Simultaneous estimation of heat transfer coefficient and thermal conductivity with application to microelectronic materials

Abstract

An estimation of unknown properties of materials arises naturally when one considers some aspects of thermal modeling, especially carried out by widely used numerical methods, e.g. Finite Element Method (FEM).

We propose a new approach of simultaneous thermal conductivity and heat transfer coefficient estimation based on thermographic measurements. A linear, steady-state distributed parameter model is used in order to describe the test sample. Thermal properties measurement is equivalent to the unknown parameter estimation of this system. The proposed method is practically applied for estimation of thermal conductivity and heat transfer coefficient of thick-film modules made on alumina (96% Al₂O₃) and DP951 ceramic substrates. In these experiments a high-resolution thermographic scanner is used.

The obtained results for thermal conductivity and heat transfer factor are fully comparable with previously published ones.

Introduction

Recently, the development of wide range of simulation tools for microelectronic applications has been observed. They support thermal design, testing and optimization of microelectronic systems such as: sensors, integrated microsystems, microdevices, etc. Performance and reliability can also be investigated using several simulation methods [1].

In most cases of thermal calculations, numerical methods are used because of the complexity of geometry and boundary condition of modeled structures. The widely exploited approach involves Finite Element Method (FEM). Many commercially available software packages are commonly used in order to address thermal problems in microelectronics.

In order to solve real cases of thermal problem every simulation tool (either analytical or numerical) needs a description of materials used for structure fabrication, including the thermal properties of those materials. The thermal conductivity which affects level of temperature in the considered microstructure, and has an influence on level of thermal stresses and finally on the reliability of devices [2], is of crucial significance.

Test samples made in thick-film and LTCC (Low Temperature Cofired Ceramic) technology are considered in the presented paper. These technologies are widely applied for power modules fabrication, microelectronic devices packaging, sensors and actuators, and passive integrated components manufacturing [3], [4]. In many of the above mentioned applications the thermal management plays very important role, and thermal modeling is often used in order to investigate module performance or optimization. In all those studies knowledge of thermal properties of the materials as well as heat transfer coefficient is necessary in order to achieve reliable results.

A value of thermal conductivity can be found in data sheets of the material, delivered from the producer, but usually it may significantly vary with the time of storage, conditions of storage, parameters of the processing or manufacturing of the final device. Thus, in order to perform a reliable simulation or modeling the thermal conductivity of the material should be at least estimated based on some experimental data. There are many publications devoted to the thermal conductivity of electronic materials' measurements or estimations. Generally, one can notice that most of them are based on the construction of a dedicated device [5] or equipment [6], [7].

The second important factor for thermal modeling describes the heat exchange between considered device and environment. It is usually called heat transfer coefficient. This parameter is used to characterize the convective and radiative (in some temperature range) way of heat dissipation. It was previously reported that this factor shows strong dependence on the surface dimensions [8]. Although the problem of natural convection of horizontal plates was quite widely investigated there are only few papers devoted to that factor related to microelectronic devices. Thus an effective and simple method of heat transfer coefficient determination related to the microelectronic systems is required.

However, the problem of measurement of thermal conductivity and heat transfer factor has been previously exploited but there are only few papers related to this problem where both of those parameters are estimated in the one procedure [9]. In this paper we propose an alternative approach for simultaneous estimation of thermal conductivity and heat transfer coefficient.

Basically, we translate the above-described problem into a problem of parameter estimation of the heat transfer model. These kinds of inverse heat conduction problems [10] have been considered for a long time and are known to be rather difficult [11]. There exist many methods developed in order to solve such problems. Most of them are iterative algorithms that improve the initial guess of parameter values based on some goal function (e.g. square error function). Problem statement and conditions can significantly influence the convergence and thus the performance of such methods.

An application of repeated least square algorithm (RLSQA) [12] for parameters estimation is discussed in the presented paper. The algorithm is non-iterative and requires only the solution to two systems of linear equations. Convergence of this method is guaranteed, meaning the algorithm is robust and reliable. Generally, the eigenvalues of the heat operator are estimated basing on measurements in the first step of the algorithm. Then, unknown parameters are calculated based on the estimates of eigenvalues.

It is well-known that the location of measurements as well as placement of sources have a big impact on the estimation quality in case of distributed parameter systems. Thus experiment design methods should be applied in this kind of issues [13], [14]. This problem is also discussed in this paper in terms of sensitivity functions.

The whole procedure concerning thick-film and LTCC technology materials is shown in this paper, step by step. We begin with heat transfer descriptions in the test sample, then a basic introduction to the RLSQA method is presented. After that, the algorithm for thermal conductivity and heat transfer coefficient is shown including the exact form of all required data. The practical application of the presented method is given, as well. Measurement equipment used in the experiments is described in detail. The examples concern the alumina (96% Al₂O₃) and DP951 ceramics widely used in the thick-film and LTCC technologies. Test sample topology as well as the results of thermographic measurements of temperature field on the top surface test samples are shown.

It is known that both examined factors can depend on other physical quantities, especially on temperature. We assume that in the examined range of temperature this dependence is not so big, and the linear model is still valid. Moreover, all measurements are carried out by three different power levels of heat source, and three different mean temperatures of the sample.

Obtained results are fully comparable with previously published. It must be emphasized that application of the presented approach in not limited to the microelectronic materials characterization and can be successfully applied in other fields of the science. We would also like to stress that our approach is a typical estimation procedure, which cannot be treated as a typical measurement, but it can significantly help in preliminary investigations of the examined materials.