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Mathematical modeling of semiconductor quantum dots based on the nonparabolic effective-mass approximation

100%
Liu J.-L.
Nanoscale Systems: Mathematical Modeling, Theory and Applications
|
2012
|
tom 1
58-79
EN
Within the effective mass and nonparabolic band theory, a general framework of mathematical models and numerical methods is developed for theoretical studies of semiconductor quantum dots. It includes single-electron models and many-electron models of Hartree-Fock, configuration interaction, and current-spin density functional theory approaches. These models result in nonlinear eigenvalue problems from a suitable discretization. Cubic and quintic Jacobi-Davidson methods of block or nonblock version are then presented for calculating the wanted eigenvalues that are clustered in the interior of the spectrum and may have small gaps and degeneracy. These are challenging issues arising from modeling a great variety of semiconductor nanostructures fabricated by advanced technology in semiconductor industry and science. Generic algorithms for many-electron simulations under this framework are also provided. Numerical results obtained within this framework are summarized to three eminent aspects, namely, accuracy of models, physical novelty, and effectivity of nonlinear eigensolvers. Concerning numerical accuracy, important details related to experimental data are also addressed.
2
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A Quantum Corrected Poisson-Nernst-Planck Model for Biological Ion Channels

100%
Liu J.-L.
Molecular Based Mathematical Biology
|
2015
|
tom 3
|
nr 1
EN
A quantum corrected Poisson-Nernst-Planck (QCPNP) model is proposed for simulating ionic currents through biological ion channels by taking into account both classical and quantum mechanical effects. A generalized Gummel algorithm is also presented for solving the model system. Compared with the experimental results of X-ray crystallography, it is shown that the quantum PNP model is more accurate than the classical model in predicting the average number of ions in the channel pore. Moreover, the electrostatic potential has been found to reach as high as 19% difference between two models around the charged vestibule which has been shown to play a significant role in the permeation of ions through ion-selective ligand-gated or voltage-activated channels. These results indicate that the QCPNP model may be considered as a more refined continuum model that can be incorporated into a multi-scale electrophysiology modeling.
3
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Poisson-Fermi Formulation of Nonlocal Electrostatics in Electrolyte Solutions

51%
Liu J.-L., Xie D., Eisenberg B.
Molecular Based Mathematical Biology
|
2017
|
tom 5
|
nr 1
116-124
EN
We present a nonlocal electrostatic formulation of nonuniform ions and water molecules with interstitial voids that uses a Fermi-like distribution to account for steric and correlation efects in electrolyte solutions. The formulation is based on the volume exclusion of hard spheres leading to a steric potential and Maxwell’s displacement field with Yukawa-type interactions resulting in a nonlocal electric potential. The classical Poisson-Boltzmann model fails to describe steric and correlation effects important in a variety of chemical and biological systems, especially in high field or large concentration conditions found in and near binding sites, ion channels, and electrodes. Steric effects and correlations are apparent when we compare nonlocal Poisson-Fermi results to Poisson-Boltzmann calculations in electric double layer and to experimental measurements on the selectivity of potassium channels for K+ over Na+.
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