S 3 ¼
∂u 3
∂x 3
, E 3 ¼ À
∂φ
∂x 3
,
ð3:135Þ
where uniform doping has been assumed. Δp and Δn are given in Eq. (1.7). p 0 and n 0
are defined by Eq. (1.8). We consider an n-type semiconductor with p ffi 0. For timeharmonic vibrations described by a common time factor exp(iω t), with successive
substitutions, we can write Eqs. (3.131) and (3.132) 2 as
c 33 u 3,33 þ e 33 φ ,33 ¼ Àρω
2 u 3 ,
e 33 u 3,33 À ε 33 φ ,33 ¼ q ÀΔn
ð
Þ,
iωq Δn
ð Þ ¼ Àqn 0 μ
n
33 φ ,33 þ qD
n
33
∂ Δn
ð Þ
∂x 2
3
:
ð3:136Þ
We look for a solution of Eq. (3.136) in the following form:
u 3
φ
Δn
8
> <
> :
9
> =
> ;
¼
A
B
C
8
> <
> :
9
> =
> ;
exp ζx 3
ð Þ,
ð3:137Þ
where A, B, C, and ζ are undetermined constants. Substituting Eq. (3.137) into
Eq. (3.136), we obtain a system of linear homogeneous equations for A, B, and C:
c 33 ζ
2
þ ρω
2
e 33 ζ
2
0
e 33 ζ
2
Àε 33 ζ
2
q
0
Àn 0 μ
n
33 ζ
2 D
n
33 ζ
2
À iω
2
6
4
3
7
5
A
B
C
8
> <
> :
9
> =
> ;
¼ 0:
ð3:138Þ
For nontrivial solutions of A, B, or C, the determinant of the coefficient matrix of
Eq. (3.138) has to vanish, which leads to the following cubic equation for ζ
2 :
À c 33 ε 33 þ e
2
33
À
Á
D
n
33 ζ
6
þ iωc 33 ε 33 þ iωe
2
33 þ c 33 qn 0 μ
n
33 À ρω
2 D
n
33 ε 33
À
Á ζ
4
þ iρω
3
ε 33 þ ρω
2 qn 0 μ
n
33
À
Á ζ
2
¼ 0:
ð3:139Þ
Equation (3.139) has six roots for ζ. ζ
(1)
¼ ζ
(2)
¼ 0 is a repeated root. It is easy to
show that ζ
(1)
¼ ζ
(2)
¼ 0 leads to
u 3
φ
Δn
8
> <
> :
9
> =
> ;
¼ B
1
ð Þ
0
x 3
0
8
> <
> :
9
> =
> ;
þ B
2
ð Þ
0
1
0
8
> <
> :
9
> =
> ;
,
ð3:140Þ
62
3 Extension of Rods