Bulk Modes in Silicon Crystal Silicon - Semantic Scholar
Bulk Modes in Silicon Crystal Silicon Gavin K. Ho
Siavash Pourkamali
Farrokh Ayazi
ParibX, Inc. Mountain View, CA USA [email protected]
Dept. of Electrical and Computer Eng. University of Denver Denver, CO USA
School of Electrical and Computer Eng. Georgia Institute of Technology Atlanta, GA USA
Abstract—Bulk dilation modes in rectangular plates of particular orientations in single crystal silicon are found to have distinct advantages over to those in isotropic materials. Finite element modeling using an anisotropic model reveals plates aligned to the ‹110› direction in (100) silicon have modes which offer excellent capacitive coupling and good isolation to the supports. Low impedance and high Q are predicted for select geometries. Experimental results are included, showing a 100MHz resonator having an unloaded Q of 90,000, which demonstrate an f-Q product of ~1×1013.
I.
A silicon plate resonator with a large lateral dimension is introduced to satisfy the requirements for low R1, high Q, and good linearity. See Figure 1. As the lateral dimension is increased, it is known that wavelike properties commonly appear in the observed modes. The modes for some isotropic plates are shown in Figure 2.
INTRODUCTION
Micromechanical resonators [1] are a promising alternative to traditional macro-scale frequency control devices. Silicon is an attractive material since it is well characterized, it has deterministic material properties, and it offers low acoustic loss. Electrostatic micromechanical resonators, also known as capacitive resonators, are attractive for a number of reasons. Since metallization is not necessary on the resonating structure, the resonator can be monolithic in the strictest sense. Very high quality factors, no built-in stress, and excellent long-term stability are possible in silicon capacitor resonators.
Figure 1. Perspective view of a capacitive plate resonator with two supports and two fixed electrodes.
The characteristic motional resistance R1 in early demonstrations of capacitive resonators was 10 kΩ to 1 MΩ. It is desirable for resonators, especially those at higher frequencies, to have lower R1. Reducing R1 (i.e., increasing output current from an input voltage) involves increasing electrode area, improving electromechanical coupling, and optimizing the mode shape of the resonator. High Q reduces the close-to-carrier phase noise. Ensuring high Q requires minimizing the magnitude of the loss mechanisms. It is necessary to minimize both the intrinsic and extrinsic loss mechanisms. The design-dependent extrinsic losses are reduced through mode isolation (from other modes) and designing for mode shapes which limit energy leakage into the surrounding medium. Good linearity is also desired in the resonator for a low noise floor. The linearity limit of a resonator is determined by material properties and physical dimensions. It is evident that a large resonator is desirable.
0.2L
0.8L
1.8L
4.1L Figure 2. Typical mode shapes in narrow, wide, and very wide rectangular plates in isotropic materials (ν=0.28).
The objective of this work includes: 1) characterizing the dilation modes of rectangular plates in isotropic and anisotropic materials and 2) understanding whether plate modes can be optimized for the metrics of interest. Finite element analysis is performed in ANSYS using SOLID186 elements and isotropic and anisotropic material models. II.
BULK DILATION MODES
A brief discussion of bulk dilation modes in capacitive resonators follows. The natural frequencies are
fn =
nva , 2L
(1)
where n is the mode number, va is the acoustic velocity and L is the length of the resonator. The motional resistance of a capacitive bulk dilation mode resonator may be simplified to
R1 =
krπ Ei ρ m d 4 , 2 2Qε 2VP A
(2)
where kr is the relative dynamic stiffness, Ei is the elastic modulus in the i direction, ρm is the mass density, d is the gap size, ε is the permittivity of the gap, VP is the polarization voltage, and A is the overlap area of the electrodes. It is evident that increasing A reduces R1 – more current is available when the transduction area is increased. Reducing kr also reduces R1. An interesting observation is that R1 is not directly dependent on frequency. Contrary to the trend in prior art, higher frequency resonators do not necessarily have higher R1.
III.
ISOTROPIC PLATES
The analysis begins by extending the findings on bulk dilation modes of isotropic plates summarized in [2]. In a plate with a small width w (e.g., 0.2L), the mode is entirely uniform along the ends. See Figure 2. As w is increased to 0.8L, some non-uniformity is observed. Lateral motion at the support location exists. For a plate with a width of 1.8L, the most prominent mode is a third mode along the x direction. For a width of 4.1L, the most prominent mode is a seventhorder mode along x. Although the frequencies of these modes are very similar, the latter three modes are in three different modal families. Focusing on the seventh-order modal family of an isotropic material with a Poisson ratio ν of 0.28, observe the attributes of the mode as the width is increased. See Figure 3. For w=3.5L, there are out-of-phase regions which cause charge cancellation. The mode in this geometry has poor coupling and a large kr. For w=4.1L, the displacement is somewhat uniform. With this geometry, the ux motion at the support is minimal. For w=4.3L, the displacement is most uniform. This is the optimal plate geometry for coupling and the kr is 1.2. Further increasing w to 4.8L leads to a more wavelike mode with substantial ux motion at the support. In these thin isotropic plates, the optimal coupling and optimal support isolation occur for different values of w. The solutions for isotropic plates having widths of [0,10]L, thickness t of L/5, and ν = 0.28 are summarized in Figure 4. kr and normalized ux (to the average uy) are plotted. The minima for kr, for w>2L, are approximately 1.2. There are geometries in which the support ux is small. However, ux is quite sensitive to the width.
kr is a comparison between the dynamic stiffness kn and the stiffness of a slender ideal 1-dimensional resonator.
kr =
kn kn ,1D
(3)
w = 3.5L
It is strongly dependant on the mode shape. One goal of this work is to understand the effect of increasing A on kr. The dynamic stiffness kn for each mode is found from the energy in the mode W and the average of the displacement of the resonator electrode surface normal to the fixed electrode uy. For a low kn (and consequently low kr), all the energy in the mode is preferably attributed to uy. kn =
2W uy
2
(4)
w = 4.1L
w = 4.3L
w = 4.8L Figure 3. Variation of mode shape within a modal family for width of 3.5L, 4.1L, 4.3L, and 4.8L (isotropic, ν=0.28).
3
Relative Dynamic Stiffness k r
2.75 2.5 2.25 2 1.75 1.5 1.25 1 0.75 0.5
3D isotropic, ν =0.28, t= L/5 0
0.5
1
1.5
2
2.5
3
3.5
4
4.5 5 5.5 Width [L]
6
6.5
7
7.5
8
8.5
9
9.5
10
2.5
3
3.5
4
4.5 5 5.5 Width [L]
6
6.5
7
7.5
8
8.5
9
9.5
10
2.5
3
3.5
4
4.5 5 5.5 Width [L]
6
6.5
7
7.5
8
8.5
9
9.5
10
0.4 0.35 0.3 | ux,supp / uy,avg |
0.25 0.2 0.15 0.1 0.05 0 -0.05 -0.1
3D isotropic, ν =0.28, t= L/5 0
0.5
1
1.5
2
1.25
Normalized Frequency, fn / f1D
1.2 1.15 1.1 1.05 1 0.95 0.9 0.85 0.8 0.75
3D isotropic, ν =0.28, t= L/5 0
0.5
1
1.5
2
Figure 4. Relative dynamic stiffness, normalized support motion, and normalized frequency in isotropic plates with ν=0.28 and a thickness of L/5.
IV.
ANISOTROPIC PLATES
The analysis for anisotropic plates is performed similarly. A very interesting observation is that uniform modes are observed in plates aligned to the ‹110› direction in (100) silicon. A thorough analysis was completed for plates with t equal to L/5 and L/2. The mode shape for L/5-thick plates with w=[3.4,4.1]L are very uniform (Figure 5). For w=3.7L, the mode has the smallest ux response at the support. The kr and normalized support ux are shown in Figure 6.
w = 3.4L
There are several advantages for these anisotropic plates. First, kr is ~1 for a large variety of widths. Next, ux is smaller compared to isotropic plate modes. The ux minima coincide with the kr minima. Also, ux has low sensitivity to w.
w = 3.7L
Increasing the thickness to L/2 reveals further advantages. See Figure 7. Compared to L/5 plates, the range of widths for which kr ≈ 1 is broader. The ux also have lower sensitivity to w. The most beneficial attribute is that the transduction area is increased without increasing die size.
w = 4.1L Figure 5. Variation of mode shape within a modal family for width of 3.4L, 3.7L, and 4.1L (anisotropic, t=L/5).
3
Relative Dynamic Stiffness k r
2.75 2.5 2.25 2 1.75 1.5 1.25 1 0.75 0.5
3D anisotropic, (100) 〈 110〉 t =L/5 0
0.5
1
1.5
2
2.5
3
3.5
4
4.5 5 5.5 Width [L]
6
6.5
7
7.5
8
8.5
9
9.5
10
3
3.5
4
4.5 5 5.5 Width [L]
6
6.5
7
7.5
8
8.5
9
9.5
10
0.4 0.35 0.3 | ux,supp / uy,avg |
0.25 0.2 0.15 0.1 0.05 0 -0.05 -0.1
3D anisotropic, (100) 〈 110〉 t =L/5 0
0.5
1
1.5
2
2.5
Figure 6. Relative dynamic stiffness and normalized support motion in anisotropic plates aligned to ‹110› in (100) silicon having a thickness of L/5.
3
Relative Dynamic Stiffness k r
2.75 2.5 2.25 2 1.75 1.5 1.25 1 0.75 0.5
3D anisotropic, (100) 〈 110〉 t =L/2 0
0.5
1
1.5
2
2.5
3
3.5
4
4.5 5 5.5 Width [L]
6
6.5
7
7.5
8
8.5
9
9.5
10
3
3.5
4
4.5 5 5.5 Width [L]
6
6.5
7
7.5
8
8.5
9
9.5
10
3
3.5
4
4.5 5 5.5 Width [L]
6
6.5
7
7.5
8
8.5
9
9.5
10
0.4 0.35 0.3 | ux,supp / uy,avg |
0.25 0.2 0.15 0.1 0.05 0 -0.05 -0.1
3D anisotropic, (100) 〈 110〉 t =L/2 0
0.5
1
1.5
2
2.5
1.25
Normalized Frequency, fn / f1D
1.2 1.15 1.1 1.05 1 0.95 0.9 0.85 0.8 0.75
3D anisotropic, (100) 〈 110〉 t =L/2 0
0.5
1
1.5
2
2.5
Figure 7. Relative dynamic stiffness, normalized support motion, and normalized frequency in anisotropic plates aligned to ‹110› in (100) silicon having a thickness of L/2.
Periodicity is evident from the results. Widths of even multiples of L provide small motion at the supports and good coupling. The optimal widths for minimizing support loss are 2.0L, 3.9L, 5.8L, 7.7L, and 9.7L. V.
EXPERIMENTATION
Resonators with L=40μm and t=L/2 were fabricated using the HARPSS-on-SOI process [3]. A resonator with a width of 9.7L and 225-nm gaps is shown in Figure 8. The resonance frequency of this resonator was measured to be 103 MHz along with a quality factor of 90000. See Figure 9. The optimized design enabled the f-Q product of ~1×1013.
CONCLUSIONS Bulk dilation modes of plates aligned to the ‹110› direction in (100) silicon were found to have improved attributes over plates in isotropic materials. In anisotropic plates, the relative dynamic stiffness is unity for many geometries. For a broad range of widths, a 1-dimensional assumption applies. The thorough exercise also revealed specific widths at which the motion of the support location is minimal. This enabled a 100-MHz resonator to demonstrate an f-Q product of 1×1013. Future work includes characterizing modes in plates with larger thickness. REFERENCES
Resonator
Anchor
[1]
Input/Output pad
[2]
[3]
Input/Output pad
L = 40 μm w = 9.7 L
t = 20 μm d = 225 nm
Figure 8. Scanning electron micrograph of a resonator fabricated using the HARPSS-on-SOI process.
f0 = 103 MHz Qu = 90000
Figure 9. Transmission frequency response of the resonator showing an unloaded Q of 90000 at a frequency of 103 MHz.
C. T.-C. Nguyen, “MEMS technology for timing and frequency control,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control, vol. 54, pp. 251-270, Feb. 2007. R. Holland, “Contour extensional resonant properties of rectangular piezoelectric plates”, IEEE Trans. Son. Ultrason., SU-15, n. 2, pp97105, Apr 1968 S. Pourkamali, G. K. Ho, and F. Ayazi, “Low-Impedance VHF and UHF Capacitive Silicon Bulk Acoustic Wave Resonators – Part I: Concept and Fabrication”, IEEE Trans. Elec. Dev., vol. 54, n. 8, pp. 2017-2023, Aug. 2007.
Siavash Pourkamali
Farrokh Ayazi
ParibX, Inc. Mountain View, CA USA [email protected]
Dept. of Electrical and Computer Eng. University of Denver Denver, CO USA
School of Electrical and Computer Eng. Georgia Institute of Technology Atlanta, GA USA
Abstract—Bulk dilation modes in rectangular plates of particular orientations in single crystal silicon are found to have distinct advantages over to those in isotropic materials. Finite element modeling using an anisotropic model reveals plates aligned to the ‹110› direction in (100) silicon have modes which offer excellent capacitive coupling and good isolation to the supports. Low impedance and high Q are predicted for select geometries. Experimental results are included, showing a 100MHz resonator having an unloaded Q of 90,000, which demonstrate an f-Q product of ~1×1013.
I.
A silicon plate resonator with a large lateral dimension is introduced to satisfy the requirements for low R1, high Q, and good linearity. See Figure 1. As the lateral dimension is increased, it is known that wavelike properties commonly appear in the observed modes. The modes for some isotropic plates are shown in Figure 2.
INTRODUCTION
Micromechanical resonators [1] are a promising alternative to traditional macro-scale frequency control devices. Silicon is an attractive material since it is well characterized, it has deterministic material properties, and it offers low acoustic loss. Electrostatic micromechanical resonators, also known as capacitive resonators, are attractive for a number of reasons. Since metallization is not necessary on the resonating structure, the resonator can be monolithic in the strictest sense. Very high quality factors, no built-in stress, and excellent long-term stability are possible in silicon capacitor resonators.
Figure 1. Perspective view of a capacitive plate resonator with two supports and two fixed electrodes.
The characteristic motional resistance R1 in early demonstrations of capacitive resonators was 10 kΩ to 1 MΩ. It is desirable for resonators, especially those at higher frequencies, to have lower R1. Reducing R1 (i.e., increasing output current from an input voltage) involves increasing electrode area, improving electromechanical coupling, and optimizing the mode shape of the resonator. High Q reduces the close-to-carrier phase noise. Ensuring high Q requires minimizing the magnitude of the loss mechanisms. It is necessary to minimize both the intrinsic and extrinsic loss mechanisms. The design-dependent extrinsic losses are reduced through mode isolation (from other modes) and designing for mode shapes which limit energy leakage into the surrounding medium. Good linearity is also desired in the resonator for a low noise floor. The linearity limit of a resonator is determined by material properties and physical dimensions. It is evident that a large resonator is desirable.
0.2L
0.8L
1.8L
4.1L Figure 2. Typical mode shapes in narrow, wide, and very wide rectangular plates in isotropic materials (ν=0.28).
The objective of this work includes: 1) characterizing the dilation modes of rectangular plates in isotropic and anisotropic materials and 2) understanding whether plate modes can be optimized for the metrics of interest. Finite element analysis is performed in ANSYS using SOLID186 elements and isotropic and anisotropic material models. II.
BULK DILATION MODES
A brief discussion of bulk dilation modes in capacitive resonators follows. The natural frequencies are
fn =
nva , 2L
(1)
where n is the mode number, va is the acoustic velocity and L is the length of the resonator. The motional resistance of a capacitive bulk dilation mode resonator may be simplified to
R1 =
krπ Ei ρ m d 4 , 2 2Qε 2VP A
(2)
where kr is the relative dynamic stiffness, Ei is the elastic modulus in the i direction, ρm is the mass density, d is the gap size, ε is the permittivity of the gap, VP is the polarization voltage, and A is the overlap area of the electrodes. It is evident that increasing A reduces R1 – more current is available when the transduction area is increased. Reducing kr also reduces R1. An interesting observation is that R1 is not directly dependent on frequency. Contrary to the trend in prior art, higher frequency resonators do not necessarily have higher R1.
III.
ISOTROPIC PLATES
The analysis begins by extending the findings on bulk dilation modes of isotropic plates summarized in [2]. In a plate with a small width w (e.g., 0.2L), the mode is entirely uniform along the ends. See Figure 2. As w is increased to 0.8L, some non-uniformity is observed. Lateral motion at the support location exists. For a plate with a width of 1.8L, the most prominent mode is a third mode along the x direction. For a width of 4.1L, the most prominent mode is a seventhorder mode along x. Although the frequencies of these modes are very similar, the latter three modes are in three different modal families. Focusing on the seventh-order modal family of an isotropic material with a Poisson ratio ν of 0.28, observe the attributes of the mode as the width is increased. See Figure 3. For w=3.5L, there are out-of-phase regions which cause charge cancellation. The mode in this geometry has poor coupling and a large kr. For w=4.1L, the displacement is somewhat uniform. With this geometry, the ux motion at the support is minimal. For w=4.3L, the displacement is most uniform. This is the optimal plate geometry for coupling and the kr is 1.2. Further increasing w to 4.8L leads to a more wavelike mode with substantial ux motion at the support. In these thin isotropic plates, the optimal coupling and optimal support isolation occur for different values of w. The solutions for isotropic plates having widths of [0,10]L, thickness t of L/5, and ν = 0.28 are summarized in Figure 4. kr and normalized ux (to the average uy) are plotted. The minima for kr, for w>2L, are approximately 1.2. There are geometries in which the support ux is small. However, ux is quite sensitive to the width.
kr is a comparison between the dynamic stiffness kn and the stiffness of a slender ideal 1-dimensional resonator.
kr =
kn kn ,1D
(3)
w = 3.5L
It is strongly dependant on the mode shape. One goal of this work is to understand the effect of increasing A on kr. The dynamic stiffness kn for each mode is found from the energy in the mode W and the average of the displacement of the resonator electrode surface normal to the fixed electrode uy. For a low kn (and consequently low kr), all the energy in the mode is preferably attributed to uy. kn =
2W uy
2
(4)
w = 4.1L
w = 4.3L
w = 4.8L Figure 3. Variation of mode shape within a modal family for width of 3.5L, 4.1L, 4.3L, and 4.8L (isotropic, ν=0.28).
3
Relative Dynamic Stiffness k r
2.75 2.5 2.25 2 1.75 1.5 1.25 1 0.75 0.5
3D isotropic, ν =0.28, t= L/5 0
0.5
1
1.5
2
2.5
3
3.5
4
4.5 5 5.5 Width [L]
6
6.5
7
7.5
8
8.5
9
9.5
10
2.5
3
3.5
4
4.5 5 5.5 Width [L]
6
6.5
7
7.5
8
8.5
9
9.5
10
2.5
3
3.5
4
4.5 5 5.5 Width [L]
6
6.5
7
7.5
8
8.5
9
9.5
10
0.4 0.35 0.3 | ux,supp / uy,avg |
0.25 0.2 0.15 0.1 0.05 0 -0.05 -0.1
3D isotropic, ν =0.28, t= L/5 0
0.5
1
1.5
2
1.25
Normalized Frequency, fn / f1D
1.2 1.15 1.1 1.05 1 0.95 0.9 0.85 0.8 0.75
3D isotropic, ν =0.28, t= L/5 0
0.5
1
1.5
2
Figure 4. Relative dynamic stiffness, normalized support motion, and normalized frequency in isotropic plates with ν=0.28 and a thickness of L/5.
IV.
ANISOTROPIC PLATES
The analysis for anisotropic plates is performed similarly. A very interesting observation is that uniform modes are observed in plates aligned to the ‹110› direction in (100) silicon. A thorough analysis was completed for plates with t equal to L/5 and L/2. The mode shape for L/5-thick plates with w=[3.4,4.1]L are very uniform (Figure 5). For w=3.7L, the mode has the smallest ux response at the support. The kr and normalized support ux are shown in Figure 6.
w = 3.4L
There are several advantages for these anisotropic plates. First, kr is ~1 for a large variety of widths. Next, ux is smaller compared to isotropic plate modes. The ux minima coincide with the kr minima. Also, ux has low sensitivity to w.
w = 3.7L
Increasing the thickness to L/2 reveals further advantages. See Figure 7. Compared to L/5 plates, the range of widths for which kr ≈ 1 is broader. The ux also have lower sensitivity to w. The most beneficial attribute is that the transduction area is increased without increasing die size.
w = 4.1L Figure 5. Variation of mode shape within a modal family for width of 3.4L, 3.7L, and 4.1L (anisotropic, t=L/5).
3
Relative Dynamic Stiffness k r
2.75 2.5 2.25 2 1.75 1.5 1.25 1 0.75 0.5
3D anisotropic, (100) 〈 110〉 t =L/5 0
0.5
1
1.5
2
2.5
3
3.5
4
4.5 5 5.5 Width [L]
6
6.5
7
7.5
8
8.5
9
9.5
10
3
3.5
4
4.5 5 5.5 Width [L]
6
6.5
7
7.5
8
8.5
9
9.5
10
0.4 0.35 0.3 | ux,supp / uy,avg |
0.25 0.2 0.15 0.1 0.05 0 -0.05 -0.1
3D anisotropic, (100) 〈 110〉 t =L/5 0
0.5
1
1.5
2
2.5
Figure 6. Relative dynamic stiffness and normalized support motion in anisotropic plates aligned to ‹110› in (100) silicon having a thickness of L/5.
3
Relative Dynamic Stiffness k r
2.75 2.5 2.25 2 1.75 1.5 1.25 1 0.75 0.5
3D anisotropic, (100) 〈 110〉 t =L/2 0
0.5
1
1.5
2
2.5
3
3.5
4
4.5 5 5.5 Width [L]
6
6.5
7
7.5
8
8.5
9
9.5
10
3
3.5
4
4.5 5 5.5 Width [L]
6
6.5
7
7.5
8
8.5
9
9.5
10
3
3.5
4
4.5 5 5.5 Width [L]
6
6.5
7
7.5
8
8.5
9
9.5
10
0.4 0.35 0.3 | ux,supp / uy,avg |
0.25 0.2 0.15 0.1 0.05 0 -0.05 -0.1
3D anisotropic, (100) 〈 110〉 t =L/2 0
0.5
1
1.5
2
2.5
1.25
Normalized Frequency, fn / f1D
1.2 1.15 1.1 1.05 1 0.95 0.9 0.85 0.8 0.75
3D anisotropic, (100) 〈 110〉 t =L/2 0
0.5
1
1.5
2
2.5
Figure 7. Relative dynamic stiffness, normalized support motion, and normalized frequency in anisotropic plates aligned to ‹110› in (100) silicon having a thickness of L/2.
Periodicity is evident from the results. Widths of even multiples of L provide small motion at the supports and good coupling. The optimal widths for minimizing support loss are 2.0L, 3.9L, 5.8L, 7.7L, and 9.7L. V.
EXPERIMENTATION
Resonators with L=40μm and t=L/2 were fabricated using the HARPSS-on-SOI process [3]. A resonator with a width of 9.7L and 225-nm gaps is shown in Figure 8. The resonance frequency of this resonator was measured to be 103 MHz along with a quality factor of 90000. See Figure 9. The optimized design enabled the f-Q product of ~1×1013.
CONCLUSIONS Bulk dilation modes of plates aligned to the ‹110› direction in (100) silicon were found to have improved attributes over plates in isotropic materials. In anisotropic plates, the relative dynamic stiffness is unity for many geometries. For a broad range of widths, a 1-dimensional assumption applies. The thorough exercise also revealed specific widths at which the motion of the support location is minimal. This enabled a 100-MHz resonator to demonstrate an f-Q product of 1×1013. Future work includes characterizing modes in plates with larger thickness. REFERENCES
Resonator
Anchor
[1]
Input/Output pad
[2]
[3]
Input/Output pad
L = 40 μm w = 9.7 L
t = 20 μm d = 225 nm
Figure 8. Scanning electron micrograph of a resonator fabricated using the HARPSS-on-SOI process.
f0 = 103 MHz Qu = 90000
Figure 9. Transmission frequency response of the resonator showing an unloaded Q of 90000 at a frequency of 103 MHz.
C. T.-C. Nguyen, “MEMS technology for timing and frequency control,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control, vol. 54, pp. 251-270, Feb. 2007. R. Holland, “Contour extensional resonant properties of rectangular piezoelectric plates”, IEEE Trans. Son. Ultrason., SU-15, n. 2, pp97105, Apr 1968 S. Pourkamali, G. K. Ho, and F. Ayazi, “Low-Impedance VHF and UHF Capacitive Silicon Bulk Acoustic Wave Resonators – Part I: Concept and Fabrication”, IEEE Trans. Elec. Dev., vol. 54, n. 8, pp. 2017-2023, Aug. 2007.
