On the relationship between the temperature coefficient of resistance and the thermal conductance of integrated metal resistors

Sensors and Actuators A 116 (2004) 137–144

On the relationship between the temperature coefficient of resistance and the thermal conductance of integrated metal resistors A. Scorzoni, M. Baroncini, P. Placidi∗ DIEI, Faculty of Engineering, Via G. Duranti 93, 06125 Perugia, Italy Received 29 October 2003; received in revised form 5 April 2004; accepted 5 April 2004 Available online 18 May 2004

Abstract The concepts of temperature coefficient of resistance (TCR) and thermal conductance (Gth ) entail devices with uniform temperature. However, Joule heated integrated metal resistors usually feature a non-constant temperature profile. After defining an effective TCR and an effective Gth , this paper describes simple relationships able to correlate these two parameters with measured quantities. These relationships are applied to the case of the heating element of a micromachined gas sensor and are exploited to derive the effective TCR and Gth of the same element when a passivation layer is added on top of it. The information presented in this paper could also provide useful rules of thumb for the verification of finite element modeling simulations. © 2004 Elsevier B.V. All rights reserved. Keywords: Integrated metal resistor; Microheater; Microsensor

1. Introduction It is well known that the resistance versus temperature relationship extracted from a calibration of a metal resistor in a uniform temperature environment can be described by a temperature dependent first-order function, i.e. R(T) = Ra [1 + TCRa (T − Ta )]

(1)

where R and Ra are the resistance values at temperatures T and Ta , respectively, Ta the ambient temperature and TCRa the temperature coefficient of resistance at Ta . Eq. (1) is often used in “reverse mode” in order to extract the resistor temperature from a simple resistance measurement [1–4]. In principle, this procedure is correct only provided that the resistor temperature is constant throughout the whole resistor length, i.e. when Joule heating is negligible. In a number of applications, however, Eq. (1) is exploited to derive the resistor temperature as caused by Joule heating. If this is the case, the temperature along the heating resistor is not uniform and significant thermal gradients are usually located at the end segments of the resistor, close to the bonding pads [5,6]. Fig. 1 shows two typical microheater layouts and a schematic temperature profile along the line. This is ∗ Corresponding author. Tel.: +39-075-585-3636; fax: +39-075-585-3654. E-mail address: [email protected] (P. Placidi).

0924-4247/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.sna.2004.04.003

confirmed both by analytical models [7,8] and by numerical simulations [9]. The common conclusion of analytical and numerical models is that between the two external regions where thermal gradients are located, a finite segment exists where the line temperature can be treated as a constant, herewith called TAA (“active area” temperature). It should be emphasized also that in the case of complex three-dimensional structures like the double spiral resistor in Fig. 1(a), especially if drawn on thermally insulating membranes, the absence of depressions—therefore of flex points—in the central region of the temperature curve can be assured only by means of a suitable design procedure. Therefore, using the previously extracted values of Ra and TCRa , the estimated temperature value will likely provide a sort of average of the temperature profile along the heating element length instead of the actual active area temperature and could cause errors in the extracted temperature. Errors as high as 45 ◦ C have been reported in [6]. As a consequence, the “reverse” extraction procedure could reasonably be employed only when the constant temperature region of the resistor accounts for the majority of the resistive region. This is usually true for straight line resistors with wide end segments (as shown in Fig. 1(a)) [5] but it is not appropriate for spiral resistors (Fig. 1(b)) typically used as sensor microheaters. A second relationship often employed when dealing with thermal properties of heated devices can be expressed as

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Fig. 1. (a) Two typical metal resistor layouts and (b) schematic temperature profile along the resistor. x is the longitudinal coordinate parallel to the line axis (x is bent at line corners) and L is the total length of the line between the two contacting pads.

follows: P = Gth,a (T)(T − Ta )

(2)

where the subscript “a” refers to ambient temperature, P (W) the dissipated power in the resistor, T the “hot” temperature of the device (supposed constant) and the thermal conductance Gth,a (T) (W/◦ C) is a “global” thermal loss coefficient which accounts for the conduction, convection and radiation heat transfer mechanisms. In general, Gth,a is a function of both the “hot” temperature T and the reference temperature Ta . Again, Eq. (2) is useful only when the device temperature is constant, which is not the case with most integrated resistors. This paper shows the feasibility of a generalization of Eqs. (1) and (2) to the case in which a constant temperature, middle region of finite length can be found in the resistor, even if the temperature throughout the whole resistor is not constant. The proposed theory is applied to a test structure whose layout and technological process is briefly illustrated in Section 2, together with the adopted measurement methods. Section 3 defines the effective TCR and shows a model able to justify the constancy of the effective TCR, while Section 4 defines the effective thermal conductance. Useful relationships between effective TCR and effective thermal conductance are derived in Section 5. Finally, Section 6 shows an example on how the proposed theory could be exploited to derive the effective thermal parameters of the microheater described in Section 2 when an additional passivation layer is introduced on top of it.

2. Test structure and measurement methods A 230 nm thick platinum metal layer was sputtered on the top of a 250 nm thin Si3 N4 membrane and photolithography defined to create a four-point probe measurement spiral resistor [6]. The dimension of the membrane realized with the bulk-Si micromachining technique, was 1.0 mm × 1.0 mm. Fig. 2 shows the layout of the adopted structure, whose inner region features dedicated voltage measuring lines terminated with contact pads (called “VT” to resemble voltage taps) and symmetrically connected to an internal winding of

Fig. 2. Schematic structure of the resistor. The voltage pads “2w” and “VT” are shown. The active area features an approximately constant temperature TAA .

the double spiral, on the edge of the “active area” (of about 0.5 mm × 0.5 mm). Through the voltage taps it is possible to measure the dissipated power PVT in the inner region. The resistor did not feature a passivation layer. The variation of the microheater resistance, as a function of applied power, was investigated by exploiting two types of dc measurements: (i) the first one implements a standard two-wire method using the “2w” pads (Fig. 2) for both voltage sense and forcing current using two separate probe pairs; (ii) the second one uses the “2w” pads to supply the test current and the “VT” pads to sense the voltage (VT measurements). Thermal simulations through SOLIDIS-ISE-TCAD helped in designing the actual geometry of the spiral resistor structure in such a way that the inner region between the two “VT” taps, features a constant “active area” temperature TAA with a maximum error TAA = ±7 ◦ C. The I–V measurements were made in air (Ta = 25.5 ◦ C) at the wafer level on a statistically relevant number of microheaters, placed in different areas of a wafer. Details on the measurements are reported elsewhere [6].

3. The effective temperature coefficient of resistance 3.1. Extraction of the parameters of the active area Heat losses in integrated resistors are caused by conduction, convection and radiation heat transfer mechanisms. The relative importance of these mechanisms depends on the temperature range and on mechanical force-induced fluid movement. In [10] a non-linear dependence was shown to hold between these power losses and the microheater active area temperature TAA (also called “peak temperature”). A “global” thermal loss coefficient of the heater Gth,VT (in W/◦ C) was defined using the following analytical form: TAA − Ta Gth,VT = Gth,VTa + B + A(TAA − Ta ) (3) TAA − Ta + Tt where Gth,VTa (in W/◦ C), A (in W/(◦ C)2 ), B (W/◦ C) and Tt (K) are suitable constants. Eq. (3) is composed of (i) a con-

A. Scorzoni et al. / Sensors and Actuators A 116 (2004) 137–144

PVT = Gth,VT (TAA − Ta )

(4)

Since, the active area features an approximately constant temperature equal to TAA , it is legitimate to use the typical temperature dependent relationship for the resistance of metals: RVT = RVTa (1 + TCRa (TAA − Ta ))

200 150

100

0

100

200 T AA (˚C)

300

400

Fig. 3. “2w” resistance vs. active area temperature for eight different test structures, located randomly in a 4 in. wafer. Symbols are calculated values obtained following the procedure described in [6] while solid lines are linear fittings.

(5)

where RVT and RVTa are the “VT resistances” at temperatures TAA and Ta , respectively, while TCRa is the temperature coefficient of resistance at temperature Ta . If the temperature Ta is known, the system composed by Eqs. (4) and (5), written for a number of different (RVT , PVT ) couples, can be solved iteratively for each sample under test and the thermal model parameters Gth,a , A, B, Tt and TCRa can be extracted, together with the active area temperature corresponding to each (RVT , PVT ) couple. This was actually done in [6] where it was possible to derive the active area temperature TAA of the structures as a function of applied current (and “VT” power), as well as the dependence of the Gth,VT on active area temperature. The extracted TCRa was (3.26 ± 0.03) × 10−3 ◦ C−1 at Ta = 25.5 ◦ C. 3.2. Definition of the effective TCR As a by-product of the previously described extraction procedure we can now plot the resistance R measured at the “2w” terminals as a function of the temperature TAA established at the “VT” terminals. From Fig. 3, we observe that the R behavior versus active area temperature is still perfectly linear (the minimum value of the correlation coefficient was 0.999993), even if the resistance is measured at the outer terminals, while the constant temperature region is located between the “VT” pads. As a consequence, we can still write a linear relationship: R = Ra [1 + TCReff,a (TAA − Ta )]

250

R (Ω )

stant term Gth, VTa accounting for the thermal conductance when TAA = Ta , (ii) a step-wise term describing the onset of convection in air [11] and (iii) a linear term. We note that in [6] the same functional relationship was adopted, but with Tt = 0 and merging Gth, VTa and B in only one parameter. In the sequel of this paper, we will realize that this assumption is valid when the considered temperature is greater than about 100 ◦ C. Therefore, the total thermal power PVT dissipated in the active area was expressed as [6]:

139

(6)

where Ra is the value of R at Ta , and TCReff ,a is an effective TCR which is assumed to be constant. Reminding the comments previously written on Eq. (1) and since the average temperature of the measured R is smaller than TAA , we conclude that TCReff,a < TCRa . By elaborating the data of Fig. 3, we extracted a value of TCReff,a = (2.80 ± 0.06) × 10−3 ◦ C−1 at Ta = 25.5 ◦ C. Please note that this finding is not related with what described in the introduction, i.e. the customary use of a linear

calibration made in a uniform temperature environment and consequent use of this calibration to extract the temperature of the resistor. While the latter method allows one to extract the actual TCRa of the metal line (but, when Joule heating is present, the extracted temperature is affected by a significant error, as reported above), once the TCReff ,a is found, the actual active area temperature of the metal line can be safely extracted. 3.3. An analytical validation for the constancy of the effective TCR With a simple analytical model, we can now justify the constancy of the effective TCR for a typical microheater geometry. Let us suppose that: (i) the temperature of the line in a given cross-section is constant; (ii) current crowding along the line is negligible. We note the latter assumption is not valid at the line corners and where abrupt changes in line width occur. In our spiral resistor bevelled corners with 45◦ angles have been used in order to minimize current crowding. Moreover, line width changes are restricted again to line corners, thus minimizing the locations where current crowding occurs. Finally, tapered cross-sectional changes have been considered at line ends and this further helps in being compliant with the assumption. Referring to Fig. 1, considering x as a longitudinal coordinate parallel to the line axis (x is obviously bent at line corners), the total resistance of the line can be written as:
L/2
L/2 ρ(T(x)) ρa [1 + TCRa T(x)] R= dx = 2 dx A(x) A(x) −L/2 0 (7) where T(x) = T(x)−Ta and we have further supposed that: (i) the line geometry is symmetrical with respect to x = 0; (ii) the cross-sectional area A and the temperature T change as a function of x; (iii) the local resistivity ρ(x) features the same temperature dependence as the whole line kept at constant temperature. The latter assumption is equivalent to ne-

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glect the local perturbation to the metal resistivity caused by grain boundaries. Since the thickness tm of the metal is constant, the cross-sectional area is defined as A(x) = tm W(x), where W(x) is the variable width of the line. Therefore   R = Ra 1 + TCRa T 1/W (8) where

Ra = 2

ρa tm




0

(T(x)/W(x)) dx  L/2 0 (1/W(x)) dx

L/2 0

1 0.8

0.5 V

0.6

1V

0.4

 L/2

T 1/W =

1.2

g (x )

140

dx W(x)

(9)

being T 1/W the weighted average of the incremental temperature with weight (1/W). The experimentally verified Eq. (6) should assume the same value of Eq. (8). By equating the two relationships, we obtain TCReff,a (TAA − Ta ) = TCRa T 1/W

1.5 V

0.2

and

(10)

0 0

2V 10

20 # of macrocell

30

40

Fig. 4. The function g(x) of Eq. (12) along half of the platinum microheater as a function of the macrocell number in the SOLIDIS-ISE mesh. The macrocells are the basic building blocks of the computational grid. The platinum resistivity was considered as a linear function of the temperature with TCRa = 3.26 × 10−3 ◦ C−1 at Ta = 25.5 ◦ C. The two minima in the final wavy behavior are located in the two outer corners of the spiral. The active area starts from around macrocell #30.

then χ=

T 1/W TCReff,a = , TCRa TAA − Ta

χ ≤ 1, χ = constant

(11)

Constancy of Eq. (11) represents a condition on the weighted average incremental temperature profile T 1/W normalized to the maximum temperature increment (TAA − Ta ). In order for χ (and TCReff ,a ) to be a constant for fixed microheater geometry and for every heating current introduced in the line, the quantity T 1/W must be proportional to (TAA − Ta ) for every active area temperature TAA . This is true, e.g. if we suppose that the shape of the temperature outline along the resistor could be considered as invariant when the active area temperature TAA varies. In other words, the function T(x) − Ta g(x) = (12) TAA − Ta

[5] for a straight line sitting on an oxidized silicon substrate. The solution was given in the quite general case of a symmetrical resistor spanning x∈[−L/2, L/2] and featuring two different line widths (width = w for x∈[−l/2, l/2] and W = aw (where a ≥ 1) for x∈[−L/2, −l/2] and x∈[l/2, L/2]). For x > 0, in each ith (i = 1, 2) section the temperature increment T must satisfy the following energy balance equation: d2 (T) − λ2i T + Si = 0 dx2

(13)

where λ2 (in m−1 ) and Si (in ◦ C/m2 ) are suitable constants related to the thermal and geometrical parameters of the two different sections [5]. Extracting T from Eq. (13) and substituting in Eq. (9), the weighted average temperature increment T 1/W can be calculated as follows:

 T 1/W =

S1 λ21

(l/2) + (1/a)(λ21 /S1 )(S2 /λ22 )((L − l)/2) + (λ21 /S1 )((1/λ21 ) − (1/aλ22 ))( (d(T))/dx)|l/2 + (1/a)(λ21 /S1 )(1/λ22 )((d(T))/dx)L/2 (l/2) + (1/a)((L − l)/2)

should be invariant as a function of the active area temperature. This is actually the case with our structure for typical active area temperatures. Fig. 4 shows four different curves g(x) numerically calculated using SOLIDIS-ISE-TCAD for different values of the applied voltage (i.e. for different values of the active area temperature TAA ). Clearly, the four curves are superimposed and this supports our experimental results. 3.4. The Schafft analytical model We want now to demonstrate that Eq. (11) is also verified in the case of a well known analytical solution proposed in

(14)

It is easy to demonstrate that for a fixed geometry and typical thermal and electrical parameters (metal and oxide thermal conductivity and thickness, metal width, TCR, resistivity and heating current) the fraction in Eq. (14) is approximately constant, while S1 /λ12 is approximately the active area increment temperature at the center of the line. As an example, Fig. 5 shows the value of χ as a function of current density for two different pairs of metal width (w) and oxide thickness (tox ). The value of the latter parameter has been increased up to 10 ␮m in order to model a really bad thermal conduction between the metal line and the silicon substrate, thus approaching the case of a metal line sitting on a silicon nitride membrane and exchanging heat with

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Similarly to the case of the effective TCR, it is also possible to define an effective thermal conductance referred to the “2w” terminals and expressed as: P TAA − Ta

1.6E-04 1.3E-04 1.0E-04 7.0E-05 0

4. The effective thermal conductance

Gth =

1.9E-04

Gth (W/˚C)

the surrounding ambient. In Fig. 5 the other parameters are kept constant and similar to those of our process (see figure caption). Clearly, χ can be considered as a constant within 5% of its value. Therefore, even in the case of huge current densities through the line and bad thermal conduction with the surrounding ambient, the analytical solution from Schafft yields a weighted average temperature of the line approximately proportional to the active area temperature.

(15)

This new parameter refers to the total power P dissipated on the microheater at the “2w” terminals. However, the temperature, we are concerned about is again the active area temperature, located between the VT taps. The effective Gth is typically dependent on TAA − Ta . In fact in the general case P is not linearly dependent on TAA − Ta , therefore, the dependence of the numerator and the denominator of Eq. (15) on TAA − Ta do not cancel out. In the sequel, we will suppose that, on a first approximation, Gth is not dependent on the ambient temperature which is referred to, provided that this temperature is not too far from typical values (25 ◦ C). For this reason the symbol for Gth does not include a subscript “a”. Fig. 6 shows the behavior of Gth as a function of the active area temperature, calculated using the previously described extraction procedure (see Section 3.1).

Fig. 5. Percent values of χ as a function of current density for different values of the metal width (w) and oxide thickness (tox ). Maximum temperature increments are about 620 ◦ C at maximum current density for both curves. Other parameters are: metal thermal conductivity 0.716 W (cm ◦ C−1 ), oxide thermal conductivity 9.6 mW (cm ◦ C−1 ), ρa = 10 ␮" cm, TCRa = 3.3 × 10−3 ◦ C−1 , metal thickness tm = 230 nm, W/w = 2, L = 1200 ␮m, l = 800 ␮m.

141

100

200 T AA (˚C)

300

400

Fig. 6. “2w” effective thermal conductance Gth vs. active area temperature for the four terminal test structure.

The effective Gth features the expected initial steep increase due to the onset of convection in air [11]. Then, for temperatures greater than about 100 ◦ C, the behavior changes to an approximately linear trend whose average, as a function of TAA − Ta , can be expressed as Gth (W/◦ C) = m(TAA − Ta ) + q, with m = 1.7579 × 10−7 W/(◦ C)2 and q = 1.1493 × 10−4 W/◦ C. This is the typical range where the operating temperature of microheaters can be found.

5. The relationship between the effective TCR and the effective thermal conductance The effective TCR takes into account that, due to heat dissipation in the surroundings, the active area temperature cannot be kept constant throughout the whole metal line. On the other hand, the parameter Gth is a practical measurement of this tendency to heat dissipation. It is therefore expected that these two quantities are tied by a mathematical relationship. To our knowledge, a general derivation of such a mathematical relationship is practically impossible, being dependent on a number of thermal parameters and on the particular geometry of the microheater. However, from Eqs. (6) and (15) it is evident that, given one of the two quantities (TCReff ,a or Gth ), the incremental temperature TAA − Ta is easily extracted for each measured (R, P) pair and the other quantity (Gth or TCReff ,a ) can be calculated. This action is straightforward starting from the knowledge of TCReff ,a . However, being Gth dependent on TAA − Ta , the extraction of the latter quantity is based on an implicit relationship (i.e. Eq. (15) again), which, in general, should be solved numerically. Moreover, by substituting for TAA − Ta from Eq. (15) to Eq. (6) we obtain   TCReff,a R = 1+ P (16) Ra Gth where R, Gth and P are all implicitly dependent on TAA −Ta . In Eq. (16), which relates the measured quantities R and P, the parameters TCReff ,a and Gth exhibit as a single pa-

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rameter. This confirms the close relationship between them. It should be noted that Eq. (16) does not express a linear relationship between R and P since in our experiment the parameter Gth is temperature dependent.

The proposed theory was applied to the characterization of platinum resistors nominally identical to those described in Section 2 and reported in Fig. 7, but with an added double passivation layer of spin-on-glass and low temperature oxide, with total thickness 1.255 ␮m. Passivation was opened by photolithography only on the “2w” pads, thus creating the typical layout of a sensor’s microheater. The process was completed by depositing a further platinum layer which was photolithography defined to create two contacts for a chemoresistive layer, which was eventually deposited through a metal mask. Finally, the microheaters were mounted on a TO-8 metal package. Since “VT” pads were not available in this structure (being covered by the passivation and by the other upper layers), I–V measurements were performed at an ambient temperature of 23 ◦ C by exploiting the “2w” pads only. For this reason, it was not possible to derive the active area temperature following the procedure of [6]. Therefore, a different method for extracting TCReff ,a and Gth had to be found. In the sequel the two different devices described in Section 2 and in this section will be referred to as the “experiment #1” and “experiment #2”, respectively. Fig. 8 displays the average “2w” measurements of the two experiments shown in the form of Eq. (16), i.e. of a (R, P) plot. From Fig. 8, we observe a larger heat dissipation of experiment #2 for equal incremental resistance. This is expected, since the microheaters share the same geometry and the increased passivation thickness boosts the heat losses by conduction through the membrane. While in experiment #1 all variables were well known, VT measurements were not available in experiment #2, therefore, it was not possible to perform a direct extraction of all parameters. However, the main differences between the two experiments are passivation thickness and the presence

#1

1.75 R/Ra

6. Extraction of the thermal parameters of a passivated test structure

2

#2

1.5 1.25 1

0

0.02

0.04 P (W)

0.06

Fig. 8. R/Ra , P plot extracted from “2w” I–V measurements of the two experiments.

of a package. Fig. 9 shows a comparison of the dc lumped element thermal models of the two experiments. In experiment #2, while the thermal losses by conduction are increased by the passivation (thus increasing Gth through an amount G0 ), the package introduces an additional term to the thermal resistance, thus decreasing Gth . The thermal resistance of the TO-8 package Gth,pack , as derived from the data sheet, is 16.7 mW/◦ C as a worst case, i.e. two orders of magnitude greater than the Gth of Fig. 6. Since these Gth are in series, supposing that G0 does not increase the order of magnitude of Gth (as demonstrated at the end of the present section), we will short-circuit Gth,pack . Moreover, we could consider G0 as approximately temperature independent, since it is mainly due to conduction through the additional passivation layer. Hence, we will treat experiment #2 by considering an additional constant term G0 to the Gth1 of experiment #1. With obvious notations: Gth2 = Gth1 + G0

(17)

In addition, considering only the linear portion of Gth1 at a sufficiently high temperature (Gth1 = m(TAA − Ta ) + q), we write Gth2 = m(TAA − Ta ) + q + G0

(18)

where all symbols and relative values have previously been defined.

• TAA

TAA

Gth

G0 •

Gth Ta

Gth,pack Ta

Fig. 7. Schematic cross-section of the test structures (a) unpackaged for wafer level measurements and (b) packaged with passivation.

0.08

experiment #1

experiment #2

Fig. 9. Static thermal models of the two experiments.

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From the discussion of Section 5, we therefore expect that the effective TCR is modified in experiment #2 with respect to #1. From Eqs. (15) and (18), we write (q + G0 )2 + 4mP2 − (q + G0 ) TAA − Ta = (19) 2m where P2 is the power P of experiment #2 and we have chosen the positive sign of the square root since TAA > Ta . It should be noted that the problem mentioned in Section 5 of the implicit dependence of TAA − Ta on the effective Gth has found here an analytical solution due to the linear dependence of the Gth on TAA − Ta within a limited temperature range. On the other hand, Eq. (6) states that TAA − Ta =

1 R2 − Ra,2 Ra,2 TCReff,a,2

(20)

Eqs. (19) and (20) are actually a system of equations (two equations for every resistance–power pairs) with unknown parameters G0 and TCReff ,a,2 . By subtracting Eq. (19) from Eq. (20) and minimizing the sum of the squares of these differences for every measured pair (R, P), we obtained optimal values of G0 = 1.52 × 10−5 W/◦ C and TCReff,a,2 = 2.66 × 10−3 ◦ C−1 . In particular, the parameter TCReff ,a,2 should be compared to the TCReff,a,1 = (2.80 ± 0.06) × 10−3 ◦ C−1 , thus showing that in our experiment a relative increase G0 /Gth1 of 8–12%, depending on the considered temperature range, corresponds to a 5% decrease of the effective TCR. Finally, the extracted value of G0 is one order of magnitude smaller than Gth1 and this supports the assumption, we made at the beginning of the present section. It should be finally underlined that the last computation was actually done by excluding the values of Gth1 corresponding to a temperature increment TAA − Ta below 100 ◦ C. This is due to two main reasons. The first, simple one is that typical operating temperatures of microheaters are greater than 100 ◦ C. The other motivation is that in the case of temperatures less than 100 ◦ C the extracted thermal conductance is affected by a non-negligible relative error, greater than 1%. This can be estimated by simple error propagation formulas [12] applied to Eqs. (15) and (5) and observing that the relative error on measured power P can be neglected being less than 1%. The worst-case error turns out to be Gth (TAA ) 2R/R (TCRa ) < + Gth (TAA ) 1 − (RVTa /RVT ) TCRa

(21)

where R/R is the relative error of the four-wire resistance measurement. When a small current is introduced in the microheater, the difference between RVT and RVTa is very small. For example, using 4 mA we found that the denominator of the first term on the RHS of Eq. (21) is about 0.05, therefore, the relative error 2R/R is multiplied by 20. Since at the same current, we estimated a relative error R/R ∼ = 0.5%, the first term on the RHS of Eq. (21) yields a relative error of 10%. Even neglecting the relative error on TCRa in Eq. (21), in order to keep the relative error on TAA − Ta

143

smaller than 1% we had to use a current greater than 10 mA, which corresponds to a relative temperature increment of about 100 ◦ C. 7. Conclusions In this paper, we discussed the relationship between the TCR and the Gth of Joule-heated integrated metal resistors. We recognized that a non-constant temperature profile could induce significant errors in the extraction of the active area temperature. In order to circumvent this problem, two parameters can be defined, namely an effective TCR and an effective Gth , able to relate the measured resistance and dissipated power to the active area temperature. These parameters can only be defined when a finite, inner segment exists where the resistor temperature can be treated as a constant. Then, we showed that the effective TCR is simply related to the effective Gth and we understood that one parameter could be derived once the other parameter and other measured quantities are known. The model was applied to the case of the heating element of a micromachined gas sensor and was exploited to derive the effective TCR and Gth of the same element when a passivation layer was added on top of it. Acknowledgements The authors would like to thank the staff of the silicon fabrication facility of CNR IMM Sezione di Bologna for careful samples preparation and for supplying the test structures. We are also indebted with Dr. G.C. Cardinali and Dr. I. De Munari for helpful discussions and suggestions. References [1] S.H.K. Fung, Z. Tang, P.C.H. Chan, J.K.O. Sin, P.W. Cheung, Thermal analysis and design of a micro-hotplate for integrated gas sensor applications, Sens. Actuators A 54 (1996) 482–487. [2] D. Lee, W. Chung, T. Kim, J. Baek, Low power micro-gas sensors, in: Proceedings of the Eurosensors, vol. IX, Stockholm, Sweden, 1995, pp. 827–830. [3] W. Chung, J. Lim, D. Lee, N. Miura, N. Yamazoe, Thermal and gas-sensing properties of planar-type micro gas sensor, Sens. Actuators B 64 (2000) 118–123. [4] A. Pike, J.W. Gardner, Thermal modeling and characterization of micropower chemoresistive silicon sensors, Sens. Actuators B 45 (1997) 19–26. [5] H.A. Schafft, Thermal analysis of electromigration test structures, IEEE Trans. Electron Dev. 34 (1987) 664–672. [6] M. Baroncini, P. Placidi, A. Scorzoni, G.C. Cardinali, L. Dori, S. Nicoletti, Accurate extraction of the temperature of the heating element in micromachined gas sensors, in: Proceedings of the 2001 IEEE International Symposium on Circuits and Systems (ISCAS), Sydney, Australia, 2001, pp. 445–448. [7] J.J. van Baar, R.J. Wiegerink, T.S.J. Lammerink, G.J.M. Krijnen, M.C. Elwenspoek, Sensitive thermal flow sensor based on a micro-machined two dimensional resistor array, in: Proceedings of the Transducers 2001, Munich, 10–14 June 2001, pp. 1436–1439.

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Biographies Andrea Scorzoni has received his doctoral degree in Electronics in 1989. Since 1983, he has been working at the CNR-IMM Institute (ex LAMEL).

In 1988, he became a Research Associate IMM. Since 1998, he is Professor of Electronics at the Faculty of Engineering of Perugia (Italy). His areas of interest include ohmic contact resistivity measurements and modeling for both VLSI applications and SiC-based integrated circuits, electromigration phenomena, reliability and degradation mechanisms of semiconductor devices, solid-state radiation sensors and gas sensors and, in general, electrical measurements and solid-state electron devices. Milena Baroncini has received the Electronic Engineering degree and the Electronic Engineering PhD degree from the University of Perugia, Italy, in 1995 and 2000, respectively. She is currently a contract researcher at the Department of Electronic and Information Engineering of the University of Perugia, Italy. Her research interests concern the thermal modeling and the design and realization of interface circuits for gas microsensors applications. Pisana Placidi has received the “laurea” degree cum laude and the PhD degree in Electronic Engineering from the University of Perugia, Italy, in 1994 and 2000, respectively. She is currently a contract researcher at Department of Electronic and Information Engineering of the University of Perugia, Italy. Her research interests are mainly connected with mixed IC design.