本篇主要在探討量子傳輸效應，藉由反覆疊代帕森和薛丁格方程
式求得穩態解。隨著半導體科技演進至奈米等級，求解帕森和飄移
擴散方程式的古典粒子傳輸模型變得不再精準，因為忽略了量子波
的概念。在程式開發中，利用有限差分法建立矩陣來發展計算簡化
的薛丁格方程式，並考慮載子散射機制所造成的影響。其中，非平
衡態格林函數和邊界條件的設定都被引用至薛丁格方程式。這個研
究的的主要特點是成功地引入載子散射機制到薛丁格方程，造成能
量在不同的能階釋放能量到更低能階的效果，並反覆疊代達穩態。
最後是波森和薛丁格方程式兩者反覆疊代計算求得載子濃度、電流
密度與元件位能。本文模擬分析許多元件結構像是穿隧結構、穿隧
共振元件、穿隧場效電晶體元件以及n-i-n結構。

This thesis studies the quantum transport effect by solving the Poisson and Schrodinger equation self-consistently. As we know, as the semiconductor scaling technology enters the nanoscale world, the classical carrier transport model by solving Poisson and drift-diffusion equation model becomes less valid, due to the ignorance of quantum wave pictures. To develop the program for modeling the nanostructure, the finite difference method with the simpli ed Schrodinger equation and considering the scattering effect is used for developing the program. Next, the nonequilibrium green function method is used for boundary condition and they are applied to solve the Schrodinger equation. The key feature of this study is successfully adding the scattering mechanism into the Schrodinger solver for energy relaxation in different energies and solve self-consistently. The last step is the Poisson equation and the Schrodinger Hamiltonian are self-consistently iterated to get the carrier density, current density, and potential of the
device. Several device structures are examined, including the tunneling structures, resonant tunneling devices, tunneling eld effect transistors, and n-i-n structures.

口試委員會審查表. . . . . . . . . . . . . . . . . . . . . . . . . i
中文摘要. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
英文摘要. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
目錄. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi
圖目錄. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
表目錄. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Prologue . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Transport Models . . . . . . . . . . . . . . . . . . . . . 3
1.2.1 The evolution of the scale of electronic devices . 3
1.2.2 Classical transport model . . . . . . . . . . . . 5
1.2.3 Sub-micrometer scale device modeling . . . . . 5
1.3 Thesis overview . . . . . . . . . . . . . . . . . . . . . . 8
2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Poisson Equation . . . . . . . . . . . . . . . . . . . . . 9
2.3 Finite Difference Method . . . . . . . . . . . . . . . . . 13
2.4 Schrodinger Equation . . . . . . . . . . . . . . . . . . . 16
2.5 Con ned States . . . . . . . . . . . . . . . . . . . . . . 17
2.6 Open boundary value problem for continous wave . . . 18
2.7 Physic and Derivation of Carrier Transport . . . . . . . 22
2.7.1 The introduction of the Schrodinger equation by
including the scattering mechanism . . . . . . . 24
2.7.2 The Derivation of the Divergence J . . . . . . . 25
2.7.3 The Derivation of the Scattering Rate . . . . . 26
2.7.4 The Scattering Rate and the Divergence J . . . 32
2.8 Chapter summary . . . . . . . . . . . . . . . . . . . . . 34
3 Device Simulation . . . . . . . . . . . . . . . . . . . . . . . . 35
3.1 Flat Band . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.2 Tunneling Device . . . . . . . . . . . . . . . . . . . . . 38
3.3 Resonant Tunneling Device - RTD . . . . . . . . . . . . 41
3.4 Tunneling Field Effect Transistors - TFET . . . . . . . 48
3.5 n-i-n Structure - Scattering Mechanism . . . . . . . . . 59
3.5.1 Schrodinger and Phonon Scattering Iteration . . 60
3.5.2 Total Carrier Density and Poisson Solver Iteration 68
3.5.3 Iteration of whole Quantum Transport Program 72
3.5.4 Different bias of n-i-n structures . . . . . . . . . 75
4 Conclusion and Future Work . . . . . . . . . . . . . . . . . . 82
4.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . 84
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

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