Submitted to IOP, J. Micromech. and Microeng. by Nan Wang et al.
Cover Page
Methods for Precisely Controlling the Residual Stress
and Temperature Coefficient of Frequency of a MEMS
resonator based on AlN cavity-SOI Platform
Nan Wang1, Jinghui Xu1,*, Xiaolin Zhang1, Guoqiang Wu1, Yao Zhu1, Wei Li2, and
Yuandong Gu1
1
Institute of Microelectronics, A*STAR (Agency for Science, Technology and
Research), 2 Fusionopolis Way, #08-02 Tower A (Innovis), Singapore 138634
2
Exploit Technologies Pte Ltd, A*STAR, 30 Biopolis Street, #09-02 Matrix,
Singapore 138671
*Corresponding Author: Jinghui Xu
E-mail: [email protected]
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Submitted to IOP, J. Micromech. and Microeng. by Nan Wang et al.
Abstract:
In this paper, we report an experimentally verified numerical model developed for
precisely predicting and controlling the initial bending of a multi-layer-stack
composite cantilever structure which is caused by the residual stress of the individual
constituting layers, as well as the cantilever’s thermal coefficient of frequency (TCF).
The developed model is exemplified using a flexural-mode cantilever resonator
according to
the process
flow of
the
aluminium
nitride (AlN) cavity
silicon-on-insulator (SOI) platform. The same AlN cavity SOI platform is also utilized
to microfabricate the exemplified cantilever, which is then used to experimentally
verify the accuracy and consistency of the numerical model. The experimental results
show a difference less than 3.5% as compared with the numerical model,
demonstrating the accuracy of the developed numerical model and the feasibility to
optimize the cantilever’s initial deflection and TCF simultaneously, achieving
minimum values for both parameters at the same time.
Index Term: AlN, MEMS resonator, Multilayer composite, Cavity-SOI platform,
Residual stress, Thermal coefficient of frequency (TCF).
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Submitted to IOP, J. Micromech. and Microeng. by Nan Wang et al.
1. Introduction
Frequency reference oscillators, which exist in almost all electronic systems, have two
major applications, namely the consumer electronics and the telecom/networking
industries. The classic crystal oscillator is facing challenges from an alternative
approach based on microelectromechanical systems (MEMS) resonator technology,
which decreases the size and cost of the fabricated resonators while enhances the level
of system integration [1, 2]. As a result, MEMS based oscillators are attracting more
research attention because of their capability of electronics integration [3]. As
compared with quartz crystal, the silicon-integrated micromechanical resonators
based on capacitive transduction mechanism [4-9] can be fully integrated with
electronics without the need of external circuitry, the size and cost of the resonators
are therefore greatly reduced. However, due to the weak capacitive electro-acoustic
coupling, most capacitive resonators suffer from high motional impedance and in
order to reduce it, the transduction gap needs to be narrow, which brings extra
challenge to fabrication process and subsequently increases the cost back. On the
contrary, piezoelectric-based [10-20] resonators can have lower motional impedance
and therefore gaining much research attention as well. Researchers have paid a lot of
effort to optimize the performance of the resonators, including optimizing the
properties of the piezoelectric material [21], controlling the thermal properties of the
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Submitted to IOP, J. Micromech. and Microeng. by Nan Wang et al.
fabricated resonator [22-26], as well as modelling and optimizing the residual stress
and the tip deflection of the stack-layer structures [27-32], because residual stress in
in various layers of constituting materials can induce large initial static deflection of
the cantilever and affect bring the challenge for the following packaging process by
increasing the package thickness. These approaches have been proven to be effective
in optimizing the performance of the fabricated resonators, with the optimization of
the residual stress and the thermal behaviour of the resonators being especially
important for the long-term reliability and thermal stability of the fabricated
resonators operating in environments with fluctuating temperatures. In addition, the
residual stress is also an important contributing factor for frequency drift. The
frequency shift caused by residual stress is typically over 100ppm, which does not
meet the application requirement of timing devices. Moreover, overall residual stress
is the major source to cause aging of the devices, which is detrimental to the devices’
long-term reliability. Therefore, optimizing the residual stress and the thermal
behaviour of the resonators is crucial to resonators for timing applications.
In this paper, we report a numerical calculation method for predicting and precisely
controlling both the residual stress and the thermal coefficient of frequency (TCF) of
a multi-layer cantilever resonator according to the process flow of our in-house
aluminium nitride (AlN) cavity silicon-on-insulator (SOI) platform. The same AlN
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Submitted to IOP, J. Micromech. and Microeng. by Nan Wang et al.
cavity SOI platform is also utilized to microfabricate the cantilever, which is then
used to experimentally verify the accuracy and consistency of the numerical model.
With its predicting accuracy experimentally verified, the numerical model can help
designers to design multi-layer cantilevers with desired residual stress and TCF, as
well as and minimum initial cantilever deflection, which subsequently lead to
expected uniformity, long-term stability and reliability, and its thermal performance.
One is to note that the model developed is focused on multi-layer cantilever structures,
with its one end fixed to the substrate and assumed not moving.
In section 2 of this manuscript, the AlN cavity-SOI platform’s consisting layer stack
and materials, which is used in both the predicting numerical model and actual
microfabrication of the cantilever resonator, as well as its brief fabrication process, is
described. Section 3 first explains the numerical method to derive the position of the
neutral axis of a multi-layer composite cantilever, which builds the foundation for
calculating the effective Young’s modulus of the composite cantilever in the later part
of this section. Using the formula derived in section 3, section 4 discusses the
calculation results of the displacement at the tip of a bending cantilever due to
residual stress, which is also verified by FEM simulation results and experimental
data. Similarly, methods to minimize the displacement at the tip of a bending
cantilever are also discussed in this section. Section 5 derives the numerical methods
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to quantitatively predict the TCF of a multi-layer composite cantilever and
subsequently shows that by varying the thickness of the top SiO2 layer, the TCF for
both temperature ranges, i.e., the entire temperature range of calculation fromo -50°C
to 300°C and the typical device-working temperature range from -40°C to 85°C, can
be quantitatively tuned and precisely controlled down to sub-ppm level, satisfying the
strict requirement for the timing device applications.
2. The AlN cavity-SOI platform
The fabrication process of the AlN Cavity-SOI platform starts with etching the
predefined cavity with a depth of 20 µm on a bare silicon wafer, which is then bonded
to another SOI wafer with 1 µm BOX and 5.0 µm structural Si to form the cavity-SOI
wafer. Then a stack of AlN seed layer/bottom Mo/AlN/top Mo is deposited
sequentially and the top Mo is patterned using the SiO2 as a hard mask. The thickness
of the four layers is 20 nm, 0.2 µm, 1 µm, and 0.2 µm, respectively. Next, in order to
isolate Aluminum connection line and top Mo electrode, a layer of SiO2 with a
thickness of 0.7 µm is deposited and patterned to serve as the inter layer dielectric.
Next, a layer of Al with thickness of 0.7 µm is deposited and patterned, followed by
the deposition of a layer of SiO2, which is going to serve the purpose as the hard mask
to etch the AlN/Mo/AlN/ stack to get the active structure released. Lastly, the device
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Submitted to IOP, J. Micromech. and Microeng. by Nan Wang et al.
is completely released by dry etching the silicon and buried oxide at the release
trench, which is also predefined.
Fig. 1(a) shows the cross section of a typical cantilever structure fabricated by the
AlN on cavity-SOI platform, which can be simply divided into three parts: 1)
substrate with a cavity on the top side, with the depth of the cavity being 20 µm; 2) Si
support layer above the cavity based on fusion bonding between thermal SiO2 and Si;
and 3) Mo/AlN/Mo stack, which is the functional layer of the platform. The exact
thickness of each layer of the whole stack is depicted in Fig. 1(b), while Fig. 1(c)
shows overview of the typical cantilever structure based on the AlN on cavity SOI
platform. Fig. 2 shows the SEM image of a fabricated cantilever resonator with its
residual stress not optimized, as evidenced by the large initial static deflection. As
discussed previously, reducing the initial static deflection by reducing the residual
stress of the composite multi-layer cantilever will help with the fabrication process by
reducing the gap between the cantilever resonator and the cavity under the cantilever,
and also ease the subsequent packaging process by reducing the package thickness.
3. Neutral Plane and Effective Young’s modulus of a multi-layer
composite structure
3.1 Neutral Plane position
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A composite multi-layer-material cantilever structure is considered as an example.
The cantilever is of beam structure, with one end of the beam fixed to the anchor and
thus not able to move during operation. The top view of the cantilever is of a
rectangular shape. The moment of each individual layer is related to its curvature, 1/ρ,
which is defined as the flexural rigidity of the ith layer in the beam relative to the
beam’s neutral axis, as given in Eq. (1) below, where Mi is the bending moment of the
ith layer, E is the effective Young’s Modulus and I is the effective moment of inertia:
1


M
EI
i
(1)
And the displacement at the tip of the cantilever due to residual stress induced
bending can be expressed by Eq. (2), where disp is the displacement at the tip of the
cantilever, L is the length of the cantilever, ρ is the radius of the curvature, and θ is the
angle of the intersection of the two lines which are tangential to the two ends of the
cantilever, respectively, as illustrated in Fig. 3.
2
L
L2 M 
L2 M 2 
 (2)
disp  L sin(  / 2)  L  /   
 1 
2
2 EI  24 E 2 I 2 
Therefore, based on Eq. (2), in order to suppress the initial static deflection of the
cantilever due to stress induced initial deformation, i.e., to make disp minimum or
close to zero, the total bending moments of all the constituting layers relative to the
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Submitted to IOP, J. Micromech. and Microeng. by Nan Wang et al.
neutral axis caused by residual stress need to be as small as possible. In other words,
we need make the moment cancel each other. To solve these moments, it is necessary
to find the location of neutral axis and effective Young’s modulus of the composite
cantilever beam.
The neutral axis is the axis in the cross section of a beam where the bending stress and
hence the strain is zero. For a multi-layer composite structure, the location of the
neutral axis is determined by both the stiffness and the geometrical dimensions of
every individual constituting material sections.
Taking a two layer composite cantilever structure, whose schematic is shown in Fig.
4, as an example, with Fig. 4(a) showing the x-y plane of the cross section and Fig.
4(b) showing the y-z plane of the cross section. Since the stress distributed along the
entire cross section has to be in equilibrium, the sum of the forces in the x-direction is
zero. Therefore, we can have:
F
x
 0   dA    1dA    2 dA
A1
A2
(3)
where Fx is the force applied along x-direction; σ, σ1, σ2 are the overall stress, stress
on the first layer of material, stress on the second layer of material, respectively;
similarly, A is the area of the cantilever. Since the radius of curvature, ρ, determines
the bending stress in any section, as given by σ = -Ey/ρ, where E is the Young’s
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Submitted to IOP, J. Micromech. and Microeng. by Nan Wang et al.
modulus, Eq. (3) can be replaced by Eq. (4) below:
0

A1
 E1
y1

dA 

A2
 E2
y2

dA
(4)
where y is the distance from neutral axis to the considered position. Note that the radii
of curvature for various layers are approximately equal to one another, since it was
assumed that the beam thickness is much less than the overall beam curvature. Based
on the assumption, Eq. (4) can be simplified to
0  E1

A1
y1dA  E2

A2
y2 dA
(5)
Assuming the neutral axis is located in the bottom material, i.e. h < h1 as shown in
Fig. 4 (b), then:
E1

h
0
y1b1dy1  E1

h1  h
0
y1b1dy1  E2

h2  h1  h
h1  h
y2b2 dy2
(6)
Therefore,
E1b1h12  E2b2 h22  2E2b2 h2 h1
h
2E1b1h1  2E2b2h2
(7)
In Eq. (7), b1, b2 is the width along z-direction of the first and second material,
respectively. Similarly, assuming the neutral axis is located in the top material, i.e.
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Submitted to IOP, J. Micromech. and Microeng. by Nan Wang et al.
h1+h2>h>h1, we can get unified neutral axis expression as in Eq. (7). Furthermore, for
multi-layer structure such as the structure of AlN cavity-SOI platform, we can get the
neutral axis formula as below, using the principle and mathematical induction
methodology:
N
h

N
Ei bi hi2
 2
i 1
 E b h (h
i 1
i i i
 hi  2    h1 )
i 2
N
2
E b h
i i i
i 1
(8)
Where h is the distance of neutral axis to the bottom surface of multi-layer structure,
Ei, bi and hi are the Young’s modulus, the width along z-direction and thickness along
y-direction of the ith layer of material, respectively, while N is the number of
constituting layers of the composite structure.
3.2 Effective Young’s Modulus
The bending stress in any multi-layer composite beam structure can be found by using
the equation of moment equilibrium at any of its internal locations. Again, using the
assumption in Eq. (3), summing up all the moments gives:
M
z
0
  ydA  M  
i
i
 1 ydA 
A1
11

A2
 2 ydA  M
(9)
Submitted to IOP, J. Micromech. and Microeng. by Nan Wang et al.
Also, since the radius of curvature, ρ, determines the bending stress, which is given by
σ = -Ey/ρ, Eq. (9) can be simplified into:
M
E1


y 2dA 
A1
E2


y 2 dA
A2
(10)
It is to be noted that the integral is the area’s second moment, which is also the area
moment of inertia, and the inertial moment is the moment around the neutral axis, i.e.
the neutral axis is the coordinate reference axis. Therefore, Eq. (10) can be simplified
to:
M
E1

I1 
E2

I2
(11)
Rearranging Eq. (11) gives:

E1I1  E2 I 2 EI

M
M
(12)
Here, E is the effective Young’s modulus of the multi-layer composite structure, I is
the effective moment of inertia of the composite beam and given by I=I1+I2.
Substituting the expression for I into the Eq. (12), we can get the effective Young’s
modulus of composite structure as:
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E
E1I1  E2 I 2
I or ( I1  I 2 )
(13)
Similar to section 3.1, extending Eq. (13) to the composite structures with N layers of
materials, the Young’s modulus of multi-layer composite structure can be expressed
as:
E
E1I1  E2 I 2    EN I N
I or ( I1  I 2    I N )
(14)
Applying the Eq. (8) and Eq. (14) to the AlN cavity-SOI platform, and substituting the
material properties listed in Table 1 [33] to these two equations, we can get the neutral
axis location at h ≈ 4.67um and the effective Young’s modulus being 129.52GPa.
Table 1: Common material properties used to calculate the neutral plane position and the
effective Young’s Modulus
Young’s
Layer
Layer
Modulus [Pa]
thickness [µm]
width [µm]
BOX
7.00×1010
1.00
100
Structural Si
1.69×1011
5.00
100
Material
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Bottom Mo
3.29×1011
0.20
100
AlN
3.40×1011
1.00
100
Top Mo
3.29×1011
0.20
100
Top SiO2
7.00×1010
1.00
100
4. Stress induced initial deformation modelling and optimization
4.1 Calculation of displacement at the tip of a bending cantilever due to residual stress
From Equation (9), the bending moment of a cantilever can be re-written as
y2
M i    i wi ydy
y1
(15)
Where Mi is the moment caused by the residual stress σi in layer i, w is the width of
the cantilever, y1, y2 are the coordinates of the bottom and top surface of the ith layer.
In the aforementioned AlN cavity-SOI platform, the residual stress mainly exists in
the bottom thermal SiO2 layer and the piezoelectric AlN layer. Using the related
residual stress measurement results for each layer shown in Table 2, the displacement
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at the tip of a bending cantilever can be calculated, with the calculation results of tip
displacement for different cantilever lengths shown in Fig. 5.
Table 2: Measured residual stress value for the constituting layers of the fabricated AlN
on cavity-SOI platform
Material
BOX
Si
Mo
AlN
Top SiO2
Measured stress [MPa]
-350
0
300
50
-10
The calculation results shown in Fig. 5 are also verified by FEM simulation and
experimental demonstration using the same stack information of the AlN on
cavity-SOI platform. For example, when the length of the cantilever is 200µm, a
5.12µm of displacement at the tip of the bending cantilever is obtained from Fig. 5.
Meanwhile, the same design and process information, including the measured residual
stress value, dimension of the cantilever (200µm × 100µm), material properties and
thickness of the constituting layers, as well as the stack information is used to perform
numerical simulation using Coventorware. The simulation model is built by extrusion
of the mask information and subsequently imported into the analyzer. Meshing is then
performed after the solid 3D model is built, with the mesh type being ‘extruded
bricks’, meshing algorithm being ‘Pave, Qmorph’ and element order being
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“parabolic”. An element size of 10µm in the planar direction and 1µm in the extrude
direction is chosen and the biasing is set to be ‘none’. The side surface of the anchor
is mechanically fixed in all directions in the boundary condition setting. Another
boundary condition is that 1V is applied to the top electrode while 0V is applied to the
bottom electrode. ‘MemMech’ is chosen to be the analysis type and the physics is set
as ‘Piezoelectric’. Simulation results show that the displacement of the bending
cantilever tip is 5.4µm, which is in excellent agreement with the calculated value.
Moreover, after the above cantilever resonator (200µm × 100µm) has been fabricated
on the AlN on cavity-SOI platform, the profile of the cantilever is inspected using a
Wyko Veeco surface profiler and the measurement results show a 5.3µm displacement
takes place on the tip of a cantilever of 200µm × 100µm based on the aforementioned
AlN on cavity-SOI platform, which matched the analytical and simulation results very
well with a relative error <3.5%.
4.2 Methods to minimize the displacement at the tip of a bending cantilever
In most of the cantilever based MEMS applications, minimum initial bending of the
cantilever is preferred, i.e., the displacement at the tip of the cantilever is preferred to
be minimum. There are several methods to optimize the initial bending of the
cantilever by reducing the displacement of the tip: To control the residual stress of
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each individual constituting layer of the composite cantilever during the deposition
process, to change the location of the neutral axis by changing the thickness of the
constituting layers, or the combination of the two methods above.
For example, assume that the residual stress of the BOX and Top SiO2 is fixed to be
-350MPa and -10MPa, respectively, as indicated in Table 2, we can tune the residual
stress of AlN stack to achieve as small deflection as possible by calculating the
multi-layer structure bending moment relative to neutral axis with the help of solving
equation (15):
M BOX  w( y22  y12 ) / 2  1.44  10 7
(16)
M TopSiO2  w( y22  y12 ) / 2 = -3.31×109
(17)
Assuming the stress of AlN and Mo as σAlN and σMo, respectively, thus
M AlN   AlN w( y22  y12 ) / 2   AlN  2.136  10 16 (18)
M Mo   Mo w( y22  y12 ) / 2   Mo  8.21  1017
(19)
Therefore, in order to avoid the initial deflection, the total residual stress induced
moment should be zero. Then we can have
M BOX  M TopSiO2  M AlN  M Mo  0
17
(20)
Submitted to IOP, J. Micromech. and Microeng. by Nan Wang et al.
and the relationship between σAlN and σMo can be expressed according to equation (21)
below by substituting all the known values into equation (20):
σAlN = -0.66GPa - 0.384 σMo
(20)
In this case, as long as the residual stress of the AlN layer and the Mo layer after
deposition can be controlled to the relationship described in equation (20), the total
bending moment of the composite cantilever is zero with respect to the neutral axis
and thus, the displacement at the tip of the cantilever is zero and zero deflection of the
cantilever based on the AlN on cavity-SOI platform can be achieved. Similarly,
controlling the relationship among the thickness of the constituting layers of the
platform can also help to achieve zero initial deflection of the cantilever and the
detailed optimization steps will not be discussed here for brevity.
5. TCF prediction and control
TCF, or the temperature coefficient of frequency, refers to the relative change in the
resonant frequency of a resonator in association with a given change in temperature.
Temperature fluctuation usually causes the variation in certain material properties,
e.g., Young’s modulus, due to thermal expansion or compression. The variation in
Young’s modulus induced by temperature change directly drifts off the resonant
frequency of a MEMS resonator since the Young’s modulus is one of the factors
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determining the resonant frequency. Therefore, for resonators for timing applications
which require extreme frequency precision, the temperature effect on the resonant
frequency needs to be well compensated and minimized.
Generally, materials such as Si exhibit negative TCF, in which the Young’s modulus
of the material decreases as temperature increases. While materials such as SiO2
exhibit positive TCF, in which Young’s modulus will increase as temperature
increases. Specifically, the TCF of MEMS resonators based on silicon only is around
-30 ppm/°C, or 3750 ppm over the entire temperature range of -40 °C to 85 °C, which
does not satisfy the requirement of timing applications and compensation of the TCF
is required[34, 35]. Since SiO2 exhibits a positive TCF and Si exhibits a negative
TCF, one of the common passive temperature compensation approaches is add a layer
of SiO2 which is stacked parallel with the structural silicon layer to reduce the overall
TCF of the composite resonator [35].
Essentially, the basic principle of passively compensating the TCF using an extra
layer of SiO2 is to keep the equivalent Young’s modulus of the multi-layer composite
resonator unchanged with the variation in temperature. Since the length and the width
of most MEMS cantilever resonators are much larger than the thickness, it is valid to
assume that the change of temperature will not change the neutral axis position.
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Therefore, the second inertial moment of all layers also will not change.
Expanding Equation (14) by considering the temperature coefficient of elasticity, we
can get the expression for the equivalent Young’s modulus as equation (21):
E
E1 I1  E2 I 2    E N I N
I or ( I1  I 2    I N )
 E01  (1   011T   01 2 T 2 ) I1 
  E0 n  (1   0 n 1T   0 n  2 T 2 ) I N

I or ( I1  I 2    I N )
 (21)
where α01-1 is the 1st order temperature coefficient of elasticity of the first layer, and
α01-2 is the 2nd order temperature coefficient of elasticity of the first layer. The related
material properties to calculate the effective Young’s modulus and subsequently TCF
are shown in Table 3 [33] below.
Table 3: Material properties to calculate TCF of a multi-layer composite cantilever
resonator based on the AlN on cavity-SOI platform
AlN
Si
SiO2
Mo
elastic
constants, cij
[GPa]
c11
c12
c13
c33
c44
c66
410.06
100.69
83.82
386.24
100.58
154.70
170
70
329
1st order TCE
Tc11
-10.65
-63
204
-117
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Tcij [10-6/K]
Tc12
Tc13
Tc33
Tc44
Tc66
-11.67
-11.22
-11.13
-10.82
-10.80
2nd order TCE
T2cij [10-9/K2]
T2c11
T2c12
T2c13
T2c33
-20.61
-19.51
-19.88
-20.03
T2c44
T2c66
-20.36
-20.39
piezoelectric
stress coef., eij
[C/m]
e15
e31
e33
relative
permittivity, εij
CTE, αij
[10-6/K]
density, ρ [kg/m3]
-52
221
-24.9
-0.48
-0.58
1.55
-
-
-
ε11
ε33
9
11
11.7
11.7
4.2
4.2
-
α11
α33
4.22+2.67×10-3(t)
2.97+2.91×10-3(t)
-
0.55
0.55
4.8
2329
2200
10200
3260
Substituting the values listed in Table 3 into equation (21), the equivalent Young’s
modulus of a multi-layer composite cantilever can be obtained as shown in Equation
(22) below
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E

E1I1  E2 I 2    E N I N
I or ( I1  I 2    I N )
1
I or ( I1  I 2    I N )

 70  109  (1  204  10 6 T  221  10  9 T 2 ) I1



  169  109  (1  63  10  6 T  52  10  9 T 2 ) I 2


2
9
6
9
I
)
T

10

50

T

10

50

1
(

10

340



3


2
9
6
9
I
)
T

10

9
.
24

T

10

117

1
(

10

329

4


  340  109  (1  50  10  6 T  50  10  9 T 2 ) I 5 


2
9
6
9
  329  10  (1  117  10 T  24.9  10 T ) I 6 


2
9
6
9
I
)
T

10

221

T

10

204

1
(

10

70

7


(22)
and the moment of inertial of the constituting layers of the 200µm × 100µm cantilever
based on the AlN on cavity-SOI platform can be calculated and listed in Table 4
below.
Table 4: calculated moment of inertial of individual layers of the AlN on cavity-SOI
platform based cantilever resonator with a length-by-width dimension of 200µm × 100µm
Layer
Calculated moment of inertial
BOX
1.746×10-21
Si structural
1.724×10-21
AlN seed
3.599×10-24
Bottom Mo
4.220×10-23
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Submitted to IOP, J. Micromech. and Microeng. by Nan Wang et al.
AlN
4.292×10-22
Top Mo
1.351×10-22
Top SiO2
1.522×10-21
Substituting the calculated moment of inertial of the constituting layers of the AlN on
cavity-SOI platform list in Table 4, we can get
E
E1 I1  E2 I 2    E N I N

I or ( I1  I 2    I N )
1
I or ( I1  I 2    I N )
122  10 12  (1  204  10 6 T  221 10 9 T 2 ) 


  291 10 12  (1  63 10 6 T  52  10 9 T 2 )



12
6
9
2
  1.22  10  (1  50  10 T  50 10 T ) 
  13.9  10 12  (1  117 10 6 T  24.9  10 9 T 2 ) 


  146 10 12  (1  50 10 6 T  50  10 9 T 2 )



12
6
9
2
  44.4  10  (1  117  10 T  24.9  10 T ) 


12
6
9
2
  107  10  (1  204 10 T  221 10 T ) 
(23)
From Equation (23) above, the 1st and the 2nd order temperature coefficient of
elasticity can be obtained and expressed as Equation (24) and Equation (25),
respectively:
24888 - 18333 - 61 - 1626.3 - 7300 - 5194.8 + 21828 = 14200 (24)
26962 - 15312 - 61 - 346.11 - 7300 - 1105.56 + 23647 = 26484 (25)
23
Submitted to IOP, J. Micromech. and Microeng. by Nan Wang et al.
Therefore, the equivalent Young’s modulus considering the temperature coefficient of
elasticity will be:
E
E1I1  E2 I 2    EN I N
I or ( I1  I 2    I N )
725E  12  14200 E  18T  26484 E  21T 2

5.60 E  21
 129  109  2.53  106 T  4729.3T 2
(26)
 129  109 (1  19.6  10 6 T  36.7  10 9 T 2 )
Using equation (26), the TCF for the 200µm × 100µm cantilever based on the AlN on
cavity-SOI platform can be calculated based on the platform fabrication process as
described in section 2 and the thickness information of the constituting layers depicted
in Fig. 1(b), and the calculation results, which is obtained by taking the resonant
frequency at 25°C as the base temperature, are shown in Fig. 6, with Fig 6(a) showing
the TCF over the entire temperature range of calculation, i.e., -50°C to 300°C, and
Fig. 6(b) showing the TCF over the typical device-working temperature range, i.e.,
-40°C to 85°C. From the data shown in Fig. 6, one can see that the the AlN on
cavity-SOI platform is not optimized for temperature compensated resonators for
timing applications, as the TCF over the entire temperature range (-50°C to 300°C) is
4700 ppm, or 13.34 ppm/°C, and the TCF for the typical device-working range (-40°C
to 85°C) is 1205ppm, or 9.65ppm/°C. Both of the calculated TCF are too high and not
24
Submitted to IOP, J. Micromech. and Microeng. by Nan Wang et al.
acceptable for timing applications which require sub-ppm level of temperature
stability. Therefore, the platform needs to be further optimized to reduce the TCF to
meet the tight requirement of applications such as timing devices.
One of the approaches to tune the platform to achieve lower TCF is to tune the
thickness of the constituting layers of the composite cantilever. Since the effective
Young’s modulus of a composite cantilever is related to the thickness of the
constituting layers, as explained in equation (14), changing the thickness of the
constituting layers will change the effective Young’s modulus and its temperature
dependence. As such, the TCF of the same cantilever resonator design, i.e., same
length and width of the cantilever can be changed.
Theoretically, the thickness of all the constituting layers can be changed since MEMS
fabrication process is not standardized. However, from the perspective of the
fabrication complexity, changing the thickness of the top SiO2 layer is the most
effective way since the deposition and patterning of the top SiO2 layer is the last
fabrication step before releasing the device. The calculated TCF against the thickness
of the top SiO2 layer is depicted in Fig. 7, with Fig. 7(a) showing the over the entire
temperature range of calculation, i.e., -50°C to 300°C, and Fig. 7(b) showing the TCF
over the typical device-working temperature range, i.e., -40°C to 85°C. The calculated
25
Submitted to IOP, J. Micromech. and Microeng. by Nan Wang et al.
TCF/°C is also depicted in the secondary axis of both Fig. 7(a) and Fig. 7(b).
Fig. 7 shows a non-linear relationship between the thickness of the top SiO2 layer and
the calculated TCF as expected, since the temperature coefficient of elasticity of the
materials are in a higher order relationship with the temperature. It is to be noted that
minima exist for all the curves shown in both Fig. 7(a) and Fig. 7(b), indicating that
the TCF of the multi-layer composite cantilever based on the AlN on cavity-SOI
platform has a lower limit at certain temperature, for both temperature ranges.
Specifically, as indicated in Fig. 7(a), minimum TCF for the calculation over the
entire temperature range of calculation (-50°C to 300°C) can be obtained when the
thickness of the top SiO2 layer is 0.535µm, reaching a value of 226ppm over the
temperature range or 0.65ppm/°C equivalently. While as shown in Fig. 7(b), the
minimum TCF for the typical device-working temperature range (-40°C to 85°C) of
36.6ppm (or 0.29ppm/°C equivalently) can be achieved when the thickness of the top
SiO2 layer is 0.64µm. Therefore, the strict requirement on the temperature stability for
timing devices applications can be met by the current AlN on cavity-SOI platform by
tuning the thickness of the top SiO2 layer to a specific value depending on the
temperature range of the intended application. Actually, if sub-ppm level of
temperature stability is required instead of a minimum TCF value, there are a range of
top oxide thickness which can meet this requirement. As shown in Fig. 7(b), the TCF
26
Submitted to IOP, J. Micromech. and Microeng. by Nan Wang et al.
values for top oxide thickness between 0.55µm and 0.7µm are below 1 ppm, giving
substantial tolerance for fabrication uniformity and accuracy and improving the
robustness of the platform for timing devices which has stringent requirement on TCF.
The designed cantilevers with three different thickness of SiO2 deposited, i.e., 0.4µm,
0.6µm, and 0.8µm, are also fabricated and tested, with their measured TCF
characteristics depicted in Fig. 8 (a), Fig. 8(b), and Fig. 8(c), respectively, to
experimentally validate the model developed. The measurement results are labelled on
the figures and also summarized in Table 5 below. Also shown in the table is the
numerically derived TCF value, which is extracted from Fig. 7(b) for the purpose of
easy comparison.
Table 5: Comparison between numerically derived TCF and experimentally measured
TCF for cantilevers with various thickness of SiO2 deposited
Thickness of SiO2 [µm]
Numerically derived TCF
(Fig. 7(b)) [ppm/°C]
Measured TCF (Fig. 8)
[ppm/°C]
0.4 µm
3.372
3
0.6 µm
0.592
0.32
0.8 µm
2.352
2.16
From Table 5, one can see that the measured data are in good agreement with the
numerically derived TCF value, confirming the accuracy of the model developed. The
slight discrepancy could be attributed to the difference between the material properties
27
Submitted to IOP, J. Micromech. and Microeng. by Nan Wang et al.
used in the numerical derivation and the material properties of the fabricated
cantilever. Nevertheless, despite the slight discrepancy, the accuracy of the derived
model is validated and the TCF of a multi-layer cantilever can be precisely controlled
by tuning the thickness of the top SiO2 layer to a specific value depending on the
temperature range of the intended application, in order to meet the strict requirement
on the temperature stability for timing devices.
6. Conclusion
In this paper, a numerical model for precisely predicting and controlling the initial
bending of a multi-layer-stack composite cantilever which is caused by the residual
stress of the individual constituting layers, as well as the cantilever’s thermal
coefficient of frequency, is developed using a flexural-mode cantilever resonator
based on the aluminium nitride (AlN) cavity silicon-on-insulator (SOI) platform as an
example. The platform is also used to fabricate the said multi-layer-stack composite
cantilever, in order to experimentally verify the accuracy and consistency of the
model developed. The obtained experimental results are in excellent agreement with
the results predicted by the numerical model with a difference less than 3.5%,
demonstrating the accuracy of the numerical model developed and the feasibility to
optimize the cantilever’s initial deflection and TCF simultaneously, achieving
28
Submitted to IOP, J. Micromech. and Microeng. by Nan Wang et al.
minimum values for both parameters at the same time.
Acknowledgement:
This work was partially supported by ETPL_14-R15GAP-0012_MEMS Timing
Device. This study is also part of technology commercialization project, MEMSing,
supported by Exploit Technologies Pte Ltd, A*STAR. The team help companies to
capture the opportunities in Internet of Things (IoT) market through our MEMS
sensor/actuator portfolio and technology service.
29
Submitted to IOP, J. Micromech. and Microeng. by Nan Wang et al.
Figure Captions:
Figure 1: (a) Cross section of a typical cantilever structure fabricated by the AlN on
cavity-SOI platform. (b) Exact thickness of each layer of the whole stack of the AlN
on cavity-SOI platform. (c) Overview of the typical cantilever structure based on the
AlN on cavity SOI platform.
Figure 2: SEM image of a multi-layer-stack cantilever resonator fabricated based on
the AlN on cavity-SOI platform. The stress of the fabricated cantilever resonator is
not optimized, as shown by the large initial static deflection.
Figure 3: Illustration of the method to calculate the displacement at the tip of the
cantilever due to residual stress induced bending.
Figure 4: (a) the x-y plane, and (b) the y-z plane of the cross section of a two layer
composite cantilever structure to exemplify the derivation process
Figure 5: The displacement at the tip of a bending cantilever for different cantilever
lengths
Figure 6: TCF for the 200µm × 100µm cantilever based on the AlN on cavity-SOI
platform (a) over the entire temperature range of calculation, i.e., -50°C to 300°C, (b)
over the typical device-working temperature range, i.e., -40°C to 85°C
30
Submitted to IOP, J. Micromech. and Microeng. by Nan Wang et al.
Figure 7: Calculated TCF against the thickness of the top SiO2 layer for the 200µm ×
100µm cantilever (a) over the entire temperature range of calculation, i.e., -50°C to
300°C, and (b) over the typical device-working temperature range, i.e., -40°C to
85°C. The overall TCF is shown on the primary axis on the left and the TCF/°C is
shown on the secondary axis on the right.
Figure 8: Measured TCF for the 200µm × 100µm cantilever based on the AlN on
cavity-SOI platform for SiO2 of different thickness: (a) 0.4µm, (b) 0.6µm, and (c)
0.8µm. The parameter of TCF/°C is also calculated based on the measurement results
and indicated on the respective figures.
31
Submitted to IOP, J. Micromech. and Microeng. by Nan Wang et al.
List of Figures:
Figure 1 (a):
Figure 1(b):
Figure 1(c):
32
Submitted to IOP, J. Micromech. and Microeng. by Nan Wang et al.
Figure 2:
Figure 3:
θ
θ/2
d
ρ
θ/2
33
Submitted to IOP, J. Micromech. and Microeng. by Nan Wang et al.
Figure 4 (a):
a
y
E2
M
M
E1
x
a
Figure 4 (b):
A2
y
y2
Neutral axis
h
y1
z
A1
34
Submitted to IOP, J. Micromech. and Microeng. by Nan Wang et al.
Figure 5:
35
Submitted to IOP, J. Micromech. and Microeng. by Nan Wang et al.
Figure 6 (a):
Figure 6 (b):
36
Submitted to IOP, J. Micromech. and Microeng. by Nan Wang et al.
Figure 7 (a):
Figure 7 (b):
37
Submitted to IOP, J. Micromech. and Microeng. by Nan Wang et al.
Figure 8(a):
Figure 8(b):
38
Submitted to IOP, J. Micromech. and Microeng. by Nan Wang et al.
Figure 8 (c):
39
Submitted to IOP, J. Micromech. and Microeng. by Nan Wang et al.
References:
[1]
van Beek J T M and Puers R 2012 A review of MEMS oscillators for frequency
reference and timing applications J. Micromech. Microeng. 22 013001
[2]
Nguyen C T C 2007 MEMS technology for timing and frequency control IEEE
Transactions on Ultrasonics, Ferroelectrics and Frequency Control 54 251-70
[3]
Lavasani H M, Abdolvand R and Ayazi F 2007 A 500MHz Low Phase-Noise
A1N-on-Silicon Reference Oscillator. In: IEEE Custom Integrated Circuits
Conference, pp 599-602
[4]
Ryder S, Lee K B, Meng X F and Lin L W 2004 AFM characterization of
out-of-plane high frequency microresonators Sens. Actuator A-Phys. 114
135-40
[5]
Wang K, Wong A C and Nguyen C T C 2000 VHF free-free beam high-Q
micromechanical resonators J. Microelectromech. Syst. 9 347-60
[6]
Pourkamali S, Hashimura A, Abdolvand R, Ho G K, Erbil A and Ayazi F 2003
High-Q single crystal silicon HARPSS capacitive beam resonators with
self-aligned sub-100-nm transduction gaps J. Microelectromech. Syst. 12
487-96
[7]
Lin Y-W, Li S-S, Xie Y, Ren Z and Nguyen C T C 2005 Vibrating micromechanical
resonators with solid dielectric capacitive transducer gaps. In: 2005 IEEE
International Frequency Control Symposium (FCS 2005): IEEE) pp 128-34
[8]
Alsaleem F M, Younis M I and Ruzziconi L 2010 An Experimental and
Theoretical Investigation of Dynamic Pull-In in MEMS Resonators Actuated
Electrostatically J. Microelectromech. Syst. 19 794-806
[9]
Zine-El-Abidine I and Yang P 2009 A tunable mechanical resonator J.
Micromech. Microeng. 19 125004
[10]
Ho G K, Abdolvand R, Sivapurapu A, Humad S and Ayazi F 2008
Piezoelectric-on-silicon lateral bulk acoustic
resonators J. Microelectromech. Syst. 17 512-20
wave
micromechanical
[11]
Piazza G, Stephanou P J and Pisano A P 2006 Piezoelectric aluminum nitride
vibrating contour-mode MEMS resonators J. Microelectromech. Syst. 15
1406-18
[12]
Piazza G, Stephanou P J and Pisano A P 2007 Single-chip multiple-frequency
AlN MEMS filters based on contour-mode piezoelectric resonators J.
40
Submitted to IOP, J. Micromech. and Microeng. by Nan Wang et al.
Microelectromech. Syst. 16 319-28
[13]
Ruiz-Diez V, Manzaneque T, Hernando-Garcia J, Ababneh A, Kucera M, Schmid
U, Seidel H and Sanchez-Rojas J L 2013 Design and characterization of
AlN-based in-plane microplate resonators J. Micromech. Microeng. 23
[14]
Olsson R H, Wojciechowski K E, Baker M S, Tuck M R and Fleming J G 2009
Post-CMOS-Compatible Aluminum Nitride Resonant MEMS Accelerometers J.
Microelectromech. Syst. 18 671-8
[15]
Weinstein D and Bhave S A 2009 Internal Dielectric Transduction in
Bulk-Mode Resonators J. Microelectromech. Syst. 18 1401-8
[16]
Miller N and Piazza G 2014 Nonlinear dynamics in aluminum nitride
contour-mode resonators Appl. Phys. Lett. 104
[17]
Henry M D, Nguyen J, Young T R, Bauer T and Olsson R H 2014 Frequency
Trimming of Aluminum Nitride Microresonators Using Rapid Thermal
Annealing J. Microelectromech. Syst. 23 620-7
[18]
Ho G K, Sundaresan K, Pourkamali S and Ayazi F 2010 Micromechanical IBARs:
Tunable High-Q Resonators for Temperature-Compensated Reference
Oscillators J. Microelectromech. Syst. 19 503-15
[19]
Wang N, Tsai J M, Hsiao F L, Soon B W, Kwong D L, Palaniapan M and Lee C
2011 Experimental Investigation of a Cavity-Mode Resonator Using a
Micromachined Two-Dimensional Silicon Phononic Crystal in a Square Lattice
IEEE Electron Device Lett. 32 821-3
[20]
Wang N, Tsai J M L, Hsiao F L, Soon B W, Kwong D L, Palaniapan M and Lee C
2012 Micromechanical Resonators Based on Silicon Two-Dimensional
Phononic Crystals of Square Lattice Journal of Microelectromechanical
Systems 21 801-10
[21]
Chen Q M, Qin L F and Wang Q M 2007 Property characterization of AlN thin
films in composite resonator structure J. Appl. Phys. 101 084103
[22]
Melamud R, Chandorkar S A, Kim B, Lee H K, Salvia J C, Bahl G, Hopcroft M A
and Kenny T W 2009 Temperature-Insensitive Composite Micromechanical
Resonators J. Microelectromech. Syst. 18 1409-19
[23]
Tazzoli A, Rinaldi M and Piazza G 2012 Experimental Investigation of
Thermally Induced Nonlinearities in Aluminum Nitride Contour-Mode MEMS
Resonators IEEE Electron Device Lett. 33 724-6
[24]
Samarao A K and Ayazi F 2012 Temperature Compensation of Silicon
41
Submitted to IOP, J. Micromech. and Microeng. by Nan Wang et al.
Resonators via Degenerate Doping IEEE Trans. Electron Devices 59 87-93
[25]
Delmas L, Sthal F, Bigler E, Dulmet B and Bourquin R 2005
Temperature-compensated cuts for length-extensional and flexural vibrating
modes in GaPO(4) beam resonators IEEE Trans. Ultrason. Ferroelectr. Freq.
Control 52 666-71
[26]
Melamud R, Kim B, Chandorkar S A, Hopcroft M A, Agarwal M, Jha C M and
Kenny T W 2007 Temperature-compensated high-stability silicon resonators
Appl. Phys. Lett. 90
[27]
DeVoe D L and Pisano A P 1997 Modeling and optimal design of piezoelectric
cantilever microactuators J. Microelectromech. Syst. 6 266-70
[28]
Tibeica C, Damian V and Muller R 2011 SiO2-METAL CANTILEVER STRUCTURES
UNDER THERMAL AND INTRINSIC STRESS. In: 2011 International
Semiconductor Conference (Cas 2011), pp 167-70
[29]
Liu R, Jiao B, Kong Y, Li Z, Shang H, Lu D, Gao C and Chen D 2013 Elimination of
initial stress-induced curvature in a micromachined bi-material
composite-layered cantilever J. Micromech. Microeng. 23
[30]
Lee S Y, Ko B and Yang W S 2005 Theoretical modeling, experiments and
optimization of piezoelectric multimorph Smart Materials & Structures 14
1343-52
[31]
Liu M W, Tong J H, Wang L D and Cui T H 2006 Theoretical analysis of the
sensing and actuating effects of piezoelectric multimorph cantilevers
Microsystem Technologies-Micro-and Nanosystems-Information Storage and
Processing Systems 12 335-42
[32]
Shi Z F, Xiang H J and Spencer B F, Jr. 2006 Exact analysis of multi-layer
piezoelectric/composite cantilevers Smart Materials & Structures 15 1447-58
[33]
Coventor I 2015 MEMS Design and Analysis.
[34]
Tabrizian R, Pardo M and Ayazi F 2012 A 27 MHz temperature compensated
MEMS oscillator with sub-ppm instability. In: Micro Electro Mechanical
Systems (MEMS), 2012 IEEE 25th International Conference on, pp 23-6
[35]
Melamud R, Chandorkar S A, Bongsang K, Hyung Kyu L, Salvia J C, Bahl G,
Hopcroft M A and Kenny T W 2009 Temperature-Insensitive Composite
Micromechanical Resonators Microelectromechanical Systems, Journal of 18
1409-19
42
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43