The one-dimensional governing equations are
T 33,3 ¼ ρ€ u 3 ,
D 3,3 ¼ q Δp À Δn
ð
Þ ,
J
p
3,3 ¼ Àq _
p,
J
n
3,3 ¼ q _
n,
ð3:191Þ
T 33,3 ¼ c 33 S 33 À e 33 E 3 ,
D 3 ¼ e 33 S 33 þ ε 33 E 3 ,
ð3:192Þ
J
p
3 ¼ qpμ
p
33 E 3 À qD
p
3 p ,3
ffi qp 0 μ
p
33 E 3 À qD
p
3 Δp
ð Þ ,3 ,
J
n
3 ¼ qpμ
n
33 E 3 þ qD
n
3 n ,3
ffi qn 0 μ
n
33 E 3 þ qD
n
3 Δn
ð Þ ,3 ,
ð3:193Þ
S 33 ¼ u 3,3 , E 3 ¼ Àφ ,3 :
ð3:194Þ
The substituon of Eqs. (3.192), (3.193) and (3.194) into Eq. (3.191) yields four
second-order ordinary differential equations for u 3 , φ, Δp, and Δn. The equations
may be linear or nonlinear depending on whether the linearized or nonlinear version
of Eq. (3.193) is used. The axial electric polarization P 3 and the effective polarization charge density ρ
P can be calculated from the following expressions:
P 3 ¼ D 3 À ε 0 E 3 ,
ρ
P
¼ ÀP 3,3 :
ð3:195Þ
We follow the simplified notation in Eq. (3.80). The boundary and continuity
conditions are
T ÀL
ð Þ ¼ F, D ÀL
ð Þ ¼ 0,
J
p
ÀL
ð Þ ¼ 0, J
n
ÀL
ð Þ ¼ 0,
ð3:196Þ
u 0
À
ð Þ ¼ u 0
þ
ð Þ, T 0
À
ð Þ ¼ T 0
þ
ð Þ,
φ 0
À
ð Þ ¼ φ 0
þ
ð Þ, D 0
À
ð Þ ¼ D 0
þ
ð Þ,
ð3:197Þ
p 0
À
ð Þ ¼ p 0
þ
ð Þ, J
p 0
À
ð Þ ¼ J
p 0
þ
ð Þ,
n 0
À
ð Þ ¼ n 0
þ
ð Þ, J
n 0
À
ð Þ ¼ J
n 0
þ
ð Þ,
ð3:198Þ
T L
ð Þ ¼ F, D L
ð Þ ¼ 0,
J
p L
ð Þ ¼ 0, J
n L
ð Þ ¼ 0:
ð3:199Þ
Δp and Δn satisfy the following global charge neutrality conditions:
3.14 Extension of a PN Junction
77