q ¼ 1.6 Â 10
À19 coul is the elementary charge. p and n are the concentrations of
holes and electrons. N
À
A and N
þ
D are the concentrations of ionized accepters and
donors from doping. J
p
i and J
n
i are the hole and electron current densities. In the
following, we will neglect those terms in Eq. (1.1) 3,4 associated with thermal
recombination and generation as well as other processes of electric, magnetic,
mechanical, and thermal origins. Constitutive relations accompanying Eq. (1.1)
describing material behaviors can be written in the following form:
T ij ¼ c
E
ijkl S kl À e kij E k ,
D i ¼ e ikl S kl þ ε
S
ik E k ,
J
p
i ¼ qpμ
p
ij E j À qD
p
ij p ,j
¼ qpμ
p
ij E j À qpD
p
ij ln p
ð
Þ ,j ,
J
n
i ¼ qnμ
n
ij E j þ qD
n
ij n ,j
¼ qnμ
n
ij E j þ qnD
n
ij ln n
ð
Þ ,j ,
ð1:2Þ
where S is the strain tensor, E the electric field vector, c
E
ijkl the elastic stiffness, e ijk the
piezoelectric constants, and ε
S
ij the dielectric constants. μ
p
ij and μ
n
ij are the carrier
mobility. D
p
ij and D
n
ij are the carrier diffusion constants. qpμ
p
ij E j and qnμ
n
ij E j are the
drift currents. ÀqD
p
ij p ,j and qD
n
ij n ,j are the diffusion currents. The piezoelectric
constitutive relations in Eq. (1.2) can be written in other forms in terms of other
material constants [1–4]. For example,
S ij ¼ s
E
ijkl T kl þ d kij E k ,
D i ¼ d ikl T kl þ ε
T
ik E k ,
ð1:3Þ
where s
E
ijkl are the elastic compliance, d ijk the piezoelectric constants, and ε
T
ij the
dielectric constants. The mobility and diffusion constants, e.g., μ
p
33 , μ
n
33 , D
p
33 , and D
n
33 ,
satisfy the Einstein relation:
μ
p
33
D
p
33
¼
μ
n
33
D
n
33
¼
q
k B T
,
ð1:4Þ
where k B is the Boltzmann constant and T the absolute temperature. The strain S and
the electric field E are related to the mechanical displacement u and the electric
potential φ through
S ij ¼ u i,j þ u j,i
À
Á
=2,
E i ¼ Àφ ,i :
ð1:5Þ
2
1 Macroscopic Theory
À19 coul is the elementary charge. p and n are the concentrations of
holes and electrons. N
À
A and N
þ
D are the concentrations of ionized accepters and
donors from doping. J
p
i and J
n
i are the hole and electron current densities. In the
following, we will neglect those terms in Eq. (1.1) 3,4 associated with thermal
recombination and generation as well as other processes of electric, magnetic,
mechanical, and thermal origins. Constitutive relations accompanying Eq. (1.1)
describing material behaviors can be written in the following form:
T ij ¼ c
E
ijkl S kl À e kij E k ,
D i ¼ e ikl S kl þ ε
S
ik E k ,
J
p
i ¼ qpμ
p
ij E j À qD
p
ij p ,j
¼ qpμ
p
ij E j À qpD
p
ij ln p
ð
Þ ,j ,
J
n
i ¼ qnμ
n
ij E j þ qD
n
ij n ,j
¼ qnμ
n
ij E j þ qnD
n
ij ln n
ð
Þ ,j ,
ð1:2Þ
where S is the strain tensor, E the electric field vector, c
E
ijkl the elastic stiffness, e ijk the
piezoelectric constants, and ε
S
ij the dielectric constants. μ
p
ij and μ
n
ij are the carrier
mobility. D
p
ij and D
n
ij are the carrier diffusion constants. qpμ
p
ij E j and qnμ
n
ij E j are the
drift currents. ÀqD
p
ij p ,j and qD
n
ij n ,j are the diffusion currents. The piezoelectric
constitutive relations in Eq. (1.2) can be written in other forms in terms of other
material constants [1–4]. For example,
S ij ¼ s
E
ijkl T kl þ d kij E k ,
D i ¼ d ikl T kl þ ε
T
ik E k ,
ð1:3Þ
where s
E
ijkl are the elastic compliance, d ijk the piezoelectric constants, and ε
T
ij the
dielectric constants. The mobility and diffusion constants, e.g., μ
p
33 , μ
n
33 , D
p
33 , and D
n
33 ,
satisfy the Einstein relation:
μ
p
33
D
p
33
¼
μ
n
33
D
n
33
¼
q
k B T
,
ð1:4Þ
where k B is the Boltzmann constant and T the absolute temperature. The strain S and
the electric field E are related to the mechanical displacement u and the electric
potential φ through
S ij ¼ u i,j þ u j,i
À
Á
=2,
E i ¼ Àφ ,i :
ð1:5Þ
2
1 Macroscopic Theory