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Semiconductor devices are contained in essentially all electronic devices, like computers, MP3 players, and mobile phones, for instance. The development of these components is also based on numerical simulations. Often, standard models, like drift-diffusion equations, are employed in industrial simulations, together with corrections taking into account quantum mechanical effects. However, these corrections are either not accurate enough or too complicate for the simulation of modern nano-scale devices, and it becomes necessary to include quantum terms in the modeling part. The solution of standard quantum models, like the Schroedinger or Wigner equation, is very time-consuming such that computationally feasible, but still accurate models need to be devised. Quantum hydrodynamic (QHD) equations seem to fulfill the request of physical accuracy and computational efficiency. QHD equations consist of conservation laws for the particle density and the current density, like in gas dynamics, including a highly nonlinear term involving the so-called Bohm potential. This term has third-order derivatives, and the QHD model is of dispersive type. The main mathematical difficulty is the treatment of the third-order differential operator and the proof of the nonnegativity of the particle density, since maximum principles cannot be applied here in general. Recently, Bresch and Desjardin (2005) have studied related Euler equations including Korteweg-type terms with third-order derivatives, describing diffusive capillarity effects. Interestingly, for a special choice of the capillarity function, the third-order Bohm potential term is recovered. Bresch and Desjardins succeeded in proving new a priori estimates, leading to general existence results, when certain viscous terms are introduced. In particular, they could show that vacuum (zero particle density) is excluded. This observation is one of the starting points of this project. The main objective is to understand the influence of various types of viscosity in the QHD equations and to employ viscous terms to stabilize numerical approximations. Moreover, we will show some properties of the QHD model without viscosity.
Short description of the task performed by Croatian partner
Non-uniqueness of solutions and bifurcation in the QHD equations. Lent-Kirkner boundary conditions. New estimates for viscous QHD models. Existence of solutions to viscous QHD models.
Ovo web sjedište koristi tehnologiju "kolačića" (eng. cookie) da bi se korisnicima prikazao dinamičan i personaliziran sadržaj, te su oni nužni za ispravan rad. Ako nastavite pregledavati ove stranice, kolačići će biti korišteni u suradnji s vašim preglednikom Weba.
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