Mathematical modeling of semiconductors: From quantum mechanics to devices
- Kantner, Markus
- Mielke, Alexander
- Mittnenzweig, Markus
- Rotundo, Nella
2010 Mathematics Subject Classification
- 35K57 80M30 81S22, 82D37
- Semiconductor modeling, drift-diffusion system, open quantum system,, Lindblad operator, reaction-diffusion systems, detailed balance condition, gradient structure, thermodynamically consistent coupling
We discuss recent progress in the mathematical modeling of semiconductor devices. The central result of this paper is a combined quantum-classical model that self-consistently couples van Roosbroeck's drift-diffusion system for classical charge transport with a Lindblad-type quantum master equation. The coupling is shown to obey fundamental principles of non-equilibrium thermodynamics. The appealing thermodynamic properties are shown to arise from the underlying mathematical structure of a damped Hamitlonian system, which is an isothermal version of so-called GENERIC systems. The evolution is governed by a Hamiltonian part and a gradient part involving a Poisson operator and an Onsager operator as geoemtric structures, respectively. Both parts are driven by the conjugate forces given in terms of the derivatives of a suitable free energy.
- Topics in Applied Analysis and Optimisation - Partial Differential Equations, Stochastic and Numerical Analysis, M. Hintermüller, J.F. Rodrigues, eds., vol. 5 of Springer-CIM Series, Springer, Springer - Cham, 2019, pp. 269--293.