128
5 Mechanical Properties
(a)
(b)
(c)
Fig. 5.37 Schematic representation of nanoscroll formation. a Strained heterostructure (blue/green) that is planar due
to large substrate thickness, b starting removal of sacrificial layer (black), c release of thin film into nanoscroll geometry
(a)
y
t
r
(b)
Fig. 5.38 a Schematic representation of a cylindrically rolled sheet with radial direction r , tangential direction t and
direction along the cylinder axis y. b SEM images of multiwall (In,Ga)As/GaAs nanoscroll rolled up over about 50 µm.
Part (b) from [426]
If bending strain occurs only in one of the tangential directions, the energy density is given by
U =
Y
2 (1 − ν 2 )
((
2
t +
2
y + 2ν ν t y ) ,
(5.87)
where y is the strain in the unbent direction (cylinder axis) as shown in Fig. 5.38a. For a strained
heterostructure made up from two layers the curvature is given by (calculated analogous to (5.85),
χ = Y 2 /Y 1 [419])
κ =
6(1 + ν) χ d 1 d 2 (d 1 + d 2 )
d
4
1 + 4χ d
3
1 d 2 + 6χ d
2
1 d
2
2 + 4χ d 1 d
3
2 + χ 2 d
4
2
,
(5.88)
which differs from (5.85) only by the factor 1 + ν in the nominator.
For cubic material and a (001) surface the energy is given as
U 100 =
C 11 − C 12
2C 11
C 11 ((
2
t +
2
y + C 12 (( t + y )
2
)
(5.89)
for a scrolling direction along 100. When the (001)-oriented film winds up along a direction hk0
having an angle φ with the [100] direction (φ = 45
◦ for 110), the strain energy is given by (C 0 is
given by (5.58))
U φ = U 100 + C 0
t − y
2
2
sin
2
(2φ) .
(5.90)