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Document Type

Open Access

Degree Program

Electrical & Computer Engineering

Degree Type

Master of Science in Electrical and Computer Engineering (M.S.E.C.E.)

Year Degree Awarded

2010

Month Degree Awarded

May

Keywords

Time Dependent, Carbon Nanotubes, Quantum System, Evolution, Eigenvalue, Diagonalization

Abstract

The famous Moore's law states: Since the invention of the integrated circuit, the number of transistors that can be placed on an integrated circuit has increased exponentially, doubling approximately every two years. As a result of the downscaling of the size of the transistor, quantum effects have become increasingly important while affecting significantly the device performances. Nowadays, at the nanometer scale, inter-atomic interactions and quantum mechanical properties need to be studied extensively. Device and material simulations are important to achieve these goals because they are flexible and less expensive than experiments. They are also important for designing and characterizing new generation of electronic device such as silicon nanowire or carbon nanotube (CNT) transistors. Several modeling methods have been developed and applied to electronic structure calculations, such as: Hartree-Fock, density functional theory (DFT), empirical tight-binding, etc. For transport simulations, most of the device community focuses on studying the stationary problem for obtaining characteristics such as I-V curves. The non-equilibrium transport problem is then often addressed by solving a multitude of time-independent Schrodinger-type equation for all possible energies. On the other hand, for many other electronic applications including high-frequency electronics response (e.g. when a time-dependent potential is applied to the system), the description of the system behavior necessitate insights on the time dependent electron dynamics. To address this problem, it is then necessary to solve a time-dependent Schrodinger-type equation. In this thesis, we will focus on solving time-dependent problems with application to CNTs. We will be identifying all the numerical difficulties and propose new effective modeling and numerical schemes to address the current limitations in time-dependent quantum simulations. we will point out that two numerical errors may occur: an integration error and the anti-commutation issue error; the direct computation above being mathematically equivalent to performing the integration of the time dependent Hamiltonian using a rectangle numerical quadrature formula along the total simulation times. After careful study and many numerical experiments, we found that the Gaussian quadrature scheme provides a good trade off between computational consumption and numerically accuracy, meanwhile unitary, stability and time reversal properties are well preserved. The new Gaussian quadrature integration scheme uses (i) much fewer points in time to approximate the integral of the Hamiltonian, (ii) ordered exponential to factorize the time evolution operator, (iii) FEM discretize techniques (iv) and at last, the FEAST eigenvalue solver to diagonalize and solve each exponential.

First Advisor

Eric Polizzi

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