在目前最新奈米級半導體元件的發展上，量子力學扮演了相當重要的角色，同時在元件模擬與設計上也同樣要予以考量。因此開發一個包含量子效應的元件模擬器是必要的。在本論文中，我們將介紹一維薛丁格(Schrodinger)計算與密度梯度模型(density gradient model) 來模擬量子效應。首先我們提出了等效電路法與高效率特徵值解法器來解薛丁格波動方程式，根據這個薛丁格等效電路模型，我們可以建立柏松─薛丁格互解方程式等效電路模型來模擬金氧半(MOS)元件在反轉區的量子效應。此外，我們也將分別使用簡化數值法與等效電路法來模擬克若尼-潘尼(Kronig-Penney)模型，以觀察半導體元件中能帶的特性。在元件模擬中，為了使所有變數都能有相同數量級而不需比例縮放參數(scaling factor)，我們提出了對數比例法(log-scale)以幫助其在牛頓疊代法的收斂。為了解決記憶體空間不足的問題，我們也提出了帶狀不完全LU(Banded incomplete LU)法來改善這問題。最後，我們將建立起密度梯度公式的等效電路模型，並利用分離模式(decoupled method)與對數比例法來解量子飄移擴散模型(quantum drift-diffusion model)。

In up-to-date development of nanoscale semiconductor devices, quantum mechanisms play an important role and have to be properly taken into account in the simulation and design. Therefore, it is necessary to develop the device simulator including quantum effects. The 1-D Schrodinger computation and density gradient model for quantum effect simulations will be introduced in this dissertation. We propose a simplified equivalent circuit model to solve the Schrodinger equation and an efficient eigenvalue and eigenvetor solver to solve the eigenvalue problem. Based on the equivalent circuit model of Schrodinger equation, the equivalent circuit model of the Poisson-Schrodinger equation can be created and can be simulated to show that the MOS device features of the quantum effects at strong inversion. Moreover, the Kronig-Penney approximation will be also applied to reveal the essential features of the energy band structure of semiconductors with the simplified numerical method and the equivalent circuit method. To make all variables in similar orders without scaling factors, we propose a log-scale method to help Newton-Raphson iterations to easily converge in device simulations. Also, we will propose a matrix solver using Banded incomplete LU method to improve the problem that the memory space is insufficient. Finally, the density gradient model will be converted to an equivalent circuit form and we use the decoupled method with a log-scale method to solve the self-consistent quantum drift-diffusion model.

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