臺灣博碩士論文加值系統

此論文的工作一直致力解讀在一個圓盤微結構附近，考慮Dresselhaus 型自
旋軌道耦合作用下自旋相關的散射效應。Dresselhaus 型自旋軌道耦合作用在這
裡主要包括了linear-k 相關以及cubic-k 相關的貢獻。
以分波方法為基礎，在散射區間內經計算後可以得到完整散射後的波函數。
透過調查入射電子平面波後，DSOI 造成空間辦別的散射效應，linear-k 與cubic-k
DSOI 貢獻的差異可以明顯地被辨識，同時得到所對應的能量耗散關係。在我們的發
現：對於linear-k Dresselhaus，電子自旋密度與機率密度分佈擁有空間對稱性
輪廓，與平面波入射角度無關。相反地，在cubic-k Dresselhaus SOI 例子中明顯
地表示出與平面波入射角度相關。
特別地，若入射平面波角度為幾個特定的角度，我們可以發現有相似的電子
自旋密度對應在cubic-k Dresselhaus 例子與linear-k Dresselhaus 例子之
間。

This thesis work has devoted to the study of spin-dependent scattering e®ects from a
circular-disk microscopic structure with Dresselhaus-type spin-orbit coupling. The Dres-
selhaus spin-orbit coupling considered here includes both contributions terms for one is
linear-k dependence and the other is cubic-k dependence.
Based on the method of partial waves, the complete scattering wave function in a
circular scattering region can be rigorously derived and obtained. Through investigating
their spatial-resolved scattering behaviors from linear and cubic Dresselhaus-type SOI
disk under the electron plane wave incidence, di®erent DSOI contributions can be appar-
ently discerned, and their corresponding detail energy dispersion relationships as well. In
our ‾ndings: for linear-k Dresselhaus case, the spin density and probability density distri-
butions own their spatial symmetry pro‾le, which is featured independence of the plane
wave incident angle. On the contrary, strong incident angle dependence is manifested for
the case of cubic-k Dresselhaus spin-orbit interaction.
In particular , for incidence plane wave in some characteristic angle, we can ‾nd similar
spin density responses between cubic-k Dresselhaus case and linear-k Dresselhaus case.

Abstract in Chinese i
Abstract in English ii
Acknowledgement iii
1 Introduction 1
1.1 Background : types of spin-orbit coupling system in solid state system . . 2
1.2 Motivation : cubic-k Dresselhaus spin-orbit interaction (SOI) . . . . . . . 3
1.3 A simple guide to thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Two dimensional Dresselhaus-type SOI electron system 5
2.1 Linear-k Dresselhaus SOI . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.1.1 Incident plane wave . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1.2 Cylindrical form representation of the eigenstates . . . . . . . . . . 12
2.2 Cubic-k Dresselhaus SOI . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2.1 Incident plane wave . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2.2 Cylindrical form representation of the eigenstates . . . . . . . . . . 17
2.3 Energy dispersion of Dresselhaus SOI system . . . . . . . . . . . . . . . . . 20
2.4 BIA in spin splitting in 2D systems . . . . . . . . . . . . . . . . . . . . . . 24
2.5 Connection between linear-k DSOI and ROSI systems . . . . . . . . . . . . 25
3 Scattering from a cylindrically symmetric potential in a Dresselhaus SOI
system 28
3.1 Cubic-k Dresselhaus SOI . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.1.1 Coupled cylindrical wave representations of the incoming wave . . . 29
3.1.2 Coupled cylindrical wave representations of the outgoing wave . . . 31
3.2 Linear-k Dresselhaus SOI . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.2.1 Coupled cylindrical wave representations . . . . . . . . . . . . . . . 36
3.3 Scattering of the scattering state . . . . . . . . . . . . . . . . . . . . . . . 39
3.4 Spin density of the scattering state . . . . . . . . . . . . . . . . . . . . . . 39
3.5 Particle continuity equation . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4 Numerical results and discussions 44
4.1 Results for linear-k Dresselhaus SOI . . . . . . . . . . . . . . . . . . . . . 44
4.2 Results for cubic-k Dresselhaus SOI and more Discussions . . . . . . . . . 48
4.3 Discussion on the connection of spin density with the plane wave direction 49
5 Future work 50
A Simplify the boundary condition problem 51
B Derivation of spin density of the total wave function after scattering 55
C Comparison of analytical results and numerical approach for cylindrical
wave case 60

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