the structural difference between direct and indirect semiconductors. The top of the valence
band of most semiconductors occurs at a value of effective momentum ( k), equal to zero.
Semiconductors in which the bottom of the conduction band is also at k = 0 are direct band
gap materials. Semiconductors in which the bottom of the conduction band occurs at other
points in momentum space are indirect band gap materials. According to the atomic
structure of the semiconductors, their outermost valence electrons are in s- or p-type
104
Optoelectronic Devices and Properties
orbitals. Although true only for elements in their atomic form, the s- or p-like character is
also retained in the crystalline semiconductors. The bands of a semiconductor are a result of
the crystal potential that originates from the equilibrium arrangement of atoms in the lattice.
If the edge of the conduction band is made up of s-type states, the semiconductor is direct
band gap. If, on the other hand, the lowest conduction band edge is made up of p-type
states, then the semiconductor is indirect band gap.
One characteristic example of semiconductor compound is indium antimonite (InSb), the
first to be discovered in 1950, presenting low bandgap, Eg = 0.17 eV, and consequently
applied in far infrared detector technology. Moreover, the invention of the semiconductor
laser and the discovery of the Gunn Effect, turned the interest to other III-V compounds
such as GaAs (Eg = 1.43 eV) and InP (Eg = 1.35 eV). GaP, which has its band gap ( Eg = 2.1 eV) in the visible part of the spectrum is important for the development of the light-emitting
diode (LED). GaP band gap is indirect, but by certain doping techniques, the radiative
efficiency can be improved.
One main advantage of the Binary compounds is that they can be combined or “alloyed” to
form Ternary or Quaternary compounds. These compounds are made up of three or four
group III and group V atoms and, by choosing different elements it is possible to create
materials of different band gaps, with various emission energies for light sources. However,
by alloying, it is possible to vary the band gap continuously and monotonically and together
with it, the band structure and the electronic and optical properties. The varying band gap
also allows the building of heterojunctions, which are important for high-performance
electronic and optoelectronic devices. The quaternary compounds of InxGayAll- x-yAs, and
Inl-xGaxAsyPl-y present band gaps which correspond to the spectral window in which silica
fibers have their lowest loss and dispersion, making their research and development very
important for optical communications. (Bhattacharya, 1997)
2.2 Electron – hole formation and recombination
The creation or annihilation of electron-hole pairs rules the operation of almost all
optoelectronic devices. Pair formation involves raising an electron in energy from the
valence band to the conduction band, resulting with a hole behind in the valence band. In
principle, any energetic particle incident on a semiconductor, which can impart energy at
least equal to the bandgap energy to a valence band electron, will create pairs. The simplest
way to create electron-hole pairs is to irradiate the semiconductor. Photons with sufficient
energy are absorbed, and these impart their energy to the valence band electrons raising
them to the conduction band. This process is, therefore, also called absorption. The reverse
process, that of electron and hole recombination, is associated with the pair giving up its
excess energy after recombination. This process may be radiative or non-radiative. In a non
radiative transition, the excess energy due to recombination is usually imparted to phonons
and dissipated as heat in the material. In a radiative transition, the excess energy is
dissipated as photons, usually having energy equal to the bandgap. This is the luminescent
process, which is classified in three cases according to the method by which the electron-
hole pairs are created: a) Photoluminescence, involves the radiative recombination of
electron-hole pairs created by injection of photons, b) cathodoluminescence, involves
radiative recombination of electron-hole pairs created by electron bombardment and c)
electroluminescence, involves radiative recombination following carriers’ injection in a p-n
junction.
Effects of Ionizing Radiation on Optoelectronic Devices
105
In the case of a semiconductor in equilibrium (i.e., without any incident photons or injection
of electrons), the carrier densities can be calculated from an equilibrium Fermi level using
Fermi-Dirac or Boltzmann statistics. When excess carriers are created by one of the
techniques described above, non-equilibrium conditions are generated and the concept of a
Fermi level is no longer valid. One can, however, define non-equilibrium distribution
functions for electrons and holes as:
(1)
n ( )
1
f E =
⎛ E-E ⎞
1+exp
fn
⎜
⎟
k T
⎝ β ⎠
(2)
p ( )
1
1 - f E =
⎛ E-E ⎞
1+exp
fp
⎜
⎟
k T
⎝ β ⎠
where Efn and Efp are the quasi-Fermi levels for electrons and holes respectively. For the
non-degenerate case (the Fermi level is several kT below Ec), the distribution functions of
equations 1, 2 can be written as:
⎛ E − E ⎞
f ( E) ≅ exp
fn
⎜
⎟ (3)
n
⎜ k
⎟
⎝
β T
⎠
⎛ E − E ⎞
f ( E) ≅ exp
fp
⎜
⎟ (4)
p
⎜ k
⎟
⎝
β T
⎠
and the non-equilibrium carrier concentrations are given by:
⎛ E − E ⎞
n = N exp
fn
c
⎜
⎟ (5)
c
⎜ k
⎟
⎝
β T
⎠
⎛ E − E ⎞
p = N exp
v
fp
⎜
⎟ (6)
v
⎜ k
⎟
⎝
β T
⎠
The quasi-Fermi levels, provide the proper tool for calculating the changes of carrier
concentration as a function of position in a semiconductor.
Generation and recombination processes involve transition of electrons across the energy
bandgap and differ for direct and indirect bandgap semiconductors. In a direct bandgap
semiconductor, the valence band maximum and the conduction band minimum occur at the
zone center ( k = 0) so that by an upward or downward transition of electrons the
momentum is conserved. Therefore, in direct bandgap semiconductors such as GaAs, an
electron raised to the conduction band, (i.e. by photon absorption) will dwell there for a
very short time and recombine again with a valence band hole to emit light of energy equal
106
Optoelectronic Devices and Properties
to the bandgap. In the case of an indirect bandgap semiconductor, where the conduction
band minima are not at k = 0, upward or downward transition of carriers results in a
momentum change so that the involvement of a phonon is needed for the conservation of
momentum. Thus, an electron dwelling in the conduction band minimum, cannot
recombine with a hole at k = 0 until a phonon with the right energy and momentum is
available. Both phonon emission and absorption processes can assist the downward
transition. In order for the right phonon collision to occur, the dwell time of the electron in
the conduction band increases. Consequently the probability of radiative recombination is
much higher in direct than indirect bandgap semiconductors justifying the use of them as
light sources such as light-emitting diodes and lasers.
Moreover impurities and defects in the crystal lattice also serve as traps and recombination
centers. It is most likely that an electron and a hole recombine non-radiatively through such
a defect center and the excess energy is dissipated into the lattice as heat. As advanced
epitaxial techniques are being developed, the purity of the crystals continues to improve.
The simulation that is presented aims in the investigation of the defects generated under
ionized radiation. .(Bhattacharya, 2003)
2.3 Radiative and non-radiative recombination
For continuous carrier generation by optical excitation or injection, a quasi-equilibrium or
steady state is produced. Electrons and holes are created and annihilated in pairs and,
depending on the injection level, a steady-state excess density is established in the crystal,
also necessary for the overall charge neutrality. When the excitation source is removed, the
density of excess carriers returns to the equilibrium values, n0 and p0 . The excess carriers
usually decay exponentially with respect to time following the - exp (-t/r), where r is defined
as the lifetime of excess carriers. The lifetime is determined by a combination of intrinsic and
extrinsic parameters, and affects the performance characteristics of most optoelectronic
devices. It should be noted that, depending on the semiconductor sample and its surface,
there can be a very strong surface recombination component which depends on the density
of surface states.
Generally, the excess carriers decay by radiative and/or non-radiative recombination, in
which the excess energy is dissipated by photons and phonons. The former is of
importance for the operation of luminescent devices. Non-radiative recombination usually
takes place via surface or bulk defects and traps and reduces the radiative efficiency of the
material. Consequently the research in materials aims in reducing these non-radiative
centers by optimizing the fabrication method or by applying passivation techniques on
these centers.
2.4 Electromagnetic description of reflection and refraction
The function and the properties of the materials used in optoelectronic devices are strongly
determined by the propagation of plane, single-frequency, electromagnetic waves. By the
Maxwell’s equations, useful expressions can be obtained for the dissipation, storage and
transport of energy, resulting from the propagation of waves in material media. Moreover
analyzing the incidence, reflection and transition of an electromagnetic wave on the border
surface between two dielectric materials, can result in a better comprehension of
optoelectronic devices.
Effects of Ionizing Radiation on Optoelectronic Devices
107
2.4.1 Maxwell’s equations
The following four laws constitute the basis for the electromagnetic theory:
Gauss’s law for electric fields:
∇ E ρ
= ε (7)
0
Gauss’s law for magnetic fields:
∇ B = 0 (8)
B
∂
Faraday’s law:
∇ × E = −
(9)
t
∂
⎛
E
∂ ⎞
Ampere – Maxwell law:
∇ × B = μ ⎜ J + ε
⎟
(10)
0
0
⎜
t ⎟
∂
⎝
⎠
The laws above can be applied in combination with the constitutive equations:
D = ε E + P and B = μ H + M
(11)
0 (
)
0
where J, is the current density, E and H are the electric and magnetic fields respectively, D
and B, are the electric and magnetic displacements and P and M, are the electric and
magnetic polarizations respectively.
Considering that in the dielectric materials there are no free charge carriers and currents, the
application of Maxwell’s equations, in combination with the constitutive equations result
in the boundary conditions standing at the boundary surface between two dielectric
materials. These conditions denote that during the transition from the dielectric 1 to the
dielectric 2, the tangential (at the boundary surface) components of E and H as well as the
normal (at the boundary surface) components of D and B, are continuous. The mathematical
expression can be written as follows:
E2t = E1t, H2t = H1t, D2n = D1n and B2n = B1n
where the indexes 1 and 2 denote the limiting values, in the case of approaching the
boundary surface from the dielectric 1 or 2.
These boundary conditions will serve for the calculation of important coefficients in the case
of reflection and transmission of light. (Yariv,1976)
2.4.2 Reflection and refraction
2.4.2.1 Angle of incidence = 0o
In figure 1, the vectors of the electric (E) and magnetic (H) fields are subscripted as “i” for
incident wave, “r” for reflected wave and “t” for transmitted wave. In this case the respective
equations in complex form, for the vectors of the fields E(electric) and H(magnetic) will be:
i(ω t− k 1 x)
i(ω t− 1
k x)
Ε = E e
Η = H e
x ≺
iy
i
iz
i
0
0
0
(
)
i(ω t− k 1 x)
i(ω t− k 1 x)
Ε = E e
Η = H e
x ≺
(12)
ry
r
rz
r
0
0
0
(
)
i(ω t− k 2 x)
i(ω t− k 2 x)
Ε = E e
Η = H e
x
ty
t
tz
t
0
0
0
(
)
108
Optoelectronic Devices and Properties
In the above equations, the unknown quantities are the amplitudes: Er0 and Et0. By these, are
defined the coefficient of reflection r=Er0/Ei0 and the coefficient of transmission t=Et0/Ei0.
These coefficients can be calculated by applying boundary conditions for the fields E and H,
and finally the quantities of Er0 and Et0 can be calculated. Finally:
n μ − n μ
1 2
2 2
r =
(13)
n μ + n μ
1 2
2 1
and
2 n μ
1 2
t =
(14)
n μ + n μ
1 2
2 1
where n1,2 and μ1,2 are the refraction coefficient and magnetic permeability respectively, of
the two dielectric materials. (Papakitsos,2002)
y axi
x s
area of
r
mate
t rial wi
rial w th n
1
area of material with n
1
area of material with 2
Ei
Er
Et
Hr
H
u
u
u
u
2
X ax
X a is
x
1
1
0
Hi
H
Ht
H
z axis
z ax
Fig. 1. EMF vectors presenting the transmission and reflection of an EM wave incident at a
boundary surface between to dielectric materials
By the coefficients above, also the quantities of reflectance, R and transmittance, T can be
I
I
calculated. These quantities are defined as:
r
R =
and
t
T =
where I
I
I
r is the intensity of
i
i
the reflected wave, Ii is the intensity of the incident wave and It is the intensity of the
transmitted wave. In the special case of angle of incidence = 0o,
2
2
n E
2 cμ
E
1
r 0
1
2
r 0
R =
=
= r
(15)
2
n E
2 cμ
Ei 0
1
i 0
1
2
2
n E
2 cμ
n μ E
n μ
and
2
t 0
2
2
2 1
t 0
2 1
T =
=
=
t (16)
2
n E
2 cμ
n μ E
n μ
1 2
r 0
1 2
1
r 0
1
From the above relations, the energy conservation is proved, since R+T=1!
Effects of Ionizing Radiation on Optoelectronic Devices
109
In the case of transparent dielectrics, where the magnetic permeability is semi equal to the
magnetic permeability of vacuum, the above relations are far simplified as follows:
2
n − n
2 n
⎛ n − n ⎞
4 n n
1
2
r =
,
1
t =
,
1
2
R = ⎜
⎟ and
1 2
T =
n + n
n + n
n +
⎝
n
2
1
2
1
2
( n + n
1
2 )
1
2 ⎠
⎧ 0 for n
n
It is important to note that
1
2
r = ⎨
but t 0 for every n
0 for n
n
1, n2.
⎩≺
≺
1
2
The physical meaning hiding here is that when a wave is heading from a material with n1 to
a material with n2 with n
n the phase of the reflected wave is unchanged in accordance
1
2
to the phase of the incident wave. On the opposite case ( n ≺ n ) the phase of the reflected
1
2
wave has a difference of π in accordance to the phase of the incident wave.
2.4.2.2 Angle of incidence = θo
In the case of a random angle of incidence, we consider the transverse wave of the electric
field (TE) and the transverse wave of the magnetic field (TM). Consequently, following the
method of figure 1, the coefficients r and t are now written as:
E
E
E
E
For the transverse wave of the electric field
<