to increase the buckling load. The effect of semiconduction represented by the n 0
term in the denominator of Eq. (4.91) tends to reduce the piezoelectric stiffening
because of the screening effect of the electrons, and hence lowers the buckling load.
When n 0 is set to zero in Eq. (4.91), we have following critical buckling load P
p
cr of
the corresponding piezoelectric dielectric beam:
P
p
cr
P
e
cr
¼ 1 þ
k
2
33
1 þ
ε 11 A
ε 33 I
L
mπ
À Á 2 :
ð4:93Þ
As a numerical example, consider a beam of radius a ¼ 25 nm, length
L ¼ 600 nm, and initial carrier concentration n 0 ¼ 10
23 /m
3 . We are mainly interested
in the carrier concentration perturbation Δn, which is normalized according to
x 3 ¼ x 3 =L, x 2 ¼ x 2 =a, Δn ¼ x 2 cos mπx 3 ,
ð4:94Þ
and is shown in Fig. 4.12 for m ¼ 1, 2, and 3. It can be seen that the electrons move
toward the top and bottom (the lateral surface) of the beam during buckling and that
Fig. 4.12 Buckling modes of normalized Δn. (a) m ¼ 1. (b) m ¼ 2. (c) m ¼ 3
110
4 Bending of Beams