Existence of weak solution and semiclassical limit for quantum drift-diffusion model

Abstract

The existence and semiclassical limit of the solution to one-dimensional transient quantum drift-diffusion model in semiconductor simulation are discussed. Besides the proof of existence of the weak solution, it is also obtained that the semiclassical limit of this solution solves the classical drift-diffusion model. The key estimates rest on the entropy inequalities derived from separation of quantum quasi-Fermi level. (2000). 35K35, 65M12, 65M20, 76Y05. Mathematics Subject Classification

A sixth-order parabolic system in semiconductors

“…Semiclassical limit shows that the quantum corrected drift diffusion model converges to the classical drift diffusion model, as discussed in [4] for O(ε 2 ) order correction and the results in this paper for O(ε 4 ) order correction, both are in the case of "good" boundary conditions. While a further interesting problem is the convergence rate analysis, whether the difference between O(ε 2 ) corrected solution and the limiting solution can be controlled by O(ε 2 )?…”

The isentropic quantum drift-diffusion model in two or three space dimensions

“…Chen ZAMP of a fourth order equation (1.2) was treated and the exponential transformation technique as in [19,20,21,7] was also used to ensure the positivity of approximate solution which is essential for the uniform estimate. More precisely, let τ > 0 such that T = N τ (without loss of generality, otherwise, let…”

“…For the one space dimensional QDDM, Jüngel and Pinnau [20,21] first obtain a positivity preserving global weak solution with large enough lattice temperature θ for the mixed Dirichlet-Neumann boundary isothermal problem, and in the sequel, a series of works [6][7][8][9] investigate this model elaborately with homogeneous Neumann and mixed Dirichlet-Neumann boundary conditions in the aspects of weak existence, semiclassical limit and long time behavior.…”

The Existence and Long-Time Behavior of Weak Solution to Bipolar Quantum Drift-Diffusion Model*

“…Similar results for periodic-boundary problem were proved in [2,3] recently. For the one space dimensional QDDM, Jüngel and Pinnau [15,16] first obtained a positivity preserving global weak solution with large enough θ for the mixed Dirichlet-Neumann boundary problem, and in the sequel, a series of works (see [6][7][8]) investigated this model elaborately with homogeneous Neumann in the aspects of weak existence and semiclassical limit. But to the authors' knowledge, there are no results on the long-time behavior for bipolar transient (isothermal or isentropic) model.…”

“…This article is organized as follows. In Section 2, we will construct the approximation problem by the method of semi-discretization in time, which was used in [14][15][16]6] as well as in [4] to deal with strongly coupled parabolic system. Section 3 contains all of the uniform entropy estimates which will be used in the proof of existence.…”