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論文名稱 Title |
纖鋅結構之塊材與量子井中的自旋分裂 Spin Splitting in Bulk Wurtzite Materials and Their Quantum Wells |
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系所名稱 Department |
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畢業學年期 Year, semester |
語文別 Language |
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學位類別 Degree |
頁數 Number of pages |
75 |
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研究生 Author |
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指導教授 Advisor |
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召集委員 Convenor |
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口試委員 Advisory Committee |
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口試日期 Date of Exam |
2011-07-15 |
繳交日期 Date of Submission |
2011-08-01 |
關鍵字 Keywords |
自旋分裂、Rashba效應、纖鋅礦、Dresselhaus 效應、量子井、原子軌道線性組合 spin slitting, LCAO, Wurtzite, Rashba effect, Dresselhaus effect, quantum well |
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統計 Statistics |
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中文摘要 |
利用原子軌道線性組合之方法研究應變下氮化鋁之自旋分裂。在雙能帶k.p 模型Hso=(αwz-γ’k2//+λ’k2z)(σxky-σykx)+H0so, 發現應變與晶格場不僅會引起一次線性k(αwz )還有三次方k 項( γ’and λ’ )。在這裡H0so=(-γ0k2//+λ0k2z)(σxky-σykx) 是對應於理想的纖鋅結構且會產生圓錐形的最小自旋分裂面。當雙軸應變增加時, 最小自旋分裂面的形狀會從雙葉雙曲面變成圓錐形,最後成了單葉雙曲面。 此外,也利用原子軌道線性組合之方法研究A面與M面纖鋅結構量子井中自旋分裂。在量子井中, 自旋分裂主要是來自於線性一次方項的貢獻, 但是在大k// 時三次方項的貢獻就不可以被忽略了。用雙能帶k.p 模型擬合LCAO的數據可以得到線性一次方項與三次方項的數值。當量子井的厚度增加時, 線性一次方項與三次方項的數值是在減少的。 |
Abstract |
The spin-splitting energies in strained bulk wurtzite aluminum nitride (AlN) are studied using the linear combination of atomic orbital method. It is found that strain and crystal field induce not only a linear-k (αwz ) but also two cubic-k terms (γ’and λ’ ) in the two-band k.p Hamiltonian Hso=(αwz-γ’k2//+λ’k2z)(σxky-σykx)+H0so, where H0so=(-γ0k2//+λ0k2z)(σxky-σykx) is for ideal wurtzite and generates a cone-shaped minimum-spin-splitting (MSS) surface. As biaxial strain increases, the shape of the MSS surface changes from a hexagonal hyperboloid of two sheets in unstrained AlN to a hexagonal cone, and eventually becomes a hyperboloid of one sheet. The spin-splitting energies of first conduction band for A-plane and M-plane wurtzite are calculated by the sp3 linear combination of atomic orbital (LCAO). The results show the spin-splitting energies are dominated by linear-k term but contribution of cubic-k terms can not be neglected for larger k//. The parameter of linear-k and cubic-k terms are evaluated from the LCAO calculated spin-splitting energies fitting to two band k•p model as increasing well width. The coefficients of linear-k and cubic-k terms decrease. |
目次 Table of Contents |
中文摘要…………………………………………………………………………………i Abstact……………………………………….…………………………………………..ii Chapter 1 Introduction...........……………………………………………………………………….01 1-1 Bulk inversion asymmetry in bulk zinc-blende……………………………………..…….....01 1-2 Bulk inversion asymmetry in bulk wurtzite………………………………………..………...06 1-3 Minimum-spin-splitting (MSS) surface in bulk wurtzite…..……………………….………..10 1-4 Spin splitting in quantum wells ……….…………………………………………………......11 Chapter 2 Methods…………………………………………………………………………..…….…..12 2-1 Atomic structure of bulk wurtzite………………………………………………………..….12 2-2 The Hamiltonian without spin-orbital interaction for ideal wurtzite………………………..12 2-3 The Hamiltonian with spin-orbital interaction for ideal bulk wurtzite……………………...16 2-4 The Hamiltonian with spin-orbital interaction for real or strained wurtzite………………..20 2-5 The Hamiltonian for wurtzite quantum wells …………………………………………….23 2-6 Two-Band k.p model in wurtzite structure…………………………………………...…….28 Chapter 3 Results……………………………………………………………………………………...30 3-1 The shape of the spin-degenerate surface in the two-band k•p model…………….………..30 3-2 Ideal bulk wurtzite…………………………………………………………………………..34 3-3 Real and strained bulk wurtzite………………………………………………………..…….40 3-4 The spin splitting of quantum wells…………………………………………………………53 Chapter 4 Conclusions........................................................................................................................65 References……………………………………………………………………………………...…..…...66 Figures Fig. 1-1 Resonant spin lifetime transistor………………………………………………………….….…..03 Fig. 1-2 Crystal structure of zinc-blende ………………………….……………………………….….….04 Fig. 1-3 Structure inversion asymmetry…….…………………………………………………….….……05 Fig. 1-4 Crystal structure of wurtzite....................…………………………………………….….……….07 Fig. 1-5 Schematic of atomic arrangement for ideal wurtzite ……..…………………….….…………….08 Fig. 1-6 Schematic of atomic arrangement for real or stained wurtzite …………………………………..09 Fig. 2-1(a) Wurtzite structure.......………………………………………………………………….….…..13 Fig. 2-1(b) Reciprocal lattice………………………………………………………………………….......13 Fig. 2-2 The ratio of lattice constants c to a as a function of the biaxial strain ...…………………....…21 Fig. 2-3 The internal parameter u as a function of a biaxial strain ……………………………………….22 Fig. 2-4(a) The top view of the reciprocal lattice……………………………………………….…….…..25 Fig. 2-4(b) A-plane quantum wells………………………………………………………………………..26 Fig. 2-4(c) M-plane quantum wells………………………………………………………………………..27 Fig. 3-1 (a) Spin-degenerate surface in the shape of cone…………………………………………...…...31 Fig. 3-1 (b) Spin-degenerate surface in the shape of hyperboloid of one sheet……………………….….32 Fig. 3-1 (c) Spin-degenerate surface in the shape of hyperboloid of two sheets………………………....33 Fig. 3-2 (a) Spin-splitting energies as a function of k// on the plane of kz=0.0 (p/c)……………………...36 Fig. 3-2 (b) Spin-splitting energies as a function of k// on the plane of kz=0.3 (p/c)……………………...36 Fig. 3-2 (c) Spin-splitting energies as a function of k// on the plane of kz=0.5 (p/c)……………………...36 Fig. 3-3 (i) Projection of minimum spin-splitting and spin-degenerate surface onto kz=0.2 (p/c)……….37 Fig. 3-3 (ii) Projection of minimum spin-splitting and spin-degenerate surface onto kz=0.4 (p/c)……....37 Fig. 3-3 (iii) Projection of minimum spin-splitting and spin-degenerate surface onto kz=0.6 (p/c)……...37 Fig. 3-4 Three-dimensional plot of minimum spin-splitting surface for a ideal AlN wurtzite..………......39 Fig. 3-5 (a) Spin-splitting energies as a function of k// under external strain 0.00% with kz=0.0 (p/c)…...42 Fig. 3-5 (b) Spin-splitting energies as function of k// under external strain 1.57% with kz=0.0 (p/c)…….42 Fig. 3-5 (c) Spin-splitting energies as function of k// under external strain 2.00% with kz=0.0 (p/c)…….42 Fig. 3-6 (a) The value of awz in two-band k.p Hamiltonian as a function of external strain…………….43 Fig. 3-6 (b) The value of g in two-band k.p Hamiltonian as functions of external strain………...……..44 Fig. 3-6 (c) The value of l in two-band k.p Hamiltonian as functions of external strain……….............45 Fig. 3-6 (d) The value of awz.g wz in two-band k.p Hamiltonian as a function of external strain………..46 Fig. 3-7 (a) The minimum spin-splitting surface for a unstrained wurtzite…...…………………………..50 Fig. 3-7 (b) The minimum spin-splitting surface for a tensile strain 2.00 % wurtzite…………………….51 Fig. 3-7 (c) The minimum spin-splitting surface for a tensile strain 3.00 % wurtzite..…………………...52 Fig. 3-8 (a) Spin-splitting energies as function of k’ for A-plane QWs with LW=4.6 A at k’z=0.0…..........56 Fig. 3-8 (b) Spin-splitting energies as function of k’ for A-plane QWs with LW=10.8 A at k’z=0.0…........56 Fig. 3-8 (c) Spin-splitting energies as function of k’ for A-plane QWs with LW=18.5 A at k’z=0.0………56 Fig. 3-9 (a) Spin-splitting energies as function of k’ for M-plane QWs with LW=5.30 A at k’z=0.0……...57 Fig. 3-9 (b) Spin-splitting energies as function of k’ for M-plane QWs with LW=10.7 A at k’z=0.0….......57 Fig. 3-9 (c) Spin-splitting energies as function of k’ for M-plane QWs with LW=18.7 A at k’z=0.0….......57 Fig. 3-10 (a) Spin-splitting energies of A-plane QWs with LW=4.6 A and k’//=0.05 (p/c)……..................58 Fig. 3-10 (b) Spin-splitting energies of A-plane QWs with LW=4.6 A and k’//=0.10 (p/c)…..…...............59 Fig. 3-10 (c) Spin-splitting energies of A-plane QWs with LW=4.6 A and k’//=0.23 (p/c)....…..................60 Fig. 3-11 (a) Spin-splitting energy of M-plane QWs with LW=5.3 A and k’//=0.01 (p/c)……....................61 Fig. 3-11 (b) Spin-splitting energy of M-plane QWs with LW=5.3 A and k’//=0.078 (p/c)…….................62 Fig. 3-11 (c) Spin-splitting energy of M-plane QWs with LW=5.3 A and k’//=0.1 (p/c)…..…...................63 Fig. 3-12 (a) The value of a IA in the two-band k.p Hamiltonian as functions of well width….…………64 Fig. 3-12 (b) The value of g IA in the two-band k.p Hamiltonian as functions of well width..………..…64 Fig. 3-12 (c) The value of l IA in the two-band k.p Hamiltonian as functions of well width…………....64 |
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