where ∇ is the two-dimensional gradient operator. The relevant components of T ij ,
D i , J
p
i , and J
n
i are
T 5
T 4
& '
¼
T 31
T 32
&
'
¼ c∇u þ e∇φ,
ð1:29Þ
D 1
D 2
& '
¼ e∇u À ε∇φ,
ð1:30Þ
J
p
1
J
p
2
( )
¼ Àqp 0 μ
p
∇φ À qD
p
∇ Δp
ð Þ,
J
n
1
J
n
2
( )
¼ Àqn 0 μ
n
∇φ þ qD
n
∇ Δn
ð Þ,
ð1:31Þ
where we have denoted c ¼ c 44 , e ¼ e 15 , ε ¼ ε 11 , μ
p
¼ μ
p
11 , μ
n
¼ μ
n
11 , D
p
¼ D
p
11 , and
D
n
¼ D
n
11 . 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∇
2 u þ e∇
2 φ ¼ ρ€ u,
e∇
2 u À ε∇
2 φ ¼ q Δp À Δn
ð
Þ ,
ð1:32Þ
∂
∂t
Δp
ð Þ ¼ p 0 μ
p
∇
2 φ þ D
p
∇
2 Δp
ð Þ,
∂
∂t
Δn
ð Þ ¼ Àn 0 μ
n
∇
2 φ þ D
n
∇
2 Δn
ð Þ,
ð1:33Þ
where ∇
2 ¼ ∂
2 =∂x
2
1 þ ∂
2 =∂x
2
2 is the two-dimensional Laplacian.
For static problems, Eqs. (1.32) and (1.33) reduce to
c∇
2 u þ e∇
2 φ ¼ 0,
e∇
2 u À ε∇
2 φ ¼ q Δp À Δn
ð
Þ ,
ð1:34Þ
0 ¼ p 0 μ
p
∇
2 φ þ D
p
∇
2 Δp
ð Þ,
0 ¼ Àn 0 μ
n
∇
2 φ þ D
n
∇
2 Δn
ð Þ:
ð1:35Þ
Equations (1.34) and (1.35) can be decoupled as follows. From Eq. (1.34) 1 ,
∇
2 u ¼ À
e
c
∇
2 φ:
ð1:36Þ
Substituting Eq. (1.36) into Eq. (1.34) 2 , we obtain
8
1 Macroscopic Theory