5.3 Elasticity
127
1100
1090
1080
1070
1060
1050 0
200 400 600 800 1000 1200 1400
Time (s)
Temperature (°C)
0.05
0.0
-0.05
-0.1
-0.15
Curvature (m )
-1
cooling
epitaxy
6K
Fig. 5.36 Curvature of the middle of a Si wafer during GaN growth on an AlN interlayer grown at low temperatures
on GaN and subsequent cooling. During growth the decrease in curvature indicates convex bowing due to compressive
stress; during cooling the wafer flattens and becomes concave due to thermally induced tensile stress. Adapted from [422]
κ =
6a 1 a 2 (a 2 − a 1 ) d 1 d 2 (d 1 + d 2 ) Y 1 Y 2
a
3
2 d
4
1 Y
2
1 + α Y 1 Y 2 + a
3
1 d
4
2 Y
4
2
(5.84)
α = a 1 a 2 d 1 d 2
−a 2 d 1 (2d 1 + 3d 2 ) + a 1 (6d
2
1 + 9d 1 d 2 + 4d
2
2 )
.
For a 2 = a 1 (1 + ) we develop κ to first order of and find (χ = Y 2 /Y 1 ) [418, 419]
κ =
6χ 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
.
(5.85)
In the case of a substrate (d s ) with a thin epitaxial layer (d f d s ), the radius of curvature is approximately (Stoney’s formula [420])
κ = 6
d f
d 2
s
Y f
Y s
.
(5.86)
Conversely, if the radius of curvature is measured [421], e.g. optically, the film curvature (and through
models also the film strain) can be determined during epitaxy as depicted in Fig. 5.36.
5.3.6 Scrolling
In some cases cylindrically scrolled structures are important, e.g. for thin-film flexible electronics,
nanotubes, nanoscrolls or nanohelixes. The scrolling of thin layers must be avoided by suitable strain
management for thin layers that are lifted off from their substrate for transfer to another flat substrate. If
the film remains attached to its substrate, a scroll can be fabricated as schematically shown in Fig. 5.37.
Such structures were first reported in [423], a review can be found in [424]. The shape of such scroll
is investigated in [425] without a priori assumptions on its shape.
127
1100
1090
1080
1070
1060
1050 0
200 400 600 800 1000 1200 1400
Time (s)
Temperature (°C)
0.05
0.0
-0.05
-0.1
-0.15
Curvature (m )
-1
cooling
epitaxy
6K
Fig. 5.36 Curvature of the middle of a Si wafer during GaN growth on an AlN interlayer grown at low temperatures
on GaN and subsequent cooling. During growth the decrease in curvature indicates convex bowing due to compressive
stress; during cooling the wafer flattens and becomes concave due to thermally induced tensile stress. Adapted from [422]
κ =
6a 1 a 2 (a 2 − a 1 ) d 1 d 2 (d 1 + d 2 ) Y 1 Y 2
a
3
2 d
4
1 Y
2
1 + α Y 1 Y 2 + a
3
1 d
4
2 Y
4
2
(5.84)
α = a 1 a 2 d 1 d 2
−a 2 d 1 (2d 1 + 3d 2 ) + a 1 (6d
2
1 + 9d 1 d 2 + 4d
2
2 )
.
For a 2 = a 1 (1 + ) we develop κ to first order of and find (χ = Y 2 /Y 1 ) [418, 419]
κ =
6χ 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
.
(5.85)
In the case of a substrate (d s ) with a thin epitaxial layer (d f d s ), the radius of curvature is approximately (Stoney’s formula [420])
κ = 6
d f
d 2
s
Y f
Y s
.
(5.86)
Conversely, if the radius of curvature is measured [421], e.g. optically, the film curvature (and through
models also the film strain) can be determined during epitaxy as depicted in Fig. 5.36.
5.3.6 Scrolling
In some cases cylindrically scrolled structures are important, e.g. for thin-film flexible electronics,
nanotubes, nanoscrolls or nanohelixes. The scrolling of thin layers must be avoided by suitable strain
management for thin layers that are lifted off from their substrate for transfer to another flat substrate. If
the film remains attached to its substrate, a scroll can be fabricated as schematically shown in Fig. 5.37.
Such structures were first reported in [423], a review can be found in [424]. The shape of such scroll
is investigated in [425] without a priori assumptions on its shape.