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研究生：	林育陞 Yu-Sheng Lin
論文名稱：	Spin Transport Properties in Magnetic Heterojunctions: Analytical derivation in Green’s function and Multi-reflection process
指導教授：	唐毓慧 Yu-Hui Tang
口試委員:
學位類別：	碩士 Master
系所名稱：	理學院 - 物理學系 Department of Physics
論文出版年：	2020
畢業學年度：	108
語文別：	中文
論文頁數：	89
中文關鍵詞：	FLST 、STT 、格林函數、電子穿隧模型、厚屏障、諧振穿隧、機制、通式、解析解、數值解
外文關鍵詞：	FLST, STT, Green's function, Electron tunneling model, Thick barrier, Resonant tunneling, Mechanism, General expression, Analytical, Numerical
相關次數：	點閱：24 下載：0
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非易失性磁性隨機存取存儲器(magneto resistive random access memory, MRAM)。目前已被廣泛的研究。實驗上可以利用磁場和電流控制電極磁矩的偏向，當自旋極化電子的電流通過電極時，電子的自旋角動量將與電極相互作用。導致電極的磁化方向感受到“扭矩”。進而產生MRAM中兩種重要的行為：讀取和寫入。目前產生不平衡的自旋電流主要有兩種方法，一種是使用鐵磁材料費米能量處不平衡的電子自旋產生。另一種則是利用自旋過濾(Spin-filter, SF)的方式。
在理論計算上，可以使用第一原理搭配電子密度理論(Density Function Theory, DFT)下去做迭代計算。這種方法花費時間相對較長，且模型複雜。而緊密束縛模型(Tight-binding model, TB)搭配格林函數(Green’s function)用來處理自旋電子學(Spintonics)這種將電子結構簡化過後的模型雖然會有過度簡化之嫌，卻因為其具有計算速度快，以及模型物理分析上的優勢。在處理一些複雜的非共線性(non-collinear) 磁性結構 (magnetic tunnel junction, MTJ) 中，緊密連結模型就扮演了一個預測結果的腳色。比如在以氧化鎂和鐵組成的MTJ (Fe/MgO/Fe)的計算中，因為MgO只有貢獻了Δ_1的能帶，緊密連結系統將可以發揮其作用。
這篇研究上，我們會首先著重討論在多層材料中非共線性系統下的平行(T_∥)和垂直(T_⊥ )自旋磁矩如何以共線性系統下的通式表達。此篇將會以自旋過濾為基底的層磁性穿隧接合 FM/I/SF/I/FM 或是四層穿隧接合FM/I/SF/NM作為例子，利用非平衡態的格林函數和單帶緊束縛模型(single band tight binding)來計算自旋電子的穿隧特性。其中五層磁性穿隧接合的結果中，考慮到SF屏障與FM電極之間的多重交互作用過程， SF屏障在T_⊥≫T_∥中發揮了龐大的作用。在先前的研究中，我們發現格林函數可以藉由戴森方程組(Dyson equation)展開，因此首先我們會分析每條路徑的重要性，接著使用分析後的結果，在厚障壁電位下給出一套更完整的T_∥和T_⊥的通式。
同時我們也會合併格林函數以及電子穿隧模型，以較為完整的方式解釋T_∥以及T_⊥的成因和機制。我們發現T_∥將由電流主導，T_⊥則是和層間交換耦合作用(interlayer exchange coupling, IEC)有關。因此，我們的推導可能可以為新型MTJ的傳輸機制提供一種新型態的方法。然而在能帶區，多重反射的模型，和以線性系統下的表達式將需要被修正。我們期待這些研究能夠為磁性材料的性質帶來一些新的理解，並期待能提供有效解釋實驗量測的結果。

Recently, the non-volatile magnetic random access memory (MRAM) has been extensively studied, where magnetic field and current can be used to control the magnetic moment of ferromagnetic (FM) material. When the spin-polarized current is injected into a noncollinear FM electrode, the spin angular momentum transfers to the FM electrode and causes the “spin torque” to switch the magnetization direction of the FM electrode. There are two important behaviors in MRAM: read and write. At present, there are mainly two methods for generating an unbalanced spin current. One is to generate an unbalanced electron spin at the Fermi energy of a ferromagnetic material. The other is to use Spin-filter.
For spintronics simulation, the Density Function Theory (DFT) based first principle calculation is usually applied to perform iterative calculation but very time consuming. On the other hand, the tight-binding (TB) model with nonequilibirum Green's function method (NEGF) is much easier to analysis the physical underlying mechanism but simplify the electronic structure. In dealing with some complex non-collinear magnetic tunneling junctions (MTJs), i.e. Fe/MgO/Fe MTJs, the TB model plays a significant role in predicting noncollinear spin torque effect.
In this study, we first focus on how the spin transfer torque (STT) and field-like spin torque (FLST) spin magnetic moments in non-collinear systems in multilayer materials are expressed in the general formula of collinear systems. For the spin-filter-based (FM / I / SF / I / FM) MTJs, we employ the single band tight binding model with NEGF method to simulate the tunneling characteristics of spin electrons. In consideration of multiple interaction processes between the SF barrier and the FM electrodes, surprisingly, the SF barrier cause FLST>>STT. The Dyson equation is applied to expand the Green's function, analysis the importance of each path, and then further derive general formula of FLST and STT under thick barrier condition.
To gain deeper understanding of noncollinear spin torque effect, we further employ the electron tunneling model to investigate the underlying mechanisms of STT and FLST in MTJs. We find that STT is dominated by spin current, while FLST is correlated to the interlayer exchange coupling. Thus, our derivation may provide a new state method for the transmission mechanism of new MTJ. We hope our research can provide new insights of noncollinear spin torque driven spintronics devices.

Contents
Chapter 1 Introduction    1
Chapter 2 Basic LLG and Green’s function theory    3
1 Landau-Lifshitz-Gilbert equation    3
1.1 LL equation    3
1.2 LLG equation    3
1.3 LLGS equation    4
2 Green’s function    5
2.1 Introduction of Green’s function    5
3 Computation model with Keldesh Green’s function    10
3.1 Tight binding chain    10
3.2 Infinite lead    11
3.3 Semi-infinite lead    12
3.4 Green’s function in isolated Green’s functions form    13
3.5 Perturbation in t order in high barrier    15
3.6 Spin torque    15
Chapter 3 Multi-reflection process in high or thick barrier with Green’s functions: FM/I/SF/I/FM    18
1 Junction structure and parameter – FM/I/SF/I/FM    18
2 Multi-reflection contribution FLST and STT on 5-Layer    22
2.1 FLST    22
2.2 STT    27
2.3 Summary of contribution STT and FLST    28
3 General expressions of FLST and STT    29
3.1 General expression FLST    29
3.2 General expression STT    30
3.3 Examination of FLST and STT in general expression    31
4 Extra rotation in spin filter effect on general expression    34
4.1 Thick barrier assumption for FLST    34
4.2 Thick barrier assumption for STT    35
Chapter 4 Physical interpretation of spin torque in high and thick barrier with electron tunneling model    36
1 Electron tunneling in continuous hetero lead case without spin    37
2 Electron tunneling in discreet hetero lead without spin    39
3 Mechanism of I and spin torque in electron tunneling model    42
3.1 Current    43
3.2 Mechanism of STT    43
3.3 Mechanism of FLST    46
4 Mechanism of I and spin torque in LLG    49
Chapter 5 Resonant tunneling    50
1 The limitation of the expansion form for Green’s function    50
2 Contribution of collective reflection path for FLST    51
3 Numerical results for FM/I/SF/I/FM junction    52
3.1 Current, Dos, FLST, STT and Transmission    52
4 General expression and angle dependence of STT and FLST    59
4.1 Discussion under thick barrier    60
4.2 Discussion under resonant barrier    60
Chapter 6 Conclusion    61
Appendix A. 4-layer structure Green’s function    62
Appendix B. Demonstration of 5-layer structure with Green’s function    64
Appendix C. Path ordering calculation by programing    66
Appendix D. Calculation detail in resonant state    68
D.1 K space projection on energy    68
D.2 K space integration    68
D.3 Adaptive integral in very elementary way    69
Appendix E. Derivation detail of STT and FLST in 5-Layer junction    71


                                

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