S 3 ¼ w ,3 À x 2 v ,33 :
ð7:108Þ
Similarly, the electric potential and electric field over a cross section are approximated by
φ ffi φ x 3 , t
ð
Þ, E 3 ¼ Àφ ,3 :
ð7:109Þ
For the carrier concentration perturbations in the semiconductor layer,
Δp ffi Δp x 3 , t
ð
Þ, Δn ffi Δn x 3 , t
ð
Þ:
ð7:110Þ
We follow the simplified notations for the relevant axial fields and material parameters in Eqs. (7.77) and (7.78). For the piezoelectric layer, the one-dimensional
constitutive relations and effective material constants in Eqs. (7.80) and (7.81) are
still applicable. The constitutive relations take the following form with the use of
Eqs. (7.108) and (7.109):
T ¼ c
1
ð Þ S À e
1
ð Þ E À λ
1
ð Þ
θ
¼ c
1
ð Þ w ,3 À x 2 v ,33
ð
Þþe
1
ð Þ
φ ,3 À λ
1
ð Þ
θ,
D ¼ e
1
ð Þ S þ ε
1
ð Þ E þ p
1
ð Þ
θ
¼ e
1
ð Þ w ,3 À x 2 v ,33
ð
ÞÀε
1
ð Þ
φ ,3 þ p
1
ð Þ
θ:
ð7:111Þ
Similarly, for the semiconductor layer, Eqs. (7.83) and (84) are still applicable. The
one-dimensional constitutive relations take the following form with the use of
Eqs. (7.108) and (109):
(2) Si
(1) PZT
x3
L
P
x2
x1
x2
b
h2
h1
Fig. 7.18 Side view and
cross section of a composite
beam of a piezoelectric
dielectric and a
nonpiezoelectric
semiconductor
204
7 Thermal Effects