ε ij
 à ¼
ε 11 0
0
0 ε 11 0
0
0 ε 33
0
B
@
1
C
A,
ð1:11Þ
where c 66 ¼(c 11 –c 12 )/2. μ ij and d ij have the same structure as ε ij . The matrices in
Eq. (1.11) are the same as those of transversely isotropic piezoelectric materials such
as polarized ceramics when the poling axis is along x 3 . Equation (1.11) yields the
following piezoelectric constitutive relations:
T 11 ¼ T 1 ¼ c 11 S 1 þ c 12 S 2 þ c 13 S 3 À e 31 E 3 ,
T 22 ¼ T 2 ¼ c 12 S 1 þ c 11 S 2 þ c 13 S 3 À e 31 E 3 ,
T 33 ¼ T 3 ¼ c 13 S 1 þ c 13 S 2 þ c 33 S 3 À e 33 E 3 ,
T 23 ¼ T 4 ¼ c 44 S 4 À e 15 E 2 ,
T 31 ¼ T 5 ¼ c 44 S 5 À e 15 E 1 ,
T 12 ¼ T 6 ¼ c 66 S 6 ,
D 1 ¼ e 15 S 5 þ ε 11 E 1 ,
D 2 ¼ e 15 S 4 þ ε 11 E 2 ,
D 3 ¼ e 31 S 1 þ S 2
ð
Þþe 33 S 3 þ ε 33 E 3 :
ð1:12Þ
To see the basic acoustoelectric coupling in ZnO, consider one-dimensional
longitudinal motions along the c-axis satisfying u 1 ¼ u 2 ¼ 0, ∂/∂x 1 ¼ 0, and ∂/
∂x 2 ¼ 0. In this case, the nontrivial fields are
u 3 ¼ u x 3 , t
ð
Þ, φ ¼ φ x 3 , t
ð
Þ,
Δp ¼ Δp x 3 , t
ð
Þ, Δn ¼ Δn x 3 , t
ð
Þ:
ð1:13Þ
From Eq. (1.5), the relevant strain and electric field components are
S 33 ¼ u 3,3 ,
E 3 ¼ Àφ ,3 :
ð1:14Þ
The relevant components of T ij , D i , J
p
i , and J
n
i are
T 33 ¼ c 33 S 3 À e 33 E 3 ¼ c 33 u 3,3 þ e 33 φ ,3 ,
D 3 ¼ e 33 S 3 þ ε 33 E 3 ¼ e 33 u 3,3 À ε 33 φ ,3 ,
ð1:15Þ
1.3 One-Dimensional Problems of Crystals of Class (6mm)
5