Δp ¼ p À p 0 , Δn ¼ n À n 0 ,
p 0 ¼ N
À
A , n 0 ¼ N
þ
D :
ð3:67Þ
We consider the case of uniform doping. p 0 and n 0 are constants. The linearized
constitutive relations are from Eqs. (3.2) and (1.9):
T 3 ¼ c 33 S 3 À e 33 E 3 ,
D 3 ¼ e 33 S 3 þ ε 33 E 3 ,
ð3:68Þ
J
p
3 ¼ qp 0 μ
p
33 E 3 À qD
p
33
∂ Δp
ð Þ
∂x 3
,
J
n
3 ¼ qn 0 μ
n
33 E 3 þ qD
n
33
∂ Δn
ð Þ
∂x 3
,
ð3:69Þ
where, from Eq. (3.6),
S 3 ¼ u 3,3 , E 3 ¼ Àφ 3 :
ð3:70Þ
Consider an electrically isolated rod. There are no concentrated free charges at its
ends, and there are no currents flowing in or out of the rod at its ends. In this case the
boundary conditions are
T 3 ÆL
ð Þ ¼ T, D 3 ÆL
ð Þ ¼ 0,
ð3:71Þ
J
p
3 ÆL
ð Þ ¼ 0, J
n
3 ÆL
ð Þ ¼ 0:
ð3:72Þ
The rod is assumed to be electrically neutral at the reference state. Therefore Δn and
Δp must satisfy the following charge neutrality conditions:
Z L
ÀL
Δndx ¼ 0,
Z L
ÀL
Δpdx ¼ 0:
ð3:73Þ
Equation (3.73) is equivalent to one condition because of the charge equation in
Eq. (3.66) and the charge boundary conditions in Eq. (3.71). To determine the
mechanical displacement and the electric potential uniquely, we set
x3
x2
x1
c
2L
T=f/A
T=f/A
Fig. 3.7 A piezoelectric
semiconductor rod of
crystals of class (6mm)
46
3 Extension of Rods