results. Since the fields are large and may be inaccurate near the left end at x 3 ¼ 0,
only the results for x 3 > 0.5 Â 10
-7 m are shown. It is seen that at the first resonance
the carrier concentration is mainly near the top and bottom surfaces and is essentially
uniform along the beam except near the right end. At the second and the third
resonances, the fields have nodal points (zeros) along x 3 and change their signs
across the nodal points. The behavior of the electric potential is similar [3].
4.5 Transient Bending
Consider an n-type ZnO beam with a circular cross section as shown in Fig. 4.8. The
left end is fixed. The right end is under the action of a transverse shear force f y
suddenly applied when the beam is at rest at t ¼ 0 [4].
Consider bending without shear deformation. The relevant equations from Sect.
4.2 are
u 2 x, t
ð Þ ffi v x 3 , t
ð
Þ, u 3 x, t
ð Þ ffi Àx 2 v ,3 x 3 , t
ð
Þ,
φ x, t
ð Þ ffi x 2 ϕ
1
ð Þ x 3 , t
ð
Þ, Δn x, t
ð Þ ffi x 2 n
1
ð Þ x 3 , t
ð
Þ,
ð4:56Þ
Fig. 4.7 Imaginary parts of the carrier concentration distribution perturbation Nx 2 at resonances.
(a) ω ¼ ω (1) . (b) ω ¼ ω (2) . (c) ω ¼ ω (3)
102
4 Bending of Beams