A 4 mm2 Double Differential Torsional MEMS Accelerometer ... - MDPI

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A 4 mm2 Double Differential Torsional MEMS Accelerometer Based on a Double-Beam Configuration Tongqiao Miao, Dingbang Xiao *, Qingsong Li, Zhanqiang Hou and Xuezhong Wu * College of Mechatronics Engineering and Automation, National University of DefenseTechnology, Changsha 410073, China; [email protected] (T.M.); [email protected] (Q.L.); [email protected] (Z.H.) * Correspondence: [email protected] (D.X.); [email protected] (X.W.); Tel.: +86-0731-8457-4958 (D.X.) Received: 28 August 2017; Accepted: 27 September 2017; Published: 2 October 2017

Abstract: This paper reports the design and simulation of a 4 mm2 double differential torsional MEMS accelerometer based on a double-beam configuration. Based on the structure of conventional torsional accelerometers, normally composed of one pair of proof masses and one torsional beam, this work explores the double differential configuration: a torsional accelerometer with two pairs of unbalanced proof masses rotating in reverse. Also, the torsional beam is designed as a double-beam structure, which is a symmetrical structure formed by two torsional beams separated by a certain distance. The device area of the novel accelerometer is more than 50 times smaller than that of a traditional double differential torsional MEMS accelerometer. The FEM simulation results demonstrate that the smaller device does not sacrifice other specifications, such as mechanical sensitivity, nonlinearity and temperature robustness. The mechanical sensitivity and nonlinearity of a ±15 g measuring range is 59.4 fF/g and 0.88%, respectively. Compared with traditional single-beam silicon structures, the novel structure can achieve lower maximum principle stress in critical regions and reduce the possibility of failure when high-g acceleration loading √ is applied along all three axes. The mechanical noise equivalent acceleration is about 0.13 mg/ Hz in the theoretical calculations and the offset temperature coefficient is 0.25 mg/◦ C in the full temperature range of −40 ◦ C to 60 ◦ C. Keywords: double differential; torsional; MEMS; accelerometer; double-beam; temperature robustness

1. Introduction Year after year the MEMS market is growing faster than the average semiconductor industry and accelerometers are widely used in most MEMS products [1–4]. MEMS accelerometers are widely used in consumer electronics, such as cell phones, portable gaming consoles, cameras and pedometers to detect vibration or tilt motion [5]. Many kinds of MEMS accelerometers have been reported recently, such as capacitive [2–4], resonant [6–8], optical [9], and piezoresistive accelerometers [10] and so on. Although a resonant accelerometer achieved µg level of bias stability in [6] and an optical accelerometer achieved tens of µg level resolution in [9], which are better than capacitive accelerometers, the structures are more complex to fabricate and the cost is higher [2–9]. The piezoresistive accelerometer has a large measurement range from 0.25 g to 25,000 g and it is easy to fabricate according to [10], but the temperature robustness is the main challenge. This paper mainly focuses on the study of capacitive accelerometers. Among capacitive accelerometers, the torsional accelerometer is one of the most popular structures for its simple design and low cost, but challenges remain in increasing their bias stability and temperature robustness [11], which are the main obstacles that need to be overcome for inertial navigation systems [12]. Some studies to solve these problems have been

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reported. The typical methods are temperature compensation with readout circuits [13] or using a thermostatic package [14]. Some other methods based on structure design and package stress isolation have also been proposed, such as a four point supporting frame [15], a highly symmetrical spring-mass structure [16], soft adhesive attachment [17], and special fabrication processes [18], but the methods are complicated and expensive [15–18]. Inspired by the high performance butterfly gyroscope [19,20] and the differential vibrating beam accelerometer [21], our team has designed and reported two types of double differential torsional MEMS accelerometers based on single-beam structures [22,23]. To further improve the performance and promote the commercial value, this paper presents a novel double-beam double differential torsional MEMS accelerometer of only 4 mm2 , which utilizes a double differential configuration composed of four proof masses split into two pairs. The double differential arithmetic is introduced for matching the double differential configuration to reduce the output drift to 0.25 mg/◦ C in the full temperature range of −40 ◦ C to 60 ◦ C. Compared with traditional single-beam silicon structures, the novel structure can achieve a lower maximum principle stress in critical regions and reduce the possibility of failure by changing the position of critical regions. Although the device area of our novel accelerometer is more than 50 times smaller than that of traditional double differential torsional MEMS accelerometer [22,23], the novel device does not sacrifice other specifications, such as sensitivity and nonlinearity, which are 59.4 fF/g and 0.88%, respectively, for a measuring range of ±15 g. The mechanical noise is analyzed theoretically and the calculated equivalent acceleration is √ about 0.13 mg/ Hz in the theoretical calculation. The fabrication tolerance is also analyzed and the fabrication process is introduced. Afterwards, a FEM simulation by the COMSOL software verifies all the results presented in this paper. The remainder of the paper is hence organized as follows: Section 2 provides a description of the structure design and theoretical consideration of the novel structure. Then, Section 3 collects the results of our simulation analysis of mechanical sensitivity, nonlinearity, shock resistant, tolerance in fabrication and temperature robustness. Next, Section 4 provides the design of the fabrication process. Finally, some concluding remarks and proposals for future works are collected in Section 5. 2. Structure Design and Theoretical Considerations 2.1. Description and Working Principle As Figure 1 presents, the double differential capacitive micro-accelerometer presented in this paper basically consists of two parts: a silicon substrate and a silicon structure bonded onto it. The overview of the device shows that the silicon structure is mainly composed of four proof masses, a double-beam and a stress-released structure. The arrow along the z-axis indicates the sensing direction of the device. The double-beam is the torsional beam of the proof masses formed by silicon dry etching. Four rotatable proof masses can be divided into two pairs and each pair forms a torsional structure. We define that the first pair of masses are mass 1 and mass 4 and the second pair of masses are mass 2 and mass 3. Mass 1 and mass 3 are etched with the same curved depth to realize the unbalanced masses of each pair. The cross section of the silicon structure is presented in Figure 1c and the geometrical parameters of the novel structure are listed in Table 1. Taking navigation application requirements into consideration, the range of this accelerometer is designed as ±15 g. An open-loop circuit is utilized to simplify the readout circuit. The capacitive gap is designed as 2 µm to realize excellent sensitivity, non-linearity and signal to noise ratio properties. Only the capacitive gap parameter needs to be changed when different sensitivities and measuring ranges are required in other applications.

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Figure Figure1.1.Design Designmodel modelof ofthe theaccelerometer: accelerometer:(a) (a)Overview Overviewofofthe thedevice; device;(b) (b)Back Backview viewofofthe thedevice; device; (c) (c)The Thecross crosssection section of of the the silicon silicon structure. structure. Table Table1.1.Geometrical Geometricalparameters parametersof ofthe thenovel novelaccelerometer. accelerometer. Description Description Wafer thickness Wafer thickness Curved depth Curved depth Width of single beam Width of single beam beams Distance between two torsional Distance between two torsional beams Effective area of each sensing capacitor Effective area of each sensing capacitor Original gap of the capacitor Original gap of the capacitor Original Originalcapacitance capacitance of of each each sensing sensing capacitor capacitor Capacitance Capacitanceof ofreference reference capacitor capacitor Mechanical Mechanical sensitivity sensitivity Fundamental Fundamentalresonance resonance frequency frequency Measuring range range Measuring

Symbol Symbol t t h hb bd d 𝐴 A0 0 𝑑 d0 0 C0𝐶0 C f𝐶𝑓 𝑆𝑚𝑒𝑐ℎ Smech f 0 𝑓0 R a𝑅𝑎

Value Value 50 μm 5040 µm μm 4011 µm μm 1135 µm μm 35 µm 2 0.72 mm 0.72 mm2 2.0 μm 2.0 µm 3.18 3.18 pFpF 6.36 6.36 pFpF 59.4 fF/g 59.4 fF/g 3269 Hz 3269 Hz ±15 ±15g g

ItItcan canbe beseen seenthat thatthe thearrow arrowalong along the the z-axis z-axis indicates indicatesthe thesensing sensingdirection direction of of device. device. When Whenan an acceleration accelerationalong alongthe thesensing sensingdirection directionisisapplied, applied,the thetwo twotorsional torsionalstructures structureswill willrotate rotatereversely reversely because because of of the the unbalanced unbalanced masses masses of of each each pair. pair. The Theworking workingprinciple principleisis shown shown in in the the form form of of simulation result in Figure 2. simulation result in Figure 2. On On the the silicon silicon substrate, substrate, there there are are four four electrodes electrodes made made to to form form four four capacitors capacitors with with silicon silicon planes of four masses. Thus, the acceleration along the sensing direction can be calculated planes of four masses. Thus, the acceleration along the sensing direction can be calculatedby bydouble double differential differential capacitance capacitance signals: signals: 𝑎 ∝ (Δ𝐶1 − Δ𝐶4 ) − (Δ𝐶2 − Δ𝐶3 ) = (Δ𝐶1 + Δ𝐶3 ) − (Δ𝐶2 + Δ𝐶4 )

(1)

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a ∝ (∆C1 − ∆C4 ) − (∆C2 − ∆C3 ) = (∆C1 + ∆C3 ) − (∆C2 + ∆C4 )

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Figure 2. FEM simulation when 1 g acceleration is applied along the sensing direction, the positive z direction, at room temperature. temperature. The color shows the displacement of the silicon structure compared with position along the direction (z with the the initial initial position alongwhen the sensing sensing direction (z direction). direction). Figure 2. FEM simulation 1 g acceleration is applied along the sensing direction, the positive z direction, at room temperature. The color shows the displacement of the silicon structure compared

Herein, Δ𝐶 , 𝑖 = position 1~4 isalong the the changed of the capacitor formed of the mass i and the with∆C the𝑖initial sensingcapacitance direction (z direction). Herein, i , i = 1 ∼ 4 is the changed capacitance of the capacitor formed of the mass i and the electrode below below it. it. The The measurement measurement of the capacitances capacitances addition Equation (1) is realized realized by by the the electrode of the addition in in Equation (1) is Herein, Δ𝐶 is the changed capacitance of the capacitor formed of the mass i and the 𝑖 , 𝑖 = 1~4 connection of the diagonal electrodes on the silicon substrate, as shown in Figure 1b. Thus, the output connection of the diagonal electrodes on the silicon substrate, as shown in Figure 1b. Thus, the output electrode below it. The measurement of the capacitances addition in Equation (1) is realized by the signals of of two two sensing sensing electrodes electrodes are are addition addition of of capacitance capacitance 11 and and 3, 3, 22 and and4, 4,respectively. respectively. signals connection of the diagonal electrodes on the silicon substrate, as shown in Figure 1b. Thus, the output The schematic of circuit is shown 3. 3. It mainly consists of the sensing element, the The schematic ofreadout readout circuit showninof inFigure Figure It1 mainly consists of the sensing element, signals of two sensing electrodes areisaddition capacitance and 3, 2 and 4, respectively. charge amplifier, the high pass filter, the second amplifier, the demodulator and the low pass filter. the chargeThe amplifier, the pass filter, the second amplifier, the consists demodulator and the low pass schematic of high readout circuit is shown in Figure 3. It mainly of the sensing element, the filter. Thus, the total output output voltage canthe besecond calculated as: the demodulator and the low pass filter. chargethe amplifier, the highvoltage pass filter, amplifier, Thus, total can be calculated as: Thus, the total output voltage can be calculated (𝐶1 + 𝐶3as: ) − (𝐶2 + 𝐶4 ) 𝑉0 = 𝑘𝑉𝑚(C1 + C3 ) − (C2 + C4 ) (𝐶 ) + 𝐶 𝐶𝑓(𝐶2 + 𝐶4 ) V0 = kVm 1 3 − 𝑉0 = 𝑘𝑉𝑚 C 𝐶𝑓f

(2)

(2) (2)

where 𝑘 is the gain of the signal amplifier and 𝐶𝑓 is the reference capacitance. 𝑘 is the of gain the signal amplifier and is the referencecapacitance. capacitance. wherewhere k is the gain theofsignal amplifier and C f 𝐶is reference 𝑓 the

Figure3.3.Schematic Schematic of the Figure thereadout readoutcircuit. circuit. Figure 3. Schematic of the readout circuit.

2.2. Theoretical Consideration

2.2. Theoretical Consideration 2.2. Theoretical Consideration The original stress-released structure is shown in Figure 4a. As the width of the stress-released The original stress-released structure is shown in Figure 4a.ofAs the width of the stress-released structure is only 2% of the length of torsional beam,inthe design4a. thethe stress-released structure has The original stress-released structure is shown Figure As width of the stress-released structure islittle only 2% of the length of torsional beam, the design of the stress-released structure has very influence on the stiffness of the torsional beam. Thus, the stress-release-structure can be very structure is only 2% of the length of torsional beam, the design of the stress-released structure has little influence onthe thesimplified stiffness of the torsional beam. Thus, the stress-release-structure ignored ignored and model is shown in Figure 4b. Since the mass of the beam is can verybe small very little influence on the stiffness of the torsional beam. Thus, the stress-release-structure can be compared to that of the proof mass in this structure, it is neglected in the theoretical analysis. ignored and the simplified model is shown in Figure 4b. Since the mass of the beam is very small compared to that of the proof mass in this structure, it is neglected in the theoretical analysis.

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and the simplified model is shown in Figure 4b. Since the mass of the beam is very small compared to Sensors 2017,2017, 17, 2264 Sensors 17, 2264 5 of 16 5 of 16 that of the proof mass in this structure, it is neglected in the theoretical analysis.

(a)

(b)

(a) (b) Figure 4. Simplification of stress-released structure: (a) Original model; (b) Simplified model.

Figure 4. Simplification of stress-released structure: (a) Original model; (b) Simplified model. Figure 4. Simplification of stress-released structure: (a) Original model; (b) Simplified model. Thus, the mathematical model of double differential torsional capacitive accelerometer based on Thus, the mathematical model4.ofBecause doubleofdifferential torsional capacitive accelerometer double-beam is shown in Figure the symmetrical characteristic of the structure, twobased Thus, the mathematical model of double differential torsional capacitive accelerometer based on on double-beam is shown Figure 4.structures Becausehave of the symmetrical characteristic thepaper structure, torsional beams and twointorsional similar force states, respectively.ofThis double-beam is shown in Figure 4. Because of the symmetrical characteristic of the structure, two chooses one of each totwo analyze the torsional stiffness the mechanical sensitivity of the double two torsional beams and torsional structures haveand similar force states, respectively. This paper torsional beams andto two torsional structures have similar states, respectively. Thisdouble paper differential torsional capacitive As Figure 5aforce presents, tosensitivity make the of structure chooses one of each analyze the accelerometer. torsional stiffness and the mechanical the symmetrical, the parameters of the device are set as: chooses one of each to analyze the torsional stiffness and the mechanical sensitivity of the double differential torsional capacitive accelerometer. As Figure 5a presents, to make the structure symmetrical,

differential torsional capacitive 5a presents, to make the (3) structure the parameters of the device are set accelerometer. as: 𝑚2 = 𝑚4 > As 𝑚1 =Figure 𝑚3 symmetrical, the parameters of the device are set as: (4) 𝑙𝑚1 = 𝑙𝑚2 = 𝑙𝑚3 = 𝑙𝑚4 = 𝑙𝑚 m2 = m4 > m1 = m3 (3) (3) 𝑚2 = 𝑚4 > 𝑚1 = 𝑚3 𝑙𝑒1 = 𝑙𝑒2 = 𝑙𝑒3 = 𝑙𝑒4 = 𝑙𝑒

(5)

lm1 lm2 ==𝑤lm4 𝑙𝑚1 𝑙𝑤 =l𝑤 𝑙𝑚3= 𝑙𝑚4 m3 𝑚2= 𝑤=== = == 𝑤=lm𝑙𝑚

(6)

(4)

𝑙𝑒1==l𝑙e2 𝑙𝑒2= 𝑙𝑒3= 𝑙𝑒4==le𝑙𝑒 le1 ==l𝑙e3 ==l𝑙𝑒e4 𝐴𝐵 𝐶𝐷

(7)

(5)

=w𝑤 ==w 𝑙𝑠𝑤 𝑘𝑙w w1𝑤1==w𝑙𝑤 𝐵𝐶 𝑒𝑤 3== 2 2= 3= 44=

(8)

(6)

𝑙𝐴𝐷 = l𝐿 𝑙=𝐴𝐵𝑙= = =+ 𝑙2𝑙 = l(𝑘+2)𝑙 𝑙e𝑒 𝑒= 𝑒 𝐶𝐷 AB 𝑠 lCD

(9)

(7) (7)

1

(a)

2

3

4

𝑙𝐵𝐶==ls𝑙𝑠==kl𝑘𝑙e 𝑒 l BC

(8) (8)

𝑙𝐴𝐷==L𝐿==ls𝑙𝑠++2l2𝑙e 𝑒==(k(𝑘+2)𝑙 l AD + 2)le𝑒

(9) (9)

(b)

Figure 5. Mathematical model of double differential torsional capacitive accelerometer based on double-beam: (a) Torque model of the front torsional beam; (b) Mechanical model of the left torsional structure.

(b)torsional beam and According to the (a) calculation formula of twist angle, the torques of the front the twist5.angles of both torsional structures can be calculated as: capacitive accelerometer based on Figure Mathematical model of of double differential differential torsional Figure 5. Mathematical model double torsional capacitive accelerometer based on double-beam: (a) Torque model of the front torsional beam; (b) Mechanical model model of the left double-beam: (a) Torque model of the front torsional beam; (b) Mechanical of torsional the left structure. torsional structure. According to the calculation formula of twist angle, the torques of the front torsional beam and the twist angles of both torsional structures can be calculated as:

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According to the calculation formula of twist angle, the torques of the front torsional beam and the twist angles of both torsional structures can be calculated as: M A = MD = φB = φC =

k k MB = M k+2 k+2 C

(10)

MB L MC L k = 2 (k + 2) GIt (k + 2) GIt k

(11)

2

herein, G is the shear modulus of silicon, and It is torsional moment of inertia which is relative with the cross section. φS can be expressed as: φS = φB = φC =

MS L 1 = √ GI t ( k + 2) k+ k

2

√2 k

2

MS L GIt

(12)

Actually, in the design of MEMS torsional accelerometers, it’s important to get the maximum torsional angle φS by optimizing structure sizes to increase the mechanical sensitivity. In this paper, the condition of maximum torsional angle φSmax and it’s maximum value are:

√

2 k = √ ,k = 2 k

φSmax =

(13)

MS L 8GIt

(14)

This means that to achieve the maximum mechanical sensitivity, the lengths of both torsional beams’ part AB (CD) and BC should keep the relationship: ls = 2le

(15)

As Figure 5b presents, when the acceleration, a is applied along the sensing direction, the two pairs of torsional structures will rotate reversely. The torque to the geometrical center, T, and the moment equilibrium equation of the left torsional structure is: T = m4 alm4 − m1 alm1 = (m4 − m1 ) alm = ∆malm

(16)

T = 2MS + FB d

(17)

herein, ∆m is the difference between the masses of the left torsional structure and FB is the force of z-axis component at the point B. According to the basic beam theory, the displacement of z-axis component at point B caused by FB can be calculated as: ZB =

7FB L3 6144EIy

(18)

herein, E is the Young’s modulus of silicon, and Iy is area moment of inertia of the front beam about the y-axis. Actually, the absolute value of the twist angle φSmax is extremely small in this structure, so the approximate calculation of the twist angle φSmax is: 7FB L3 3072dEIy

(19)

8 384d2 EIy + 14L2 GIt = 7L3

(20)

φSmax ≈ sinφSmax =

kφ =

T φSmax

ZB d 2

=




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From the Equation (20), the torsional stiffness of the double differential torsional accelerometer based on the double-beam structure is related to the distance between two torsional beams d and the out-of-plane bending stiffness EIy , which is different from the traditional double differential torsional structure based on single-beam. With the Taylor Series at the point φSmax = 0, the changed capacitances can be calculated as:
E w le 3 + 12lm 2 le E wle lm ∆C1 = ∆C3 ≈ φSmax + φSmax 2 (21) d0 2 12d0 2
E w le 3 + 12lm 2 le E wle lm φSmax + φSmax 2 (22) ∆C2 = ∆C4 ≈ − d0 2 12d0 2 ∆C = (∆C1 + ∆C3 ) − (∆C2 + ∆C4 ) =

4E wle lm φSmax d0 2

(23)

where E is the permittivity of the gas in the gap. Thus, the mechanical sensitivity of the double differential torsional capacitance accelerometer based on double-beam is obtained: Smech =

2d0 2

7E wle lm 2 L3 ∆m
384d2 EIy + 14L2 GIt

(24)

From Equation (24), the mechanical sensitivity of the double differential torsional accelerometer based on a double-beam is related to the distance between two torsional beams d, which is essentially different from the traditional double differential torsional accelerometer based on a single-beam. Also, with the decrease of the distance d between two torsional beams, the double-beam torsional stiffness will decrease from the Equation (20) and the mechanical sensitivity will rise under the same condition, which means that we can improve the mechanical sensitivity by changing the distance between two torsional beams when we pursue a smaller device area. Actually, the novel device is much smaller than traditional double differential torsional accelerometers and the smaller device area will lead to a higher noise floor. Thus, it’s necessary to calculate the noise floor theoretically. The primary source of mechanical noise for the device is the Brownian motion of gas molecules surrounding the proof mass and the Brownian motion of the proof mass suspension on anchors. The theoretical mechanical noise of the accelerometer in this paper can be calculated with the equation [4]: s TNEA =

4K B Twn QM

(25)

herein, Q is quality factor of the structure, M is the total mass of the proof masses in the accelerometer, K B is the Boltzmann constant, T is the absolute temperature and wn is the natural angular frequency. All the above parameters can be calculated easily based on√Table 1 and basic theory. The calculated mechanical noise equivalent acceleration is about 0.13 mg/ Hz. From Equation (25), the proof mass can be enlarged to decrease the mechanical noise of this device. 3. Simulation Analysis 3.1. Simulation Analysis of Mechanical Sensitivity and Nonlinearity Nonlinearity is the systematic deviation from the straight line that defines the nominal input-output relationship. The nonlinearity of the accelerometer can be calculated as:

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Vj∗ − Vj

K= |Vmax − Vmin |

(26) max

where Vj is the average output voltage value of each acceleration, Vj∗ is the corresponding voltage value of on the fitting lines, Vmax and Vmin are the maximum and minimum voltages of output respectively. According to the Equation (2), the nonlinearity of accelerometer can also be calculated as: Cj∗ − Cj

K= |Cmax − Cmin |

(27) max

where Cj is the average output capacitance value of each acceleration, Cj∗ is the corresponding capacitance value of on the fitting lines, Cmax and Cmin are the maximum and minimum capacitances of the output, respectively. As for the design of the novel double differential torsional capacitance accelerometer, it’s very important to balance the mechanical sensitivity and nonlinearity. In this paper, we establish a FEM model using COMSOL according to the dimensions in Table 1. As the Sensors 2017, 17, 2264 16 nonlinearity is mainly related to the dimensions of the double-beam when we pursue 8aofsmaller device area, it’s necessary to perform the FEM analysis for the relationship between the mechanical device area, it’s necessary to perform the FEM analysis for the relationship between the mechanical sensitivity and and nonlinearity of of thethe measuring ±15g)g)bybychanging changing distance between sensitivity nonlinearity measuringrange range ((±15 thethe distance between two two torsional beams in the of 20 to 50 µmμm under the same results are torsional beams in range the range of µm 20 μm to 50 under the sameconditions. conditions. The The simulation simulation results shown Figurein6a.Figure When6a. theWhen distance betweenbetween two torsional beams dbeams is decreased, the mechanical areinshown the distance two torsional d is decreased, the mechanical sensitivity willAlso, be increased. Also, the nonlinearity will be increased when the mechanical sensitivity will be increased. the nonlinearity will be increased when the mechanical sensitivity sensitivity is increased, which shows theof importance balancing the mechanical and is increased, which shows the importance balancingofthe mechanical sensitivitysensitivity and nonlinearity. Thus,sensitivity the mechanical sensitivity andofnonlinearity of the novel device areto designed be and Thus,nonlinearity. the mechanical and nonlinearity the novel device are designed be 59.4 to fF/g 59.4 fF/g and 0.88%, respectively, when the distance between two torsional beams d is 35 μm. In order 0.88%, respectively, when the distance between two torsional beams d is 35 µm. In order to verify the to verify the correctness of the theoretical calculation, we compare the results of theoretical correctness of the theoretical calculation, we compare the results of theoretical calculation in Equation calculation in Equation (27) and our simulation. As shown in Figure 6b, the theoretical calculation (27) and our simulation. As shown in Figure 6b, the theoretical calculation and simulation results of and simulation results of mechanical sensitivity are in good agreement and the maximum relative mechanical are in good the maximum relative error is 3% in whole range error issensitivity 3% in the whole range agreement of d, whichand demonstrates the theoretical calculation of the mechanical of d, which demonstrates the theoretical calculation of mechanical sensitivity is sound. sensitivity is sound.

(a)

(b)

Figure 6. The simulation results of mechanical sensitivity and nonlinearity: (a) The simulation results Figure 6. The simulation results of mechanical sensitivity and nonlinearity: (a) The simulation results of relationship between the nonlinearity and the mechanical sensitivity; (b) The theoretical calculation of relationship between the nonlinearity and the mechanical sensitivity; (b) The theoretical calculation and simulation results of mechanical sensitivity. and simulation results of mechanical sensitivity.

3.2. Simulation Analysis of Shock Resistance

3.2. Simulation Analysis of Shock Resistance

The post-impact response of the MEMS accelerometer has received attention in recent years [24,25]. In fact, shocks still represent an important issue as has for the reliability of inertial MEMSyears devices The post-impact response of the MEMS accelerometer received attention in recent [24,25]. and failures linked to cracks spreading in high stressed regions of the MEMS can suddenly occur [26]. and In fact, shocks still represent an important issue as for the reliability of inertial MEMS devices Focusing physical effects ofinshocks dropsregions on the MEMS it can be shown failures linkedontothe cracks spreading high and stressed of the accelerometer, MEMS can suddenly occur [26]. that the mechanical side of the problem is by far the most prominent one and the electrical side can Focusing on the physical effects of shocks and drops on the MEMS accelerometer, it can be shown that be disregarded [27]. This is due to the fact the inertial and possible contact forces turn out to exceed by orders of magnitude the electrical ones, when more than 105 g is induced by the shock loading [24]. As far as the reliability of the sensor subject to shock-like loading is concerned, the value of the maximum principle stress 𝜎𝑝 is considered the triggering factor for possible failure mechanisms [28]. Because of the re-entrant corners at the connection with the anchor and the substrate, 𝜎𝑝 is actually

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the mechanical side of the problem is by far the most prominent one and the electrical side can be disregarded [27]. This is due to the fact the inertial and possible contact forces turn out to exceed by orders of magnitude the electrical ones, when more than 105 g is induced by the shock loading [24]. As far as the reliability of the sensor subject to shock-like loading is concerned, the value of the maximum principle stress σp is considered the triggering factor for possible failure mechanisms [28]. Because of the re-entrant corners at the connection with the anchor and the substrate, σp is actually located in critical regions very close to the end of the cross-sections of the supporting beams [29]. What turned out from previous experiments is that standard accelerometers can sustain high-g acceleration levels without malfunctioning, since the stress field in critical regions results to be much below the characteristic tensile strength of silicon (typically higher than 1 GPa) [27]. As for traditional single torsional beam structures, the critical regions are near the symmetry line of the differential masses, where the moment of force is the maximum. However, in the design of the double-beam structure presented in this paper, the position of critical regions can be moved away from the symmetry line of differential masses by increasing the distance between two torsional beams, which can reduce σp and improve the shock resistance of the novel device. Thus, it’s necessary to perform a FEM simulation of the maximum principle stress σp in critical regions by changing the distance between two torsional beams. Sensors 2017, 17, 2264 9 of 16 In this paper, we performed a FEM simulation for the novel device by changing the distance a FEM simulation of the 𝜎𝑝 inacceleration critical regions byloading changing the distance between two torsional beams d maximum when aprinciple high-gstress input (with a peak value of between two torsional beams. about 5500 g) is applied along all three axes. The input acceleration loading is already considered In this paper, we performed a FEM simulation for the novel device by changing the distance between two torsional beams d when a high-g input acceleration loading (with a peak value of about and depicted in [30]. Meanwhile, the torsional stiffness remains the same by changing the width g) is applied along all three axes. The input acceleration loading is already considered and of single beam5500 b, which means the mechanical sensitivity of the novel accelerometer remains at depicted in [30]. Meanwhile, the torsional stiffness remains the same by changing the width of single 59.4 fF/g. In order the results moresensitivity comparable andaccelerometer meaningful, we athave done beamto b, make which means the mechanical of the novel remains 59.4 fF/g. In the same FEM order to make the results more comparable and meaningful, we have done the same FEM simulation simulation for the single-beam accelerometer which has the same mechanical sensitivity. Considering for the single-beam accelerometer which has the same mechanical sensitivity. Considering the the feasibility offeasibility fabrication processing and accuracy ofmechanical the mechanical of fabrication processing andthe the accuracy of the sensitivity, sensitivity, the parameter dthe parameter d ranges 25 μm to changes 45 μm and every changes5every μm. Except thetorsional torsional beams, the two ranges from 25 µm tofrom 45 µm and µm.5Except forfor the beams, the two structures structures are the same. The FEM simulation models of the two accelerometers are shown in Figure are the same. The 7. FEM simulation models of the two accelerometers are shown in Figure 7.

(a)

(b)

Figure 7. The simulation models of two accelerometers: (a) Simulation model of double-beam simulation models (b) of Simulation two accelerometers: (a) Simulation model of double-beam torsional accelerometer; model of single-beam torsional accelerometer.

Figure 7. The accelerometer; (b) Simulation model of single-beam torsional accelerometer.

torsional

As the simulation results of the time evolution of the maximum principle stress in critical regions are not easily observed, we have selected 20 feature points of each line to draw the variation trend in Figure 8. As shown in Figure 8a,b, when the high-g input acceleration is applied along x-axis or yAs the simulation results of the time evolution of the maximum principle stress in critical regions axis, the double-beam structure has much lower 𝜎𝑝 than the single-beam structure in the whole are not easily observed, webehave selected pointsstructure of each the variation trend range of d. It can seen from Figure 8c20 thatfeature the double-beam has line lower to 𝜎𝑝 draw in a particular of d when the high-g8a,b, input acceleration along z-axis.acceleration Thus, the double-beam structurealong x-axis or in Figure 8. As range shown in Figure when theis applied high-g input is applied can achieve lower 𝜎𝑝 in critical regions and lower possibility of failure mechanisms than singley-axis, the double-beam structure has lower the single-beam structure p than beam structure. In conclusion, this much paper chooses the σ distance between two torsional beams as 35 μm in the whole to decrease in critical regions8c andthat realize better shock resistance.structure has lower σp in a particular range of d. It can be seen𝜎𝑝 from Figure the double-beam

range of d when the high-g input acceleration is applied along z-axis. Thus, the double-beam structure can achieve lower σp in critical regions and lower possibility of failure mechanisms than single-beam structure. In conclusion, this paper chooses the distance between two torsional beams as 35 µm to decrease σp in critical regions and realize better shock resistance.

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(a)

(b)

(c) Figure 8. Time evolution of the maximum principal stress in critical regions of double-beam torsional Figure 8. Time evolution of the maximum principal stress in critical regions of double-beam torsional accelerometer and single-beam torsional accelerometer when the high-g acceleration is applied along accelerometer and single-beam torsional accelerometer when the high-g acceleration is applied along all three axes in trend line: (a) Simulation results when the high-g acceleration is applied along x-axis; all three axes in trend line: (a) Simulation results when the high-g acceleration is applied along x-axis; (b) Simulation results when the high-g acceleration is applied along y-axis; (c) Simulation results (b) Simulation results when the high-g acceleration is applied along y-axis; (c) Simulation results when when the high-g acceleration is applied along z-axis. the high-g acceleration is applied along z-axis.

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3.3. Simulation Analysis of Tolerance in Fabrication 3.3. Simulation Analysis of Tolerance in Fabrication Symmetry design is the key to the implementation of a double differential torsional MEMS accelerometer. are unavoidable in practical processing. source SymmetryHowever, design is fabrication the key to imperfections the implementation of a double differential torsionalAMEMS of uncertaintiesHowever, which cannot be ignored at the filmare level is the variation in the over-etch [31]. [32], accelerometer. fabrication imperfections unavoidable in practical processing. A In source an etch variation defect is one where the thickness of the device structure does not meet the design of uncertainties which cannot be ignored at the film level is the variation in the over-etch [31]. In [32], expectations due defect to the etch variations caused by fluctuations of structure temperature, concentration an etch variation is one where the thickness of the device doesetchant not meet the design and other reasons. variation caused of aboutby10% in the geometry can be expected As for expectations due toOverall, the etchavariations fluctuations of temperature, etchant[33–36]. concentration the novel inOverall, this paper, the torsional double-beam the key to thebemechanical sensitivity, so and other device reasons. a variation of about 10% in theisgeometry can expected [33–36]. As for it’s novel necessary toindeliberately set torsional some imperfections their the mechanical the device this paper, the double-beamtois analyze the key to the influence mechanicaltosensitivity, so it’s sensitivity.to deliberately set some imperfections to analyze their influence to the mechanical sensitivity. necessary perform a FEM FEM analysis analysis for for the the tolerance tolerance in in fabrication fabrication by by making making the the double-beam double-beam Here, we perform have different width. The novel device is set as a baseline and the range of machining error is set as ±11μm. ± µm. The Thesimulation simulation results results are are shown shown in in Figure Figure 9. They They show show that the effect of two beams’ widths on mechanical mechanicalsensitivity sensitivity rate of change the and sametheand the mechanical sensitivity on rate of change is theissame mechanical sensitivity deviation deviation increases increases as the machining error increases. The mechanical sensitivity rate ofischange is4.2% fromto −4.2% as the machining error increases. The mechanical sensitivity rate of change from − 3.6% to 3.6% the maximum machining is ±0.2 μm. However, the maximum machining when the when maximum machining error is ±error 0.2 µm. However, when thewhen maximum machining error is error is ±1 the mechanical sensitivity rate of change from −16.8% 18.9%, a huge ± 1 µm, theμm, mechanical sensitivity rate of change is from is −16.8% to 18.9%,towhich is awhich huge is error for error for the novelThus, device. in achieve order to mechanical achieve mechanical sensitivity the fabrication the novel device. in Thus, order to sensitivity accuracy,accuracy, the fabrication process processbe should be designed carefully the compensation mask compensation should be implemented to decrease should designed carefully and theand mask should be implemented to decrease the the machining of torsional the torsional beam. machining errorerror of the beam.

Figure 9. FEM simulation of the tolerance in fabrication by making the the double-beam having different width. Figure 9. FEM simulation of the tolerance in fabrication by making double-beam having different width.

3.4. Simulation Analysis of Temperature Robustness 3.4. Simulation Analysis of Temperature Robustness The temperature drift is one of the most important factors contributing to the change of The temperature driftthe is one of the most to the change of the the capacitance between silicon plane andimportant substrate.factors The contributing proposed structure achieves capacitance between the silicon substrate. The proposed structure the temperature self-calibration by theplane designand of double differential configuration and achieves symmetrical temperature self-calibration by the design of double differential configuration and symmetrical structure, greatly improving the temperature robustness of the device. The packaged device is shown structure, greatly improving the temperature robustness of the device. The packaged device is shown in Figure 10. in Figure 10. The bonded accelerometer die, composed of a silicon structure and a silicon substrate, is attached The bondedpackage accelerometer of aAs silicon structure andand a silicon substrate, is produce attached on the ceramic bottomdie, by composed an Au film. all the materials connections will on the ceramic package bottom by an Au film. As all the materials and connections will produce

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thermal stress with withthe thetemperature temperature changes, the temperature performance simulation should be thermal stress changes, thethe temperature performance simulation should be done thermal stress with the temperature changes, temperature performance simulation should be done for the whole system. In this paper, we have performed a FEM simulation of the device within for the system. In this we have performed a FEM simulation of the within the done forwhole the whole system. In paper, this paper, we have performed a FEM simulation of device the device within ◦ ◦ ◦ the temperature range from −40 ℃ to 60 ℃ while the reference temperature is 20 ℃. The constraint temperature range from −40−40 C℃ to to 60 60 C℃ while thethe reference temperature isis2020 C. the temperature range from while reference temperature ℃. The The constraint condition is shown in Figure 11a. InInorder order totomake make the whole system have free expansion, point is1 condition is is shown shownin inFigure Figure11a. 11a.In orderto makethe thewhole whole system have free expansion, point condition system have free expansion, point 11 is fixed completely while thethe other points are constrained constrained in special special directions. The deformation deformation at high high is fixed completely while other points are constrained in special directions. The deformation at fixed completely while the other points are in directions. The at and low temperature is shown in Figure 11b and the simulation results of temperature draft are highlow andtemperature low temperature is shown in Figure 11b and the simulation results of temperature and is shown in Figure 11b and the simulation results of temperature draftdraft are shown in Figure Figure 11c. 11c. It can canItbe be seen that, that, in the theinfull full temperature range,range, four capacitors capacitors have the the same are shown in Figure can be seen thetemperature full temperature four capacitors have the shown in 11c. It seen that, in range, four have same change trend. Also, each capacitor has about 2100 mg output draft which is a huge error for the MEMS same change Also,capacitor each capacitor has 2100 aboutmg 2100 mg output draft is which is error a huge for the change trend.trend. Also, each has about output draft which a huge forerror the MEMS accelerometer. MEMS accelerometer. accelerometer.

Figure 10. Schematic of the packaged device. Figure 10. 10. Schematic Schematic of of the the packaged packaged device. device. Figure

(a) (a)

(b) (b) Figure 11. Cont.

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(c) ◦ C: to Figure11. 11.FEM FEMsimulation simulation of the device attemperature the temperature range−from (a) The Figure of the device at the range from 40 ◦ C −40 to 60℃ (a) 60 The℃: constrain constrain condition of the simulation; (b) The deformation at high and low temperature; The condition of the simulation; (b) The deformation at high and low temperature; (c) The simulation (c) results simulation results of temperature draft.of temperature draft.

However, the design of double differential configuration in this paper eliminates the common However, the design of double differential configuration in this paper eliminates the common mode variation signal of each capacitor by twice difference of two pairs of differential masses. Thus, mode variation signal of each capacitor by twice difference of two pairs of differential masses. we choose the double differential capacitance calculated by Equation (32) as the output in this paper Thus, we choose the double differential capacitance calculated by Equation (32) as the output in and the temperature draft is reduced to 25 mg. The simulation results demonstrate the advantage of this paper and the temperature draft is reduced to 25 mg. The simulation results demonstrate the the double differential configuration in temperature self-calibration. advantage of the double differential configuration in temperature self-calibration. 4. Fabrication Fabrication Process Process 4. For aa traditional traditional double double differential differential torsional torsional MEMS MEMS accelerometer, accelerometer, wet wet etching etching isis utilized utilized to to For fabricate the suspended silicon beam with an accurate sloping cross sectional shape [22,23]. However, fabricate the suspended silicon beam with an accurate sloping cross sectional shape [22,23]. However, theprocess processenlarges enlargesthe thedevice devicearea areaand and reduces reducesthe the utilization utilization ratio ratio of of the the silicon silicon chip. chip. In In this this paper, paper, the novelprocess processbased basedon onaapre-buried pre-buriedmask maskand anddry dryetching etchingtechniques techniquesisisused usedto tofabricate fabricatethe thesilicon silicon aanovel structure, which makes the device more than 50 times smaller than a traditional double differential structure, which makes the device more than 50 times smaller than a traditional double differential torsional MEMS MEMS accelerometer, accelerometer, greatly greatly promoting promoting the thepotential potentialcommercial commercialvalue. value. The The detailed detailed torsional processingprocedure procedureofofthe thenovel novelaccelerometer accelerometerisisshown shownin inFigure Figure12, 12,where wherethe theprocess processof ofthe thenovel novel processing device is: (a) the starting material, a (100) silicon-on-insulator (SOI) wafer with device layer thickness device is: (a) the starting material, a (100) silicon-on-insulator (SOI) wafer with device layer thickness of66µm; μm;(b) (b)first firstdry dryetching etching the electrode substrate 2 μm to form capacitive (c) second of the electrode substrate forfor 2 µm to form the the capacitive gap;gap; (c) second dry dry etching the electrode substrate for 4 μm to form the silicon electrode; (d) growing of a 0.5 μm etching the electrode substrate for 4 µm to form the silicon electrode; (d) growing of a 0.5 µm thick thick silicon dioxide layer; (e) etching the silicon dioxide on the electrode substrate; (f) bonding the silicon dioxide layer; (e) etching the silicon dioxide on the electrode substrate; (f) bonding the SOI SOI wafer of electrode substrate (100) SOIwith wafer withlayer device layer thickness μm; (g) wafer of electrode substrate with a with (100) aSOI wafer device thickness of 50 µm; of (g)50 removing removing the handler layer and the buried oxide layer to fabricate to silicon structure; (h) growing of the handler layer and the buried oxide layer to fabricate to silicon structure; (h) growing of a 400 nm a 400 nm thick silicon dioxide; (i) first etching the silicon dioxide layer for half the thickness, thick silicon dioxide; (i) first etching the silicon dioxide layer for half the thickness, approximately approximately 200 the nm,pre-buried completingprocess the pre-buried process in this step;silicon (j) second etching silicon 200 nm, completing in this step; (j) second etching dioxide for 400 nm dioxide for 400 nm to etch the silicon structure; (k) first dry etching the silicon structure for 10 μm; (l) to etch the silicon structure; (k) first dry etching the silicon structure for 10 µm; (l) third etching of third etching of the dioxide for out 200the nmper-buried to reveal out the(m) per-buried mask; (m) second dry the silicon dioxide forsilicon 200 nm to reveal mask; second dry etching the silicon etching the silicon structure for 40 μm; (n) scouring off the silicon dioxide to complete the fabrication. structure for 40 µm; (n) scouring off the silicon dioxide to complete the fabrication. Asititshown shown in Figure 12,fabrication the fabrication is composed mainly composed of three parts: the As in Figure 12, the processprocess is mainly of three parts: the fabrication fabrication of the electrodethe substrate, the bonding process and the fabrication of the silicon structure. of the electrode substrate, bonding process and the fabrication of the silicon structure. In order In order to reduce the alignment error caused by the bonding process in Figure 12f, the bonding to reduce the alignment error caused by the bonding process in Figure 12f, the bonding process is process is designed before the fabrication of the silicon structure. Considering that the electrode substrate will be destroyed when the silicon structure is fabricated, the growing of a protection layer

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Sensors 2017,before 17, 2264the designed

14 of be 16 fabrication of the silicon structure. Considering that the electrode substrate will destroyed when the silicon structure is fabricated, the growing of a protection layer in Figure 12d is in Figure to 12d is designed to protect theAlso, silicon we design theprocess pre-buried dioxide designed protect the silicon structure. westructure. design theAlso, pre-buried dioxide in Figure 12i process in Figure 12i to fabricate the differential silicon masses with different thicknesses. to fabricate the differential silicon masses with different thicknesses.

Figure 12. The fabrication process of the accelerometer. Figure 12. The fabrication process of the accelerometer.

5. Conclusions Conclusions 5. 2 double differential torsional MEMS This paper paper reports reports the the design design and and simulation simulation of of aa 44 mm This mm2 double differential torsional MEMS accelerometer based based on onaadouble-beam double-beamconfiguration. configuration.The Thedevice device area novel accelerometer accelerometer area of of thethe novel accelerometer is is more than 50 times smaller than that of the traditional double differential torsional MEMS more than 50 times smaller than that of the traditional double differential torsional MEMS accelerometer. The Thedesign design model is established the working principle is analyzed in detail. accelerometer. model is established andand the working principle is analyzed in detail. Then Then the model of mechanical sensitivity is obtained theoretically to guide the structure design. the model of mechanical sensitivity is obtained theoretically to guide the structure design. Also, the Also, the nonlinearity, shock resistance and temperature robustness of the novel device are analyzed nonlinearity, shock resistance and temperature robustness of the novel device are analyzed by FEM by FEM simulation to demonstrate the excellent features of theAfterwards, design. Afterwards, the fabrication simulation to demonstrate the excellent features of the design. the fabrication tolerance tolerance is and analyzed and the fabrication is introduced. is analyzed the fabrication process is process introduced. Above all, the smaller device promotes the potential commercial commercial value value without without sacrificing sacrificing the the Above all, the smaller device promotes the potential specifications of of mechanical mechanical sensitivity, sensitivity, nonlinearity specifications nonlinearity and and temperature temperature robustness, robustness, which which are are 59.4 59.4 fF/g, fF/g, ◦ C respectively. The mechanical noise is analyzed in theory and the calculated 0.88% and 0.25 mg/ 0.88% and 0.25 mg/℃ respectively. The mechanical noise is analyzed in theory and the calculated

equivalent acceleration is about 0.13 mg/√Hz in theoretical calculation. Compared with traditional single-beam silicon structure, the novel structure can achieve lower maximum principle stress in

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√ equivalent acceleration is about 0.13 mg/ Hz in theoretical calculation. Compared with traditional single-beam silicon structure, the novel structure can achieve lower maximum principle stress in critical regions and reduce the possibility of failure mechanisms when high-g acceleration loading is applied along all three axes. The next step of present investigation is to finish the manufacturing and testing of the novel device to verify the design method and simulation results reported in this paper. This approach is expected to provide meaningful guidance for the design of a novel device and the method can also be utilized in the design of other MEMS devices. Acknowledgments: The authors would like to thank to the Laboratory of Microsystems, National University of Defense Technology, China, for technical support and access to equipment. This work was supported by the National Natural Science Foundation of China (Grant No. 51335011 and 51505489). Author Contributions: Dingbang Xiao and Xuezhong Wu conceived the study concepts. Qingsong Li and Zhanqiang Hou designed the simulation method. Tongqiao Miao and Qingsong Li carried out and analyzed the simulation results. Tongqiao Miao wrote the paper. Besides, Dingbang Xiao and Xuezhong Wu offered sound advices on the language. Conflicts of Interest: The authors declare no conflict of interest.

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