42
Compact Models for Integrated Circuit Design
n are the respective total concentrations due to generation, then Δp = p – p o
and Δn = n – n o and the net recombination rate due to direct recombination
is given by
U
n
p
n
p
=
=
∆
∆
τ
τ
(2.48)
where:
τ n and τ p are the excess carrier electron and hole lifetime, respectively
For band-to-band recombination, the excess carrier lifetime for an electron is
equal to that of a hole since the single phenomenon annihilates an electron
and a hole simultaneously.
For indirect bandgap semiconductors such as silicon and germanium
(Figure 2.11c), the probability of direct recombination is very low. Physically,
this means that the minimum energy gap between E c and E v does not occur at
the same point in the momentum space as shown in Figure 2.11c. In this case,
for an electron to reach the VB, it must experience a change of momentum
as well as energy to satisfy the conservation principle. This can be achieved
by recombination processes through intermediate trapping levels, called the
indirect recombination as shown in Figure 2.12.
Impurities that form electronic states deep in the energy gap assist the
recombination of electrons and holes in the indirect bandgap semiconductors. Here the word deep indicates that the states are far away from the band
edges and near the center of the energy gap. These deep states are commonly
referred to as recombination centers or traps. Such recombination centers are
usually unintentional impurities, which are not necessarily ionized at room
temperature. These deep level impurities have concentrations far below the
concentration of donor or acceptor impurities, which have shallow energy
levels. Gold (Au) is a deep level impurity intentionally used in silicon to
increase the recombination rate. This recombination via deep level impurities or traps is often referred to as the indirect recombination. The process
shown in Figure 2.12 consists of (1) an electron capture by an empty center,
(2) electron emission from an occupied center, (3) hole capture by an occupied center, and (4) hole emission by an empty center.
E c 1
2
1 = Electron capture
2 = Electron emission
3 = Hole capture
4 = Hole emission
2 + 4 or 4 + 2: generation
1 + 3 or 3 + 1: recombination
3
4
E t
E f
E v
E i
FIGURE 2.12
Generation and recombination in an indirect bandgap semiconductor; E t is the trap level deep
into the bandgap; 1, 2, 3, and 4 represent the generation and recombination processes.
Compact Models for Integrated Circuit Design
n are the respective total concentrations due to generation, then Δp = p – p o
and Δn = n – n o and the net recombination rate due to direct recombination
is given by
U
n
p
n
p
=
=
∆
∆
τ
τ
(2.48)
where:
τ n and τ p are the excess carrier electron and hole lifetime, respectively
For band-to-band recombination, the excess carrier lifetime for an electron is
equal to that of a hole since the single phenomenon annihilates an electron
and a hole simultaneously.
For indirect bandgap semiconductors such as silicon and germanium
(Figure 2.11c), the probability of direct recombination is very low. Physically,
this means that the minimum energy gap between E c and E v does not occur at
the same point in the momentum space as shown in Figure 2.11c. In this case,
for an electron to reach the VB, it must experience a change of momentum
as well as energy to satisfy the conservation principle. This can be achieved
by recombination processes through intermediate trapping levels, called the
indirect recombination as shown in Figure 2.12.
Impurities that form electronic states deep in the energy gap assist the
recombination of electrons and holes in the indirect bandgap semiconductors. Here the word deep indicates that the states are far away from the band
edges and near the center of the energy gap. These deep states are commonly
referred to as recombination centers or traps. Such recombination centers are
usually unintentional impurities, which are not necessarily ionized at room
temperature. These deep level impurities have concentrations far below the
concentration of donor or acceptor impurities, which have shallow energy
levels. Gold (Au) is a deep level impurity intentionally used in silicon to
increase the recombination rate. This recombination via deep level impurities or traps is often referred to as the indirect recombination. The process
shown in Figure 2.12 consists of (1) an electron capture by an empty center,
(2) electron emission from an occupied center, (3) hole capture by an occupied center, and (4) hole emission by an empty center.
E c 1
2
1 = Electron capture
2 = Electron emission
3 = Hole capture
4 = Hole emission
2 + 4 or 4 + 2: generation
1 + 3 or 3 + 1: recombination
3
4
E t
E f
E v
E i
FIGURE 2.12
Generation and recombination in an indirect bandgap semiconductor; E t is the trap level deep
into the bandgap; 1, 2, 3, and 4 represent the generation and recombination processes.