Geometry optimization for micro-pressure sensor considering dynamic interference Zhongliang Yu, Yulong Zhao, Lili Li, Bian Tian, and Cun Li Citation: Review of Scientific Instruments 85, 095002 (2014); doi: 10.1063/1.4895999 View online: http://dx.doi.org/10.1063/1.4895999 View Table of Contents: http://scitation.aip.org/content/aip/journal/rsi/85/9?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Design optimization of high pressure and high temperature piezoresistive pressure sensor for high sensitivity Rev. Sci. Instrum. 85, 015001 (2014); 10.1063/1.4856455 Incorporation of beams into bossed diaphragm for a high sensitivity and overload micro pressure sensor Rev. Sci. Instrum. 84, 015004 (2013); 10.1063/1.4775603 The design and analysis of beam-membrane structure sensors for micro-pressure measurement Rev. Sci. Instrum. 83, 045003 (2012); 10.1063/1.3702809 Monolithic rheometer plate fabricated using silicon micromachining technology and containing miniature pressure sensors for N 1 and N 2 measurements J. Rheol. 47, 1249 (2003); 10.1122/1.1595095 Micromachined ultrasharp silicon and diamond-coated silicon tip as a stable field-emission electron source and a scanning probe microscopy sensor with atomic sharpness J. Vac. Sci. Technol. B 16, 3185 (1998); 10.1116/1.590348
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REVIEW OF SCIENTIFIC INSTRUMENTS 85, 095002 (2014)
Geometry optimization for micro-pressure sensor considering dynamic interference Zhongliang Yu, Yulong Zhao,a) Lili Li, Bian Tian, and Cun Li State Key Laboratory for Manufacturing Systems Engineering, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, China
(Received 20 May 2014; accepted 6 September 2014; published online 24 September 2014) Presented is the geometry optimization for piezoresistive absolute micro-pressure sensor. A figure of merit called the performance factor (PF) is defined as a quantitative index to describe the comprehensive performances of a sensor including sensitivity, resonant frequency, and acceleration interference. Three geometries are proposed through introducing islands and sensitive beams into typical flat diaphragm. The stress distributions of sensitive elements are analyzed by finite element method. Multivariate fittings based on ANSYS simulation results are performed to establish the equations about surface stress, deflection, and resonant frequency. Optimization by MATLAB is carried out to determine the dimensions of the geometries. Convex corner undercutting is evaluated. Each PF of the three geometries with the determined dimensions is calculated and compared. Silicon bulk micromachining is utilized to fabricate the prototypes of the sensors. The outputs of the sensors under both static and dynamic conditions are tested. Experimental results demonstrate the rationality of the defined performance factor and reveal that the geometry with quad islands presents the highest PF of 210.947 Hz1/4 . The favorable overall performances enable the sensor more suitable for altimetry. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4895999] I. INTRODUCTION
Microelectromechanical systems (MEMS) technologies have drawn great attention in recent years, due to the small size, low weight, low power consumption, high robustness, and high performance.1 The first micromechanical device based on modern micromachining techniques was developed in the 1960s.2 Significant commercial production of MEMS started in the 1980s with the appearance of pressure sensors. In the 1990s, microaccelerometers, microfluidic devices, and optical MEMS achieved batch production successively. Micromirrors are now entering the volume production stage. MEMS switches are the next predictable products. Among these devices, the pressure sensor is one of the best developed MEMS devices in use today.3 Especially, the piezoresistive pressure sensor has been applied widely due to its excellent linearity, fine sensitivity, and simple and direct signal transduction mechanism between the mechanical and the electrical domains.4–6 These merits bring about the high robustness, because of which a number of piezoresistive pressure sensors are desired for micro-pressure measurements in aerospace engineering.7–11 According to the relationship between pressure and height, the aircraft altimetry can be obtained through measuring pressure. Due to the extremely low pressure in high altitude, high sensitivity is needed to ensure the accuracy of orbital correction. A high overload resistance is also required for a micro-pressure sensor to suffer atmosphere on the earth. To develop a micro-pressure sensor with high sensitivity and overload resistance is of importance and necessity for aerospace. Moreover, the vibration interfera) Author to whom correspondence should be addressed. Electronic mail:
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0034-6748/2014/85(9)/095002/8/$30.00
ence should be taken into account to improve the accuracy of pressure measurements. In some degree the chip geometry determines the sensitivity, overload resistance, and dynamic properties.9, 10, 12, 13 The lower the pressure range is, the thinner the diaphragm is needed to maintain high sensitivity. However, excessively thin membrane may induce large deflection and instability leading to unfavorable performances of a sensor such as linearity, resonant frequency, safety factor, and etcetera.14 Therefore, the geometry design and optimization of a sensor chip are important. To improve the performances of the sensors, a variety of sensing geometries have been designed. For instance, the geometry with a center boss on the diaphragm and an annular groove formed on the back surface; the beam-diaphragm geometry by introducing beams on the flat membrane of the twin isles geometry forming a shape like dumbbell; the ribbed and bossed geometry via incorporating rib into the diaphragm; the beam-membrane geometry through etching a cross beam on the flat diaphragm.8, 15–17 However, the existing schemes discussed feature unsatisfactory performances in sensitivity and overload resistance. Besides, the dynamic interference is not involved. To improve the overall performance, both the geometry design and the optimization method are critical. In this investigation, three geometries are put forward for the measuring range of 500 Pa. By incorporating beams into the diaphragm, stresses are expected to be concentrated. High overload resistance is in prospect due to the introduction of islands to limit the displacement. Nevertheless, the introduced islands will make the vibration interference protrude. Taking the vibration influences into account is important for improving the accuracy of micro-pressure sensors under dynamic conditions. To describe the comprehensive performance of a sensor including sensitivity, resonant frequency, and
85, 095002-1
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acceleration interference, a performance factor is established. To get the optimal geometry, a general optimization method is raised. As no theoretical formula is needed to be derived, the method can be widely applied rather than limited to the geometries proposed. The advantages are obvious especially when the theoretical formula is hard to be deduced. By converting the nonlinear problem into linear, the optimization is much more efficient. II. PERFORMANCE FACTOR DEFINITION
As absolute micro-pressure sensors always have to be exposed to atmosphere, the introduction of islands is indispensible for suffering the high overload. However, the introduced mass decreases the natural frequency and increases the acceleration interference. To describe the comprehensive performances of a sensor including sensitivity, resonant frequency, and acceleration interference, a figure of merit called the performance factor (PF) is defined as a quantitative assessment:
PF = SNR ·
4
f =
Ufp · 4 f, Ufa
1 π (σ − σy )Ui , 2 44 x
(1)
(2)
where Uf is the full scale output voltage, Ui is the input voltage, and π 44 is the shearing piezoresistance coefficient. σ x , σ y are the longitudinal and transversal surface stresses at the central point of a resistor. The full scale output Uf can also be expressed as Uf = S · R · Ui ,
SNR = SNR =
(3)
Ufp Ufa
Ufp Ufa
=
≈
σdp Uip σda Uia
Sp Rp Uip Sa Ra Uia
= K1
σdp σda
= K1 K2
(4)
,
Sp Sa
,
(5)
where σ dp , σ da are the difference of longitudinal and transversal surface stresses at the center of a resistor under pressure and acceleration interference applied, respectively. Sp , Sa ; Rp , Ra ; Uip , Uia are the sensitivity, measuring range, and input voltage under the pressure and acceleration interference exerted separately. K1 is defined as Uip /Uia , and usually the input voltages Uip , Uia are equal, so K1 is 1. K2 is defined as Rp /Ra . Since the measuring range Rp , Ra are 500 Pa and 15 g in this investigation, K2 can be calculated as 3.4. Namely, both K1 , K2 can be regarded as constant here. According to Eqs. (4) and (5), the PF in Eq. (1) can be rewritten as PF =
where Ufp , Ufa are the full scale output voltages under pressure and acceleration interference applied, respectively. SNR is the signal-to-noise ratio defined as Ufp /Ufa . f is the resonant frequency, and the fourth root of it is extracted to make the effect weight between f and SNR in the same order of magnitude. The relationship between full scale output and stress can be expressed as follows (take the resistor oriented in direction on a (100) n-type silicon wafer, for example):18, 19 Uf ≈
where S is the sensitivity and R is the measuring range. Based on Eqs. (2) and (3), the SNR can be written as
Sp σdp Ufp · 4 f = K1 K2 · 4 f ≈ K1 · 4 f. Ufa Sa σda
(6)
The defined PF in Eq. (6) is favorable to reflecting an overall performance including sensitivity and dynamic properties. III. GEOMETRY ANALYSIS A. Geometry establishment
Since absolute micro-pressure sensors have to bear atmosphere on the earth, which is hundreds of times higher than measuring range, the silicon geometry can be easily fractured under such a high overload. In view of the situation, the typical bossed-diaphragm (E-type) should be taken into account. Due to the mass bulk’s support, the membrane may stand atmosphere without breaking. However, the introduction of mass bulk partly sacrifices the effective stress that reflects sensitivity, and makes the sensor sensitive to vibration interfering signal. Attempt to increase the sensitivity, bandwidth and decrease the dynamic interfering signal, three geometries, namely, beam-membrane-mono-island (BMMI), beam-membrane-dual-island (BMDI), and beam-membranequad-island (BMQI) are raised as shown in Fig. 1. Sensitive
FIG. 1. The schematic diagram of the front and cross-sectional views of the three geometries.
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beams are located on the membrane and islands are placed in the rear cavity. B. Geometry optimization
In order to optimize and determine the geometry dimensions, formulas need to be established. As the existence of beams on the membrane, theoretical formulas are difficult to derive, while approximate ones can be drawn by the combination of FEM calculation and multivariate fitting. For convenience of illustration, the front views and the cross-sectional views in Fig. 1 are marked with dimension variables. L1 , L2 , L4 refer to the effective width of membranes; H1 , H2 , H4 refer to the thickness of membranes; I1 , I2 , I4 refer to the top width of islands; D2 , D4 refer to the distance of two opposite islands; W1 , W2 , W4 refer to the width of beams; B1 , B2 , B4 refer to the thickness of beams; and t2 , t4 refer to the length of sensitive beams at each side. The numbers 1, 2, 4 in the subscripts of variables represent the BMMI, BMDI, BMQI geometries, respectively, and this principle applies throughout. The mechanical stress and the maximum deflection of typical flat diaphragm geometry are the power functions of each variable.20 Similarly, the functional forms of the three geometries might be the same. The differential stress of BMMI is assumed as follows: σdp1 = K · B1a · H1m · I1n · Lr1 · W1s ,
(7)
where σ dp1 is the difference of x and y direction stress at the center of a resistor (as shown in Fig. 1) when a 500 Pa pressure is applied. B1 , H1 , I1 , L1 , W1 are the independent dimension variables chosen from the variables described above; K, a, m, n, r, s are the undetermined constants. To ascertain the constants, the variation of σ dp1 with variables should be studied by ANSYS loop computation using the standard (100) silicon wafer material properties described in Ref. 21. In the calculation, three values for each variable are assigned. Therefore, 243 loops are needed to cover the entire variable space. Based on the results, multivariate fitting by MATLAB is carried out. For the simplification, nonlinear fitting is transformed to linear via taking the logarithm of Eq. (7): ln(σdp1 ) = ln(K) + a · ln(B1 ) + m · ln(H1 ) + n · ln(I1 ) + r · ln(L1 ) + s · ln(W1 ).
(8)
The parameters after logarithm can be regarded as new variables. Hence, a multiple linear regression problem that costs much less is raised and easily solved. To get the constant K in Eq. (7), a nature exponential of the constant item ln(K) in Eq. (8) should be taken. The fitted equation concerning the differential stress σ dp1 is obtained: σdp1 = 39.48159
L3.40878 1 . B10.52501 H11.40483 I10.67321 W10.611
(9)
In the same manner, the equations regarding the variables σ dp2 , σ dp4 , σ da1 , σ da2 , σ da4 , ω1 , ω2 , ω4 , σ overload1 , σ overload2 , σ overload4 , f1 , f2 , f4 can be established. The variables σ dp2 , σ dp4 refer to the same meaning as σ dp1 ; σ da1 , σ da2 , σ da4 refer to the difference of x and y direction stress at the center of a resistor
under 15 g acceleration applied along the normal direction of membrane; ω1 , ω2 , ω4 refer to the maximum deflection at the center of membrane under the pressure of 500 Pa; σ overload1 , σ overload2 , σ overload4 refer to the maximum von Mises stress under an atmospheric pressure of 100 kPa; and f1 , f2 , f4 refer to the resonant frequency. The numbers in subscripts are used for the discrimination of different geometries as stated previously. Based on the equations above, the performance factors PF1 , PF2 , PF4 can be got according to Eq. (6): PF1 = 8.74875
W10.03367 B10.06486 H10.09918 L0.53685 1 , I11.21308 W20.02749 B20.03479 L0.95913 2 , H20.0975 I21.56925 t20.15535
(11)
L1.64431 W40.00268 4 . B40.02583 H40.06344 I42.0849 t40.2659
(12)
PF2 = 0.10174
PF4 = 3.31608 × 10−2
(10)
During establishment of the equations for BMDI and BMQI, three values for each variable are assigned, thus 729 loops are needed to involve the whole variable space. Specifications about the equations discussed are that the ranges of all the variables are constrained by actual demands. Besides, the international system of units is adopted throughout. To search for the optimal overall performances, geometry optimization models are built and listed in Table I, where the unit micrometer is adopted for the dimension variables. In the table, σ b is the ultimate strength of single crystal silicon; n1 , n2 , n4 are the safety factors. According to the small deflection theory, the nonlinearity below 1% FS can be achieved if the maximum deflection of the flat diaphragm is kept under one fifth of the film thickness.22 For rough reference, the same evaluation of maximum deflection as a constraint is employed in the models. Through taking the natural logarithms of objective functions and constraints, equivalent linear optimization problems that apparently simplify computation are raised. MATLAB is used to search for the optimal solutions of the three geometries, and the value of dimension variables is got via taking nature exponentials of the optimization results. TABLE I. The optimization models about the performance factors of the geometries. BMMI Max (PF1 ) subject to 20 ≤ B1 ≤ 30 10 ≤ H1 ≤ 20 2300 ≤ I1 ≤ 3000 5000 ≤ L1 ≤ 5700 100 ≤ W1 ≤ 200 ω1 ≤ 0.2H1 σ overload1 ≤ σ b /n1 σ b = 7 Gpa n1 = 15
BMDI
BMQI
Max (PF2 ) subject to 20 ≤ B2 ≤ 50 20 ≤ H2 ≤ 30 1500 ≤ I2 ≤ 1600 5000 ≤ L2 ≤ 5700 100 ≤ W2 ≤ 200 100 ≤ t2 ≤ 1200 ω2 ≤ 0.2H2 σ overload2 ≤ σ b /n2 σ b = 7 Gpa n2 = 11
Max (PF4 ) subject to 30 ≤ B4 ≤ 50 20 ≤ H4 ≤ 30 1500 ≤ I4 ≤ 1600 5000 ≤ L4 ≤ 5700 100 ≤ W4 ≤ 200 100 ≤ t4 ≤ 400 ω4 ≤ 0.2H4 σ overload4 ≤ σ b /n4 σ b = 7 Gpa n4 = 4
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FIG. 2. The analysis and calculation models about convex corner undercutting.
Under the given constraints and safety factors in Table I, the sizes of geometries are determined. The three geometries feature the identical 7000 μm × 7000 μm overall dimensions, 20 μm thickness membranes, 200 μm width, and 30 μm thickness beams. The lengths of sensitive beams of BMMI and BMQI are 1700 μm and 200 μm. The lengths of sensitive beams of BMDI at each side and the center are 980 μm and 750 μm. The top widths of islands are 2300 μm, 1500 μm, 1500 μm corresponding to BMMI, BMDI, BMQI successively. The uniform geometry dimensions are acquired by adjusting the constraints and safety factors in Table I, which makes it obvious for PF comparison later. Certainly, the constraints and safety factors can be modified according to the specific optimization situations. C. Convex corner undercutting analysis
Under the determined dimensions above, the islands of BMDI and BMQI lack enough space for compensation, thus
the convex corner (CC) undercutting occurs. The final dimensions of the islands are roughly estimated using the fast etching planes stated in Ref. 23, to which the etching concentration and temperature have been referred. For simplification, the fast etching planes are assumed as {4 1 1} all the time. According to the literature, the etching depth is 0.544 times of side length of the compensation structure. Based on the relationship, the etching depth when the compensation structure consumed can be calculated. In Figs. 2(a) and 2(e), h21 , h41 representing the midway etching depth of BMDI, BMQI are 204 μm, 108.8 μm. By this time, complete corners have formed simultaneously on both the bottom and midway root planes of islands. Points A20m , A40m , A21m , A41m represent the vertices of corners on the bottom and midway root planes of BMDI, BMQI. This moment, A20m A21m , A40m A41m coincides with the intersection lines of {1 1 1} planes. With the etching going on, A20m A21m , A40m A41m will move parallel to these intersection lines, although the corners are undercut by etching front lines in directions. This is because the corners
FIG. 3. Simulation results about stress distribution and stress path of the three geometries.
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TABLE II. The simulation and calibration results of the sensors.
Ufp (mV/500Pa) Ufa (mV/15g) f (Hz) PF (Hz1/4 )
E-type ANSYS
BMMI ANSYS
BMMI Experiments
BMDI ANSYS
BMDI Experiments
BMQI ANSYS
BMQI Experiments
15.066 1.132 4802.1 110.8192
19.996 1.644 6936.6 110.9798
17.482 1.448 7375 111.8827
29.307 1.674 6242.7 155.6142
24.171 1.351 6950 163.3561
29.311 1.148 10686 259.6504
26.693 1.274 10275 210.947
from the bottom to the midway root are exposed for the same etching time. As for the corners from the midway root plane to the final root plane, they will be undercut by {4 1 1} planes. That is, the vertices of corners from the midway root to the final root planes are always on the intersection lines of {4 1 1} planes. In Figs. 2(a) and 2(e), the points A20 , A40 ; A21 , A41 ; and A22 , A42 represent the final vertices of corners on the bottom, midway root, and final root planes of BMDI and BMQI. A20 , A 40 ; A 20 , A 40 are the projections of A20 , A40 on the midway root and final root planes of BMDI and BMQI. h22 , h42 are the etching depths between the midway root and final root planes of BMDI and BMQI which are 146 μm, 241.2 μm due to the total etching depth of 350 μm. α is the angle between the lines of A20 A21 , A40 A41 , and {1 0 0} planes and is 45◦ , and β is the angle between the lines of A21 A22 , A41 A42 , and {1 0 0} planes and is 75.96◦ . According to the conditions, the length of A21 A 20 , A22 A 20 , A41 A 40 , A42 A 40 can be calculated. In Figs. 2(b) and 2(f), dc20 , dc40 are the vertical distances from points O20 , O40 to the etching front lines A20 M20 , A40 M40 , and they can be calculated by multiplying h22 , h42 with 1.46 on the basis of Ref. 23. According to the geometrical relationship, A20 N20 , C20 N20 , C40 N40 are the side lengths of island bottom of BMDI and BMQI and are 302 μm, 171 μm, 284 μm. Based on these bottom dimensions and the length of A21 A 20 , A22 A 20 , A41 A 40 , A42 A 40 calculated above, the dimensions of the midway root and final root of islands can be estimated. In Figs. 2(c) and 2(g), A 21 N 21 , C 21 N 21 , C 41 N 41 are the side lengths of the midway root of BMDI, BMQI and are 302 μm, 460 μm, 373 μm, and A22 N22 , C22 N22 , C42 N42 are the side length of the final root
of BMDI, BMQI and are 457 μm, 245 μm, 423 μm. The estimated island models after CC undercutting are shown in Figs. 2(d) and 2(h).
FIG. 4. The pressure calibration system.
FIG. 5. The acceleration and modal calibration setup.
D. Performances evaluation
To evaluate the performances of the three sensors with the calculated dimensions (considering the influence of CC undercutting), ANSYS is used. The simulation results about the distribution of von Mises stress on sensitive beams and the stress path along the x-axis from the center to the edge of sensitive beams under the pressure of 500 Pa are shown in Fig. 3. To assess the influence of vibration on pressure measurements, both acceleration analysis and modal analysis are carried out. In the simulation for acceleration analysis, a maximum acceleration of 15 g is exerted due to the human extreme limit. Both the static and dynamic analysis results above are listed in Table II. Besides, the E-type is analyzed as well. For obvious comparison, the dimensions of E-type are equal with the ones of BMMI except for the beam. In the table, the full scale outputs under pressure and acceleration are derived by Eq. (2) based on the simulated differential stress. In this study, the concentration of ion implantation is 3 × 1014 cm−3 less than 1 × 1017 cm−3 , so π 44 in Eq. (2) is 138 × 10−7 cm2 /N.24
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FIG. 6. Experimental results of the output voltage versus applied pressure.
FIG. 7. Experimental results of the output voltage versus applied acceleration.
V. RESULTS AND DISCUSSION
IV. EXPERIMENTS
To test the static characterization of the sensors, a complete experimental setup is built as shown in Fig. 4. The compressor acts as a pressure source. The sensors are calibrated with a reference pressure monitor (FLUKE A100K), excited by a 3 V DC power supply (RIGOL DP1116A), and the outputs are measured by a multi-meter (KEITHLEY 2000). To assess the dynamic performances approximately, another three sensor dies corresponding to BMMI, BMDI, BMQI with a through-hole on the glass base are utilized. The hole makes the pressures inside and outside the cavity equal, thus the applied atmospheric pressure is equivalent to zero, and the sensor chips are only affected by vibration acceleration, which is convenient for dynamic experiments. A stable centrifugal machine is used for acceleration calibration along the normal direction of the membrane of the three sensors as shown in Fig. 5. Through changing the rational speed, acceleration up to 15 g is imposed. The natural frequency is calibrated through testing the three sensors with a hole. By fixing one of the three tested sensors and a reference sensor on a shaker, as shown in Fig. 5, a peak concerning the voltage ratio will be generated when a sine sweep frequency passes through.
The pressure calibration results are plotted in Fig. 6, where the output voltage as a function of pressure is presented. The pressure is varying from 20 Pa to 500 Pa at room temperature. In the figure, the calibrated data of 5-round journey are described with different kinds of lines fixed by least square fitting. The standard errors of testing points are marked by error bars, and the maximum standard errors are enlarged. The acceleration calibration results are drawn in Fig. 7, in which the acceleration up to 15 g with an interval of 2.5 g is imposed. The measuring results are described by least square method as well, and the maximum standard errors of the testing points are enlarged. In the modal calibration, the peaks induced by the resonance are drawn in Fig. 8. Although slightly different damping factors and minor fabrication variations are existent, the acceleration and modal experiments are still significant for evaluating the dynamic performances of the sensors under near-vacuum condition. The experimental results above are listed in Table II. In the table, the full scale outputs about pressure and acceleration are calculated based on the least square method. The PF
FIG. 8. The modal calibration results.
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is derived from Eq. (6), and it reveals that the BMQI features the highest comprehensive performance. The PF of BMQI is 29.1% higher than BMDI and 88.5% higher than BMMI. The simulation results show that the sensitivity of BMMI is 32.7% higher than the E-type one, and the natural frequency is 44.4% higher. However, the PF of BMMI has been dragged down to be nearly the same by the 45.2% higher acceleration interference. The estimated performance factors are consistent with the actually calibrated ones, so they can be used for evaluating the overall performance of the absolute micro-pressure sensors. The performance factor PF is significant for assessing the comprehensive performance including dynamic interference. VI. CONCLUSIONS
This work attempts to explore the assessment for the performance of absolute micro-pressure sensors considering vibration influences. A performance factor has been proposed to try to reflect the comprehensive performance. To validate the rationality of the defined performance factor, three geometries have been proposed, optimized, and actually fabricated. Experimental results have demonstrated that the performance factor is of significance for evaluating the overall property of a sensor. ACKNOWLEDGMENTS
This work is supported by the National Science Foundation (NSF) for Distinguished Young Scholars of China (No. 51325503), Young Scientists Fund of the National Natural Science Foundation of China (NNSFC) (No. 51305336), National High Technology Research and Development Program of China (863 Program) (No. 2013AA041108), the Program for Changjiang Scholars and Innovative Research Team in University of China (No. IRT1033), and Postdoctoral Science Foundation of China (No. 2013M532036).
TABLE III. The determination coefficients of the related equations. Equation related
R2
σ dp1 σ da1 ω1 σ overload1 f1
0.96793 0.96261 0.98129 0.97664 0.97572
Equation related σ dp2 σ da2 ω2 σ overload2 f2
R2 0.96030 0.96423 0.96007 0.99241 0.96476
Equation related σ dp4 σ da4 ω4 σ overload4 f4
R2 0.96686 0.97484 0.978214 0.98751 0.98308
APPENDIX A: THE DETERMINATION COEFFICIENT
To validate the rationality of the hypothesis about the functional forms, the coefficients of determination R2 are calculated and listed in Table III which has verified the goodness of fit, thus ANSYS simulation results can be almost represented by the equations established previously. APPENDIX B: FABRICATION
The sensors are all fabricated by the bulk micromachining, using the standard double side polished n-type (100) silicon wafers. The resistivity of the silicon wafers is 6000∼8000 · cm and the thickness is 400 μm. The specific fabrication process flow is illustrated in Fig. 9. (a) Photolithography is employed to pattern piezoresistors on the front side of the silicon wafer, after SiO2 layers are grown on both sides of the substrate by thermal oxidation. Ion implantation of boron is carried out with a concentration of 3 × 1014 cm−3 forming a sheet resistance of 220 . (b) Heavy boron ion diffusion is performed to consolidate the connections of piezoresistors. (c) The passivation layers of Si3 N4 , SiO2 are deposited by means of low pressure chemical vapor deposition (LPCVD) and plasma enhanced chemical vapor deposition (PECVD) successively. (d) Contacts are then photo-patterned and etched on the front side
FIG. 9. The schematic of the main process flow.
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to sputter Au. Ohmic contacts between Au wires and piezoresistors are reinforced by sintering process. (e) For creating cavities, forming islands, and reducing the height of islands, KOH etching is used on the backside of the wafers after patterned. The etching is carried out in the pure aqueous KOH solution with a KOH content of 30 wt. % at the temperature of 80 ± 1 ◦ C, resulting in an etching rate of 1.0 μm/min. (f) The anti-adsorption electrodes made of Cr are sputtered on the glass. Then the backside of the wafer is attached to the Pyrex 7740 glass under vacuum condition by anodic bonding process to create absolute pressure sensors. (g) Inductively coupled plasma (ICP) etch is involved to form beams on the front side. The fabricated sensor dies are shown in Fig. 10 where SEM images of the three sensor chips are displayed. The CCD photographs of the undercut islands in the rear cavity of BMDI and BMQI are presented in Fig. 11. The calculated CC undercutting conforms well to the actual fabrication except for some minor difference on the root, which is caused by the assumed unchanging fast etching planes. 1 J.
W. Judy, Smart Mater. Struct. 10(6), 1115–1134 (2001). N. Tufte, D. Long, and P. W. Chapman, J. Appl. Phys. 33(11), 3322– 3327 (1962). 3 W. P. Eaton and J. H. Smith, Smart Mater. Struct. 6(5), 530–539 (1997). 4 A. A. Barlian, W.-T. Park, J. R. Mallon, Jr., A. J. Rastegar, and B. L. Pruitt, IEEE Proc. 97(3), 513–552 (2009). 5 S. M. Sze, Semiconductor sensors (John Wiley and Sons Inc, New York, 1994), pp. 17–80. 6 L. Chang, Foundations of MEMS (China Machine Press, Beijing, 2008), pp. 223–224. 7 H. S. Ko, C. W. Liu, and C. Gau, Sens. Actuators, A 138(1), 261–267 (2007). 8 J. K. Reynolds, D. Catling, R. C. Blue, N. I. Maluf, and T. Kenny, Sens. Actuators, A 83(1–3), 142–149 (2000). 9 P. Mackowiak, M. Schiffer, X. Xin, E. Obermeier, and N. Ha-Duong, in Proceedings of the 12th Electronics Packaging Technology Conference (EPTC 2010) (2010), pp. 757–761. 10 B. Tian, Z. Yulong, J. Zhuangde, and H. Bin, Rev. Sci. Instrum. 83(4), 045003 (2012). 11 A. Berns, U. Buder, E. Obermeier, A. Wolter, and A. Leder, Sens. Actuators, A 132(1), 104–111 (2006). 12 Z. Guiming, Z. Libo, J. Zhuangde, Y. Shuming, Z. Yulong, H. Enze, H. Rahman, W. Xiaopo, and L. Zhigang, J. Phys. D 44(42), 425402 (2011). 13 C. Gradolph, A. Friedberger, G. Muller, and J. Wilde, Sens. Actuators, A 150(1), 69–77 (2009). 14 L. Lin, H.-C. Chu, and Y.-W. Lu, J. Microelectromech. Syst. 8(4), 514–522 (1999). 15 M. Shimazoe, Y. Matsuoka, A. Yasukawa, and M. Tanabe, Sens. Actuators 2(3), 275–282 (1982). 16 M. H. Bao, L. Z. Yu, and Y. Wang, Sens. Actuators, A 21(1–3), 137–141 (1990). 17 R. H. Johnson, S. Karbassi, U. Sridhar, and B. Speldrich, Sens. Actuators, A 35(2), 93–99 (1992). 18 W. Quan, D. Jianning, and W. Wenxiang, Sens. Actuators, A 120(2), 468– 473 (2005). 19 S. K. Clark and K. D. Wise, IEEE Trans. Electron Devices 26(12), 1887– 1896 (1979). 20 W. C. Young, Roark’s formulas for stress and strain, 6th ed. (McGraw-Hill, New York, 1986), pp. 440–464. 21 M. A. Hopcroft, W. D. Nix, and T. W. Kenny, J. Microelectromech. Syst. 19(2), 229–238 (2010). 22 S. Timoshenko and S. woinosky-Krieger, Theory of plates and shells, 2nd ed. (McGraw Hill Classic Textbook, New York, 1987). 23 W. Fan and D. C. Zhang, J. Micromech. Microeng. 16(10), 1951–1957 (2006). 24 O. N. Tufte and E. L. Stelzer, J. Appl. Phys. 34(2), 313–318 (1963). 2 O.
FIG. 10. Photos of the fabricated sensor dies. (a) The sensor die and SEM images of BMMI; (b) The sensor die and SEM images of BMDI; (c) The sensor die and SEM images of BMQI.
utilizing reactive ion etching (RIE). In order to activate the boron ion electrically and make dopant uniform, the annealing technology is executed at 1100 ◦ C for 30 min under nitrogen atmosphere. For the connections of resistors and formations of bonding pads, metallization process is performed
FIG. 11. The CCD photograph of the undercut islands on the bottom and root planes. (a) The undercut island of BMDI; (b) The undercut island of BMQI.
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REVIEW OF SCIENTIFIC INSTRUMENTS 85, 095002 (2014)
Geometry optimization for micro-pressure sensor considering dynamic interference Zhongliang Yu, Yulong Zhao,a) Lili Li, Bian Tian, and Cun Li State Key Laboratory for Manufacturing Systems Engineering, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, China
(Received 20 May 2014; accepted 6 September 2014; published online 24 September 2014) Presented is the geometry optimization for piezoresistive absolute micro-pressure sensor. A figure of merit called the performance factor (PF) is defined as a quantitative index to describe the comprehensive performances of a sensor including sensitivity, resonant frequency, and acceleration interference. Three geometries are proposed through introducing islands and sensitive beams into typical flat diaphragm. The stress distributions of sensitive elements are analyzed by finite element method. Multivariate fittings based on ANSYS simulation results are performed to establish the equations about surface stress, deflection, and resonant frequency. Optimization by MATLAB is carried out to determine the dimensions of the geometries. Convex corner undercutting is evaluated. Each PF of the three geometries with the determined dimensions is calculated and compared. Silicon bulk micromachining is utilized to fabricate the prototypes of the sensors. The outputs of the sensors under both static and dynamic conditions are tested. Experimental results demonstrate the rationality of the defined performance factor and reveal that the geometry with quad islands presents the highest PF of 210.947 Hz1/4 . The favorable overall performances enable the sensor more suitable for altimetry. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4895999] I. INTRODUCTION
Microelectromechanical systems (MEMS) technologies have drawn great attention in recent years, due to the small size, low weight, low power consumption, high robustness, and high performance.1 The first micromechanical device based on modern micromachining techniques was developed in the 1960s.2 Significant commercial production of MEMS started in the 1980s with the appearance of pressure sensors. In the 1990s, microaccelerometers, microfluidic devices, and optical MEMS achieved batch production successively. Micromirrors are now entering the volume production stage. MEMS switches are the next predictable products. Among these devices, the pressure sensor is one of the best developed MEMS devices in use today.3 Especially, the piezoresistive pressure sensor has been applied widely due to its excellent linearity, fine sensitivity, and simple and direct signal transduction mechanism between the mechanical and the electrical domains.4–6 These merits bring about the high robustness, because of which a number of piezoresistive pressure sensors are desired for micro-pressure measurements in aerospace engineering.7–11 According to the relationship between pressure and height, the aircraft altimetry can be obtained through measuring pressure. Due to the extremely low pressure in high altitude, high sensitivity is needed to ensure the accuracy of orbital correction. A high overload resistance is also required for a micro-pressure sensor to suffer atmosphere on the earth. To develop a micro-pressure sensor with high sensitivity and overload resistance is of importance and necessity for aerospace. Moreover, the vibration interfera) Author to whom correspondence should be addressed. Electronic mail:
[email protected]
0034-6748/2014/85(9)/095002/8/$30.00
ence should be taken into account to improve the accuracy of pressure measurements. In some degree the chip geometry determines the sensitivity, overload resistance, and dynamic properties.9, 10, 12, 13 The lower the pressure range is, the thinner the diaphragm is needed to maintain high sensitivity. However, excessively thin membrane may induce large deflection and instability leading to unfavorable performances of a sensor such as linearity, resonant frequency, safety factor, and etcetera.14 Therefore, the geometry design and optimization of a sensor chip are important. To improve the performances of the sensors, a variety of sensing geometries have been designed. For instance, the geometry with a center boss on the diaphragm and an annular groove formed on the back surface; the beam-diaphragm geometry by introducing beams on the flat membrane of the twin isles geometry forming a shape like dumbbell; the ribbed and bossed geometry via incorporating rib into the diaphragm; the beam-membrane geometry through etching a cross beam on the flat diaphragm.8, 15–17 However, the existing schemes discussed feature unsatisfactory performances in sensitivity and overload resistance. Besides, the dynamic interference is not involved. To improve the overall performance, both the geometry design and the optimization method are critical. In this investigation, three geometries are put forward for the measuring range of 500 Pa. By incorporating beams into the diaphragm, stresses are expected to be concentrated. High overload resistance is in prospect due to the introduction of islands to limit the displacement. Nevertheless, the introduced islands will make the vibration interference protrude. Taking the vibration influences into account is important for improving the accuracy of micro-pressure sensors under dynamic conditions. To describe the comprehensive performance of a sensor including sensitivity, resonant frequency, and
85, 095002-1
© 2014 AIP Publishing LLC
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acceleration interference, a performance factor is established. To get the optimal geometry, a general optimization method is raised. As no theoretical formula is needed to be derived, the method can be widely applied rather than limited to the geometries proposed. The advantages are obvious especially when the theoretical formula is hard to be deduced. By converting the nonlinear problem into linear, the optimization is much more efficient. II. PERFORMANCE FACTOR DEFINITION
As absolute micro-pressure sensors always have to be exposed to atmosphere, the introduction of islands is indispensible for suffering the high overload. However, the introduced mass decreases the natural frequency and increases the acceleration interference. To describe the comprehensive performances of a sensor including sensitivity, resonant frequency, and acceleration interference, a figure of merit called the performance factor (PF) is defined as a quantitative assessment:
PF = SNR ·
4
f =
Ufp · 4 f, Ufa
1 π (σ − σy )Ui , 2 44 x
(1)
(2)
where Uf is the full scale output voltage, Ui is the input voltage, and π 44 is the shearing piezoresistance coefficient. σ x , σ y are the longitudinal and transversal surface stresses at the central point of a resistor. The full scale output Uf can also be expressed as Uf = S · R · Ui ,
SNR = SNR =
(3)
Ufp Ufa
Ufp Ufa
=
≈
σdp Uip σda Uia
Sp Rp Uip Sa Ra Uia
= K1
σdp σda
= K1 K2
(4)
,
Sp Sa
,
(5)
where σ dp , σ da are the difference of longitudinal and transversal surface stresses at the center of a resistor under pressure and acceleration interference applied, respectively. Sp , Sa ; Rp , Ra ; Uip , Uia are the sensitivity, measuring range, and input voltage under the pressure and acceleration interference exerted separately. K1 is defined as Uip /Uia , and usually the input voltages Uip , Uia are equal, so K1 is 1. K2 is defined as Rp /Ra . Since the measuring range Rp , Ra are 500 Pa and 15 g in this investigation, K2 can be calculated as 3.4. Namely, both K1 , K2 can be regarded as constant here. According to Eqs. (4) and (5), the PF in Eq. (1) can be rewritten as PF =
where Ufp , Ufa are the full scale output voltages under pressure and acceleration interference applied, respectively. SNR is the signal-to-noise ratio defined as Ufp /Ufa . f is the resonant frequency, and the fourth root of it is extracted to make the effect weight between f and SNR in the same order of magnitude. The relationship between full scale output and stress can be expressed as follows (take the resistor oriented in direction on a (100) n-type silicon wafer, for example):18, 19 Uf ≈
where S is the sensitivity and R is the measuring range. Based on Eqs. (2) and (3), the SNR can be written as
Sp σdp Ufp · 4 f = K1 K2 · 4 f ≈ K1 · 4 f. Ufa Sa σda
(6)
The defined PF in Eq. (6) is favorable to reflecting an overall performance including sensitivity and dynamic properties. III. GEOMETRY ANALYSIS A. Geometry establishment
Since absolute micro-pressure sensors have to bear atmosphere on the earth, which is hundreds of times higher than measuring range, the silicon geometry can be easily fractured under such a high overload. In view of the situation, the typical bossed-diaphragm (E-type) should be taken into account. Due to the mass bulk’s support, the membrane may stand atmosphere without breaking. However, the introduction of mass bulk partly sacrifices the effective stress that reflects sensitivity, and makes the sensor sensitive to vibration interfering signal. Attempt to increase the sensitivity, bandwidth and decrease the dynamic interfering signal, three geometries, namely, beam-membrane-mono-island (BMMI), beam-membrane-dual-island (BMDI), and beam-membranequad-island (BMQI) are raised as shown in Fig. 1. Sensitive
FIG. 1. The schematic diagram of the front and cross-sectional views of the three geometries.
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beams are located on the membrane and islands are placed in the rear cavity. B. Geometry optimization
In order to optimize and determine the geometry dimensions, formulas need to be established. As the existence of beams on the membrane, theoretical formulas are difficult to derive, while approximate ones can be drawn by the combination of FEM calculation and multivariate fitting. For convenience of illustration, the front views and the cross-sectional views in Fig. 1 are marked with dimension variables. L1 , L2 , L4 refer to the effective width of membranes; H1 , H2 , H4 refer to the thickness of membranes; I1 , I2 , I4 refer to the top width of islands; D2 , D4 refer to the distance of two opposite islands; W1 , W2 , W4 refer to the width of beams; B1 , B2 , B4 refer to the thickness of beams; and t2 , t4 refer to the length of sensitive beams at each side. The numbers 1, 2, 4 in the subscripts of variables represent the BMMI, BMDI, BMQI geometries, respectively, and this principle applies throughout. The mechanical stress and the maximum deflection of typical flat diaphragm geometry are the power functions of each variable.20 Similarly, the functional forms of the three geometries might be the same. The differential stress of BMMI is assumed as follows: σdp1 = K · B1a · H1m · I1n · Lr1 · W1s ,
(7)
where σ dp1 is the difference of x and y direction stress at the center of a resistor (as shown in Fig. 1) when a 500 Pa pressure is applied. B1 , H1 , I1 , L1 , W1 are the independent dimension variables chosen from the variables described above; K, a, m, n, r, s are the undetermined constants. To ascertain the constants, the variation of σ dp1 with variables should be studied by ANSYS loop computation using the standard (100) silicon wafer material properties described in Ref. 21. In the calculation, three values for each variable are assigned. Therefore, 243 loops are needed to cover the entire variable space. Based on the results, multivariate fitting by MATLAB is carried out. For the simplification, nonlinear fitting is transformed to linear via taking the logarithm of Eq. (7): ln(σdp1 ) = ln(K) + a · ln(B1 ) + m · ln(H1 ) + n · ln(I1 ) + r · ln(L1 ) + s · ln(W1 ).
(8)
The parameters after logarithm can be regarded as new variables. Hence, a multiple linear regression problem that costs much less is raised and easily solved. To get the constant K in Eq. (7), a nature exponential of the constant item ln(K) in Eq. (8) should be taken. The fitted equation concerning the differential stress σ dp1 is obtained: σdp1 = 39.48159
L3.40878 1 . B10.52501 H11.40483 I10.67321 W10.611
(9)
In the same manner, the equations regarding the variables σ dp2 , σ dp4 , σ da1 , σ da2 , σ da4 , ω1 , ω2 , ω4 , σ overload1 , σ overload2 , σ overload4 , f1 , f2 , f4 can be established. The variables σ dp2 , σ dp4 refer to the same meaning as σ dp1 ; σ da1 , σ da2 , σ da4 refer to the difference of x and y direction stress at the center of a resistor
under 15 g acceleration applied along the normal direction of membrane; ω1 , ω2 , ω4 refer to the maximum deflection at the center of membrane under the pressure of 500 Pa; σ overload1 , σ overload2 , σ overload4 refer to the maximum von Mises stress under an atmospheric pressure of 100 kPa; and f1 , f2 , f4 refer to the resonant frequency. The numbers in subscripts are used for the discrimination of different geometries as stated previously. Based on the equations above, the performance factors PF1 , PF2 , PF4 can be got according to Eq. (6): PF1 = 8.74875
W10.03367 B10.06486 H10.09918 L0.53685 1 , I11.21308 W20.02749 B20.03479 L0.95913 2 , H20.0975 I21.56925 t20.15535
(11)
L1.64431 W40.00268 4 . B40.02583 H40.06344 I42.0849 t40.2659
(12)
PF2 = 0.10174
PF4 = 3.31608 × 10−2
(10)
During establishment of the equations for BMDI and BMQI, three values for each variable are assigned, thus 729 loops are needed to involve the whole variable space. Specifications about the equations discussed are that the ranges of all the variables are constrained by actual demands. Besides, the international system of units is adopted throughout. To search for the optimal overall performances, geometry optimization models are built and listed in Table I, where the unit micrometer is adopted for the dimension variables. In the table, σ b is the ultimate strength of single crystal silicon; n1 , n2 , n4 are the safety factors. According to the small deflection theory, the nonlinearity below 1% FS can be achieved if the maximum deflection of the flat diaphragm is kept under one fifth of the film thickness.22 For rough reference, the same evaluation of maximum deflection as a constraint is employed in the models. Through taking the natural logarithms of objective functions and constraints, equivalent linear optimization problems that apparently simplify computation are raised. MATLAB is used to search for the optimal solutions of the three geometries, and the value of dimension variables is got via taking nature exponentials of the optimization results. TABLE I. The optimization models about the performance factors of the geometries. BMMI Max (PF1 ) subject to 20 ≤ B1 ≤ 30 10 ≤ H1 ≤ 20 2300 ≤ I1 ≤ 3000 5000 ≤ L1 ≤ 5700 100 ≤ W1 ≤ 200 ω1 ≤ 0.2H1 σ overload1 ≤ σ b /n1 σ b = 7 Gpa n1 = 15
BMDI
BMQI
Max (PF2 ) subject to 20 ≤ B2 ≤ 50 20 ≤ H2 ≤ 30 1500 ≤ I2 ≤ 1600 5000 ≤ L2 ≤ 5700 100 ≤ W2 ≤ 200 100 ≤ t2 ≤ 1200 ω2 ≤ 0.2H2 σ overload2 ≤ σ b /n2 σ b = 7 Gpa n2 = 11
Max (PF4 ) subject to 30 ≤ B4 ≤ 50 20 ≤ H4 ≤ 30 1500 ≤ I4 ≤ 1600 5000 ≤ L4 ≤ 5700 100 ≤ W4 ≤ 200 100 ≤ t4 ≤ 400 ω4 ≤ 0.2H4 σ overload4 ≤ σ b /n4 σ b = 7 Gpa n4 = 4
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FIG. 2. The analysis and calculation models about convex corner undercutting.
Under the given constraints and safety factors in Table I, the sizes of geometries are determined. The three geometries feature the identical 7000 μm × 7000 μm overall dimensions, 20 μm thickness membranes, 200 μm width, and 30 μm thickness beams. The lengths of sensitive beams of BMMI and BMQI are 1700 μm and 200 μm. The lengths of sensitive beams of BMDI at each side and the center are 980 μm and 750 μm. The top widths of islands are 2300 μm, 1500 μm, 1500 μm corresponding to BMMI, BMDI, BMQI successively. The uniform geometry dimensions are acquired by adjusting the constraints and safety factors in Table I, which makes it obvious for PF comparison later. Certainly, the constraints and safety factors can be modified according to the specific optimization situations. C. Convex corner undercutting analysis
Under the determined dimensions above, the islands of BMDI and BMQI lack enough space for compensation, thus
the convex corner (CC) undercutting occurs. The final dimensions of the islands are roughly estimated using the fast etching planes stated in Ref. 23, to which the etching concentration and temperature have been referred. For simplification, the fast etching planes are assumed as {4 1 1} all the time. According to the literature, the etching depth is 0.544 times of side length of the compensation structure. Based on the relationship, the etching depth when the compensation structure consumed can be calculated. In Figs. 2(a) and 2(e), h21 , h41 representing the midway etching depth of BMDI, BMQI are 204 μm, 108.8 μm. By this time, complete corners have formed simultaneously on both the bottom and midway root planes of islands. Points A20m , A40m , A21m , A41m represent the vertices of corners on the bottom and midway root planes of BMDI, BMQI. This moment, A20m A21m , A40m A41m coincides with the intersection lines of {1 1 1} planes. With the etching going on, A20m A21m , A40m A41m will move parallel to these intersection lines, although the corners are undercut by etching front lines in directions. This is because the corners
FIG. 3. Simulation results about stress distribution and stress path of the three geometries.
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TABLE II. The simulation and calibration results of the sensors.
Ufp (mV/500Pa) Ufa (mV/15g) f (Hz) PF (Hz1/4 )
E-type ANSYS
BMMI ANSYS
BMMI Experiments
BMDI ANSYS
BMDI Experiments
BMQI ANSYS
BMQI Experiments
15.066 1.132 4802.1 110.8192
19.996 1.644 6936.6 110.9798
17.482 1.448 7375 111.8827
29.307 1.674 6242.7 155.6142
24.171 1.351 6950 163.3561
29.311 1.148 10686 259.6504
26.693 1.274 10275 210.947
from the bottom to the midway root are exposed for the same etching time. As for the corners from the midway root plane to the final root plane, they will be undercut by {4 1 1} planes. That is, the vertices of corners from the midway root to the final root planes are always on the intersection lines of {4 1 1} planes. In Figs. 2(a) and 2(e), the points A20 , A40 ; A21 , A41 ; and A22 , A42 represent the final vertices of corners on the bottom, midway root, and final root planes of BMDI and BMQI. A20 , A 40 ; A 20 , A 40 are the projections of A20 , A40 on the midway root and final root planes of BMDI and BMQI. h22 , h42 are the etching depths between the midway root and final root planes of BMDI and BMQI which are 146 μm, 241.2 μm due to the total etching depth of 350 μm. α is the angle between the lines of A20 A21 , A40 A41 , and {1 0 0} planes and is 45◦ , and β is the angle between the lines of A21 A22 , A41 A42 , and {1 0 0} planes and is 75.96◦ . According to the conditions, the length of A21 A 20 , A22 A 20 , A41 A 40 , A42 A 40 can be calculated. In Figs. 2(b) and 2(f), dc20 , dc40 are the vertical distances from points O20 , O40 to the etching front lines A20 M20 , A40 M40 , and they can be calculated by multiplying h22 , h42 with 1.46 on the basis of Ref. 23. According to the geometrical relationship, A20 N20 , C20 N20 , C40 N40 are the side lengths of island bottom of BMDI and BMQI and are 302 μm, 171 μm, 284 μm. Based on these bottom dimensions and the length of A21 A 20 , A22 A 20 , A41 A 40 , A42 A 40 calculated above, the dimensions of the midway root and final root of islands can be estimated. In Figs. 2(c) and 2(g), A 21 N 21 , C 21 N 21 , C 41 N 41 are the side lengths of the midway root of BMDI, BMQI and are 302 μm, 460 μm, 373 μm, and A22 N22 , C22 N22 , C42 N42 are the side length of the final root
of BMDI, BMQI and are 457 μm, 245 μm, 423 μm. The estimated island models after CC undercutting are shown in Figs. 2(d) and 2(h).
FIG. 4. The pressure calibration system.
FIG. 5. The acceleration and modal calibration setup.
D. Performances evaluation
To evaluate the performances of the three sensors with the calculated dimensions (considering the influence of CC undercutting), ANSYS is used. The simulation results about the distribution of von Mises stress on sensitive beams and the stress path along the x-axis from the center to the edge of sensitive beams under the pressure of 500 Pa are shown in Fig. 3. To assess the influence of vibration on pressure measurements, both acceleration analysis and modal analysis are carried out. In the simulation for acceleration analysis, a maximum acceleration of 15 g is exerted due to the human extreme limit. Both the static and dynamic analysis results above are listed in Table II. Besides, the E-type is analyzed as well. For obvious comparison, the dimensions of E-type are equal with the ones of BMMI except for the beam. In the table, the full scale outputs under pressure and acceleration are derived by Eq. (2) based on the simulated differential stress. In this study, the concentration of ion implantation is 3 × 1014 cm−3 less than 1 × 1017 cm−3 , so π 44 in Eq. (2) is 138 × 10−7 cm2 /N.24
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FIG. 6. Experimental results of the output voltage versus applied pressure.
FIG. 7. Experimental results of the output voltage versus applied acceleration.
V. RESULTS AND DISCUSSION
IV. EXPERIMENTS
To test the static characterization of the sensors, a complete experimental setup is built as shown in Fig. 4. The compressor acts as a pressure source. The sensors are calibrated with a reference pressure monitor (FLUKE A100K), excited by a 3 V DC power supply (RIGOL DP1116A), and the outputs are measured by a multi-meter (KEITHLEY 2000). To assess the dynamic performances approximately, another three sensor dies corresponding to BMMI, BMDI, BMQI with a through-hole on the glass base are utilized. The hole makes the pressures inside and outside the cavity equal, thus the applied atmospheric pressure is equivalent to zero, and the sensor chips are only affected by vibration acceleration, which is convenient for dynamic experiments. A stable centrifugal machine is used for acceleration calibration along the normal direction of the membrane of the three sensors as shown in Fig. 5. Through changing the rational speed, acceleration up to 15 g is imposed. The natural frequency is calibrated through testing the three sensors with a hole. By fixing one of the three tested sensors and a reference sensor on a shaker, as shown in Fig. 5, a peak concerning the voltage ratio will be generated when a sine sweep frequency passes through.
The pressure calibration results are plotted in Fig. 6, where the output voltage as a function of pressure is presented. The pressure is varying from 20 Pa to 500 Pa at room temperature. In the figure, the calibrated data of 5-round journey are described with different kinds of lines fixed by least square fitting. The standard errors of testing points are marked by error bars, and the maximum standard errors are enlarged. The acceleration calibration results are drawn in Fig. 7, in which the acceleration up to 15 g with an interval of 2.5 g is imposed. The measuring results are described by least square method as well, and the maximum standard errors of the testing points are enlarged. In the modal calibration, the peaks induced by the resonance are drawn in Fig. 8. Although slightly different damping factors and minor fabrication variations are existent, the acceleration and modal experiments are still significant for evaluating the dynamic performances of the sensors under near-vacuum condition. The experimental results above are listed in Table II. In the table, the full scale outputs about pressure and acceleration are calculated based on the least square method. The PF
FIG. 8. The modal calibration results.
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is derived from Eq. (6), and it reveals that the BMQI features the highest comprehensive performance. The PF of BMQI is 29.1% higher than BMDI and 88.5% higher than BMMI. The simulation results show that the sensitivity of BMMI is 32.7% higher than the E-type one, and the natural frequency is 44.4% higher. However, the PF of BMMI has been dragged down to be nearly the same by the 45.2% higher acceleration interference. The estimated performance factors are consistent with the actually calibrated ones, so they can be used for evaluating the overall performance of the absolute micro-pressure sensors. The performance factor PF is significant for assessing the comprehensive performance including dynamic interference. VI. CONCLUSIONS
This work attempts to explore the assessment for the performance of absolute micro-pressure sensors considering vibration influences. A performance factor has been proposed to try to reflect the comprehensive performance. To validate the rationality of the defined performance factor, three geometries have been proposed, optimized, and actually fabricated. Experimental results have demonstrated that the performance factor is of significance for evaluating the overall property of a sensor. ACKNOWLEDGMENTS
This work is supported by the National Science Foundation (NSF) for Distinguished Young Scholars of China (No. 51325503), Young Scientists Fund of the National Natural Science Foundation of China (NNSFC) (No. 51305336), National High Technology Research and Development Program of China (863 Program) (No. 2013AA041108), the Program for Changjiang Scholars and Innovative Research Team in University of China (No. IRT1033), and Postdoctoral Science Foundation of China (No. 2013M532036).
TABLE III. The determination coefficients of the related equations. Equation related
R2
σ dp1 σ da1 ω1 σ overload1 f1
0.96793 0.96261 0.98129 0.97664 0.97572
Equation related σ dp2 σ da2 ω2 σ overload2 f2
R2 0.96030 0.96423 0.96007 0.99241 0.96476
Equation related σ dp4 σ da4 ω4 σ overload4 f4
R2 0.96686 0.97484 0.978214 0.98751 0.98308
APPENDIX A: THE DETERMINATION COEFFICIENT
To validate the rationality of the hypothesis about the functional forms, the coefficients of determination R2 are calculated and listed in Table III which has verified the goodness of fit, thus ANSYS simulation results can be almost represented by the equations established previously. APPENDIX B: FABRICATION
The sensors are all fabricated by the bulk micromachining, using the standard double side polished n-type (100) silicon wafers. The resistivity of the silicon wafers is 6000∼8000 · cm and the thickness is 400 μm. The specific fabrication process flow is illustrated in Fig. 9. (a) Photolithography is employed to pattern piezoresistors on the front side of the silicon wafer, after SiO2 layers are grown on both sides of the substrate by thermal oxidation. Ion implantation of boron is carried out with a concentration of 3 × 1014 cm−3 forming a sheet resistance of 220 . (b) Heavy boron ion diffusion is performed to consolidate the connections of piezoresistors. (c) The passivation layers of Si3 N4 , SiO2 are deposited by means of low pressure chemical vapor deposition (LPCVD) and plasma enhanced chemical vapor deposition (PECVD) successively. (d) Contacts are then photo-patterned and etched on the front side
FIG. 9. The schematic of the main process flow.
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to sputter Au. Ohmic contacts between Au wires and piezoresistors are reinforced by sintering process. (e) For creating cavities, forming islands, and reducing the height of islands, KOH etching is used on the backside of the wafers after patterned. The etching is carried out in the pure aqueous KOH solution with a KOH content of 30 wt. % at the temperature of 80 ± 1 ◦ C, resulting in an etching rate of 1.0 μm/min. (f) The anti-adsorption electrodes made of Cr are sputtered on the glass. Then the backside of the wafer is attached to the Pyrex 7740 glass under vacuum condition by anodic bonding process to create absolute pressure sensors. (g) Inductively coupled plasma (ICP) etch is involved to form beams on the front side. The fabricated sensor dies are shown in Fig. 10 where SEM images of the three sensor chips are displayed. The CCD photographs of the undercut islands in the rear cavity of BMDI and BMQI are presented in Fig. 11. The calculated CC undercutting conforms well to the actual fabrication except for some minor difference on the root, which is caused by the assumed unchanging fast etching planes. 1 J.
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FIG. 10. Photos of the fabricated sensor dies. (a) The sensor die and SEM images of BMMI; (b) The sensor die and SEM images of BMDI; (c) The sensor die and SEM images of BMQI.
utilizing reactive ion etching (RIE). In order to activate the boron ion electrically and make dopant uniform, the annealing technology is executed at 1100 ◦ C for 30 min under nitrogen atmosphere. For the connections of resistors and formations of bonding pads, metallization process is performed
FIG. 11. The CCD photograph of the undercut islands on the bottom and root planes. (a) The undercut island of BMDI; (b) The undercut island of BMQI.
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