semiconductor with p ffi 0. As an approximation, we neglect the charge on the righthand side of Eq. (3.131) 2 . This decouples the problem into two one-way coupled
subproblems. One consists of the equations of piezoelectricity for the mechanical
displacement u 3 and the electric potential φ. The other is the continuity equation for
the electron concentration perturbation Δn. The two subproblems are treated one at a
time below. The piezoelectricity subproblem is governed by
∂T 3
∂x 3
¼ ρ€ u 3 ,
∂D 3
∂x 3
¼ 0:
ð3:150Þ
Using Eqs. (3.133) and (3.135), we can write Eq. (3.150) as
c 33 u 3,33 þ e 33 φ ,33 ¼ ρ€ u 3 ,
e 33 u 3,33 À ε 33 φ ,33 ¼ 0:
ð3:151Þ
The boundary conditions are
u 3 ¼ 0, D 3 ¼ 0, x 3 ¼ 0,
T 3 ¼ f , D 3 ¼ 0, x 3 ¼ L:
ð3:152Þ
The initial conditions are
u 3 ¼ 0, _
u 3 ¼ 0, φ ¼ 0, t ¼ 0:
ð3:153Þ
From Eq. (3.151) 2 and the boundary conditions on the electric displacement in
Eq. (3.152), we have
D 3 ¼ e 33 u 3,3 À ε 33 φ ,3 0:
ð3:154Þ
Then
T 3 ¼ c 33 u 3,3 þ e 33 φ ,3 ¼ c 33 u 3,3 þ e 33
e 33
ε 33
u 3,3 ¼ b c 33 u 3,3 ,
ð3:155Þ
where
b c 33 ¼ c 33 þ
e
2
33
ε 33
ð3:156Þ
x3
x1
c
L
F
Fig. 3.21 A piezoelectric
semiconductor rod under the
sudden application of an end
force
68
3 Extension of Rods