Àε∇
2 φ ¼ q Δp À Δn
ð
Þ ,
ð1:37Þ
where
ε ¼ ε þ
e
2
c
:
ð1:38Þ
We rewrite Eq. (1.35) as
∇
2 Δp
ð Þ ¼ À
p 0 μ
p
D
p ∇
2 φ,
∇
2 Δn
ð Þ ¼
n 0 μ
n
D
n ∇
2 φ:
ð1:39Þ
Subtracting the two equations in Eq. (1.39) from each other, we have
∇
2 φ ¼ À
q
εk
2
∇
2 Δp À Δn
ð
Þ ,
ð1:40Þ
where
k
2
¼
p 0 μ
p
D
p þ
n 0 μ
n
D
n
q
ε
:
ð1:41Þ
The substitution of Eq. (1.40) into Eq. (1.37) yields a single equation for Δp À Δn:
∇
2 Δp À Δn
ð
Þ¼k
2
Δp À Δn
ð
Þ :
ð1:42Þ
Once Δp À Δn is obtained from Eq. (1.42), φ, u, Δp, and Δn can be determined from
Eqs. (1.40), (1.36), and (1.39).
1.5 Thermal Couplings
Treatments of thermal effects in nonpiezoelectric semiconductors can be found in
[9]. For a thermopiezoelectric semiconductor, the three-dimensional phenomenological theory couples the linear theory of thermopiezoelectricity [4, 10, 11] and the
macroscopic theory of semiconductors. It consists of the following equation of
1.5 Thermal Couplings
9