Fundamentals

1.1 Working Principles and Thermodynamics of Heterogeneous Photocatalysis

1.1.1 Conductors, Insulators and Semiconductors

The valence bond theory is useful in explaining the structure and the geometry of the molecules but it does not provide direct information on bond energies and fails to explain the magnetic properties of certain substances. The molecular orbitals (MO) theory solves these drawbacks. It is based on the assumption that the electrons of a molecule can be represented by wave functions, ψ, called molecular orbitals, characterized by suitable quantic numbers that determine their form and energy. The combination of two atomic orbitals gives rise to two molecular orbitals, indicated as ψ+ and ψ-. If an electron occupies the ψ+ molecular orbital, a stable bond is formed between two nuclei, and so this is called bonding orbital. Conversely, when an electron occupies the ψ- orbital it is called anti-bonding orbital and the presence of an electron promotes the dissociation of the molecule. Molecular orbitals of a solid consisting of n equal atoms are obtained by means of a linear combination of atomic orbitals. The number of molecular orbitals formed is equal to that of the atomic ones. By increasing the number of atoms the difference between the energetic levels decreases and a continuous band of energy is formed for high values of n.

The width of the various bands and the separation among them depends on the internuclear equilibrium distance between adjacent atoms. If the energetic levels of isolated atoms are not so different, progressive enlargement of the bands may lead to their overlapping by decreasing the internuclear distance. The most external energetic band full of electrons is called the valence band (VB).

The energy band model for electrons can be applied to all crystalline solids and allows one to establish if a substance is a conductive or an insulating material. Indeed, the properties of a solid are determined by the difference in energy between the different bands and the distribution of the electrons contained within each band.

If the valence band is partially filled or it is full and overlapped with a band of higher energy, electrons can move, allowing conduction (conductors), as in the case of metals that have relatively few valence electrons that occupy the lowest levels of the most external band.

In contrast, the valence band is completely filled in ionic or covalent solids but it is separated by a high energy gap from the subsequent empty band. In this situation no electrons can move even if high electric fields are applied and the solid is an insulator. If the forbidden energy gap is not very high, some electrons could pass into the energetic empty band by means of thermal excitation and the material behaves as a weak conductor, i.e., as a semiconductor. The empty band, which allows the movement of the electrons, is called the conduction band (CB).

The energy difference between the lowest conduction band edge and the highest valence band edge is called the band gap (EG). A material is generally considered a semiconductor when EG ≤ 3 eV, whereas it is considered a wide band gap semiconductor when its band gap value ranges between 3 and 4 eV.

Figure 1.1 shows the position of the energy bands of different types of materials.

1.1.2 Properties of Semiconductor Materials

The de Broglie relation associates a wavelength with the electron as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1)

where p is the momentum and h is the Planck constant. It shows that the wavelength is inversely proportional to the momentum of a particle and that the frequency is directly proportional to the particle's kinetic energy:

f = E/h (1.2)

The wave number corresponds to the number of repeating units of a propagating wave per unit of space. It is defined as:

[bar.v] = 1/λ (1.3)

In the case of non-dispersive waves the wave number is proportional to the frequency, f:

[bar.v] = f/v (1.4)

where v is the propagation velocity of the wave. For electromagnetic waves propagating in vacuum the following relation is obtained:

[bar.v] = f/c (1.5)

where c is the velocity of light.

The wave vector is a vector related to a wave and its amplitude is equal to the wave number while its direction is that of the propagation of the wave:

[bar.k] = 2π/λ (1.6)

Therefore, the electron momentum can be expressed as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.7)

where [??] is the reduced Plank constant also known as the Dirac's constant ([??] = h/2π).

The electron energy is therefore:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.8)

where the coefficient m is the inertial mass of the wave-particle. As m varies with the wave vector, it is called effective mass, m*, defined as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.9)

A semiconductor is called a direct band gap semiconductor if the energy of the top of the valence band lies below the minimum energy of the conduction band without a change in momentum, whereas it is called an indirect band gap semiconductor if the minimum energy in the conduction band is shifted by a difference in momentum (Figure 1.2).

The probability f(E) that an energetic level of a solid is occupied by electrons can be determined by the Fermi–Dirac distribution function. It applies to fermions (particles with half-integer spin, including electrons, photons and neutrons, which must obey the Pauli exclusion principle) and states that a given allowed level of energy E is function of temperature and of the Fermi level, E0F according to the equation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.10)

where kB is the Boltzmann constant. The level E0F represents the probability of 50% of finding an electron in it. For intrinsic semiconductors and for insulating materials, E0F falls inside the energetic gap and its value depends on the effective mass of electrons present at the end of the conduction band (m*e), on the effective mass of electrons at the beginning of the valence band (m*h), and on the amplitude of the band gap (EG) according to the following equation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.11)

The value of E0F is equivalent to the electrochemical potential of the electron, i.e., it can be considered as the work necessary to transport an electron from an...