Analytical study and compensation for temperature drifts of a bulk silicon MEMS capacitive accelerometer

Sensors and Actuators A 239 (2016) 174–184

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Sensors and Actuators A: Physical journal homepage: www.elsevier.com/locate/sna

Analytical study and compensation for temperature drifts of a bulk silicon MEMS capacitive accelerometer Jiangbo He a,∗ , Jin Xie a , Xiaoping He b , Lianming Du b , Wu Zhou c a b c

School of Mechanical Engineering, Southwest Jiaotong University, Chengdu 610031, China Institute of Electronic Engineering, China Academy of Engineering Physics, Mianyang 621900, China University of Electronic Science and Technology of China, Chengdu 611731, China

a r t i c l e

i n f o

Article history: Received 2 September 2015 Received in revised form 20 December 2015 Accepted 14 January 2016 Available online 19 January 2016 Keywords: MEMS Capacitive accelerometer Temperature drift Stiffness temperature dependence Stiffness asymmetry Adhesive die attaching

a b s t r a c t An analytical study and a compensation structure for temperature drifts of bias and scale factor of a bulk silicon MEMS capacitive accelerometer are presented. The analytical model for the temperature drift of bias (TDB) and temperature drift of scale factor (TDSF) is established based on the analysis results of thermal deformation and stiffness temperature dependence. The model shows that TDB is only caused by thermal deformation, while TDSF consists of two parts caused by stiffness temperature dependence and thermal deformation, respectively. The two parts are positive and negative, respectively, but the second part has greater absolute value. First part of TDSF can be reduced by high doping. TDB and second part of TDSF can be both reduced by soft adhesive die attaching or increasing substrate thickness. In silicon structure, TDB can be reduced by middle-locating anchors for moving electrodes in sensitive direction or decreasing the stiffness asymmetry of springs, while second part of TDSF can be reduced by middle-locating anchors for fixed electrodes in sensitive direction. By middle-locating anchors both for moving electrodes and fixed electrodes, a temperature compensation structure is designed to reduce TDB and second part of TDSF. Consequently, TDSF is reduced by making the two parts of TDSF cancel each other. Experimental results show that TDB is suppressed from 1.85 mg/◦ C to 0.52 mg/◦ C, while TDSF from −162.7 ppm/◦ C to −50.8 ppm/◦ C. © 2016 Elsevier B.V. All rights reserved.

1. Introduction High-precision MEMS accelerometers are increasingly needed in numerous applications including self-contained navigation, seismometer for oil-exploration and earthquake prediction, and platform stabilization in space [1]. Among the utilized sensing schemes, e.g., capacitance, resonance, tunneling current, MEMS capacitive accelerometers are more attractive and promising for high precision devices due to high sensitivity, large readout bandwidth, and good noise performance [2]. However, MEMS capacitive accelerometers are inevitably subject to the effects of temperature, characterized by the temperature drift of bias (TDB) and temperature drift of scale factor (TDSF). Temperature drifts affect accuracy, and are the obstacle to be overcome for inertial navigation systems [3].

∗ Corresponding author at: No.111, Erhuanlu, Beiyiduan, Chengdu 610031, China. Fax: +86 28 66363899. E-mail address: [email protected] (J. He). http://dx.doi.org/10.1016/j.sna.2016.01.026 0924-4247/© 2016 Elsevier B.V. All rights reserved.

The study on temperature drifts includes compensation and theoretical analysis. Compensation methods can be divided into two categories: active or passive. Active compensation is to tune output electronically [4] or to keep temperature constant [5]. Active compensation typically involves the additional circuitry and complexity. Passive compensation based on package stress isolation has also been proposed to cancel out the effects of temperature, such as soft adhesive attaching [6] and wire-bonding mounting [7], fourpoint supporting frame [8], and centrally locating both moving and fixed elements [9]. The published theoretical reports on the temperature drifts are mainly based on finite element method (FEM). In our previous work, effects of randomness of fabrication errors and mismatch of coefficient of thermal expansion (CTE) between silicon and substrate on TDB were studied by multiphysics FEM [10]. Using a sensor and package interaction model, the warp of sensor and its effect on TDB were studied [11]. Li et al. [12] and Zhang et al. [8] studied the deflection of electrode plate induced by the thermal deformation of MEMS capacitance accelerometers by FEM, but did not establish the models for TDB or TDSF further. Zhang et al. [13] studied changes of capacitive gaps induced by thermal deformation by FEM,

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Fig. 3. Open-loop differential capacitive principle. ±Va is preload AC voltage. CA and CB represent capacitors on the bottom and top sides, respectively. Table 1 Material properties and dimensions for packaging module.

Fig. 1. SEM of the accelerometer.

Fig. 2. Schematic diagram for ceramic package and die of the accelerometer.

but did not establish the models for TDB or TDSF further too. Temperature coefficient of elastic modulus (TCEM) of silicon commonly affects performances of MEMS devices, such as the frequency temperature drift in MEMS resonators [14,15]. However, there were no published reports on the effect of TCEM on performances of MEMS capacitive accelerometers. FEM is time consuming and cannot clearly indicate main factors affecting temperature drifts. In this work, an analytical study on TDB and TDSF is proposed based on the analysis results of thermal deformation and stiffness temperature dependence mainly caused by TCEM of silicon. Then based on the theoretical results, a compensation structure for accelerometers is designed to reduce TDB and TDSF. 2. Fabrication and detection principle The capacitive accelerometer discussed in this work is shown in Fig. 1 and 2. In the die of the accelerometer, anchors and sensing elements, such as proof mass, comb capacitors and folded beams, are made of single crystal silicon. The substrate under sensing elements is made of Pyrex 7740 glass. The die is mounted in a ceramic package using epoxy adhesives. The crystal orientation of single crystal silicon in the accelerometer is shown in Fig. 2. The [1 0 0] direction is oriented on the out-of-plane direction. The [1 1 0] direction is oriented on the sensitive direction and the axial direction of the

Layer

Ceramic [21]

Young’s moduls (Gpa) Poisson’s ratio CTE (ppm/◦ C) Thickness (␮m)

400 0.22 6.5 1000

Adhesive 2.5 0.38 60 4

Glass [21] 63 0.2 3.25 500

folded beam. Moving electrodes on proof mass and fixed electrodes on anchors compose four comb capacitors. The proof mass is connected to anchors by four folded beams, which compose springs of the accelerometer. The accelerometer is fabricated based on a bulk silicon process, which was detailed before in Ref. [10]. In short, the process begins with a Cr/Au metallization process on a Pyrex 7740 glass wafer. Then, boron doping by diffusion is performed on the front side of a (100) silicon wafer. On the boron doping side of the silicon wafer, deep reactive ion etching is employed to define sensing elements and anchors. Next, the silicon wafer was flipped and bonded to glass wafer by anodic bonding. Finally, the undoped silicon was completely dissolved in alkaline liquor, leaving the boron doped silicon on the glass substrate. The detection of the accelerometer is based on the open-loop differential capacitive principle, as shown in Fig. 3. The acceleration force makes the proof mass move, and is balanced by elastic force generated by springs. The displacement of proof mass changes the capacitance of the accelerometer. The detected variation amplitude of the capacitance difference between capacitors CA and CB via modulation and demodulation with preload AC voltage was used to generate the output voltage Vout = M

CA − CB , CA + CB

(1)

where M is the coefficient that depends on circuit parameters. 3. Thermal deformation analysis As shown in Fig. 2, the substrate has a much larger volume than silicon structure. As such, it is assumed that the deformation of substrate is not constrained by the silicon structure, and anchors follow the deformation of the substrate top surface. The thermal deformation model is divided two modules, called as silicon structure module and packaging module (consisting of substrate, adhesive and ceramic package). The thermal deformation of packaging module is computed firstly. Then, the deformation of the substrate top surface is input into silicon structure module to compute the deformation of silicon structure. 3.1. Packaging module In packaging module, the adhesive layer is much thinner than other two layers and the elastic modulus of the adhesive layer is much lower than those of other two layers, as shown in Table 1. As such, the packaging module can be simplified as an adhesive bonded bimaterial structure, as shown in Fig. 4. This kind of layered

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J. He et al. / Sensors and Actuators A 239 (2016) 174–184

Fig. 4. Adhesive bonded bimaterial structure.

Fig. 5. Axial displacement on the top boundary of glass substrate with temperature change of −40 ◦ C.

structures is commonly used in electronic and MEMS packaging. There have been several analytical models developed to understand the stress in the bimaterial structure. In these models, the transverse deformations of adherends were modeled based on Euler–Bernoulli beam theory [16,17] or Timoshenko beam theory (TBT) [18–20]. The TBT models are more accurate when the adherends have relatively high thicknesses. In this work, a TBT model based on the Weissgraeber’s work [20] is taken to compute the deformation of packaging module. Governing equations for adherends were modeled by axial stretching theory and TBT, which were described in Eqs. (1)–(14) in literature [20]. The thin adhesive layer was modeled as a weak interface, which resulted in simple relations between displacements in adherends and stresses in adhesive layer [20]. However, thermal expansion must be added into axial displacements in adherends when the thermal stress is modeled [16,18]. Based on the governing equation of adhesive layer described in literature [20], the governing equation of adhesive layer for modeling thermal stress is 0 = Ga

0 =



1 d (w1 + w2 ) (u2t − u1b ) + (˛2 − ˛1 ) xT + t 2 dx



,

(2)

Ea (1 − va ) w2 − w1 t (1 + va ) (1 − 2va )

where u1b and u2t denote axial displacements on the bottom boundary of adherend 1 and the top boundary of adherend 2, respectively, w1 and w2 denote transverse displacements in adherend 1 and adherend 2, respectively, ˛1 and ˛2 denote CTEs of adherend 1 and adherend 2, respectively, Ea , Ga and a denote Young’s modulus, Shear modulus and Poisson’s ratio of adhesive, respectively, T denotes the temperature change. Combing Eq. (2) and governing equations for adherends, displacements in adherends can be solved by eigenvalue method [20]. In this work, materials for adherend 1 and adherend 2 are glass and ceramic, respectively. Material properties and dimensions are listed in Table 1. The length of the bimaterial structure is 1700 ␮m. The axial displacement on the top boundary of glass substrate with temperature change of −40 ◦ C is shown in Fig. 5. It can be seen that the displacement curve has good linearity, especially in the interval [−1000 ␮m, 1000 ␮m]. Fitting the displacement curve using linear regression, the displacement can be expressed as: u = ˛eq xT,

(3)

Fig. 6. Schematic diagram for the dimensions of MEMS accelerometer. Subscripts m and f represent the moving and fixed electrodes, respectively, and i and i + 1 represent the index of electrodes.

where ˛eq is called as equivalent CTE. In this work, the equivalent CTE is about 4.2 ppm/◦ C, and greater than the CTE of glass. This can be explained by the fact that ceramic has a higher CTE than glass, which means that the packaging exacerbates the deformation of substrate. Although the equivalent CTE is extracted according to the displacement under the temperature change of −40 ◦ C, the equivalent CTE is independent of temperature change. The reason is that material properties are assumed to be constant in analytical models for bimaterial structure [18–20]. As such, the displacement is proportional to the temperature change. From the definition of the equivalent CTE, it is known that the equivalent CTE represents the linear relationship between displacement function and temperature change, so the equivalent CTE is independent temperature change. 3.2. Silicon structure module In Fig. 6, the proof mass is connected to anchors by two flexible springs. Anchors follow the deformation of the substrate top surface as described at the beginning of Section 3. From the conclusion in Section 3.1, it is known that the thermal deformation of the substrate top surface is characterized by the equivalent CTE, which is approximate 4.2 ppm/◦ C. The spring and proof mass are made of silicon, and silicon has an isotropic CTE at room temperature of approximate 2.6 ppm/◦ C [22]. As such, the thermal mismatch generates deformation in springs and proof mass. However, because the proof-mass is much stiffer than flexible springs, the deformation mainly occurs in flexible springs. The deformation in proof mass is very small and can be neglected. As such, the deformation of proof mass follows the thermal expansion of silicon. Generally, two flexible springs have different stiffness due to randomness of fabrication errors [10]. When the temperature changes, the equilibrium condition of proof mass can be expressed as



KA uc − ˛s TLm + ˛eq TLa







+KB uc + ˛s TLm − ˛eq TLa = 0,

(4)

where uc denotes the displacement of proof mass, KA and KB denote the spring stiffness connecting proof mass, T denotes the temperature change, ˛s denotes the CTE of silicon, La denotes the distance

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CTE on the stiffness can be predicted by computing the temperature coefficient of stiffness TCS =

Fig. 7. Schematic diagram for dimensions of a folded beam.

from anchors to the midline, and Lm denotes the half length of proof mass. Solving Eq. (4), the displacement of proof mass can be expressed as uc =

 KA − KB  ˛s TLm − ˛eq TLa . KA + KB

(5)

The difference between Lm and La is generally very small. As such, the displacement of proof mass can be approximately expressed as uc ≈

 KA − KB  ˛s − ˛eq TLa . KA + KB

(6)

The displacement of proof mass makes moving electrodes move. In addition, moving electrodes have displacements relative to the center of proof mass due to thermal expansion. Fixed electrodes are directly located on anchors, so their displacements follow the deformation of substrate top surface. The displacement differences between moving electrodes and fixed electrodes result in changes in gaps. With CTEs and dimensions shown in Fig. 6, changes in gaps and their average value can be expressed as dAi = −lmi ˛s T + lf i ˛eq T + uc dBi = −lmi ˛s T + lf i ˛eq T − uc DAi = −lf i ˛eq T + lm

i+1 ˛s T − uc

,

(7)

d =

i=1

2N







= d˛s + (lf + Lf ) ˛eq − ˛s



T , (8)

N−1

D =

where TCS denotes the temperature coefficient of stiffness. In Eq. (10), the temperature coefficient of stiffness equals the sum of TCEM and CTE, and is independent of dimensions. As such, for the total stiffness of springs in the accelerometer, its temperature coefficient also equals the sum of TCEM and CTE. Undoped silicon has a TCEM of approximate −64 ppm/◦ C at room temperature, so temperature coefficient of stiffness for undoped silicon is approximate −61.4 ppm/◦ C. However, doping can reduce TCEM [15]. For TCEM of doped single crystalline silicon, J.J. Hall studied TCEM of phosphorus-doped single crystalline silicon with 2e19/cm3 [24]. A. Jaakkola extracted TCEM of borondoped single crystalline silicon with 6e18/cm3 and 3e19/cm3 [25]. E.J. Ng extracted TCEM of boron-doped single crystalline silicon with 4.1e18/cm3 , 1.4e20/cm3 and 1.7e20/cm3 [26]. In this work, boron is doped into silicon by diffusion, as described in Section 2. Therefore, the boron concentration varies continually along the diffusion depth. As such, TECM varies continually along the diffusion depth, so TECM under a wide range is needed to analyze the effect of doping on the temperature coefficient of stiffness. In addition, the continually varied TECM makes the silicon like functional grade material. Therefore, the temperature coefficient of stiffness should be calculated based on composite material mechanics. These works needs to be studied in future. In summary, due to the aforementioned high boron-doping process, temperature coefficient of stiffness is a negative value, which is greater than −61.4 ppm/◦ C in this work. 5. Temperature drifts

x= (dAi + dBi )

(DAi + DBi )

i=1

2 (N − 1)





= D˛s − (lf + Lf ) ˛eq − ˛s



T

where d and D denote the narrow and wide gap before temperature change, respectively, lm i , lm i + 1 , and lf i denote locations of electrodes, Lf denote the half length of an anchor for fixed electrodes, and lf denote the location of first fixed electrode, as shown in Fig. 6.

−ma , K (1 + TCST )

A spring in the accelerometer is a simple support structure composed of a folded beam. Its dimensions are shown in Fig. 7. The stiffness equation of a spring is [23] Ew3 h , 2L3

(9)

where E denote the Young’s modulus of the used silicon, w denote the width of the beam, h denote the height of the beam, and L denote the length of the beam. Young’s modulus and dimensions are all functions of temperature. Dimensions of the beam change with temperature due to thermal expansion. Effects of TCEM and

(11)

where x denotes the displacement, m denotes the total mass of proof mass and moving electrodes, and K denotes the total stiffness, which equals the sum of KA and KB . After temperature change and with the displacement of proof mass, gaps of CA become (d + dAi + x) and (D + DAi − x), while those of CB become (d + dBi − x) and (D + DBi + x). Ignoring the fringe effect, CA and CB can be expressed as CA = 2εa ˝

CB = 2εa ˝

 N   i=1 N  i=1

4. Stiffness temperature dependence

KU =

(10)

After temperature change, the displacement of proof mass caused by input acceleration can be expressed as

DBi = −lf i ˛eq T + lm i+1 ˛s T + uc N 

1 dKU = TCEM + ˛s , KU dT

N−1



i=1 N−1

,

 1 1 + (d + d + yAi ) (D + D + zAi )  1 1 + (d + d + yBi ) (D + D + zBi )

(12)

i=1

where yAi = dAi − d + x, yBi = dBi − d − x, zAi = DAi − D − x, zBi = DBi − D + x, and ˝ represents the overlapping area in a pair of moving and fixed electrodes, N denotes the fixed electrode number in a comb capacitor. Substituting Eq. (12) into Eq. (1), Vout becomes a multivariable function of yAi , yBi , zAi and zBi . Based on the multivariable Taylor Expansion Theorem [27], Vout is simplified as



Vout =

2

2



−M (D + D) − (d + d)  (x + uc ) [(d + d)  + (D + D)] (d + d) (D + D)

,

(13)

where  is equal to (N − 1)/N. N is generally much greater than 1, so  can be assumed to be 1. Substituting Eq. (11) into Eq. (13), Vout becomes a linear function of a. In the function, the constant

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Table 2 Material properties and dimensions for accelerometer. Property

Value

Units

Folded beam width (w) Folded beam length (L) Folded beam height (h) Narrow capacitive gap (d) Wide capacitive gap (D) Electrode width (wf ) Distance from anchors of proof mass to midline (La ) Distance from the first fixed electrode to midline (lf ) Silicon CTE (˛s ) Equivalent CTE (˛eq ) Silicon Young’s modulus in [1 1 0] direction (E) Temperature coefficient of stiffness (TCS) Total mass of proof mass and moving electrodes (m) Fixed electrodes number in a comb structure (N)

5 480 42 5 25 6.5 960 45 2.6 4.2 169 −61.4 < TCS < 0 59.4 21

␮m ␮m ␮m ␮m ␮m ␮m ␮m ␮m ppm/◦ C ppm/◦ C GPa ppm/◦ C ␮g –

term is the bias expressed in units of voltage, and the coefficient of first-degree term is the scale factor. Bias expressed in units of voltage is generally transformed into the bias expressed in units of acceleration by being divided by scale factor. The bias and scale factor after temperature change are p0 |T =T = − k1 |T =T =



K (1 + TCST ) uc Kuc ≈− , m m

(14)

1 1 − d + d D + D

(15)



Mm , K (1 + TCST )

where p0 and k1 denote the bias expressed in units of acceleration and scale factor, respectively. TCST, d and D are all small, so scale factor is simplified as k1 |T =T =

(D − d) (1 − TCST ) − dDK

 d d2

−

D D2

 Mm K

.

(16)

5.1. Analytical formulas for TDB and TDSF TDB represents the change in bias caused by temperature change. From Eq. (14), it is known that the bias before temperature changing is zero. Therefore, TDB can be expressed as TDB =

p0 |T =T Kuc =− . T mT

(17)

Substituting Eq. (6) into Eq. (17), TDB is expressed as TDB =

 KA − KB  ˛eq − ˛s La . m

(18)

TDSF represents the change in scale factor induced by temperature change. Based on the expression for scale factor, Eq. (16), TDSF is expressed as TDSF =

k1 |T =T − k1 |T =0 k1 |T =0 T

= −TCS

−

d/ (dT ) − D/ (DT ) −1

,

(19)

where equals D/d. Substituting Eq. (8) into Eq. (19), TDSF can be expressed as



TDSF = −TCS + −˛s −





2 + 1 (lf + Lf ) ( − 1) d



˛eq − ˛s





,

(20)

To calculate TDB and TDSF, material properties and dimensions of the accelerometer need to be substituted into Eqs. (18) and (20) respectively. Material properties and dimensions are listed in Table 2. In this work, the designed width of all folded beams is 5 ␮m. However, when calculating the spring stiffness (KA and KB in Eq. (18)), the beam width difference induced by the randomness of fabrication errors must be considered. The width difference

Fig. 8. Effects of spring asymmetry on TDB. wA and wB denote the beam width of KA and KB , respectively. Because the problem is symmetric, only the situation, wA ≤ wB , is considered.

between different beams exhibits up to 0.5 ␮m [10], so the actual beam width lies in [4.75 ␮m, 5.25 ␮m] randomly in this work. The effect of springs asymmetry induced by the beam width difference on TDB will be analyzed in detail in Section 5.2.1. According to the definition in Section 4, TCS represents the temperature coefficient of stiffness, and is a negative value greater than −61.4 ppm/◦ C. In Eq. (20), Lf represents the half length of an anchor for fixed electrodes, as shown in Fig. 6. According to dimensions in a comb capacitor, as shown in Fig. 6, Lf can be expressed as Lf =

(N − 1) (d (1 + ) + 2wf ) , 2

(21)

where wf denotes the width of an electrode, and N denotes the fixed electrode number in a comb capacitor. 5.2. Analysis of TDB Based on Eq. (18), effects of springs asymmetry, Young’s modulus of adhesive, substrate thickness and locations of anchors for moving electrodes on TDB will be discussed in this section. 5.2.1. Effect of springs asymmetry on TDB According to Eq. (18), TDB is directly proportional to the stiffness difference, (KA − KB ). Therefore, TDB is affected by spring asymmetry, and this coincides with the conclusion obtained by multiphysics FEM [10]. Springs asymmetry is caused by the beam width difference induced by the randomness of fabrication errors. As proposed in Section 5.1, the actual beam width lies in [4.75 ␮m, 5.25 ␮m] randomly. As such, the sign and absolute value of the stiffness difference, (KA − KB ), is random. Consequently, the sign and absolute of TDB is also random. For analyzing the effect of springs asymmetry on TDB, beam widths at two ends of proof mass are set to be different in this work. Effects of springs asymmetry on TDB are shown in Fig. 8. It can be seen that when beam widths difference increases from 0.1 ␮m to 0.5 ␮m, TDB increases from 1.21 mg/◦ C to 6.05 mg/◦ C sharply. Therefore, it is necessary to improve the spring symmetry by reducing the randomness of fabrication errors. 5.2.2. Effects of Young’s modulus of adhesive and substrate thickness on TDB In Eq. (18), TDB is directly proportional to CTE difference (˛eq − ˛s ). ˛s denotes the CTE of silicon, while ˛eq is called as equivalent CTE according to the definition in Section 3.1. The equivalent CTE characterize the thermal deformation of substrate top surface affected by packaging. Therefore, TDB is affected not only by substrate but also by packaging. Effects of Young’s modulus of adhesive and substrate thickness on TDB are shown in Fig. 9. It can be seen that TDB decreases significantly as Young’s modulus of adhesive decreases or substrate thickness increases. Soft adhesive die attaching (adhesives with low elastic modulus) is a commonly

J. He et al. / Sensors and Actuators A 239 (2016) 174–184

Fig. 9. Effects of Young’s modulus of adhesive and substrate thickness on TDB. Ea and hs denote the elastic modulus of adhesive and substrate thickness, respectively. wA and wB are set to be 5.2 ␮m and 4.8 ␮m, respectively.

179

Fig. 10. Effects of Young’s modulus of adhesive and substrate thickness on the second part of TDSF. Ea and hs denote the elastic modulus of adhesive and substrate thickness, respectively. wA and wB are set to be 5.2 ␮m and 4.8 ␮m, respectively. “TDSF2 ” denotes the second part of TDSF.

used method to isolate MEMS packaging stress [6,28]. In this work, soft adhesive die attaching can isolate the packaging exacerbating effect on the substrate deformation. Increasing substrate thickness improves the substrate immunity to packaging effect. As such, soft adhesive die attaching and increasing substrate thickness can both reduce the equivalent CTE. Due to the fact that the equivalent CTE is greater than the CTE of silicon, so lower equivalent CTE results in lower and TDB.

5.2.3. Effects of locations of anchors for moving electrodes on TDB In Eq. (18), TDB is proportional to La , which is determined by the location of the anchor for moving electrodes in sensitive direction. In literature [9], based on a novel fabrication process combining bulking and surface micromachining, all anchors (both for moving electrodes and fixed electrodes) are middle-located both in sensitive and other directions to reduce the temperature effect on MEMS accelerometer. However, it is concluded in this work that TDB can be reduced as long as anchors for moving electrodes are middle-located in sensitive direction. This is benefit for middlelocating anchors without developing novel fabrication process. In this work, a temperature compensation structure with middlelocated anchors for moving electrodes is proposed. In Section 6, compensation structure will be discussed in detail.

Fig. 11. Effects of d and on the second part of TDSF. “TDSF2 ” denotes the second part of TDSF.

5.3.2. Effects of Young’s modulus of adhesive and substrate thickness on the second part of TDSF According to the second term of Eq. (20), the second part of TDSF is proportional to CTE difference (˛eq − ˛s ). This is same with TDB, as discussed in Section 5.2.2. As such, soft adhesive die attaching and increasing substrate thickness can also both reduce the second part of TDSF. Effects of Young’s modulus of adhesive and substrate thickness are shown in Fig. 10. In Fig. 10, it can be seen that the second part of TDSF decreases significantly as Young’s modulus of adhesive decreases or substrate thickness increases.

5.3. Analysis of TDSF Based on Eq. (20), TDSF will be divided into two parts, and the two parts will be compared in this section. Then, effects of Young’s modulus of adhesive, substrate thickness and locations of anchors for fixed electrodes on the second part of TDSF will be discussed.

5.3.1. Comparison between the first part and second part of TDSF According to Eq. (20), TDSF consists of two parts. The first part equals the opposite of TCS. According to the definition in Section 4, TCS represents the temperature coefficient of stiffness, and is a negative value, which is greater than −61.4 ppm/◦ C. Therefore, the first part of TDSF is positive, but lower than 61.4 ppm/◦ C. The second of TDSF can be computed by substituting CTEs and dimensions listed in Table 2 into the second term of Eq. (20). The result shows that the second part of TDSF is about −202 ppm/◦ C. As a whole, TDSF is negative and locates in [−202 ppm/◦ C, −138.6 ppm/◦ C]. The two parts of TDSF cancel each other, which can be employed to compensate TDSF, as discussed in detail in Section 6. This kind of temperature compensation technology has been employed to reduce the frequency drift in MEMS resonators [29].

5.3.3. Effects of locations of anchors for fixed electrodes on the second part of TDSF According to the second term of Eq. (20), the second part of TDSF is proportional to (lf + Lf ). As shown in Fig. 6, lf and Lf denote the location of first fixed electrode and the half length of an anchor for fixed electrodes, respectively. (lf + Lf ) represents center locations of anchors for fixed electrodes in sensitive direction. As such, the second part of TDSF can be reduced as long as anchors for fixed electrodes are middle-located in sensitive direction. Same as TDB, this is benefit for middle-locating anchors without developing novel fabrication process. In the accelerometer shown in Fig. 1, Lf is much longer than lf , so locations of anchors are mainly determined by Lf . According to Eq. (21), lower Lf can be achieved by reducing N, d, and wf , which denote fixed electrodes number in a comb capacitor, the narrow capacitive gap, the ratio of wide capacitive gap to narrow capacitive gap and the width of a electrode, respectively. However, lower N is against the detection due to lower sensing capacitance. Lower wf results in lower stability against adhesion. Effects of d and on the second part of TDSF are shown in Fig. 11. It can be seen that the second part of TDSF decreases as d increases. However, higher d results in lower sensing capacitance and scale

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J. He et al. / Sensors and Actuators A 239 (2016) 174–184

Fig. 12. Global SEM photo of compensation structure.

Fig. 14. Schematic diagram for deformation analysis of compensation structure.

trodes using long floating silicon beams. In the sensitive direction, long floating silicon beams for moving and fixed electrodes are very rigid, so they are called as rigid connector for moving electrodes (RCME) and rigid connector for fixed electrodes (RCFE), respectively. However, RCMEs and RCFEs bend in out-of-plane direction due to residual stress induced by born diffusion. To suppress the bending, which may result in adhesion, flexible and floating beams are designed to support RCMEs and RCFEs, called as flexible connector of moving electrodes (FCME) and flexible connector of fixed electrodes (FCFE), respectively. The length of proof mass in compensation structure is same with that in original structure shown in Fig. 1, while the width is increased by 100 ␮m to locate anchors for moving electrodes. The total mass of proof mass and moving electrodes, capacitive gaps and springs in compensation structure are same with those in original structure in order to ensure performances of compensation structure to be same with those of the original structure except temperature drifts. Fig. 13. Local SEM photos of compensation structure. RCME—rigid connector of proof mass; FCME—flexible connector of proof mass; RCFE—rigid connector of fixed electrodes; FCFE—flexible connector of fixed electrodes.

6.2. Interpretation for the temperature insensitivity of compensation structure

factor, so is against detection. The relationship between the second part of TDSF and has peaks, such as the point where = 4 and d = 4 ␮m. However, is generally greater than 5 to guarantee the linearity of accelerometers [30]. As whole, middle-locating anchors for fixed electrodes by reducing N, d, and t all affect other performances. In Section 6, a temperature compensation structure without effects on other performances is proposed to reduce the second part of TDSF.

In compensation structure, FCMEs and FCFEs are much more flexible than RCMEs and RCFEs, respectively, so their effects on deformations of RCMEs and RCFEs are small. As such, FCMEs and FCFEs can be neglected when analyzing the deformation of compensation structure, as shown in Fig. 14. When the temperature changes, the equilibrium condition of proof mass in compensation structure can be expressed as

6. A temperature compensation structure

KB uc + ˛s TLm − ˛s T (La − la ) − ˛eq Tla = 0

6.1. Description

where la is the distance from anchors for moving electrodes to the midline. With Eq. (22), the displacement of proof mass can be solved as

According to conclusions in Sections 5.2 and 5.3, a temperature compensation structure is designed, as shown in Figs 12 and 13. TDB and the second part of TDSF are reduced by middle-locating anchors for moving electrodes and fixed electrodes, respectively. Consequently, TDSF is reduced by making two parts of TDSF cancel each other. Anchors are middle-located only in sensitive direction, and this benefits anchors locating without developing novel fabrication process. In compensation structure, springs, proof mass and moving electrodes are connected to anchors for moving electrodes using long floating silicon beams. Anchors for fixed electrodes are shortened in sensitive direction and also connected to fixed elec-









KA uc − ˛s TLm + ˛s T (La − la ) + ˛eq Tla +

uc =

 KA − KB  ˛s − ˛eq Tla . KA + KB

,

(22)

(23)

Combing Eqs. (17) and (23), TDB for compensation structure can be expressed as TDB =

 KA − KB  ˛eq − ˛s la . m

(24)

Fig. 14 shows that displacements of fixed electrodes involve deformations of both anchors and RCMEs, which follow the deformation of substrate top surface and thermal expansion of silicon,

J. He et al. / Sensors and Actuators A 239 (2016) 174–184

181

respectively. Same as original structure, moving electrodes move with the displacement of proof mass and have displacements relative to the center of proof mass due to thermal expansion. Displacement differences between moving electrodes and fixed electrodes result in gaps changes. With CTEs and dimensions shown in Fig. 14, gaps changes and their average value can be expressed as









dAi = −lm i ˛s T + lf i − lg ˛s T + lg ˛eq T + uc dBi = −lm i ˛s T + lf i − lg ˛s T + lg ˛eq T − uc









DAi = − lf i − lg ˛s T − lg ˛eq T + lm

i+1 ˛s T

− uc

DBi = − lf i − lg ˛s T − lg ˛eq T + lm

i+1 ˛s T

+ uc













d = d˛s + lf + lg

D = D˛s − lf + lg

˛eq − ˛s



˛eq − ˛s



,

(25)

T .

(26)

T

where lg is the length of anchors for fixed electrodes in compensation structure. With Eqs. (19) and (26), the second part of TDSF for compensation structure can be expressed



TDSF = −TCS + −˛s −





2 + 1

lf + lg

( − 1) d





˛eq − ˛s





.

(27)

Fig. 15. FEM models for thermal deformations. The symmetric model and submodel are employed to reduce time. Because suspended elements, for instance proof mass, do not affect the deformation of substrate, they are deleted in submodel #1. Submodel #1 is simulated firstly. Then, the displacement of substrate is extracted and input into submodel #2 (as boundary condition) to complete the simulation of submodel #2.

As described in Section 6.1, except locations of anchors, other parameters in compensation structure are same with those in original structure, such as the total mass of proof mass and moving electrodes, capacitive gaps and springs. As such, the only difference between TDB of original structure and that of compensation structure, which are represented by Eqs. (18) and (24) respectively, is that La is greater than la . In original structure and compensation structure, La and la are 960 ␮m and 190 ␮m, respectively. Therefore, TDB of compensation structure must be lower than that of original structure. Formulas of TDSF for original structure and compensation structure are Eqs. (20) and (27), respectively. Their only difference is that Lf is greater than lg . In original structure and compensation structure, Lf and lg are 450 ␮m and 110 ␮m, respectively. Therefore, the second part of TDSF for compensation structure, which is represented by the second term of Eq. (27), must be lower than that for original structure. Consequently, the reduced second part leads to a lower TDSF for the compensation structure because the first part of TDSF and the second part of TDSF compensate each other. 6.3. Verification by FEM FEM is employed to study differences in TDB and TDSF between original structure and compensation structure. The obtained result will also be employed to compare with experimental results in Section 7. The FEM simulation is carried out using the Comsol Multiphysics 4.3b software package [31]. The simulation process is: firstly, the thermal deformation is simulated by FEM, and the displacements of proof mass and average variation in gaps are extracted according to the simulation results; secondly, TDB is computed by substituting the displacement of proof mass into Eq. (17), and the second part of TDSF is computed by substituting average change in gaps into the second term in Eq. (19). The FEM models for thermal deformations of the original structure and compensation structure are shown in Fig. 15. Beam widths are the same with those in Fig. 8. The temperature change is set to be −40 ◦ C. Material properties of glass substrate, adhesive layer and ceramic package are listed in Table 1. The single crystal silicon is anisotropic. Its crystal orientation in this work is shown in Fig. 2. According to this orientation and the research results in [32], elastic constants of single crystal silicon used in this work can be determined.

Fig. 16. FEM comparisons of TDB and the second part of TDSF between original structure and compensation structure. “TDSF2 ” denotes the second part of TDSF.

FEM results of TDB and the second part of TDSF are shown in Fig. 16. It can be seen that FEM results of TDB and the second part of TDSF for compensation structure are both much lower than those of original structure. FEM results verify the conclusion that compensation structure has lower TDB and second part of TDSF. TDB for compensation structure is about 12.9% of that for the original structure. The second part of TDSF is negative and not dependent on the difference between wA and wB . This coincides with the conclusion obtained in Section 5.2. FEM results of the second part of TDSF for the original structure and compensation structure are −193 ppm/◦ C

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Fig. 17. Testing equipments for accelerometers. DC supply and digital voltmeter provide the power input and display for output voltage, respectively.

and −77 ppm/◦ C, respectively. Taking the first part of TDSF into account, FEM results of TDSF for the original structure and compensation structure locate in [−193 ppm/◦ C, −141.6 ppm/◦ C] and [−77 ppm/◦ C, −15.6 ppm/◦ C], respectively. 7. Experiment 7.1. Testing process Accelerometers with the original structure and compensation structure are fabricated by the same fabrication process, as described in Section 2. The global and local SEM photos of compensation structure are shown in Figs. 12 and 13. The die is attached to the ceramic package by epoxy, and the input, output, and ground voltage pads are wire-bonded to the copper trace on the ceramic package. A packaged accelerometer is mounted on a printed circuit board (PCB) for testing, as shown in Fig. 17. In order to measure TDB and TDSF, bias and scale factor under different ambient temperature must be measured. For this purpose, the PCB with accelerometer is mounted inside a temperature chamber. The chamber can maintain the ambient temperature with ±0.5 ◦ C and change temperature from −55 ◦ C to 100 ◦ C. Bias and scale factor under different temperature are measured according to the output in ±1 g gravity field. If biases and scale factors under T0 and T1 are measured, TDB and TDSF between T0 and T1 are TDB =

p0 (T1 ) − p0 (T0 ) , T1 − T0

TDSF =

k1 (T1 ) − k1 (T0 ) . (T1 − T0 ) k1 (T0 )

(28) (29)

7.2. Experimental results and discussions 7.2.1. Temperature drifts In this work, temperature drift curves of bias and scale factor for five accelerometers with the original structure are measured, as shown in Fig. 18. In Fig. 18(a), all temperature curves of bias are monotonic. However, some of them increase, whereas others decrease. This can be explained by the randomness of fabrication errors. In Eq. (18), TDB is directly proportional to the stiffness difference (KA − KB ). Due to the randomness of fabrication errors, the sign of (KA − KB ) is random, as discussed in Section 5.2.1. As such, TDB can be positive or negative. In Fig. 18(b), all temperature drift curves of scale factor decrease as temperature increases. This coincides with the conclusion that TDSF is negative in Section 5.3.1. Though CTE in this work is assumed to be constant, CTE is actually dependent on temperature. For instance, the CTE of silicon has an increment of about 0.3 ppm/◦ C when temperature increases from 25 ◦ C to 80 ◦ C [33]. The temperature dependence of

Fig. 18. Temperature dependence of bias and scale factor. The room temperature (20 ◦ C) is set to be the reference temperature.

CTE could induce nonlinearity in temperature drift curves, as shown in Ref. [10]. However, temperature drift curves shown in Fig. 18 are approximately linear. This may be due to the temperature dependence for the CTE of ceramic. From Eq. (18), it is known that TDB and TDSF are both directly proportional to the CTE difference (˛eq − ˛s ). ˛s denotes the CTE of silicon, while ˛eq is equivalent CTE defined in Section 3.1. The equivalent CTE characterize the thermal deformation of substrate top surface. From the conclusion in Section 3.1, it is known that the increasing of CTE for glass and ceramic both exacerbate the deformation of glass top surface. Therefore, the equivalent CTE increases with the increasing of CTE both for glass and ceramic. The CTE of glass almost do not increase with temperature, but it is greater than that of silicon until 180 ◦ C [34]. The CTE of ceramic has an increment of about 0.8 ppm/◦ C when temperature increases from 25 ◦ C to 80 ◦ C [35]. As whole, the equivalent CTE have positive temperature dependence. It could make the CTE difference have low temperature dependence that both CTE of silicon and equivalent CTE have the positive temperature dependence. The monotonic temperature curve allows TDB and TDSF to be evaluated according to outputs at two measuring points. In this work, 5 ◦ C and 55 ◦ C are selected as two measuring points. TDBs and TDSFs of 20 accelerometers with the original structure are demonstrated in Fig. 19. In Fig. 19(a), the average absolute value of TDB for the original structure is 1.85 mg/◦ C. However, TDBs are very random. For instance, the maximum value and minimum value of TDB for the original structure are about 4 mg/◦ C and 0.3 mg/◦ C, respectively. This can also be explained by the randomness of fabrication errors. Due to the randomness of fabrication errors, the absolute value of TDB is random, as discussed in Section 5.2.1. This randomness could ideally result TDB to be equal to zero. In Fig. 19(b), the randomness of TDSF is much lower than that of TDB. This can be explained by the fact that the stiffness difference

J. He et al. / Sensors and Actuators A 239 (2016) 174–184

Fig. 19. TDBs and TDSFs of 20 accelerometers with the original structure. TDB is represented by absolute value.

(KA − KB ) does not affect TDSF. The average value of TDSF for the original structure is −162.7 ppm/◦ C, and locates in [−200 ppm/◦ C, −138.6 ppm/◦ C] obtained by FEM in Section 6.3. The difference between −162.7 ppm/◦ C and the simulation result of the second part of TDSF obtained in Section 6.3 is about 27 ppm/◦ C. This means that that high boron doping decrease TCEMS from −64 ppm/◦ C to about −30 ppm/◦ C. 7.2.2. Comparison of TDB and TDSF between the original structure and compensation structure TDBs and TDSFs of 20 accelerometers with compensation structure are demonstrated in Fig. 20. In Fig. 20(a), the average absolute value of TDB is 0.52 mg/◦ C for compensation structure, which is about 28% of that of the original structure. This finding provides evidence for the conclusion that compensation structure considerably reduces TDB. The simulation result of TDB ratio between compensation structure and original structure is 12.9% in Section 6.3, while the experimental result here is 28%. This means that some other factors may affect TDB, and this issue needs to be further examined in future. In Fig. 20(b), the average value of TDSF is −50.8 ppm/◦ C for compensation structure, which is about 31% of that of the original structure. This result proves that compensation structure can considerably reduce TDSF. The negative TDSF for compensation structure means that the absolute value of the second part of TDSF is still greater than that of the first part of TDSF in compensation structure. In future work, more precise structural designing is needed to make the first part of TDSF and the second part of TDSF compensate each other to approach a TDSF close to zero. 8. Conclusion and future work TDB and TDSF of a bulk silicon MEMS capacitive accelerometer are studied analytically in this work. TDB is only caused by thermal deformation, and can be reduced by soft adhesive die-attaching, increasing substrate thickness or middle-locating anchors for moving electrodes in sensitive direction. TDSF is composed of two parts, and they have opposite signs. The first part is mainly determined

183

Fig. 20. TDBs and TDSFs of 20 accelerometers with compensation structure.

by the TCEM of silicon. The second part can be reduced by soft adhesive die-attaching, increasing substrate thickness or middlelocating anchors for fixed electrodes in sensitive direction. By middle-locating anchors both for moving electrodes and fixed electrodes, a temperature compensation structure is designed to reduce TDB and TDSF. Experimental results show that TDB is suppressed from 1.85 mg/◦ C to 0.52 mg/◦ C, while TDSF from −162.7 ppm/◦ C to −50.8 ppm/◦ C. The future work will focus on following two aspects. First, the soft adhesive die attaching will be developed. Second, by precise structural designing, two parts of TDSF will compensate each other to make TDSF close to zero. Acknowledgements This research was partially supported by National Natural Science Foundation of China (Grant No. 51175437). Authors would like to acknowledge Mr. Zhigui Shi and Ms. Juan Liu for their assistance in fabrication and experiments. References [1] M. Bao, Micromechanical transducers, pressure sensors, accelerometers and gyroscopes, in: S. Middelhoek (Ed.), Handbook of Sensors and Actuators, Elsevier, Amsterdam, 2000. [2] B. Tang, et al., Process development of an all-silicon capacitive accelerometer with a highly symmetrical spring-mass structure etched in TMAH + Triton-X-100, Sens. Actuators A: Phys. 217 (2014) 105–110. [3] Y. Dong, et al., Ultra-high precision mems accelerometer, in: Transducers’11, Beijing, China, 2011, pp. 695–698. [4] H. Ko, et al., Highly programmable temperature compensated readout circuit for capacitive microaccelerometer, Sens. Actuators A: Phys. 158 (2010) 72–83. [5] C. Falconi, Marco Fratini, CMOS microsystems temperature control, Sens. Actuators B: Chem. 129 (2008) 59–66. [6] P. Zwahlen, et al., Navigation grade MEMS accelerometer, in: MEMS 2010, Hong Kong, China, 2010, pp. 631–634. [7] S. Schröder, et al., Stress-minimized packaging of inertial sensors using wire bonding, in: Transducers’13, Barcelona, Spain, 2013, pp. 1962–1965. [8] Y. Zhang, et al., A SOI sandwich differential capacitance accelerometer with low-stress package, in: IEEE-NEMS 2014, Hawaii, USA, 2014, pp. 341–345. [9] A. Geisberger, S. Schroeder, J. Dixon, et al., A high aspect ratio MEMS process with surface micromachined polysilicon for high accuracy inertial sensing, in: Transducers’13, Barcelona, Spain, 2013, pp. 18–21. [10] G. Dai, et al., Thermal drift analysis using a multiphysics model of bulk silicon MEMS capacitive accelerometer, Sens. Actuators A: Phys. 172 (2011) 369–378.

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Biographies Jiangbo He received the B.S. degree (mechanical engineering) and M.S. degree (mechanical engineering) from Southwest Jiaotong University, China, in 2009 and 2011, respectively. He is currently working toward the Ph.D. degree in the department of mechanical engineering, Southwest Jiaotong University, China. His research interests include micromechanics and micro sensors and actuators. Jin Xie received the B.S. degree (mechanical engineering), M.S. degree (mechanical engineering) and Ph.D. degree (mechanical engineering) from Southwest Jiaotong University, China, in 1982, 1989 and 2007, respectively. From 2003, he was a professor in School of Mechanical Engineering, Southwest Jiaotong University, China. His research areas are mechanism, nonlinear dynamics in mechanical engineering, robot, micromechanics. Xiaoping He works in China Academy of Engineering Physics as a professor. Her interests lie in micromechanics and micro sensors and actuators. Lianming Du works in China Academy of Engineering Physics as a professor. His interests lie in integrated circuit design and micro sensors and actuators. Wu Zhou received the Ph.D. degree from Southwest Jiaotong University, China, in 2011. Now, he is an associate professor in School of Mechanical, Electronic, and Industrial Engineering, University of Electronic Science and Technology of China. His interests lie in micro and nanomechanics, micro sensors and microfabrication.