J
p
3 ¼ qp 0 μ
p
33 E 3 À qD
p
33 Δp
ð Þ ,3
¼ Àqp 0 μ
p
33 φ ,3 À qD
p
33 Δp
ð Þ ,3 ,
J
n
3 ¼ qn 0 μ
n
33 E 3 þ qD
n
33 Δn
ð Þ ,3
¼ Àqn 0 μ
n
33 φ ,3 þ qD
n
33 Δn
ð Þ ,3 :
ð1:16Þ
In Eq. (1.16), the linearized constitutive relations in Eq. (1.9) are used and uniform
doping with constant p 0 and n 0 is assumed. With successive substitutions, the
relevant ones of Eq. (1.1) take the following form:
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 q Δp À Δn
ð
Þ ,
ð1:17Þ
∂
∂t
Δp
ð Þ ¼ p 0 μ
p
33 φ ,33 þ D
p
33 Δp
ð Þ ,33 ,
∂
∂t
Δn
ð Þ ¼ Àn 0 μ
n
33 φ ,33 þ D
n
33 Δn
ð Þ ,33 :
ð1:18Þ
For static problems, Eqs. (1.17) and (1.18) reduce to
c 33 u 3,33 þ e 33 φ ,33 ¼ 0,
e 33 u 3,33 À ε 33 φ ,33 ¼ q Δp À Δn
ð
Þ ,
ð1:19Þ
0 ¼ p 0 μ
p
33 φ ,33 þ D
p
33 Δp
ð Þ ,33 ,
0 ¼ Àn 0 μ
n
33 φ ,33 þ D
n
33 Δn
ð Þ ,33 :
ð1:20Þ
Equation (1.19) can be written as
u 3,33 ¼ À
e 33
c 33
φ ,33 ,
Àε 33 φ ,33 ¼ q Δp À Δn
ð
Þ ,
ð1:21Þ
where
ε 33 ¼ ε 33 þ
e
2
33
c 33
:
ð1:22Þ
ε 33 is a piezoelectrically modified dielectric constant. From Eq. (1.20), we can write
Δp
ð Þ ,33 ¼ À
p 0 μ
p
33
D
p
33
φ ,33 ,
Δn
ð Þ ,33 ¼ À
n 0 μ
n
33
D
n
33
φ ,33 :
ð1:23Þ
6
1 Macroscopic Theory