為了找尋適當的方法描述自旋霍爾效應中多變的實驗結果，在本篇論文中，我們著重在外加電場下重電洞半導體系統產生k-cubic Rashba 系統的自旋霍爾電流，並利用兩種方法計算自旋霍爾電導率的值：密度矩陣(Density matrices)和與時間相關的自旋動力學，前者能計算出呈現弛豫時間機制的自旋霍爾電流，相較於實驗[J. Wunderlich, et al, Phys. Rev. Lett. 94, 047204 (2005)]，我們發現即使系統是處在純淨極限(clean limit)的條件下其內秉自旋霍爾電導率仍大於實驗值，之後我們轉而計算後者並利用海森堡的方法去計算與時間相關的自旋動力學，不同於先前的文獻將時間趨近於零假設預設為系統衰退至平衡態，我們發現時間項在微擾展開至電場的一次項後可得相近的式子，同時還可直接計算與時間相關的自旋霍爾電流，有趣的是，我們也發現在k-cubic Rashba 系統中每一個不同動量的重電洞粒子會有不同的自旋進動比例，此外，若假設觀測值是平均於一週期的自旋進動則自旋霍爾電導率會漸進於久保公式(Kubo formula)，其中平均的設定乃基於自旋進動一週期的時間與載子在材料中傳輸時間的比較，從某個意義上來說，我們是重現了久保公式中產生的自旋霍爾電流但結果仍然大於實驗值，k-cubic Rashba 系統中的自旋動力學最終對自旋電流沒有顯著的影響。或許如同先前文獻中所示，實驗自旋霍爾電導率會降低是由於樣品厚度使自旋電流流入其他方向所致。

In order to search proper method describing the various experimental results of spin-Hall effect, in this thesis, we focus on the spin-Hall current response to applied electric field in the heavy-hole semiconductor system, which is k-cubic Rashba system. We calculate the value of spin hall conductivity by using two methods: the density matrix method and time-dependent spin dynamics method. The former enables us to calculate the spin-Hall current in the presence of relaxation mechanisms. Compare our result with experiment [J. Wunderlich, et al, Phys. Rev. Lett. 94, 047204 (2005)], we find that, even the system is in the clean limit, the intrinsic spin-Hall conductivity is still larger than the experimental value. We turn to the latter case and calculate the time dependent spin dynamics by using Heisenberg method. Unlike the previous literatures, the time decaying to equilibrium is assumed to be very closed to zero, we find that the time component in the perturbation expansion to first order of electric field can have a closed form and we can exactly calculate the time-dependence of spin–Hall current. Interestingly, we find that each heavy hole particle with different momentum have different spin precession rate in the k-cubic Rashba system. Furthermore, the spin-Hall conductivity would asymptotically approach Kubo formula if we assume that the observed value should be the average in a periodic time of spin precession. The justification of the average is based on the comparison of spin precession time in a period and carrier transportation time across the sample. In this sense, our result reproduces the Kubo formula, and the resulting spin-Hall current is still larger than the experimental value. The spin dynamics in the k-cubic Rashba system has no significant effect on the spin current. The reduced spin-Hall conductivity in the experiment may be due to the thickness of the sample in which the spin current can flow into different direction, as shown in the previous literature.

論文審定書 i
致謝 ii
中文摘要 iii
英文摘要 iv
目錄 vi
圖次 vii
表次 viii
第一章 介紹與研究動機 01
1.1 自旋電子學的發展歷史與優勢 01
1.2 內秉自旋霍爾效應的介紹 03
第二章 內秉自旋霍爾效應的哈密頓方程式 10
第三章 內秉自旋霍爾效應中的弛豫時間 13
3.1 自旋電流的定義 13
3.2 密度矩陣的應用 14
3.3 自旋電流算符的演算 22
3.4 自旋霍爾電導率的計算 30
第四章 自旋算符的動力學系統 36
4.1 薛丁格算符在外加電場下的漸進式展開 36
4.2 含時間項的自旋算符演算 38
4.3 自旋霍爾電導率的計算 50
第五章 結論與預測 57
附錄 A 58
參考文獻 59

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