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Simulation of THz Oscillations in Semiconductor Devices Based on Balance Equations Tobias Linn1

· Kai Bittner2 · Hans Georg Brachtendorf2 · Christoph Jungemann1

Received: 16 January 2020 / Revised: 23 July 2020 / Accepted: 10 September 2020 / Published online: 22 September 2020 © The Author(s) 2020

Abstract Instabilities of electron plasma waves in high-mobility semiconductor devices have recently attracted a lot of attention as a possible candidate for closing the THz gap. Conventional moments-based transport models usually neglect time derivatives in the constitutive equations for vectorial quantities, resulting in parabolic systems of partial differential equations (PDE). To describe plasma waves however, such time derivatives need to be included, resulting in hyperbolic rather than parabolic systems of PDEs; thus the fundamental nature of these equation systems is changed completely. Additional nonlinear terms render the existing numerical stabilization methods for semiconductor simulation practically useless. On the other hand there are plenty of numerical methods for hyperbolic systems of PDEs in the form of conservation laws. Standard numerical schemes for conservation laws, however, are often either incapable of correctly handling the large source terms present in semiconductor devices due to built-in electric fields, or rely heavily on variable transformations which are specific to the equation system at hand (e.g. the shallow water equations), and can not be generalized easily to different equations. In this paper we develop a novel well-balanced numerical scheme for hyperbolic systems of PDEs with source terms and apply it to a simple yet non-linear electron transport model. Keywords Hyperbolic Balance Laws · THz Oscillations in Semiconductors · Well-balanced numerical Scheme · Isothermal hydrodynamic model Mathematics Subject Classification 65M08 · 76M12 · 78M12

This project has been funded by Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) Ref. No.: JU406/14-1 and the Austrian Science Fund (FWF): I3130-N30.

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Tobias Linn [email protected] Kai Bittner [email protected] Hans Georg Brachtendorf [email protected]

1

Institute of Electromagnetic Theory, RWTH Aachen University, Kackertstr. 15-17, 52072 Aachen, Germany

2

University of Applied Sciences of Upper Austria, 4232 Hagenberg, Austria

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Journal of Scientific Computing (2020) 85:6

1 Introduction In recent years instabilities of electron plasma waves in high-electron-mobility transistors (HEMT) have attracted a lot of attention as a possible solution for closing the THz gap [4,7,8]. By solving the full Boltzmann Transport Equation (BTE) the behavior of such devices can be simulated in a very accurate way [13]. As transient simulations of the BTE however are computationally very expensive, especially for two or three dimensional devices, simpler transport models are needed. By using balance equations derived from the BTE together with appropriate closure relations, mod