J
p
i ¼ qp 0 μ
p
ij E j À qD
p
ij P j ,
J
n
i ¼ qn 0 μ
n
ij E j þ qD
n
ij N j ,
ð5:4Þ
S ij ¼ u i,j þ u j,i
À
Á
=2,
E i ¼ Àφ ,i ,
ð5:5Þ
where we have denoted the carrier concentration perturbation gradients by
P j ¼ Δp
ð Þ ,j , N j ¼ Δn
ð Þ ,j :
ð5:6Þ
From Eqs. (1.8) and (1.7), we have
p 0 ¼ N
À
A , n 0 ¼ N
þ
D ,
ð5:7Þ
which are constants for uniform doping, and
p ¼ p 0 þ Δp, n ¼ n 0 þ Δn:
ð5:8Þ
5.2 Hierarchy of Two-Dimensional Equations
Consider the plate of uniform thickness 2h shown in Fig. 5.1. The x 1 and x 2 axes are
in the plate middle plane where x 3 ¼ 0. The metal electrodes are assumed to be very
thin with negligible mechanical effects.
We expand the mechanical displacement vector, the electric potential, and the
carrier concentration perturbations into power series in x 3 :
x3
x1
x1
x2
Electrodes
2h
n
s
Fig. 5.1 Plan view and
cross section of a thin
piezoelectric semiconductor
plate
114
5 Extension and Bending of Plates