Adiabatic approximation of the Schrödinger–Poisson system with a partial confinement: The stationary case

Asymptotic quantum transport models of a two-dimensional gas are presented. The models are the stationary versions of those introduced in a previous paper by Ben Abdallah, Mehats, Pinaud. The starting point is a singular perturbation of the three-dimensional stationary Schrodinger–Poisson system posed on bounded domain. The electron injection in the device is modeled thanks to open boundary conditions. Under a small density assumption, the asymptotics lead to a full two-dimensional first-order approximation of the initial model. An intermediate model, called the “2.5D adiabatic model” in Ben Abdallah, Mehats, Pinaud is then introduced. It shares the same structure as the limit but is shown to be a second-order approximation of the three-dimensional model.

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

[2]  Philippe Bechouche,et al.  Semi-Classical Limit of a Schrödinger Equationfor a Stratified Material , 2000 .

[3]  George A. Hagedorn,et al.  A Time-Dependent Born–Oppenheimer Approximation with Exponentially Small Error Estimates , 2001 .

[4]  Stefan Teufel,et al.  Adiabatic perturbation theory in quantum dynamics , 2003 .

[5]  Pierre Degond,et al.  A coupled Schrödinger drift-diffusion model for quantum semiconductor device simulations , 2002 .

[6]  Olivier Pinaud,et al.  Adiabatic Approximation of the Schrödinger-Poisson System with a Partial Confinement , 2005, SIAM J. Math. Anal..

[7]  Gauge fields and extrapotentials in constrained quantum systems , 2000, quant-ph/0001059.

[8]  R. C. T. da Costa,et al.  Quantum mechanics of a constrained particle , 1981 .

[9]  Kevin F. Brennan,et al.  Quantum Semiconductor Structures , 1992 .

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

[11]  A. Gossard,et al.  Observation of intersubband scattering in a 2-dimensional electron system , 1982 .

[12]  J. H. Davies,et al.  The physics of low-dimensional semiconductors , 1997 .

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

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

[15]  Pierre Degond,et al.  Coupling one-dimensional time-dependent classical and quantum transport models , 2002 .

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

[17]  F. Castella,et al.  L2 Solutions to the Schrödinger–Poisson System: Existence, Uniqueness, Time Behaviour, and Smoothing Effects , 1997 .

[18]  J. Harris,et al.  Two‐dimensional electron gas structures with mobilities in excess of 3×106 cm2 V−1 s−1 , 1987 .

[19]  Eric Polizzi,et al.  Self-consistent three-dimensional models for quantum ballistic transport in open systems , 2002 .

[20]  Craig S. Lent,et al.  The quantum transmitting boundary method , 1990 .

[21]  Hopkins,et al.  Single-particle subband spectroscopy in a parabolic quantum well via transport experiments. , 1993, Physical review. B, Condensed matter.

[22]  F. Stern,et al.  Electronic properties of two-dimensional systems , 1982 .

[23]  P. Markowich,et al.  A Wigner‐function approach to (semi)classical limits: Electrons in a periodic potential , 1994 .

[24]  Olivier Pinaud,et al.  A mathematical model for the transient evolution of a resonant tunneling diode , 2002 .

[25]  Naoufel Ben Abdallah,et al.  On a Vlasov–Schrödinger–Poisson Model , 2005 .

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

[27]  Naoufel Ben Abdallah,et al.  A Hybrid Kinetic-Quantum Model for Stationary Electron Transport , 1998 .

[28]  K. West,et al.  Electron mobilities exceeding 107 cm2/V s in modulation‐doped GaAs , 1989 .

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

[30]  Stefan Teufel,et al.  Adiabatic Decoupling and Time-Dependent Born–Oppenheimer Theory , 2001 .

[31]  Hopkins,et al.  Direct observation of resonant subband-Landau-level coupling in a transport experiment. , 1992, Physical review. B, Condensed matter.