10.13 Excess-Carrier Profiles
333
10.13.1 Generation at Surface
First, the generation of excess carriers in a semi-infinite piece of semiconductor shall occur only at the
surface at x = 0 (strong absorption limit). The generation is zero everywhere else and the excitation is
incorporated via the boundary condition p(x = 0) = p 0 . The general solution for the homogeneous
equation (10.63), i.e. G = 0, is
p(x) = C 1 exp
−
x
L p
+ C 2 exp
x
L p
,
(10.64)
with the diffusion length L p =
D p τ p . Taking the boundary condition p(x → ∞) = 0 the solution
is (C 2 = 0)
p(x) = p 0 exp
−
x
L p
.
(10.65)
In order to connect p 0 with the total generation rate per unit area G tot , we calculate
G tot =
∞
0
p(x)
τ p
dx = p 0
L p
τ p
= p 0
D p
τ p
.
(10.66)
If a slab of finite thickness d is considered, the boundary condition on the back surface comes into play.
Assuming a contact that extracts all excess charge carriers, p(d) = 0. In conjunction with (10.64),
we find
p(x) =
p 0
2
1 + coth
d
L p
exp
−
x
L p
+
1 − coth
d
L p
exp
x
L p
,
(10.67)
with
G tot =
d
0
p(x)
τ p
dx = p 0
L p
τ p
tanh
d
2 L p
.
(10.68)
Generally, the excited electrons and holes diffuse into the bulk at different speed due to their different
diffusion length (or mobility). As consequence, a spatial separation of electrons and hole densities
occurs which leads to an electric field, the Dember field [1045–1047]. The illuminated surface becomes
positive compared to the dark case since D n > D p . A treatment of the non-neutral diffusion situation
is given in [1048]. As shown in Fig. 10.35, the non-equal electron and hole densities create locally
a non-zero charge density δρ and an associated potential δφ; for a silicon material parameters (n i =
10
10 cm
−3 ), the Dember voltage is calculated as V Dem = δφ(0) = 1.84 meV for a generation rate of
G = 2 × 10
10 /(cm
2 s). Extensions of the theory have been given considering traps [1049] and the effect
of finite sample thickness and surface recombination [1050] (under certain conditions, the sign of the
field can be reverse).
333
10.13.1 Generation at Surface
First, the generation of excess carriers in a semi-infinite piece of semiconductor shall occur only at the
surface at x = 0 (strong absorption limit). The generation is zero everywhere else and the excitation is
incorporated via the boundary condition p(x = 0) = p 0 . The general solution for the homogeneous
equation (10.63), i.e. G = 0, is
p(x) = C 1 exp
−
x
L p
+ C 2 exp
x
L p
,
(10.64)
with the diffusion length L p =
D p τ p . Taking the boundary condition p(x → ∞) = 0 the solution
is (C 2 = 0)
p(x) = p 0 exp
−
x
L p
.
(10.65)
In order to connect p 0 with the total generation rate per unit area G tot , we calculate
G tot =
∞
0
p(x)
τ p
dx = p 0
L p
τ p
= p 0
D p
τ p
.
(10.66)
If a slab of finite thickness d is considered, the boundary condition on the back surface comes into play.
Assuming a contact that extracts all excess charge carriers, p(d) = 0. In conjunction with (10.64),
we find
p(x) =
p 0
2
1 + coth
d
L p
exp
−
x
L p
+
1 − coth
d
L p
exp
x
L p
,
(10.67)
with
G tot =
d
0
p(x)
τ p
dx = p 0
L p
τ p
tanh
d
2 L p
.
(10.68)
Generally, the excited electrons and holes diffuse into the bulk at different speed due to their different
diffusion length (or mobility). As consequence, a spatial separation of electrons and hole densities
occurs which leads to an electric field, the Dember field [1045–1047]. The illuminated surface becomes
positive compared to the dark case since D n > D p . A treatment of the non-neutral diffusion situation
is given in [1048]. As shown in Fig. 10.35, the non-equal electron and hole densities create locally
a non-zero charge density δρ and an associated potential δφ; for a silicon material parameters (n i =
10
10 cm
−3 ), the Dember voltage is calculated as V Dem = δφ(0) = 1.84 meV for a generation rate of
G = 2 × 10
10 /(cm
2 s). Extensions of the theory have been given considering traps [1049] and the effect
of finite sample thickness and surface recombination [1050] (under certain conditions, the sign of the
field can be reverse).