Existence and stability of steady-state of one-dimensional quantum hydrodynamic system for semiconductors |
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Authors: | Feimin Huang Hai-Liang Li Akitaka Matsumura |
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Institution: | a Institute of Applied Mathematics, AMSS, Academia Sinica, Beijing 100080, PR China b Department of Mathematics, Capital Normal University, Beijing 100037, PR China c Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, Toyonaka 560-0043, Osaka, Japan |
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Abstract: | A transient quantum hydrodynamic system for charge density, current density and electrostatic potential is considered in spatial one-dimensional real line. The equations take the form of classical Euler-Poisson system with additional dispersion caused by the quantum (Bohm) potential and used, for instance, to account for quantum mechanical effects in the modelling of charge transport in ultra submicron semiconductor devices such as resonant tunnelling trough oxides gate and inversion layer energy quantization and so on.The existence and uniqueness and long time stability of steady-state solution with spatial different end states and large strength is proven in Sobolev space. To guarantee the existence and stability, we propose a stability condition which can be viewed as a quantum correction to classical subsonic condition. Furthermore, since the argument for classical hydrodynamic equations does not apply here due to the dispersion term, we also show the local-in-time existence of strong solution in terms of a reformulated system for the charge density and the electric field consisting of two coupled semilinear (spatial) fourth-order wave type equations. |
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Keywords: | Quantum hydrodynamic models Existence and exponential stability of steady-state Global existence of smooth solutions Semiconductor model |
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