Page 151 - A Comprehensive Guide to Solar Energy Systems

152 A CoMPrehensIVe GuIde To soLAr enerGy sysTeMs

momentum. If there are no possible interactions satisfying the conservation laws , the en-
ergy of photons cannot be absorbed by the material and the material is transparent. If
there are possible interactions by which the material particles can increase their energy
(the original energy plus photon energy is one of the possible energy states of the particle
in the material), the photon can be absorbed. suppose that Φ(x) is the flux of photons in
the distance x from the surface, the change of the photon flux dΦ in the path dx due to
absorption can be expressed as
α
dΦ=−αΦdx, d Φ= −Φdx, (8.2)
where α is a proportionality constant called the absorption coefficient that depends on the
photon energy, that is, α = α(λ).
If we integrate eq. (8.1) for a constant photon energy (illumination with monochro-
matic light), the transmitted photon flux decreases exponentially with the coordinate, x,
 x 
Φ x (; ) 0 λ)exp[ − ( =Φ exp  −  , (8.3)
αλ x )]
λ = Φ (
0
λ 
 x ()
L
Φ(x;λ)=Φ 0 (λ)exp[−α(λ)x]=Φ 0 exp−
xxL(λ),
where x L is the so-called absorption length. The absorption length gives the distance from
the surface, at which the photon flux (light intensity) decreases to 36.8% of the flux Φ 0
entering (semiconductor) at the material surface. on the path from the surface to the ab-
sorption length, 63.2% of incident photon flux is absorbed. Practically all photons are ab-
sorbed in a layer of thickness 3x L .
The interactions of the photons with the material particles may be
• interactions with the lattice,
• interactions with free electrons, or
• interactions with bonded electrons.

Interactions with the lattice (nucleus) are possible for low energy photons (deep infra-
red), and interactions with free charge carriers (electrons) are important when the carrier
concentration is high. Both interactions result in increasing the kinetic energy of particles
and, consequently, in an increase of the materials temperature.
Interactions with bonded electrons are the most important. If the photon energy is
higher than the energy of the bond, the electron can “liberalize” and move in the material,
the unoccupied place behaves like a positive charged “hole”. In this way, the incident light
generates some excess carriers (electron/hole pairs).
In crystalline semiconductors, all possible energies of bonded valence electrons create
the valence band, and all possible energies of free electrons create the conduction band.
The dependence of energy on momentum is called the structure of the band. The differ-
ence between the maximum energy in the valence band W v and the minimum energy in
the conduction band W c , W g = W c − W v , is the so-called bandgap. At the temperature T > 0
some of bonds may be incomplete due to interactions with the lattice or impurities or
defects. In thermal equilibrium, free electron concentration n 0 and hole concentration p 0
are connected by