在Dresselhaus-type 介觀量子環中的持續性電流和自旋密度

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(1)國立交通大學電子物理研究所碩士論文在 Dresselhaus-type 介觀量子環中的持續性電流和自旋密度 PERSISTENT CURRENT AND SPIN DENSITY IN A MESOSCOPIC DRESSELHAUS-TYPE QUANTUM RING. 研究生：吳智偉指導教授：朱仲夏教授. 中華民國九十八年七月.

(2) 在 Dresselhaus-type 介觀量子環中的持續性電流和自旋密度 PERSISTENT CURRENT AND SPIN DENSITY IN A MESOSCOPIC DRESSELHAUS-TYPE QUANTUM RING. 研究生：吳智偉. Student：Chih-Wei Wu. 指導教授：朱仲夏教授. Advisor：Prof. Chon Saar Chu. 國立交通大學電子物理研究所碩士論文 A Thesis Submitted to Department of Electrophysics College of Science National Chiao Tung University in Partial Fulfillment of the Requirements for the Degree of Master in Electrophysics. July 2009 Hsinchu, Taiwan, Republic of China. 中華民國九十八年七月.

(3) 在 Dresselhaus-type 介觀量子環中的持續性電流和自旋密度. 研究生：吳智偉. 指導教授：朱仲夏教授. 國立交通大學電子物理研究所. 摘要本論文的工作主要是去探究以及了解在有限寬的介觀環裡 Dresselhaus 自旋軌道交互作用效應的物理意涵，包含有和沒有外加磁通量兩種情況。而自旋密度和持續性電流是我們關注的物理量。特別地，我們藉由個別獨立打開 Dresselhaus 自旋軌道交互作用和磁通量，去仔細分析本徵態和能譜來得到物理意涵。在打開 Dresselhaus 自旋軌道交互作用強度的情況下，我們證明所有的能階都是 Kramer’s type 的雙重簡併，以及藉由能階排斥原理和相對基底本徵態比重來瞭解能譜的趨勢。在弱自旋軌道交互作用場的範疇下，能譜和 Dresselhaus 自旋軌道交互作用強度成二次方關係，此結果與我們的微擾分析相同。在漸增磁通量的情況下，我們呈現了 Kramer 簡併態的分裂，以及確認分裂後能譜的物理原因。對於在打開 Dresselhaus 自旋軌道交互作用或磁通量後的本徵態，我們發展了一套有系統的方法去得到它的通式。我們也計算了一個本徵態所對應的自旋密度和淨 z 方向投影自旋量。當系統中有 N 個電子時，總合每個本徵態就能得到全部的總量。. i.

(4) Persistent current and spin density in a mesoscopic Dresselhaus-type quantum ring Student: Chih-Wei Wu. Advisor: Chon-Saar Chu Department of Electrophysics National Chiao-Tung University. Abstract The work of this thesis is to explore and to understand physical insights on the effects of Dresselhaus spin-orbit interaction (DSOI) in a finite width mesoscopic ring, both with and without a magnetic flux. The physical quantifies of interest are the spin density and persistent current. Specifically, our insight is obtained from a detail analysis of the eigenstates and the energy spectrum as the DSOI is turned on and, independently, when the magnetic flux is turned on. For the case of turning on of the DSOI strength, we demonstrate that all energy levels are doubly degenerate, of the Kramer’s type, and the trend of the energy spectrum is understood by level repulsions and the relative weighting between the constituent bases state ket. The energy spectrum depends on the DSOI quadratically in the weak SOI field regime, and this is consistent with our perturbation result. For the case of increasing the magnetic flux, we show that the Kramer degenerate states split and the physical reason for the subsequent order in the energy splitting is identified. For the eigenstate in the presence of the DSOI and the magnetic flux, we have developed a systematic way to produce its general form. We have calculated the spin density and the net spin Sz of an eigenstate. Summing these gives us the total when there are N electrons in the system.. ii.

(5) 致謝. 自我實現，一直是我成長的基本動力，我有夢，就會勇敢去追。回想起二年前，在教書之餘，抓緊時間挑燈夜讀，為的不僅只是提升專業知識、攻讀學位，更重要的是自我實現的完成。如今，正值收割豐碩果實的時候，我心存感激，由衷感謝朱老師仲夏的耐心指導，不管是在研究上或做人處事的態度上都讓我獲益良多，而研究室的學長同學們也在各方面給予我莫大的幫助，尤其是律堯和志宣，慶幸有他們的協助，我才能如期完成研究。在交大七百多個日子裡，很感謝研究室的夥伴們陪我一同度過，志雄、律堯、榮興、志宣、吉偉、昆宜、冠誼，重訓室的肌肉男們陪我一起成長，瑞甫、冠旭、家權、阿草、霸子…，第二餐廳裡老陳快餐的歐巴桑偷偷免費幫我加滷蛋，以及我的女友怡佩，一路上的支持與鼓勵，讓我沒有後顧之憂。謝謝你們，謝謝!. iii.

(6) Contents Abstract in Chinese. i. Abstract in English. ii. Acknowledgement. iii. Contents. iv iii. List of Figures. vi iii. 1 Introduction. 1. 1.1. Spin-orbit coupling in solid state systems . . . . . . . . . . . . . . . . . . .. 2. 1.2. Introduction to persistent current . . . . . . . . . . . . . . . . . . . . . . .. 5. 1.2.1. One-dimension ring with an external magnetic flux . . . . . . . . .. 5. 1.2.2. Persistent current in mesoscopic rings with spin-orbit coupling . . .. 8. 2 Finite width mesoscopic Dresselhaus spin-orbit (DSO) ring 2.1. 2.2. 15. Model of a mesoscopic ring with Dresselhaus spin-orbit coupling . . . . . . 15 2.1.1. Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15. 2.1.2. Dimensionless . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19. 2.1.3. Eigenenergy and Eigenstate . . . . . . . . . . . . . . . . . . . . . . 20. Numerical results and discussion . . . . . . . . . . . . . . . . . . . . . . . . 22 2.2.1. Energy spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22. 2.2.2. Spin density and net magnetic dipole moment . . . . . . . . . . . . 27 iv.

(7) CONTENTS 3 Finite width mesoscopic Dresselhaus spin-orbit (DSO) ring with a magnetic flux 3.1. 31. Theoretical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.1.1. Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31. 3.1.2. Dimensionless . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33. 3.1.3. Eigenenergy and Eigenstate . . . . . . . . . . . . . . . . . . . . . . 34. 3.2. Expression for charge current density . . . . . . . . . . . . . . . . . . . . . 35. 3.3. Numerical results and discussion . . . . . . . . . . . . . . . . . . . . . . . . 37 3.3.1. Energy spectrum - Removal of degeneracy . . . . . . . . . . . . . . 37. 3.3.2. Persistent charge current . . . . . . . . . . . . . . . . . . . . . . . . 41. 4 Conclusion and future work. 45. 4.1. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45. 4.2. Future works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47. A Solve the finite width mesoscopic ring Hamiltonian with DSO coupling 48 B Solve the finite width mesoscopic ring Hamiltonian with DSO coupling after considering magnetic flux. 52. C Prove state |ψa i and |ψb i do not mixing. 55. D Demonstrate that the energy level split into the higher energy part and the lower energy part. 57. E The table of the eigenstates. 59. v.

(8) List of Figures 1.1. (a) Electro-optic light modulator. The polarizer makes the input light polarized at an angle of 45o with respect to the y axis; (b) Spin-polarized FET proposed by Datta et al. [4] in 1990. The iron contacts playing the roles of polarizer and analyzer are made of ferromagnetic materials. These figures were plotted by Datta et al. [4] in 1990. . . . . . . . . . . . . . . .. 1.2. Band structure of a linear lattice with lattice constant L. The figure was obtained from Kronig-Penney model. [8] . . . . . . . . . . . . . . . . . . .. 1.3. 6. The single-electron energy state En (Φ) of the normal mental one-dimensional ring as a function of flux proposed by Buttiker et al.[9] . . . . . . . . . . .. 1.4. 3. 7. A mesoscopic ring with Rashba spin-orbit coupling threaded by magnetic flux. The external magnetic flux is in z-direction and the angle θ between z and r parameterizes the strength of the RSO coupling. This figure was plotted by Splettstosser et al. in 2003. [10] . . . . . . . . . . . . . . . . . .. 1.5. 9. It is the energy spectrum for the ideal one-dimension ring with a deltabarrier impurity. Parameters are impurity strength constant A = 0.1 and title angle cos θ = 2/5. The energy levels are shifted in flux direction by 1/ cos θ. Solid lines correspond to spin-up states in the local spin-frame basis while dashed line to spin-down states. This figure was plotted by Splettstosser et al. in 2003. [10] . . . . . . . . . . . . . . . . . . . . . . . .. vi. 9.

(9) LIST OF FIGURES 1.6. Persistent charge current-flux characteristics for a set of values for the spinorbit coupling strength. The total number of electrons is 4N in panel (a). 4N + 2 in panel (b), and 2N + 1 in panel (c) with the impurity strength constant A=0.1. These figures were plotted by Splettstosser et al. in 2003. [10] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. 1.7. In even number electrons 4N + 2 case, comparison of the persistent spin current (dash line) with the charge current (solid line) of the ring. The spin current is for spin projection onto the local spin frame and parameters are the impurity strength constant A = 0.5, the tilt angle cos θ = 0.66. This figure was plotted by Splettstosser et al. in 2003. [10] . . . . . . . . . . . . 13. 1.8. Persistent spin currents of the ring with odd number electrons 2N + 1 (dotted line) and even number electrons 4N + 2 (dashed line). Parameters are impurity strength constant A = 0.5 and tilt angle cos θ = 0.9 corresponding to a small spin-orbit coupling strength. The figure was plotted by Splettstoesser et al. [10] . . . . . . . . . . . . . . . . . . . . . . . . . . . 13. 2.1. The figure is the cross section of the finite width mesoscopic ring, where r1 and r2 are the inner and outer radii.. 2.2. . . . . . . . . . . . . . . . . . . . . . 16. The energy spectrum for the quantum ring which is determined from the determinant equation Eq. (2.23) and Eq. (2.24), and it is shown the effects caused by the DSOI on the first seven quantum ring energy levels in units e These of Ef , as a function of the dimensionless DSO coupling constant β. doubly degeneracy states come from the superposition of the different unperturbed eigenstates respectively. The physical parameters are obtained o. for the case of GaAs: β = 25eV · A, m∗ ≈ 0.067me , λf = 40nm, r1 = 10nm, o. r2 = 70nm, d = 50 A.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23. vii.

(10) LIST OF FIGURES 2.3. The relative weighting between the constituent bases state ket plays an important role to effect the trend of the energy spectrum. When the Dresselhaus strength is small: (a) c1 |E1,3,↑ i has more weight than c2 |E1,2,↑ i in state |ψa i, (b) c4 |E1,−3,↓ i has more weight than c3 |E1,−2,↑ i in state |ψb i, (c) c6 |E1,3,↓ i has more weight than c5 |E1,4,↑ i in state |ψc i, (d) c7 |E1,−3,↑ i has more weight than c8 |E1,−4,↓ i in state |ψd i. . . . . . . . . . . . . . . . . 25. 2.4. In the small Dresselhaus strength region, 0 ≤ βe ≤ 0.1, we can get the trend of the energy level which is represented in the up (down) quadratic form. The physical parameters are the same with Fig. 2.2.. 2.5. . . . . . . . . . . . . 26. It is the z-projection spin density for each states as a function of the dimensionless radial length which the angular is fixed to zero.(a) |ψa,m i states, (b) |ψb,m i states, (c) |ψc,m i states, (d) |ψd,m i states. . . . . . . . . . . . . . 28. 2.6. It is the z-projection spin density as a function of the radial ratio. ρ−r1 r2 −r1. which the angular is fixed to zero.(a) radial ratio in different range, (b) radial ratio is set as big as possible. . . . . . . . . . . . . . . . . . . . . . . 29 3.1. (a) Several lowest dimensionless eigenenergies E as a function of the magnetic flux Φ/Φ0 for the quantum ring without DSOI, (b) with DSO coupling o. constant β = 25eV · A. Where the legend denotes the state in the form of ( η , m ), η is a, b, c or d and m is the dominated magnetic quantum number. The other physical parameters are obtained for the case of GaAs: o. m∗ ≈ 0.067me , λf = 40nm, r1 = 10nm, r2 = 70nm, d = 50 A. 3.2. . . . . . . . 38. Left hand side is the the several lowest dimensionless eigenenergies E as a function of the dimensionless DSO coupling strength β˜ from 0 to 1, right hand side is the eigenenergies as a function of the magnetic flux Φ/Φ0 from 0 to 0.6. The physical parameters are the same with Fig. 3.1.. viii. . . . . . . . 39.

(11) LIST OF FIGURES 3.3. Electron numbers from N = 1 to N = 4, left hand side is the total persistent charge current of the quantum ring as a function of magnetic flux Φ/Φ0 , right hand side is Sz, Lz, Sz + Lz and Lz(β˜ = 0.01) as a function of magnetic flux Φ/Φ0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42. 3.4. Electron numbers from N = 5 to N = 8, left hand side is the total persistent charge current of the quantum ring as a function of magnetic flux Φ/Φ0 , right hand side is Sz, Lz, Sz + Lz and Lz(β˜ = 0.01) as a function of magnetic flux Φ/Φ0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43. ix.

(12) Chapter 1 Introduction Today, more and more new artificial technology is built upon charge-based electronics where spin of an electron, however, was neglected in the technology application before. This stream of neglecting spin in the application sector may have set a different schedule in the past decades when spintronics were proposed and realized. In order to control electrons on spin transport and spin accumulation in semiconductors, we should invoke the spin-orbit interaction (SOI). According to the physical origin of the SOI, the SOI can be divided into intrinsic and extrinsic types, the more detail will be discussed in the next section. In other way, the quantum ring is a popular device which can be designed as calculation units, therefore, many scientists get into this area and discover many interesting physical phenomenon. Although persistent current with Rashba spin-orbit interaction [1] in one-dimensional ring has been discussed generally, few researches focus on Dresselhaus spin-orbit interaction [2] in finite width ring. We propose two important points of view about our motivation. First, one-dimensional ring is an ideal case which can not be performed by experiment exactly, and it is impossible to discuss the spin density in r-direction,too. Therefore, we suggest a finite width ring-shaped potential pattern to study the spin accumulation and the persistent current realistically. Second, Dresselhaus effect is as important as the Rashba effect in semiconductor, more and more research focus on the effect of Dresselhaus. 1.

(13) CHAPTER 1. INTRODUCTION linear and cubic term but there are still some interesting issues worth to be studied. For example, the energy spectrum of the DSO quantum ring has mixing degeneracy states which was discussed first by G.E.Marques et al. in 2008 [3], but in their research, there are still many physical origins should be made clear step by step. In order to concentrate on the Dresselhaus spin-orbit interaction, we will not include other intrinsic and extrinsic SOI. Neglecting Rashba-type SOI is justified in a symmetric quantum well. In our results, we will present detail about solving coupling Hamiltonians to get degeneracy spectrum to Dresselhaus strength and discuss the particle density, spin accumulation and persistent charge current.. 1.1. Spin-orbit coupling in solid state systems. Recently, more researches concentrate on the study of creating spintronic devices in semiconductor without magnetic materials which are designed by the interaction between spin-orbit coupling and quantum confinement. It can be realized by the spin-orbit interaction (SOI) to control electron spins. Datta and Das proposed an efficient model of spin-transistor in which the electron spin can transport and precess via SOI from a ferromagnetic source injecting into semiconductor to reaching a ferromagnetic drain [4]. The system is shown in Fig. 1.1. In this kind of spin-transistor, we can detect the polarization of the electron spin which depends on the strength of spin-orbit coupling by tuning an applying gate voltage. Besides, the spin polarization parallel to the polarization of the drain can pass through the channel, therefore, one can tune the voltage to modulate the current flow for ’on-’ or ’off-’ state. Before we concentrate on the spintronics devices in semiconductor, we should understand the physical phenomenon clearly in nanoscale especially. Electron spin, the only internal degree of freedom of electrons, follows naturally from the Dirac equation when Dirac tried to put wave function in a covariant form, when space and time appear on equal footing. A nonrelativistic limit of the Dirac equation gives rise to the spin-orbit. 2.

(14) CHAPTER 1. INTRODUCTION. Figure 1.1: (a) Electro-optic light modulator. The polarizer makes the input light polarized at an angle of 45o with respect to the y axis; (b) Spin-polarized FET proposed by Datta et al. [4] in 1990. The iron contacts playing the roles of polarizer and analyzer are made of ferromagnetic materials. These figures were plotted by Datta et al. [4] in 1990.. 3.

(15) CHAPTER 1. INTRODUCTION interaction term, a term that has found great success in atomic energy spectra. The form of this spin-orbit interaction, in vacuum, is [5]. Hso = −. e~ ~ σ · (E × p) = σ · (∇V × p) 2 2 4m0 c 4m20 c2. (1.1). where m0 is the free electron mass, ~ is the Planck constant and c is the velocity of light. The physical interpretation of Hso is given below. An electron moving in an electric potential region sees, in its frame of reference, an effective magnetic field couples with the electron spin through the magnetic moment of the electron spin. Therefore, the SOI is established via this effective magnetic field which depends on the orbital motion of the electron, and this physics holds in semiconductor too when V (r) becomes the periodic potential of the host lattice and also the impurities. The k · p model is the well-known method to describe electronic state calculations in semiconductor, when we investigate physical effect in the vicinity of the band edges. Furthermore, within the envelope function approximation (EFA), the energy band can be characterized by effective masses. The SOI in semiconductors requires, first of all, an effective electric field in the material. Such effective electric field can find contribution from the build-in crystal field when the crystal has bulk inversion asymmetric (BIA) the so-called Dresselhaus SOI [2], or structural inversion asymmetry (SIA), the so-called Rashba SOI [1]. The BIA is found in zincblende structure and the SIA in asymmetric quantum wells (QWs) or heterostructures. However within the effective mass approximation, the effect of all the fast-varying atomic potential has been incorporated into the effective mass. Slower varying V (r), with variation length scale much greater than the lattice spacing, is found to contribute to SOI with a much greater SO coupling constant λ. For a central potential V (r) depends on only r without angular dependence in vacuum, the SO coupling is. 4.

(16) CHAPTER 1. INTRODUCTION. ~ ~ 1 dV ~2 1 dV L λvac 1 dV σ · (∇V × p) = σ · (r × p) = ·σ = − L · σ (1.2) 2 2 2 2 2 2 4m0 c 4m0 c r dr 4m0 c r dr ~ ~ r dr where L is the orbital angular momentum,λvac = −~2 /(4m20 c2 ) ≈ −3.72 × 10−6 ˚ A2 . But in a semiconductor, also for a central potential, the SO coupling can be expressed in the form of. Hso = −. , where λ ≈. p2 3. λ 1 dV L·σ ~ r dr h. 1 Eg2. −. (1.3). 1 (Eg +∆0 )2. i .. For a 2DEG, the SOI becomes Hso = − λ~ ρ1 dVdρ(ρ) Lz · σz . Here P is the momentum matrix element between s- and p-orbital, Eg is the energy band gap, and ∆0 represents the SOI energy split to the spin split-off hole band [? ]. Of particular interest is that λ = 120˚ A2 in InAs, which is seven order of magnitude greater than λvac [6]. Roughly speaking, this large enhancement of SO coupling constant can be understood in the following. With λvac ∝ InAs,. m0 m∗. ∼. 2 1 ; m0 c 0.023 Eg. ∼. 0.5M eV 0.418eV. 1 m20 c2. =. 1 1 , m0 m0 c2. ;leading to. λ λvac. we can see that. λ λvac. ∼. m0 m0 c2 . m∗ Eg. For. ∼ 52 × 106 . Comparing to, we see that. the above hand waving argument has captured the essential physical origin of the great enhancement.. 1.2 1.2.1. Introduction to persistent current One-dimension ring with an external magnetic flux. In 1983, Buttiker et al. proposed a model [7] which is a one-dimensional normal mental ring threaded by a flux with elastic scattering at zero temperature. In this case, the single-electron states of this ring can be acquired from the band structure with the potential V (x) = V (x + L), which the potential V (x) around the loop with perimeter L. 5.

(17) CHAPTER 1. INTRODUCTION. Figure 1.2: Band structure of a linear lattice with lattice constant L. The figure was obtained from Kronig-Penney model. [8]. corresponds to the time-varying vector potential of the system. The physical behavior of an electron passing through is the same as an electron in a linear lattice with periodic potential. The potential variation in one period is consistent with the change of the vector potential in one route around the ring. We can see a similar phenomenon in Kronig-Penney model.[8] The band structure of a linear lattice with lattice constant L is shown in Fig. 1.2. The single-electron energy state of the ring system is an analogy of the band structure through the rule. Φ Φ0. k = − (2π/L) ,. where the electron flux quantum Φ0 = hc/e. The electronic energy states En (Φ) periodic in flux Φ with period Φ0 are shown in Fig. 1.3. The same periodic behavior occurs in the persistent current. As we known, the current carried by each state En (Φ) for a time-independent flux at £ ¤ n zero temperature is in = − Le ~1 ∂E . According to the analogy between the wave vector ∂k n (Φ) . Then the total persistent and magnetic flux, the current is obtained as in = −c ∂E∂Φ P P n (Φ) can be obtained by summing over all occupied charge current I = in = −c ∂E∂Φ. n. n. 6.

(18) CHAPTER 1. INTRODUCTION. Figure 1.3: The single-electron energy state En (Φ) of the normal mental one-dimensional ring as a function of flux proposed by Buttiker et al.[9]. states up to the Fermi energy at zero temperature. If the flux is time-dependent and increases linearly with time, there exists induced electric field E =. V L. 1 dΦ = − cL coercing the state through the Brillouin zone [9]. Under dt. this situation, we use the analogy between wave vector and magnetic flux again, and an electron will perform Bloch oscillation by circulating around the ring. The time for the electron to complete one cycle is T =. ∆k (eE/~). =. 2π/L (eE/~). =. 2π~ eV. with V = EL. For one cycle,. the variation of flux is equal to one flux quantum Φ0 = hc/e corresponds to the change of wave vector equal to the reciprocal wave vector G = 2π/L. The electromotive field produces an oscillating current with frequency ν = eV /h, i.e., a Josephson frequency with a single electronic charge. This persistent current will exist even when the field is turned off or the vector potential is fixed, and the current will keep at the value of flux to which it is fixed. [9]. 7.

(19) CHAPTER 1. INTRODUCTION. 1.2.2. Persistent current in mesoscopic rings with spin-orbit coupling. A mesoscopic ring system with spin-orbit coupling effect was discussed by Splettstosser et al. in 2003 [10]. The model is a mesoscopic isolated quasi-one-dimension ring with Rashba spin-orbit (RSO) coupling threaded by magnetic flux at low temperature. The external magnetic field provides a z-direction flux and the effective magnetic field coming from RSO coupling is in the radial r-direction. The tilt angle θ between z and r is presented as the strength of the RSO coupling shown in Fig. 1.4. If the particle number is not too large, the effect of RSO coupling on persistent charge current with even number or odd number of electrons is quite different from each other. In the other way, the prominent effect of RSO coupling on persistent spin current is the existence of the spin current for even electron number in the ring which vanishes in the absence of RSO coupling. The energy spectrum for the ring with an idealized impurity was obtained by Splettstosser et al. and shown in Fig. 1.5. The author expressed the impurity by a delta-function barrier V0 δ (φ) and the Zeeman energy is negligible in the local spin frame. They demonstrated the energy spectrum from the Hamiltonian of electrons in x-y plane. The electrons are assumed to move in a ring confined by a parabolic radial potential Vc (r) = 12 mω 2 (r − a)2 in the Hamiltonian:. (p − eA)2x + (p − eA)2y H= + Vc (r) + Hso = H0 + Hso 2m where Hso =. α ~. h. (1.4). i σx (p − eA)y − σy (p − eA)x is the Rashba spin-orbit coupling term.. Briefly speaking, the author obtained the approximated secular equation of the system by projecting the Hamiltonian on the eigenstates of the Hamiltonian without RSO coupling term and modeling the impurity by its energy-dependent transmission amplitude. Then they solved the secular equation to get the energy spectrum. When comparing with 8.

(20) CHAPTER 1. INTRODUCTION. Figure 1.4: A mesoscopic ring with Rashba spin-orbit coupling threaded by magnetic flux. The external magnetic flux is in z-direction and the angle θ between z and r parameterizes the strength of the RSO coupling. This figure was plotted by Splettstosser et al. in 2003. [10]. Figure 1.5: It is the energy spectrum for the ideal one-dimension ring with a delta-barrier impurity. Parameters are impurity strength constant A = 0.1 and title angle cos θ = 2/5. The energy levels are shifted in flux direction by 1/ cos θ. Solid lines correspond to spin-up states in the local spin-frame basis while dashed line to spin-down states. This figure was plotted by Splettstosser et al. in 2003. [10]. 9.

(21) CHAPTER 1. INTRODUCTION the energy levels after calculation for the same ring without RSO coupling, we can find that the energy levels for the system with RSO coupling are shifted in the flux direction by 1/ cos θ. For the reason of the energy splitting coming from RSO coupling, 1/ cos θ corresponds to the strength of the coupling. The shift in flux direction is different by the electron spin polarization state. On the basis of local spin frame, spin-up states move to the left (solid line) while spin-down state to the right (dash line). At zero temperature, the persistent charge current carried by each state n is in = n (Φ) , and n stands for a set of quantum numbers used to label the eigenstates including − ∂E∂Φ. the spin projection in the local spin frame. Therefore, the total persistent charge current P gs in = − ∂E , where I is obtained by summing over all occupied states, i.e., I = ∂Φ n=occupied. Egs is the ground state energy of the system. The flux dependence of the persistent charge current is individually different according to the number of electrons Ne . Here three cases are discussed for Ne is not too large:(i)Ne = 4N , (ii) Ne = 4N + 2, and (iii)Ne = 2N + 1 where N is a positive integer. If Ne is large enough, the persistent charge current has a general behavior independent of Ne . In case (i), where Ne = 4N , the numbers of spin-up and spin-down electrons are both even. The current flux characteristics are shown in Fig. 1.6. The solid line is for the system in the absence of RSO coupling. The effect of RSO coupling on the charge current is getting more obvious as 1/ cos θ increases. For each spin polarization, the currentcharacteristics of even number spin electrons are shifted in flux by ±1/2 cos θ. The total persistent charge current of the system is the superposition of the two shifted current flux characteristics. Case (ii), where Ne = 4N + 2, is obtained from the 4N case by shifting flux by Φ0 /2 and corresponds to an odd number of spin-up and spin-down electrons. The current is the analogous superposition of the two shifted current flux characteristics for odd number spin electrons. In these two case (i) and (ii), the strength of RSO coupling corresponding to 1/ cos θ can be measured. Further more, we can observe the behavior that the minimum 10.

(22) CHAPTER 1. INTRODUCTION. Figure 1.6: Persistent charge current-flux characteristics for a set of values for the spinorbit coupling strength. The total number of electrons is 4N in panel (a). 4N + 2 in panel (b), and 2N + 1 in panel (c) with the impurity strength constant A=0.1. These figures were plotted by Splettstosser et al. in 2003. [10] 11.

(23) CHAPTER 1. INTRODUCTION distance between two jumps of the current within one period interval is related to 1/ cos θ. For case (iii), Ne = 2N + 1. As strength of RSO coupling 1/ cos θ increases, the coupling suppresses the rounding more. It is because that the rounding comes from the impurity, RSO plays a role of decreasing the effect of impurity. This suppression makes the current flux characteristics from sawtooth shapes with jumps. This jumps come from the level crossing in the energy for the system with opposite spin in this odd number of electrons. The persistent spin current of each eigenstate (q, σ) in the quantization axis of local (qσ). spin frame is Iθ. = − 1e ∂E∂Φq,σ , where Eq,σ is an eigenenergy of local spin frame. The. eigenstates of the system in the local spin frame are ei(q+1/2)φ |±i with |±i denoting the eigenspinors of spin operator σ bz and q being an integer. This spin current for each state is the charge current for each state multiplied by the magnetization over charge e factor where the total spin current is obtained by summing the contribution from each state P (qσ) like the case in charge current Iθ = Iθ at zero temperature. This spin current is the qσ. projection onto the quantization axis of local spin frame. The other projections of spin current can be derived from Iθ through Iz = Iθ cos θ and Ir = Iθ sin θ, where Iz is for z projection and Ir is for the radial r projection. For even number electrons 4N + 2, the effect of RSO coupling on the persistent spin current can be observed through comparison of the spin current with the charge current. In this ring for even number of electrons, the spin current vanishes without RSO coupling. However, with RSO coupling, the spin current of the ring with electrons 4N + 2 is shown with dashed line in Fig. 1.7. by Splettstoesser et al. [10]. Evidently, the spin current has nonzero value at zero temperature and its characteristic curve exhibits dramatically different flux dependence from the charge characteristic curve (solid line) of the same ring. It shows a clear signature of RSO coupling. The RSO coupling produces a relative shift of energy band in flux direction and hence enables a finite persistent spin current for an even number of electrons. For odd number of electrons (2N + 1), the spin current is supposed to be proportional 12.

(24) CHAPTER 1. INTRODUCTION. Figure 1.7: In even number electrons 4N + 2 case, comparison of the persistent spin current (dash line) with the charge current (solid line) of the ring. The spin current is for spin projection onto the local spin frame and parameters are the impurity strength constant A = 0.5, the tilt angle cos θ = 0.66. This figure was plotted by Splettstosser et al. in 2003. [10]. Figure 1.8: Persistent spin currents of the ring with odd number electrons 2N + 1 (dotted line) and even number electrons 4N + 2 (dashed line). Parameters are impurity strength constant A = 0.5 and tilt angle cos θ = 0.9 corresponding to a small spin-orbit coupling strength. The figure was plotted by Splettstoesser et al. [10]. 13.

(25) CHAPTER 1. INTRODUCTION the charge current in the absence RSO coupling. In this ring with RSO coupling, the spin current of the ring with odd number electrons is shown with dotted curve in Fig. 1.8. The spin current flux characteristic curve has different flux dependence from the charge characteristic curve. According to the result, the proportion relation does not exist because of the presence of RSO coupling. The persistent spin current just like a magnetization current, could be detectable via tuning an external electric field. By making a Lorentz transform to the rest frame of spin, the electrostatic potential for a point at a distance z is obtained as ϕ (z) ≈. µ0 gµB Iθ 4π. sin θ za2 .. Where the distance z is the vertical length from the center of the ring and is summed to be much smaller than the radius a of the ring. Besides, in the expression of potential, g is the gyromagnetic ratio, µ0 is the vacuum permeability, and µB is the Bohr magneton. This potential expression is the same as the one resulting from the persistent spin current in Heisenberg rings derived by Kopietz et al. [11]. If it is possible to measure the spin current through electric field generated by its transported magnetization, the existence and magnitude of RSO coupling in conducting rings can be finally verified.. 14.

(26) Chapter 2 Finite width mesoscopic Dresselhaus spin-orbit (DSO) ring Recently, it is possible to grow self-assembled annulus semiconductor structures in a large range of inner and outer radii by using molecular beam epitaxy. Typical samples show a circular cross section with an inner radius about 10 nm, and the outer radius ranges between 30 and 70 nm [12–16]. This kind of structures has been studied by their potential applications as spintronic and quantum computing. [14, 15, 17, 18]. In this chapter, we will start the system from the Hamiltonian to get the energy spectrum and spin density.. 2.1. Model of a mesoscopic ring with Dresselhaus spinorbit coupling. 2.1.1. Hamiltonian. The geometric model is an isolated finite width mesoscopic ring with Dresselhaus spinorbit (DSO) coupling without any external electromagnetic field. The cross section of the structure is shown in Fig. 2.1. Assuming r1 (r2 ) and d are the inner (outer) radii and height, the center of the ring is at origin.. 15.

(27) CHAPTER 2. FINITE WIDTH MESOSCOPIC DRESSELHAUS SPIN-ORBIT (DSO) RING. Figure 2.1: The figure is the cross section of the finite width mesoscopic ring, where r1 and r2 are the inner and outer radii.. The total single-particle Hamiltonian in a mesoscopic ring that has bulk inversion asymmetry (BIA) and formed out of a quantum well with the well thickness along z and z along [001] has the form:. H = H0 + HDSO. H0 =. (2.1). p2 → + V (− ρ , z) ∗ 2m. (2.2). − → → →. HDSO = − σ · h− k. (2.3). − → → denotes the effective magnetic field arising from the DSO coupling, m∗ is the Here h − k → electron effective mass, V (− ρ , z) is a hard-wall confinement potential, β is the Dresselhaus → SO coupling constant, and − σ = (σx , σy , σz ) is the vector of Pauli spin matrices. For the case of a narrow quantum well, the electrons occupy only the lowest subband, we can − → → over the well thickness d to give average h − k. 16.

(28) CHAPTER 2. FINITE WIDTH MESOSCOPIC DRESSELHAUS SPIN-ORBIT (DSO) RING. ¡ ®¢ hxk = βkx ky2 − kz2 ,. (2.4). ¡ 2 ® ¢ kz − kx2 ,. (2.5). hyk = βky. ¡ ¢ hzk = β hkz i kx2 − ky2 ∼ = 0,. (2.6). hkz i = 0,. (2.7). 2 ® ³ π ´2 kz ∼ . = d. (2.8). If we just consider the k-linear BIA contribution, the effective HDSO assumes the form. ® ® HDSO = H1D = β kz2 [σy ky − σx kx ] = −β kz2 [σ+ k+ + σ− k− ] ,. (2.9). where σ± = 21 (σx ± iσy ) and k± = kx ± iky . The effect of the operator σ± with subscript + (−) is to increase (decrease) the spin of the electron by unit of ~. The operator k± can i h ∂ ∂ ≡ −ie±iϕ p± be expressed in the form k± = −ie±iϕ ∂ρ ± i ρ1 ∂ϕ ρ,ϕ which effect is to change the orbital angular momentum by ±~. This is evident by the factor e±iφ in k± . In matrix form.   H∼ =.  p2 2m∗. +V. −β < kz2 > k−. −β < p2 2m∗. .  p2 2m∗. kz2. > k+   +V = +V iβ( πd )2 e−iϕ p− ρ,ϕ. iβ( πd )2 eiϕ p+ ρ,ϕ p2 2m∗.   . (2.10). +V. In the following, we will guess the form of the eigenstate of H with the choice of ¯ 0 ® quantum numbers parallel that of the unperturbed eigenstate ¯En,m,σ . Here n, m, σ are, respectively, the quantum numbers for the radial, azimuthal and spin degrees of freedom. Hints for the form of the eigenstate of H can be obtained from applying HDSO repeatedly ¯ 0 ® . upon ¯En,m,σ 17.

(29) CHAPTER 2. FINITE WIDTH MESOSCOPIC DRESSELHAUS SPIN-ORBIT (DSO) RING ¯ 0 ® Consider, for example, applying HDSO to ¯En,m,↑ , we get, apart from a proportionality constant −β hkz2 i, the essence of HDSO is carried by the operator σ+ k+ + σ− k− , ¯ 0 ® ¯ 0 ® HDSO ¯En,m,↑ ∝ (σ+ k+ + σ− k− ) ¯En,m,↑ ¯ 0 ® = σ− k− ¯En,m,↑ ¯ E ¯ (1) ∝ ¯En,m−1,↓ ,. (2.11). ¯ E ¯ (1) where the superscript (1) denotes the state ¯En,m,σ to have a different radial dependence ¯ 0 ® from that of ¯En,m,σ . Applying HDSO the second time gives ¯ ¯ E E ¯ (1) ¯ (1) HDSO ¯En,m−1,↓ ∝ (σ+ k+ + σ− k− ) ¯En,m−1,↓ ¯ E ¯ (1) = σ+ k+ ¯En,m−1,↓ ¯ E ¯ (2) ∝ ¯En,m,↑ .. (2.12). From this we see that only |En,m,↑ i and |En,m−1,↓ i are coupled by the action of HDSO . We no longer use the superscript in the states |En,m,σ i because the eigenstate of H should include arbitrary number of times the HDSO is applied. This amounts to give the form for the total eigenfunction where f1 (ρ), f2 (ρ) have to be solved explicitly.   Ψ=.  imϕ. f1 (ρ)e. f2 (ρ)ei(m−1)ϕ.  . (2.13). For a hard-wall confinement potential    0 for r1 < ρ < r2 and |z| < − → V ( ρ , z) =   ∞ otherwise. d 2. (2.14). the Schrodinger equations inside the ring is given by   .  p2 2m∗. iβ( πd )2 eiϕ p+ ρ,ϕ. iβ( πd )2 e−iϕ p− ρ,ϕ. p2 2m∗.  .  imϕ. f1 (ρ)e. f2 (ρ)ei(m−1)ϕ 18. .   =E.  imϕ. f1 (ρ)e. f2 (ρ)ei(m−1)ϕ.   . (2.15).

(30) CHAPTER 2. FINITE WIDTH MESOSCOPIC DRESSELHAUS SPIN-ORBIT (DSO) RING. 2.1.2. Dimensionless. In order to get more clearly formula and prepare for the Numerical analysis, we set some parameters to dimensionless the Hamiltonian: (a) Fermi wave length lf , (b) Fermi wave vector kf = e (e) p± ∇ = kf ∇, ρ,ϕ. 1 , lf. (c) Fermi energy Ef =. ~2 kf2 , 2m∗. (d) Del operator in polar coordinates π 2 e = E , (g) βe = β ( d ) ·kf , (h) ρer = ρ . = kf pe± , (f) E ρ,ϕ Ef Ef r1. After calculation, we get the dimensionless couple equations ( The detail process is shown in Appendix A )..   . i h (m−1) e imϕ ∂ f2 (e e 1 (e e 2 [f1 (e ρ ) − f (e ρ ) = −Ef ∇ ρr )eimϕ ] − iβe ρr )eimϕ r 2 r ∂ ρe ρe h i ¤ £  m i(m−1)ϕ ∂ i(m−1)ϕ e e 2 f2 (e e 2 (e  ∇ − iβe ρr )e f (e ρ ) + ρe f1 (e ρr ) = −Ef ρr )ei(m−1)ϕ ∂ ρe 1 r. (2.16). Since these differential equations have the forms of Bessel equation and the recurrence relation, we can choose the Bessel functions as the spinor part eigenfunctions. Therefore,.   . e m (γ ρer )eimϕ −γ 2 AJm (γ ρer )eimϕ − iβe [−γBJm (γ ρer )] eimϕ = −EAJ.  e m−1 (γ ρer )ei(m−1)ϕ  −γ 2 BJm−1 (γ ρer )ei(m−1)ϕ − iβe [γAJm−1 (γ ρer )] ei(m−1)ϕ = −EBJ (2.17). where A and B are the coefficients of the two eigenfunction components. Comparing with above couple equations, we find that the ratio of the coefficients R=. B A. = ±i, then we get two branches of wave functions.   Ψ=.  imϕ. Jm (γ ρer )e. iJm−1 (γ ρer )ei(m−1)ϕ. .   ,Ψ = .  imϕ. Jm (γ ρer )e. −iJm−1 (γ ρer )ei(m−1)ϕ. 19.  .. (2.18).

(31) CHAPTER 2. FINITE WIDTH MESOSCOPIC DRESSELHAUS SPIN-ORBIT (DSO) RING. 2.1.3. Eigenenergy and Eigenstate. e E e E e = 0 and γ 2 −βγ− e = 0 when R is equal to +i and We get two energy dispersion γ 2 +βγ− µ µ ¶ ¶ q q 1 1 2 2 e e e e e e −i respectively. Then, if we set γ1 = 2 β + β + 4E and γ2 = 2 −β + β + 4E , we can obtain four-eigenstates ( first kind of Bessel function ): .  imϕ. Jm (γ1 ρer ) e.  (I) . .    , (II) .  imϕ. Jm (−γ1 ρer ) e.  ,. −iJm−1 (γ1 ρer ) ei(m−1)ϕ iJm−1 (−γ1 ρer ) ei(m−1)ϕ     imϕ imϕ Jm (γ2 ρer ) e Jm (−γ2 ρer ) e     (III)   , (IV )  . i(m−1)ϕ i(m−1)ϕ iJm−1 (γ2 ρer ) e −iJm−1 (−γ2 ρer ) e. (2.19). But (I) (II) and (III) (IV ) have the same forms, there are just two independent bases. Therefore, we chose (I) and (III) to be our eigenstates. Besides, the index of m should be integers (including positive and negative), then we have to introduce the second kind of Bessel function ”Neumann” to be eigenstates.  Nm (γ1 ρer ) eimϕ.  (I) . . . .    , (II) . Nm (−γ1 ρer ) eimϕ.  ,. iNm−1 (−γ1 ρer ) ei(m−1)ϕ −iNm−1 (γ1 ρer ) ei(m−1)ϕ     imϕ imϕ Nm (γ2 ρer ) e Nm (−γ2 ρer ) e     (III)  .  , (IV )  −iNm−1 (−γ2 ρer ) ei(m−1)ϕ iNm−1 (γ2 ρer ) ei(m−1)ϕ. (2.20). In order to get more clear physical meaning, we change Bessel functions to Hankel (1). (2). functions: Hα (x) = Jα (x) + iNα (x), Hα (x) = Jα (x) − iNα (x). Thus, we get the total eigenfunction: .   Ψ = a. (1) Hm (1) −iHm−1. (γ1 ρer ) eimϕ. . .    + b. (2) Hm (2) −iHm−1. (γ1 ρer ) eimϕ. (γ1 ρer ) ei(m−1)ϕ (γ1 ρer ) ei(m−1)ϕ    (1) (2) imϕ imϕ Hm (γ2 ρer ) e Hm (γ2 ρer ) e     + c  + d , (1) (2) iHm−1 (γ2 ρer ) ei(m−1)ϕ iHm−1 (γ2 ρer ) ei(m−1)ϕ . 20.   (2.21).

(32) CHAPTER 2. FINITE WIDTH MESOSCOPIC DRESSELHAUS SPIN-ORBIT (DSO) RING where a, b, c and d are the coefficients which can be decided by confinement potential and normalization. Since the annulus is defined by the hard-wall confining potential Eq. (2.14), we can apply the boundary conditions Ψ (ρ = r1 ) = 0, Ψ (ρ = r2 ) = 0. These conditions give the equation set:.         .  (2) Hm (γ2 r1 )    (1) (2) (1) (2)  −Hm−1 (γ1 r1 ) −Hm−1 (γ1 r1 ) Hm−1 (γ2 r1 ) Hm−1 (γ2 r1 )    (1) (2) (1) (2)  Hm (γ1 r2 ) Hm (γ1 r2 ) Hm (γ2 r2 ) Hm (γ2 r2 )    (1) (2) (1) (2) −Hm−1 (γ1 r2 ) −Hm−1 (γ1 r2 ) Hm−1 (γ2 r2 ) Hm−1 (γ2 r2 ) (1). Hm (γ1 r1 ). (2). Hm (γ1 r1 ). (1). Hm (γ2 r1 ).  a   b   =0 c    d. (2.22). If there exists non-trivial solution for a, b, c and d, the determinant of the matrix M must vanish, i.e.,. det (M ) = 0,. (2.23). where . (1) Hm. (2) Hm. (1) Hm. (2) Hm. (γ1 r1 ) (γ1 r1 ) (γ2 r1 ) (γ2 r1 )    −H (1) (γ1 r1 ) −H (2) (γ1 r1 ) H (1) (γ2 r1 ) H (2) (γ2 r1 ) m−1 m−1 m−1 m−1  M = (2) (1) (2)  H (1) (γ r ) Hm (γ1 r2 ) Hm (γ2 r2 ) Hm (γ2 r2 )  m 1 2  (1) (2) (1) (2) −Hm−1 (γ1 r2 ) −Hm−1 (γ1 r2 ) Hm−1 (γ2 r2 ) Hm−1 (γ2 r2 ) The more detail process is shown in Appendix A.. 21.      .   . (2.24).

(33) CHAPTER 2. FINITE WIDTH MESOSCOPIC DRESSELHAUS SPIN-ORBIT (DSO) RING. 2.2 2.2.1. Numerical results and discussion Energy spectrum. Physical parameters used in the calculation are obtained for the case of GaAs: with o. DSO coupling constant β = 25eV · A, effective mass m∗ ≈ 0.067me , Fermi wave length λf = 40nm. Structures parameters are inner radius r1 = 10nm, outer radius r2 = 70nm o. and the height of the ring d = 50 A. [3]. Fig. 2.2 presents the energy spectrum for the quantum ring which is determined from the determinant equation Eq. (2.23) and Eq. (2.24), where γ1 and γ2 contain the energy E. We want to show the effects caused by the DSOI on the first seven quantum ring energy e Due to levels in units of Ef , as a function of the dimensionless DSOI coupling constant β. the confinement potential and in the absence of the DSO interaction, we know that there exist discrete energy levels in y-axis. And each of them have degenerate states, namely, ¯ 0 ¯ 0 ® ® 0 ¯E ¯ n,±m,↑ and En,±m,↓ for a given eigenenergy En,m,σ . When we introduce, however, the DSOI strength, we can see that the unperturbed energy level splitting into two parts and most of them decreasing as βe increasing monotically. Specially, some eigenenergies have the trend which arise up first and then go down when βe increasing. In this section, we will focus on the physical insight of the splitting energy in our system. First, we propose a useful regulation to understand the development of the states. If we just consider the confinement potential in the quantum ring without DSOI, we can get the discrete quantum energy levels and each of them has different degeneracy states ¯ 0 ® ¯ 0 ® ¯ 0 ® respectively, for instance, there are four degeneracy states ¯E1,3,↑ , ¯E1,3,↓ , ¯E1,−3,↑ and ¯ 0 ® 0 ¯E 1,−3,↓ in the same eigenenergy E1,±3 . However, if we add the DSO interaction to our system, we find that each original degeneracy energy level split into two groups of energy levels, and the new energy level exists new degeneracy states which is the superposition of the original eigenstates in H0 . The phenomenon can be understood by the operator form of the HDSO = −β hkz2 i (σ+ k+ + σ− k− ) which has been presented in Eq. (2.11) to Eq. (2.12) in section 2.1. In HDSO , the operators k+ and k− are like the angular moment 22.

(34) CHAPTER 2. FINITE WIDTH MESOSCOPIC DRESSELHAUS SPIN-ORBIT (DSO) RING. 1. |ψ ⟩ =c |E. ⟩ +c |E. |ψ ⟩ =c |E. ⟩ +c |E. a. 1. 1,3, ↑. b. 3. 1,−2, ↑. 2. 4. 1,2, ↓. ⟩. 1,−3, ↓. ⟩. m=±3 | ψ c ⟩ = c5 | E1,4, ↑ ⟩ + c6 | E1,3, ↓ ⟩. | ψ d ⟩ = c7 | E1,−3, ↑ ⟩ + c8 | E1,−4, ↓ ⟩. f. Energy ( E ). 0.8. 0.6 m=±2. 0.4 m=±1 m=0 0.2. 0. 0. 0.2. 0.4. 0.6. 0.8. 1. β˜ Figure 2.2: The energy spectrum for the quantum ring which is determined from the determinant equation Eq. (2.23) and Eq. (2.24), and it is shown the effects caused by the DSOI on the first seven quantum ring energy levels in units of Ef , as a function e These doubly degeneracy states come of the dimensionless DSO coupling constant β. from the superposition of the different unperturbed eigenstates respectively. The physical o parameters are obtained for the case of GaAs: β = 25eV · A, m∗ ≈ 0.067me , λf = 40nm, o. r1 = 10nm, r2 = 70nm, d = 50 A.. 23.

(35) CHAPTER 2. FINITE WIDTH MESOSCOPIC DRESSELHAUS SPIN-ORBIT (DSO) RING operator, the operators σ+ and σ− are like spinor operator. Each of them can create or destroy the quantum number m or ↑↓ respectively. Today, we use HDSO to operate the unperturbed states repeatedly in the quantum ring, the angular moment quantum number and the spin quantum number are created or destroyed respectively to become new superposition states |ψa i = c1 |E1,3,↑ i + c2 |E1,2,↑ i, |ψb i = c3 |E1,−2,↑ i + c4 |E1,−3,↓ i, |ψc i = c5 |E1,4,↑ i + c6 |E1,3,↓ i and |ψd i = c7 |E1,−3,↑ i + c8 |E1,−4,↓ i. Therefore, we can use this simple method to understand the development of states in DSOI systems. Second, we find that the relative weighting between the constituent bases state ket plays an important role to effect the trend of the energy spectrum when the DSOI strength increases slowly. For example, we start from the unperturbed energy level in m = ±3. We can understand that the probability of the states |E1,3,↑ i and |E1,−3,↓ i have more weight in state |ψa i and |ψb i from Fig. 2.3 when βe is small. Physically, it is because that the 0 0 splitting energy levels start from E1,±3 , and the effect coming from energy levels E1,2,↓ , 0 E1,−2,↑ can be seen as the correction of the energy. Therefore, we can distinguish which. e one of the elements of superposition states is domination in small β. Third, we deduce that the level repulsion is the reason of the two splitting energy levels arising up and going down respectively when βe is small. Because of the first and second analysis, we know that the splitting energy level consists of the dominate energy level and the upper (lower) energy level. The upper or lower energy level play a repulsive role to effect the trend of the spectrum. For example, the original eigenstates in the 0 eigenenergy E1,±3 split into two new degeneracy state |ψa i = c1 |E1,3,↑ i + c2 |E1,2,↑ i and 0 0 |ψb i = c3 |E1,−2,↑ i + c4 |E1,−3,↓ i. The lower energy E1,±2 repulses E1,±3 to arise up energy.. Fourth, in the next chapter, we will get the result about each of the perturbed states ( |ψa i, |ψb i, |ψc i and |ψd i ) is not mixing by considering external magnetic flux. Therefore, we can use nondegenerate perturbation theory to analyze perturbation problems. Fifth, in the small Dresselhaus strength region, we can get the trend of the energy level which is represented in the quadratic form. The reason can be demonstrated by two methods shown below. In nondegenerate perturbation theory, we can obtain the result, 24.

(36) CHAPTER 2. FINITE WIDTH MESOSCOPIC DRESSELHAUS SPIN-ORBIT (DSO) RING. 1. 1. 2. |c | E. 0.8. 2. 1,2, ↓. 2. ⟩|. ⟩|2. 0.4. 0.2. 0 0. |c | E. ⟩|2. 4. 1,−3, ↓. 0.6. 0.2. 0.4. 0.6. 0.8. 0.4. 0.2. 0 0. 1. 0.2. 0.4. β˜. 0.8. 1. 1 (c). 1,4, ↑. |c | E. 0.8. 6. 1,3, ↓. 2. ⟩|. 2. ⟩|. ⟩|. |c | E. ⟩|2. 8. 1,−3, ↑ 1,−4, ↓. 0.6. 2. 2. 0.6. 0.8. 2. |c | E 7. (d). 2. 5. |c7 | E1,−3, ↑ ⟩| , |c8 | E1,−4, ↓ ⟩|. |c | E 2. 0.6 β˜. 1. |c5 | E1,4, ↑ ⟩| , |c6 | E1,3, ↓ ⟩|. ⟩|. 1,−2, ↑. 2. 0.6. 0.8. 2. |c | E 3. (b). 2. (a). 1,3, ↑. 2. |c1 | E1,3, ↑ ⟩| , |c2 | E1,2, ↓ ⟩|. 1. |c3 | E1,−2, ↑ ⟩| , |c4 | E1,−3, ↓ ⟩|. |c | E. 0.4. 0.2. 0 0. 0.2. 0.4. 0.6. 0.8. 0.4. 0.2. 0 0. 1. β˜. 0.2. 0.4. 0.6. 0.8. 1. β˜. Figure 2.3: The relative weighting between the constituent bases state ket plays an important role to effect the trend of the energy spectrum. When the Dresselhaus strength is small: (a) c1 |E1,3,↑ i has more weight than c2 |E1,2,↑ i in state |ψa i, (b) c4 |E1,−3,↓ i has more weight than c3 |E1,−2,↑ i in state |ψb i, (c) c6 |E1,3,↓ i has more weight than c5 |E1,4,↑ i in state |ψc i, (d) c7 |E1,−3,↑ i has more weight than c8 |E1,−4,↓ i in state |ψd i.. 25.

(37) CHAPTER 2. FINITE WIDTH MESOSCOPIC DRESSELHAUS SPIN-ORBIT (DSO) RING. 0.838 0.836 0.834. m=±3. Energy(Ef). 0.832 0.83 0.828 0.826 0.824 0.822. 0. 0.02. 0.04. 0.06. 0.08. 0.1. β˜. Figure 2.4: In the small Dresselhaus strength region, 0 ≤ βe ≤ 0.1, we can get the trend of the energy level which is represented in the up (down) quadratic form. The physical parameters are the same with Fig. 2.2.. 0. hn| H |ni +. ¯ ¯ X ¯hm| H 0 |ni¯2 m. ε0n − ε0m. ∼ =0+. ¯ ¯ X ¯hm| H 0 |ni¯2 m. ε0n − ε0m. (2.25). 0 where the first order perturbation is equal to zero hn| H |ni ∼ = 0 and the second order. perturbation is not. In other way, let us simplifythe problem  to a 2 × 2 matrix, then the  εn B  Hamiltonian H0 + HDSO can be represented as   , where B is denoted as offB εm diagonal term proportional to DSO coupling strength. If we calculate the determination of this matrix when β is small, we can get the approximate eigenenergies.. εn + εm εn − εm ε= ± 2 2. Ã. 1 1+ 2. µ. 2 εn − εm. !. ¶2 B2. (2.26). This result is the same as our numerical analysis which is shown in Fig. 2.4 Finally, we also find that there are two degeneracy eigenstates which are cause of the time-reversal symmetry in each new splitting energy level. If we use −(m − 1) to instead 26.

(38) CHAPTER 2. FINITE WIDTH MESOSCOPIC DRESSELHAUS SPIN-ORBIT (DSO) RING of m in our Schrodinger equation Eq. (2.16), we can get the same couple Schrodinger equations which are meant to be Kramer degeneracy.. 2.2.2. Spin density and net magnetic dipole moment. For a given eigenenergy En,m , a set coefficients of eigenfunction can be determined from Rr + the equation set Eq. (2.22). With the normalization condition r12 ψm ψm ρdρ = 1, we can obtain the uniquely determined eigenfunction Eq. (2.21).. .  If the general form of the eigenfunction is defined as ψm (e ρr , φ) = .  imφ. fm (e ρr ) e. gm−1 (e ρr ) ei(m−1)φ.  ,. the components of the spinor part eigenfunction are. (1) (2) (1) (2) fm (e ρr ) = aHm (γ1 ρer ) + bHm (γ1 ρer ) + cHm (γ2 ρer ) + dHm (γ2 ρer ). (2.27). ³ ´ (1) (2) (1) (2) gm−1 (e ρr ) = i −aHm−1 (γ1 ρer ) − bHm−1 (γ1 ρer ) + cHm−1 (γ2 ρer ) + dHm−1 (γ2 ρer ) (2.28) 2 2 Then, the particle density of each state m is Nm (e ρr ) = fm (e ρr ) + gm−1 (e ρr ).. The spin density can be derived by the same method just like particle density. The z + 2 2 z-projection spin density for each m state is Sm (e ρr ) = ψm σz ψm = fm (e ρr ) − gm−1 (e ρr ).. By summing the contributions from ground state to the mth state, one can acquire the P z total spin density at zero temperature Sz = Sm (e ρr ). The spin density as a function of m. dimensionless radial position ρer = ρ/r1 of the ring with each state is shown in Fig. 2.5 at o. DSO constant β = 25eV · A . In Fig. 2.5, we can see that when we input single-electron in each state, the net zprojection spin is not zero, which can be seen as a magnetic dipole moment. But, because of the cancelation relation between states, there will left very few z-projection spins. Therefore, if we input odd-number of electrons in our system, we can get finite net total 27.

(39) CHAPTER 2. FINITE WIDTH MESOSCOPIC DRESSELHAUS SPIN-ORBIT (DSO) RING. 0.04 |ψ. (a). |ψ. |ψ a,2⟩. 0.02. 0.04 ). −2. ). |ψ a,1⟩. −2. |ψ a,0⟩. 0. ρ*<Sz> (625µ m. ρ*<Sz> (625µ m. 0.06. ⟩. a,3. −0.02. −0.04. 2. |ψ. 3. 4 ρ˜r. 5. 6. −0.04 1. 7. (c). |ψ c,2⟩. 0.025. |ψ c,1⟩. 0.02 0.015. 0 −0.005 −0.01 −0.015 −0.02. 2. 3. 4 ρ˜r. 5. 6. 7. |ψ. (d). ⟩. d,3. |ψ d,2⟩ |ψ d,1⟩. 0.01 0.005 0 −0.005. −0.025 −0.03 1. |ψ b,0⟩. 0.03. ⟩. c,3. ). ρ*<Sz> (625µ m. −2. ). 0.005. |ψ b,1⟩. −0.02. −2. 0.01. (b). 0. ρ*<Sz> (625µ m. −0.06 1. 0.02. ⟩. b,3. |ψ b,2⟩. 2. 3. 4 ρ˜r. 5. 6. −0.01 1. 7. 2. 3. 4 ρ˜r. 5. 6. 7. Figure 2.5: It is the z-projection spin density for each states as a function of the dimensionless radial length which the angular is fixed to zero.(a) |ψa,m i states, (b) |ψb,m i states, (c) |ψc,m i states, (d) |ψd,m i states.. 28.

(40) CHAPTER 2. FINITE WIDTH MESOSCOPIC DRESSELHAUS SPIN-ORBIT (DSO) RING. 0.8 (a) 0.6 0.4 ρ*<Sz>. 0.2 0 −0.2. (r1,r2)=(1,3). −0.4. (r1,r2)=(3,5). −0.6. (r1,r2)=(5,7) (r1,r2)=(1,7). −0.8 0. 0.2. 0.4 0.6 (ρ−r1)/(r2−r1). 0.8. 1. 40 (b) 30 20 ρ*<Sz>. 10 0 −10 −20 (r1,r2)=(90,99). −30 −40 0. (r1,r2)=(90,102) 0.2. 0.4 0.6 (ρ−r1)/(r2−r1). 0.8. 1. 1 Figure 2.6: It is the z-projection spin density as a function of the radial ratio rρ−r which 2 −r1 the angular is fixed to zero.(a) radial ratio in different range, (b) radial ratio is set as big as possible.. 29.

(41) CHAPTER 2. FINITE WIDTH MESOSCOPIC DRESSELHAUS SPIN-ORBIT (DSO) RING z-projection spin. In the other way, we change the inner and outer radii for different range to observe the spin polarization phenomenon. Fig. 2.6 are the z-projection spin density as a function of radial ratio. ρ−r1 . r2 −r1. We know that when the inner and outer radii become bigger in the same. time, the ring width are more like a horizontal channel in the ring system. Therefore, we can see the effect of the different ring width between Fig. 2.6 (a) and Fig. 2.6 (b), then we make a simple conclusion: When the ring width is big enough, the spin accumulation phenomenon would be more observable.. 30.

(42) Chapter 3 Finite width mesoscopic Dresselhaus spin-orbit (DSO) ring with a magnetic flux 3.1. Theoretical model. 3.1.1. Hamiltonian. In this chapter, we start the system from the single-particle Hamiltonian which describes an electron in the mesoscopio ring piercing by a magnetic flux Φ with DSO coupling. The geometric model is almost the same with Fig. 2.1 in chapter 2. Therefore, we can write down the Hamiltonian quickly:. .   H'. p2 2m∗. −β <. +V kz2. > k−. −β < p2 2m∗. kz2. > k+  . +V. (3.1). In order to consider adding the magnetic flux in our system, we should correct the momentum: (a) p → p − eA; (b) k → k −. eA , ~. 31. where A is the vector potential chosen as.

(43) CHAPTER 3. FINITE WIDTH MESOSCOPIC DRESSELHAUS SPIN-ORBIT (DSO) RING WITH A MAGNETIC FLUX the axial gauge. And the corresponding magnetic flux is. Φ Φ0. =. 2πeAϕ ρ h. =. eAϕ ρ ~. in units of. flux quantum Φ0 = he . In the following, just like in chapter 2, we guess the form of the eigenstate of H with ¯ 0 ® the choice of quantum numbers parallel that of the unperturbed eigenstate ¯En,m,σ . Hints for the form of the eigenstate of H can be obtained from applying HDSO repeatedly upon ¯ 0 ® ¯En,m,σ . This amounts to give the form for the total eigenfunction where f1 (ρ), f2 (ρ) have to be solved explicitly..   Ψ=.  f1 (ρ)eimϕ i(m−1)ϕ.  .. (3.2). f2 (ρ)e. By rewriting the Hamiltonian in polar coordinates and a hard-wall confinement potential defining the annulus is.    0 for r1 < ρ < r2 and |z| < − → V ( ρ , z) =   ∞ otherwise. d 2. (3.3). we can get the coupling Schrodinger equations.          . ~2 2m∗. · ³ ´ ³ ´2 ¸ ¡ π ¢ 2 iϕ ³ ∂ Φ Φ 2 imϕ k − 2 ρΦ0 kϕ + ρΦ f (ρ) e + iβ + e 1 d ∂ρ 0. i ∂ ρ ∂ϕ. = Ef1 (ρ)eimϕ · ³ ´ ³ ´2 ¸ ¡ π ¢ 2 −iϕ ³ ∂ 2  Φ Φ ~ 2 i(m−1)ϕ  k − 2 k + f (ρ) e + iβ e −  ϕ 2 2m∗ ρΦ0 ρΦ0 d ∂ρ       = Ef (ρ)ei(m−1)ϕ 2. +. Φ ρΦ0. i ∂ ρ ∂ϕ. ´. −. f2 (ρ) ei(m−1)ϕ. Φ ρΦ0. ´. f1 (ρ) eimϕ. (3.4). 32.

(44) CHAPTER 3. FINITE WIDTH MESOSCOPIC DRESSELHAUS SPIN-ORBIT (DSO) RING WITH A MAGNETIC FLUX. 3.1.2. Dimensionless. In order to get more clearly formula and prepare for the Numerical analysis, we set some parameters to dimensionless the Hamiltonian: (a) Fermi wave length lf , (b) Fermi wave vector kf = e (e) p± ∇ = kf ∇, ρ,ϕ. 1 , lf. (c) Fermi energy Ef =. ~2 kf2 , 2m∗. (d) Del operator in polar coordinates π 2 e = E , (g) βe = β ( d ) ·kf , (h) ρer = ρ . = kf pe± , (f) E ρ,ϕ Ef Ef r1. After calculation, we get the dimensionless couple equations (The detail process is shown in Appendix B.).          . ½ 1 ∂ ρe ∂ ρe. ½. ³. ρe∂∂ρe. ´ −. 1 ρe2. ³ m−. e 1 (e = −Ef ρr )eimϕ ³ ´ ³ ∂ 1 1 ∂ ρ e − m−1− ρe ∂ ρe ∂ ρe ρe2.          = −Ef e 2 (e ρr )ei(m−1)ϕ. Φ Φ0. Φ Φ0. ´2 ¾. ´2 ¾. f1 (e ρr ) eimϕ − iβe. ³. ∂ ∂ ρe. f2 (e ρr ) ei(m−1)ϕ − iβe. − ρ1e(m −. ³. ∂ ∂ ρe. +. 1 ρe. Φ Φ0. ´ − 1) f2 (e ρr ) eimϕ. ³ m−. Φ Φ0. ´´. f1 (e ρr ) ei(m−1)ϕ. (3.5). Let q = m −.   . 1 ∂ ρe ∂ ρe. ρe∂∂ρe. ´. Eq. (3.5) become. ´ e 1 (e − f1 (e ρr ) e − − 1) f2 (e ρr ) eimϕ = −Ef ρr )eimϕ n ³ ´ o ³ ´ 2  q ∂ i(m−1)ϕ e e 2 (e  1 ∂ ρe ∂ − (q−1) f2 (e ρr ) e − iβ ∂ ρe + ρe f1 (e ρr ) ei(m−1)ϕ = −Ef ρr )ei(m−1)ϕ ρe ∂ ρe ∂ ρe ρe2 n. ³. Φ , Φ0. q2 ρe2. o. imϕ. − iβe. ³. ∂ ∂ ρe. 1 (q ρe. (3.6). Since these differential equations have the forms of Bessel equation and the recurrence relation, we can choose the Bessel functions as the spinor part eigenfunctions. Therefore,   . e q (γ ρer ) −γ 2 AJq (γ ρer ) − iβe [−γBJq (γ ρer )] = −EAJ.  e q−1 (γ ρer )  −γ 2 BJq−1 (γ ρer ) − iβe [γAJq−1 (γ ρer )] = −EBJ where A and B are the coefficients of the two eigenfunction components.. 33. (3.7).

(45) CHAPTER 3. FINITE WIDTH MESOSCOPIC DRESSELHAUS SPIN-ORBIT (DSO) RING WITH A MAGNETIC FLUX Comparing with above couple equations, we find that the ratio of the coefficients R=. B A. = ±i, then we get two branches of wave functions .  Jq (γ ρer )eimϕ.  Ψ=. i(m−1)ϕ.  Jq (γ ρer )eimϕ.   ,Ψ = . iJq−1 (γ ρer )e. 3.1.3.  i(m−1)ϕ.  .. (3.8). −iJq−1 (γ ρer )e. Eigenenergy and Eigenstate. e E e E e = 0 and γ 2 −βγ− e = 0 when R is equal to +i and We get two energy dispersion γ 2 +βγ− µ µ ¶ ¶ q q 1 1 e and γ2 = e , −βe + βe2 + 4E −i respectively. Then, if we set γ1 = 2 βe + βe2 + 4E 2 we can obtain the total eigenfunction .   Ψ = a. (1) Hq. (γ1 ρer ) eimϕ.    + b. (1) −iHq−1. .  (2) Hq. (γ1 ρer ) eimϕ. (2) −iHq−1. (γ1 ρer ) ei(m−1)ϕ (γ1 ρer ) ei(m−1)ϕ    (2) (1) imϕ imϕ    Hq (γ2 ρer ) e  Hq (γ2 ρer ) e +c    + d (2) (1) iHq−1 (γ2 ρer ) ei(m−1)ϕ iHq−1 (γ2 ρer ) ei(m−1)ϕ .   ,. (3.9). where a, b, c and d are the coefficients. Since the annulus is defined by the hard-wall confinement potential, we can apply the boundary conditions Ψ (ρ = r1 ) = 0, Ψ (ρ = r2 ) = 0. These conditions give the equation set: . (1) Hq. (2) Hq. (1) Hq. (2) Hq. (γ1 r1 ) (γ1 r1 ) (γ2 r1 ) (γ2 r1 )    −H (1) (γ1 r1 ) −H (2) (γ1 r1 ) H (1) (γ2 r1 ) H (2) (γ2 r1 ) q−1 q−1 q−1 q−1   (2) (1) (2)  H (1) (γ r ) Hq (γ1 r2 ) Hq (γ2 r2 ) Hq (γ2 r2 )  q 1 2  (1) (2) (1) (2) −Hq−1 (γ1 r2 ) −Hq−1 (γ1 r2 ) Hq−1 (γ2 r2 ) Hq−1 (γ2 r2 ). . .  a     b      = 0  c      d. (3.10). If there exists non-trivial solution for a, b, c and d, the determinant of the matrix M must vanish, i.e.,. det (M ) = 0,. (3.11) 34.

(46) CHAPTER 3. FINITE WIDTH MESOSCOPIC DRESSELHAUS SPIN-ORBIT (DSO) RING WITH A MAGNETIC FLUX where . (1). (2). Hq (γ1 r1 )  Hq (γ1 r1 )   −H (1) (γ1 r1 ) −H (2) (γ1 r1 ) q−1 q−1  M = (2)  H (1) (γ r ) Hq (γ1 r2 )  q 1 2  (1) (2) −Hq−1 (γ1 r2 ) −Hq−1 (γ1 r2 ).  (2) Hq (γ2 r1 )   (1) (2) Hq−1 (γ2 r1 ) Hq−1 (γ2 r1 )    (1) (2) Hq (γ2 r2 ) Hq (γ2 r2 )    (1) (2) Hq−1 (γ2 r2 ) Hq−1 (γ2 r2 ) (1). Hq (γ2 r1 ). (3.12). The more detail process is shown in Appendix B.. 3.2. Expression for charge current density. The Hamiltonian of the system including magnetic flux in polar coordinate is ½. · ³ ´ ³ ´2 ¸¾ ∂2 Φ ∂ + − − ∂ϕ2 + 2i Φ0 ∂ϕ + ΦΦ0 H= h ³ í . ∂ Φ ∂ 1 2 iϕ −iϕ iϕ −iϕ +iβ hkz i (e σ+ + e σ− ) ∂ρ + (e σ+ − e σ− ) ρ i ∂ϕ + Φ0 ∂2 ∂ρ2. ~2 − 2m ∗. 1 ∂ ρ ∂ρ. 1 ρ2. (3.13). We start from time-dependent Schrodinger equation. Hψ = i~. ∂ψ . ∂t. (3.14). Then, we put the complex conjugate wave function to operator Eq. (3.14): ½. · ³ ´2 ¸¾ ³ ´ ∂2 Φ ∂ = − ψ + − ∂ϕ2 + 2i Φ0 ∂ϕ + ΦΦ0 ψ h ³ í ∂ ∂ +iβ hkz2 i ψ + (eiϕ σ+ + e−iϕ σ− ) ∂ρ + (eiϕ σ+ − e−iϕ σ− ) ρ1 i ∂ϕ + ΦΦ0 ψ +. i~ ∂ψ ∂t. ~2 + − 2m ∗ψ. ∂2 ∂ρ2. 1 ∂ ρ ∂ρ. 1 ρ2. (3.15). At the same time, we change the wave function in Eq. (3.15) to become opposite complex conjugate form: ½. · ³ ´2 ¸¾ ³ ´ ∂2 Φ ∂ = − − ∂ϕ2 − 2i Φ0 ∂ϕ + ΦΦ0 ψ+ + h ³ í ∂ ∂ + (e−iϕ σ− − eiϕ σ+ ) ρ1 −i ∂ϕ + ΦΦ0 ψ + −iβ hkz2 i ψ (e−iϕ σ− + eiϕ σ+ ) ∂ρ + −ψi~ ∂ψ ∂t. ~2 − 2m ∗ψ. ∂2 ∂ρ2. 1 ∂ ρ ∂ρ. 1 ρ2. 35. (3.16).

(47) CHAPTER 3. FINITE WIDTH MESOSCOPIC DRESSELHAUS SPIN-ORBIT (DSO) RING WITH A MAGNETIC FLUX By combining Eq. (3.15) and Eq. (3.16), we can obtain the azimuthal component of the current density comparing with the continuity equation. · µ ¶ ¸ ¤ −i~ ∂ψ + Φ i 2 ® £ iϕ + + ∂ψ + −iϕ + Jϕ = ψ − ψ − 2i ψ ψ − β k e ψ σ ψ − e ψ σ ψ . + − z 2m∗ ρ ∂ϕ ∂ϕ Φ0 ~ (3.17). In other way, the general form of the wave function can be expressed briefly by defining its two components:. .   Ψ≡. imϕ. fq (e ρr ) e. gq−1 (e ρr ) ei(m−1)ϕ.  . (3.18). where fq (e ρr ) = aHq(1) (γ1 ρer ) + bHq(2) (γ1 ρer ) + cHq(1) (γ2 ρer ) + dHq(2) (γ2 ρer ). (3.19). h i (1) (2) (1) (2) gq−1 (e ρr ) = i −aHq−1 (γ1 ρer ) − bHq−1 (γ1 ρer ) + cHq−1 (γ2 ρer ) + dHq−1 (γ2 ρer ). (3.20). They give the terms in azimuthal component of current density :. ψ+. ∂ψ ∂ψ + ∗ −ψ = 2imfq∗ fq + 2i (m − 1) gq−1 gq−1 ∂ϕ ∂ϕ. (3.21). ∗ gq−1 ψ + ψ = fq∗ fq + gq−1. µ ψ + σ+ ψ =. (3.22) ¶. ∗ fq∗ e−imϕ gq−1 e−i(m−1)ϕ. µ ψ + σ− ψ =. . ∗ gq−1 e−i(m−1)ϕ.  imϕ. fq e  0 1   ∗ −iϕ (3.23)    = fq gq−1 e i(m−1)ϕ 0 0 gq−1 e. ¶ fq∗ e−imϕ. . . . . fq eimϕ  0 0   ∗ iϕ    = gq−1 fq e (3.24) 1 0 gq−1 ei(m−1)ϕ 36.

(48) CHAPTER 3. FINITE WIDTH MESOSCOPIC DRESSELHAUS SPIN-ORBIT (DSO) RING WITH A MAGNETIC FLUX If we set veβ =. ~ Jϕ = ∗ m r1. β hkz2 i·m∗ r1 , ~2. ½. we can get the azimuthal component of the current density:. ¾ ¤ £ ∗ ¤ 1£ ∗ ∗ ∗ qfq fq + (q − 1) gq−1 gq−1 − ie vβ fq gq−1 − gq−1 fq ρe. (3.25). The total charge current density at zero temperature is the sum of the contribution from ground state to the N th state where N is the total electron number. In the absence of magnetic flux, the distribution of the total charge current density is zero even in the presence of DSO coupling.. 3.3 3.3.1. Numerical results and discussion Energy spectrum - Removal of degeneracy. Physical parameters used in the below calculation are obtained for the case of GaAs: with o. DSO coupling constant β = 25eV · A, effective mass m∗ ≈ 0.067me . Structures parameters are inner radius r1 = 10nm and outer radius r2 = 70nm. The energy spectrum which is plotted in the range of −0.6 <. Φ Φ0. < 0.6 adopted the same parameters in chapter 2 before.. Fig. 3.1 is the energy spectrum displayed as a function of the magnetic flux. Φ Φ0. in our. system which is obtained from the determinant equation det (M ) = 0, and β˜ is turned off in (a) turned on in (b). The dimensionless energy E is eigenenergy in units of Ef =. ~2 kf2 . 2m. In Fig. 3.1 (b), we see that each of the discrete eigenenergy splits into two parts (arising up and going down) when we turn on external magnetic flux, and it is interesting that the bigger m become, the wider energy splitting would be. In order to observe the development of the energy levels and eigenstates easily, we combined Fig. 2.2 and Fig. 3.1 (b) artificial. In x-axis, left hand side is the dimensionless DSO coupling strength β˜ from 0 to 1, and right hand side is magnetic flux Φ/Φ0 from 0 to 0.6 ( keep β˜ = 1). Then, we will discuss some issue about the eigenstates and the effect of the external magnetic flux in this section. 37.

(49) CHAPTER 3. FINITE WIDTH MESOSCOPIC DRESSELHAUS SPIN-ORBIT (DSO) RING WITH A MAGNETIC FLUX. 1 (a) 0.9. Energy ( Ef ). 0.8 0.7 0.6 0.5 0.4 0.3 0.2 −1.5. −1. −0.5. 0 Φ/Φ0. 0.5. 1. (a,0) (b,−1) (c,0) (a,1) (c,1) (a,2) (d,−1) (b,−2) (c,2) (d,−2) (c,3) (d,−3) (c,4) (d,−4). (b) 0.6. Energy(Ef). 0.5 0.4 0.3 0.2 0.1 0. −0.4. 1.5. −0.2. 0 Φ/Φ0. 0.2. 0.4. 0.6. Figure 3.1: (a) Several lowest dimensionless eigenenergies E as a function of the magnetic flux Φ/Φ0 for the quantum ring without DSOI, (b) with DSO coupling constant β = o. 25eV · A. Where the legend denotes the state in the form of ( η , m ), η is a, b, c or d and m is the dominated magnetic quantum number. The other physical parameters are obtained for the case of GaAs: m∗ ≈ 0.067me , λf = 40nm, r1 = 10nm, r2 = 70nm, o. d = 50 A.. 38.

(50) CHAPTER 3. FINITE WIDTH MESOSCOPIC DRESSELHAUS SPIN-ORBIT (DSO) RING WITH A MAGNETIC FLUX. Figure 3.2: Left hand side is the the several lowest dimensionless eigenenergies E as a function of the dimensionless DSO coupling strength β˜ from 0 to 1, right hand side is the eigenenergies as a function of the magnetic flux Φ/Φ0 from 0 to 0.6. The physical parameters are the same with Fig. 3.1.. 39.