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 (G th ) 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 G th , 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 G th 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) = R a [1 + TCR a (T - T a )] (1) where R and R a are the resistance values at temperatures T and T a , respectively, T a the ambient temperature and TCR a the temperature coefficient of resistance at T a . Eq. (1) is often used in “reverse mode” in order to extract the resistor temperature from a simple resistance measure- ment [1–4]. In principle, this procedure is correct only pro- vided 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 bond- ing 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: placidi@diei.unipg.it (P. Placidi). 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, here- with called T AA (“active area” temperature). It should be em- phasized also that in the case of complex three-dimensional structures like the double spiral resistor in Fig. 1(a), espe- cially if drawn on thermally insulating membranes, the ab- sence of depressions—therefore of flex points—in the cen- tral region of the temperature curve can be assured only by means of a suitable design procedure. Therefore, using the previously extracted values of R a and TCR a , 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 conse- quence, 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 appropri- ate 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 0924-4247/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.sna.2004.04.003