Abstract
We derive rigorously a set of boundary conditions for heterogenous devices using a description via the quantum hydrodynamic system provided by the Madelung transformations. In particular, we show that the generalized enthalpy should be constant at the interface between classical and quantum domains. This condition provides a set of boundary conditions, which we use to prove the existence and the uniqueness of regular steady solutions of the quantum hydrodynamic system. Finally, we analyse the linear stability of the system supplied with our boundary conditions and we test numerically our model on a toy device.
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References
Ascher U, Christiansen J, Russell RD (1979) A collocation solver for mixed order systems of boundary value problems. Math Comput 33(146):659–679
Antonelli P, Marcati P (2009) On the finite energy weak solutions to a system in quantum fluid dynamics. Commun Math Phys 287(2):657–686
Antonelli P, Marcati P (2012) The quantum hydrodynamics system in two space dimensions. Arch Ration Mech Anal 203(2):499–527
Brezzi F, Gasser I, Markowich PA, Schmeiser C (1995) Thermal equilibrium states of the quantum hydrodynamic model for semiconductors in one dimension. Appl Math Lett 8(1):47–52
Bader G, Ascher U (1987) A new basis implementation for a mixed order boundary value ODE solver. SIAM J Sci Stat Comput 8:483–500
Calderón-Muñoz WR, Jena D, Sen M (2009) Hydrodynamic instability of confined two-dimensional electron flow in semiconductors. J Appl Phys 106(1):014506
Calderón-Muñoz WR, Jena D, Sen M (2010) Temperature influence on hydrodynamic instabilities in a one-dimensional electron flow in semiconductors. J Appl Phys 107(7):074504
Di Michele F, Marcati P, Rubino B (2013) Steady states and interface transmission conditions for heterogeneous quantum-classical 1-d hydrodynamic model of semiconductor devices. Phys D 243:1–13
Gasser I, Jüngel A (1997) The quantum hydrodynamic model for semiconductors in thermal equilibrium. Z Angew Math Phys 48(1):45–59
Gyi MT, Jüngel A (2000) A quantum regularization of the one-dimensional hydrodynamic model for semiconductors. Adv Differ Equ 5(4–6):773–800
Gualdani MP, Jüngel A (2004) Analysis of the viscous quantum hydrodynamic equations for semiconductors. Eur J Appl Math 15(5):577–595
Jüngel A, Li H (2004) Quantum Euler–Poisson systems: existence of stationary states. Arch Math (Brno) 40:435–456
Jüngel A, Pinnau R (2003) Convergent semidiscretization of a nonlinear fourth order parabolic system. M2AN Math Model Numer Anal 37(2):277–289
Marcati P, Pan R (2001) On the diffusive profiles for the system of compressible adiabatic flow through porous media. SIAM J Math Anal 4:790–826
Marcati P, Mei M, Rubino B (2005) Optimal convergence rates to diffusion waves for solutions of the hyperbolic conservation laws with damping. J Math Fluid Mech suppl. 2:S224–S240
Nishibata S, Suzuki M (2008) Initial boundary value problems for a quantum hydrodynamic model of semiconductors: asymptotic behaviors and classical limits. J Differ Equ 244(4):836–874
Zhang B, Jerome J (1996) On a steady-state quantum hydrodynamic model for semiconductors. Nonlinear Anal 26(4):845–856
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Communicated by Armin Iske.
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Di Michele, F., Marcati, P. & Rubino, B. Stationary solution for transient quantum hydrodynamics with bohmenian-type boundary conditions. Comp. Appl. Math. 36, 459–479 (2017). https://doi.org/10.1007/s40314-015-0235-2
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DOI: https://doi.org/10.1007/s40314-015-0235-2