43
Review of Basic Device Physics
Let us consider the following example where an impurity like Au is
introduced that provides a trapping level or a set of allowed states at energy E t .
The trap level E t is assumed to act like an acceptor (it can be neutral or negatively charged). Recombination is accomplished by trapping an electron and
a hole. (The analysis can be easily extended to the case where the trap acts
like a donor, that is, positively charged or neutral charge states.) The indirect
recombination process was originally proposed by Shockley and Read [22]
and independently suggested by Hall [23] and, therefore, is often referred to
as the Shockley–Read–Hall (SRH) recombination. By considering the transition processes shown in Figure 2.12, Shockley, Read, and Hall showed that
for low-level injection, the net recombination rate is given by
U
v N pn n
n p n
E E kT
th
t
i
i
t
i
=
−
(
)
+ +
−
(
)
σ
2
2 cosh
(2.49)
where:
v th is the carrier thermal velocity (≈ 1 × 10 7 cm sec –1 )
σ is the carrier capture cross section (≈10 –15 cm 2 )
N t is the density of trap centers
v th σN t is the capture probability or capture cross section
From Equation 2.49 we observe the following:
1. The “driving force” or the rate of recombination is proportional to
pn n i
−
(
)
2 , that is, the deviation from the equilibrium condition
2. U = 0 when np n i
=
(
)
2 , that is, equilibrium condition
3. U is maximum when E t = E i , that is, trap levels near the mid-band
are the most efficient recombination centers
Thus, for the simplicity of understanding, let us consider the case when
E t = E i . Then from Equation 2.49, the net recombination rate is given by
U
v N pn n
n p n
th
t
i
i
=
−
(
)
+ +
σ
2
2
(2.50)
For an n-type semiconductor with low-level injection, n >> p + 2n i ; denoting
p = p n as the total excess minority carrier concentration and p
n n
n
i
o =
(
)
2
as
the equilibrium minority carrier concentration, we get after simplification of
Equation 2.50
U v N p p
p
th
t
n
n
p
=
−
(
)=
σ
τ
o
∆
(2.51)
Review of Basic Device Physics
Let us consider the following example where an impurity like Au is
introduced that provides a trapping level or a set of allowed states at energy E t .
The trap level E t is assumed to act like an acceptor (it can be neutral or negatively charged). Recombination is accomplished by trapping an electron and
a hole. (The analysis can be easily extended to the case where the trap acts
like a donor, that is, positively charged or neutral charge states.) The indirect
recombination process was originally proposed by Shockley and Read [22]
and independently suggested by Hall [23] and, therefore, is often referred to
as the Shockley–Read–Hall (SRH) recombination. By considering the transition processes shown in Figure 2.12, Shockley, Read, and Hall showed that
for low-level injection, the net recombination rate is given by
U
v N pn n
n p n
E E kT
th
t
i
i
t
i
=
−
(
)
+ +
−
(
)
σ
2
2 cosh
(2.49)
where:
v th is the carrier thermal velocity (≈ 1 × 10 7 cm sec –1 )
σ is the carrier capture cross section (≈10 –15 cm 2 )
N t is the density of trap centers
v th σN t is the capture probability or capture cross section
From Equation 2.49 we observe the following:
1. The “driving force” or the rate of recombination is proportional to
pn n i
−
(
)
2 , that is, the deviation from the equilibrium condition
2. U = 0 when np n i
=
(
)
2 , that is, equilibrium condition
3. U is maximum when E t = E i , that is, trap levels near the mid-band
are the most efficient recombination centers
Thus, for the simplicity of understanding, let us consider the case when
E t = E i . Then from Equation 2.49, the net recombination rate is given by
U
v N pn n
n p n
th
t
i
i
=
−
(
)
+ +
σ
2
2
(2.50)
For an n-type semiconductor with low-level injection, n >> p + 2n i ; denoting
p = p n as the total excess minority carrier concentration and p
n n
n
i
o =
(
)
2
as
the equilibrium minority carrier concentration, we get after simplification of
Equation 2.50
U v N p p
p
th
t
n
n
p
=
−
(
)=
σ
τ
o
∆
(2.51)