105
Electronic Properties of Strain-Engineered Semiconductors
The deformation may be described in terms of a symmetrical strain tensor as
1
2
u
u
u
u
u
u
u
im
i
m
m
i
mi
i
im
=
∂
∂
+
∂
∂

 

  =
= δ
(4.32)
If the distance between two lattice points of an undeformed Si substrate is
dl and for a deformed Si substrate (strained Si substrate) is dl′ , then we get
2
2
2
2
2
1
3
dl dx dy dz
dx i
i
∑
=
+
+
=
=
(4.33)
and
2
2
2
1
3
2
2
2
1
3
1
3
dl
dx
dx du
dx
dx du du
i
i
i
i
i
i
i
i
i
i
∑
∑
∑
(
)
(
)
′ =
′ =
+
=
+
+
=
=
=
(4.34)
The tensor du i determines the variation of distances between the lattice
points. Neglecting the quantity
2
du i and expressing du i in terms of du m ,
1
3
du
u
u
du
i
i
m
m
m
∑
=
∂
∂
=
(4.35)
and combining Equations (4.33 to 4.35), we obtain
2
dl′ as
2
2
2
2
1
3
, 1
3
dl
dx
dl
u
x
dx dx
i
i
m
i
m
i m
∑
∑
′ =
′ =
+
∂
∂
=
=
(4.36)
Taking into account Equation (4.32) we obtain for
2
dl′
2
2
2
, 1
3
2
,
dl
dl
u
x
u
x
dx dx
dl
u dx dx
i
m
m
i
i
m
i m
im
i
m
i m
∑
∑
′ =
+
∂
∂
+
∂
∂

 

 
=
+
=
(4.37)
We may write from Equation (4.37)
2
2
2
,
2
2
dl
dl
u dx dx
dl
udx
i
im
i
m
i m
i
i
i
∑
∑
′ =
+
δ
=
+
(4.38)
The variation of volume upon deformation is given by
1
1
1
1
(1)
( 2)
(3)
(1)
( 2)
(3)
V dx dy dz dx
u dy
u dz
u
V
u
u
u
(
) (
) (
)
(
)
′ = ′ ′ ′ =
+
+
+
=
+
+
+
(4.39)