Finite Element Modeling of Thermo-Electric Materials
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FINITE ELEMENT MODELING OF THERMO-ELECTRIC MATERIALS D. C. van Duyn and P. de Vries (email: [email protected]) Delft University of Technology, Electrical Engineering Department, P.O. Box 5031, 2600 GA Delft, The Netherlands.
ABSTRACT This paper deals with the modeling of materials in which electric, thermal and cross effects between these are important. Also the effect of a magnetic field is included. A basic model and its FEM implementation are discussed. Some numerical test result are given.
INTRODUCTION In recent years a full coupled numerical solution of the thermal and electrical problem has become more and more important. Fields of interest include sensor design (e.g. optimum design of silicon thermopile sensors [ 1,2] and thermal offset causes in a spinning current Hall plate [3]), simulation of hot carrier effects, and thermal management in bipolar and VLSI devices and recently SO1 structures. Our special interest is in the field of sensors [4]. The objective of this paper is to discuss some advances in the development of a general purpose program capable of covering a wide range of modeling problems where thermal and electric effects are important. Following the usual nomenclature these effects can be classified as electric, thermal, thermoelectric, thermomagnetic and galvanomagnetic effects. Nowadays, such calculations become more and more feasible with the availability of very fast (vector and parallel) computers. The package discussed in this paper runs on a CONVEX C3XX, providing sufficient power to tackle large problems, and is based on the data and memory management structure of the software package SEPRAN [5]. Our final goal is to do inverse modeling of the thermoelectric material parameters and evaluation of (small) thermal parasitic effects in sensors, therefore, emphasis is on an accurate implementation strategy and the physics included. The main topics of this paper are as follows. The first section discusses the physical model we desire to implement. The second section discusses an implementation strategy which provides a good compromise between accuracy and efficiency. The third section discusses some simple test simulation results.
PHYSICAL MODEL The purpose of this section is to derive a framework on which the implementation strategy can be based. We use the rather elaborate theory of thermodynamics of irreversible processes (TIP) [6], which enables us to draw powerful conclusions about the structure of the model and its impact on the numerical approach. TIP relies on the assumption that the system can be separated into a number of virtually independent thermodynamic subsystems between which energy
and momentum relaxation occurs. For a semiconductor (simplified) this is schematically pictured in Figure 1. In this case we will neglect momentum relaxation effects and concentrate on the energy relaxation between the main subsystems. This results in the so-called drift diffusion model with extensions for thermal, thermoelectric, galvanomagnetic and thermomagnetic effects. Fluxes and t
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