the p 0 and n 0 in Eq. (1.8). Equation (3.12) implies that pn ¼ p
0 n
0 which is a constant.
Within the macroscopic theory we are using, p
0 and n
0 are integration constants and
can be determined from, e.g., boundary conditions. Therefore, in general, p
0 and n
0
depend on the electromechanical loads and structural parameters in addition to
material properties. The substitution of Eqs. (3.9) and (3.12) into Eq. (3.5) 2 gives
the following equation governing the electric potential:
Àε 33 φ ,33
¼ q p
0 exp À
q
k B T
φ
À n
0 exp
q
k B T
φ
þ N
þ
D x 3
ð Þ À N
À
A x 3
ð Þ
!
:
ð3:13Þ
For small φ, we make the following approximation in Eq. (3.12):
p ffi p
0 1 À
q
k B T
φ
,
n ffi n
0 1 þ
q
k B T
φ
,
ð3:14Þ
which can describe small carrier concentration perturbations and low electric potential. Substituting Eq. (3.14) into Eq. (3.13), we obtain a linear equation for the
potential:
φ ,33 ¼ À
q
ε 33
p
0
À n
0
À p
0
þ n
0
À
Á q
k B T
φ þ N
þ
D x 3
ð Þ À N
À
A x 3
ð Þ
!
,
ð3:15Þ
which can be further written as
φ ,33 À k
2
φ ¼ À
q
ε 33
p
0
À n
0
þ N
þ
D x 3
ð Þ À N
À
A x 3
ð Þ
Â
Ã
,
ð3:16Þ
where
k
2
¼
q
ε 33
p
0
þ n
0
À
Á q
k B T
¼
1
λ
2
D
,
λ
2
D ¼
ε 33 k B T
p 0 þ n 0
ð
Þ q 2 :
ð3:17Þ
The general solution to Eq. (3.16) can be written as
φ ¼ C 2 cosh kx 3 þ C 3 sinh kx 3 þ
q
k
2
ε 33
p
0
À n
0
À
Á þ φ
p x 3
ð Þ,
ð3:18Þ
where C 2 and C 3 are integration constants. φ
p is a particular solution of the following
nonhomogeneous equation:
34
3 Extension of Rods