Group classification of an energy transport model for semiconductors with crystal heating
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Group classification of an energy transport model for semiconductors with crystal heating Mariangela Ruscica · Rita Tracinà
Received: 6 December 2013 / Revised: 20 March 2014 / Accepted: 15 April 2014 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2014
Abstract We consider an energy transport model for semiconductors describing the electrothermal effects. The unipolar model involves four equations: the continuity and the energy balance equations for the electrons, the thermal diffusion equation for the lattice and the Poisson equation for the electric potential. The model can be derived by moment method from the Boltzmann transport equation for electrons in semiconductors. For this model, we perform a symmetry group classification by the infinitesimal Lie method and exact solutions are found. Keywords
Group classification · Energy transport model · Exact solutions
Mathematics Subject Classification
35Q99 · 35C06
1 Introduction In the last years, the improvement of technology and design about semiconductor devices has been coupled to the development of mathematical models, which describe several phenomena like the influence of the thermal heating of the carriers and crystal lattice that may strongly influence the behavior of semiconductor devices and even lower their performance (Selberherr 1984). The non-isothermal device models were introduced by Stratton (1962), employing drift– diffusion equations model and heat flux models for the lattice temperature.
Communicated by Jose Alberto Cuminato. M. Ruscica · R. Tracinà (B) Dipartimento di Matematica e Informatica, Università di Catania, Viale A. Doria 6, 95125 Catania, Italy e-mail: [email protected] M. Ruscica e-mail: [email protected]
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M. Ruscica, R.Tracinà
A thermodynamic approach to extend the drift–diffusion equations was developed by Wachutka (1990) and later it was generalized in Albinus et al. (2002) using maximum entropy principle. Then, the energy transport model coupled to the lattice heat equation was introduced. The energy transport models are macroscopic models that can be derived by Boltzmann equation, through the moment method or the Hilbert expansion method (Ben and Degond 1996). It takes into account the thermal effects related to the flow of electrons through the crystal, unlike the drift–diffusion models are based on the assumption of isothermal motion. In the energy transport model developed in Chen et al. (1992) and Romano (2007), the lattice temperature is supposed constant, but the charge carriers temperature is a variable. This model is based on a set of three equations: the balance equations for the density and energy charge carriers, coupled Poisson equation for the electric potential. In the energy transport model with heating crystal (Romano and Rusakov 2010; Romano and Zwierz 2010; Brunk and Jungel 2011), another moment equation has to be introduced, regarding the effects derived by a nonconstant lattice temperature. The increasing lattice temperature, due to the collision of the charge carrie
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