4
J. Oliver–Meseguer and A. Leyva–Pérez
atoms but also on the relationship surface/volume, the size and shape and the oxidation state, among other parameters. In the case of metallic nanoparticles, the most
relevant characteristic is the plasmon. Bulk plasmons are quantum waves from electrons that are produced when the electrons are perturbed with an incident beam,
changing the equilibrium position and vibrating in a given characteristic frequency.
In other words, the plasmon is a collective oscillation of the conductive electrons
that are shining by an appropriate wavelength.
Metal clusters are extremely small particles with a diameter less than 1 nm.
Because of that, they are located between the bulk and atomic states of the corresponding metal. Due to the possibility to observe finite-size effects on the physical
properties of metal clusters and to understand their microscopic origins, they have
attracted physicists over the last four decades. Development of new experimental and
theoretical methods has led to a discovery of a variety of remarkable size-specific
phenomena and physicochemical properties. During this development, the community has come to be convinced that metal clusters are promising functional units
of novel materials and has tried to develop cluster-based materials using “small is
different” and “every atom counts” claims [18, 67].
1.1.3 Jellium Model
The characteristic features of small metal clusters, like the electronic shell structure
and band gap, can be well understood in terms of quantum motion of the delocalized
valence electrons and positively charged ionic core. This concept, known as the
Jellium model for neutral metal clusters, can also be applied to charged systems
[58].
This model was developed initially for clusters in the gas phase, and the cluster is
replaced by an electronic structure in layers that consists of spherical Jellium (UEG—
Uniform Electron Gas or HEG—Homogeneous Electron Gas) positively charged and
surrounded by electrons. It is considered that the electrons move in a medium field
potential occupying, according to the Aufbau principle, energy levels. This model
represents a good approximation since it preserves most of the physicochemical
characteristics of the clusters. The total energy, as a function of the cluster size, can
be calculated by the approximation that represents the covered electronic levels and
corresponds with the most stable clusters that possess the “magic numbers.”
Density functional theory calculations to determine the geometric structures of
the metal clusters are often used to confirm the experimental data and approximations
used by the Jellium model. In general, these studies demonstrate also that there is
an oscillation in the stability and the electronic properties of the clusters in function
of the atom numbers, neutral clusters with odd number of atoms being more stable
than ionic clusters with even atoms.
For all this, the Jellium model provides a good approximation to the electronic
behavior of the clusters, describing the dependency of the photoemission energy with
the number of atoms in the cluster and following Eq. 1.1:
J. Oliver–Meseguer and A. Leyva–Pérez
atoms but also on the relationship surface/volume, the size and shape and the oxidation state, among other parameters. In the case of metallic nanoparticles, the most
relevant characteristic is the plasmon. Bulk plasmons are quantum waves from electrons that are produced when the electrons are perturbed with an incident beam,
changing the equilibrium position and vibrating in a given characteristic frequency.
In other words, the plasmon is a collective oscillation of the conductive electrons
that are shining by an appropriate wavelength.
Metal clusters are extremely small particles with a diameter less than 1 nm.
Because of that, they are located between the bulk and atomic states of the corresponding metal. Due to the possibility to observe finite-size effects on the physical
properties of metal clusters and to understand their microscopic origins, they have
attracted physicists over the last four decades. Development of new experimental and
theoretical methods has led to a discovery of a variety of remarkable size-specific
phenomena and physicochemical properties. During this development, the community has come to be convinced that metal clusters are promising functional units
of novel materials and has tried to develop cluster-based materials using “small is
different” and “every atom counts” claims [18, 67].
1.1.3 Jellium Model
The characteristic features of small metal clusters, like the electronic shell structure
and band gap, can be well understood in terms of quantum motion of the delocalized
valence electrons and positively charged ionic core. This concept, known as the
Jellium model for neutral metal clusters, can also be applied to charged systems
[58].
This model was developed initially for clusters in the gas phase, and the cluster is
replaced by an electronic structure in layers that consists of spherical Jellium (UEG—
Uniform Electron Gas or HEG—Homogeneous Electron Gas) positively charged and
surrounded by electrons. It is considered that the electrons move in a medium field
potential occupying, according to the Aufbau principle, energy levels. This model
represents a good approximation since it preserves most of the physicochemical
characteristics of the clusters. The total energy, as a function of the cluster size, can
be calculated by the approximation that represents the covered electronic levels and
corresponds with the most stable clusters that possess the “magic numbers.”
Density functional theory calculations to determine the geometric structures of
the metal clusters are often used to confirm the experimental data and approximations
used by the Jellium model. In general, these studies demonstrate also that there is
an oscillation in the stability and the electronic properties of the clusters in function
of the atom numbers, neutral clusters with odd number of atoms being more stable
than ionic clusters with even atoms.
For all this, the Jellium model provides a good approximation to the electronic
behavior of the clusters, describing the dependency of the photoemission energy with
the number of atoms in the cluster and following Eq. 1.1: