potentials in each region and to identify the dipole responsible of the scattered field. The geometric factors
By analogy with the problem of the sphere treated above, the cross sections can be expressed in the form:
An important case is that of an excitation by an electric field polarized collinearly with one of the principal axes of the ellipsoid. We can write:
with the volume of the ellipsoid. Depending on the direction of polarization
The surface plasmon frequency for an ellipsoid:
If the case of prolate ellipsoid (a = c > b), then Lx = Lz and Ly depend only on its aspect ratio . is written as a function of the ellipticity
Figures 8 and 9 illustrate the evolution of SPR as a function of the shape of the ellipsoid in the case of gold and silver nanoparticles, respectively. The cross sections are calculated according to Eq. (41) for a prolate ellipsoid whose volume is equivalent to that of a 20-nm nanoparticle embedded in vacuum with
9. Mie theory
The quasi-static description developed previously is strictly valid only in the limit where
The coefficients and are defined by
where and are the Riccati-Bessel functions of order
10. Optical response of several spherical particles: generalized Mie theory
Finally, we study a situation that can be encountered experimentally: the optical response of a group of close particles. This situation is illustrated in Figure 11. The incident field on a particle i is the sum of the applied field and the set of scattered fields by the other particles.
The scattered field by a particle j depends on and . The optical response of the set will be obtained by solving this problem in a self-consistent way (coupled equations):
In the case of spherical particles, this problem can be solved accurately but at the cost of a large numerical effort. The idea is to apply the results of the Mie theory developed previously since it allows to express the field scattered by a particle as a function of the incident field on this same particle. Nevertheless, for each of the particles considered, the theory requires to define a reference whose origin (center of the particle) and the orientation are fixed and linked to this particle [33, 34, 35]. Figure 12 shows the extinction spectra of gold dimer nanoparticles as a function of the interparticle distance
In this chapter, we demonstrated the capabilities of the Mie theory and quasi-static approximation to calculate the optical response of metallic nanoparticles. The quasi-static approximation is typically valid for nanoparticle diameter D ≤ λ/10 (λ is the wavelength of the incident light). For other dimensions of size, we calculated the optical response in the framework of Mie theory. We recalled the differences between extinction, absorption and scattering cross section of metallic nanoparticles. The spectrum of the extinction cross section present a resonance attributed to the collective oscillation of electrons in the conduction band: surface plasmon resonance (SPR). SPR spectral profile is very sensitive to the size, morphology and environment of the metallic nanoparticles. It is concluded that as the size of the spherical nanoparticle increases, the extinction magnitude increases whereas the spectral position of the surface plasmon resonance (SPR) is redshifted by using Mie theory. However, the peak position of SPR in quasi-static approximation is independent of size. We used the parameter g for confined nanoparticles to introduce the collision between electrons and the surface of nanoparticles. As the g factor increases, then there are more damping of oscillations. We concluded that the extinction cross-sectional magnitude is decreased and the width at half peak of SPR is increased. We investigated the effect of shape using dipolar approximation. We found that the surface plasmon resonance depends strongly on the polarization of the electromagnetic incident wave on the nanoparticle. As the aspect ratio decreases, the spectral position is shifted toward higher wavelength (when the light is polarized along major axis) and shifted toward smaller wavelength (when the light is polarized along minor axis). The mode along the long axis gives a larger extinction cross section because it is equivalent to a larger volume of material probed in relative to the sphere. We studied the extinction cross section of metallic nanoparticle embedded in different matrices. We remarked that the SPR is redshifted and accompanied by a large enhancement of its absorption cross section with increasing dielectric constant of the matrix. In the case of interacting systems, we have seen that the distance between nanoparticles is a crucial parameter. In the case of gold dimer nanoparticles, the smaller the interparticle distance, the more the dipolar resonance is redshifted with the appearance of higher order resonances.
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