JOURNAL OF APPLIED PHYSICS 105, 123516 共2009兲
Mechanics of noncoplanar mesh design for stretchable electronic circuits
J. Song,1,a兲 Y. Huang,2,3 J. Xiao,3 S. Wang,3 K. C. Hwang,4 H. C. Ko,5 D.-H. Kim,5
M. P. Stoykovich,6 and J. A. Rogers5,7,b兲
1
Department of Mechanical and Aerospace Engineering, University of Miami, Coral Gables,
Florida 33146, USA
2
Department of Civil and Environmental Engineering, Northwestern University, Evanston,
Illinois 60208, USA
3
Department of Mechanical Engineering, Northwestern University, Evanston, Illinois 60208, USA
4
Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China
5
Department of Materials Science and Engineering, University of Illinois, Urbana, Illinois 61801, USA
6
Department of Chemical and Biological Engineering, University of Colorado, Boulder,
Colorado 80309, USA
7
Department of Mechanical Science and Engineering, Department of Electrical and Computer Engineering,
Department of Chemistry, and Frederick Seitz Materials Research Laboratory, University of Illinois,
Urbana, Illinois 61801, USA
共Received 17 February 2009; accepted 7 May 2009; published online 22 June 2009兲
A noncoplanar mesh design that enables electronic systems to achieve large, reversible levels
stretchability 共⬎100%兲 is studied theoretically and experimentally. The design uses semiconductor
device islands and buckled thin interconnects on elastometric substrates. A mechanics model is
established to understand the underlying physics and to guide the design of such systems. The
predicted buckle amplitude agrees well with experiments within 5.5% error without any parameter
fitting. The results also give the maximum strains in the interconnects and the islands, as well as the
overall system stretchability and compressibility. © 2009 American Institute of Physics.
关DOI: 10.1063/1.3148245兴
I. INTRODUCTION
Stretchable electronics is emerging as a technology that
could be valuable for various applications such as flexible
displays,1 electronic eye camera,2 conformable skin sensors,3
smart surgical gloves,4 and structural health monitoring
devices.5 There exist two approaches to achieve stretchability: 共i兲 coplanar stretchable interconnects 共bonded to a substrate兲 between rigid device islands,6 and 共ii兲 wavy layout
共i.e., small wave兲 throughout the whole circuit system.7 Both
are fabricated on elastometric substrates, and provide some
degree of stretchability 共e.g., 10%兲. None offers strainindependent electronic performance at large strains, which is
of interest in practical applications.2,4 Here, we present a
noncoplanar mesh design, which is based on the
interconnect-island6 concept to accomplish much higher
stretchability 共i.e., up to 100%兲. Figure 1 schematically illustrates the fabrication of circuits with noncoplanar mesh design on compliant substrates.2,8 The silicon 共or other semiconductor material兲 islands, on which the active devices or
circuits are fabricated, are chemically bonded to a prestrained 共e.g., 50%兲 elastometric substrate of a material such
as poly共dimethylsiloxane兲 共PDMS兲, while interconnects are
loosely bonded.8 Releasing the prestrain leads to compression, which causes the interconnects to buckle and move out
of the plane of the substrate to form arc-shaped structures.
The poor adhesion of interconnects 共to PDMS兲 and their narrow geometries and low stiffness 共compared to device is-
lands兲 cause the out-of-plane deformation to localize only to
interconnects, and therefore the strain in islands is very
small. Figure 2 shows a scanning electron micrograph of
silicon structure 共island: 20⫻ 20 ␮m2, 50 nm thick; interconnect: 20⫻ 4 ␮m2, 50 nm thick兲 on a 1 mm thick PDMS
substrate. The inset clearly shows that the islands remain flat
while interconnects buckle.
Kim et al.8 developed a full-scale finite element model
for the noncoplanar mesh design. But, it is rather complex
and does not lead to simple, analytical solutions to be used in
the design and optimization of these systems. This paper
a兲
Author to whom correspondence should be addressed. Electronic mail:
jsong8@miami.edu.
b兲
Electronic mail: jrogers@uiuc.edu.
0021-8979/2009/105共12兲/123516/6/$25.00
FIG. 1. 共Color online兲 Schematic illustration of the process for fabricating
electronics with noncoplanar mesh designs on a complaint substrate.
105, 123516-1
© 2009 American Institute of Physics
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123516-2
J. Appl. Phys. 105, 123516 共2009兲
Song et al.
above equation also agrees well with the full-scale finite element analysis.8
tot
The total energy Ubridge
of the interconnect consists the
bending
membrane
.
bending energy Ubridge and the membrane energy Ubridge
bending
The bending energy Ubridge can be obtained from Eq. 共1兲
3
/ 12 as
and the bending rigidity Ebridgehbridge
bending
=
Ubridge
=
FIG. 2. A scanning electron micrograph of a noncoplanar silicon mesh
structure 共island: 20⫻ 20 ␮m2, 50 nm thick; interconnect: 20⫻ 4 ␮m2, 50
nm thick兲 on a PDMS substrate.
aims at establishing a mechanics theory for the mechanical
response of noncoplanar mesh design and predicting the
maximum strains in the interconnects as well as in the islands, which are both important to determine the stretchability of the system. For simplicity, silicon is used for both
islands and interconnects, though the mechanics theory can
be applied to other systems that involve multiple materials.
The paper is outlined as follows. The mechanics models of
the interconnects and the islands are described in Secs. II and
III, respectively. Stretchability/compressibility of various
nonplanar mesh designs are discussed in Sec. IV.
II. MECHANICS MODEL OF INTERCONNECTS
The width of interconnects is much smaller than the
width of island such that the rotation at the ends of interconnects is very small. This is verified by the finite element
analysis 共see Sec. III and also Kim et al.8兲 as well as by the
analytical solution given in the Appendix. Therefore, the
“bridge”-like interconnect is modeled as a beam with
clamped ends 共as shown in Fig. 3兲 since its thickness
共hbridge ⬃ 50 nm兲 is much smaller than any other characteristic length 共width: wbridge ⬃ 4 ␮m; length: Lbridge ⬃ 20 ␮m兲.
The beam, however, undergoes large rotation once the interconnect buckles. Let X denote the initial, strain-free configuration of the beam 共top figure in Fig. 1兲, and x the buckled
configuration 共bottom figure in Fig. 1兲. The distance between
0
to Lbridge in these two configuraislands changes from Lbridge
tions, as shown in Fig. 3, and the buckle amplitude A is to be
determined. The out-of-plane displacement w of the interconnect takes the form
w=
冉
冊 冉
冊
2␲x
2␲X
A
A
1 + cos
=
1 + cos 0
,
2
Lbridge
2
Lbridge
冕
0
Lbridge
/2
0
−Lbridge
/2
dX
共2兲
where Ebridge is the Young’s modulus of the interconnect.
membrane
The membrane strain ␧bridge
, which determines the
membrane energy, is related to the out-of-plane displacement
w in Eq. 共1兲 and in-plane displacement u by9
membrane
␧bridge
=
冉 冊
du 1 dw
+
dX 2 dX
2
共3兲
.
membrane
via the
The membrane force N is then related to ␧bridge
membrane
. The
tension rigidity Ebridgehbridge as N = Ebridgehbridge␧bridge
force equilibrium
dN
= 0,
dX
共4兲
requires a constant membrane force, which gives the in-plane
displacement
u=
冉
冊 冉
冊
0
− Lbridge
Lbridge
4␲
␲A2
sin
X
−
X.
0
0
0
16Lbridge
Lbridge
Lbridge
L
0
共5兲
/2
0
du = Lbridge − Lbridge
Here the conditions u共0兲 = 0 and 兰−Lbridge
0
/2
bridge
have been imposed. Equation 共3兲 then gives a constant membrane strain
membrane
=
␧bridge
0
− Lbridge
Lbridge
␲ 2A 2
−
.
0
0
2
4共Lbridge兲
Lbridge
共6兲
The membrane energy in the interconnect is given by
membrane
=
Ubridge
冕
0
Lbridge
/2
0
−Lbridge
/2
1
membrane 2
兲 dX
Ebridgehbridge共␧bridge
2
冋
1
␲ 2A 2
0
= EbridgehbridgeLbridge
0
2
4共Lbridge
兲2
0
Lbridge − Lbridge
共1兲
0
Lbridge
册
2
0
2Lbridge
␲
冑
0
Lbridge
− Lbridge
0
Lbridge
共7兲
.
Minimization of total energy
tot
⳵Ubridge
/ ⳵A = 0 gives the amplitude
A=
FIG. 3. 共Color online兲 Schematic diagram of mechanics model for the interconnect region of a noncoplanar mesh structure.
2
3
Ebridgehbridge
␲ 4A 2
,
0
12
兲3
共Lbridge
+
which satisfies vanishing displacement and slope at the two
0
/ 2. The sinusoidal buckle profile in the
ends X = ⫾ Lbridge
冉 冊
3
1 Ebridgehbridge
d 2w
2
12
dX2
in
− ␧c ,
the
interconnect
共8兲
2
0
/ 关3共Lbridge
兲2兴 is the critical buckling strain
where ␧c = ␲2hbridge
0
for Euler buckling of a doubly clamped beam. For 共Lbridge
0
− Lbridge兲 / Lbridge ⬍ ␧c, the interconnect does not buckle, and
membrane
therefore has no bending. The membrane strain is ␧bridge
0
0
0
0
= −共Lbridge − Lbridge兲 / Lbridge. Once 共Lbridge − Lbridge兲 / Lbridge exceeds ␧c, the interconnect buckles such that the membrane
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123516-3
J. Appl. Phys. 105, 123516 共2009兲
Song et al.
FIG. 4. 共Color online兲 The maximum strain in the interconnects vs the
prestrain for different interconnect lengths.
strain remains a constant −␧c 关see from Eqs. 共6兲 and 共8兲兴, and
the bending strain 共curvature ⴱ hbridge / 2兲 increases
bending
0
= 2␲共hbridge / Lbridge
兲
with
the
deformation,
␧bridge
0
0
⫻冑关共Lbridge − Lbridge兲 / Lbridge兴 − ␧c. The maximum 共compressive兲 strain in the interconnect is the sum of membrane and
bending strains, and is given by
max
=
␧bridge
hbridge
2␲ 0
Lbridge
⬇ 2␲
冑
冑
0
Lbridge
− Lbridge
hbridge
0
Lbridge
0
Lbridge
0
Lbridge
− Lbridge
0
Lbridge
0
Lbridge
=
␧pre
.
1 + ␧pre
− ␧c + ␧c
,
共9兲
共10兲
The maximum strain in the interconnect in Eq. 共9兲 is then
related to the prestrain in the substrate by
max
= 2␲
␧bridge
hbridge
0
Lbridge
冑
␧pre
.
1 + ␧pre
10 ␮m and thickness hbridge = 50 nm 关or equivalently hbridge
0
= 25, 50, and 100 nm and length Lbridge
= 20 ␮m兴. Therefore
thin and long interconnects give small strain, and thus increase the stretchability.
III. MECHANICS MODEL OF ISLANDS
0
0
− Lbridge兲 / Lbridge
where the approximation holds for 共Lbridge
2
0
2
Ⰷ hbridge / 共Lbridge兲 .
Once the prestrain ␧pre in the substrate is relaxed, the
bridge length is reduced to Lbridge. Therefore the prestrain is
0
− Lbridge兲 / Lbridge, which can be rewritten
given by ␧pre = 共Lbridge
as
0
− Lbridge
Lbridge
FIG. 5. 共Color online兲 Distribution of the strain ␧xx in islands 共20
⫻ 20 ␮m2兲 when the interconnect relaxes from 20 to 17.5 ␮m.
共11兲
The initial length of the interconnect in experiments is
0
= 20 ␮m and the thickness hbridge = 50 nm, which give
Lbridge
a critical buckling strain ␧c = 0.0021%. The measured bridge
length is Lbridge = 17.5 ␮m after relaxation, which corresponds to prestrain ␧pre = 14.3% in the substrate. Equation 共8兲
then predicts the amplitude A = 4.50 ␮m, which agrees well
with the experimentally measured amplitude 4.76 ␮m. The
maximum strain in the interconnect is 0.56%. This value is
smaller than the fracture strain of silicon 共⬃1%兲, and is
much smaller than the prestrain ␧pre. For 1% interconnect
strain, the prestrain can reach 68.1%.
max
given in
The maximum strain in the interconnect ␧bridge
Eq. 共9兲 or Eq. 共11兲 is proportional to the interconnect thick0
max
. Figure 4 shows ␧bridge
ness and to length ratio, hbridge / Lbridge
0
versus the prestrain ␧pre for the length Lbridge = 40, 20, and
The finite element method is used to study the silicon
island on PDMS substrate. The island is modeled as a plate
since its thickness hisland is much smaller than the length
0
. The buckled interconnects give the axial force N
Lisland
3
= Ebridgehbridge␧c and bending moment M = 共Ebridgehbridge
/ 12兲
0
3
0
2
2
冑
⫻关2␲ A / 共Lbridge兲 兴 ⬇ ␲Ebridgehbridge / 共3Lbridge兲 ␧pre / 共1 + ␧pre兲
over the width wbridge on each edge of the island.
The PDMS substrate is modeled by a unit cell that is a
0
+ Lbridge in the 共X-Y兲 plane but
square with edge length Lisland
very thick in the island thickness direction 共Z兲. The displacements are continuous across the island/substrate interface,
and the rest of the top surface is traction free. Periodic
boundary conditions are imposed on the lateral surfaces 共X-Z
and Y-Z planes兲 of the substrate.
Figure 5 shows the strain distribution ␧xx in a Si island
0
= 20 ␮m, and
共Eisland = 130 GPa, ␯island = 0.27, length Lisland
10
thickness hisland = 50 nm兲 on an PDMS substrate 共Esubstrate
= 2 MPa and ␯substrate = 0.48兲.11 The axial force and bending
moment result from the buckled interconnect 共Ebridge
0
= 130 GPa, Lbridge
= 20 ␮m, hbridge = 50 nm, wbridge = 4 ␮m,
and Lbridge = 17.5 ␮m after relaxation兲. The maximum strain
occurs at the interconnect/island boundary.
The strain due to the axial force is negligible as compared to that due to the bending moment. Dimensional analymax
sis gives the maximum strain in the silicon island as ␧island
2
2
= ␣6共1 − ␯island兲M / 共Eislandhisland兲, where ␣ is a nondimensional function of dimensionless elastic properties
Esubstrate / Eisland, ␯substrate, and ␯island, and nondimensional
0
/ Lisland. Finite element analylengths wbridge / Lisland and Lbridge
ses show that ␣ is essentially a constant of unity. For large
variations in island and substrate elastic properties and inter0
, ␣ only deviates from
connect width wbridge and length Lbridge
one by a few percent. The maximum strain in the silicon
island is then given by
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123516-4
J. Appl. Phys. 105, 123516 共2009兲
Song et al.
2
兲M
6共1 − ␯island
max
␧island
⬇
2
Eislandhisland
= 2␲
2
3
共1 − ␯island
兲Ebridgehbridge
2
0
Eislandhisland
Lbridge
冑
␧pre
,
1 + ␧pre
共12兲
which can also be related to the maximum strain in interconnect by
max
=
␧island
2
2
兲Ebridgehbridge
共1 − ␯island
2
Eislandhisland
max
␧bridge
.
共13兲
Therefore, a stiff and thick island reduces its strain.
max
that the noncoplanar mesh
The maximum prestrain ␧pre
design can accommodate is obtained by equating the strains
failure
and
in Eqs. 共11兲 and 共12兲 to the failure strains 共⬃1%兲 ␧bridge
failure
␧island of interconnect and island materials, respectively,
max
⬍
␧pre
a2
1 − a2
where
a=
if
a ⬍ 1,
共14兲
冋
册
0
2
Eislandhisland
Lbridge
failure
failure
min ␧bridge
,
␧island
.
2
2
2␲hbridge
共1 − ␯island
兲Ebridgehbridge
共15兲
For a ⱖ 1 共e.g., long interconnects兲, the maximum prestrain
is then governed by the failure of PDMS substrate.
IV. STRETCHABILITY/COMPRESSIBILITY OF
NONCOPLANAR MESH DESIGN
0
0
The length of a unit cell changes from Lbridge
+ Lisland
to
0
Lbridge + Lisland after the prestrain ␧pre is relaxed, where
0
/ 共1 + ␧pre兲 is obtained from Eq. 共10兲, and the
Lbridge = Lbridge
change in island length is negligible because the interconnect
buckles to accommodate the release of prestrain. The length
0
once the system is
of the unit cell becomes Lbridge
⬘ + Lisland
subject to an applied strain ␧applied, where the length of interconnect Lbridge
is related to ␧applied by
⬘
␧applied =
⬘ − Lbridge
Lbridge
0
Lbridge + Lisland
共16兲
.
The stretchability/compressibility characterizes how
much the noncoplanar mesh design can accommodate further
deformation. It is defined as the critical applied strain that
leads to failure of interconnect or island, which occurs when
the maximum strains in interconnect or island reach the corfailure
failure
and ␧island
of the associated
responding failure strains ␧bridge
materials, respectively. The stretchability is determined by
the condition at which the buckled interconnect returns to a
flat state, at which point it cannot accommodate any additional stretching. This condition is obtained from Lbridge
⬘
0
= Lbridge
as
␧stretchability =
0
− Lbridge
Lbridge
Lbridge +
0
Lisland
=
␧pre
1 + 共1 + ␧pre兲
0
Lisland
. 共17兲
0
Lbridge
The result clearly shows that long interconnects, short islands, and large prestrains increase the stretchability. For the
FIG. 6. Stretchability and compressibility vs the prestrain for the noncoplanar mesh design 共island: 20⫻ 20 ␮m2, 50 nm thick; interconnect: 20
⫻ 4 ␮m2, 50 nm thick兲 when the failure strains of interconnect and island
are 1%.
0
0
limit of long interconnect Lbridge
Ⰷ Lisland
, the stretchability is
the prestrain ␧pre. For the other limit of short interconnect
0
0
0
0
Lbridge
Ⰶ Lisland
, the stretchability is 共Lbridge
/ Lisland
兲关␧pre / 共1
+ ␧pre兲兴.
The maximum strain in Eq. 共9兲 for the interconnect now
max
0
0
0
= 2␲共hbridge / Lbridge
兲冑共Lbridge
− Lbridge
,
becomes ␧bridge
⬘ 兲 / Lbridge
while Eq. 共12兲 still holds for the maximum strain in island.
The compressibility is reached when they reach the correfailure
failure
and ␧island
, or when then
sponding failure strains ␧bridge
neighbor islands start to contact. This gives the compressibility
␧compressibility = min
冤
共1 + ␧pre兲a2 − ␧pre
1 + 共1 + ␧pre兲
0
Lisland
0
Lbridge
,
1
1 + 共1 + ␧pre兲
0
Lisland
0
Lbridge
冥
, 共18兲
where a is given in Eq. 共15兲. For long interconnects 共large a兲,
the
compressibility
is
␧compressibility = 1 / 关1 + 共1 + ␧pre兲
0
0
/ Lbridge
兲兴, corresponding to the contact of neighbor
⫻共Lisland
islands. For short interconnects 共small a兲, the compressibility
is
␧compressibility = 关共1 + ␧pre兲a2 − ␧pre兴 / 关1 + 共1 + ␧pre兲
0
0
⫻共Lisland / Lbridge兲兴, corresponding to the failure of interconnect or island.
Figure 6 shows the stretchability and compressibility
0
= 20 ␮m, hisland = 50 nm,
versus the prestrain for Lisland
0
Lbridge = 20 ␮m, hbridge = 50 nm, and wbridge = 4 ␮m. The failure strains of the interconnect and the island are assumed, for
failure
failure
= ␧island
= 1%. The stretchability insimplicity, to be ␧bridge
creases with the prestrain, but the compressibility decreases.
Therefore, prestrain cannot be adjusted to give both maximum stretchability and maximum compressibility.
One way to achieve large stretchability and compress0
of interconnect. As
ibility is to increase the length Lbridge
shown in Fig. 7 for the same set of properties as Fig. 6 and
the prestrain ␧pre = 50%, both stretchability and compressibility increase with the length of interconnect, though the compressibility increases much faster than the stretchability. The
dot on the curve for compressibility separates the failure of
interconnect or island 共left of the dot兲 from the contact of
neighbor islands 共right of the dot兲.
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123516-5
J. Appl. Phys. 105, 123516 共2009兲
Song et al.
bending rigidity of the interconnect, w is the out-of-plane
0
0
displacement that satisfies w共−Lbridge
/ 2兲 = w共Lbridge
/ 2兲 = 0, N
is the compressive force at the ends, and M = kw⬘共
0
0
/ 2兲 = −kw⬘共Lbridge
/ 2兲 is the moment at the ends. The
−Lbridge
buckle profile is given by
冉 冊
2␰X
− cos ␰
0
Lbridge
,
1 − cos ␰
cos
w=A
FIG. 7. Stretchability and compressibility vs the length of interconnect for
the noncoplanar mesh design when the prestrain is 50%. The dot on the
curve for compressibility separates the failure of interconnect or island 共left
of the dot兲 from the contact of neighbor islands 共right of the dot兲.
V. CONCLUDING REMARKS
A mechanics model has been established for stretchable
electronics with noncoplanar mesh designs. The results predict analytically the buckling amplitude, which agrees well
with experiments 共5.5% error兲 without any parameter fitting.
The maximum strains in the interconnect and island are also
obtained analytically, and are used to predict the stretchability and compressibility. The above model can be extended
multilayer interconnects and islands by replacing the corresponding tension and bending rigidities.
where the amplitude A is to be determined, ␰ is determined
0
from tan ␰ = −2EI/ kLbridge
␰, and ␲ / 2 ⱕ ␰ ⱕ ␲. The limit k
→ ⬁ gives ␰ → ␲, and the buckle profile degenerates to Eq.
共1兲 for a doubly clamped beam. The other limit k → 0 gives
␰ → ␲ / 2, and the buckle profile becomes w
0
兲, which satisfies w = w⬙ = 0 at two ends X
= A cos共␲X / Lbridge
0
= ⫾ Lbridge / 2 of a simply supported beam.
The bending and membrane energies in interconnects are
obtained from Eqs. 共2兲 and 共7兲, respectively. Minimization of
total energy gives the amplitude
冑
0
A = Lbridge
APPENDIX: ROTATION AT THE ENDS OF
INTERCONNECTS
Section II assumes that interconnects are clamped, which
gives vanishing rotation at the ends. The effect of nonvanishing rotation can be accounted for by the rotational springs
with the spring constant k such that the bending moment M
and the rotation ␪ at the ends of interconnects are related by
M = k␪. For each given M, the finite element analysis of islands and substrate in Sec. III gives the rotation ␪, and the
ratio M / ␪ gives the spring constant k. For example, k
= 1.58␮N for the elastic properties and thicknesses of islands
and substrate in Fig. 5.
0
with rotational
For an interconnect of length Lbridge
0
springs at its ends 共X = ⫾ Lbridge / 2兲, the equilibrium gives
3
/ 12 is the
EI共d2w / dX2兲 = −Nw + M, where EI = Ebridgehbridge
2共1 − cos ␰兲2
2␰2 − ␰ sin共2␰兲
冑
0
Lbridge
− Lbridge
0
Lbridge
− ␧cr ,
共A2兲
2
0
␧cr = hbridge
/ 关3共Lbridge
兲2兴关2␰ + sin共2␰兲兴␰2 / 关2␰ − sin共2␰兲兴
where
is the critical buckling strain. The maximum compressive
strain in the interconnect is the sum of membrane and bending strains, and it is given by
2␰hbridge
max
=
␧bridge
ACKNOWLEDGMENTS
We thank for various helps of T. Banks in processing by
use of facilities at Frederick Seitz Materials Research Laboratory. This materials is based on work supported by the
National Science Foundation under Grant No. ECCS0824129, NSFC, and the US Department of Energy, Division
of Materials Sciences under Award No. DE-FG0207ER46471, through the Materials Research Laboratory and
Center for Microanalysis of Materials 共Grant No. DE-FG0207ER46453兲 at the University of Illinois at UrbanaChampaign. J.S. acknowledges the financial support from the
College of Engineering at the University of Miami.
共A1兲
0
Lbridge
冑
+ ␧cr
⬇
2␰hbridge
0
Lbridge
2␰
2␰ − sin共2␰兲
冑
冑
2␰
2␰ − sin共2␰兲
0
Lbridge
− Lbridge
冑
0
Lbridge
0
Lbridge
− Lbridge
0
Lbridge
− ␧cr
,
共A3兲
0
0
where the approximation holds for 共Lbridge
− Lbridge兲 / Lbridge
0
2
Ⰷ hbridge / 共Lbridge兲 .
0
= 20 ␮m, hbridge = 50 nm,
For Ebridge = 130 GPa, Lbridge
and Lbridge = 17.5 ␮m after relaxation, Eq. 共A2兲 gives the amplitude A = 4.61 ␮m, which agrees well with 4.50 ␮m given
by Eq. 共8兲. The maximum strain in the interconnect is 0.49%
from Eq. 共A3兲, which is smaller than 0.56% given by Eq.
共11兲. Therefore, the rotation at the ends of interconnect can
be neglected.
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