Q ,3 ¼ 0,
M ,3 À Q ¼ 0,
D
1
ð Þ
3,3 À D
0
ð Þ
2 ¼ qI p
1
ð Þ
À n
1
ð Þ
,
J
n 1
ð Þ
3,3 À J
n 0
ð Þ
2
¼ 0,
ð4:32Þ
Q ¼ c 44 A v ,3 þ ψ
ð
Þþe 15 Aϕ
1
ð Þ ,
M ¼ c 33 Iψ ,3 þ e 33 Iϕ
1
ð Þ
,3 ,
ð4:33Þ
D
0
ð Þ
2 ¼ e 15 A v ,3 þ ψ
ð
ÞÀε 11 Aϕ
1
ð Þ ,
D
1
ð Þ
3 ¼ e 33 Iψ ,3 À ε 33 Iϕ
1
ð Þ
,3 ,
ð4:34Þ
J
n 0
ð Þ
2
¼ Àqn 0 μ
n
11 Aϕ
1
ð Þ
þ qD
n
11 An
1
ð Þ ,
J
n 1
ð Þ
3
¼ Àqn 0 μ
n
33 Iϕ
1
ð Þ
,3 þ qD
n
33 In
1
ð Þ
,3 ,
ð4:35Þ
c 44 A v ,33 þ ψ ,3
À
Á þ e 15 Aϕ
1
ð Þ
,3 ¼ 0,
c 33 Iψ ,33 þ e 33 Iϕ
1
ð Þ
,33 À c 44 A v ,3 þ ψ
ð
ÞÀe 15 Aϕ
1
ð Þ
¼ 0,
e 33 Iψ ,33 À ε 33 Iϕ
1
ð Þ
,33 À e 15 A v ,3 þ ψ
ð
Þþε 11 Aϕ
1
ð Þ
¼ qI Àn
1
ð Þ
,
Àqn 0 μ
n
33 Iϕ
1
ð Þ
,33 þ qD
n
33 In
1
ð Þ
,33 þ qn 0 μ
n
11 Aϕ
1
ð Þ
À qD
n
11 An
1
ð Þ
¼ 0:
ð4:36Þ
The boundary conditions are
v 0
ð Þ ¼ 0, ψ 0
ð Þ ¼ 0, M L
ð Þ ¼ 0, Q L
ð Þ ¼ f y ,
ð4:37Þ
D
1
ð Þ
3 0
ð Þ ¼ 0, J
n 1
ð Þ
3
0
ð Þ ¼ 0, D
1
ð Þ
3 L
ð Þ ¼ 0, J
n 1
ð Þ
3
L
ð Þ ¼ 0,
ð4:38Þ
where we have assumed an electrically isolated beam. For bending without extension, the carrier concentration perturbation is simply Δn ¼ x 2 n
(1) , an odd function of
x 2 that satisfies the charge neutrality condition automatically.
With some algebra, it can be found that the general solution of Eq. (4.36) is
ϕ
1
ð Þ
¼C 1 sinh λ 1 x 3 þ C 2 cosh λ 1 x 3 þ C 3 sinh λ 3 x 3 þ C 4 cosh λ 3 x 3
À
D
n
11 A
D
n
33 ε 33 I
e 33 c 44 À e 15 c 33
c 44
C 5
a 2
,
ð4:39Þ
ψ ¼ À
e 33
c 33
C 1 sinh λ 1 x 3 þ C 2 cosh λ 1 x 3 þ C 3 sinh λ 3 x 3 þ C 4 cosh λ 3 x 3
ð
Þ
þ
C 5
2
x
2
3 þ C 6 x 3 þ C 7 þ
D
n
11 Ae 33
D
n
33 ε 33 Ic 33
e 33 c 44 À e 15 c 33
c 44
C 5
a 2
,
ð4:40Þ
96
4 Bending of Beams