Current Coupling of Drift-Diffusion Models and Schrödinger--Poisson Systems: Dissipative Hybrid Models

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

“…The final aim is to be able to predict from numerical simulations the I-V characteristic curve for devices which involve an unusual coupling between spectral quantities associated with the quantum mechanics and nonlinear effects due to the electrostatic mean field. Two types of models were considered : purely quantum ones based on Schrödinger-Poisson systems or Wigner-Poisson systems (see for example [1,2,3,4,5,6,30,34,35]); and quantum hydrodynamic or drift-diffusion models (see for example [9,10,11,13,36]). The second ones which assume local thermal equilibrium or local entropy maximizing states are well suited for situations where quantum effects, averaged by the statistics over a large number of particles, only bring small corrections to classical mechanics.…”

Stability of the standing waves of the concentrated NLSE in dimension two

“…On the other hand, noting that q( ω) = 0 and that q(•) is smooth and strictly increasing on ( ω, +∞), one can take the inverse q(ω) of the function ω(q) := 4 e −2γ−4πβq 2σ , q > 0, (12) and plug it into (10), to obtain the energy as a function of q, i.e.,…”

“…The behaviors of ω(q) and E(q) are depicted in Figure 2(a) and 2(a), respectively. This alternative form can be useful in computation since ( 13) is more manageable than (10). Furthermore,…”

“…For instance, in semiconductor theory the effect of electronic charge accumulation in a resonant tunneling in a double barrier heterostructure [20] is typically studied using a concentrated NLSE. More recently, other applications have been suggested: the spontaneous formation of quantum coherent non-dissipative patterns in semiconductor heterostructures with nonlinear properties [11]; the dynamics of the mixed states of statistical physics [23]; the appearance of quantum turbulence in the probability density [9]; the scattering in nuclear physics models for the disexcitation of isomeric states and also the production of weakly bounded states in heavy nuclei close to the instability; the analysis of resonant tunneling diodes, which exhibits intrinsic instability [31] or the fabrication of semiconductor superlattices, for the estimate of the time decay rates for the solutions to the Schrödinger-Poisson equations in the repulsive case [10,25].…”

Uniqueness for Dissipative Schrödinger-Poisson Systems

“…This has implications for dissipative hybrid models considered in [4] which use a mixed description by a drift-diffusion model and a dissipative Schrödinger-Poisson system. In more detail, one divides the device ∆ = [a 0 , b 0 ] into two regions Ω c = (a 0 , a) ∪ (b, b 0 ) and Ω q = (a, b), which are called "classical zone" and "quantum zone", respectively.…”