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Analysis characterisation and optimisation of temperature coefficient parameters in capacitive pressure sensors
Sensors and Actuators A 93 (2001) 44±47
Analysis characterisation and optimisation of temperature coef®cient parameters in capacitive pressure sensors G. Blasquez*, C. Douziech, P. Pons CNRS-LAAS, 7 Avenue Roche, 31077 Toulouse Cedex 4, France Received 7 December 2000; received in revised form 19 February 2001; accepted 3 March 2001
Abstract The literal expression of the temperature coef®cient is derived in the case of plane sensing cells. It states that the thermal behaviour of the capacitance depends on the cell dimensions and the temperature coef®cients of the ®xed plate and the internal cavity. Measurements carried out in a set of sensors, constituted of aluminium ®xed plates and thin silicon membranes bonded to Pyrex substrates, show that the amplitude of the capacitance thermal coef®cient is very sensitive to the bonded length and the ®xed plate thickness. Analyses of experimental results indicate that, very often, thermo-mechanical deformations of the internal cavity determine the amplitude of the capacitance temperature coef®cient. Nevertheless, they also demonstrate the feasibility of sensing cells in which the capacitance temperature coef®cient is very small and practically equal to the temperature coef®cient of the ®xed plate. In these cells, the fabrication of the ®xed plate from ultra thin ®lms and/or metals featuring small thermal expansion coef®cients should give capacitance temperature coef®cient lower than 44 ppm/8C. Finally it is shown that analyses of isomorphous sensors allow to separate the contributions of the ®xed plate and the cell structure and in addition to withdraw the effects of packaging and parasitic capacitors. # 2001 Elsevier Science B.V. All rights reserved. Keywords: Capacitive sensors; Temperature sensitivity; Temperature coef®cient; Structural parameters; Silicon membrane
1. Introduction In a recent paper [1], it has been shown that temperature causes nanometric deformations of the structure of pressure sensors fabricated by bonding a silicon membrane to a Pyrex substrate. The numerical evaluation of capacitance variations induced by these reversible deformations yielded amplitudes in the order of some hundreds of ppm/8C for thick membranes. Moreover, it has been proved that the bond width and the substrate thickness determine largely the thermal behaviour. In the following, the behaviour of sensors with a thinner membrane and therefore, a higher sensitivity to pressure, is investigated. This study relies on the analysis of the behaviour of a family of isomorphous sensors and on the general expressions that link the structural parameters to the capacitance and to the temperature coef®cient. It highlights the role played by the ®xed plate and con®rms the importance of the bond. Finally, it de®nes the minimum of the temperature coef®cient. * Corresponding author. Tel.: 33-561-336388; fax: 33-561-336208. E-mail address: [email protected] (G. Blasquez).
From a methodological point of view, the combined use of a case study and analytical calculations leads to a simple description of the respective roles of the sensor's structural parameters. Likewise, it allows to assess their interactions. It leads to conclusions that can be applied to sensors that are structurally and technologically different from those considered here. 2. Fundamentals A simple model of a capacitive pressure sensor is shown in Fig. 1. It consists of: (a) a rigid and insulating substrate of thickness hp; (b) a conductive membrane of thickness hs bonded to the substrate and (c) a rigid plate deposited inside a cavity of depth de. In the following, the bond width is denoted ab. At rest, the membrane and the ®xed plate form a planar capacitor whose capacitance is equal to C. If fringe electric ®elds remain negligible, the relation between the design parameters and C can be written as C e0
0924-4247/01/$ ± see front matter # 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 4 - 4 2 4 7 ( 0 1 ) 0 0 6 2 6 - 4
A de
dm
(1)
G. Blasquez et al. / Sensors and Actuators A 93 (2001) 44±47
45
Fig. 1. Basic structure and dimensions of a sensing cell.
where dm and A are, respectively the thickness and the area of the fixed plate. Temperature T modi®es all the structural parameters and, therefore, C. Over a large temperature range, the thermal behaviour of C can only be accurately characterised by its relative partial derivative t C
1 dC C dT
(2)
The operator t() is sometimes called the differential or incremental temperature coef®cient. For simplicity, t(C) will be called the temperature coef®cient of C. The analytical calculation of t(C) from Eqs. (1) and (2) leads to the following formula: t C t A
de
dm t dm dm
de
de t de dm
(3)
where t(A), t(dm) and t(de) denote, respectively the differential temperature coefficients of A, dm and de. Variations of A and dm are due to the ®xed plate deformations, de variations are mainly caused by deformations of the membrane and the insulating substrate [1].
Fig. 2. Examples of temperature coefficient features: (~) S1; (*) S2. Full lines represent the best fitting lines given by the least square method. Table 2 Capacitance temperature coefficients given by the least square method
K0i (ppm/8C) K1i (ppm/8C)
t1
t2
t3
t4
227 0.12
483 2.67
44 0.03
174 1.30
Consequently their stray capacitances due to fringe ®elds are almost negligible. Fig. 2 illustrates the behaviour of their temperature coef®cients. Data can be ®tted by a straight line of the form ti C k0i k1i T
(4)
where k0i and k1i are the model coefficients specific to the sensor Si. The estimates of k0i and k1i of all the considered sensors are given in Table 2. They were computed from measurements carried out in the temperature range ( 308C; 1508C), the least square method and Eqs. (3) and (4). For simplicity, the temperature coefficients ti(C) are denoted ti with i 1 to 4 in this table.
3. Experiments The group of sensors considered in the following sections has a 7740 Pyrex substrate, a silicon membrane and a ®xed aluminium plate (A 6:15 mm2). Their pertinent dimensions are given in Table 1. Morphologically, these sensors resemble to those described in [1]. The main differences lie in the membrane thickness (50 mm instead of 300 mm), metalisation (aluminium replacing a Ti/Au bilayer) and the bonding temperature (4508C instead of 4008C). To minimise electric fringing ®elds these sensors have in addition a guard electrode around the ®xed plate. Table 1 Sensor dimensions Si
hp (mm)
hs (mm)
de (mm)
dmi (mm)
abi (mm)
S1 S2 S3 S4
1.5 1.5 1.5 1.5
50 50 50 50
1.8 1.8 1.8 1.8
1 1 0.1 0.1
1 0.25 1 0.25
4. Data analysis 4.1. Estimation of t(A) In order to perform a preliminary analysis of the experimental results, it is easier to estimate the order of magnitude of t(dm) and t(A). In principle, these temperature coef®cients depend on the thermal and mechanical deformations of the ®xed plate and the substrate. In [2] it has been shown that the temperature coef®cient of 1 mm thick aluminium ®lms, deposited on silica and enduring substantial mechanical deformations, is close to the thermal expansion coef®cient aAl of bulk aluminium (aAl 25 ppm/8C near room temperature). In pressure sensors, the mechanical deformations of the ®xed plate and the Pyrex substrate are signi®cantly lower than in the preceding case. Consequently, we infer that the behaviour of the ®xed plate is mainly determined by thermal deformations and therefore that t dm aAl and t A 2aAl represent reasonable estimates of t(dm) and t(A).
46
G. Blasquez et al. / Sensors and Actuators A 93 (2001) 44±47
4.2. t(C) minimum From the estimates proposed in the preceding Section 4.1, the analysis of the data in Table 2 yields: t de @ t A in S1, S2 and S4 and t A @ t de in S3. In other terms, thermal variations are mainly due to thermo-mechanical deformations of the membrane and the substrate in S1, S2, and S4 and to the deformation of the ®xed plate in S3. Let us note that if the thermo-mechanical deformations of the membrane and the substrate are negligible for any temperature and if dm @ de, then t C t A:
(5)
This reduced expression represents the minimum of the capacitance temperature coef®cient. Data in Table 2 and the considerations in Section 4.1 indicate that this limit has been approached in the sensor S3. 4.3. Critical dimensions The accurate evaluation of the in¯uence of the sensor dimensions on the thermal behaviour requires to perform comparative analyses of isomorphous sensors. In the following, we develop this technique in order to gain a better understanding of the thermal behaviour and to exhibit some features of t(de) which is the most in¯uencing factor in many sensors. If t(de) predominates, and if dm varies while ab remains constant, it results from Eqs. (1) and (3) that the ratios of: capacitances, k0i, and k1i must be very similar. From the data relative to the couples (S2, S4) and (S1, S3) it can easily be shown that C2 k02 k12 2:1; 2:8; 2:1 (6) C4 k04 k14 C1 k01 k11 2:1; 5:2; 4:0 (7) C3 k03 k13 These evaluations corroborate the preliminary results derived in Section 4.2: t de @ t A in S1, S2 and S4 and t A @ t de in S3. Let us note that these deductions are independent of the assumptions done in Section 4.1. In addition, if t(de) predominates, and if dm varies, Eq. (3) speci®es that the thermal variations due to the thermomechanical deformations of the membrane and the substrate are greatly ampli®ed as soon as dm has the same order of magnitude that de. Moreover, if dm remains constant, and if ab varies, Table 2 shows that the temperature coef®cients can change drastically. Numerically, the comparisons of the data relative to the couples (S1, S2) and (S3, S4) give C2 k02 k12 1; 2:1; 22:2 (8) C1 k01 k11 C4 k04 k14 1; 4:0; 43:3 (9) C3 k03 k13
As expected from Eq. (1), ab variations have no in¯uence on C. However, the effects of ab variations on the temperature coef®cients are impressive. The underlying mechanisms have been analysed in [1] by simulation of the particular case of a sensor having a thick membrane. The data presented here extend the validity of the conclusions derived in [1] to the case of sensors realised with thin membranes. 4.4. Packaged sensors To re®ne the preceding considerations, it should be taken into consideration that the sensor sensing cells are always housed in a metal or plastic case. The latter's thermal deformations can in¯uence the intrinsic features of the cells. Also, interconnects between cases and sensing cells induce parasitic capacitances that are temperature dependent. Nevertheless, the difference of the temperature coef®cients of the packaged sensors in which capacitances and the packagings are identical, does not depend on the case and interconnects effects. In the following, we apply this re®ned analysis to the couples (S1, S2) and (S3, S4) in order to specify if t(de) depends on dm. To begin with, we note that the two ®rst terms of Eq. (3) are identical in (S1, S2) and (S3, S4). Thus, theoretically the differences t2 t1 and t4 t3 are only a function of the t(de) terms. The literal expression of the difference ratio can be written as t2 t4
t1 de t3 de
dm4 t2 de dm2 t4 de
t1 de t3 de
(10)
Evaluation of Eq. (10) from the data given in Tables 1 and 2 leads to t2 t4
t1 256 t3 130
2:55T 1:27T
(11)
As a ®rst approximation, Eq. (11) is equal to 2 for any T. Computation of the ratio de dm4 = de dm2 yields 2.1. Consequently, the differences t2 de t1 de and t4 de t3 de are almost equal. This implies that the temperature coef®cients ti(de) are independent of dmi. From this particular case, we see that t(de) and dm can be considered as independent variables in Eq. (3) if dm < 10 3 hp. 5. Conclusions The thermal behaviour is primarily determined by the deformations of the membrane, the insulating substrate and the ®xed plate. Secondly, it is in¯uenced by packaging and parasitic capacitances. In order to elucidate the in¯uence of the sensing cell dimensions on the sensor temperature coef®cient, it is appropriated to analyse the behaviour of isomorphous cells. This method applied to devices with a Pyrex substrate, a thin silicon membrane and a ®xed aluminium plate has
G. Blasquez et al. / Sensors and Actuators A 93 (2001) 44±47
con®rmed that the temperature coef®cient is a decreasing function of the bond width. In addition, it has revealed that the thermal variations due to structural deformations can be greatly ampli®ed by the thickness of the ®xed plate. Finally, it has shown that the thermal sensitivity is minimal as soon as the capacitance temperature coef®cient is equal to the temperature coef®cient of the ®xed plate. The considerations developed in this paper suggest that it is possible to fabricate sensors whose temperature coef®cient is lower than 44 ppm/8C. In order to obtain these very stable devices, it is necessary to realise the ®xed plate from ultra thin ®lms of metals which feature small thermal expansion coef®cients. In these ®lms, the temperature coef®cient tends probably to the substrate thermal expansion coef®cient (3.2 ppm/8C near room temperature). Nevertheless, their mechanical and corrosion resistances can be limited and reliability problems may affect sensors of this type. On a more general level, the greater part of what has been developed here can easily be applied or extended to numerous capacitive devices irrespective of their technology and/ or their function [3±9]. References [1] G. Blasquez, X. Chauffleur, P. Pons, C. Douziech, P. Favaro, P. Menini, Intrinsic thermal behaviour of pressure sensors, Sens. Actuators A85 (2000) 65±69. [2] W. Fang, C. Lo, On the thermal expansion coefficients of thin films, Sens. Actuators A84 (2000) 310±314.
47
[3] W. Alwang, Sensors for measuring pressure, Sensors, Vol. 7, VCH, Weinheim, 1994, pp. 513±576. [4] W.H. Ko, Capacitive sensors, Sensors, Vol. 7, VCH, Weinheim, 1994, pp. 105±144. [5] M. Esahi, Sensors for measuring acceleration, Sensors, Vol. 7, VCH, Weinheim, 1994, pp. 332±358. [6] R. Lerch, Sensors for measuring sound, Sensors, Vol. 7, VCH, Weinheim, 1994, pp. 577±576. [7] B. Kloek, M. Degrauwe, Approach to microsystem design, Sensors, Vol. 8, VCH, Weinheim, 1995, pp. 23±50. [8] D. Schaudel, Microsensors and actuators for industrial process applications: technology systems and their manufacture, Sensors, Vol. 1, VCH, Weinheim, 1996, pp. 173±196. [9] M.N. Horenstein, J.A. Perreault, T.G. Bifano, Differential capacitive position sensor for planar MEMS structures with vertical motion, Sens. Actuators A80 (2000) 53±61.
Biographies G. Blasquez received the ``Doctorat d'Etat'' degree in Physics from the Paul Sabatier University, Toulouse, France, in 1973. He has joined the CNRS in1966 where he is now ``Directeur de Recherches''. His research expertise lies in semiconductor devices including technology, design, modelling and characterisation. C. Douziech received the ``D.E.A.'' degree in Electrical Engineering from the Paul Sabatier University, Toulouse, France, in 1997. He is currently pursuing the ``Doctorat'' degree in the field of silicon sensors. In 2000, he joined the ALTIS Semiconductor Co., where he is working in the test of integrated circuits. P. Pons received the ``Doctorat'' degree in Electrical Engineering from the Paul Sabatier University, Toulouse, France, in 1990. He has joined the CNRS in1993 where he is now ``Charge de Recherches''. His research expertise lies in microtechnologies, silicon sensors and coplanar millimetric circuits.
Analysis characterisation and optimisation of temperature coef®cient parameters in capacitive pressure sensors G. Blasquez*, C. Douziech, P. Pons CNRS-LAAS, 7 Avenue Roche, 31077 Toulouse Cedex 4, France Received 7 December 2000; received in revised form 19 February 2001; accepted 3 March 2001
Abstract The literal expression of the temperature coef®cient is derived in the case of plane sensing cells. It states that the thermal behaviour of the capacitance depends on the cell dimensions and the temperature coef®cients of the ®xed plate and the internal cavity. Measurements carried out in a set of sensors, constituted of aluminium ®xed plates and thin silicon membranes bonded to Pyrex substrates, show that the amplitude of the capacitance thermal coef®cient is very sensitive to the bonded length and the ®xed plate thickness. Analyses of experimental results indicate that, very often, thermo-mechanical deformations of the internal cavity determine the amplitude of the capacitance temperature coef®cient. Nevertheless, they also demonstrate the feasibility of sensing cells in which the capacitance temperature coef®cient is very small and practically equal to the temperature coef®cient of the ®xed plate. In these cells, the fabrication of the ®xed plate from ultra thin ®lms and/or metals featuring small thermal expansion coef®cients should give capacitance temperature coef®cient lower than 44 ppm/8C. Finally it is shown that analyses of isomorphous sensors allow to separate the contributions of the ®xed plate and the cell structure and in addition to withdraw the effects of packaging and parasitic capacitors. # 2001 Elsevier Science B.V. All rights reserved. Keywords: Capacitive sensors; Temperature sensitivity; Temperature coef®cient; Structural parameters; Silicon membrane
1. Introduction In a recent paper [1], it has been shown that temperature causes nanometric deformations of the structure of pressure sensors fabricated by bonding a silicon membrane to a Pyrex substrate. The numerical evaluation of capacitance variations induced by these reversible deformations yielded amplitudes in the order of some hundreds of ppm/8C for thick membranes. Moreover, it has been proved that the bond width and the substrate thickness determine largely the thermal behaviour. In the following, the behaviour of sensors with a thinner membrane and therefore, a higher sensitivity to pressure, is investigated. This study relies on the analysis of the behaviour of a family of isomorphous sensors and on the general expressions that link the structural parameters to the capacitance and to the temperature coef®cient. It highlights the role played by the ®xed plate and con®rms the importance of the bond. Finally, it de®nes the minimum of the temperature coef®cient. * Corresponding author. Tel.: 33-561-336388; fax: 33-561-336208. E-mail address: [email protected] (G. Blasquez).
From a methodological point of view, the combined use of a case study and analytical calculations leads to a simple description of the respective roles of the sensor's structural parameters. Likewise, it allows to assess their interactions. It leads to conclusions that can be applied to sensors that are structurally and technologically different from those considered here. 2. Fundamentals A simple model of a capacitive pressure sensor is shown in Fig. 1. It consists of: (a) a rigid and insulating substrate of thickness hp; (b) a conductive membrane of thickness hs bonded to the substrate and (c) a rigid plate deposited inside a cavity of depth de. In the following, the bond width is denoted ab. At rest, the membrane and the ®xed plate form a planar capacitor whose capacitance is equal to C. If fringe electric ®elds remain negligible, the relation between the design parameters and C can be written as C e0
0924-4247/01/$ ± see front matter # 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 4 - 4 2 4 7 ( 0 1 ) 0 0 6 2 6 - 4
A de
dm
(1)
G. Blasquez et al. / Sensors and Actuators A 93 (2001) 44±47
45
Fig. 1. Basic structure and dimensions of a sensing cell.
where dm and A are, respectively the thickness and the area of the fixed plate. Temperature T modi®es all the structural parameters and, therefore, C. Over a large temperature range, the thermal behaviour of C can only be accurately characterised by its relative partial derivative t C
1 dC C dT
(2)
The operator t() is sometimes called the differential or incremental temperature coef®cient. For simplicity, t(C) will be called the temperature coef®cient of C. The analytical calculation of t(C) from Eqs. (1) and (2) leads to the following formula: t C t A
de
dm t dm dm
de
de t de dm
(3)
where t(A), t(dm) and t(de) denote, respectively the differential temperature coefficients of A, dm and de. Variations of A and dm are due to the ®xed plate deformations, de variations are mainly caused by deformations of the membrane and the insulating substrate [1].
Fig. 2. Examples of temperature coefficient features: (~) S1; (*) S2. Full lines represent the best fitting lines given by the least square method. Table 2 Capacitance temperature coefficients given by the least square method
K0i (ppm/8C) K1i (ppm/8C)
t1
t2
t3
t4
227 0.12
483 2.67
44 0.03
174 1.30
Consequently their stray capacitances due to fringe ®elds are almost negligible. Fig. 2 illustrates the behaviour of their temperature coef®cients. Data can be ®tted by a straight line of the form ti C k0i k1i T
(4)
where k0i and k1i are the model coefficients specific to the sensor Si. The estimates of k0i and k1i of all the considered sensors are given in Table 2. They were computed from measurements carried out in the temperature range ( 308C; 1508C), the least square method and Eqs. (3) and (4). For simplicity, the temperature coefficients ti(C) are denoted ti with i 1 to 4 in this table.
3. Experiments The group of sensors considered in the following sections has a 7740 Pyrex substrate, a silicon membrane and a ®xed aluminium plate (A 6:15 mm2). Their pertinent dimensions are given in Table 1. Morphologically, these sensors resemble to those described in [1]. The main differences lie in the membrane thickness (50 mm instead of 300 mm), metalisation (aluminium replacing a Ti/Au bilayer) and the bonding temperature (4508C instead of 4008C). To minimise electric fringing ®elds these sensors have in addition a guard electrode around the ®xed plate. Table 1 Sensor dimensions Si
hp (mm)
hs (mm)
de (mm)
dmi (mm)
abi (mm)
S1 S2 S3 S4
1.5 1.5 1.5 1.5
50 50 50 50
1.8 1.8 1.8 1.8
1 1 0.1 0.1
1 0.25 1 0.25
4. Data analysis 4.1. Estimation of t(A) In order to perform a preliminary analysis of the experimental results, it is easier to estimate the order of magnitude of t(dm) and t(A). In principle, these temperature coef®cients depend on the thermal and mechanical deformations of the ®xed plate and the substrate. In [2] it has been shown that the temperature coef®cient of 1 mm thick aluminium ®lms, deposited on silica and enduring substantial mechanical deformations, is close to the thermal expansion coef®cient aAl of bulk aluminium (aAl 25 ppm/8C near room temperature). In pressure sensors, the mechanical deformations of the ®xed plate and the Pyrex substrate are signi®cantly lower than in the preceding case. Consequently, we infer that the behaviour of the ®xed plate is mainly determined by thermal deformations and therefore that t dm aAl and t A 2aAl represent reasonable estimates of t(dm) and t(A).
46
G. Blasquez et al. / Sensors and Actuators A 93 (2001) 44±47
4.2. t(C) minimum From the estimates proposed in the preceding Section 4.1, the analysis of the data in Table 2 yields: t de @ t A in S1, S2 and S4 and t A @ t de in S3. In other terms, thermal variations are mainly due to thermo-mechanical deformations of the membrane and the substrate in S1, S2, and S4 and to the deformation of the ®xed plate in S3. Let us note that if the thermo-mechanical deformations of the membrane and the substrate are negligible for any temperature and if dm @ de, then t C t A:
(5)
This reduced expression represents the minimum of the capacitance temperature coef®cient. Data in Table 2 and the considerations in Section 4.1 indicate that this limit has been approached in the sensor S3. 4.3. Critical dimensions The accurate evaluation of the in¯uence of the sensor dimensions on the thermal behaviour requires to perform comparative analyses of isomorphous sensors. In the following, we develop this technique in order to gain a better understanding of the thermal behaviour and to exhibit some features of t(de) which is the most in¯uencing factor in many sensors. If t(de) predominates, and if dm varies while ab remains constant, it results from Eqs. (1) and (3) that the ratios of: capacitances, k0i, and k1i must be very similar. From the data relative to the couples (S2, S4) and (S1, S3) it can easily be shown that C2 k02 k12 2:1; 2:8; 2:1 (6) C4 k04 k14 C1 k01 k11 2:1; 5:2; 4:0 (7) C3 k03 k13 These evaluations corroborate the preliminary results derived in Section 4.2: t de @ t A in S1, S2 and S4 and t A @ t de in S3. Let us note that these deductions are independent of the assumptions done in Section 4.1. In addition, if t(de) predominates, and if dm varies, Eq. (3) speci®es that the thermal variations due to the thermomechanical deformations of the membrane and the substrate are greatly ampli®ed as soon as dm has the same order of magnitude that de. Moreover, if dm remains constant, and if ab varies, Table 2 shows that the temperature coef®cients can change drastically. Numerically, the comparisons of the data relative to the couples (S1, S2) and (S3, S4) give C2 k02 k12 1; 2:1; 22:2 (8) C1 k01 k11 C4 k04 k14 1; 4:0; 43:3 (9) C3 k03 k13
As expected from Eq. (1), ab variations have no in¯uence on C. However, the effects of ab variations on the temperature coef®cients are impressive. The underlying mechanisms have been analysed in [1] by simulation of the particular case of a sensor having a thick membrane. The data presented here extend the validity of the conclusions derived in [1] to the case of sensors realised with thin membranes. 4.4. Packaged sensors To re®ne the preceding considerations, it should be taken into consideration that the sensor sensing cells are always housed in a metal or plastic case. The latter's thermal deformations can in¯uence the intrinsic features of the cells. Also, interconnects between cases and sensing cells induce parasitic capacitances that are temperature dependent. Nevertheless, the difference of the temperature coef®cients of the packaged sensors in which capacitances and the packagings are identical, does not depend on the case and interconnects effects. In the following, we apply this re®ned analysis to the couples (S1, S2) and (S3, S4) in order to specify if t(de) depends on dm. To begin with, we note that the two ®rst terms of Eq. (3) are identical in (S1, S2) and (S3, S4). Thus, theoretically the differences t2 t1 and t4 t3 are only a function of the t(de) terms. The literal expression of the difference ratio can be written as t2 t4
t1 de t3 de
dm4 t2 de dm2 t4 de
t1 de t3 de
(10)
Evaluation of Eq. (10) from the data given in Tables 1 and 2 leads to t2 t4
t1 256 t3 130
2:55T 1:27T
(11)
As a ®rst approximation, Eq. (11) is equal to 2 for any T. Computation of the ratio de dm4 = de dm2 yields 2.1. Consequently, the differences t2 de t1 de and t4 de t3 de are almost equal. This implies that the temperature coef®cients ti(de) are independent of dmi. From this particular case, we see that t(de) and dm can be considered as independent variables in Eq. (3) if dm < 10 3 hp. 5. Conclusions The thermal behaviour is primarily determined by the deformations of the membrane, the insulating substrate and the ®xed plate. Secondly, it is in¯uenced by packaging and parasitic capacitances. In order to elucidate the in¯uence of the sensing cell dimensions on the sensor temperature coef®cient, it is appropriated to analyse the behaviour of isomorphous cells. This method applied to devices with a Pyrex substrate, a thin silicon membrane and a ®xed aluminium plate has
G. Blasquez et al. / Sensors and Actuators A 93 (2001) 44±47
con®rmed that the temperature coef®cient is a decreasing function of the bond width. In addition, it has revealed that the thermal variations due to structural deformations can be greatly ampli®ed by the thickness of the ®xed plate. Finally, it has shown that the thermal sensitivity is minimal as soon as the capacitance temperature coef®cient is equal to the temperature coef®cient of the ®xed plate. The considerations developed in this paper suggest that it is possible to fabricate sensors whose temperature coef®cient is lower than 44 ppm/8C. In order to obtain these very stable devices, it is necessary to realise the ®xed plate from ultra thin ®lms of metals which feature small thermal expansion coef®cients. In these ®lms, the temperature coef®cient tends probably to the substrate thermal expansion coef®cient (3.2 ppm/8C near room temperature). Nevertheless, their mechanical and corrosion resistances can be limited and reliability problems may affect sensors of this type. On a more general level, the greater part of what has been developed here can easily be applied or extended to numerous capacitive devices irrespective of their technology and/ or their function [3±9]. References [1] G. Blasquez, X. Chauffleur, P. Pons, C. Douziech, P. Favaro, P. Menini, Intrinsic thermal behaviour of pressure sensors, Sens. Actuators A85 (2000) 65±69. [2] W. Fang, C. Lo, On the thermal expansion coefficients of thin films, Sens. Actuators A84 (2000) 310±314.
47
[3] W. Alwang, Sensors for measuring pressure, Sensors, Vol. 7, VCH, Weinheim, 1994, pp. 513±576. [4] W.H. Ko, Capacitive sensors, Sensors, Vol. 7, VCH, Weinheim, 1994, pp. 105±144. [5] M. Esahi, Sensors for measuring acceleration, Sensors, Vol. 7, VCH, Weinheim, 1994, pp. 332±358. [6] R. Lerch, Sensors for measuring sound, Sensors, Vol. 7, VCH, Weinheim, 1994, pp. 577±576. [7] B. Kloek, M. Degrauwe, Approach to microsystem design, Sensors, Vol. 8, VCH, Weinheim, 1995, pp. 23±50. [8] D. Schaudel, Microsensors and actuators for industrial process applications: technology systems and their manufacture, Sensors, Vol. 1, VCH, Weinheim, 1996, pp. 173±196. [9] M.N. Horenstein, J.A. Perreault, T.G. Bifano, Differential capacitive position sensor for planar MEMS structures with vertical motion, Sens. Actuators A80 (2000) 53±61.
Biographies G. Blasquez received the ``Doctorat d'Etat'' degree in Physics from the Paul Sabatier University, Toulouse, France, in 1973. He has joined the CNRS in1966 where he is now ``Directeur de Recherches''. His research expertise lies in semiconductor devices including technology, design, modelling and characterisation. C. Douziech received the ``D.E.A.'' degree in Electrical Engineering from the Paul Sabatier University, Toulouse, France, in 1997. He is currently pursuing the ``Doctorat'' degree in the field of silicon sensors. In 2000, he joined the ALTIS Semiconductor Co., where he is working in the test of integrated circuits. P. Pons received the ``Doctorat'' degree in Electrical Engineering from the Paul Sabatier University, Toulouse, France, in 1990. He has joined the CNRS in1993 where he is now ``Charge de Recherches''. His research expertise lies in microtechnologies, silicon sensors and coplanar millimetric circuits.