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EFFECT OF THICKNESS ANISOTROPY ON DEGENERATE MODES IN OXIDE MICRO-HEMISPHERICAL SHELL RESONATORS
L. D. Sorenson, P. Shao, and F. Ayazi School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA, USA ABSTRACT
Wirebond
The effect of thickness anisotropy on the degenerate elliptical resonance modes of micro-hemispherical shell resonators (µHSRs) created using the thermal oxidation process is investigated. This anisotropy arises from the variation in wet thermal oxide growth according to the exposed crystal planes of the single-crystal-silicon hemispherical mold used to generate the µHSRs. It is shown that, despite the presence of thickness anisotropy, the degenerate resonance modes of oxide µHSRs can exhibit zero intrinsic frequency split depending on the particular resonance mode and symmetry of the thickness anisotropy imparted from the underlying silicon wafer. Measured results verified by simultaneous electrical excitation on the 0° and 45° axes demonstrate less than 94 Hz intrinsic m=3 frequency split for a 1240 µm oxide µHSR (limited by measurement conditions), which is to the authors’ knowledge the smallest as-fabricated frequency split reported to date for any µHSR.
0° Axis
22.5° Axis 45° Axis
Figure 1: Optical micrograph of a 1240 µm diameter, 2 µm thick ALD TiN-coated thermally-grown oxide micro-hemispherical shell resonator (µHSR) assembled with multi-axis silicon electrode pillars for capacitive testing created using the process detailed in [1].
INTRODUCTION
There is increasing interest in axisymmetric 3D micromachined shell resonators with applications including rotation sensors, due to their potential for high quality factor (Q) at low frequencies and high insensitivity to environmental vibrations [1–4]. Acting as a gyroscope, the principle axes of vibration of a µHSR (such as that shown in Fig. 1) precess due to Coriolis forces when the device experiences external rotation [5]. The precession occurs due to transfer of energy between frequency-matched degenerate modes of the shell. Deviations from axisymmetry in the shell may split the frequencies of the ideally-degenerate modes, resulting in reduced coupling between the modes and impacting the precession. These types of axial asymmetry can arise for a
(110)
number of reasons; for example, deviations in the shape of the mold, misalignment of the wafer crystalline axes, random but unevenly distributed surface roughness, spatial variations in the material properties, the shape, balance, and size of the anchoring stem, and variations in material growth and deposition may all contribute to elastic axial asymmetry. Therefore, understanding of the origins and effects of these contributions is critical to achieving high performance µHSRs. In this work, the effect of thickness anisotropy arising due to crystalline axis dependency in the wet thermal oxide growth on hemispherical single-crystal silicon molds is considered (as exemplified in Fig. 2).
(110)
(010)
(110)
(100)
(001)
(101)
(100)
(111) (100)
(111)
(111)
(111) (011)
(a)
(011)
(101)
(101)
(010)
(110)
(010)
120° symmetry
90° symmetry
(011) (111)
(001) (110)
(b)
(001) Wafer
(111) Wafer
Figure 2: (a, b) Top view of the relative thickness map for 20 hours of wet thermal oxide growth at 1100°C in ideal hemispherical micromolds on (001) and (111) single-crystal silicon wafers, respectively, estimated using a Deal-Grove model [6]. The insets show the corresponding isometric views to illustrate the 3D nature of oxide growth in hemispherical micromolds. Important silicon symmetry planes are indicated for reference. The growth of oxide on (001) wafers is expected to show 90° rotational symmetry, while the (111) wafer exhibits 120° rotational symmetry.
978-1-4673-5655-8/13/$31.00 ©2013 IEEE
169
MEMS 2013, Taipei, Taiwan, January 20 – 24, 2013
shell and potentially splitting the frequency of the degenerate modes. Further, although not considered in detail here, the wafer surface may not be perfectly aligned to the desired crystal plane in practice, tilting the axis of symmetry of a thermal oxide shell. To explore techniques to mitigate these thickness anisotropy effects, finite element modeling was employed to understand the sensitivity of the low-frequency flexural bending modes of the shell to thickness anisotropy.
[100]
[001]
[100]
(a)
FINITE ELEMENT MODEL
The finite element method is often employed for its ability to work with complex geometry. However, in the case of small deviations from ideal geometry, especially in the case of a thin shell (Fig. 4(a)), it can be difficult or impossible to modify the geometry or finite element mesh to account for such deviations. For example, one may map the oxide growth variations onto the mesh by moving each node a few nanometers in the appropriate direction. However, this would require introduction of an atypical preprocessing step to implement the mesh modification algorithm. Shell elements with a variable thickness parameter could be employed, but this complicates modeling of the shell-support stem interface (Fig. 4(a)). A more convenient option is mapping the oxide thickness variations according to Figure 2 onto the Young’s modulus by deriving the resulting effect on the modal frequencies while maintaining a constant geometric thickness of the shell. To justify the equivalence of modulating the thickness and Young’s modulus, start from the square of the angular eigenfrequencies of the flexural bending modes of a
Silicon Hemispherical Mold Oxide Growth vs. Silicon Crystal Plane as a Function of Plane Index 3.375
(111)
Oxide Growth after 20 hours
3.37 3.365
(011)
3.36 3.355
y = -0.0428x2 + 0.166x + 3.2118
3.35 3.345 3.34
P
3.335
(001) 3.33
(b)
1
1.1
1.2
X Y Z R
1.3 1.4 1.5 Plane Index P (dimensionless)
1.6
1.7
1.8
Figure 3: (a) Schematic diagram depicting a few of the crystal plane normals of silicon which are exposed during isotropic etching of the mold; (b) wet thermal silicon dioxide growth after 20 hours vs. crystal orientation, as determined for (001), (011), and (111) silicon wafers by the Deal-Grove model [6] and interpolated using a plane index P. A maximum of 36 nm (1.08%) difference was obtained between the 3 crystal planes
HEMISPHERICAL MOLD OXIDATION
After defining the hemispherical silicon mold with an isotropic SF6 etch as in [1], the entire set of silicon crystal planes will be exposed (Fig. 3(a)). Following the Deal-Grove model [6], the growth of thermal oxide in this mold can be accurately predicted for specific planes under varied processing conditions using empirically-derived parameters from growth on (001), (110), and (111) silicon wafers. Rate constants for these planes are available in [7], and they generally follow an Arrhenius temperature dependency allowing for prediction over a wide range of growth temperatures. However, little literature exists on oxide growth between these crystal planes, and that which does exist presents difficulties in application such as being limited to thin or native oxide layers [8], [9]. The growth conditions ultimately determine the structure of the μHSR shown in Figure 1. To create large and balanced μHSRs, oxide layers with a few µm of thickness must be employed to survive the XeF2 release process [1]. Therefore, we propose interpolation of the available growth data through a dimensionless crystal plane index P (Fig. 3(b)). This approach can be generalized as knowledge of oxide growth kinetics is improved for uncommon orientations. Figure 2 was generated using the thickness interpolation of Figure 3(b). These thickness modulations introduce anisotropy into the structure, perturbing the rotational symmetry of the
(a)
(b)
(c) Figure 4: (a) SEM view of a free-standing thermal oxide shell of a μHSR prior to assembly; (b) isometric view of rotationally symmetric and uniform mesh created over a constant thickness shell with support stem with color indicating silicon plane index P; (c) top view of (b).
170
hemispherical shell [10]:
where
I m 0 f
tan 2m 2 sin3
m
(1)
d ,
J m 0 f m2 1 sin2 2m cos sin tan 2m
h , . (4) h It can also be shown that there is a similar contribution due to a Young’s modulus perturbation with an additional factor of ½: m 1 E , . (5) m 2 E Thus, there is an equivalency between small perturbations in thickness and Young’s modulus over the area of the shell, which can be exploited to study the impact of thickness modulations on the frequency splits without impacting the geometry or mesh of the resonator. This is achieved by entering the following expression in place of the normal Young’s modulus of the material in the finite element software: h , (6) E E E , E1 2 . h It is interesting to note that small radial deviations from the ideal geometry can also be accounted for in a manner similar to (6), but with a factor of -4 instead of 2. Further, any perturbations to the other variables in (1) can be accounted for, or they may be used as an alternative to m
2 2 E h 2 I m , m2 m 2 m 2 1 3 21 R 4 J m
(2)
d ,(3)
2
ωm is the angular frequency of the mθ flexural bending mode, m is the number of complete periods of the vibration pattern around the rim of the shell in the azimuthal direction θ, E is the Young’s modulus, ν is the Poisson’s ratio, ρ is the mass density, h is the thickness of the shell (assumed constant), R is the radius, φ is the longitudinal coordinate, and φf denotes the final angle determining the extent of the hemispherical shell. The vibration eigenmodes will be spatially separated by 90°/m to maintain orthogonality. Starting from (1), it can be shown that at an arbitrary point on the midsurface of the shell, the contribution to change in the overall frequency of the shell from a small thickness perturbation (Δh
L. D. Sorenson, P. Shao, and F. Ayazi School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA, USA ABSTRACT
Wirebond
The effect of thickness anisotropy on the degenerate elliptical resonance modes of micro-hemispherical shell resonators (µHSRs) created using the thermal oxidation process is investigated. This anisotropy arises from the variation in wet thermal oxide growth according to the exposed crystal planes of the single-crystal-silicon hemispherical mold used to generate the µHSRs. It is shown that, despite the presence of thickness anisotropy, the degenerate resonance modes of oxide µHSRs can exhibit zero intrinsic frequency split depending on the particular resonance mode and symmetry of the thickness anisotropy imparted from the underlying silicon wafer. Measured results verified by simultaneous electrical excitation on the 0° and 45° axes demonstrate less than 94 Hz intrinsic m=3 frequency split for a 1240 µm oxide µHSR (limited by measurement conditions), which is to the authors’ knowledge the smallest as-fabricated frequency split reported to date for any µHSR.
0° Axis
22.5° Axis 45° Axis
Figure 1: Optical micrograph of a 1240 µm diameter, 2 µm thick ALD TiN-coated thermally-grown oxide micro-hemispherical shell resonator (µHSR) assembled with multi-axis silicon electrode pillars for capacitive testing created using the process detailed in [1].
INTRODUCTION
There is increasing interest in axisymmetric 3D micromachined shell resonators with applications including rotation sensors, due to their potential for high quality factor (Q) at low frequencies and high insensitivity to environmental vibrations [1–4]. Acting as a gyroscope, the principle axes of vibration of a µHSR (such as that shown in Fig. 1) precess due to Coriolis forces when the device experiences external rotation [5]. The precession occurs due to transfer of energy between frequency-matched degenerate modes of the shell. Deviations from axisymmetry in the shell may split the frequencies of the ideally-degenerate modes, resulting in reduced coupling between the modes and impacting the precession. These types of axial asymmetry can arise for a
(110)
number of reasons; for example, deviations in the shape of the mold, misalignment of the wafer crystalline axes, random but unevenly distributed surface roughness, spatial variations in the material properties, the shape, balance, and size of the anchoring stem, and variations in material growth and deposition may all contribute to elastic axial asymmetry. Therefore, understanding of the origins and effects of these contributions is critical to achieving high performance µHSRs. In this work, the effect of thickness anisotropy arising due to crystalline axis dependency in the wet thermal oxide growth on hemispherical single-crystal silicon molds is considered (as exemplified in Fig. 2).
(110)
(010)
(110)
(100)
(001)
(101)
(100)
(111) (100)
(111)
(111)
(111) (011)
(a)
(011)
(101)
(101)
(010)
(110)
(010)
120° symmetry
90° symmetry
(011) (111)
(001) (110)
(b)
(001) Wafer
(111) Wafer
Figure 2: (a, b) Top view of the relative thickness map for 20 hours of wet thermal oxide growth at 1100°C in ideal hemispherical micromolds on (001) and (111) single-crystal silicon wafers, respectively, estimated using a Deal-Grove model [6]. The insets show the corresponding isometric views to illustrate the 3D nature of oxide growth in hemispherical micromolds. Important silicon symmetry planes are indicated for reference. The growth of oxide on (001) wafers is expected to show 90° rotational symmetry, while the (111) wafer exhibits 120° rotational symmetry.
978-1-4673-5655-8/13/$31.00 ©2013 IEEE
169
MEMS 2013, Taipei, Taiwan, January 20 – 24, 2013
shell and potentially splitting the frequency of the degenerate modes. Further, although not considered in detail here, the wafer surface may not be perfectly aligned to the desired crystal plane in practice, tilting the axis of symmetry of a thermal oxide shell. To explore techniques to mitigate these thickness anisotropy effects, finite element modeling was employed to understand the sensitivity of the low-frequency flexural bending modes of the shell to thickness anisotropy.
[100]
[001]
[100]
(a)
FINITE ELEMENT MODEL
The finite element method is often employed for its ability to work with complex geometry. However, in the case of small deviations from ideal geometry, especially in the case of a thin shell (Fig. 4(a)), it can be difficult or impossible to modify the geometry or finite element mesh to account for such deviations. For example, one may map the oxide growth variations onto the mesh by moving each node a few nanometers in the appropriate direction. However, this would require introduction of an atypical preprocessing step to implement the mesh modification algorithm. Shell elements with a variable thickness parameter could be employed, but this complicates modeling of the shell-support stem interface (Fig. 4(a)). A more convenient option is mapping the oxide thickness variations according to Figure 2 onto the Young’s modulus by deriving the resulting effect on the modal frequencies while maintaining a constant geometric thickness of the shell. To justify the equivalence of modulating the thickness and Young’s modulus, start from the square of the angular eigenfrequencies of the flexural bending modes of a
Silicon Hemispherical Mold Oxide Growth vs. Silicon Crystal Plane as a Function of Plane Index 3.375
(111)
Oxide Growth after 20 hours
3.37 3.365
(011)
3.36 3.355
y = -0.0428x2 + 0.166x + 3.2118
3.35 3.345 3.34
P
3.335
(001) 3.33
(b)
1
1.1
1.2
X Y Z R
1.3 1.4 1.5 Plane Index P (dimensionless)
1.6
1.7
1.8
Figure 3: (a) Schematic diagram depicting a few of the crystal plane normals of silicon which are exposed during isotropic etching of the mold; (b) wet thermal silicon dioxide growth after 20 hours vs. crystal orientation, as determined for (001), (011), and (111) silicon wafers by the Deal-Grove model [6] and interpolated using a plane index P. A maximum of 36 nm (1.08%) difference was obtained between the 3 crystal planes
HEMISPHERICAL MOLD OXIDATION
After defining the hemispherical silicon mold with an isotropic SF6 etch as in [1], the entire set of silicon crystal planes will be exposed (Fig. 3(a)). Following the Deal-Grove model [6], the growth of thermal oxide in this mold can be accurately predicted for specific planes under varied processing conditions using empirically-derived parameters from growth on (001), (110), and (111) silicon wafers. Rate constants for these planes are available in [7], and they generally follow an Arrhenius temperature dependency allowing for prediction over a wide range of growth temperatures. However, little literature exists on oxide growth between these crystal planes, and that which does exist presents difficulties in application such as being limited to thin or native oxide layers [8], [9]. The growth conditions ultimately determine the structure of the μHSR shown in Figure 1. To create large and balanced μHSRs, oxide layers with a few µm of thickness must be employed to survive the XeF2 release process [1]. Therefore, we propose interpolation of the available growth data through a dimensionless crystal plane index P (Fig. 3(b)). This approach can be generalized as knowledge of oxide growth kinetics is improved for uncommon orientations. Figure 2 was generated using the thickness interpolation of Figure 3(b). These thickness modulations introduce anisotropy into the structure, perturbing the rotational symmetry of the
(a)
(b)
(c) Figure 4: (a) SEM view of a free-standing thermal oxide shell of a μHSR prior to assembly; (b) isometric view of rotationally symmetric and uniform mesh created over a constant thickness shell with support stem with color indicating silicon plane index P; (c) top view of (b).
170
hemispherical shell [10]:
where
I m 0 f
tan 2m 2 sin3
m
(1)
d ,
J m 0 f m2 1 sin2 2m cos sin tan 2m
h , . (4) h It can also be shown that there is a similar contribution due to a Young’s modulus perturbation with an additional factor of ½: m 1 E , . (5) m 2 E Thus, there is an equivalency between small perturbations in thickness and Young’s modulus over the area of the shell, which can be exploited to study the impact of thickness modulations on the frequency splits without impacting the geometry or mesh of the resonator. This is achieved by entering the following expression in place of the normal Young’s modulus of the material in the finite element software: h , (6) E E E , E1 2 . h It is interesting to note that small radial deviations from the ideal geometry can also be accounted for in a manner similar to (6), but with a factor of -4 instead of 2. Further, any perturbations to the other variables in (1) can be accounted for, or they may be used as an alternative to m
2 2 E h 2 I m , m2 m 2 m 2 1 3 21 R 4 J m
(2)
d ,(3)
2
ωm is the angular frequency of the mθ flexural bending mode, m is the number of complete periods of the vibration pattern around the rim of the shell in the azimuthal direction θ, E is the Young’s modulus, ν is the Poisson’s ratio, ρ is the mass density, h is the thickness of the shell (assumed constant), R is the radius, φ is the longitudinal coordinate, and φf denotes the final angle determining the extent of the hemispherical shell. The vibration eigenmodes will be spatially separated by 90°/m to maintain orthogonality. Starting from (1), it can be shown that at an arbitrary point on the midsurface of the shell, the contribution to change in the overall frequency of the shell from a small thickness perturbation (Δh
