臺灣博碩士論文加值系統

在二維線性系統與絕對零度的條件下，研究半導體中的內稟自旋霍爾效應(Rashba系統以及Rashba-Dresslhaus系統)與自旋電流。根據自旋連續方程式，自旋密度隨時間變化的量等於傳統自旋電流密度的散度加上自旋矩密度，因為實驗中尚未觀察到自旋矩密度，本文主要專注討論傳統自旋電流密度。我們利用久保公式(Kubo formula)來計算施加一垂直磁場在此二維線性系統中的傳統自旋霍爾電導率。我們發現垂直二維線性系統的磁場與自旋霍爾電導率的確相關，但意外地發現磁場的方向並不影響自旋霍爾電導率的正負方向，只與磁場的絕對值大小有關，系統的時間反演對稱確實被磁場破壞，我們發現二維平面中的自旋分量的方向會隨著磁場方向的不同而反向，但因為自旋的大小必須保持不變所以(z)方向的自旋分量的方向卻反常地維持不變，最後得到自旋霍爾電導率與磁場方向無關只與磁場的大小有關。我們也發現在磁場趨於無限大時，自旋霍爾電導率也會趨於零，原因是磁場的增加使得能隙也跟著變大。

In the two-dimensional systems, we investigate the intrinsic spin-Hall effect (Rashba
system and Rashba-Dresslhaus system) and spin current in the semiconductors at
zero temperature. According to the spin continuity equation, the rate of change in spin density equals the divergence of the conventional spin current denstiy plus the spin torque density. However, the spin torque density has not yet been found in any experiment. In this sense, this paper only focuses on the discusssion of the conventional spin current denstiy. We use the Kubo formula to compute the conventional spin-Hall conductivity in the two-dimensional systems with a perpendicular magnetic field. We show the magnetic field in the system indeed changes the magnitude of spin-Hall conductivity but surprisingly the change in direction of the magnetic field does not change the sign of spin-Hall conductivity. The spin-Hall conductivity depends only on the absolute value of the magnetic field. The time-reversal symmetry is indeed been broken in this system. We show that the in-plane spin changes sign when the orientation of the applied magnetic field is changed. Furthermore, because the magnitude of spin must be preserved, the spin z component has an anomalous result that it depends on the magnitude of the magnetic field. The resulting spin-Hall conductivity depends only on the magnitude of the magnetic field. We also find that and the spin-Hall conductivity tends to zero when magnetic field comes infinite, which stems from the fact that the increasing magnetic field would increase the energy gap.

論文審定書 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .i
中文摘要 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ii
英文摘要 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .iii
目錄 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .iv
圖次 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .v
第一章 介紹與研究動機 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1
1.1
自旋霍爾效應的發展簡史 . . . . . . . . . . . . . . . . . . . . . . . .1
1.2
自旋軌道作用與磁場 . . . . . . . . . . . . . . . . . . . . . . . . . . .4
第二章 基本定理與模型 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.1
從狄拉克方程式到包立方程式 . . . . . . . . . . . . . . . . . . . . . .8
2.2
k · p 微擾 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3
Kane 及 Rashba 模型 . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.4
外加垂直磁場的哈米爾頓方程 . . . . . . . . . . . . . . . . . . . . . . 15
第三章 自旋霍爾電導率 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.1
久保公式 (Kubo formula) 中的電導率 . . . . . . . . . . . . . . . . . 18
3.2
傳統霍爾自旋電導率 . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.3
k-線性系統傳統自旋電導率 . . . . . . . . . . . . . . . . . . . . . . . 23
3.4
出平面磁場與傳統自旋電導率 . . . . . . . . . . . . . . . . . . . . . . 27
第四章 外加磁場下的自旋霍爾電導率 . . . . . . . . . . . . . . . . . . . . . . . . 34
4.1
Rashba 系統. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.2
Rashba - Dresselhaus 系統 . . . . . . . . . . . . . . . . . . . . . . . . 38
第五章 結論 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
參考文獻 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
附錄 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

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