similar. (a) shows the electron concentration perturbation at an early instant. Then,
approximately at the largest deflection in (d), the electrons form a distribution that
does not change much along the beam except near the right end. Δn is linear in x 2 ,
with more electrons at the top surface and less at the bottom.
4.6 Buckling
Consider a ZnO beam with a circular cross section as shown in Fig. 4.11. It is under a
compressive axial force P. We study the buckling of the beam in the y-z plane
[5]. Initially P is applied while the two ends are electrically shorted. P produces a
compressive axial force only without an axial electric field and carrier concentration
perturbation. Then the compressed beam is disturbed for its buckling when the two
ends are electrically open.
We consider an n-type piezoelectric semiconductor beam in static bending
without shear deformation. From the relevant equations in Sect. 4.2, we have
Q ,3 À Pv ,33 ¼ 0,
D
1
ð Þ
3,3 À D
0
ð Þ
2 ¼ qI Àn
1
ð Þ
,
J
n 1
ð Þ
3,3 À J
n 0
ð Þ
2
¼ 0,
ð4:81Þ
M ¼ Àc 33 Iv ,33 þ e 33 Iϕ
1
ð Þ
,3 ,
ð4:82Þ
Q ¼ M ,3 ¼ Àc 33 Iv ,333 þ e 33 Iϕ
1
ð Þ
,33 ,
ð4:83Þ
D
0
ð Þ
2 ¼ Àε 11 Aϕ
1
ð Þ ,
D
1
ð Þ
3 ¼ Àe 33 Iv ,33 À ε 33 Iϕ
1
ð Þ
,3 ,
ð4:84Þ
J
n 0
ð Þ
2
¼ Àqn 0 μ
n
11 Aϕ
1
ð Þ
þ qD
n
11 An
1
ð Þ ,
J
n 1
ð Þ
3
¼ Àqn 0 μ
n
33 Iϕ
1
ð Þ
,3 þ qD
n
33 In
1
ð Þ
,3 ,
ð4:85Þ
where, in Eq. (4.81) 1 , we have added a term ÀPv ,33 [6] which describes the
contribution of the axial force to bending and is responsible for buckling. The
substitution of Eqs. (4.83), (4.84), and (4.85) into Eq. (4.81) yields the following
three equations for v, ϕ
(1) , and n
(1) :
x1, x
x2, y
L
a
c
x3, z
P
P
Fig. 4.11 A ZnO beam in
compression
108
4 Bending of Beams