where C 5 through C 7 are undetermined constants. We have five remaining boundary
conditions in Eq. (6.90). To determine C 1 through C 7 , we need two more conditions
as follows. Δp and Δn must satisfy the following global charge neutrality conditions:
Z L
0
Δpdx 3 ¼ 0,
Z L
0
Δndx 3 ¼ 0:
ð6:100Þ
Only one of Eq. (6.100) is independent. The other is implied by integrating
Eq. (6.87) between 0 and L and using b
D 0
ð Þ ¼ b
D L
ð Þ ¼ 0, which gives
Z L
0
q Δp À Δn
ð
Þ dx 3 ¼ 0:
ð6:101Þ
To determine the electric potential uniquely, we set
φ 0
ð Þ ¼ 0:
ð6:102Þ
Equations (6.102) and one of Eq. (6.100) are the two additional conditions needed.
As an example, we consider PZTs and BaTiO 3 for the piezoelectric dielectric
material and silicon for the nonpiezoelectric semiconductor. For geometric parameters we consider a beam with L ¼ 600 nm, h ¼ 10 nm, c ¼ 15 nm, and b ¼ 50 nm.
The reference carrier concentrations are p 0 ¼ n 0 ¼ 10
23 /m
3 . The end force
F ¼ 20 Â 10
À12 N ¼ 20 pN. Some of these parameters are varied separately in
the figures below.
Figure 6.10 shows the main effect of interest, i.e., mechanically induced redistribution of mobile charges in the bending of a composite beam of piezoelectric
dielectrics and nonpiezoelectric semiconductors. Thus the composite beam effectively behaves like a homogeneous piezoelectric semiconductor beam. Different
from the static bending of a ZnO beam in Sect. 4.3 where the mobile charges are
driven to the top and bottom surfaces of the beam, in the composite beam here the
mobile charges are driven to the two ends of the beam. This is because of the specific
orientations of the materials involved. The fields in the figure are linear in F. They
become stronger or larger when F increases.
The change of number of carriers in the differential element in Fig. 6.9, i.e., Δp
(2cbdz) where dz ¼ dx 3 , may be used as a somewhat crude measure of the strength of
the coupling between F and Δp. For a differential element of the beam near its right
end, with dz ¼ 10 nm, we plot Δp(2cbdz) in Fig. 6.11 versus the thickness ratio h/c
of the piezoelectric and semiconductor layers when h + c is fixed. The curves for
different material combinations all have a maximum roughly near h ¼ c, which is
reasonable.
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6 Composite Structures
conditions in Eq. (6.90). To determine C 1 through C 7 , we need two more conditions
as follows. Δp and Δn must satisfy the following global charge neutrality conditions:
Z L
0
Δpdx 3 ¼ 0,
Z L
0
Δndx 3 ¼ 0:
ð6:100Þ
Only one of Eq. (6.100) is independent. The other is implied by integrating
Eq. (6.87) between 0 and L and using b
D 0
ð Þ ¼ b
D L
ð Þ ¼ 0, which gives
Z L
0
q Δp À Δn
ð
Þ dx 3 ¼ 0:
ð6:101Þ
To determine the electric potential uniquely, we set
φ 0
ð Þ ¼ 0:
ð6:102Þ
Equations (6.102) and one of Eq. (6.100) are the two additional conditions needed.
As an example, we consider PZTs and BaTiO 3 for the piezoelectric dielectric
material and silicon for the nonpiezoelectric semiconductor. For geometric parameters we consider a beam with L ¼ 600 nm, h ¼ 10 nm, c ¼ 15 nm, and b ¼ 50 nm.
The reference carrier concentrations are p 0 ¼ n 0 ¼ 10
23 /m
3 . The end force
F ¼ 20 Â 10
À12 N ¼ 20 pN. Some of these parameters are varied separately in
the figures below.
Figure 6.10 shows the main effect of interest, i.e., mechanically induced redistribution of mobile charges in the bending of a composite beam of piezoelectric
dielectrics and nonpiezoelectric semiconductors. Thus the composite beam effectively behaves like a homogeneous piezoelectric semiconductor beam. Different
from the static bending of a ZnO beam in Sect. 4.3 where the mobile charges are
driven to the top and bottom surfaces of the beam, in the composite beam here the
mobile charges are driven to the two ends of the beam. This is because of the specific
orientations of the materials involved. The fields in the figure are linear in F. They
become stronger or larger when F increases.
The change of number of carriers in the differential element in Fig. 6.9, i.e., Δp
(2cbdz) where dz ¼ dx 3 , may be used as a somewhat crude measure of the strength of
the coupling between F and Δp. For a differential element of the beam near its right
end, with dz ¼ 10 nm, we plot Δp(2cbdz) in Fig. 6.11 versus the thickness ratio h/c
of the piezoelectric and semiconductor layers when h + c is fixed. The curves for
different material combinations all have a maximum roughly near h ¼ c, which is
reasonable.
160
6 Composite Structures