Structural Designing of a MEMS Capacitive Accelerometer for Low Temperature Coefﬁcient and High Linearity

sensors

Article Structural Designing of a MEMS Capacitive Accelerometer for Low Temperature Coefﬁcient and High Linearity

Jiangbo He 1,* ID , Wu Zhou 2,* ID , Huijun Yu 2, Xiaoping He 3 and Peng Peng 2

1 School of Mechanical Engineering, Xihua University, Chengdu 610039, China 2 School of Mechanical and Electrical Engineering, University of Electronic Science and Technology of China, Chengdu 611731, China; [email protected] (H.Y.); [email protected] (P.P.) 3 Institute of Electronic Engineering, China Academy of Engineering Physics, Mianyang 621900, China; [email protected] * Correspondence: [email protected] (J.H.); [email protected] (W.Z.); Tel.: +86-28-6183-0227 (W.Z.)

Received: 14 December 2017; Accepted: 18 February 2018; Published: 22 February 2018

Abstract: The low temperature coefficient and high linearity of the input-output characteristics are both required for high-performance microelectromechanical systems (MEMS) capacitive accelerometers. In this work, a structural designing of a bulk MEMS capacitive accelerometer is developed for both low temperature coefficient and high linearity. Firstly, the contrary effect of the wide-narrow gaps ratio (WNGR) on the temperature coefficient of the scale factor (TCSF) and linearity error is discussed. Secondly, the ability of an improved structure that can avoid the contrary effect is illustrated. The improved structure is proposed in our previous work for reducing the temperature coefficient of bias (TCB) and TCSF. Within the improved structure, both the TCSF and linearity error decrease with increasing WNGR. Then, the precise designing of the improved structure is developed for achieving lower TCB, TCSF, and linearity error. Finally, the precise structural designing is experimentally verified.

Keywords: MEMS; capacitive accelerometer; temperature coefﬁcient; linearity error

1. Introduction High-precision MEMS capacitive accelerometers are increasingly needed in numerous applications including inertial navigation [1], gravity measurement [2], vibration measurements [3], and so on. The low temperature coefficient and high linearity of the input-output characteristics are both required for the high-performance MEMS capacitive accelerometers, especially for those applied in inertial navigation [4,5]. Temperature coefficient can be induced by the thermal stress and temperature dependence of material properties [6,7]. The thermal stress is induced by the mismatch of thermal expansion among device layer, substrate, and package [8]. Linearity error (or nonlinearity) is the inherent drawback of capacitive accelerometers because the gap variation is used to detect the acceleration [9]. Even if the closed-loop detection principle is employed, linearity error still exists due to uncertainties, such as manufacturing errors [10]. Although the temperature coefficient and linearity error can both be compensated actively [11,12], more complex circuitry is required. The temperature coefficient and linearity error are both related to the device dimensions. Thus, the careful designing of device dimensions is necessary for both low temperature coefficient and linearity error. In our previous work [6], the temperature coefficient of the bias (TCB) and temperature coefficient of the scale factor (TCSF) are studied in detail. An improved structure is also designed for both low TCB and TCSF (TCB < 1.03 mg/◦C and TCSF < 66 ppm/◦C). However, TCB and TCSF are still high for high-performance accelerometers. For example, TCB is much higher than the results in

Sensors 2018, 18, 643; doi:10.3390/s18020643 www.mdpi.com/journal/sensors SensorsSensors2018 2018,, 18 18,, 643 x FOR PEER REVIEW 22 of of 13 13

TCSF are still high for high-performance accelerometers. For example, TCB is much higher than the ◦ ◦ ◦ severalresults literaturesin several (100literaturesµg/ C (100 in [ 13μg/°C] and in [14 [13]], 200 andµ g/[14],C 200 in [4μ]g/°C and in [15 [4]], 300andµ [15],g/ C 300 in [μ16g/°C]). As in such,[16]). moreAs such, improvement more improvement of design isof required design is to required achieve lowerto achieve TCB lower and TCSF. TCB Inand addition, TCSF. In the addition, linearity the is notlinearity considered is not in considered the previous in work. the previous In this work, work the. In temperature this work, coefficientthe temperature and linearity coefficient error and are designedlinearity simultaneously.error are designed Firstly, simultaneously. the contrary Firstly, effect ofthe the contrary wide-narrow effect of gaps the ratiowide-narrow (WNGR) gaps on TCSF ratio and(WNGR) linearity on errorTCSF isand discussed. linearity Then, error ais precise discussed. design Then, to improve a precise the design structure to improve is developed the structure for both is lowdeveloped temperature for both coefficients low temperature and linearity coefficients error. and linearity error. 2. Fabrication and Detection Principle 2. Fabrication and Detection Principle The structure of the MEMS capacitive accelerometer studied in this work is shown in Figure1. The structure of the MEMS capacitive accelerometer studied in this work is shown in Figure 1. Anchors and sensing elements (proof mass, comb capacitors, and springs) are made of single crystal Anchors and sensing elements (proof mass, comb capacitors, and springs) are made of single crystal silicon. The bottom substrate is made of Pyrex 7740 glass. The accelerometer is fabricated based on silicon. The bottom substrate is made of Pyrex 7740 glass. The accelerometer is fabricated based on a a bulk silicon process, which was detailed before in [7]. In short, the process begins with a Cr/Au bulk silicon process, which was detailed before in [7]. In short, the process begins with a Cr/Au metallization process on a Pyrex 7740 glass wafer. Then, boron doping by diffusion is performed metallization process on a Pyrex 7740 glass wafer. Then, boron doping by diffusion is performed on on a (100) silicon wafer, and then DRIE is employed to define sensing elements and anchors. Next, a (100) silicon wafer, and then DRIE is employed to define sensing elements and anchors. Next, the the silicon wafer was flipped and bonded to the glass wafer by anodic bonding. Finally, the undoped silicon wafer was flipped and bonded to the glass wafer by anodic bonding. Finally, the undoped silicon is completely dissolved to leave the sensing elements and anchors. silicon is completely dissolved to leave the sensing elements and anchors.

(a) (b)

Figure 1. The structure of the MEMS accelerometer. (a) Scanning electron microscopy (SEM); Figure 1. The structure of the MEMS accelerometer. (a) Scanning electron microscopy (SEM); (b) Scheme (b) Scheme diagram of dimensions for a comb capacitor. diagram of dimensions for a comb capacitor. The capacitances of the accelerometer are detected by the self-balancing bridge principle, as depictedThe capacitancesin Figure 2. The of theinertial accelerometer force makes are th detectede proof mass by themove self-balancing and is balanced bridge by the principle, spring asforce. depicted The moving in Figure of2 .the The proof inertial mass force changes makes the the capacitance proof mass of move the accele and isrometer. balanced Drivers by the provide spring force.AC signals The moving to the of fixed the proof electrodes. mass changes The movable the capacitance electrodes of theare accelerometer.connected to the Drivers output provide terminal AC signalsthrough to the fixedreading electrodes. circuitry. The Inmovable response electrodes to a sensed are connectedacceleration, to thefeedback output is terminal provided through from thethe readingoutput circuitry.terminal Into responseboth drivers to a to sensed adjust acceleration, the amplitude feedback of the is AC provided signals from on the the fixed output electrodes, terminal toas bothdepicted drivers in toFigure adjust 2b. the The amplitude objective of the of ACthe signalsfeedback on theis to fixed null electrodes, any AC assignal depicted on the in Figure movable2b. Theelectrodes. objective The of thedetailed feedback principle is to null realizing any AC the signal ampl onitude themovable adjusting electrodes. of the AC The signals detailed can principlebe found realizingin the literature the amplitude [17]. In adjustingorder to null of the any AC AC signals signal canon the be foundmovable in theelectrodes, literature the [17 following]. In order equation to null anymust AC be signal satisfied on the movable electrodes, the following equation must be satisfied

1 11 1 C VCV12− AVout V out− C CVV A+ VV outout 0= 0(1) (1) 1 A M MM2 A M

where VA is the initial amplitude of the AC signal, C1 and C2 are the capacitance, M is a feedback coefficient depending on circuit parameters. Solving Equation (1) leads to

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where VA is the initial amplitude of the AC signal, C1 and C2 are the capacitance, M is a feedback Sensorscoefﬁcient 2018, 18, depending x FOR PEER on REVIEW circuit parameters. Solving Equation (1) leads to 3 of 13

C1 − C2 Vout = M CCVA (2) C12+ C VMout 1 2 V A (2) CC12

Figure 2. Scheme diagram of the self-balancing differential capacitive principle. (a) Schematic block Figure 2. Scheme diagram of the self-balancing differential capacitive principle. (a) Schematic block diagram; (b) Graphs of waveforms on the fixed plate. diagram; (b) Graphs of waveforms on the ﬁxed plate.

3. Contrary Effect of the WNGR on TCSF and Linearity Error 3. Contrary Effect of the WNGR on TCSF and Linearity Error

3.1.3.1. Analytical Analytical Model Model for for the the Linearity Linearity Error Error AccordingAccording to to the the electrostatic electrostatic theory, theory, capacitancescapacitances of the accelerometer can bebe expressedexpressed asas 2N 21N  2(N−1)εΩ C1 = 2NεΩ + C1 dxd−x DxD+x ( − ) (3) = 2NεΩ + 2 N 1 εΩ (3) C2 2Nd+x 21ND−x  C  2 dx Dx where d and D denote the narrow and wide gaps shown in Figure1, respectively, x denotes the wheredisplacement d and D of proofdenote mass, the narrowN denotes and the wide number gaps of shown ﬁxed electrodes in Figure in 1, a combrespectively, capacitor, x Ωdenotesrepresents the displacementthe overlapping of areaproof in mass, a pair ofN movabledenotes andthe number ﬁxed electrodes. of fixed electrodes in a comb capacitor, Ω representsSubstituting the overlapping Equation area (3) into in a (1) pair and of expanding movable and output fixedV electrodes.out using Taylor's theorem leads to Substituting Equation (3) into (1) and expanding output Vout using Taylor's theorem leads to (η − 1)x (η − 1)x3 V = + 3 MV (4) out 11xx2 3 A VMVηd η d out23 A (4) dd where η denotes WNGR (the ratio of D to d). whereThe η denotes displacement WNGR of (the proof ratio mass of D caused to d). by the input acceleration can be expressed as The displacement of proof mass caused by the input acceleration can be expressed as −ma x = ma (5) x K (5) K where m, a, and K denote the proof mass, acceleration, and the stiffness of springs, respectively. where m, a, and K denote the proof mass, acceleration, and the stiffness of springs, respectively. Substituting Equation (5) into (4), the output Vout can be expressed as a function of the acceleration Substituting Equation (5) into (4), the output Vout can be expressed as a function of the acceleration 3 Vout = k1a +3 k3a Vkakaout 13− k = − η 1 m MV 1  1ηdm K A (6) η−1 m 3 kkMV1 = − MVA 3 ηdK2d3 K A (6) 3  1 m kMV3  23 A  d K

In Equation (6), k1 denotes the scale factor (or sensitivity), and k3 causes linearity error. According to the definition in [18] the linearity error represents the systematic deviation from the straight line

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In Equation (6), k1 denotes the scale factor (or sensitivity), and k3 causes linearity error. According to the deﬁnition in [18] the linearity error represents the systematic deviation from the straight line that deﬁnes the nominal input-output relationship. The linearity error over full range can be expressed as

max|Vout − Vout_linear| ∆non = × 100 (7) Voutmax where Vout_linear denotes the output obtained from the nominal input-output relationship. From Equation (6), it is known that “k1a” represents the nominal input-output relationship. Substituting Equation (6) into (7) leads to

k a3 k a3 1 ma 2 = 3 max ≈ 3 max = max ∆non 3 (8) k1amax + k3amax k1amax η Kd

According to Equation (8), it is known that the linearity error decreases with the increasing of the narrow gap. However, increasing the narrow gap is against the detection due to the lower capacitance. Equation (8) also shows that higher WNGR results in lower linearity error, and even nulls the linearity error. In conclusion, high WNGR is the better way to achieve low linearity error. However, high WNGR may also inﬂuence the temperature coefﬁcient. In next section, it is shown that WNGR has the contrary effect on TCSF and linearity error.

3.2. Illustration of the Contrary Effect According to our previous work [6], TCSF of the MEMS accelerometer can be expressed as

2 η + 1 l f η + 1 w f TCSF = −TCS + −αs − + (N − 1) + αeq − αs (9) η2 − η d 2 d where TCS denotes the temperature coefficient of stiffness, αs denotes the coefficient of thermal expansion (CTE) of silicon, αeq is called as equivalent CTE, N denotes the fixed electrode number in a comb capacitor, and lf denotes the distance from the first fixed electrode to the midline, wf denotes the width of an electrode, as shown in Figure1. TCS equals the sum of the temperature coefficient of elastic modulus and CTE. However, the temperature coefficient of elastic modulus is much higher than CTE [19]. Thus, TCS is mainly determined by the temperature coefficient of elastic modulus. Due to the ◦ high boron doped silicon, TCS is about 30 ppm/ C[6]. The equivalent CTE αeq represents the thermal deformation on the substrate top surface. Because the thermal stress enhances the thermal deformation on the substrate top surface, the equivalent CTE is higher than the CTE of Pyrex 7740 glass. From Equation (9), it is known that the relationship between TCSF and WNGR is nonlinear. The nonlinear relationship is shown in Figure3. For comparison, the relationship between the linearity error and WNGR is also shown in Figure3. Except for the equivalent CTE, the employed parameters (listed in Table1) are the same as those in the previous work [ 6]. In this work, the minimum equivalent CTE is employed and equal to the CTE of the glass substrate (about 3.25 ppm/◦C). The minimum equivalent CTE can be achieved by using the very soft adhesive for the die-attach [6]. From Figure3, it can be seen that WNGR must be high for low linearity error. For example, a WNGR higher than 5.5 results in a linearity error lower than 0.5%. On the other side, TCSF achieves the minimum with the WNGR of 4. When WNGR is higher than 4, TCSF increases with increasing WNGR. Therefore, the effect of WNGR on TCSF and linearity error is contrary. Sensors 2018, 18, x FOR PEER REVIEW 5 of 13

Table 1. Parameters for the MEMS accelerometer.

Parameter Value Units Parameter Value Units

Mass (m) 59.4 μg Electrode width (wf) 6.5 μm Fixed electrodes number in a Distance from the first fixed 21 45 μm comb structure (N) electrode to midline (lf)

Stiffness of springs (K) 16 N/m Equivalent CTE (αeq) 3.25 ppm/°C

maximum acceleration (amax) 20 g Silicon CTE (αs) [6] 2.6 ppm/°C Narrow gap (d) 5 μm TCS [6] −30 ppm/°C Sensors 2018, 18WNGR, 643 (η) 5 5 of 13

-40 2 Nonlinearity -50 TCSF 1.5

-60 1 -70

TCSF(ppm/°C) 0.5 -80 (%) Linearityerror

-90 0 2345678 η

Figure 3. The effect of WNGR (η) on TCSF and linearity error.

3.3. Structural Designing for AvoidingTable 1. Parameters the Contrary for Effect the MEMS accelerometer. In this section, the ability of an improved structure (shown in Figure 4) that can avoid the Parameter Value Units Parameter Value Units contrary effect is illustrated. The improved structure is proposed in our previous work for achieving µ µ low TCB andMass TCSF. (m For) the improved 59.4 structure, TCSFg is expressedElectrode width as [6]. (w f ) 6.5 m Distance from the ﬁrst Fixed electrodes number in a 21TCSF TCSF TCSFﬁxed electrode to 45 µm comb structure (N) STD TS (10) midline (lf) ◦ Stiffness of springs (K) 16TCSF N/mSTD  TCS Equivalent CTE (αeq) 3.25 ppm/ (11)C ◦ maximum acceleration (amax) 20 g Silicon CTE (αs)[6] 2.6 ppm/ C 2 ◦ Narrow gap (d) 5 µm1 llfg TCS [6] −30 ppm/ C TCSF TS s 2  eq  s (12) WNGR (η) 5  d

3.3.where Structural lg is the Designing length of foranchors Avoiding for thefixed Contrary electrodes, Effect TCSFSTD and TCSFTS denote the TCSF induced by the temperature dependence of the spring stiffness and thermal stress, respectively. InThe this improved section, thestructure ability ofdoes an improvednot change structure the detecting (shown principle in Figure 4and) that electrode can avoid layout, the contrary so the effectlinearity is illustrated. error is still The expressed improved by Equation structure is(8). proposed In this work, in our two previous coefficients work are for defined achieving to represent low TCB andthe effect TCSF. of For WNGR the improved on linearity structure, error and TCSF TCSF is expressed as [6]. 1 TCSF =TCSFSTD + TCSFTS (10) l_ error  (13) TCSFSTD = 2−TCS1 (11)   TCSF 2 2 η +1 l f+ lg TCSF = −αs − αeq − αs (12) TS η2 − η d where lg is the length of anchors for fixed electrodes, TCSFSTD and TCSFTS denote the TCSF induced by the temperature dependence of the spring stiffness and thermal stress, respectively. The improved structure does not change the detecting principle and electrode layout, so the linearity error is still expressed by Equation (8). In this work, two coefficients are defined to represent the effect of WNGR on linearity error and TCSF

1 βl_error = η 2 (13) = η +1 βTCSF η2−η

KB/2 KB/2

d lg Sensors 2018, 18, 643 6 of 13 wf Sensors 2018, 18, x FOR PEER REVIEW 6 of 13 2lf 2la

KB/2 KB/2

d KA/2 KA/2 lg (a) wf (b) 2lf 2la

Figure 4. The improved structure for MEMS accelerometer.lg (a) Overall layout; (b) Scheme diagram of dimensions for the comb capacitor.

According to Equations (8) and (10), the linearity error decreases with decreasing βl_error, and TCSF decreases with decreasing βTCSF. The relationship between βl_error and WNGR is shown in Figure 5, and so is the relationship between βTCSF and WNGR. It can be seen that βl_error and βTCSF both KA/2 KA/2 decrease with increasing WNGR. As such, the improved structure avoids the contrary effect of WNGR on TCSF and (linearitya) error. (b) Then,Figure as long4. The as improved precise structure parameters for MEMS are empl accelerometer.oyed, the (verya) Overall low layout;temperature (b) Scheme coefficient diagram andof Figure 4. The improved structure for MEMS accelerometer. (a) Overall layout; (b) Scheme diagram of linearitydimensions error can bothfor the be comb obtained capacitor. with the improved structure. In the next section, a precise design for thedimensions improved for structure the comb is capacitor. developed.

According to Equations (8) and (10), the linearity error decreases with decreasing βl_error, and 3 0.6 TCSF decreases with decreasing βTCSF. The relationship between βl_error and WNGR is shown in 系列βl_error2 Figure 5, and so is the relationship2.5 between βTCSF and WNGR.β It can be0.5 seen that βl_error and βTCSF both 系列TCSF1 decrease with increasing WNGR. As such, the improved structure avoids the contrary effect of WNGR on TCSF and linearity2 error. 0.4 Then, as long as precise parameters are employed, the very low temperature coefficient and

TCSF 1.5 0.3 β linearity error can both be obtained with the improved structure. In the nextl_error section, a precise design β for the improved structure1 is developed. 0.2

0.5 3 0.1 0.6

系列βl_error2 2.5 0.5 0 βTCSF 0 2468系列1 2 η 0.4

FigureFigureTCSF 1.5 5.5. TheThe effecteffect ofof WNGRWNGR ononβ βl_error andand ββTCSF. . 0.3

β l_error TCSF l_error β 1 0.2 4. PreciseThen, Design as long for as both precise Low parameters Temperature are Coefficient employed, and the veryLinearity low temperatureError coefficient and linearity error can both be obtained0.5 with the improved structure. In the next0.1 section, a precise design 4.1. Design for Low TCSF and Linearity Error for the improved structure is developed. 0 0 From Equation (10), it is known that TCSF consists of TCSFSTD and TCSFTS. TCSFSTD and TCSFTS 4. Precise Design for both Low Temperature2468 Coefficient and Linearity Error cancel each other. TCSFSTD is equal to the opposite ofη TCS. TCS is about −30 ppm/°C, as explained in Section 3.2. Thus, if TCSFTS is −30 ppm/°C, TCSF is null. However, the absolute value of TCSFTS is 4.1. Design for Low TCSF and Linearity Error much higher than 30 ppm/°C [6].Figure As such, 5. The TCSF effectTS of must WNGR be reduced. on βl_error and TCSF βTCSFTS .is directly proportional to the CTEFrom difference Equation (α (10),eq − α its). is According known that to TCSF the study consists in Section of TCSF 3.2,STD αandeq is TCSF higherTS. than TCSF αSTDs. Thus,and TCSF TS ◦ decreasescancel4. Precise each with other. Design decreasing TCSF forSTD both αiseq. equalLowThe minimum Temperature to the opposite αeq is Coefficient about of TCS. 3.25 TCS ppm/°Cand is aboutLinearity and− 30is Errorachieved ppm/ C, by as the explained soft die- in ◦ attach.Section Therefore, 3.2. Thus, the if TCSF minimumTS is − α30eq is ppm/ employedC, TCSF in this isnull. work. However, the absolute value of TCSFTS is 4.1. Design for Low TCSF and◦ Linearity Error much higher than 30 ppm/ C[6]. As such, TCSFTS must be reduced. TCSFTS is directly proportional to the CTE difference (α − α ). According to the study in Section 3.2, α is higher than α Thus, From Equation (10),eq it iss known that TCSF consists of TCSFSTD and eqTCSFTS. TCSFSTD ands. TCSFTS TCSF decreases with decreasing α . The minimum α is about 3.25 ppm/◦C and is achieved by the cancelTS each other. TCSFSTD is equaleq to the opposite ofeq TCS. TCS is about −30 ppm/°C, as explained in soft die-attach. Therefore, the minimum α is employed in this work. Section 3.2. Thus, if TCSFTS is −30 ppm/°C,eq TCSF is null. However, the absolute value of TCSFTS is Besides, TCSF also varies with the narrow gap and WNGR. The effect of the narrow gap and much higher thanTS 30 ppm/°C [6]. As such, TCSFTS must be reduced. TCSFTS is directly proportional to WNGR on TCSF and linearity error is shown in Figure6. The length of anchors for fixed electrodes the CTE differenceTS (αeq − αs). According to the study in Section 3.2, αeq is higher than αs. Thus, TCSFTS decreases with decreasing αeq. The minimum αeq is about 3.25 ppm/°C and is achieved by the soft die- attach. Therefore, the minimum αeq is employed in this work.

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Besides, TCSFTS also varies with the narrow gap and WNGR. The effect of the narrow gap and WNGR on TCSFTS and linearity error is shown in Figure 6. The length of anchors for fixed electrodes lg is 110μm, and other parameters are listed in Table 1. These employed parameters are the same as those in the previous work except the αeq. For the narrow gap of 5 μm employed in the previous work, it can be seen that WNGR should be 4.4 to make TCSFTS close to −30 ppm/°C. On the other side, with the narrow gap of 4.5 μm and WNGR of 6.5, a TCSFTS close to −30 ppm/°C can also be achieved. With the two sets of the narrow gap and WNGR, both low linearity errors can be achieved, and they are about 0.42% and 0.4%, respectively. Because lower narrow gap results in higher capacitances, the narrow gap of 4.5 μm and WNGR of 6.5 are selected in this work. Another important problem is the increased length of the comb capacitor induced by the dimension variation. From Figure 4, the length of the comb capacitor is expressed as

LcombNddww12  f  f (14) Sensors 2018, 18, 643 7 of 13 For the narrow gap of 5 μm and WNGR of 5 employed in the previous work, Lcomb is 866.5 μm. On the other side, for the narrow gap of 4.5 μm and WNGR of 6.5, Lcomb is 941.5 μm. The increasing of Llgcombis is 110 veryµm, small and compared other parameters to the overall are listed size in ofTable the die1. These(3200 μ employedm × 3200 μ parametersm). As such, are the the increasing same as ofthose Lcomb in does the previousnot result work in a huge except impact the α eqon. the die yield.

1.2 -20 d=5μm 1 -25 d=4.5μm d=4μm 0.8 -30 0.6 (ppm/ºC)

TS -35 d=5μm 0.4 Linearity error (%) error Linearity TCSF -40 d=4.5μm 0.2 d=4μm -45 0 3456789 3456789 η η (a) (b)

Figure 6. The effect of the narrow gap (d) and WNGR (η) on the TCSFTS and linearity error. (a) TCSFTS; Figure 6. The effect of the narrow gap (d) and WNGR (η) on the TCSF and linearity error. (a) TCSF ; (b) Linearity error. TS TS (b) Linearity error. 4.2. Designing for Low TCB For the narrow gap of 5 µm employed in the previous work, it can be seen that WNGR should Besides TCSF, TCB is another important◦ parameter affecting the temperature performance of be 4.4 to make TCSFTS close to −30 ppm/ C. On the other side, with the narrow gap of 4.5 µm and MEMS accelerometers. According to the previous◦ work [6], TCB is expressed as WNGR of 6.5, a TCSFTS close to −30 ppm/ C can also be achieved. With the two sets of the narrow gap and WNGR, both low linearity errors can be achieved,KK and they are about 0.42% and 0.4%, respectively. TCB ABl (15) Because lower narrow gap results in higher capacitances,m eq the s narrow a gap of 4.5 µm and WNGR of 6.5 are selected in this work. where la is the distance from the anchor for movable electrodes to the midline as depicted in Another important problem is the increased length of the comb capacitor induced by the Figure 4, KA and KB denote the stiffness of the springs connecting proof mass. dimension variation. From Figure4, the length of the comb capacitor is expressed as TCB also decreases with decreasing equivalent CTE. The minimum αeq is equal to the CTE of the substrate. If the silicon substrate is employed, the minimum αeq is equal to the CTE of silicon. As a Lcomb = (N − 1) d + dη + 2w f + w f (14) result, TCB disappears. TCSFTS also disappears. However, TCSF can’t be close to zero because TCSFSTD still exists. In this work, the glass substrate is employed to achieve null TCSF by making For the narrow gap of 5 µm and WNGR of 5 employed in the previous work, Lcomb is 866.5 µm. TCSFTS and TCSFSTD cancel each other. TCB is reduced by other methods, such as shorter la. In this On the other side, for the narrow gap of 4.5 µm and WNGR of 6.5, Lcomb is 941.5 µm. The increasing of work, the anchor for movable electrodes is redesigned, as depicted in Figure 7. Through the Lcomb is very small compared to the overall size of the die (3200 µm × 3200 µm). As such, the increasing redesigning of the anchor, la is reduced from 190 μm (employed in the previous work) to 90 μm. of Lcomb does not result in a huge impact on the die yield. Changing the stiffness or mass may also reduce TCB. However, this results in the variation of linearity error.4.2. Designing As such, forthe Low spring TCB stiffness and mass remain the same as those in the previous work.

Besides TCSF, TCB is another important parameter affecting the temperature performance of MEMS accelerometers. According to the previous work [6], TCB is expressed as

K − K TCB = A B α − α l (15) m eq s a

where la is the distance from the anchor for movable electrodes to the midline as depicted in Figure4, KA and KB denote the stiffness of the springs connecting proof mass. TCB also decreases with decreasing equivalent CTE. The minimum αeq is equal to the CTE of the substrate. If the silicon substrate is employed, the minimum αeq is equal to the CTE of silicon. As a result, TCB disappears. TCSFTS also disappears. However, TCSF can’t be close to zero because TCSFSTD still exists. In this work, the glass substrate is employed to achieve null TCSF by making TCSFTS and TCSFSTD cancel each other. TCB is reduced by other methods, such as shorter la. In this work, the anchor for movable electrodes is redesigned, as depicted in Figure7. Through the redesigning Sensors 2018, 18, x FOR PEER REVIEW 9 of 13

where p0 and k1 denote the bias and scale factor, respectively. In this work, the scale factor and bias under 5 °C, 15 °C, 25 °C, 35 °C, 45 °C, 55 °C, 65 °C, 75 °C, and 85 °C was measured, so TCSF and TCB under 15 °C, 25 °C, 35 °C, 45 °C, 55 °C, 65 °C, and 75 °C are obtained. The temperature dependences of TCSF and TCB are shown in Figure 9. TCB decreases with the temperature and the relationship is approximately linear. This temperature dependence may be induced by the temperature dependence of the CTE of silicon. According to Equation (15), it is known that TCB has a linear relationship with the CTE difference (αeq − αs). αeq is equal to CTE of the Pyrex 7740 glass. The result in the literature [20] shows that CTE of the Pyrex 7740 glass is almost constant from 0 °C to 100 °C. CTE of silicon increases almost linearly from 0 °C to 100 °C, but is still lower than CTE of the Pyrex 7740 glass [21]. As a result, the CTE difference (αeq − αs) decreases with increasing temperature linearly. As such, TCB decreases with increasing temperature. SensorsThe2018 ,relationship18, 643 between TCSF and temperature is also approximately linear. However, 8TCSF of 13 increases with increasing temperature. From Equation (10), the temperature dependence of TCSF can be induced by temperature dependences of (αeq − αs) and TCS. However, TCS is almost constant from of the anchor, la is reduced from 190 µm (employed in the previous work) to 90 µm. Changing the stiffness0 °C to 100 or mass°C [19]. may As also such, reduce the approximately TCB. However, li thisnear results relationship in the variation between ofTCSF linearity and temperature error. As such, is thealso spring caused stiffness by the temperature and mass remain dependence the same of as the those CTE in of the silicon. previous work.

FigureFigure 7.7. TheThe microscopemicroscope imageimage ofof thethe accelerometer.accelerometer.

Because the stiffness difference (KA − KB) is induced by random fabrication errors [6], it is difﬁcult to estimate the accurate TCB. However, the improvement of TCB can be estimated based on Equation ◦ (15). The CTE difference (αeq − αs) is about 0.65 ppm/ C in this work, while that in the previous work is about 1.6 ppm/◦C[6]. As a result, the estimated TCB in this work is about 20% of that in the previous work. Integrating the designing works on the TCSF, TCB, and linearity error, the modiﬁcations on the improved structure are summarized as follows.

(1) In order to achieve the minimum αeq, a soft adhesive is employed for the die-attach. (2) The narrow gap d is decreased from 5 µm to 4.5 µm. (3) In order to make WNGR be 6.5, the wide gap D is modiﬁed to be 29.3 µm. µ (4) TheFigure distance 8. Testing from equipment the anchor for accelerometers. for moving electrodes DC supply to and midline digital lvoltmea is decreasedter provide from the power 190 m to 90inputµm. and display for output voltage, respectively. The other designing parameters are the same as those employed in the previous work. The parameter and performance differences made by these modiﬁcations between this work and the previous work are listed in Table2.

Table 2. The parameter and performance differences between this work and the previous work.

Parameter In This Work In Previous Work Units Narrow gap (d) 4.5 5 µm Wide gap (D) 29.3 25 µm Distance from anchors for moving 90 190 µm electrodes to midline (la) ◦ Equivalent CTE (αeq) 3.25 4.2 ppm/ C Linearity error 0.4% 0.42% TCSF almost zero 37 ppm/◦C TCB The estimated TCB in this work is about 20% of that in the previous work Sensors 2018, 18, x FOR PEER REVIEW 9 of 13

where p0 and k1 denote the bias and scale factor, respectively. In this work, the scale factor and bias under 5 °C, 15 °C, 25 °C, 35 °C, 45 °C, 55 °C, 65 °C, 75 °C, and 85 °C was measured, so TCSF and TCB under 15 °C, 25 °C, 35 °C, 45 °C, 55 °C, 65 °C, and 75 °C are obtained. The temperature dependences of TCSF and TCB are shown in Figure 9. TCB decreases with the temperature and the relationship is approximately linear. This temperature dependence may be induced by the temperature dependence of the CTE of silicon. According to Equation (15), it is known that TCB has a linear relationship with the CTE difference (αeq − αs). αeq is equal to CTE of the Pyrex 7740 glass. The result in the literature [20] shows that CTE of the Pyrex 7740 glass is almost constant from 0 °C to 100 °C. CTE of silicon increases almost linearly from 0 °C to 100 °C, but is still lower than CTE of the Pyrex 7740 glass [21]. As a result, the CTE difference (αeq − αs) decreases with increasing temperature linearly. As such, TCB decreases with increasing temperature. The relationship between TCSF and temperature is also approximately linear. However, TCSF increases with increasing temperature. From Equation (10), the temperature dependence of TCSF can be induced by temperature dependences of (αeq − αs) and TCS. However, TCS is almost constant from 0 °C to 100 °C [19]. As such, the approximately linear relationship between TCSF and temperature is also caused by the temperature dependence of the CTE of silicon.

Sensors 2018, 18, 643 9 of 13

5. Experiments

5.1. Temperature Coefﬁcients In this work, accelerometers with the modiﬁed dimensions were fabricated. The microscope image of the accelerometer is shown in Figure7. In order to reduce TCSF and TCB, the accelerometer die is attached to the ceramic package by a soft adhesive with Young's modulus lower than 10 MPa. The input, output, and ground pads are wire-bonded to the copper trace on the ceramic package. The testing principle of the self-balancing bridge is implemented in the printed circuit board (PCB) using the discrete components. The packaged accelerometer is mounted on the PCB for testing, as shown in Figure8. Figure 7. The microscope image of the accelerometer.

Figure 8. Testing equipmentequipment forfor accelerometers.accelerometers. DC supplysupply and digital voltmetervoltmeter provide the power input andand displaydisplay forfor outputoutput voltage,voltage, respectively.respectively.

In order to measure TCSF and TCB, the scale factor and bias under different temperatures must be measured. The measuring method is detailed before in [6]. With the measuring results of the scale factor and bias under different temperatures, TCSF and TCB are calculated by the following equations:

k (T + ∆T) − k (T − ∆T) TCSF = 1 1 (16) 2∆Tk1(T)

p (T + ∆T) − p (T − ∆T) TCB = 0 0 (17) 2∆T where p0 and k1 denote the bias and scale factor, respectively. In this work, the scale factor and bias under 5 ◦C, 15 ◦C, 25 ◦C, 35 ◦C, 45 ◦C, 55 ◦C, 65 ◦C, 75 ◦C, and 85 ◦C was measured, so TCSF and TCB under 15 ◦C, 25 ◦C, 35 ◦C, 45 ◦C, 55 ◦C, 65 ◦C, and 75 ◦C are obtained. The temperature dependences of TCSF and TCB are shown in Figure9. TCB decreases with the temperature and the relationship is approximately linear. This temperature dependence may be induced by the temperature dependence of the CTE of silicon. According to Equation (15), it is known that TCB has a linear relationship with the CTE difference (αeq − αs). αeq is equal to CTE of the Pyrex 7740 glass. The result in the literature [20] shows that CTE of the Pyrex 7740 glass is almost constant from 0 ◦C to 100 ◦C. CTE of silicon increases almost linearly from 0 ◦C to 100 ◦C, but is still lower than CTE of the Pyrex 7740 glass [21]. As a result, the CTE difference (αeq − αs) decreases with increasing temperature linearly. As such, TCB decreases with increasing temperature. The relationship between TCSF and temperature is also approximately linear. However, TCSF increases with increasing temperature. From Equation (10), the temperature dependence of TCSF can be induced by temperature dependences of (αeq − αs) and TCS. However, TCS is almost constant from 0 ◦C to 100 ◦C[19]. As such, the approximately linear relationship between TCSF and temperature is also caused by the temperature dependence of the CTE of silicon. Sensors 2018, 18, x FOR PEER REVIEW 10 of 13

15 250

10 200 5 150 SensorsSensors 20182018,, 1818,, 643x FOR PEER REVIEW 1010 ofof 1313 0 100

-5 (μg/ºC) TCB 15 250 TCSF (ppm/ºC) TCSF -10 50 10 200 -15 5 0 5 2035506580 150 5 2035506580 0 Temperature (°C) Temperature (°C) (a) 100 (b)

-5 (μg/ºC) TCB

TCSF (ppm/ºC) TCSF -10 Figure 9. Temperature dependences of TCSF50 and TCB. (a) TCSF; (b) TCB.

-15More results of TCSF and TCB under room temperature0 (25 °C) are shown in Figure 10. The five 5 2035506580 results of TCSF are all very low. The max TCSF is −16.1 ppm/°C,5 2035506580 and the average TCSF is about Temperature (°C) Temperature (°C) −9.8 ppm/°C. The significant improvement on TCSF is also verified by the comparison in Table 3. The max TCB is 294 μg/°C,(a and) the average TCB is 179 μg/°C. TCB is in the same(b) order of those found in the literature [4,13–16].Figure The9. Temperature comparison dependence in Table 3s ofalso TCSF verifies and TCB. the significant(a) TCSF; (b improvement) TCB. on TCB. Though TCSFFigure and TCB 9. Temperature are both reduced, dependences there of ex TCSFists deviation and TCB.( betweena) TCSF; ( bthe) TCB. experimental results and theMore theoretical results of results TCSF and estimated TCB under in sect roomion temper4. For atureexample, (25 °C)the are average shown TCSF in Figure of − 9.810. ppm/°CThe five ◦ deviatesresultsMore of from resultsTCSF the are oftheoretical TCSFall ve andry low.result TCB The underthat max is roomclose TCSF temperatureto iszero. −16.1 The ppm/°C, (25averageC) areand of shown thethe measured average in Figure TCSF TCB 10. is The aboutabout five ◦ 34.4%results−9.8 ppm/°C. of of that TCSF in The the are significant previous all very work. low.improvement The However, max on TCSF the TCSF estimation is −is 16.1also ppm/verified showsC, that by and the the thecomparison theoretical average TCB TCSFin Table estimated is about3. The ◦ in−max9.8 this TCB ppm/ work is 294isC. about The μg/°C, significant 20% and of thethat improvementaverage estimated TCB in isthe on 179 TCSFprevious μg/°C. is also TCBwork. verified is These in the by deviationssame the order comparison indicate of those inthat found Table there 3in. ◦ ◦ existThethe literature max some TCB other is[4,13–16]. 294 factorsµg/ TheaffectingC, and comparison the TCSF average and in Table TCBTCB, is 3su 179alsochµ as verifiesg/ theC. temperature TCB the issignificant in the dependence same improvement orderof of those the on circuit. found TCB. Therefore,in theThough literature more TCSF [ 4factors,13 and–16 TCB].must The are be comparison bothconsidered reduced, in in Table therethe 3future alsoexists verifies study. deviation the significantbetween the improvement experimental on results TCB. and the theoretical results estimated in section 4. For example, the average TCSF of −9.8 ppm/°C 0 deviates from the theoretical result that is close to zero.300 The average of the measured TCB is about 250 34.4%-5 of that in the previous work. However, the estimation shows that the theoretical TCB estimated in this） work is about 20% of that estimated in the previous200 work. These deviations indicate that there exist some other factors affecting TCSF and TCB, such as the temperature dependence of the circuit. -10 150 Therefore, more factors must be considered in the future study. 100 -15 (μg/°C) TCB

TCSF (ppm/°C 0 30050 -20 2500 -5 12345 ） 12345 200 -10 150 (a) (b) 100 -15 (μg/°C) TCB Figure 10. Measured results of TCSF and TCB for five accelerometers under room temperature◦ TCSF (ppm/°C Figure 10. Measured results of TCSF and TCB for ﬁve accelerometers50 under room temperature (25 C). (25 °C). (a) TCSF; (b) TCB. -20(a) TCSF; (b) TCB. 0 12345 Table 3. Comparison of measured TCSF and TCB between this12345 work and the previous work. Table 3. Comparison of measured TCSF and TCB between this work and the previous work. Parameter In This Work In Previous Work Units Parameter(a) In This Work In Previous Work(b) Units average: −9.8 average: −50.8 Figure 10. MeasuredTCSF results ofaverage: TCSF and−9.8 TCB for five average: accelerometers−50.8 ppm/°C under room ◦ temperature TCSF max: −16.1 max: −62.6 ppm/ C (25 °C). (a) TCSF; (b) TCB. max: −16.1 max: −62.6 average: 179 average: 520 average: 179 average: 520 ◦ TCBTCB μg/°Cµg/ C Table 3. Comparison of measuredmax:max: TCSF 294 294 and TCB between max: max: 1033 this 1033 work and the previous work. Parameter In This Work In Previous Work Units Though TCSF and TCB are both reduced, there exists deviation between the experimental results average: −9.8 average: −50.8 ◦ and the theoretical resultsTCSF estimated in Section4. For example, the averageppm/°C TCSF of −9.8 ppm/ C max: −16.1 max: −62.6 deviates from the theoretical result that is close to zero. The average of the measured TCB is about 34.4% of that in the previous work.average: However, 179 the estimation average: shows 520 that the theoretical TCB estimated TCB μg/°C in this work is about 20% of that estimatedmax: 294 in the previous max: work. 1033 These deviations indicate that there

Sensors 2018, 18, x FOR PEER REVIEW 11 of 13

5.2. Linearity Error The linearity error is measured by precision centrifuge testing for linear accelerometers. In short, an accelerometer is mounted on the centrifuge, and the sensitive direction is along with the radius. Then, the accelerometer is tested under different centrifugal accelerations. The distribution of centrifugal accelerations is [0 g, 5 g, 10 g, 15 g, 20 g, 15 g, 10 g, 5 g, 0 g]. Then, the accelerometer is Sensorsremounted2018, 18 ,to 643 invert the sensitive direction, so the output under the negative accelerations11 ofare 13 tested. When the output under different accelerations is obtained, a least-squares linear fit of the input-output data is made. Finally, the linearity error is calculated by Equation (7). exist some other factors affecting TCSF and TCB, such as the temperature dependence of the circuit. The measuring results of linearity error are shown in Figure 11. The average linearity error for Therefore, more factors must be considered in the future study. the five accelerometers designed in this work is about 0.84%. On the other side, the average linearity 5.2.error Linearity for five Erroraccelerometers designed in the previous work is about 1.16%. The measured linearity error in this work is 72% of that in the previous work. As such, the lower linearity error is achieved throughThe the linearity increasing error isof measuredWNGR. by precision centrifuge testing for linear accelerometers. In short, an accelerometerHowever, there is mounted exist deviations on the centrifuge,between the and measur the sensitiveed results direction and the theoretical is along with results. the Firstly, radius. Then,the measured the accelerometer linearity error is tested of 0.84% under is higher different than centrifugal the theoretical accelerations. linearity error The of distribution 0.4% listed ofin centrifugalTable 2. Secondly, accelerations the result is [0 in g, Table 5 g, 102 shows g, 15 g,that 20 the g, 15estimated g, 10 g, 5linearity g, 0 g]. error Then, for the the accelerometer accelerometers is remountedin this work to is invert slightly the lower sensitive than direction, that in the so previous the output work. under These the negative deviations accelerations may be made are tested.by the Whenfringe thecapacitance. output under The differentresults in accelerations the literature is [22] obtained, show athat least-squares the fringelinear capacitance fit of the also input-output influences datathe linearity is made. error. Finally, Because the linearity the linearity error is error calculated model by established Equation (7).in this work does not consider the fringeThe capacitance, measuring resultsthe estimated of linearity linearity error areerror shown may in underestimate Figure 11. The the average actual linearity linearity error error. for the In fiveaddition, accelerometers the decreased designed narrow in this gap work decreases is about 0.84%.the proportion On the other of fringe side, the capacitance average linearity in the errortotal forcapacitance. five accelerometers As a result, designed the influence in the of previous fringe capa workcitance is about on the 1.16%. linearity The measurederror is decreased. linearity In error other in thiswords, work the is linearity 72% of thaterror in is the decreased previous by work. the decrea As such,sing theof the lower narrow linearity gap. errorAs a result, is achieved the theoretical through theestimated increasing result of underestimates WNGR. the actual improvement in the linearity error.

1.5 1.5

1.0 1.0

0.5 0.5 Linearity eroor (%) Linearity eroor Linearityeroor (%)

0.0 0.0 12345 12345

(a) (b)

Figure 11. Comparison of the experimental results of linearity error. (a) Experimental results of Figure 11. Comparison of the experimental results of linearity error. (a) Experimental results of linearity linearity error of the accelerometers designed in this work; (b) Experimental results of linearity error error of the accelerometers designed in this work; (b) Experimental results of linearity error of the of the accelerometers designed in the previous work. accelerometers designed in the previous work.

6. Conclusions However, there exist deviations between the measured results and the theoretical results. Firstly, the measuredIn this work, linearity a structural error of design 0.84% for is a higher bulk MEMS than the capacitive theoretical accelerometer linearity error is proposed of 0.4% listedfor both in Tablelow temperature2. Secondly, thecoefficient result in and Table linearity2 shows error. that theFirs estimatedtly, the contrary linearity effect error of for WNGR the accelerometers on TCSF and inlinearity this work error is is slightly illustrated lower and than is avoided that in the by previousan improved work. structure. These deviations Within the may improved be made structure, by the fringeboth TCSF capacitance. and the The linearity results error in the decrease literature with [22] increasing show that theWNGR. fringe Then, capacitance the precise also inﬂuencesdesign of the linearityimproved error. structure Because is developed the linearity for error achieving model much established lower inTCB this and work TCSF does with not the consider precondition the fringe of capacitance,high linearity. the Finally, estimated the structural linearity errordesign may for underestimatelow TCB, TCSF the and actual linearity linearity error error. is experimentally In addition, theverified. decreased Experimental narrow gapresults decreases show that the the proportion new structural of fringe design capacitance results in ina linearity the total error capacitance. close to As0.84%, a result, a TCB the close inﬂuence to 179 μ ofg/°C, fringe and capacitance a TCSF close on to the −9.8 linearity ppm/°C. error is decreased. In other words, the linearity error is decreased by the decreasing of the narrow gap. As a result, the theoretical estimated result underestimates the actual improvement in the linearity error.

6. Conclusions In this work, a structural design for a bulk MEMS capacitive accelerometer is proposed for both low temperature coefficient and linearity error. Firstly, the contrary effect of WNGR on TCSF and linearity error is illustrated and is avoided by an improved structure. Within the improved structure, Sensors 2018, 18, 643 12 of 13 both TCSF and the linearity error decrease with increasing WNGR. Then, the precise design of the improved structure is developed for achieving much lower TCB and TCSF with the precondition of high linearity. Finally, the structural design for low TCB, TCSF and linearity error is experimentally verified. Experimental results show that the new structural design results in a linearity error close to 0.84%, a TCB close to 179 µg/◦C, and a TCSF close to −9.8 ppm/◦C. In future, more imperfect factors, such as the fringe capacitance and the temperature dependence of the circuit, need to be coupled into the models for temperature coefficients and linearity error to enable a more precise design. In addition, more parameters could be integrated into research, such as mechanical noise, bias, and so on.

Acknowledgments: This work was supported by NSAF (grant number U1530132); the National Natural Science Foundation of China (grant number 51505068); the key scientific research fund of Xihua University (grant number Z1620212). Author Contributions: J.H. and W.Z. conceived and designed the idea and experiments; H.Y. and X.H. performed the experiments; J.H. and P.P. analyzed the data; J.H. wrote the paper. Conflicts of Interest: The authors declare no conflict of interest.

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