Acta mater. 48 (2000) 4901–4915
www.elsevier.com/locate/actamat
MICROBRIDGE TESTING OF SILICON OXIDE/SILICON NITRIDE
BILAYER FILMS DEPOSITED ON SILICON WAFERS
Y. -J. SU, C. -F. QIAN†, M. -H. ZHAO‡ and T. -Y. ZHANG§
Department of Mechanical Engineering, Hong Kong University of Science and Technology, Clear Water
Bay, Kowloon, Hong Kong, China
( Received 17 April 2000; received in revised form 17 July 2000; accepted 17 July 2000 )
Abstract—The present work further develops the microbridge testing method to characterize mechanical
properties of bilayer thin films. We model the substrate deformation with three coupled springs and consider
residual stress in each layer to formulate deflection versus load under large deformation, resulting in a closedform formula. If the mechanical properties of one layer are available, the closed formula is able to simultaneously evaluate the Young’s modulus and residual stress of the other layer, and the bending strength of
the bilayer films from the microbridge test. The analytic results are confirmed by finite element calculations.
Using a load and displacement sensing nanoindenter system equipped with a microwedge probe, we conduct
microbridge tests on low-temperature silicon oxide/silicon nitride bilayer films prepared by the microelectromechanical technique. The experimental results verify the proposed method, yielding the Young’s modulus
of 41.00±3.60 GPa, the residual stress of ⫺180.88±7.90 MPa and the bending strength of 0.903±0.111 GPa
for the low-temperature silicon oxide films.  2000 Acta Metallurgica Inc. Published by Elsevier Science
Ltd. All rights reserved.
Keywords: Thin films; Mechanical properties; Stress–strain relationship measurements; Semiconductor;
Multilayers
1. INTRODUCTION
As applications of thin films dramatically increase in
microelectronic devices and microelectromechanical
systems (MEMS), characterizing, understanding and
controlling the mechanical properties of thin films
have become one of the more important challenges
in modern technology. One of the major difficulties
encountered in studying the mechanical behavior of
thin films is that they are not amenable to testing by
conventional means because of their size and configuration. Therefore, new methods have been
developed, such as nanoindentation [1–7], uniaxial
tensile testing of free-standing films [8], beam bending [9–13], and bulge testing [14–17]. Review papers
on mechanical properties of thin films are also available [18–21]. Recently, we developed a novel
microbridge testing method, which can simultaneously measure the Young’s modulus, residual
* Visiting scholar from Beijing University of Chemical
Technology, Beijing, People’s Republic of China.
† Visiting scholar from Zhengzhou Research Institute of
Mechanical Engineering, Zhengzhou, People’s Republic of
China.
‡ To whom all correspondence should be addressed. Tel.:
⫹852-2358-7192; Fax: ⫹852-2358-1543.
E-mail address: [email protected] (T.-Y. Zhang)
stress and bending strength of a single-layer thin film
[12, 13]. The experimental results on single-layer silicon nitride films deposited on silicon substrates have
confirmed this testing method.
However, characterization of mechanical properties
of multilayer films is more challenging. Mearini and
Hoffman [8] conducted tensile tests on free-standing
thin films of Al/Al2O3/Al and Al2O3/Al/Al2O3. For
free-standing multilayer films under tension, the
Young’s modulus of the composite could be the
thickness average of Young’s modulus of the individual constituents. However, the measured Young’s
modulus of the composite was significantly lower
than the thickness average of Young’s modulus of the
individual constituents, and the difference was attributed to the low density of the Al2O3 and/or
microcracks [8]. Tabata et al. [22] characterized the
mechanical properties of bilayer thin films of lowpressure chemical vapor deposition (LPCVD)
polysilicon/LPCVD SiNx or plasma-enhanced chemical vapor deposition (PECVD) SiNx/LPCVD SiNx
using the bulge test on rectangular membranes. They
simply assumed that both the Young’s modulus and
the residual stress of the bilayer would equal the
thickness average of the corresponding properties of
the individual constituents, i.e.,
1359-6454/00/$20.00  2000 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved.
PII: S 1 3 5 9 - 6 4 5 4 ( 0 0 ) 0 0 2 9 0 - 1
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SU et al.: MICROBRIDGE TEST
Ebi ⫽
E1h1 ⫹ E2h2
,
h1 ⫹ h2
(1)
sbi ⫽
s1h1 ⫹ s2h2
,
h1 ⫹ h2
(2)
where h, E and s denote the thickness, Young’s
modulus and the residual stress, respectively, and the
subscripts “1”, “2” and “bi” stand for layer 1, layer
2 and the bilayer, respectively. Firstly, the residual
stress and the Young’s modulus of single-layer films
of LPCVD SiNx were measured to be 1.0 and 290
GPa. Then the residual stress and Young’s modulus
were estimated from the bulge test on the bilayer
membranes and equations (1) and (2) to be ⫺0.18
and 160 GPa for the LPCVD polysilicon films and
0.11 and 210 GPa for the PECVD SiNx. Using equations (1) and (2), Maier-Schneider et al. [23] determined the Young’s modulus and residual stress of
LPCVD polysilicon thin films by conducting bulge
tests of bilayer thin films of polysilicon/SiNx. The
residual stress of the polysilicon films was strongly
influenced by film thickness, annealing temperature
and ion implantation, whereas the Young’s modulus
was stable and ranged from 151 to 166 GPa.
Johansson et al. [24] analyzed the elastic deflection
and the maximum stress of a bilayer microcantilever
beam induced by residual stresses and applied loading. From cantilever tests of bilayer beams of Al, Ti,
TiN, or SiO2 coated on (100)⬍100> and (100)⬍110>
Si, they found that the experimental data of the bending strength scattered greatly, typically with error bars
in the range of 15–35%. For uncoated beams, the
average bending strength was 6 GPa for ⬍100>
beams and 4 GPa for ⬍110> beams. Most of the coatings had a strength-reducing effect, particularly the
brittle, thin coatings of TiN or Ti films reduced the
strength to about 1 GPa. Hong et al. [16] performed
microcantilever beam tests on SiNx/SiO2 films and
analyzed the mechanics for bilayer beam bending.
They introduced the effective stiffness of a bilayer
beam per unit width of
Dbi ⫽
E1h31 ⫹ E2h32 E1E2h1h2(h1 ⫹ h2)2
⫹
12
4(E1h1 ⫹ E2h2)
(h1 ⫹ h2)3
⫽
Ebend,
12
(3)
where Ebend denotes the bending-equivalent Young’s
modulus. It should be noted that the plane-stress condition along the beam width is used in the derivation
of equation (3). The bending test yielded the effective
stiffness from which the Young’s modulus for SiNx
films was determined to be 235 GPa with the available Young’s modulus of 65 GPa for SiO2 films. With
the effective stiffness, the residual stresses in the films
could be estimated from the curvature measurement.
For the standard and low stress SiNx films, the average residual stresses were 1.15 GPa and 440 MPa,
respectively [16]. Using equation (3) and regarding
layer 1 as a substrate, Hashimoto et al. [25] measured
the Young’s modulus of Co–Ta–Zr thin films by conducting three-point bending on film/substrate beams.
By measuring the slopes of the load–deflection for
pure substrate and bilayer beams, they estimated the
Young’s moduli for both layers and found that for the
Co–Ta–Zr films with thicknesses of less than 4.5 µm,
the Young’s modulus increased as the thickness
decreased. To measure the mechanical properties of
compositionally modulated Au–Ni thin films, Baker
and Nix [2] conducted microcantilever tests on Au–
Ni/SiO2 bilayer beams. They introduced a transformed beam that has the elastic constants of SiO2 with
a moment of inertia of the transformed cross section
in order to produce the same deflection stiffness as
the real beam. Replacing the Young’s modulus in
equation (3) by the flexural modulus E∗ ⫽ E/(1⫺n2),
where n is the Poisson ratio, the transformed beam
model is equivalent to equation (3). To improve the
accuracy of the assessment of the elastic properties
of Au–Ni modulated thin films from the bending test,
Baker and Nix [2] proposed an empirical correction
to eliminate the effect of substrate deformation. In
their correction, the real beam supported by a deformable substrate is equivalent to an ideal beam supported by a rigid substrate of somewhat greater, but
unknown, length. Thus, they replaced the beam
length, L, by L ⫹ LC, where LC was the required
length correction, and hence ignored higher-order
terms in LC and finally reduced the bending compliance, C, to the following form
C ⫽ L3/(3Dbi) ⫹ K2L2,
(4)
where K2 reflected the substrate deformation and was
determined experimentally. Using equations (3) and
(4) and the Young’s modulus of 64 GPa and the Poisson ratio of 0.16 for SiO2, Baker and Nix [2] assessed
the flexural modulus of the Au–Ni modulated films
from the bending test. The flexural modulus of the
Au–Ni modulated films varied with the composition
wavelength and had a minimum of about 70 GPa near
the wavelength of 2 nm. When measuring the elastic
modulus of TbDyFe films by the cantilever beambending test, Mencik et al. [5] used the same empirical approach of adding a length correction to eliminate the error induced by substrate deformation.
There are many publications regarding the mechanical analysis of multilayer films in the literature. The
following are a few examples. Townsend et al. [26]
presented a general theory for the elastic interactions in
a composite plate of layers with different relaxed planar
dimensions and provided solutions for the case of equivalent elastic properties. Klein and Miller [27] followed
the theoretical framework of Townsend et al. [26] and
SU et al.: MICROBRIDGE TEST
developed applicable equations for bilayer systems,
especially for systems made up of a thick substrate and
a thin-film stack. Alshits and Kirchner [28] studied the
elasticity of multilayers, in particular, for strips, coatings
and sandwiches. Finot and Suresh [29] examined the
small and large deformation of multilayer films and
studied the effects of layer geometry, plasticity and
compositional gradients. Fang and Wickert [30] investigated the methods for the determination of the means
and the gradients of residual stresses in micromachined
cantilevers. They considered the stress state in the stillbonded film and found that the analytic simply supported or clamped boundary condition could not completely reflect the real boundary. In situ observations of
the buckling behavior of polysilicon films indicate that
the deformation near the boundary could not be predicted by either the simply supported or clamped boundary condition [31]. The findings [30, 31] indicate that
substrate deformation may not be ignored in load–
deflection tests of thin films. Thus, when Zhang et al.
[12, 13] proposed the microbridge testing method, they
modeled the substrate deformation using three coupled
springs and the spring compliances were calculated by
finite element analysis (FEA). A closed-form formula
has been derived and verified by FEA calculations and
experimental observations. The theoretical and experimental results indicate that ignoring the deformation of
the silicon substrate may induce a large error in the
determination of the elastic constant and the residual
stress, even when the degree of the microbridge deformation is small. Using the formula and the experimental
data of single-layer silicon nitride films, Zhang et al.
[12, 13] simultaneously obtained the Young’s modulus
of 219±14 GPa, the residual stress of 306±62 MPa and
the bending strength of 9.0±1.0 GPa for the single-layer
silicon nitride films.
Surface and interface stresses are not taken into
account in the above mentioned experimental and theoretical work [2, 5, 8, 12, 13, 16, 22–31]. The surface
or interface stress can be thought of as a stretched or
compressed membrane that lies in the plane of the surface or interface and exerts lateral forces on the surrounding phases. Therefore, surface and interface
stresses may play an important role in the material
behavior of films, and have attracted many researchers
recently. For example, Berger et al. in situ measured
the surface stress changes during the self-assembly of
alkanethiols on gold [32]. Cammarata et al. proposed a
simple model for interface stresses with application to
misfit dislocation generation in epitaxial thin films [33].
Spaepen gave a review of the current understanding of
the effect of interfaces on the intrinsic stresses in
polycrystalline thin films, where “intrinsic” refers to
stresses that are not the result of directly applied loads
or of differential thermal expansion between a film and
its substrate or between different parts of the film [34].
For simplicity, we will not consider surface and interface stresses in the present work as the first step of the
development of microbridge testing methods for bilayer
thin films.
4903
This paper extends the microbridge testing method
to mechanically characterize bilayer thin films. The
load–deflection relationship for a microbridge sample
of bilayer thin films is formulated with consideration
of the substrate deformation, residual stress and tension–bending coupling deformation of the films. Silicon oxide/silicon nitride films deposited on silicon
substrates are used as a model system to verify the
proposed method.
2. ANALYSIS
Figure 1 is a schematic depiction of the
microbridge test, where the microbridge is a low-temperature silicon oxide (LTO)/silicon nitride bilayer
thin film with two ends bonded on a (100) silicon
substrate (or base). The silicon base has a support
angle of 54.74°, which is caused by anisotropic etching during the sample fabrication. During the
microbridge test, the deflection is measured against
the applied load. Without applied load, the deflection
is assumed to be zero, which means that the state
without any applied load is taken as a reference-state,
even so residual stresses may deflect the bilayer
beam. Based on this reference-state we conduct the
following analysis. The deflection induced by residual
stresses is analyzed and estimated in Appendix A.
The results indicate that the residual deflection is negligible in comparison with the deflection caused by
the applied load.
2.1. Deformation of a bilayer beam
The microbridge sample of bilayer thin films is
simply regarded as a bilayer beam with two layers
bonded perfectly along the interface. It is assumed
that residual stresses along the beam width and thickness directions are completely released and uniformly
distributed along the beam length direction in each
layer. Consistent with the residual stress, the planestress condition is also applied to the beam width and
thickness directions. The coordinate system is set up
such that the x-axis is along the midplane of the
Fig. 1. Schematic depiction of the microbridge test for bilayered thin films.
4904
SU et al.: MICROBRIDGE TEST
bilayer beam, as shown in Fig. 2(a) and (b). The displacements u(x, z) and w(x, z) along the x-axis and zaxis in the two layers are expressed respectively by
w1(x, z) ⫽ w2(x, z) ⫽ w(x),
∂w
u1(x, z) ⫽ u(x)⫺z , (z1⬍z⬍z2),
∂x
∂w
u2(x, z) ⫽ u(x)⫺z , (⫺z2⬍z⬍z1),
∂x
h2 ⫹ h1
h2⫺h1
z1 ⫽
and z2
.
2
2
(5)
冉 冊
冉 冊
e1,x ⫽
∂u 1 ∂w
∂w
⫺z 2 ,
⫹
∂x 2 ∂x
∂x
e2,x ⫽
∂u 1 ∂w
∂w
⫺z 2 , (⫺z2⬍z⬍z1).
⫹
∂x 2 ∂x
∂x
2
2
2
冋
冋
冉 冊
冉 冊
册
册
s1,x ⫽ E1
∂u 1 ∂w 2 ∂2w
⫺z 2 ⫹ sr1,
⫹
∂x 2 ∂x
∂x
s2,x ⫽ E2
∂u 1 ∂w 2 ∂2w
⫺z 2 ⫹ sr2, (⫺z2⬍z⬍z1),
⫹
∂x 2 ∂x
∂x
(7)
(z1⬍z⬍z2),
where sr denotes residual stress. The resultant force
and moment are consequently calculated
From equation (5), we have strains of
2
Using the constitutive equations and the strains in
equation (6) leads to the stresses
冕
z2
Nx ⫽ s1,x dz ⫹
(z1⬍z⬍z2),
(6)
z1
冕
z1
s2,x dz
⫺z2
冋
冉 冊册
∂u 1 ∂w
⫽A
⫹
∂x 2 ∂x
冕
2
⫺B
冕
冉 冊册
z2
∂2w
⫹ Nr ,
∂x2
z1
M ⫽ s1,xz dz ⫹
z1
(8)
s2,xz dz
⫺z2
冋
∂u 1 ∂w
⫽B
⫹
∂x 2 ∂x
2
⫺D
(9)
∂2w
⫹ Mr ,
∂x2
where A, B, D, Nr and Mr are given by
A ⫽ E1h1 ⫹ E2h2,
B ⫽ 12h1h2(E1⫺E2),
1
D ⫽ 12
[E1h1(h21 ⫹ 3h22) ⫹ E2h2(3h21 ⫹ h22)], (10)
Nr ⫽ sr1h1 ⫹ sr2h2,
Mr ⫽ 12h1h2(sr1⫺sr2).
A and B are called the tension stiffness and the tension–bending coupling stiffness, respectively, and Nr
and Mr are termed the residual force and the residual
moment, respectively. The tension stiffness and the
residual force are equivalent to equations (1) and (2),
respectively. The residual moment is null when the
residual stresses in the two layers are the same. Following the equilibrium equations and using equations
(8) and (9), we finally have
Dbi
Fig. 2. (a) Coordinate system for a bilayer beam; (b) mechanics
analysis of the bilayer beam; and (c) mechanics analysis of the
substrate.
∂4w
∂2w
⫺Nx 2 ⫽ q.
4
∂x
∂x
(11)
Dbi is given in equation (3) and has relations with A,
B and D through Dbi ⫽ D⫺B2/A. The governing equation for bending a bilayer beam has the same form
as that for bending a single-layer beam. Thus, equ-
SU et al.: MICROBRIDGE TEST
ation (11) can be solved in the same way as that for
a single-layer beam.
For a deformable substrate, the beam is subjected
to a moment M0 and a deflection w0 at the beam ends
(i.e. x ⫽ 0 and x ⫽ l, where l is the bridge length).
In this case, if the tranverse load q is a line load Q
per unit width applied at the beam center, the solution
to equation (11) is given by
Q sinh(kl/2)
Q
w⫽⫺
x
sinh(kx) ⫹
Nxk sinh(kl)
2Nx
冋
册
M̄0 sinh(kx) ⫹ sinh[k(l⫺x)]
⫺
⫺1 ⫹ w0
Nx
sinh(kl)
(12)
for 0ⱕxⱕl/2,
w⫽⫺
⫺
Q sinh(kl/2)
Q
sinh[k(l⫺x)] ⫹
(l⫺x)
Nxk sinh(kl)
2Nx
冋
册
M̄0 sinh(kx) ⫹ sinh[k(l⫺x)]
⫺1 ⫹ w0
Nx
sinh(kl)
for l/2ⱕxⱕl,
where
k ⫽ √Nx/Dbi, M̄0 ⫽ M0⫺(Nx⫺Nr)(B/A)⫺Mr.
(13)
It should be noted that if Nx⬍0, k is a pure imaginary number and complex calculations are required.
At the end x ⫽ 0 or x ⫽ l, the slope of the deflection
and the displacement along the x-axis are calculated
from equations (12) and (8), and given respectively by
4905
冤冥 冤
u
SNN SNP SNM
d ⫽ SPN SPP SPM
q
冥冤 冥
SMN SMP SMM
N
P
(16)
M
is applied herein, where Sij are the compliances. A
positive strain energy requires that the compliances
Sij be symmetric. For simplicity, the compliances Sij
are assumed to depend only on the base properties
and geometry and the beam thickness. The substrate
is much larger than the bridge beam. At each side of
the bridge the substrate has a truncated wedge shape
in the cross section perpendicular to the width direction, where we use semi-infinite elements in the farthest layer from the wedge tip to make the substrate
large. However, there is a finite dimension in the
width direction of the substrate, which is a finite gap,
W0, between two adjacent bridges, as marked on Fig.
3. In the previous work [13], we concluded that when
W0/WBⱖ4, where WB is the bridge width, the bridge
gap is large enough to approximately neglect the
interaction between bridges. In the following FEA,
W0/WB is chosen to be 16. To avoid undercutting of
the substrate, the bilayer beams were fabricated to
extend on the supporting substrate by about 5–10 µm,
as illustrated by l0 in Fig. 13. The difference in l0
induces a relative error of less than 1% for the deflection at the bridge center [13]. Thus, we take l0 ⫽ 0
in the FEA for simplicity. Using a Reuss average
Young’s modulus of 159 GPa and a Poisson ratio of
0.25 for silicon [35], Zhang et al. [13] have calculated
the spring compliances for different substrate support
angles and film thicknesses. For the present samples
with the substrate support angle of 54.74° and the
total film thickness of 1.96 µm, Sij are given below
SMN ⫽ 3.40⫻10⫺3 µm/mN,
q0 ⫽
k
Q[cosh(kl/2)⫺1]
⫹ M̄0 tanh(kl/2), (14)
2Nx cosh(kl/2)
Nx
SPM ⫽ 1.54⫻10⫺2 µm/mN,
SMM ⫽ 1.57⫻10⫺2 1/mN,
SNN ⫽ 2.87⫻10⫺2 (µm)2/mN,
冕冉 冊
1/2
u0 ⫽
1
2
∂w
∂x
2
B
l(Nx⫺Nr)
.
dx ⫹ q0⫺
A
2A
SNP ⫽ 3.00⫻10⫺2 (µm)2/mN,
(15)
0
2.2. Substrate deformation
Figure 2(c) shows the mechanics analysis of the
substrate (or base) deformation, which is modelled by
three coupled springs. Three generalized forces, i.e.,
two forces N and P along the x-axis and z-axis,
respectively, and a moment, M, act on the three
coupled springs and induce correspondingly three
generalized displacements, u, d, and q. A linear
constitutive equation
SPP ⫽ 9.39⫻10⫺2 (µm)2/mN,
SNM ⫽ SMN, SMP ⫽ SPM, SPN ⫽ SNP.
At the connecting point of the film and substrate,
the displacement continuity and force equilibrium
require
u ⫽ u0, d ⫽ w0, q ⫽ q0,
N ⫽ Nx⫺Nr, P ⫽ Q/2, M ⫽ ⫺(M0⫺Mr).
(17)
Combining equations (14)–(17) with equation (12)
yields the final solution.
4906
SU et al.: MICROBRIDGE TEST
Fig. 3. Finite element mesh of the microbridge.
2.3. Load–deflection relationship
Q
SNN(Nx⫺Nr) ⫹ SNP ⫺SNM(M0⫺Mr)
2
B
l(Nx⫺Nr)
⫽ I ⫹ q0⫺
,
A
2A
Letting x ⫽ l/2 in equation (12) leads to the bridge
deflection at the center where the lateral load is
applied
Q tanh(kl/2)
Ql
⫹
2Nxk
4Nx
M̄0
1
⫺
⫺1 ⫹ w0,
Nx cosh(kl/2)
w⫽⫺
冋
册
(18)
冋冉
冊
Q2
1
a2 ⫹ b2 ⫹ kl ⫹ 4(a
8kN2x
2
⫺b) sinh(kl/2) ⫹ 2abkl cosh(kl) ⫹ (a2
I⫽
(21)
册
(22)
⫹ 2ab⫺b2 ⫹ 4b) sinh(kl) ⫹ b2 sinh(2kl)
where w0, M0 and Nx are coupled in the following
equations.
Q
w0 ⫽ SPN(Nx⫺Nr) ⫹ SPP ⫺SPM(M0⫺Mr), (19)
2
with
a⫽⫺
sinh(kl/2)
M̄0k
M̄0k
⫺
,b⫽
.
sinh(kl) Q sinh(kl)
Q sinh(kl)
(23)
(M0 ⫺ Mr)[SMMNx ⫹ k tanh(kl/2)]
1
⫽SMNNx(Nx⫺Nr) ⫹ SMPQNx
2
⫹
冉冊
册
kl
kB
(N ⫺Nr)tanh
A x
2
冋
1
1
⫺1 ,
⫹ Q
2 cosh(kl/2)
(20)
Note that the tension stiffness A, the tension–bending
stiffness B and residual force Nr all appear in equation
(21), because equation (21) is the major equation to
determine the force Nx induced by the neutral plane
stretch due to large deformation. In this sense, the
neutral plane stretch is like a uniform tension and
thus, the equivalent modulus and residual stress by
the thickness average have to be used here.
From the above formulas for the deformable sub-
SU et al.: MICROBRIDGE TEST
4907
strate, it is a straightforward matter to arrive at
expressions of the deflection at the beam center for
the simply supported or built-in bridge ends. Letting
all the compliances Sij or all the compliances Sij and
(M0⫺Mr) be zero reduces the above formulas to those
for built-in ends or simply supported ends. Thus, the
center deflection of the bridge on a rigid substrate is
given by
Q tanh(kl/2)
2Nxk
1
Ql
B(Nx⫺Nr)
⫹
⫺1 ⫹
ANx
cosh(kl/2)
4Nx
for simply supported ends
w⫽⫺
冋
册
(24)
and
w⫽
冋
冉 冊册
kl
kl Q
⫺tanh
for built-in ends, (25)
4
4 Nxk
where
l(Nx⫺Nr) ⫽ 2AI ⫹ 2Bq0.
(26)
For small deformation, the neutral plane stretch of
the bridge induced by the lateral load is negligible
and hence Nx ⫽ Nr. Thus, the above formulas for
large deflection can be easily reduced to those for
small deformation,
w⫽
再
Q SPPNrk tanh(kl/2) kl
⫺
⫹
Nr k
2
2
4
冋
册
冎
SMPNr ⫹ [1/cosh(kl/2)⫺1]
2[SMMNr/k ⫹ tanh(kl/2)]
Young’s modulus, i.e., equation (3), is applicable for
bending under small deformation. One, therefore,
cannot simply use just an equivalent modulus to calculate the bending problem under large deformation,
wherein bending couples with neutral plane stretch.
for a deformable substrate,
2.4. Finite element analysis
⫺
1
⫺1 ⫹ SPMNr
cosh(kl/2)
冋
册
kl tanh(kl/2) Q
⫺
w⫽
4
2
Nrk
(27)
(28)
for simply supported ends,
w⫽
Fig. 4. Finite element analysis of the substrate deformation; (a)
deflection, and (b) normalized deflection difference.
冋
冉 冊册
kl
kl Q
⫺tanh
for built-in ends. (29)
4
4 Nr k
Equations (27)–(29) indicate that for small deformation, the central deflection w is a linear function
of the applied lateral load Q per unit width. In this
case, k ⫽ √Nr/Dbi, the load–deflection relationship is
independent of the tension stiffness A and the tension–bending stiffness B, and hence the bending-equivalent modulus is appropriate. No matter whether a
substrate is rigid or deformable, the thickness-average
Young’s modulus, i.e., equation (1), is applicable for
pure tensile testing, while the bending-equivalent
With the commercial software ABAQUS, we carried
out FEA to verify the above formulas. Figure 3 shows
the FEA mesh with three-dimensional solid elements.
A residual stress in each layer only along the length
direction of the bridge is applied as an initial condition
to all the elements comprising the bridge and its extension on the substrate. Three different boundaries, i.e.,
simply supported ends, built-in ends and deformable
substrate, are studied in FEA, using the input data
l ⫽ 100 µm, h1 ⫽ 1 µm, h2 ⫽ 1 µm, E1 ⫽ 200 GPa,
E2 ⫽ 60 GPa, sr1 ⫽ 300 MPa and sr2 ⫽ ⫺100 MPa.
Figures 4(a) and (b) show the FEA results. Figure 4(a)
indicates the deflection w at the bridge center as a function of the lateral load Q per unit width, while Fig.
4(b) gives the normalized difference in the deflection,
whereby the FEA result for the deformable substrate,
wd, is taken as a reference. As expected, the analytic
solution almost coincides with the FEA results, as
shown in Fig. 4(a) and (b). The maximum normalized
4908
SU et al.: MICROBRIDGE TEST
difference of the analytic solution is less than 2.2%,
indicating that the analytic solutions are almost as
accurate as the FEA results. Figure 4(b) also illustrates
that under small loads, the deflection difference with
the built-in ends is about 4% lower, while the deflection
difference is roughly 60% for the simply supported
ends. As the load increases, the deflection difference
almost approaches a constant for built-in ends. Under
large loads, e.g., Q ⫽ 0.3 mN/µm, however, the builtin ends generate about three times the normalized difference in the deflection in comparison with that produced by the analytic solution. In this case, the builtin bridge ends lead to underestimation of the bridge
deflection by 8%. For the simply supported bridge ends,
however, the normalized difference in the deflection
starts from about 60% and decreases with increasing Q
until Q ⫽ 0.12 mN/µm, and then it decreases with
increasing Q. Under large loads, the deflection with the
simply supported ends is closer than the deflection with
the built-in ends to the analytic deflection. Nevertheless, as discussed by Zhang et al. [13], a small difference in the deflection may lead to a large deviation in
Young’s modulus and residual stress.
3. SAMPLE PREPARATION AND TEST PROCEDURE
The microbridge samples were fabricated on two
4-in. p-type (100) silicon wafers with a thickness of
525 µm. Before deposition of thin films, the silicon
wafers went through a HF dip to remove the native
oxide. Silicon nitride films were formed on both
sides by LPCVD at 840°C. The gas pressure was
170 mTorr and the gas flow ratio between SiCl2H2
and NH3 was 6:1. The deposition rate was 3.5
nm/min and the final thickness of the films was
approximately 0.87 µm. Wafer I was only deposited
with the silicon nitride film and served as a reference. For wafer II, low-temperature silicon oxide
(LTO) films of 1.08 µm thickness were deposited
on the silicon nitride thin film by a commercial
LPCVD system at 425°C. The gas pressure was 110
mTorr and the gas flow ratio between SiH4 and O2
was 4:5. The deposition rate was 11.5 nm/min. After
that, wafer II was annealed at 900°C in an O2 atmosphere for 30 min to produce dense LTO/silicon
nitride films. In order to have a silicon nitride film
on wafer I that is as similar as possible to that on
wafer II, wafer I was annealed in N2 gas at 900°C
for 30 min. Next, all the nitride and bilayer thin
films on the two wafer backsides were patterned by
the photolithography technique and subsequently
etched away by plasma etching. Consequently, the
exposed silicon was etched in tetramethyl
ammonium hydroxide (TMAH) solution at a temperature of 80°C to make rectangular windows with
the designed dimensions. The TMAH solution was
utilized because it maximized the difference in etching rates between the (100) silicon and LTO and/or
silicon nitride. The etching rate of Si in the ⬍100>
direction was about 3.2 µm/min, while the etching
rate of LTO or silicon nitride was about 1 Å/min.
Finally, all the silicon nitride and bilayer thin films
on the two wafer frontsides were patterned using
photolithography and then dry-etched by plasma to
complete the fabrication of the microbridge structures. After the fabrication of the samples, we measured the thickness of LTO and silicon nitride with
an ellipsometer. The thickness of the silicon nitride
thin film on wafer I was 0.8756 µm, and the thicknesses of silicon nitride thin film and LTO on wafer
II were 0.8756 and 1.08 µm, respectively. All
microbridge samples had a width of 15.8 µm. The
gap between two adjacent bridges is about 66 µm
to meet W0/WBⱖ4. The bridge length was measured
sample by sample and ranged from 85 to 110 µm
for the bilayer films and from 75 to 89 µm for the
single-layer films. All single-layer and bilayer
samples were used in the microbridge tests and each
individual length was used in the fitting process.
The microbridge test was conducted on a Nanoindenter II system equipped with a wedge indenter. The
wedge indenter was made of diamond and had a width
of 20 µm, which is wider than the sample width so that
the one-dimensional analysis holds. The measurement
resolutions of the load and the vertical displacement
were respectively 0.25 µN and 0.3 nm. In the present
study, the load was applied continuously to a sample
at a rate of 30 nm/s until the sample fractured. All tests
were conducted at room temperature. Following the
approach for single-layer films [13], we first conducted
the microbridge test on 31 single-layer samples and
evaluated the mechanical properties for the silicon
nitride films. Then, we tested 28 LTO/silicon nitride
bilayer samples. The fracture morphology of the bilayer
samples was examined using a JEOL 6300 scanning
electron microscope (SEM).
Assuming that the mechanical properties of the silicon nitride films after the deposition of the LTO films
remain unchanged, we evaluated the Young’s modulus E2 and residual stress sr2 for the LTO films by
fitting experimental load–deflection curves with the
theoretical solution using the least square technique
to minimize the following positive function
冘
n
S⫽
[wei (Qi)⫺wti(Qi, sr2, E2)]2,
(30)
i⫽1
where n represents the number of data, and wei (Qi)
and wit(Qi, sr2, E2) are the experimentally observed
and theoretical predicted deflections, respectively.
Once the Young’s moduli and residual stresses for
both layers are known, the stress in each layer will
be determined by the following equations
s1(x, z) ⫽ E1
s2(x, z) ⫽ E2
冋冉 冊
冋冉 冊
B
⫺z
A
B
⫺z
A
册
册
∂2w Nx⫺Nr
⫹
⫹ sr1, (31)
∂x2
A
∂2w Nx⫺Nr
⫹
⫹ sr2,
∂x2
A
SU et al.: MICROBRIDGE TEST
where
Qk sinh(kl/2)
∂2w
sinh(kx)
⫽⫺
∂x2
Nx sinh(kl)
2
M̄0k sinh(kx) ⫹ sinh[k(l⫺x)]
⫺
.
Nx
sinh(kl)
In the next section we will calculate the maximum
stress in the LTO layer and the maximum stress in
the silicon nitride layer. Comparing the maximum
stress in the silicon nitride film with its bending
strength evaluated from the single-layer silicon nitride
samples leads to an understanding of the failure
behavior of the bilayer films.
4. RESULTS AND DISCUSSION
By conducting the microbridge test on the 31 single-layer silicon nitride film samples and using the
method for single-layer films [13], we determined the
Young’s modulus, residual stress and bending
strength. The means and associated standard deviations of Young’s modulus, residual stress and bending strength for the single-layer silicon nitride films
annealed at 900°C are 199.28±8.63 GPa,
315.99±30.76 MPa, and 6.87±0.61 GPa, respectively.
The Young’s modulus of 199.28±8.63 GPa and
residual stress of 315.99±30.76 MPa are correspondingly more or less the same as the Young’s modulus
of 202.57±15.80 GPa and residual stress of
291.07±56.17 MPa for the single-layer silicon nitride
films annealed at 1100°C [13]. However, the two
groups of samples differ greatly in their bending
strengths. Figure 5 shows the bending strength for
each individual sample of the single-layer silicon
nitride films annealed at 900°C, where one solid circle denotes one sample. Although the evaluated bending strength scatters from 5 to 8 GPa, its mean of
6.87±0.61 GPa is almost half of the bending strength
of 12.26±1.69 GPa for the single-layer silicon nitride
Fig. 5. The evaluated bending strength of the 31 samples of
the silicon nitride films annealed at 900°C, where one solid
circle stands for one sample.
4909
films annealed at 1100°C [13]. The reduction in bending strength may be due to surface defects, which
were formed during sample fabrication. Annealing at
1100°C could greatly reduce the surface defects compared with annealing at 900°C, and thus enhance the
bending strength. Nevertheless, the bending strength
of 6.87 GPa will be used as a reference for evaluating
the bending strength of the bilayer films. Since the
single-layer or bilayer film thickness is much smaller
than the substrate thickness, we may use the semiinfinite thick substrate approximation to discuss
whether there is a great influence of the LTO layer
on the residual stress in the silicon nitride layer of
the bilayer films. At this semi-infinite thick substrate
approximation, no residual stresses would be induced
in the substrate, no matter how large the difference
in the thermal expansion coefficients is between the
film and the substrate. Since all tests were conducted
at room temperature, we took the stress-free state at
room temperature as reference, where the LTO layer
and silicon nitride layer had different stress-free
dimensions from the substrate. We may first stretch
or compress the silicon nitride layer by apply a biaxial stress, sSiN, such that the silicon nitride layer
fits the substrate perfectly. In such circumstances, we
bond the silicon nitride layer to the substrate. After
the bonding, we release the applied bi-axial applied
stress. The constraint of the substrate to the silicon
nitride film generates a residual stress field in the film.
In the same way, we may bond two silicon nitride
films to two silicon substrates, one film on one substrate. If the two silicon nitride layers are identical
and the two silicon substrates are the same, the
residual stresses in the two silicon nitride layers
should be the same. After that, we would bond a LTO
layer on a silicon nitride/silicon substrate and leave
the other as reference. Similarly, we stretch or compress the LTO layer by applying a bi-axial stress,
sLTO, such that the LTO fits the silicon nitride layer
as well as the substrate perfectly. After bonding the
LTO layer to the silicon nitride layer, we release the
applied bi-axial stress ␴LTO. The constraint of the silicon nitride film to the LTO film actually comes from
the substrate, which generates a residual stress field
in the LTO film. From the virtual operation, we may
understand that the residual stress in the silicon
nitride layer of the bilayer films could have the same
value as that in the single-layer silicon nitride films.
Therefore, the Young’s modulus of 199.28±8.63 GPa
and residual stress of 315.99±30.76 MPa evaluated
from the single-layer silicon nitride films will be used
to extract the mechanical properties of the LTO films
from the microbridge tests on the bilayer films.
During the microbridge test, the samples of
LTO/silicon nitride bilayer films deformed elastically
until fracture occurred. Figure 6 illustrates the deflection
during loading and unloading. The fact that the unload–
deflection curve coincides with the load–deflection
curve indicates the completely elastic behavior of the
bilayer bridge and its silicon base underneath.
4910
SU et al.: MICROBRIDGE TEST
Fig. 6. Experimental loading and unloading curves of a bilayer
bridge.
Fig. 8. The evaluated Young’s moduli of the LTO film for
the 28 bilayer samples, where one solid circle stands for one
sample.
Fig. 7. Comparison of the theoretical and experimental load–
deflection curves.
Fig. 9. The evaluated residual stresses of the LTO film for the
28 bilayer samples, where one solid circle stands for one
sample.
Fitting the entire experimental load–deflection
curve with equation (18), we evaluate the Young’s
modulus E2 and the residual stress sr2 for the LTO
films for each specimen. As an example, Fig. 7 illustrates an experimental load–deflection curve and the
theoretical fitting. The fitting of the analytic solution
to the experimental curve is perfect and from the fitting we extract the values of the Young’s modulus
and the residual stress of the LTO film. The evaluated
Young’s moduli and residual stresses of the LTO film
are plotted separately against the bridge length in Figs
8 and 9. It is seen that the Young’s modulus and
residual stress are statistically independent of the
bridge length. This is conceivable because the
Young’s modulus and residual stress of the LTO film
should depend only on the film and substrate
materials and the sample fabrication process. The
mean Young’s modulus and residual stress with the
corresponding standard deviations are 41.00±3.60
GPa and ⫺180.88±7.90 MPa, respectively. The LTO
Young’s modulus of 41 GPa is very close to Weihs
et al.’s value of 44 GPa for their LTO films measured
from mechanical deflection of cantilever microbeams
[9]. The residual stress of ⫺180.88 MPa in the LTO
films is about a half of the residual stress of ⫺358
MPa, measured by the wafer curvature method, in the
wet-thermal silicon oxide films [7].
If we do not consider the substrate deformation and
fit the experimental data in Fig. 7 with equation (24)
for simply supported ends, we have the Young’s
modulus of 13.54 GPa and residual stress of ⫺73.68
MPa for the LTO film. The Young’s modulus and
residual stress evaluated by equation (24) for simply
supported ends are about one-quarter and one-third of
the corresponding values evaluated by equation (18)
for a deformable substrate. Fitting the experimental
data in Fig. 7 with equation (25) for built-in ends
results in the Young’s modulus and the residual stress
of 18.21 GPa and ⫺118.2 MPa, respectively, for the
LTO film, amounts which are roughly half of the corresponding values evaluated by equation (18).
Clearly, ignoring the substrate deformation would
undervalue the Young’s modulus by about 60–80%
and the residual stress by about 40–70%. Similar
SU et al.: MICROBRIDGE TEST
4911
results are obtained for all 28 samples. The mean
values with the standard deviations of the Young’s
modulus and residual stress are, respectively,
14.43±3.49 GPa and ⫺94.12±27.71 MPa if the formula for built-in ends are used, whereas the mean
values with the standard deviations of the Young’s
modulus and residual stress are, respectively,
6.37±4.25 GPa and ⫺45.11±22.78 MPa if the formula
for simply supported ends is used.
As mentioned above, the stress in each layer at the
fracture load is given by equation (31), from which
we can identify the maximum tensile stress in each
layer and its location. As an example, Fig. 10 shows
the stress distribution at the fracture load in the bridge
length direction, where su2 and s2l denote the stresses
along the upper and lower surfaces of the LTO layer,
respectively, while su1 and sl1 are along the upper and
lower surfaces of the silicon nitride layer, respectively. Due to the symmetry, the stress distribution is
plotted in Fig. 10 for a half bridge length from one
bridge end x/l ⫽ 0 to the bridge center x/l ⫽ 0.5. Figure 10 demonstrates that the maximum tensile stress
of the silicon nitride layer occurs at the lower surface
of the bridge center, whereas the maximum tensile
stress of the LTO film occurs at the upper surface
of the bridge ends. Consequently, we calculate the
maximum tensile stress at the fracture load for every
sample. Figures 11 and 12 show the maximum tensile
stresses under the fracture loads in the LTO and silicon nitride layers, respectively. For the microbridge
length range of 85–110 µm, these maximum tensile
stresses are independent of the bridge length. The
mean maximum tensile stresses are respectively
0.903±0.111 GPa in the LTO layer and
0.983±0.087 GPa in the silicon nitride layer. Failure
will occur once the maximum tensile stress in either
layer reaches its bending strength. As mentioned
before, the bending strength for the silicon nitride film
is 6.87 GPa which is much larger than 0.983 GPa.
Therefore, we can draw the conclusion that the LTO
layer at one of the bridge ends fractures first and then
the bilayer microbridge fails. This conclusion is supported by SEM observations of failed bilayer
microbridges, as shown in Fig. 13, indicating the fracture occurs at one end of the bridges. Thus, the
maximum tensile stresses in Fig. 11 represents a distribution of the bending strength of the LTO film with
the mean of 0.903±0.111 GPa. Figure 14(a) and (b)
Fig. 10. Stresses in upper and lower surfaces of each layer
distributed along the bridge length.
Fig. 13. SEM picture showing that the fracture of the bilayer
microbridges occurred at one end of the bridges.
Fig. 11. The maximum tensile stresses in the LTO film under
the fracture loads for the 28 bilayer samples, where one solid
circle stands for one sample.
Fig. 12. The maximum tensile stresses in the silicon nitride
film under the fracture loads for the 28 bilayer samples, where
one solid circle stands for one sample.
4912
SU et al.: MICROBRIDGE TEST
Fig. 15. The slope of deflection to load for the bilayer samples
under small deformation, where one solid circle stands for one
sample.
Fig. 14. (a) SEM picture showing the fracture surface of a
bilayer bridge, where the sample was left as it was. (b) SEM
picture showing the fracture surface of a bilayer bridge, where
the sample was etched by HF solution for 5 s.
are SEM pictures of the morphology of the fracture
surface at the bridge end, where the fractured sample
in Fig. 14(a) is as it was, while the sample in Fig.
14(b) was etched by HF solution for about 5 s. Figure
14(a) and (b) show a flat interface between LTO and
silicon nitride films. The bonding between LTO and
silicon nitride seems good because one cannot detect
the interface without etching. Figure 14(b) illustrates
some surface cracks in the LTO film, indicating again
that the failure starts from the LTO film.
When the load is small, e.g. smaller than 0.007
mN/µm, the load–deflection relationship is approximately linear. In the present study, we linearly
approximate the load–deflection relationship within
the deflection range from zero to a third of the
bridge thickness, or in other words, we approximately treat the load–deflection curve in this range
as a straight line. Then, the slope p for each specimen is determined from the experimental data. Figure 15 shows the slope p as a function of the film
length l. As described in the previous paper for single-layer bridges, the slope of deflection over load
is approximately proportional to the bridge length
when klⱖ9 [13]. This relationship holds for bilayer
bridges. It is seen in Fig. 15 that p is statistically
proportional to l, because kl for all specimens in the
present study is larger than nine. We cannot evaluate both the Young’s modulus and residual stress
from a single load–deflection line or a single slope
p. However, we can evaluate both the Young’s
modulus and residual stress from many load–
deflection lines or many p’s for different bridge
lengths because both the Young’s modulus and
residual stress are independent of the bridge length.
Thus, we apply equation (30) together with equation
(27) to fit all the experimental data within the range
of deflection less than a third of the film thickness,
or apply the least square technique to fit all the p’s
in Fig. 15. It turns out that the evaluated Young’s
modulus and residual stress are respectively 36.81
GPa and ⫺147.76 MPa. The evaluated Young’s
modulus from small deformation is almost the same
as the average value obtained from large deformation, while the evaluated residual stress from
small deformation is higher than the average value
from large deformation by about 30 MPa. The difference between the values of large and small deformation models for the bilayer films is similar to that
for the single-layer films [13]. Again, if the substrate deformation is not considered, the evaluated
Young’s modulus and residual stress would be
389.32 GPa and ⫺138.43 MPa for the simply supported ends, and 32.48 GPa and ⫺138.43 MPa for
built-in ends. Note that for small deflection, if an
analytical model with simply supported ends is
used, the Young’s modulus of the film could be
overvalued by about one order of magnitude. The
built-in ends undervalue the Young’s modulus by
10 GPa. Figure 16 magnifies the small deflection
region in Fig. 9, where the small deflection calculation uses equation (27) and the same evaluated
Young’s modulus and residual stress. The result
SU et al.: MICROBRIDGE TEST
4913
on to study surface and interface stresses using the
microbridge testing method.
Acknowledgements—This work is fully supported by an RGC
grant (HKUST6013/98E) from the Research Grants Council of
the Hong Kong Special Administrative Region, People’s
Republic of China. The experiments were conducted at the
Microelectronics Fabrication Facility and the Advanced Engineering Materials Facility, HKUST.
APPENDIX A RESIDUAL DEFLECTION
Fig. 16. Comparison of the theoretical prediction and the
experimental load–deflection curve under small deformation.
shows that when the deflection is less than 0.6 µm,
the small deflection theorem works well.
5. CONCLUDING REMARKS
The microbridge testing method is superior in the
simultaneous characterization of the Young’s modulus,
residual stress and bending strength of thin films. The
present study shows that it can also be applied to multilayer films after modifying the previous analytic solution [13]. The analytic solution provides a closed-form
formula of the load–deflection relationship under large
or small deformation for films deposited on deformable
or rigid substrates with tensile or compressive residual
stress in each layer. Under large deformation, bridge
deflection couples with the neutral plane stretch such
that both the tensile-equivalent and bending-equivalent
Young’s moduli are involved. Under small deformation, one may use the bending-equivalent Young’s
modulus and the residual force, or the tensile-equivalent
residual stress to reduce the bilayer bending problem
to that of single-layer beams. The analytic solution has
been verified by the FEA results. As a typical elastic
system, the annealed LTO/silicon nitride bilayer films
have been used to demonstrate the proposed testing
method. To apply the theoretical framework to experiments, one has to assume that the Young’s modulus
and residual stress of a layer remain unchanged upon
the deposition of the second layer. This assumption
may hold for such a film/substrate system, wherein the
substrate is much thicker or much more rigid than the
film. As mentioned in the “Introduction”, surface and
interface stresses may play an important role and have
not been considered in the present study. When the two
layers have almost the same thickness and more or less
the same surface stress, the effect of surface and interface stresses on the bending behavior of bilayer
microbridges may be approximately ignored. Following
the same methodology as described here, we will go
In the text, we analyze the bilayer bridge under an
applied lateral load, wherein the deflection refers to
the state without applied load. This analysis is appropriate when the residual deflection is negligible. Here,
we estimate the residual deflection induced by the
residual stresses in the two layers. In this case, the
stress-free state is taken as reference, where layers 1
and 2 have different original stress-free lengths from
the bridge length. We may stretch layers 1 and 2 by
applying stresses s01 and s02, respectively, in the xdirection, up to the bridge length, bound them along
the interface, and then bound the bilayer ends to the
substrate. After that, we release the applied stresses
to introduce residual stresses in the two layers, as well
as residual moment and residual deflection of the
bilayer beam.
Using the same coordinate system as that shown in
Fig. 2 and letting the displacement along the x-axis
at the middle of the beam be zero, we express the
displacements in the bilayer beam as
w(1)(x, z) ⫽ w(2)(x, z) ⫽ w(x),
u(1)(x, z) ⫽ u(x)⫺z
u(2)(x, z) ⫽ u(x)⫺z
∂w s01
⫹ (x⫺l/2),
∂x
E1
(z1ⱮzⱮz2),
∂w s02
⫹ (x⫺l/2), (⫺z2ⱮzⱮz1).
∂x
E2
(A1)
Stresses associated with the displacements are
deduced from Hook’s law and given by
冋
冋
册
册
s(1)x ⫽ E1
∂u ∂2w
⫹ s01,
⫺z
∂x ∂x2
s(2)
x ⫽ E2
∂u ∂2w
⫹ s02, (⫺z2ⱮzⱮz1).
⫺z
∂x ∂x2
(z1ⱮzⱮz2),
(A2)
Consequently, we have the resultant force and
moment in the bilayer beam
∂u ∂2w
Nx ⫽ A ⫺B 2 ⫹ N0,
∂x
∂x
(A3)
∂u
∂2w
M ⫽ B ⫺D 2 ⫹ M0,
∂x
∂x
(A4)
where
N0 ⫽ s01h1 ⫹ s02h2,
(A5)
4914
SU et al.: MICROBRIDGE TEST
M0 ⫽ 12(z22⫺z21)(s01⫺s02) ⫽ 12h1h2(s01⫺s02).
The equilibrium equations of membrane force,
moment and shear force are respectively given by
∂N
⫽ 0,
∂x
(A6)
∂2M
⫽ 0,
∂x2
(A7)
∂M
.
∂x
(A8)
T⫽
Since only the resultant force Nx along the x-direction
exists, equation (A6) indicates that Nx does not
change with x. Combining equations (A3) and (A4)
leads to
∂2w B
M ⫽ ⫺Dbi 2 ⫹ (Nx⫺N0) ⫹ M0.
∂x
A
(A9)
(A10)
where c1 and c2 are constants to be determined. The
solution satisfies the following symmetric condition
at x ⫽ l/2,
∂w
∂3w
⫽ 0, T⬅⫺Dbi 3 ⫽ 0.
∂x
∂x
(A11)
Then, the deflection and the slope are expressed at
the end of x ⫽ 0
l2
d ⫽ c1 ⫹ c2, q ⫽ ⫺lc2.
4
(A12)
Substituting equation (A10) into equation (A3), we
solve the displacement, u,
u⫽⫺
l
(2Bc2 ⫹ Nx⫺N0).
2A
冊 冉
(A13)
Using equations (A10), (A3) and (A8), we have the
moment and shear force
B
M ⫽ ⫺2Dbic2 ⫹ (Nx⫺N0) ⫹ M0, T ⫽ 0.
A
(A14)
Both the moment, M, and the shear force, T, are independent of x.
Since the substrate is deformable, we use the spring
constitutive equation, i.e., equation (16) in the text,
and the linking relationship, i.e., equation (17) in the
text, to determine Nx, c1 and c2. Note P ⫽ T ⫽ 0 in
this case. Thus, we have
冊
B
l
lB
SNN⫺ SNM ⫹
N ⫹ 2DbiSNM ⫹ SNM c2
A
2A x
A
l BSNM 0
0
⫽
N ⫹ SNMM ,
⫺
2A
A
B
SMN⫺ SMM Nx ⫹ (2DbiSMM ⫹ l)c2
(A15)
A
BSMM 0
⫽⫺
N ⫹ SMMM0,
A
BSPM
l2
c1 ⫽ SPN⫺
Nx ⫹ 2DbiSPM⫺ c2
A
4
BSPM 0
⫹
N ⫺SPMM0.
A
冉
冉
冊
冊
冉
冊 冉
冊
The residual deflection at the beam center is only
given by c1. We estimate the value of the residual
deflection at the beam center with the data used in
the text and have
w ⫽ ⫺0.86⫻10⫺4 (µm)
Substituting equation (A9) into equation (A7) and
then solving it yields a deflection of
w ⫽ c1 ⫹ c2(x⫺l/2)2,
冉
(A16)
The residual deflection is about four orders of magnitude smaller than the deflection caused by the applied
load, indicating it is negligible.
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