3. ANTECEDENTES
3.2 Antecedentes nacionales
In this section, a phenomenological mechanism and parameters perceived to be behind the optical absorption of metal nanoparticles in general are proposed. While the theories of the existing models have laid enough foundation for the understanding of the optical
properties of metal NP their perceived phenomenological mechanisms and parameters are still in conflict and are unjustifiably less representative of the NP size dependent behavior. The view held in this study is that the conflicts and the discrepancies of the existing models are due to the parameters employed and the assumed phenomenological mechanisms. The emergence of new potential applications of the metal nanoparticles require accurate predictions and correct interpretation of the cause of NP size effect in order to remove uncertainties. The primary aim, therefore, is about the mechanisms and parameters for explaining the drastic change in the optical properties from bulk metal to nanosize and their influence in nanosize. Although silver is the metal for the experimental study, the existing data for the other two metals (Au and Cu) have been used in the analysis, testing and validation of the model presented in this section.
3.5.2. The Proposed Modeling Scheme
The proposed modeling scheme is as a result of the re-examination of the mathematical formalism, assumptions and parameters employed in the existing models. This, led to the identification of areas which are inconsistent with the qualitative description of the influence of NP size and the hybrid effect of the d-band and conduction band electrons in explaining the damping effect of the d-band electrons on the SPR absorption of the metal NPs. As a result, an alternative mathematical formalism, that yields equations for qualitative and quantitative analysis of the optical absorption of metal nanoparticles below 50nm, has been proposed (Ochoo et al., 2012).
3.5.2.1 Phenomenological theory of the proposed Model
The general perception has been that when electromagnetic field is applied to a small particle the conduction band electrons are displaced as a unit, figure 3.5(a), leading to a dipole moment in which a single particle can be represented by a single dipole moment (Jain et al., 2006). This picture assumes that the whole center of the negative charge is displaced in one direction once the electrons cloud of the conduction band electrons is displaced by the changing electric field (E) of light.
(a) (b)
Displaced Conduction band Field E Electron cloud Inner band electrons cloud
Electrons cloud Nucleus Field E
Figure 3.5. The displacement of electron clouds by electromagnetic field
(a) A single displaced electrons cloud (b) Two displaced electrons clouds
In a dynamic system, and when the conduction band electrons are not so much delocalized to be screened from the positive ions, the one unit displacement may not be possible at all frequencies. That is, if only part of the conduction band electrons (CB) are free as in the Drude model (Kittel, 1986) while part of the cloud and the inner band electrons (IB) experience strong positive ion effect, then the inner band electrons are expected to interact with the conduction band electrons and cause damping in their
oscillatory motion. Depending on how strong the IB and CB electrons couple in this motion, the cloud associated with the inner band electrons would be out of phase with the cloud of the near-free conduction bandelectrons at some frequencies, especially at higher frequencies of the driving external force. Because of this, the proposed scheme introduces two concentric electron clouds (IB and CB), whose centers of negative charge would be out of phase at certain frequencies, figure 3.5(b). This would depend on the metal type, the kinds of interband transitions taking place and the electron density, and would be expected to have effect on the overall polarizability, hence, the absorbed energy and the absorption peak locations. At low frequencies, near electrostatic state, the centers of the two negative charge clouds would be in phase or near phase motion, therefore, making an almost one heavy cloud pushed by the light’s electric field. The displacement, therefore, would be expected to be highest, leading to increased restoring force and, hence, the absorption. Thus, the polarization (P) can be approximated to the electrostatic state, equation 3.3.
3.3 where Eintis the induced electric field resulting from the polarization of the particle
As the frequency starts increasing, the IB electrons cloud and the CB electrons would start leaving each other (out of phase), with the cloud of the inner band electrons having smaller displacements compared to the CB electrons. The out of phase oscillation is expected to cause damping on the oscillation of the CB electrons cloud, lower the polarizability and even cause a split in the absorption spectrum, depending on the phase difference. As the frequency increases further, the CB electrons cloud is expected to
follow the oscillating force closely than the IB electrons cloud, which is likely to be incoherently out of phase with the conduction band electrons. Thus, depending on the coupling strength between the CB and IB electrons clouds, the IB electrons would draw a substantial portion of energy from the CB, damping the SPR heavily and even channeling part of the energy to the interband transition of the d-band electrons. It would then be the conduction band electrons that contribute significantly to the dynamic polarization of the nanoparticles. Thus, the overall particle polarization is determined largely by the magnitude of the frequency dependent damping, brought about by the inner band electrons. In the absence of external damping factors, it can permit the mathematical formulation for the dynamic state to be expressed in the form of equation 3.4.