which leads to the following expression for ξ 2 :
ξ
2
2 ¼ ξ
2
1 À
ρω
2
c 44
¼ ξ
2
1 1 À
v
2
v 2
T
> 0,
ð6:13Þ
where
v
2
¼
ω
2
ξ
2
1
, v
2
T ¼
c 44
ρ
:
ð6:14Þ
The following fields are needed in the boundary and continuity conditions:
φ ¼ B exp Àξ 1 x 2 À h
ð
Þ
½
þ
e 15
ε 11
A exp Àξ 2 x 2 À h
ð
Þ
½
&
'
 exp i ξ 1 x 1 À ωt
ð
Þ
½
,
T 23 ¼ À Ac 44 ξ 2 exp Àξ 2 x 2 À h
ð
Þ
½
þe 15 Bξ 1 exp Àξ 1 x 2 À h
ð
Þ
½
f
g
 exp i ξ 1 x 1 À ωt
ð
Þ
½
,
D 2 ¼ ε 11 Bξ 1 exp Àξ 1 x 2 À h
ð
Þ
½
exp i ξ 1 x 1 À ωt
ð
Þ
½
:
ð6:15Þ
Electric fields can also exist in the free space of x 2 < Àh, which are governed by
∇
2 φ ¼ 0, x 2 < Àh,
φ ! 0, x 2 ! À1:
ð6:16Þ
A surface wave solution to Eq. (6.16) is
φ ¼ C exp ξ 1 x 2 þ h
ð
Þ
½
exp i ξ 1 x 1 À ω t
ð
Þ
½
,
ð6:17Þ
where C is an undetermined constant. From Eq. (6.17), in the free space, we have
D 2 ¼ Àε 0 ξ 1 C exp ξ 1 x 2 þ h
ð
Þ
½
exp i ξ 1 x 1 À ω t
ð
Þ
½
:
ð6:18Þ
For the semiconductor plate, since the normal of the plate is along x 2 , the plate
equations in Chap. 5 cannot be used directly. Fortunately we only need the zeroorder equations of the plate which will be quickly derived in the following in the
coordinate system in Fig. 6.1. The zero-order equations govern extensional motions
described by u 1 (x 1 , x 3 , t) and u 3 (x 1 , x 3 , t). The dominating stress components are T 11 ,
T 33 , and T 13 . Therefore we make the following stress relaxation for thin plates:
T 2j ¼ 0, j ¼ 1, 2, 3:
ð6:19Þ
6.1 Surface Waves
143
ξ
2
2 ¼ ξ
2
1 À
ρω
2
c 44
¼ ξ
2
1 1 À
v
2
v 2
T
> 0,
ð6:13Þ
where
v
2
¼
ω
2
ξ
2
1
, v
2
T ¼
c 44
ρ
:
ð6:14Þ
The following fields are needed in the boundary and continuity conditions:
φ ¼ B exp Àξ 1 x 2 À h
ð
Þ
½
þ
e 15
ε 11
A exp Àξ 2 x 2 À h
ð
Þ
½
&
'
 exp i ξ 1 x 1 À ωt
ð
Þ
½
,
T 23 ¼ À Ac 44 ξ 2 exp Àξ 2 x 2 À h
ð
Þ
½
þe 15 Bξ 1 exp Àξ 1 x 2 À h
ð
Þ
½
f
g
 exp i ξ 1 x 1 À ωt
ð
Þ
½
,
D 2 ¼ ε 11 Bξ 1 exp Àξ 1 x 2 À h
ð
Þ
½
exp i ξ 1 x 1 À ωt
ð
Þ
½
:
ð6:15Þ
Electric fields can also exist in the free space of x 2 < Àh, which are governed by
∇
2 φ ¼ 0, x 2 < Àh,
φ ! 0, x 2 ! À1:
ð6:16Þ
A surface wave solution to Eq. (6.16) is
φ ¼ C exp ξ 1 x 2 þ h
ð
Þ
½
exp i ξ 1 x 1 À ω t
ð
Þ
½
,
ð6:17Þ
where C is an undetermined constant. From Eq. (6.17), in the free space, we have
D 2 ¼ Àε 0 ξ 1 C exp ξ 1 x 2 þ h
ð
Þ
½
exp i ξ 1 x 1 À ω t
ð
Þ
½
:
ð6:18Þ
For the semiconductor plate, since the normal of the plate is along x 2 , the plate
equations in Chap. 5 cannot be used directly. Fortunately we only need the zeroorder equations of the plate which will be quickly derived in the following in the
coordinate system in Fig. 6.1. The zero-order equations govern extensional motions
described by u 1 (x 1 , x 3 , t) and u 3 (x 1 , x 3 , t). The dominating stress components are T 11 ,
T 33 , and T 13 . Therefore we make the following stress relaxation for thin plates:
T 2j ¼ 0, j ¼ 1, 2, 3:
ð6:19Þ
6.1 Surface Waves
143