X 1
n¼0
a n x
n
3 ¼ a þ b
2
ffiffiffi
π
p
X 1
n¼0
À1
ð Þ
n
λ
2nþ1
n! 2n þ 1
ð
Þ
x
2nþ1
3
:
ð3:59Þ
We plot erf(λx 3 ) in Fig. 3.4 which converges everywhere. Its behavior is suitable for
describing the doping profiles of certain PN junctions with symmetry or
antisymmetry. tan
À1 (λx 3 ) and tanh(λx 3 ) have similar behaviors but are with a finite
domain of convergence.
As a numerical example, consider a rod with
L ¼ 6 μm, λ ¼ 10
7 m
À1 ,
α 1 ¼ 0:9 Â 10
21 m
À3 , β 1 ¼ À0:1 Â 10
21 m
À3 ,
α 2 ¼ 0:9 Â 10
21 m
À3 , β 2 ¼ 0:1 Â 10
21 m
À3
:
ð3:60Þ
In this case a ¼ 0 according to Eq. (3.57) and N
þ
D À N
À
A ¼ berf λx 3
ð Þ is an odd
function. Instead of Eqs. (3.48), (3.49), and (3.50), to make use of the symmetry or
antisymmetry of the doping profile implied by Eq. (3.60), we simply impose
u 3 0
ð Þ ¼ 0, φ 0
ð Þ ¼ 0,
p
0
¼ N
À
A 0
ð Þ ¼ α 1 ,
n
0
¼ N
þ
D 0
ð Þ ¼ α 2 ¼ α 1 :
ð3:61Þ
In Fig. 3.5, (a)–(c) show the effects of λ on the normalized doping and mobile charge
ρ
e /q whose definition is in Eq. (3.42), the effective polarization charge ρ
P /q given by
Eq. (3.21), the total charge (ρ
e + ρ
P )/q, the electric field, and the electric potential. ρ
e
and ρ
P are with opposite signs, and ρ
e dominates. In addition to the distributed ρ
P , for
a finite rod, in general, there may also exist concentrated effective polarization
charges at the two ends of the rod. These concentrated charges depend on the values
Fig. 3.4 Basic behavior of
error function
42
3 Extension of Rods