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− AVout 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 Sensors 2018, 18, 643 3 of 13 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)#W C1 = 2N#W + C1 dxd−x DxD+x ( − ) (3) = 2N#W + 2 N 1 #W (3) C2 2Nd+x 21ND−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 Wdenotesrepresents the displacementthe overlapping of areaproof in mass, a pair ofN movabledenotes andthe number fixed electrodes.