Modelling of integrated Peltier elements D.D.L. Wijngaards, E. Cretu, S.H. Kong and R.F. Wolffenbuttel Electronic Instrumentation Laboratory, Delft University of Technology / DIMES, Department of Electrical Engineering, Mekelweg 4, NL-2628 CD Delft, The Netherlands Phone: +31.15.278.1602, Fax: +31.15.278.5755, E-mail: d.d.l.wijngaards@its.tudelft.nl ABSTRACT On-chip integration of Peltier devices creates a number of additional parasitics, compared to conventional Peltier devices. The most important are thermal conduction through the supporting membrane and contact resistance. Analytical analysis shows that each parasitic decreases device performance, but only the contact resistance creates a significant shift in the optimum device current where the temperature reduction is maximised. To obtain an energetically correct lumped-element model, as presented in this paper, the electro-thermal Peltier effect must be linked to the thermo-electric Seebeck effect, which acts as a feedback parameter. The influence of this Seebeck voltage on a non-ideal current source, driving the Peltier device, is investigated and it is shown that the influence can be ignored if the current source has a reasonable impedance. Keywords: Peltier effect, on-chip cooling, thermoelectric refrigeration, thermal stabilisation, polySiGe 1 INTRODUCTION The conventional Peltier device finds use in a range of applications, from thermal stabilisation [1,2] to micro refrigeration [3]. The operation of such devices has long been studied and is well documented in literature. An on- chip integrated counterpart could also be used for various applications, like fully integrated dew-point sensors, thermally stabilised optical detectors and radiation detectors or thermally stabilised on-chip references. However, transformation of a conventional Peltier device, as schematically shown in Fig. 1a, into an integrated version, shown in Fig. 1b, is not straightforward. This paper addresses the consequences of this transformation, focusing on the practical behaviour of two major performance-limiting parameters, the parasitic thermal conduction through the supporting membrane and contact resistance. In the second part of this paper a new and simple lumped element model of a Peltier device is presented, which obeys the principle of conservation of energy. This model effectively links the temperature T c of the thermally stabilised region to the ambient temperature T a rather then to that of the heat sink temperature T s . In Fig. 1b, T s is the substrate temperature close to the Peltier element. 2 ANALYTICAL MODEL The behaviour of the conventional type of Peltier device, shown in Fig. 1a, is described by two simple equations, determining the heat transfer rate, q [W] and the maximum temperature reduction ∆T max = (T s – T c ) max : 2 ( ) ( ) 2 p n c s c IR q TI KT T α α = - - - - (1) ( ) ( ) 2 2 2 max 2 2 p n c c s c T zT T T KR α α - - = = (2) The heat transfer rate is directly dependent on the parameters T s , T c and the electrical current I applied. Furthermore the heat transfer depends on the total thermal conductance, K, and the electrical resistance of the element, R, which consist of multiple parameters and are defined as p p n n n p A A K L L λ λ = + and p p n n n p L L R A A ρ ρ = + (3) q I I n-type material p-type material metal Silicon substrate (at T s ) Si 3 N 4 membrane (at T c ) Al interconnect n-type material p-type material I I (a) (b) q Figure 1: Schematic drawings of (a) a discrete Peltier device and (b) an integrated Peltier device.