φ
p
,33 À k
2
φ
p
¼ À
q
ε 33
N
þ
D x 3
ð Þ À N
À
A x 3
ð Þ
Â
Ã
:
ð3:19Þ
Once φ is known, p and n can be obtained from Eq. (3.14). For the mechanical
displacement, from Eqs. (3.8) and (3.6) 2 , we have, after the integration with respect
to x 3 once,
u 3 ¼
1
c 33
C 1 x 3 À e 33 φ
ð
ÞþC 4 ,
ð3:20Þ
where C 4 is an integration constant. The polarization and the effective polarization
charge density can be calculated from the following well-known expressions:
P 3 ¼ D 3 À ε 0 E 3 ,
ρ
P
¼ ÀP k,k ¼ ÀP 3,3 :
ð3:21Þ
In the studies of other systems of mobile charges, e.g., plasmas and electrolytes,
Eq. (3.5) 2 is called the Poisson’s equation, Eq. (3.12) the Boltzmann distribution,
Eq. (3.13) the Poisson-Boltzmann equation, Eq. (3.16) the Debye-Hückel equation,
and λ D the Debye-Hückel length.
3.2 Linear Doping
Consider the simple case of linear doping described by
N
À
A x 3
ð Þ ¼ b 1 x 3 þ c 1 ,
N
þ
D x 3
ð Þ ¼ b 2 x 3 þ c 2 :
ð3:22Þ
Hence
N
þ
D x 3
ð Þ À N
À
A x 3
ð Þ ¼ bx 3 þ c,
b ¼ b 2 À b 1 , c ¼ c 2 À c 1 :
ð3:23Þ
In this case a particular solution of Eq. (3.19) is simply
φ
p
¼
q
ε 33 k
2
bx 3 þ c
ð
Þ:
ð3:24Þ
Therefore, the direct contributions of linear doping as a particular solution to the
electric potential and hence p as well as n according to Eq. (3.14) are also linear. The
corresponding drift and diffusion currents are constants. Since the electric potential
in Eq. (3.14) has to be small, the above observations are true only when |x 3 | is small.
3.2 Linear Doping
35