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Global model generation for a capacitive silicon accelerometer by finite-element analysis
ELSEVIER
Sensors and Actuators
A 67 i 1998) 153-158
Global model generation for a capacitive silicon accelerometer by finite-element analysis
Abstract A method to evaluate capacitance based on parameter extractionfrom finite-element analysisas well as a @obal model for a novei silicon microfabricated accelerometer are presented. Mechanical simulations have been performed and results are coupled with the capacltanceevaluation method to compute the static and dynamic response of the acceIerometer. Using both mechanical data and capacitance dza, a global model for the accelerometer is generated with models written in Hardware Description Language for -4nalogue device (HDL-Am) An improvement of the capacitive displacement detection rechnique is obtained. 0 1998 Etsevier Science S.A. All rights reserved. Kq~wrds: .4ccelcrometers; FEM simulation: Capacitise transducers; Hardware description language
1, Introduction A new bulk-microfab~cat~d silicon acceleromererwas recently developed at the Laboratoire de Physique et de M;ICtrologiedesOscillateurs (LPMO) , BesanFon,France [ I 1. This new accelerometerdesignpresentsvery low cross-sensitivities in addition to providing a rouIe for barch-fabrication processing of the entire device. The accelerometer structure is a silicon seismic mass suspendedby four thin beams (Fig. 1). The sensingaxis of the sensoris in the wafer plane and perpendicularto the beams.The structure wasfabricated [ 11 by double-sidedetching of a ( 100) silicon wafer, taking advantage of the rectangular symmetry of the ( 100) planes. The fabrication of thin suspendingbeamswith vertical side walls used a non-conventional alignment of 45” from the (1 IO} wafer Aat. However, this alignment leads to a large undercut with respectto the mask,which hasto be accounted for in order to obtain the desiredbeamwidths. This procedure makesfabrication of two perpendiculardeviceson one wafer in the same KO~-solution
etch step possible.
In order
to
designthe electrodesfor capacitive detection of the seismicmassdisplacement,it is necessaryto investigate the capacitance response[ 2,3] of the device. For this purpose,electricfield simulations were performed and the capacitance responsewascalculated. * Corresponding author. TeI.: +bl-2I-693-66-06: 70; E.-mail: yannick.ansel~ims.dmt.spfl.ch
Fax: i-41-Zl-693-66-
0924-4247/98/s 19.00 0 1998 Elsevier Science S.A. All rights reserved. PNS0924-4247(98)0001
1-9
Fig. 1. SEM photograph of the symmetrical silicon accelerometer.
2. Mechanical simulation of the accelerometer To describe the mechanical behaviour of the fabricaied structure, a finite-element model wasgenerated(Fig. 2) with
154
Y. Ansel er al. /Semors
and Actuators
A 67 (1998)
153-1.58
2.OE-5 l.OE-5 1
.,,,g Fig. 2. Finite-element
model of the accelerometer.
the finite-element software ANSYSa [ 41. Static simulations were performed on the model in order to evaluate the acceleration sensitivity of the sensor (Fig. 3). Anisotropic ANSYS@ SOLID 64elements are used to model the structure of the sensor. Stress-stiffening and large-deflection effects are taken into account. Finite-element results show a low mechanical cross-sensitivity perpendicularto the waferplane and non-linear static response was observed for large deflection. First measurements of the static displacements of the seismic mass due to a micro weight (measured with an optical comparator on the first prototype) agree with the simulation results: a mechanical simulated sensitivity of 0.154 p,rn mN- ’ is obtained for a device with measured sensitivity of 0.164 p,m n-&J- ‘. Dynamic finite-element simulations were also performed in order to calculate the mode shapes and the resonance frequencies of the structure (Fig. 4).
3. Mechanical
model generation of the accelerometer
The accelerometer device modelling has been split into two elements. The first element is modelled as a mechanical harmonic resonator directly obtained from the structural 0.004
( ,,,, Frepueng ‘“‘,I
I.OE2 l.OE3 I.OE4 Fig. 4. Frequency response of the accelerometer’s finite-element model for 1.O mN excitation force: upper graph, magnitude response [m] ; lowergraph, phase response [ deg]
finite-,eIement simulations (Figs. 3 and 4). The mathematical model used to describe the mechanical behaviour of the accelerometer is a general linear differential equation: aofta,-
df +...+a,dt
d”f dt”
=b,sfb,g
dx
+...+b,dt”
d”x
(1)
The indicesm and n areeven numberswhere m/2 denotes the number of zeroes in the transfer function and 11/2 the number of poles. For example, in the caseof a mass-spring systern (m =0 and n =2), the parameterb0 representsthe spring constant, b, the viscous damping and b2 the seismic mass.From the network theory point of view, this model is a passiveone-port element.The naturesof the signal going through and acrossthe pinsof the one-port model are respectively the forcefand the displacementx correspondingto the ‘mechanical2 nature in HardwareDescription Languagefor Analogue device (HDL-A@‘) . The harmonic simulation has been done with an excitation force applied on the centre of gravity of the mechanicalstructure of the accelerometer.The displacementof this centreof gravity is picked up and implementedinto a mechanicalbehavioural model. The product of the applied force and the displacementis the mechanical energy received by the accelerometer.The parametersa, and bi of the mechanicalmodel arecalculatedautomatically from finite-element simulationresults.In order to integrate in this model static non-linear behaviour results, the amplitude responseof the modelis modulatedin the caseof large deflection (Fig. 3). The modulation of the amplitude responseis implementedinto the accelerometerHDL-Am model.
4. Capacitive detection simulation and evaluation
0.0 0.5 1.0 Fig. 3. Static finite-element response of the accelerometer: upper graph, applied force [N]; lower graph, displacement response [m]
A physical parameterextractor tool (PXT) basedon the numerical integration of nodal and element degreesof freedom hasbeendeveloped [ 51, and interfaces with ANSYSS. In the case of static finite-element analysis, the extensive variable (current, force, torque, flow rate, heat-flow rate, etc.)
Y. Ansel
et nl. /Sensors
and Acruarors
is calculated for a given intensive variable (voltage, translation velocity, rotational velocity, pressure, temperature, etc.). A physical macro-parameter (resistance, capacitance, reactance, displaced volume, electrostatic force, etc.) can also be calculated for given boundary conditions. In the case of capacitance extraction, ANSYS@ element type SOLID122, a 3D 20-node electrostatic element with a VOLT degree of freedom at each node is used to model the device. Known voltages are applied to the boundary of the electrodes of the model. After solution, the electric flux density D is available at corner nodes for postprocessing. PXT calculates the capacitance, C, between electrodes by evaluating the following integral over equipotential electrode surfaces of the device:
A 67 (1998)
If the potentials and the charges can be measured or calculated, the three capacitances can be calculated with Eq. (3). Calculation of the capacitance is based on the assumption of a simplified field approximation (SFA) which gives a rough estimate of the capacitance values. The ideal parallelplate formula is applied when an upper electrode made of IZ segments moves a distance of x (x= 0: exact alignment between the upper and lower electrodes) : Cul =EaiF ‘zcr(:-x), cus=
QUP + Qlo Qlo + Qsi Qsi + QUP =
Vlo-Vsi 0 Vsi-Vlo [
(4)
(5)
(dkir)+(e/ESiOJ
CIS=&SioZ-
where S is the electrode surfaces, V is the voltage and D is the electric flux density. The capacitive detection is based on measuring the position on a movable electrode relative to a fixed or reference electrode [ 61. A fine-tooth-comb design for the two electrodes gives a capacitive position transducer in the direction perpendicular to the comb segments. The low mechanical crosssensitivity of the accelerometer renders such a transducer useful because the variation of the gap between the electrodes is small compared to the principal sensing axis variation. Three electrode sets are taken into account in the evaluation of the capacitance: the fixed upper metal electrode deposited on glass, the lower movable metal electrode deposited on a silicon oxide layer and the silicon electrode (the seismic mass). The electric potential and the charge of the upper electrode, the lower electrode and the silicon electrode are respectively Vup, Vlo, Vsi, Qup, Qlo and Qsi. The three interconnected capacitors Cul, Cus and Cls between the electrodes are shown on Fig. 5. The link between these nine quantities is given by
Olxlmin(b,c)
nab
(2) s
155
153-158
nab e
where gair and ESiO 7 are the dielectric permittivity of the air and the silicon oxide, CIand b are the length and the width of one segment, c the distance between two segments, d the air gap and e the silicon oxide layer thickness (Fig. 6). Calculation of the capacitance based on the 3D finite-element method performed with the ANSYSa Electrostatic FieldAnalyses package allows better accuracy to be achieved. The charges are extracted using the PXT tool after electrostatic analysis (Fig. 7). Assuming b = c is the displacement range, Fig. 8 displays the SFA and the FEM calculated capacitance Cul + Cus as a function of the translation x of the seismic mass. In this case, the SFA overestimates the capacitance value because Cul and Cus are calculated independently (only two electrodes are taken into account in SFA) FEM results are more accurate but time consuming. The precision is linked to the fineness
(3) Vup-Vsi Vsi-Vup
0
Fig. 5. Schematic
0 Vlo-Vup vup-VI0
of the capacitor
Ii I Cls Cus Cul
circuit.
Fig. 6. Capacitive
detection
architecture.
Fig. 7. Cross section of the finite-element model and the electric Aux density repreaented by arrows when n = 10 km, b = c = 10 pm, d = 5 p,m, e = 1 pm, r~=5.Vup=lOV,Vlo=OVandVsi=OV.
156
Y. Ansel et al. /Sensors
rind Actrrarors
6E-11
‘--
-.
--
'.
153-158
function. The natures of the input variables are ‘mechanical2 and ‘electrical’ for the output variables. The piecewise linear capacitance data extracted from the finite-element simulations are implemented into a two-port model [ 71. This twoport rnodel written in HDL-A@ is acting like a transducer converting the displacement of the seismic mass of the accelerometer into a capacitance. Eq. (7) links the output current i and the output voltage u of this two-port model:
CapacitanceCul+Cus [F] 7E-11
A 67 (1998)
'.
(7)
0
1
2
3
4
5
x
M-4
6
7
8
9
IO
Fig. 8. SFA and FEM capacitance calculation results when a=5 mm, b = c = 10 pm, d = 5 p,m, e = 1 pm and n = 400 vs. electrode displacement x.
of the mesh and thus to the computing time. In this case, the results vary less than i: 10% when the meshing size of the electrostatic finite-element model is halved. FEM results show that a good linear response of the capacitive transducer is obtained around the middle of the displacement range. The first derivative dCldx of the capacitance with the respect to the displacement x is the sensitivity evaluation criterion of the capacitive transducer. FEM shows that the maximum s of this function is obtained in the middle of the displacement range, that is, s= max( dC/&) = dC/ ~LY=bm Fig. 9 displays s for a constant electrode area and separation distance. In this case, it shows that the capacitive transducer is optimized when b/d = 2 for a5 p,rn linearrange. 5. Capacitance model generation
The electrical part of the model is a variable capacitor load implemented as a piecewise linear lookup table or as a sine Capacitive sensitivity {F&n] I.6512
I
1.4512
In Ithe case of the piecewise linear capacitance model, a space period function of the capacitance versus the displacement (enables the lookup table to be defined over one period. The v.ariable capacitance is periodic to a good approximation due to the high number of electrode segments (around 400). Nevertheless, the derivative of the piecewise linear capacitance is not continuous. This is why a sine-function model of the capacitance is an alternative to the lookup-table model. 6. Global model of the accelerometer
Fig, 10 is a schematic view of the global model accelerometer architecture. The resultant behavioural model of the whole accelerometer is used in an analogue circuit simulator to perform simulations together with the detection electronics. Generation of behavioural HDL-Aa models of devices characterized using finite elements results in rapid system simulation [ 81. This method prevents the convergence problems often encountered when coupling different field solvers. In order to illustrate the global model of the accelerometer, the electric circuit shown in Fig. 10 is employed to detect accelerations. The Cus and Cls capacitors between the variable-capacitance capacitors Cul are implemented as constantcapacitor models. A constant bias voltage and a resistor are connected in series with the accelerometer’s model. Capacitance change induces current change in the resistor and a resistor’s bias signal, which is integrated to get a signal proportional to the acceleration. Network simulations are performed on ELDO@, a SPICE-like simulator with a transient force signal. Simulation results are displayed in Fig. 11: the periodic response of the accelerometer is pointed out.
1.2E-12 IE-12 6E-13 6E-13 4E-13 2E-13
1
2
5
10
b/d (log scale) Fig. 9. Capacitive sensitivity vs. b/d ratio when a = 5 mm, b = c, d = 5 Wm. e=l pmandn(b+c)=8mm.
Accelerometer mechanical model
Capacitivetransducer model
Electronic integrator model
Fig. 10. Schematic of the accelerometer circuit. HDL-A models are thickrimmed boxes.
Y. Amel
0.03
Applied force (input)
et al. /Sensors
and Actuators
A 67 (1998)
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technology (PICS = Projet International Scientifique) .
[N] /
157
de Cooperation
I
References Cl1 G.
2.OE-4
Voltage response ~ooutput) [V] r--x
I
0.5 1.0 0.0 Fig. 11. Global simulation result of the accelerometer circuit: upper graph, applied force [N]; lower graph, transient output voltage response of the accelerometer circuit after integration [VI.
In order to improve the global model, feedback forces generated by the potentials on the seismic mass can be calculated by ANSYSO and implemented in the model. Including electrostatic feedback force in the model, a possible closed-loop accelerometer electronic circuit can be tested in order to increase the acceleration range of the sensor. Nevertheless, the presented model does not take into account possible free charges in the silicon oxide layer, which can disrupt the accelerometer detection operation. Experimental results of the accelerometer with the optimized geometrical parameters of the capacitive transducer are under investigation.
Schropfer, S. Ballandras, M. de Labachelerie, P. Blind, Y. Ansel, Fabrication of a new highly-symmetrical, in-plane accelerometer structure by anisotropic etching of (100) silicon, J. Micromech. Microeng. 7 (1997) 71-78. 121 M.H.W. Bonse, C. Mul, J.W. Spronck, Finite-element modelling as a tool for designing capacitive position sensors, Sensors and Actuators A 46-47 ( 1995) 266-269. [31 M.H.W. Bonse, F. Zhu, J.W. Spronck, A new two-dimensionalcapacitive position transducer, Sensors and Actuators A 4142 ( 1994) 2932. [41 Swanson Analysis Systems, ANSYS User’s Manual, version 5.3. Y. Ansel, Ph. Lerch, Ph. Renaud, Global modeling I51 B. Romanowicz, and simulation of microsystems, Proc. Micromechanics Europe (MME ‘96). Barcelona, Spain, 21-22 Oct., 1996, pp. 196-199. of capacitance techniques in transducer [61 W.Chr. Heerens, Application design, J. Phys. E: Sci. Instrum. 19 ( 1986) 897-906. M. Laudon, Ph. Renaud, G. Schropfer, [71 Y. Ansel, B. Romanowicz, Capacitive detection method evaluation for silicon accelerometer by physical parameter extraction from finite element simulation., SISPAD ‘97 Int. Conf. Simulation of Semiconductor Processes and Devices, Boston, MA, USA, 8-9 Sept., 1997. 181 B. Romanowicz, A. Vachoux, Y. Ansel, M. Laudon, C. Amacker, Ph. Renaud, G. Schropfer, VHDL-1076.1 Modeling examples for microsystem simulation, Proc. 2nd Workshop on Libraries, Component Modelling and Quality Assurance, Toledo, Spain, 23-25 Apr., 1997, pp. 89-103.
Biographies Yunnick Ansel received the diploma degree in microtechnol-
7. Conclusions A mechanical simulation of a silicon accelerometer was performed and results were successfully compared to measurements. Electrostatic simulations were also performed to improve the capacitive transducer of the accelerometer. Capacitance was calculated from the results of finite-element analysis with a parameter-extraction tool. The computed capacitive sensitivity expected for the optimized electrode design is around 1.4 pF km-’ for the mentioned case. As a conclusion, finite-element analysis is used to increase the performance of the studied accelerometer and global modelling is suitable for the improvements of detection electronics.
Acknowledgements This work has been funded by the Swiss National Science Foundation (FNSRS) and the Swiss Foundation forResearch in Microtechnology (FSRM) in the frame of an international exchange between French and Swiss institutions of micro-
ogy from the Ecole Nationale Superieure de Mecanique et des Microtechniques (ENSMM), Besancon, France, in 1994. He is currently pursuing the Ph.D. degree at the Institute of Microsystems, Ecole Polytechnique Fed&ale de Lausanne (EPFL), Switzerland. His main focus of research is in optimization of inertial sensors using simulation tools like FEM and EDA. Barr Romanowicz was born on 13 Sept., 1967, in Poland. He
received his diploma in microengineering from the EPF’L in 1993. He is pursuing the Ph.D. degree at the Microsystems Institute of the Microengineering Department of the EPFL. His main focus of research is in methodologies and software tools including finite elements, parameter extraction and hardware description languages related to the simulation of microsystems. Philippe Renaud received a DiplPhys. from the University of Neuchatel, and a Ph.D. in solid-state physics from the University of Lausanne. In 1988-1989 he was apost-doctoral fellow, working on superconductivity and STM, at the University of California, Berkeley. In 1990-1991 he conducted research at the IBM Zurich Laboratory with Dr H. Rohrer. In
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Y. Ansel er al. /Serwors
md Actunrors
1992-1993 he was a scientific collaborator in the Sensor and Actuator team of CSEM, Neuchgtel. Since 1993, he has been a professor at the Microsystems Institute, Department of Microengineering, EPFL. Gei-old SchrDpfer studied physics in Giessen, Germany, and Milwaukee, USA, from 1990 to 1995. After working from
A 67 (1998)
153458
1994 to 1995 on magnetoresistive sensors at the Institut fiir Mikrostrukturtechnologie und Optoelektronik e.V. (IMO), Wet&r, Germany, he received his diploma in physics from the University of Giessen. In 1996 he joined the LPMO, Besancon, France, to produce a Ph.D. thesis in the field of silicon accelerometers. His research interests include physical sensors and silicon micromachining.
Sensors and Actuators
A 67 i 1998) 153-158
Global model generation for a capacitive silicon accelerometer by finite-element analysis
Abstract A method to evaluate capacitance based on parameter extractionfrom finite-element analysisas well as a @obal model for a novei silicon microfabricated accelerometer are presented. Mechanical simulations have been performed and results are coupled with the capacltanceevaluation method to compute the static and dynamic response of the acceIerometer. Using both mechanical data and capacitance dza, a global model for the accelerometer is generated with models written in Hardware Description Language for -4nalogue device (HDL-Am) An improvement of the capacitive displacement detection rechnique is obtained. 0 1998 Etsevier Science S.A. All rights reserved. Kq~wrds: .4ccelcrometers; FEM simulation: Capacitise transducers; Hardware description language
1, Introduction A new bulk-microfab~cat~d silicon acceleromererwas recently developed at the Laboratoire de Physique et de M;ICtrologiedesOscillateurs (LPMO) , BesanFon,France [ I 1. This new accelerometerdesignpresentsvery low cross-sensitivities in addition to providing a rouIe for barch-fabrication processing of the entire device. The accelerometer structure is a silicon seismic mass suspendedby four thin beams (Fig. 1). The sensingaxis of the sensoris in the wafer plane and perpendicularto the beams.The structure wasfabricated [ 11 by double-sidedetching of a ( 100) silicon wafer, taking advantage of the rectangular symmetry of the ( 100) planes. The fabrication of thin suspendingbeamswith vertical side walls used a non-conventional alignment of 45” from the (1 IO} wafer Aat. However, this alignment leads to a large undercut with respectto the mask,which hasto be accounted for in order to obtain the desiredbeamwidths. This procedure makesfabrication of two perpendiculardeviceson one wafer in the same KO~-solution
etch step possible.
In order
to
designthe electrodesfor capacitive detection of the seismicmassdisplacement,it is necessaryto investigate the capacitance response[ 2,3] of the device. For this purpose,electricfield simulations were performed and the capacitance responsewascalculated. * Corresponding author. TeI.: +bl-2I-693-66-06: 70; E.-mail: yannick.ansel~ims.dmt.spfl.ch
Fax: i-41-Zl-693-66-
0924-4247/98/s 19.00 0 1998 Elsevier Science S.A. All rights reserved. PNS0924-4247(98)0001
1-9
Fig. 1. SEM photograph of the symmetrical silicon accelerometer.
2. Mechanical simulation of the accelerometer To describe the mechanical behaviour of the fabricaied structure, a finite-element model wasgenerated(Fig. 2) with
154
Y. Ansel er al. /Semors
and Actuators
A 67 (1998)
153-1.58
2.OE-5 l.OE-5 1
.,,,g Fig. 2. Finite-element
model of the accelerometer.
the finite-element software ANSYSa [ 41. Static simulations were performed on the model in order to evaluate the acceleration sensitivity of the sensor (Fig. 3). Anisotropic ANSYS@ SOLID 64elements are used to model the structure of the sensor. Stress-stiffening and large-deflection effects are taken into account. Finite-element results show a low mechanical cross-sensitivity perpendicularto the waferplane and non-linear static response was observed for large deflection. First measurements of the static displacements of the seismic mass due to a micro weight (measured with an optical comparator on the first prototype) agree with the simulation results: a mechanical simulated sensitivity of 0.154 p,rn mN- ’ is obtained for a device with measured sensitivity of 0.164 p,m n-&J- ‘. Dynamic finite-element simulations were also performed in order to calculate the mode shapes and the resonance frequencies of the structure (Fig. 4).
3. Mechanical
model generation of the accelerometer
The accelerometer device modelling has been split into two elements. The first element is modelled as a mechanical harmonic resonator directly obtained from the structural 0.004
( ,,,, Frepueng ‘“‘,I
I.OE2 l.OE3 I.OE4 Fig. 4. Frequency response of the accelerometer’s finite-element model for 1.O mN excitation force: upper graph, magnitude response [m] ; lowergraph, phase response [ deg]
finite-,eIement simulations (Figs. 3 and 4). The mathematical model used to describe the mechanical behaviour of the accelerometer is a general linear differential equation: aofta,-
df +...+a,dt
d”f dt”
=b,sfb,g
dx
+...+b,dt”
d”x
(1)
The indicesm and n areeven numberswhere m/2 denotes the number of zeroes in the transfer function and 11/2 the number of poles. For example, in the caseof a mass-spring systern (m =0 and n =2), the parameterb0 representsthe spring constant, b, the viscous damping and b2 the seismic mass.From the network theory point of view, this model is a passiveone-port element.The naturesof the signal going through and acrossthe pinsof the one-port model are respectively the forcefand the displacementx correspondingto the ‘mechanical2 nature in HardwareDescription Languagefor Analogue device (HDL-A@‘) . The harmonic simulation has been done with an excitation force applied on the centre of gravity of the mechanicalstructure of the accelerometer.The displacementof this centreof gravity is picked up and implementedinto a mechanicalbehavioural model. The product of the applied force and the displacementis the mechanical energy received by the accelerometer.The parametersa, and bi of the mechanicalmodel arecalculatedautomatically from finite-element simulationresults.In order to integrate in this model static non-linear behaviour results, the amplitude responseof the modelis modulatedin the caseof large deflection (Fig. 3). The modulation of the amplitude responseis implementedinto the accelerometerHDL-Am model.
4. Capacitive detection simulation and evaluation
0.0 0.5 1.0 Fig. 3. Static finite-element response of the accelerometer: upper graph, applied force [N]; lower graph, displacement response [m]
A physical parameterextractor tool (PXT) basedon the numerical integration of nodal and element degreesof freedom hasbeendeveloped [ 51, and interfaces with ANSYSS. In the case of static finite-element analysis, the extensive variable (current, force, torque, flow rate, heat-flow rate, etc.)
Y. Ansel
et nl. /Sensors
and Acruarors
is calculated for a given intensive variable (voltage, translation velocity, rotational velocity, pressure, temperature, etc.). A physical macro-parameter (resistance, capacitance, reactance, displaced volume, electrostatic force, etc.) can also be calculated for given boundary conditions. In the case of capacitance extraction, ANSYS@ element type SOLID122, a 3D 20-node electrostatic element with a VOLT degree of freedom at each node is used to model the device. Known voltages are applied to the boundary of the electrodes of the model. After solution, the electric flux density D is available at corner nodes for postprocessing. PXT calculates the capacitance, C, between electrodes by evaluating the following integral over equipotential electrode surfaces of the device:
A 67 (1998)
If the potentials and the charges can be measured or calculated, the three capacitances can be calculated with Eq. (3). Calculation of the capacitance is based on the assumption of a simplified field approximation (SFA) which gives a rough estimate of the capacitance values. The ideal parallelplate formula is applied when an upper electrode made of IZ segments moves a distance of x (x= 0: exact alignment between the upper and lower electrodes) : Cul =EaiF ‘zcr(:-x), cus=
QUP + Qlo Qlo + Qsi Qsi + QUP =
Vlo-Vsi 0 Vsi-Vlo [
(4)
(5)
(dkir)+(e/ESiOJ
CIS=&SioZ-
where S is the electrode surfaces, V is the voltage and D is the electric flux density. The capacitive detection is based on measuring the position on a movable electrode relative to a fixed or reference electrode [ 61. A fine-tooth-comb design for the two electrodes gives a capacitive position transducer in the direction perpendicular to the comb segments. The low mechanical crosssensitivity of the accelerometer renders such a transducer useful because the variation of the gap between the electrodes is small compared to the principal sensing axis variation. Three electrode sets are taken into account in the evaluation of the capacitance: the fixed upper metal electrode deposited on glass, the lower movable metal electrode deposited on a silicon oxide layer and the silicon electrode (the seismic mass). The electric potential and the charge of the upper electrode, the lower electrode and the silicon electrode are respectively Vup, Vlo, Vsi, Qup, Qlo and Qsi. The three interconnected capacitors Cul, Cus and Cls between the electrodes are shown on Fig. 5. The link between these nine quantities is given by
Olxlmin(b,c)
nab
(2) s
155
153-158
nab e
where gair and ESiO 7 are the dielectric permittivity of the air and the silicon oxide, CIand b are the length and the width of one segment, c the distance between two segments, d the air gap and e the silicon oxide layer thickness (Fig. 6). Calculation of the capacitance based on the 3D finite-element method performed with the ANSYSa Electrostatic FieldAnalyses package allows better accuracy to be achieved. The charges are extracted using the PXT tool after electrostatic analysis (Fig. 7). Assuming b = c is the displacement range, Fig. 8 displays the SFA and the FEM calculated capacitance Cul + Cus as a function of the translation x of the seismic mass. In this case, the SFA overestimates the capacitance value because Cul and Cus are calculated independently (only two electrodes are taken into account in SFA) FEM results are more accurate but time consuming. The precision is linked to the fineness
(3) Vup-Vsi Vsi-Vup
0
Fig. 5. Schematic
0 Vlo-Vup vup-VI0
of the capacitor
Ii I Cls Cus Cul
circuit.
Fig. 6. Capacitive
detection
architecture.
Fig. 7. Cross section of the finite-element model and the electric Aux density repreaented by arrows when n = 10 km, b = c = 10 pm, d = 5 p,m, e = 1 pm, r~=5.Vup=lOV,Vlo=OVandVsi=OV.
156
Y. Ansel et al. /Sensors
rind Actrrarors
6E-11
‘--
-.
--
'.
153-158
function. The natures of the input variables are ‘mechanical2 and ‘electrical’ for the output variables. The piecewise linear capacitance data extracted from the finite-element simulations are implemented into a two-port model [ 71. This twoport rnodel written in HDL-A@ is acting like a transducer converting the displacement of the seismic mass of the accelerometer into a capacitance. Eq. (7) links the output current i and the output voltage u of this two-port model:
CapacitanceCul+Cus [F] 7E-11
A 67 (1998)
'.
(7)
0
1
2
3
4
5
x
M-4
6
7
8
9
IO
Fig. 8. SFA and FEM capacitance calculation results when a=5 mm, b = c = 10 pm, d = 5 p,m, e = 1 pm and n = 400 vs. electrode displacement x.
of the mesh and thus to the computing time. In this case, the results vary less than i: 10% when the meshing size of the electrostatic finite-element model is halved. FEM results show that a good linear response of the capacitive transducer is obtained around the middle of the displacement range. The first derivative dCldx of the capacitance with the respect to the displacement x is the sensitivity evaluation criterion of the capacitive transducer. FEM shows that the maximum s of this function is obtained in the middle of the displacement range, that is, s= max( dC/&) = dC/ ~LY=bm Fig. 9 displays s for a constant electrode area and separation distance. In this case, it shows that the capacitive transducer is optimized when b/d = 2 for a5 p,rn linearrange. 5. Capacitance model generation
The electrical part of the model is a variable capacitor load implemented as a piecewise linear lookup table or as a sine Capacitive sensitivity {F&n] I.6512
I
1.4512
In Ithe case of the piecewise linear capacitance model, a space period function of the capacitance versus the displacement (enables the lookup table to be defined over one period. The v.ariable capacitance is periodic to a good approximation due to the high number of electrode segments (around 400). Nevertheless, the derivative of the piecewise linear capacitance is not continuous. This is why a sine-function model of the capacitance is an alternative to the lookup-table model. 6. Global model of the accelerometer
Fig, 10 is a schematic view of the global model accelerometer architecture. The resultant behavioural model of the whole accelerometer is used in an analogue circuit simulator to perform simulations together with the detection electronics. Generation of behavioural HDL-Aa models of devices characterized using finite elements results in rapid system simulation [ 81. This method prevents the convergence problems often encountered when coupling different field solvers. In order to illustrate the global model of the accelerometer, the electric circuit shown in Fig. 10 is employed to detect accelerations. The Cus and Cls capacitors between the variable-capacitance capacitors Cul are implemented as constantcapacitor models. A constant bias voltage and a resistor are connected in series with the accelerometer’s model. Capacitance change induces current change in the resistor and a resistor’s bias signal, which is integrated to get a signal proportional to the acceleration. Network simulations are performed on ELDO@, a SPICE-like simulator with a transient force signal. Simulation results are displayed in Fig. 11: the periodic response of the accelerometer is pointed out.
1.2E-12 IE-12 6E-13 6E-13 4E-13 2E-13
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2
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b/d (log scale) Fig. 9. Capacitive sensitivity vs. b/d ratio when a = 5 mm, b = c, d = 5 Wm. e=l pmandn(b+c)=8mm.
Accelerometer mechanical model
Capacitivetransducer model
Electronic integrator model
Fig. 10. Schematic of the accelerometer circuit. HDL-A models are thickrimmed boxes.
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Voltage response ~ooutput) [V] r--x
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0.5 1.0 0.0 Fig. 11. Global simulation result of the accelerometer circuit: upper graph, applied force [N]; lower graph, transient output voltage response of the accelerometer circuit after integration [VI.
In order to improve the global model, feedback forces generated by the potentials on the seismic mass can be calculated by ANSYSO and implemented in the model. Including electrostatic feedback force in the model, a possible closed-loop accelerometer electronic circuit can be tested in order to increase the acceleration range of the sensor. Nevertheless, the presented model does not take into account possible free charges in the silicon oxide layer, which can disrupt the accelerometer detection operation. Experimental results of the accelerometer with the optimized geometrical parameters of the capacitive transducer are under investigation.
Schropfer, S. Ballandras, M. de Labachelerie, P. Blind, Y. Ansel, Fabrication of a new highly-symmetrical, in-plane accelerometer structure by anisotropic etching of (100) silicon, J. Micromech. Microeng. 7 (1997) 71-78. 121 M.H.W. Bonse, C. Mul, J.W. Spronck, Finite-element modelling as a tool for designing capacitive position sensors, Sensors and Actuators A 46-47 ( 1995) 266-269. [31 M.H.W. Bonse, F. Zhu, J.W. Spronck, A new two-dimensionalcapacitive position transducer, Sensors and Actuators A 4142 ( 1994) 2932. [41 Swanson Analysis Systems, ANSYS User’s Manual, version 5.3. Y. Ansel, Ph. Lerch, Ph. Renaud, Global modeling I51 B. Romanowicz, and simulation of microsystems, Proc. Micromechanics Europe (MME ‘96). Barcelona, Spain, 21-22 Oct., 1996, pp. 196-199. of capacitance techniques in transducer [61 W.Chr. Heerens, Application design, J. Phys. E: Sci. Instrum. 19 ( 1986) 897-906. M. Laudon, Ph. Renaud, G. Schropfer, [71 Y. Ansel, B. Romanowicz, Capacitive detection method evaluation for silicon accelerometer by physical parameter extraction from finite element simulation., SISPAD ‘97 Int. Conf. Simulation of Semiconductor Processes and Devices, Boston, MA, USA, 8-9 Sept., 1997. 181 B. Romanowicz, A. Vachoux, Y. Ansel, M. Laudon, C. Amacker, Ph. Renaud, G. Schropfer, VHDL-1076.1 Modeling examples for microsystem simulation, Proc. 2nd Workshop on Libraries, Component Modelling and Quality Assurance, Toledo, Spain, 23-25 Apr., 1997, pp. 89-103.
Biographies Yunnick Ansel received the diploma degree in microtechnol-
7. Conclusions A mechanical simulation of a silicon accelerometer was performed and results were successfully compared to measurements. Electrostatic simulations were also performed to improve the capacitive transducer of the accelerometer. Capacitance was calculated from the results of finite-element analysis with a parameter-extraction tool. The computed capacitive sensitivity expected for the optimized electrode design is around 1.4 pF km-’ for the mentioned case. As a conclusion, finite-element analysis is used to increase the performance of the studied accelerometer and global modelling is suitable for the improvements of detection electronics.
Acknowledgements This work has been funded by the Swiss National Science Foundation (FNSRS) and the Swiss Foundation forResearch in Microtechnology (FSRM) in the frame of an international exchange between French and Swiss institutions of micro-
ogy from the Ecole Nationale Superieure de Mecanique et des Microtechniques (ENSMM), Besancon, France, in 1994. He is currently pursuing the Ph.D. degree at the Institute of Microsystems, Ecole Polytechnique Fed&ale de Lausanne (EPFL), Switzerland. His main focus of research is in optimization of inertial sensors using simulation tools like FEM and EDA. Barr Romanowicz was born on 13 Sept., 1967, in Poland. He
received his diploma in microengineering from the EPF’L in 1993. He is pursuing the Ph.D. degree at the Microsystems Institute of the Microengineering Department of the EPFL. His main focus of research is in methodologies and software tools including finite elements, parameter extraction and hardware description languages related to the simulation of microsystems. Philippe Renaud received a DiplPhys. from the University of Neuchatel, and a Ph.D. in solid-state physics from the University of Lausanne. In 1988-1989 he was apost-doctoral fellow, working on superconductivity and STM, at the University of California, Berkeley. In 1990-1991 he conducted research at the IBM Zurich Laboratory with Dr H. Rohrer. In
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1992-1993 he was a scientific collaborator in the Sensor and Actuator team of CSEM, Neuchgtel. Since 1993, he has been a professor at the Microsystems Institute, Department of Microengineering, EPFL. Gei-old SchrDpfer studied physics in Giessen, Germany, and Milwaukee, USA, from 1990 to 1995. After working from
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1994 to 1995 on magnetoresistive sensors at the Institut fiir Mikrostrukturtechnologie und Optoelektronik e.V. (IMO), Wet&r, Germany, he received his diploma in physics from the University of Giessen. In 1996 he joined the LPMO, Besancon, France, to produce a Ph.D. thesis in the field of silicon accelerometers. His research interests include physical sensors and silicon micromachining.