Computing the steady states for an asymptotic model of quantum transport in resonant heterostructures

In this article, we propose a rapid method to compute the steady states, including bifurcation diagrams, of resonant tunneling heterostructures in the far from equilibrium regime. Those calculations are made on a simplified model which takes into account the characteristic quantities which arise from an accurate asymptotic analysis of the nonlinear Schrodinger-Poisson system. After a summary of the existing theoretical results, the asymptotic model is explicitly adapted to physically realistic situations and numerical results are shown in various cases.

[1]  Bernard Helffer,et al.  Semi-Classical Analysis for the Schrödinger Operator and Applications , 1988 .

[2]  R. Landauer,et al.  Spatial variation of currents and fields due to localized scatterers in metallic conduction , 1988, IBM J. Res. Dev..

[3]  Galán,et al.  Self-oscillations of domains in doped GaAs-AlAs superlattices. , 1995, Physical review. B, Condensed matter.

[4]  Anton Arnold,et al.  Numerically Absorbing Boundary Conditions for Quantum Evolution Equations , 1998, VLSI Design.

[5]  F. Nier,et al.  Nonlinear Asymptotics for Quantum Out-of-Equilibrium 1D Systems: Reduced Models and Algorithms , 2004 .

[6]  B. Simon,et al.  Universal lower bounds on eigenvalue splittings for one dimensional Schrödinger operators , 1985 .

[7]  F. Nier,et al.  Far from equilibrium steady states of 1D-Schrödinger–Poisson systems with quantum wells I , 2008 .

[8]  André Martinez,et al.  Semiclassical asymptotics for the spectral function of long-range Schrödinger operators , 1989 .

[9]  Bernard Helffer,et al.  Résonances en limite semi-classique , 1986 .

[10]  Ferry,et al.  Scattering states and distribution functions for microstructures. , 1987, Physical review. B, Condensed matter.

[11]  Joachim Rehberg,et al.  Current Coupling of Drift-Diffusion Models and Schrödinger-Poisson Systems: Dissipative Hybrid Models , 2005, SIAM J. Math. Anal..

[12]  J. Craggs.  Applied Mathematical Sciences , 1973 .

[13]  William R. Frensley,et al.  Boundary conditions for open quantum systems driven far from equilibrium , 1990 .

[14]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[15]  Olivier Pinaud,et al.  Multiscale simulation of transport in an open quantum system: Resonances and WKB interpolation , 2006, J. Comput. Phys..

[16]  F. Nier.  A variational formulation of schrödinger-poisson systems in dimension d ≤ 3 , 1993 .

[17]  F. Chevoir,et al.  Scattering-assisted tunneling in double-barrier diodes: Scattering rates and valley current. , 1993, Physical review. B, Condensed matter.

[18]  Olivier Pinaud,et al.  Transient simulations of a resonant tunneling diode , 2002 .

[19]  J. Descloux,et al.  SIMULATION OF SOME QUANTUM MODELS FOR SEMICONDUCTORS , 2002 .

[20]  Schrödinger-Poisson systems in dimension d ≦ 3: The whole-space case , 1993 .

[21]  S. Laux,et al.  Analysis of quantum ballistic electron transport in ultrasmall silicon devices including space-charge and geometric effects , 2004 .

[22]  R. Landauer.  Spatial variation of currents and fields due to localized scatterers in metallic conduction , 1988 .

[23]  Pierre Degond,et al.  Quantum Energy-Transport and Drift-Diffusion Models , 2005 .

[24]  A. Arnold.  MATHEMATICAL CONCEPTS OF OPEN QUANTUM BOUNDARY CONDITIONS , 2001 .

[25]  N. Abdallah,et al.  QUANTUM PHYSICS; PARTICLES AND FIELDS 4241 On a multidimensional Schrodinger-Poisson scattering model for semiconductors , 2000 .

[26]  Zhangxin Chen.  A finite element method for the quantum hydrodynamic model for semiconductor devices , 1996 .

[27]  Pierre Degond,et al.  A 1D coupled Schrödinger drift-diffusion model including collisions , 2005 .

[28]  Barry Simon,et al.  Analysis of Operators , 1978 .

[29]  F. Nier.  The dynamics of some quantum open systems with short-range nonlinearities , 1998 .

[30]  Andreas Unterreiter,et al.  The Stationary Current { VoltageCharacteristics of the Quantum DriftDi usion ModelRen , 1999 .

[31]  Bernardo Cockburn,et al.  Quantum hydrodynamic simulation of hysteresis in the resonant tunneling diode at 300 K , 1995, Journal of Computational Electronics.

[32]  Carlo Presilla,et al.  On Schrödinger Equations with Concentrated Nonlinearities , 1995 .

[33]  André Martinez,et al.  Destruction of the Beating Effect¶for a Non-Linear Schrödinger Equation , 2002 .

[34]  Mark S. C. Reed,et al.  Method of Modern Mathematical Physics , 1972 .

[35]  R. Landauer,et al.  Generalized many-channel conductance formula with application to small rings. , 1985, Physical review. B, Condensed matter.

[36]  Didier Lippens,et al.  Effect of cathode spacer layer on the current‐voltage characteristics of resonant tunneling diodes , 1990 .

[37]  C. Presilla,et al.  Transport properties in resonant tunneling heterostructures , 1996, cond-mat/9607088.

[38]  Israel Michael Sigal,et al.  Introduction to Spectral Theory , 1996 .

[39]  P. Degond,et al.  On a one-dimensional Schrödinger-Poisson scattering model , 1997 .