量子漂移-扩散等温模型解的长时间性质

Long-time Property of the Solution for Isothermal Quantum Drift-diffusion Model

研究一维单极量子漂移-扩散等温模型,它是用来模拟超小半导体器件发生量子效应的宏观量子模型之一,反映了电子浓度与静电场位势之间的非线性关系.量子漂移-扩散模型与经典漂移-扩散模型的区别在于前者包含了量子校正项.从数学的角度讲,此模型是由一个非线性四阶抛物方程与一个泊松方程耦合而成的方程组.研究此模型的困难在于非线性四阶抛物方程缺少极大值原理.利用对数索伯列夫不等式与能量估计的方法,在周期边界条件下,证明了当时间趋于无穷大时此模型的解以指数函数的速度趋于它的平均值.

The one-dimensional unipolar isothermal quantum drift-diffusion model is investigated, which is one of the quantum macroscopic models used to simulate the quantum effects in superminiature semiconductor devices. The model reflects the nonlinear relation between the electron density and the electrostatic potential. The difference between the quantum and the classical drift-diffusion models is that the former includes the quantum correction term. Mathematically, the model is a system which consists of a nonlinear fourth order parabolic equation coupled with a Poisson equation. The difficulty of studying the model lies in the lack of maximum principle for the fourth order parabolic equation. By the logarithmic Sobolev inequality and the energy estimate method, this paper shows that the solution of the model exponentially approaches its mean value as time increases to infinity.