自旋霍爾效應設置下在微結構附近產生的殘餘電阻偶極和電子自旋偶極的共振現象

全文

(1)國立交通大學電子物理研究所碩士論文自旋霍爾效應設置下在微結構附近所產生的殘餘電阻偶極和自旋偶極的共振現象 RESONANT GENERATION OF RESIDUAL RESISTIVITY DIPOLE AND SPIN DIPOLE AROUND MICROSTRUCTURES IN A SPIN-HALL CONFIGURATION. 研究生：陳冠伊指導教授：朱仲夏教授. 中華民國九十六年七月.

(2) 自旋霍爾效應設置下在微結構附近所產生的殘餘電阻偶極和電子自旋偶極的共振現象 RESONANT GENERATION OF RESIDUAL RESISTIVITY DIPOLE AND SPIN DIPOLE AROUND MICROSTRUCTURES IN A SPIN-HALL CONFIGURATION. 研究生：陳冠伊. Student：Kuan-Yi Chen. 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 2007 Hsinchu, Taiwan, Republic of China. 中華民國九十六年七月.

(3) ! Ծ௽ᓅᅟਏᔈ೛࿼Πӧ༾่ᄬߕ߈ౢғ‫ූޑ‬Ꭹႝߔଽཱུ ‫ک‬ႝηԾ௽ଽཱུ‫ޑ‬Ӆਁ౜ຝ! !. ࣴ‫ز‬ғǺഋ߷Ҳ! ! ! ! ! ! ! ! !!!!!!!!! ࡰᏤ௲௤ǺԙҸহ௲௤! !. ୯ҥҬ೯εᏢ! ႝη‫ނ‬౛ࣴ‫!܌ز‬ ! !. ᄔा! ! ! ! ԜፕЎ‫ޑ‬ขᗺ‫ܫ‬ӧ΋ঁԖѦуႝ൑‫ޑ‬༾่ᄬߕ߈ǴԾ௽ॉၰҬϕբҔჹႝ಻ ‫ک‬ႝηԾ௽‫୴ޑ‬ᑈ‫ޑ‬ቹៜǶ೸ၸ΋ঁᕉ‫ޑ׎‬༾่ᄬ‫܌‬ёૈ೷ԋ‫ޑ‬ໆηӅਁਏᔈǴ ‫ॺך‬຾΋‫؁‬௖૸‫ܫ‬εԾ௽ॉၰҬϕբҔ‫ޑ‬ёૈ‫܄‬Ƕ! ! ! നख़ा‫ޑ‬ว౜ࢂǴа۳΋‫ޔ‬೏۹ౣ‫ޑ‬ǴҗႝՏఊࡋౢғ‫ޑ‬Ծ௽ॉၰጠӝёа ٬΋ঁ༾่ᄬߕ߈೷ԋёᢀ‫ޑ‬ႝηԾ௽୴ᑈǶᙖҗ‫ע‬ӧ΋ঁѦႝ൑ύߚѳᑽႝԾ ϩթ္‫܌‬Ԗႝη‫ޜޑݢ‬໔ᐒ౗уᕴǴ‫ॺך‬ёаӧໆηቸၰጄൎϣ‫ޑ‬༾่ᄬߕ߈ள ‫ډ‬΋ঁ‫ރ‬՟ଽཱུ‫ޑ‬ႝηԾ௽ϩթǴΨ൩ࢂԾ௽ଽཱུǶՔᒿ๱ໆηණ৔‫ޑ‬ӅਁǴԜ ҬϕբҔёаග‫ॺךٮ‬΋ঁፓ௓Ծ௽ଽཱུ‫ޑ‬Б‫ݤ‬Ƕ੝ձ‫ࢂޑ‬Ǵᒿ๱‫س‬಍຤ԯૈ‫ޑ‬ ‫ׯ‬ᡂǴԾ௽ଽཱུ཮߄౜рБӛ‫ޑ‬ϸᙯ‫ک‬மࡋ‫ܫޑ‬εǶ! ! ! ᙖҗЇΕ΋ঁᆶԾ௽࣬ᜢ‫ߚޑ‬ѳᑽႝηϩթǴ‫׷‬਑ङඳᚇ፦‫܌‬ౢғ‫ޑ‬Ѧ‫ޚ‬Ծ ௽ॉၰጠӝਏᔈς࿶х֖ӧ‫ޑॺך‬Եቾ္य़ǶԜਏᔈӧӅਁਔჹԾ௽ଽཱུ‫ޑ‬অ҅ ٠ؒԖॶள‫ݙ‬ཀ‫ޑ‬ଅ᝘Ƕ! ! ! ! ! ! ! i.

(4) RESONANT GENERATION OF RESIDUAL RESISTIVITY DIPOLE AND SPIN DIPOLE AROUND MICROSTRUCTURES IN A SPIN-HALL CONFIGURATION. Student: Kuan-Yi Chen. Advisor: Prof. Chon-Saar Chu. Department of Electrophysics National Chiao Tung University. Abstract This thesis focuses on spin-orbit interaction effect on charge and spin accumulation around a microstructure in an external electric field. Furthermore, a ring type microstructure is introduced to explore possible amplification of the spin-orbit interaction effect from quantum resonances allowed by such microstructures. A major finding in this thesis is that the mostly overlooked spin-orbit interaction arising from in-plane potential gradient can contribute to significant spin accumulation around a microstructure. By considering quantum-mechanical scattering of individual electrons in a nonequilibrium distribution acted upon by an external electric field, we obtain a spin dipole, or spin accumulation of a dipole-like spatial variation, around a microstructure within the ballistic range. The subsequent quantum scattering resonance provides and additional knob for the manipulation of spin dipole. Specifically, the spin dipole manifests both sign reversal and large amplification when the Fermi energy crosses a resonance. We have included the extrinsic spin-orbit interaction effects from background spinorbit scatterers by invoking a spin dependent nonequilibrium electron distribution. The correction of the spin dipole is found to be insignificant in the resonant region.. ii.

(5) ठᖴ! ! གᖴ೭‫ٿ‬ԃӭٰԙԴৣ‫ޑ‬௲ᏤǴЀ‫ࢂځ‬ᔅԆ‫่݀زࣴע‬ว߄ӧයт΢ǴаϷჹԜ ፕЎനࡕ໘ࢤ‫ޑ‬ཀ‫ـ‬Ϸঅ‫ׯ‬Ǵᡣ‫ך‬ᕇ੻‫ؼ‬ӭǶΨགᖴঞ‫ד‬໢Ꮲߏǵ✆‫ځ‬։Ꮲߏ๏ Α‫ࡐך‬ӭᝊ຦‫ޑ‬ཀ‫ـ‬ǴᗋԖჴᡍ࠻უՔॺഉ‫ࡋך‬ၸࣴ‫ز‬ғ‫ޑ‬ғఱǴᏢߏႠૐǵ༫ ථьǵӹ‫׊‬ǵߓ҇ǴᏢ‫ۊ‬లᆢǵలীǴགᖴգॺ๏‫ך‬೚ӭ࿶ᡍ΢‫ޑ‬ᔅշǶЀ‫ځ‬ག ᖴᏢߏԴЦǵᑫঢǵᗋԖᏢ‫࠹ד׌‬ӧനࡕ‫ޑ‬Вη္‫ޑ‬Ⴔᓰ‫࣬ک‬٩ࣁ‫ڮ‬ǴᗋԖ໋ࣻ ‫ޑ‬ഉՔ‫ک‬Ѻ਻Ƕനགᖴ‫ݿࢂޑ‬༰‫ޑ‬ਭ୻ǵЍ࡭‫ک‬ᡏፊǴᗋԖ‫ޑۂۂ‬ᜢЈǶགᖴ‫܌‬ ԖᜢЈ‫ޑך‬ΓǶ!. iii.

(6) Contents. Abstract in Chinese. i. Abstract in English. ii. Acknowledgement. iii. 1 Introduction. 1. 1.1. Introduction to Landauer’s residual resistivity dipole . . . . . . . . . . . .. 3. 1.2. Spin-orbit coupling in solid-state systems . . . . . . . . . . . . . . . . . . .. 4. 1.3. Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 6. 1.4. A guiding tour to this thesis . . . . . . . . . . . . . . . . . . . . . . . . . .. 7. 2 Landauer’s residual resistivity dipole around a ring-shaped microstructure 2.1. 9 Quantum-mechanical scattering in two dimensions . . . . . . . . . . . . . .. 9. 2.1.1. Method of partial waves . . . . . . . . . . . . . . . . . . . . . . . . 10. 2.1.2. Phase shift and the unitarity relation . . . . . . . . . . . . . . . . . 12. 2.1.3. Asymptotic expansion of scattering wave and the differential cross section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14. 2.2. Nonequilibrium distribution of electrons: Theory of the Boltzmann kinetic equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2.1. Relaxation time approximation . . . . . . . . . . . . . . . . . . . . 15. 2.2.2. The linear response to the uniform electric field . . . . . . . . . . . 16 iv.

(7) CONTENTS 2.3. Landauer’s residual resistivity dipole . . . . . . . . . . . . . . . . . . . . . 16 2.3.1. Charge accumulation due to nonequilibrium incident distribution . 16. 2.3.2. Screening effect and potential induced by charge accumulation . . . 18. 2.3.3. Thomas-Fermi screening in two dimensions . . . . . . . . . . . . . . 19. 2.3.4. Definition of the strength of the residual resistivity dipole . . . . . . 21. 2.4. Resonance generation of RRD around a ring-shaped structure . . . . . . . 22. 2.5. Brief summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27. 3 Residual resistivity dipole and spin dipole in the presence of spin-orbit interaction arising from in-plain potential gradient of a microstructure 28 3.1. Spin dependent asymmetric scattering in the presence of SOI . . . . . . . . 29 3.1.1. Two useful relations . . . . . . . . . . . . . . . . . . . . . . . . . . 31. 3.2. Residual resistivity dipole in the presence of local structure SOI . . . . . . 32. 3.3. Spin dipole due to spin-independent nonequilibrium incident distribution . 33 3.3.1. 3.4. Second-order correction to the spin dipole strength at resonance . . 35. Resonance of charge and spin dipole in the presence of the SOI from the microstructure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37. 3.5. 3.4.1. Resonant asymmetric skew scattering . . . . . . . . . . . . . . . . . 37. 3.4.2. Resonant RRD in the presence of the SOI . . . . . . . . . . . . . . 39. 3.4.3. Resonance of spin dipole . . . . . . . . . . . . . . . . . . . . . . . . 40. Brief summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44. 4 Spin dipole correction due to effect of extrinsic spin-orbit interaction 4.1. 46. Differential cross section in terms of spin density operator . . . . . . . . . . 46 4.1.1. Density operator formalism . . . . . . . . . . . . . . . . . . . . . . 46. 4.1.2. Spin dependent scattering cross section in terms of spin density operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47. 4.2. The spin dependent nonequilibrium distribution . . . . . . . . . . . . . . . 50. 4.3. The correction to the spin dipole due to the term h(k) · σ . . . . . . . . . 54 v.

(8) CONTENTS 4.4. Brief summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58. 5 Numerical calculation for smooth potential variation. 61. 5.1. Variable phase approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61. 5.2. Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65. 5.3. Brief summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66. 6 Conclusion and future work. 70. A Derivation of charge accumulation. 72. B Derivation of spin accumulation. 75. C Derivation of transport cross section. 78. D Derivation of transverse transport cross section. 81. E Asymmetric Mott skew scattering. 84. F Asymptotic expansion of residual resistivity dipole and charge dipole strength. 89. G Asymptotic expansion of spin dipole and spin dipole strength. 92. H Collision integrals in detail. 95. vi.

(9) List of Figures 1.1. An illustration of sample geometry for cross sectional STP experiments. [Superlattices and Microstructures 23, 699 (1998)]. . . . . . . . . . . . . .. 1.2. A. STM-topogragh, B. STP potential image and C. Cross sectional cut of B. [Superlattices and Microstructures 23, 699 (1998)] . . . . . . . . . . . .. 1.3. 3. 3. System configuration: A ring-shaped potential embedded in a two dimensional electron gas (2DEG). An electric field E sets up a current in the 2DEG. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 6. 1.4. Radial profile of the central ring-shaped repulsive potential.. 7. 2.1. Qualitative sketch of g(k) in the presence of a driving field E. The shifted. . . . . . . . .. Fermi circle is towards negative x-axis because negatively charged electrons are driven opposite to the direction of E. . . . . . . . . . . . . . . . . . . . 17 2.2. Energy spectra of an infinite circular well and phase shifts versus the chemical potential for a ring-shaped step potential barrier with parameters: V0 = 1 the potential height, a = 6 the inner radius, and b = 9 the outer radius. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25. 2.3. kσtr versus the chemical potential with parameters of a ring-shaped step potential barrier: the potential height V0 = 1, the inner radius a = 6, and the outer radius b = 9. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26. vii.

(10) LIST OF FIGURES 3.1. Radial variation of Sz (ρ) along φρ = π/2 at the two resonant energies (a) μ = 0.3312, and (b) μ = 0.3328 in the resonance pair labeled (1,5). The green dashed curve is ps /k ∗ ρ; the red dot-dashed curve includes the 2nd-order correction ℘s . The geometry of the ring: the potential height V0 = 0.75, the inner radius a = 14, and the outer radius b = 20.5. . . . . . 36. 3.2. Qualitative sketch of the phase shift in the presence SOI: for the up-spin partial waves, the solid curves in the lower plot is for positive l, and the dashed curves are for negative l; or equivalently, for partial waves with positive l, the solid curves are for up-spin waves, and the dashed curves are for down-spin waves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38. 3.3. + versus system energy: the red curve denotes the summation over posiσ⊥ + + tive l, ∞ l=0 sin[2(δl − δl+1 )]; the blue curve stands for the summation over + + + negative l, ∞ l=0 sin[2(δ−l − δ−(l+1) )]; the total σ⊥ is denoted by the black. curve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.4. kσtr (the upper plot) and kσ⊥ (the lower plot) versus the chemical potential μ with parameters of a ring-shaped step potential barrier: the potential height V0 = 1, the inner radius a = 6, and the outer radius b = 9. The red line in the lower plot is kσ⊥ for a disk-shaped potential barrier with its radius equal to 9. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41. 3.5. (a) kσ⊥ versus the Fermi energy in units of E∗ = 77.1 meV. Parameters of a ring-shaped step potential barrier: the potential height V0 = 0.75, the inner radius a = 14, and the outer radius b = 20.5. (b)-(d) Blowups of the resonance pair labeled in the same series n = 1 and l = 4, 5, 6, respectively. The dot-dashed (red) curves are the partial summations of kσ⊥ including only terms involving δ5σ . The dashed (blue) curves are kσ⊥ for V0 = 0.751. The smallest abscissa in (b)-(d) is 0.0001. . . . . . . . . . . . . . . . . . . . 42. viii.

(11) LIST OF FIGURES 3.6. Spatial distribution Sz (ρ) at the a resonant energy ε = 0.3312 with the geometry of the ring: the potential height V0 = 0.75, the inner radius a = 14, and the outer radius b = 20.5. The electric field E is applied along positive x-direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45. 4.1. Qualitative sketch of k-dependent spin polarization due to the extrinsic SOI for a negative SO coupling constant λ. Spin polarization of the upper semicircle (in the positive y-plane) is up (aligned in positive z-direction); while spin polarization of the lower semicircle (in the negative y-plane) is down (aligned in positive z-direction). . . . . . . . . . . . . . . . . . . . . . 54. 4.2. ex ˆ for The relation between spin current direction and spin dipole pex s = ps y. negative SO coupling constant λ. pex s < 0 is shown in this figure. The plus sign below the ring stands for spin up accumulation , and vice versa. 4.3. . . . 56. (a) kσ⊥ versus energy with parameters the same as in Fig. 3.5-(a),and (b) a blow-up of |γμ |kσtr versus energy. . . . . . . . . . . . . . . . . . . . . . . 58. 4.4. (a) Radial variation of Sz (ρ), Szex (ρ), and (b) Blowup of the radial radiation Szex (ρ) along φρ = π/2 at the resonant energy μ = 0.331178. At resonance, Szex so tiny that it is negligible. Parameters are on top of the figure. . . . . 59. 4.5. (a) Radial variation of Sz (ρ), Szex (ρ), and (b) a blow-up of the radial radiation Szex (ρ) along φρ = π/2 at the energy away from μ = 0.331000. Away from resonance, Szex is comparable to Sz , so nearly no spin polarization in the region away from the ring structure. Other parameters are the same as Fig. 4.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60. 5.1. Variable phase function δl (ρ) vs. radial position without SOI for a step potential of a ring. Dimensionless parameters are ε = 3, V0 = 1, a = 6, b = 9. We can see the variation of phase is only within the range of the ring potential between a and b. . . . . . . . . . . . . . . . . . . . . . . . . 65. ix.

(12) LIST OF FIGURES 5.2. Variable phase function δl (ρ) vs. radial position without SOI for a gaussian radial potential profile. Dimensionless parameters are ε = 3, V0 = 1, ρ0 = 8, h = 3. We can see the variation of phase is only within the range of the ring potential between a and b. . . . . . . . . . . . . . . . . . . . . . 66. 5.3. Phase shifts versus energy. We can see the splits of energies at which the phase crosses nπ/2 (n is odd) from below. In the lower diagram, the solid curves are phase shifts for positive l, and the dashed curves are for negative −l, both with σ = +1. Parameters are: ρ0 = 8, h = 3, V0 = 1, and dimensionless spin-orbit coupling constant λ = −0.055. . . . . . . . . . 67. 5.4. kσtr versus energy. Parameters are ρ0 = 8, h = 3, V0 = 1 . . . . . . . . . . 68. 5.5. kσ⊥ versus energy. Parameters are ρ0 = 8, h = 3, V0 = 1. . . . . . . . . . . 69. E.1 Scattering of a plane wave with spin up polarization for both total wave and scattered wave: (a) off-resonance scattering. (b) scattering resonance contributed by partial waves labeled l = 0. . . . . . . . . . . . . . . . . . . 85 E.2 Scattering of a plane wave with spin up polarization for both total wave and scattered wave: asymmetric scattering contributed by partial waves labeled (a) l = 1 and (b) l = −1. . . . . . . . . . . . . . . . . . . . . . . . 86 E.3 Scattering of a plane wave with spin up polarization for both total wave and scattered wave: asymmetric scattering contributed by partial waves labeled (a) l = 2 and (b) l = −2. . . . . . . . . . . . . . . . . . . . . . . . 87 E.4 Scattering of a plane wave with spin up polarization for both total wave and scattered wave: asymmetric scattering contributed by partial waves labeled (a) l = 3 and (b) l = −3. . . . . . . . . . . . . . . . . . . . . . . . 88. x.

(13) Chapter 1 Introduction The main stream of present day technology is built upon charge-based electronics where charge rather than spin of electrons is endeared as the carrier of information. The spin of an electron, however, is essentially neglected in the technology arena. This trend of neglecting spin in the application sector may have set a different course in the past decades when prototypes of new, spin-based electronics, or spintronics, were proposed and realized. Earlier spin-based devices made use of the physics of giant magnetoresistance (GMR) in a system consisting of magnetic materials [1]. More recent research takes on the greater challenge of achieving spintronics in all semiconductor configurations: without magnetic materials and applying magnetic fields. To achieve an all electric control 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 intrinsic type, such as Rashba [2], or Dresselhaus [3] SOI are associated with the lacking of structural inversion symmetry, or spatial inversion symmetry, respectively. On the other hand, the extrinsic SOI is due to the presence of SOI scatterers, or Mott scattering by nonmagnetic impurities in the system [4, 5]. Recently, building on these SOI mechanisms, a physical phenomenon called spin-Hall. 1.

(14) CHAPTER 1. INTRODUCTION effect (SHE) has attracted intensive attentions. This is a phenomenon where a dc charge current can give rise to transverse spin current, in the lateral direction, and spin accumulation at lateral edges of a strip-like sample [6–8]. This development has promoted great hope in meeting the challenge of all-electric generation of spin transport and accumulation. Experiment contribution of intrinsic SHE has been established in p-doped 2D semiconductor quantum wells [9], and of extrinsic SHE has been observed in n-doped 3D semiconductor thin film strips of GaAs [10]. That the electron spin can precess when it moves in an electric or magnetic field, complicates our lifes because any component of spin is not a conserved quantity. Hence connecting the spin current to the spin accumulation is an issue that demands great caution. As such, a proper definition was a great issue and remains to be entirely settled [11–13]. Since experimentally it is as yet a tough issue concerning the direct measurement of the spin current, the spin accumulation is the only alternative for the detection of SHE. However, the spin accumulation on the lateral edges of a sample should not be the only signature of SHE. Similar to Landauer’s picture of residual resistivity dipole (RRD) around , where a dipole-like charge accumulation is generated by a dc charge current around a local scatterer, a nonequilibrium spin dipole might be expected to form in a spin-Hall configuration. That a spin dipole accumulated around a non-SOI scatterer in a Rashba-type two dimensional electron gas (2DEG) has indeed be found recently [14]. On the other hand, even though there are many recent works on spin-dependent quantum scattering around microstructures [15–19], the SOI due to in-plane potential gradient has essentially been neglected. In this thesis, we will show that this in-plane potential gradient SOI can contribute to significant spin accumulation when quantum resonance is invoked.. 2.

(15) CHAPTER 1. INTRODUCTION. Figure 1.1: An illustration of sample geometry for cross sectional STP experiments. [Superlattices and Microstructures 23, 699 (1998)].. Figure 1.2: A. STM-topogragh, B. STP potential image and C. Cross sectional cut of B. [Superlattices and Microstructures 23, 699 (1998)]. 1.1. Introduction to Landauer’s residual resistivity dipole. The charge accumulation around a local scatterer that we will discuss in this section is referred to as the residual resistivity dipole (RRD), first predicted by R. Landauer in 3.

(16) CHAPTER 1. INTRODUCTION 1957 [20]. According to Landauer, local scatterers are expected to give rise to spatial variations in the electric field. This forms a microscopic point of view for the set-up of the macroscopic potential drop from these individual electric dipole fields. In other words, the Landauer RRD is the microscopic origin of the macroscopic resistance. Theoretical formalism for the calculation of the RRD is built upon the asymptotic scattering wave function around a target scatterer [21, 22]. Experiment confirmation of the Landauer RRD has to wait until 1996, when the electric potential around a defect on a bismuth surface was picked up by a scanning tunneling potentiometry (STP) [23].. 1.2. Spin-orbit coupling in solid-state systems. 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 interaction term, a term that has found great success in atomic energy spectra. The form of this spin-orbit interaction, in vacuum, is [24]. HSO = −. e σ · (E × p) = σ · (∇V × p) , 2 2 4m0 c 4m20 c2. (1.1). where m0 is the free electron mass, the Planck constant and c 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 which couples with the electron spin through the magnetic moment of the electron spin. Through this effective magnetic field, which certainly depends on the orbital motion of the electron, the SOI is established. This physics holds in semiconductor too, when V (r) becomes the periodic potential of the host lattice and also the impurities. Electronic state calculation in semiconductor is best described by the k·p model, when the states in the vicinity of the band edges is of our interest. Furthermore, within the envelope function approximation (EFA), the energy band can be characterized by effective 4.

(17) CHAPTER 1. INTRODUCTION 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, or structural inversion asymmetry (SIA), the so-called Rashba SOI. The BIA is found in zincblende structure and the SIA in asymmetric quantum wells (QWs) or heterostructures. 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) = V (r) in vacuum, the SO coupling is 1 dV 2 1 dV L λvac 1 dV σ · (∇V × p) = σ · (r × p) = · σ = − L·σ 4m20 c2 4m20 c2 r dr 4m20 c2 r dr r dr where L is the orbital angular momentum, σ is the vector Pauli matrices and λvac = 2. −2 /(4m20 c2 ) ≈ −3.72 × 10−6 ˚ A. But in a semiconductor, also for a central potential V (r) = V (r), the SO coupling is. HSO = −. λ 1 dV L · σ, r dr. where P2 1 1 λ≈ − . 3 Eg2 (Eg + Δ0 )2 For a 2DEG, the SOI becomes. HSO = −. λ 1 dV (ρ) L z σz . ρ dρ. Here P is the momentum matrix element between s- and p-orbitals, Eg is the energy band gap, and Δ0 represents the SOI energy split to the spin split-off hole band [8, 25]. Of. 5.

(18) CHAPTER 1. INTRODUCTION 2 particular interest is that λ = 120 ˚ A in InAs, which is seven order of magnitude greater. than λvac [25, 26]. Very roughly speaking, this large enhancement of SO coupling constant can be understood in the following. With λvac ∝. 1 m20 c2. =. 1 1 , m0 m0 c2. we can see that. λ m0 m0 c2 ∼ ∗ . λvac m Eg For InAs, m0 /m∗ ∼. 2 1 ; mE0gc 0.023. ∼. 0.5 MeV ; 0.418 eV. leading to. λ ∼ 52 × 106 . λvac Comparing to. 2. 120 ˚ A 2 3.73×10−6 ˚ A. = 32 × 106 , we see that the above hand waving argument has. captured the essential physical origin of the great enhancement.. 1.3. Motivation. It is this great enhancement in the SO coupling constant that causes us to take the inplane potential SOI very seriously and to study its effect. As a result, we propose a. Figure 1.3: System configuration: A ring-shaped potential embedded in a two dimensional electron gas (2DEG). An electric field E sets up a current in the 2DEG.. 6.

(19) CHAPTER 1. INTRODUCTION. Figure 1.4: Radial profile of the central ring-shaped repulsive potential.. ring-shaped potential pattern. To concentrate on the effect of the in-plane potential gradient SOI from the microstructure, we will not include other intrinsic and extrinsic SOI. Neglecting Rashba-type SOI is justified in a symmetric quantum well. In our results, particulary near the quantum resonances, to remain intact even when Dresselhaus SOI would be included. Extrinsic SOI effect we will discuss in this thesis is found to be a small correction to our conclusion.. 1.4. A guiding tour to this thesis. In Chapter 2, we will review the physics and formalism of a charge RRD around a local scatterer. Concepts such as nonequilibrium distribution of incident electrons, obtained from the Boltzmann kinetic equation, scattering wave functions of these electrons, and Thomas-Fermi screening of these pile-up of charges to the electric potential field of the electric dipole. As a preview to the resonant behavior, we will also consider the resonant behavior of the RRD around a ring-shaped microstructure. No in-plane potential gradient SOI is included in this preview session. In Chapter 3, we reconsider the ring-shaped microstructure, but with in-plane po-. 7.

(20) CHAPTER 1. INTRODUCTION tential gradient SOI included. The 2D quantum-mechanical scattering becomes spindependent and asymmetric. The resonant behavior in the charge RRD and the spin accumulation is presented. In Chapter 4, the effect of the extrinsic SOI to the resonant spin accumulation around the ring-shaped microstructure is presented. Correction is shown to be small. In Chapter 5, we consider a smooth potential profile of a ring-shaped microstructure. The resonant behavior is shown to remain intact. Finally in chapter 6, we present our conclusion and possible future work.. 8.

(21) Chapter 2 Landauer’s residual resistivity dipole around a ring-shaped microstructure In this chapter, we will present the physics of Landauer’s residual resistivity dipole (RRD) and formalism leading to such an entity. The distribution of the charge pile up of the RRD around a local scatterer is obtained by summing up the density distribution of electrons scattering from a target scatterer. The distribution of the electrons is set up by an external field. The coulomb interaction between electrons causes screening to occur and results in an induced potential variation around a scatterer. In the lowest order in 1/(kρ) where k is the Fermi wave number and and ρ the distance from the scatterer, the induced potential takes on a dipole field from which we can define a RRD strength. We present the energy dependence of the RRD which, as expected, exhibits resonant behavior in the case of a ring-shaped microstructure.. 2.1. Quantum-mechanical scattering in two dimensions. For a 2D central potential, the cylindrical symmetry governs that the scattering wave function be most conveniently expressed in polar coordinates. The scattering wave function from an incident plane wave contains a scattered wave component representing cylindrical. 9.

(22) CHAPTER 2. LANDAUER’S RESIDUAL RESISTIVITY DIPOLE AROUND A RING-SHAPED MICROSTRUCTURE outgoing wave co-centered with the central potential. Without the central potential, the free particle Schrödinger equation is given by. (∇2 + k 2 )ψ = 0, with the particle energy ε = 2 k 2 /2m∗ . Expressed in polar coordinates, the equation is 1 ∂ ∂ψ 1 ∂ 2ψ ρ + 2 2 + k2ψ = 0 . ρ ∂ρ ∂ρ ρ ∂φ. (2.1). Applying the factored form ψ(ρ) = R(ρ) Φ(φ), it can be decoupled separately into the radial and azimuthal coordinates, d2 R 1 dR l2 2 + + k − 2 R = 0; dρ2 ρ dρ ρ 2 dΦ = −l2 Φ , dφ2 where the radial equation is the Bessel equation. Therefore the free electron wave function in two dimensions can be written in the form ψ(ρ, φ) = Jl (kρ) eilφ , where Jl (x) is the Bessel function of the first kind with l = 0, ±1, ±2, . . . the quantum number of the azimuthal motion, along the coordinate φ which is the angle between ρ ˆ. and x. 2.1.1. Method of partial waves. To perform the quantum scattering calculation from a cylindrical central potential, we ˆ , in terms of the cylindrical waves, the first expand the incident plane wave, along x. 10.

(23) CHAPTER 2. LANDAUER’S RESIDUAL RESISTIVITY DIPOLE AROUND A RING-SHAPED MICROSTRUCTURE Jacobi-Anger expansion [27], given by. e. ikx. =e. ikρ cos φ. =. +∞ . il Jl (kρ) eilφ .. (2.2). l=−∞. Since Jl (kρ) is a standing wave along the radial direction, it is convenient, for our purpose, to express it in terms of two radial propagating waves, the Hankel functions,. 1 (1) (2) Hl (kρ) + Hl (kρ) , Jl (kρ) = 2 (1). (2). where Hl (x) and Hl (x) are defined as (1). Hl (x) = Jl (x) + iYl (x) ; (2). Hl (x) = Jl (x) − iYl (x) . The radial propagating nature of these Hankel functions is most transparent in their asymptotic form, i.e. the region where kρ 1,. 2 i(kρ−lπ/2−π/4) ; e kρ→∞ πkρ. 2 −i(kρ−lπ/2−π/4) (2) lim Hl (kρ) = . e kρ→∞ πkρ lim. (1) Hl (kρ). =. (1). (2). It is clear that Hl , Hl. (2.3). can be viewed as circular waves propagating radial outwards. from, inwards towards the scattering center, respectively. The cylindrical symmetry of the scattering potential causes waves to be coupled only within the same l. This is essentially the conservation of orbital angular momentum, which is true for a central potential but without SOI. The total wave function, including the incident and the scattered waves, is written in the form. Ψ(ρ) =. +∞ . il Rl (ρ) eilφ ,. (2.4). l=−∞. 11.

(24) CHAPTER 2. LANDAUER’S RESIDUAL RESISTIVITY DIPOLE AROUND A RING-SHAPED MICROSTRUCTURE where Rl is yet to be determined. Substituting Eq. (2.4) into the Schrödinger equation Eq. (2.1), the radial equation becomes 1 d dRl (ρ) l2 2m∗ 2 ρ + k − 2 − 2 V (ρ) Rl (ρ) = 0. ρ dρ dρ ρ . (2.5). Boundary conditions will be established in the next section for the solving of Rl (ρ).. 2.1.2. Phase shift and the unitarity relation. For a scattering center, whatever goes in should be eventually go out. Hence physically, the only thing that can be changed is the phase of the outgoing cylindrical waves. Therefore phase shift in the asymptotic outgoing cylindrical wave, where V (ρ) = 0, should carry information about V (ρ). Therefore, the asymptotic form of Rl (ρ) should be of the form. lim Rl (ρ) ∝. kρ→∞. 2 lπ π cos kρ − − + δl , πkρ 2 4. or more transparently, lim Rl (ρ) = √. kρ→∞. 2iδ i(kρ−lπ/2−π/4)

(25) 1 e le + e−i(kρ−lπ/2−π/4) . 2πkρ. (2.6). where δl is the phase shift of the outgoing wave. Clearly δl = 0 reduces Rl (ρ) to Jl (kρ) as expected. Note that δl is the sole quantity to be determined since the amplitude of the two cylindrical wave components in Eq. (2.6) should be the same, for unitary requirement. On the same token, δl must be real. We need to find a boundary condition for the determination of δl . For an arbitrary radial potential profile, we need to integrate the radial equation for Rl (ρ) and matches it with the expected form, Eq. (2.6), outside the potential. The radial wave function outside the range of the potential can be expressed in the form Rlout (ρ) = eiδl [cos δl Jl (kρ) − sin δl Yl (kρ)] , 12. (2.7).

(26) CHAPTER 2. LANDAUER’S RESIDUAL RESISTIVITY DIPOLE AROUND A RING-SHAPED MICROSTRUCTURE where the superscript ’out’ stands for ’outside’. Eq. (2.7) coincides with Eq. (2.6) in its asymptotic region. The boundary condition for the radial differential equation, via integrating Eq. (2.5), is that, the logarithmic derivative of Rl , L [Rl (ρ)] ≡ ρ. d ln Rl (ρ) , dρ. (2.8). must be continuous at any ρ. Since the phase shift δl is defined in the potential-vanishing region, we have. kρb [cos δl Jl (kρb ) − sin δl Yl (kρb )] ρ dRl , = cos δl Jl (kρb ) − sin δl Yl (kρb ) Rl dρ ρ=ρ− b. where ρb is the ’outer edge’ of the scattering potential, beyond which the potential is vanishing, and Jl (x) ≡ dJl (x)/dx. After some rearrangement, the phase shift is cast into the form. tan δl =. kρb Jl (kρb ) − βl Jl (kρb ) , kρb Yl (kρb ) − βl Yl (kρb ). (2.9). where. ρ dRl βl ≡ Rl dρ ρ=ρ− b. is the logarithmic derivative of the radial functions evaluated at ρ smaller but infinitesimally close to ρb . Thus far the discussion is general, without being limited to any radial potential profile inside ρb . Taking a unity amplitude for the plane wave, the total wave function outside the range of the scattering potential is given by. Ψout k (ρ). =. +∞ . il eiδl [cos δl Jl (kρ) − sin δl Yl (kρ)] eilφ .. (2.10). l=−∞. ˆ , and the incident energy is ε, giving a free particle wave The incident wave is along x √ number k = 2m∗ ε/. 13.

(27) CHAPTER 2. LANDAUER’S RESIDUAL RESISTIVITY DIPOLE AROUND A RING-SHAPED MICROSTRUCTURE. 2.1.3. Asymptotic expansion of scattering wave and the differential cross section. The phase shift is related to the differential cross section. Separating out the plane wave, in the total wave function in Eq. (2.10), we have. Ψk (ρ) = eik·ρ +. +∞ (1) 1 l 2iδl i e − 1 Hl (kρ) eil(φρ −φk ) , 2 l=−∞. (2.11). where incident wave direction k = (k, φk ), rather than being restricted to kˆ = xˆ. In the asymptotic region, the above equation is in the form eikρ Ψk (ρ) ≈ eikρ + √ f (φ), ρ with φ ≡ φρ − φk , and. fk (φ) =. +∞ 2i iδl e sin δl eilφ , πk l=−∞. (2.12). is the two dimensional scattering amplitude [28]. The scattering differential cross sections is givne by D(φ) ≡ |f (φ)|2 .. 2.2. Nonequilibrium distribution of electrons: Theory of the Boltzmann kinetic equation. In this section we will give a brief review on the Boltzmann equation which we will use to obtain the nonequilibrium electron distribution when an electric field is applied. In a semiclassical regime, the state of a N particle system can be given by the set of coordinates (pi , qi ) , where i = 1, 2, 3, · · · , dN . This constitutes a state point in the phase space. The Boltzmann equation aims to calculate the single? particle distribution f (k, r) by tracing. 14.

(28) CHAPTER 2. LANDAUER’S RESIDUAL RESISTIVITY DIPOLE AROUND A RING-SHAPED MICROSTRUCTURE its time evolution in the phase space, given by. ∂f ∂f ∂f ∂f ˙ , +k· + r˙ · = ∂t ∂k ∂r ∂t coll. (2.13). The left-hand side originates form the tracing of an infinitesimal phase space volume, taking its volume changes in due course of its motion in the phase space, and requires that the number of states enclosed remain the same. The right hand-side is the jump into or out of the infinitesimal phase space volume due to collision with background impurities and is given by. ∂f (k) = − {Wk,k f (k) [1 − f (k )] − Wk ,k f (k ) [1 − f (k)]},. ∂t coll k. (2.14). where Wk,k is the transition rate at which electrons are scattered from a state k to another state k , and its dimension is inverse of time. For the case when the scattering by the background impurities preserve, then we have Wk,k = Wk ,k , and Eq. (2.14) can be simplified to. ∂f (k) =− Wk,k [f (k) − f (k )].. ∂t coll k. 2.2.1. (2.15). Relaxation time approximation. If Wk,k is assumed to be momentum independent, then we have. ∂f (k) g(k) =− ,. ∂t coll τ. (2.16). where g(k) ≡ f (k) − f0 (k) is the nonequilibrium part of the distribution and τ the relaxation time. Eq. (2.16) describes the tendency of the nonequilibrium distribution f (k) to evolve back to its equilibrium f0 (k) in a time scale τ .. 15.

(29) CHAPTER 2. LANDAUER’S RESIDUAL RESISTIVITY DIPOLE AROUND A RING-SHAPED MICROSTRUCTURE. 2.2.2. The linear response to the uniform electric field. In a spatially homogeneous and time-independent system, according to Eq. (2.13) and Eq. (2.16), the Boltzmann equation reads, to the lowest order in the electric field E, or for weak E,. −eE ·. 1 ∂f0 g =− , ∂k τ. where e > 0. The equilibrium distribution f0 takes on the Fermi-Dirac distribution,. f0 (k) =. where εk =. 1 e(εk −μ)/kB T. 2 k2 2m∗. +1. ,. (2.17). is the electron energy, μ the chemical potential of the system, kB the. Boltzmann constant, and T the temperature. Therefore, the nonequilibrium distribution is,. g(k) =. eτ ∂f0 E · k, m∗ ∂ε. (2.18). which describes a edge-broadened, shifted Fermi sphere under the external applied electric filed E.. 2.3. Landauer’s residual resistivity dipole. 2.3.1. Charge accumulation due to nonequilibrium incident distribution. The total nonequilibrium distribution in a uniform applied electric is. f (k) = f0 (k) + g(k),. 16.

(30) CHAPTER 2. LANDAUER’S RESIDUAL RESISTIVITY DIPOLE AROUND A RING-SHAPED MICROSTRUCTURE. Figure 2.1: Qualitative sketch of g(k) in the presence of a driving field E. The shifted Fermi circle is towards negative x-axis because negatively charged electrons are driven opposite to the direction of E.. where f0 (k) is the equilibrium distribution. At very low temperature, f0 can be approximated by a Heaviside step function,. f0 (k) = θ(μ − εk ),. (2.19). and thus the nonequilibrium solution to the Boltzmann equation, according to Eq. (2.18), is. g(k) = −. eτ δ(εk − μ)E · k, m∗. (2.20). ˆ is the applied electric field. where E = E0 x ˆ and the direction of the position ρ ˆ at Taking into account the incident direction k which the scattering wave function is referred, we have. Ψk (ρ) =. +∞ . il Rl (ρ)eil(φρ −φk ) ,. (2.21). l=−∞. 17.

(31) CHAPTER 2. LANDAUER’S RESIDUAL RESISTIVITY DIPOLE AROUND A RING-SHAPED MICROSTRUCTURE where Rl (ρ) is assumed to be determined from the radial Schrödinger equation Eq. (2.5). The total charge density distribution can therefore be evaluated by summing over the scattering wave functions of all these electrons 1 n(ρ) = 2 × 2 4π. dk f (k) |Ψk (ρ)|2 = neq (ρ) + δn(ρ),. (2.22). where 1 neq (ρ) = 2 2π. dk f0 (k) |Ψk (ρ)|2. is the contribution due to equilibrium distribution, and 1 δn(ρ) = 2 2π. dk g(k) |Ψk (ρ)|2. is due to the nonequilibrium part of the distribution, which is also linear in E. The multiplying factor 2 in Eq. (2.22) is due to spin degeneracy. Integrating over k gives us a dipole-like distribution with the dipole aligned parallel to E, 2eτ E0 kμ ∗ Rl (ρ)Rl+1 (ρ), cos φρ Im π l=0 ∞. δn(ρ) = −. where kμ =. 2.3.2. √. (2.23). 2m∗ μ/. The detailed derivation of Eq. (2.23) is presented in Appendix A.. Screening effect and potential induced by charge accumulation. The pile up of charges in the previous section invite screening by the electrons in the system which then results in a local drop in the electric potential. Although the local pile up of charges is expressed completely by Eq. (2.22), the only contribution to the potential ’drop’ between electrodes is from the dipole-like distribution δn(ρ). This is because the neq , being cylindrical symmetry, does not contribute to the 18.

(32) CHAPTER 2. LANDAUER’S RESIDUAL RESISTIVITY DIPOLE AROUND A RING-SHAPED MICROSTRUCTURE potential drop. Only the δn(q) term contributes to the potential drop. This potential drop is identified to be a dipole-like form, leading Landauer to the RRD picture. In the following, we use the well-known method of Thomas-Fermi (T-F) screening [29] to obtain the potential induced by this local pile up of charges.. 2.3.3. Thomas-Fermi screening in two dimensions. It is found to be more convenient to discuss the T-F screening in 2D system in the integral form than in the differential form, given by the Laplace equation. Let σ(ρ) be the total charge distribution, including the external charge σ ext (ρ) and the screening charge σ ind (ρ). The total potential φ(ρ) is given by 1 φ(ρ) = 4π0. . σ(ρ ) dρ |ρ − ρ |. (2.24). By performing a Fourier transform on Eq. (2.24), and incorporating the convolution theorem, we obtain. φ(q) =. 1 1 σ(q) , 20 q. (2.25). where φ(q), σ(q) stand for the Fourier transforms of φ(ρ), σ(ρ), respectively, at wave vector q. The equilibrium electron number density is given by 1 n0 = 2 × 2 4π. . 1 dkf0 (k) = 2 2π. dk. 1 e(ε−μ)/kB T. +1. ,. and the factor 2 is due to the spin degeneracy. This density is neutralized by a corresponding positively charged background. In the presence of φ(ρ), the electron density becomes 1 n(ρ) = 2 × (2π)2. dk. 1 e(ε−eφ(ρ)−μ)/kB T. +1. 19. ..

(33) CHAPTER 2. LANDAUER’S RESIDUAL RESISTIVITY DIPOLE AROUND A RING-SHAPED MICROSTRUCTURE From the assumption of the neutrality, the positive charge number density is n0 , and the induced charge density is σ(ρ) = −en(ρ), is written as σ ind (ρ) = −en(ρ) + en0 = −e [n0 (μ + eφ(ρ)) − n0 (μ)] .. Assume that φ(ρ) is weak enough and can be regarded as a perturbation, then. σ ind (ρ) ≈ −e2. ∂n0 ∂n0 φ(ρ) = e2 φ(ρ) . ∂μ ∂ε. (2.26). The total potential can be written as. φ(ρ) = φext (ρ) + φind (ρ). (2.27). which contains both the external and induced parts of potential. From Eq. (2.25), the Fourier transform of Eq. (2.27) in q space is. φ(q) =.

(34) 1 ext σ (q) + σ ind (q) . 2ε0 q. (2.28). By substituting Eq. (2.26) for σ ind in Eq. (2.28), one can relate the external charge or number density to the total electric potential. φ(q) =. with k0 ≡. 1/2ε0 q ext −e σ (q) = next (q), 1 + k0 /q 2ε0 (q + k0 ). e2 ∂n0 2ε0 ∂μ. (2.29). the reciprocal of screening length. In the spatial asymptotic regime of. the external charge location, the screening is dominated by q = 0 behavior. While q = 0, we obtain . ∂n0 φ(ρ) = − e ∂μ. −1 next (ρ) .. (2.30). 20.

(35) CHAPTER 2. LANDAUER’S RESIDUAL RESISTIVITY DIPOLE AROUND A RING-SHAPED MICROSTRUCTURE In 2D system, the density of states is ∂n0 m∗ . = ∂μ π2 Therefore, the final induced potential by the charge accumulation is given by. φ(ρ) = −. 2.3.4. π2 ext n (ρ) . em∗. (2.31). Definition of the strength of the residual resistivity dipole. From Eq. (2.23) and Eq. (2.30), the RRD potential is given by 2E0 kμ τ ∗ δφ(ρ) = cos φ Im Rl (ρ)Rl+1 (ρ). ρ m∗ l=0 ∞. (2.32). Therefore, we can identify the RRD from Eq. (2.32). From Eq. (2.6) we can see that Bessel function is proportional to ρ−1/2 in the asymptotic region, and therefore the potential in Eq. (2.32) is proportional to ρ−1 . Hence Eq. (2.32) indicates a 2D electric dipole potential. After averaging away the Friedel oscillation and in the asymptotic region,. δφ(ρ) ∼ pc. cos φρ , ρ˜. (2.33). where pc corresponds to the RRD strength written as pc = −. E0 τ kμ σtr . 2πm∗. (2.34). Here ρ˜ = k ∗ ρ is a dimensionless ρ, kμ is the wave number corresponding to μ, and ∞ 4 2 σtr = sin (δl − δl+1 ), kμ l=0. (2.35). 21.

(36) CHAPTER 2. LANDAUER’S RESIDUAL RESISTIVITY DIPOLE AROUND A RING-SHAPED MICROSTRUCTURE is exactly the transport cross section of the scattering potential, being defined as σtr ≡. π. −π. dφ (1 − cos φ) D(φ).. (2.36). Eq. (2.36) describes the extent in which the electron population is deflected by the potential and plays the role of reflectance in 1d system. The forward scattering is strong, or reflection is small, when σtr is small. Meanwhile, the larger σtr is, the more the particles are scattered into every directions.. 2.4. Resonance generation of RRD around a ring-shaped structure. Units In the following expressions, all the physical quantities are dimensionless in units ac√ cording to a typical carrier concentration ne . The wave number is in units of k ∗ = 2πne , where ne = 7.4 × 1011 cm−2 is the typical carrier concentration; The system chemical potential is in units of E ∗ = 2 k ∗2 /2m∗ , and length is in 1/k ∗ .. 22.

(37) CHAPTER 2. LANDAUER’S RESIDUAL RESISTIVITY DIPOLE AROUND A RING-SHAPED MICROSTRUCTURE Ring-shaped step potential We consider a ring-shaped square potential barrier, described by step functions,. V (ρ) = V0 [θ(ρ − a) − θ(ρ − b)],. (2.37). where a, b and V0 are inner, outer radii and the potential energy height, respectively. No in-plane potential gradient SOI is included here. With this simple model of a ring-shaped potential Eq. (2.37), the radial wave function can be exactly solved, for ε > V0 , ⎧ ⎪ ⎪ ρ≤a Cl Jl (kρ), ⎪ ⎪ ⎨ Rl (ρ) = Al [Jl (κρ) − Bl Yl (κρ)] , a<ρ≤b , ⎪ ⎪ ⎪ ⎪ ⎩ eiδl [cosδl Jl (kρ) − sinδl Yl (kρ)] , ρ > b with κ =. . (2.38). 2m∗ |ε − V0 |/. For ε < V0 , the Bessel functions Jl (κρ), Yl (κρ) in the region. a < ρ ≤ b ought to be substituted by the modified Bessel functions, Il (κρ), Kl (κρ), respectively. The undetermined coefficient Bl and δl can be obtained via matching the boundary condition of the radial derivatives ρ (d ln Rl /dρ) at the inner and outer edges of the scattering potential. At ρ = a,. Bl =. κaJl (κa) − γl Jl (κa) , κaYl (κa) − γl Yl (κa). where γl ≡ kaJl (ka)/J(ka); at ρ = b, we obtain −1. δl = tan. kbJl (kb) − βl Jl (kb) , kbYl (kb) − βl Yl (kb). (2.39). where βl ≡ κb [Jl (κb) − Bl Yl (κb)]/ [Jl (κb) − Bl Yl (κb)] . One can obtain the amplitudes of partial waves Al and Cl by simply matching the function values Rl at each boundary. At ρ = b,. Al =. eiδl [cos δl Jl (kb) − sin δl Yl (kb)] ; Jl (κb) − Bl Yl (κb) 23.

(38) CHAPTER 2. LANDAUER’S RESIDUAL RESISTIVITY DIPOLE AROUND A RING-SHAPED MICROSTRUCTURE at ρ = a,. Cl =. Al [Jl (κa) − Bl Yl (κa)] ; Jl (ka). The radial function Rl (ρ) is completely solved. In the following we review conventional resonance phenomena involving atomic scattering, and then discuss the resonant behavior of the strength of RRD. We now discuss properties of phase shift through the expression Eq. (2.39). First of all, δ is well defined within an interval of π. The partial wave analysis can also be of help to explain resonance phenomena of the total scattering cross section. From the optical theorem, or directly integrating the differential cross section D(φ) with respect to φ, the total cross section is given by. σtot. +∞ 1 = sin2 δl . k l=−∞. (2.40). For a ring-shaped potential barrier, the dependence of σtot on the Fermi energy exhibits resonant peak structures. At a certain resonance energy, there shows an enhancement peak of σtot . It can be shown that the resonant peak is due solely to on phase shift term in Eq. (2.40). The resonance occurs only when the phase shift of the dominant partial wave crosses nπ/2 (n is odd integer) with positive dδl /dε [30]. Furthermore, the resonant energy is found to correspond to the bound state energy set up within a cylindrical disc of radius a, the inner radius. Fig. 2.2 shows, in the upper diagram, energy levels of a infinite circular disc well with radius a. The energy levels are labeled by (n, l) where n is the radial quantum number and l the azimuthal quantum number. Levels of the same l are indicated by the same color. The lower diagram of Eq. (2.2) shows the μ dependence of the phase shift δl . The resonant structures in δl are connected by arrows, and resonances of the same l have the same color. It can be seen that, even the energy levels in the upper diagram do not align exactly with the resonant energies in the lower diagram, they do correspond well in the color and all shows red shift 24.

(39) CHAPTER 2. LANDAUER’S RESIDUAL RESISTIVITY DIPOLE AROUND A RING-SHAPED MICROSTRUCTURE Energy spectra of a circular infinite well, radius = 6. 0. 0.5. 1. 1.5. 2. 2.5. 3. l=0 l=1 l=2 l=3 l=4 l=5. l. δ /π. 0.5. 0. phase shifts for a shell potential V = 1, a = 6, b = 9 0. −0.5 0. 0.5. 1. 1.5. μ / E*. 2. 2.5. 3. Figure 2.2: Energy spectra of an infinite circular well and phase shifts versus the chemical potential for a ring-shaped step potential barrier with parameters: V0 = 1 the potential height, a = 6 the inner radius, and b = 9 the outer radius.. in the resonant energies. Thus we have established the physical origin of the resonances, and we can label them with a set of resonance numbers (n, l), associated with the quantum numbers in a disk-shaped infinite well. The physical origin of the resonances looks like to be that when the resonance occurs, the incident wavelength of certain electron partial waves match some characteristic resonant lengths that fits the geometry of the ring. As for the RRD, Eq. (2.3) shows that the RRD strength exhibits dip structures at resonant energies. For the purpose of clarifying the physical picture, we choose a ring-shaped structure of a smaller radius with parameters: inner, outer radii a = 6, b = 9, respectively. The potential height V0 = 1. All of parameters are in the chosen units mentioned previously. In the Fig. 2.3, we plot the quantity kσtr versus the chemical potential of the. 25.

(40) CHAPTER 2. LANDAUER’S RESIDUAL RESISTIVITY DIPOLE AROUND A RING-SHAPED MICROSTRUCTURE v0 = 1 , a = 6 , b = 9. l. δ /π. 0.5 l=0 l=1 l=2 l=3 l=4 l=5. 0. −0.5 0. 0.5. 1. 1.5. 2. 2.5. 2. 2.5. 25. kσtr. 20 (1,4). 15 (2,0) (1,3) (2,1). 10. (1,2). 5 (1,0) 0 0. (1,5) (2,2). (1,1) 0.5. 1. μ / E*. 1.5. Figure 2.3: kσtr versus the chemical potential with parameters of a ring-shaped step potential barrier: the potential height V0 = 1, the inner radius a = 6, and the outer radius b = 9.. system because the dimensionless quantity kσtr can represent the charge dipole strength. The dip structures at resonances imply that when resonance occurs, the pile up of charges is smaller. This can be explained by the transport differential cross section σtr and the reminiscent of the resonance behavior in one dimensional scattering on a double barrier structure. When σtr is small, according to the definition of it Eq. (2.36), the forward scattering is strong, that is, the particles encounters little impediment so that few particles are deflected into any other directions but the forward one. The small impediment also implies a small pile up of charges, small potential drop, and most importantly, low resistance around the ring-shaped structure. √ In Fig. 2.3, the overall trend of μ in the RRD strength near the low μ regime, i.e.. 26.

(41) CHAPTER 2. LANDAUER’S RESIDUAL RESISTIVITY DIPOLE AROUND A RING-SHAPED MICROSTRUCTURE μ ≤ V0 , is because the nonequilibrium part of distribution g(k) is linear in k, and thus the strength is a square root in energy, except at resonance structures.. 2.5. Brief summary. Thus far in this chapter, we reviewed the theoretical formalism for Landuaer’s RRD and analyzed its resonance around a repulsive, ring-shaped step potential within the ballistic range. The resonance dips structures in the kσtr characterizing RRD strength which has an analog of 1D resonant tunneling through a double barrier. The position of the resonant dips below for electron energy lower than the potential barrier height is found to correspond with the quantized energy levels, labeled by quantum numbers (n, l), for a circular disk of radius a.. 27.

(42) Chapter 3 Residual resistivity dipole and spin dipole in the presence of spin-orbit interaction arising from in-plain potential gradient of a microstructure The main focus in this chapter is to explore the significance of the spin-orbit interaction arising from the in-plane potential gradient. Towards this end, we consider a ring-shaped potential profile as an explicit example. The SOI effect from the in-plane potential gradient to the Landauer RRD and the spin accumulation in driving electric field will be studied. We first present the spin dependent quantum scattering calculation and then make use of the symmetry properties in the phase shifts to arrive at an analytical forms of the charge RRD and the spin dipole that shows explicitly the dipole features as well as their orientations.. 28.

(43) CHAPTER 3. RESIDUAL RESISTIVITY DIPOLE AND SPIN DIPOLE IN THE PRESENCE OF SPIN-ORBIT INTERACTION ARISING FROM IN-PLAIN POTENTIAL GRADIENT OF A MICROSTRUCTURE. 3.1. Spin dependent asymmetric scattering in the presence of SOI. The Hamiltonian of the system, together with a spin-orbit potential arising from the in-plane gradient of the scattering potential in 2D, is written as. H=. p2 λ 1 dV (ρ) + V (ρ) + σz Lz , ∗ 2m ρ dρ. (3.1). whereV (ρ) = V (ρ) is the cylindrical symmetric potential from the ring-shaped microstructure. The third term of H causes spin dependent scattering because of the presence of the Pauli matrix σz but it does not cause spin flipping if we choose our basis spin states in the zˆ direction. Taking on a step-like profile, the ring-shaped potential V (ρ) = V0 [θ(ρ − a) − θ(ρ − b)] gives rise to the SOI term. HSO =. λ1 [δ(ρ − a) − δ(ρ − b)] σz Lz . ρ. (3.2). ˆ and in The total wave function for an electron incident along arbitrary direction k spin state χσ is written as Ψkσ (ρ) = ψkσ (ρ)χσ , where ψkσ (ρ). =. +∞ . il Rlσ (ρ)eil(φρ −φk ) .. (3.3). l=−∞. Here σz χσ = σχσ . Substituting Eq. (3.3) into the Schrödinger equation, the radial differential equation reads 1 d dRlσ (ρ) l2 2m∗ 2 ρ + k − 2 − 2 [V (ρ) + lσHSO (ρ)] Rlσ (ρ) = 0. ρ dρ dρ ρ . (3.4). The radial function Rlσ (ρ) as well as its coefficients in the three radial regions are all spin. 29.

(44) CHAPTER 3. RESIDUAL RESISTIVITY DIPOLE AND SPIN DIPOLE IN THE PRESENCE OF SPIN-ORBIT INTERACTION ARISING FROM IN-PLAIN POTENTIAL GRADIENT OF A MICROSTRUCTURE dependent, that is, depend on the index σ, and are given by ⎧ ⎪ ⎪ Clσ Jl (kρ), ρ ≤ a, ⎪ ⎪ ⎨ Rlσ (ρ) = Aσl [Jl (κρ) − Blσ Yl (κρ)] , a < ρ ≤ b, ⎪ ⎪ ⎪ ⎪ ⎩ eiδlσ [cosδ σ Jl (kρ) − sinδ σ Yl (kρ)] , ρ > b . l l For incident energy ε, we have ε =. 2 k2 , 2m∗. (3.5). 2 2. and in the barrier region, we have ε = V0 + 2mk∗ .. Since HSO is nonzero only at the inner and the outer radii, namely, ρ = a, b, respectively, the boundary condition that bring forth the spin degeneracy is given by. ρ dRlσ = Rlσ dρ ρ=a+. ρ dRlσ = Rσ dρ + l. ρ=b. ρ dRlσ 2m∗ V0 λlσ + ; Rlσ dρ ρ=a− 2. ρ dRlσ 2m∗ V0 λlσ − . Rσ dρ − 2 l. (3.6). ρ=b. The coefficient Blσ is given by, Blσ =. κaJl (κa) − γlσ Jl (κa) , κaYl (κa) − γlσ Yl (κa). where γlσ =. kaJl (ka) 2m∗ V0 λlσ . + Jl (ka) 2. The phase shift is obtained to be δlσ. −1. = tan. kbJl (kb) − βlσ Jl (kb) , kbYl (kb) − βlσ Yl (kb). where βlσ =. κb [Jl (κb) − Blσ Yl (κb)] 2m∗ V0 λlσ . − Jl (κb) − Blσ Yl (κb) 2. Finally, by imposing the condition that wave function is continuous, we obtain the coef-. 30.

(45) CHAPTER 3. RESIDUAL RESISTIVITY DIPOLE AND SPIN DIPOLE IN THE PRESENCE OF SPIN-ORBIT INTERACTION ARISING FROM IN-PLAIN POTENTIAL GRADIENT OF A MICROSTRUCTURE ficients Aσl and Clσ , given by σ. Aσl =. eiδl [cos δlσ Jl (kb) − sin δlσ Yl (kb)] ; Jl (κb) − Blσ Yl (κb). Clσ =. Aσl [Jl (κa) − Blσ Yl (κa)] . Jl (ka). and. We note in passing that in the low energy situation, when ε < V0 , Bessel functions Jl (κρ), Yl (κρ) are replaced by modified Bessel functions Il (κρ), Kl (κρ), respectively. The effect of HSO is to lift the spin degeneracy such that the differential cross section becomes Dσ (φ) = |fσ (φ)|2 , where the spin-dependent scattering amplitude is given by. fσ (φ) ≡. 3.1.1. +∞ 2i iδlσ e sin δlσ eilφ . πk l=−∞. (3.7). Two useful relations. Two important symmetry properties int he phase shift and the radial wave function are to be presented in this section. The first symmetry can be seen from Eq. (3.6), when the spin dependent term, the second term on the right-hand side, involves a product lσ of the orbital and the spin quantum number. Thus we must have the symmetry −σ σ δ−l = δl−σ , or δ−l = δlσ .. (3.8). Another symmetry relation is a direct consequence of the δlσ symmetry. The radial wave function is related to δlσ via Bessel functions. For instance, outside the range of the potential V (ρ), the radial function is given by σ. Rlσ (ρ) = eiδl [cosδlσ Jl (kρ) − sinδlσ Yl (kρ)] .. 31.

(46) CHAPTER 3. RESIDUAL RESISTIVITY DIPOLE AND SPIN DIPOLE IN THE PRESENCE OF SPIN-ORBIT INTERACTION ARISING FROM IN-PLAIN POTENTIAL GRADIENT OF A MICROSTRUCTURE The Bessel functions has a symmetry relation for l Z−l (x) = (−1)l Zl (x),. where Zl (x) stands for any one of the two kinds of Bessel functions. Thus imposing both the symmetry properties in the phase shift and the Bessel function to Rlσ (ρ), we have −σ σ R−l (ρ) = (−1)l Rl−σ (ρ), or R−l (ρ) = (−1)l Rlσ (ρ).. (3.9). This two symmetry relations Eq. (3.8) and Eq. (3.9) are useful for putting our expression, in the next section, for the RRD and spin dipole into elegant and physically most transparent forms.. 3.2. Residual resistivity dipole in the presence of local structure SOI. Following similar procedure for obtaining the charge accumulation in Eq. (2.18), except here the spin degeneracy is lifted, we write down the total charge accumulation n(ρ) as follows 1 n(ρ) = 4π 2. dk f (k). . Ψ†kσ (ρ)Ψkσ (ρ). σ. = neq (ρ) + δn(ρ), where eτ E0 kμ σ∗ δn(ρ) = − Rlσ (ρ)Rl+1 (ρ) . cos φρ Im π σ l=0 ∞. 32. (3.10).

(47) CHAPTER 3. RESIDUAL RESISTIVITY DIPOLE AND SPIN DIPOLE IN THE PRESENCE OF SPIN-ORBIT INTERACTION ARISING FROM IN-PLAIN POTENTIAL GRADIENT OF A MICROSTRUCTURE To obtain Eq. (3.10), we have to impose the symmetry relation given by Eq. (3.8) and Eq. (3.9). With the T-F screening incorporated, the total potential becomes E0 kμ τ σ∗ cos φ Im Rlσ (ρ)Rl+1 (ρ), ρ m∗ σ l=0 ∞. δφ(ρ) =. (3.11). which is a dipole field form. The strength pc of the charge dipole, again, is obtained from the asymptotic region,. pc = −. E0 τ kμ σtr , 2πm∗. (3.12). where σtr is the transport cross section with removed spin degeneracy 1 σtr ≡ 2. . π −π. dφ (1 − cos φ). . Dσ (φ). σ. ∞ 2 2 σ σ . sin δl − δl+1 = kμ σ l=0. 3.3. Spin dipole due to spin-independent nonequilibrium incident distribution. In addition to Landauer’s RRD, the presence of asymmetric Mott scattering [4], caused by the SOI in the ring-shaped potential, should give rise to interesting spin accumulation. The results of resonant scattering for the incident plane wave are in Appendix E. Following similar procedure for the RRD calculation, the spin density, in units of /2, is given by 1 Sz (ρ) = 4π 2. dk f (k). . σ Ψ†kσ (ρ)Ψkσ (ρ). σ. = Szeq (ρ) + δSz (ρ),. 33.

(48) CHAPTER 3. RESIDUAL RESISTIVITY DIPOLE AND SPIN DIPOLE IN THE PRESENCE OF SPIN-ORBIT INTERACTION ARISING FROM IN-PLAIN POTENTIAL GRADIENT OF A MICROSTRUCTURE where. Szeq (ρ). 1 = 2 4π. dkf0 (k). . σ |ψkσ (ρ)|2 .. σ. That the equilibrium spin accumulation must be zero, Szeq = 0, identically can be shown by direct calculation. Simply that the equilibrium spin distribution is isotropic for each spin causes exact cancellation between opposite spin. The total spin accumulation is then determined solely by the nonequilibrium part 1 Sz (ρ) = δSz (ρ) = 2 4π. dk g(k). . σ |Ψkσ (ρ)|2 .. σ. Again, making use of the symmetry properties in Eq. (3.8) and Eq. (3.9), we get. Sz (ρ) = nE sin φρ Re. ∞ σ∗ σ Rlσ (ρ)Rl+1 (ρ), σ. (3.13). l=0. by the use of relation Eq. (3.9). The factor nE =. eE0 τ kμ π. has the dimension of density,. linear in E0 , and with an energy dependence via kμ . The momentum relaxation time τ is assumed to be a constant, independent of energy. The angular dependence, sin φ, of the spin density Sz (ρ) indicates a dipole-like distribution, aligned perpendicular to the driving field E. This expression of spin accumulation holds within a radial distance shorter than the mean free path. The screening of spin is assumed to be negligible because of the spin-spin interaction between electrons. As in the charge dipole case, we can define a spin dipole strength from the asymptotic form of the spin accumulation,. Sz (ρ) ∼ ps. sin φρ , ρ˜. (3.14). where ρ˜ ≡ k ∗ ρ is the dimensionless radial coordinate, and ps is the spin dipole strength.. 34.

(49) CHAPTER 3. RESIDUAL RESISTIVITY DIPOLE AND SPIN DIPOLE IN THE PRESENCE OF SPIN-ORBIT INTERACTION ARISING FROM IN-PLAIN POTENTIAL GRADIENT OF A MICROSTRUCTURE The expression for ps is ps = −. n ˜E kμ σ⊥ , 2π. (3.15). where 1 σ⊥ ≡ 2. . +π. dφ sin φ −π. . σDσ (φ). (3.16). σ. is essentially the transverse moment of the net spin differential cross section, which we may call the transverse transport cross section. In terms of phase shifts, ∞ 1 σ σ⊥ = σ sin[2(δlσ − δl+1 )]. kμ σ l=0. The constant factor n ˜E = state. m∗ π2. m∗ eE0 l0∗ , π2. (3.17). where l0∗ = k ∗ τ /m∗ , is the product of the 2d density of. and the work done by the electric field within the range of mean free path eE0 l0∗ .. The minus sign appearing in Eq. (3.15) indicates that the direction of electron motion is opposite to E. As the electric field is driving in positive x-direction, the electron is driven in the negative x-direction and therefore the orientation of the spin dipole is opposite to that of kσ⊥ .. 3.3.1. Second-order correction to the spin dipole strength at resonance. The radial dependence of Sz for typical physical parameters is shown in Fig. 3.1. The exact numerical result is given by the solid curve, the asymptotic form, Eq. (3.14), is given by the dashed curve, whereas the dot-dashed curve denotes results that goes beyond the usual asymptotic from, and has included a correction term. It is clearly shown that the. 35.

(50) CHAPTER 3. RESIDUAL RESISTIVITY DIPOLE AND SPIN DIPOLE IN THE PRESENCE OF SPIN-ORBIT INTERACTION ARISING FROM IN-PLAIN POTENTIAL GRADIENT OF A MICROSTRUCTURE. Figure 3.1: Radial variation of Sz (ρ) along φρ = π/2 at the two resonant energies (a) μ = 0.3312, and (b) μ = 0.3328 in the resonance pair labeled (1,5). The green dashed curve is ps /k ∗ ρ; the red dot-dashed curve includes the 2nd-order correction ℘s . The geometry of the ring: the potential height V0 = 0.75, the inner radius a = 14, and the outer radius b = 20.5.. corrected curve fits better. Thus sin φρ ℘s Sz ∼ ps + , ρ˜ ρ˜. (3.18). where. ℘s =. n ˜E ∗ k σ⊥ , 4π. (3.19). σ⊥ =.

(51) 1 σ σ (2l + 1) cos 2(δlσ − δl+1 ) . k σ l=0. (3.20). with ∞. 36.

(52) CHAPTER 3. RESIDUAL RESISTIVITY DIPOLE AND SPIN DIPOLE IN THE PRESENCE OF SPIN-ORBIT INTERACTION ARISING FROM IN-PLAIN POTENTIAL GRADIENT OF A MICROSTRUCTURE It is still reasonable to use ps for the representation of the spin dipole strength.. 3.4. Resonance of charge and spin dipole in the presence of the SOI from the microstructure. In this section we discuss in detail the resonant features in the presence of SOI. First of all, the SOI lifts the spin degeneracy at the resonance. That is, to every resonance in the absence of SOI, we will have two resonances when SOI is introduced. This is shown in Fig. 3.2, when the phase shifts are plotted against electron energy μ. In the upper part of Fig. 3.2, the resonances denoted by the profile where the phase shift jumps from π/2 to −π/2, are for zero SOI (λ=0). In the lower part of Fig. 3.2, when λ = 0, the number of curves are doubled, representing the spin-split, and subsequently the number of resonant energies are doubled. From the insight we obtained in Chapter 2, each resonance associated with quantum numbers (n, l) occurs because the incident wavelength of the l’th and the −l’th partial waves match certain characteristic resonant length (labeled by n) fitting the geometry of the ring. In the absence of the SOI, there is spin degeneracy in the resonant lengths of the l and the −l partial waves with the same radial mode n. When the SOI is introduced, the spin degeneracy is lifted and hence the resonant energy in the phase shift is doubled. We can see from Fig. 3.2 that the energies at which the phase shifts jump from π/2 to −π/2 are different for the l’th and the −l’th partial waves.. 3.4.1. Resonant asymmetric skew scattering. Asymmetric scattering near a resonant energy shows interesting characteristics. As the asymptotes of the scattered wave is related to the differential scattering cross section, it becomes spin dependent here, and the resonant energies are spin-split. If we want to define a quantity representing the extent of lateral deflection in the scattering wave, the. 37.

(53) CHAPTER 3. RESIDUAL RESISTIVITY DIPOLE AND SPIN DIPOLE IN THE PRESENCE OF SPIN-ORBIT INTERACTION ARISING FROM IN-PLAIN POTENTIAL GRADIENT OF A MICROSTRUCTURE λ=0. 0.5. δ /π. l=0 l=1 l=2 l=3. l. 0. −0.5. μ λ<0. δσ /π. 0.5. l. 0. −0.5. μ Figure 3.2: Qualitative sketch of the phase shift in the presence SOI: for the up-spin partial waves, the solid curves in the lower plot is for positive l, and the dashed curves are for negative l; or equivalently, for partial waves with positive l, the solid curves are for up-spin waves, and the dashed curves are for down-spin waves.. following moment is a good choice η σ⊥. ≡. π. −π. dφ sin φDη (φ). (3.21). where η = ± stands for the spin state, along zˆ, of the incident electron. Expressed in terms of phase shifts, we have η σ⊥. 1 η η η sin[2(δlη − δl+1 = )] − sin[2(δ−l − δ−(l+1) )] . k l=0 ∞. 38. (3.22).

(54) CHAPTER 3. RESIDUAL RESISTIVITY DIPOLE AND SPIN DIPOLE IN THE PRESENCE OF SPIN-ORBIT INTERACTION ARISING FROM IN-PLAIN POTENTIAL GRADIENT OF A MICROSTRUCTURE If this integral is positive, the incoming electron will be scattered to its left, or positive ˆ is the incident direction. One can see from Eq. (3.22), that y-direction, assuming x + − σ⊥ = −σ⊥ ,. (3.23). which implies that incident electrons with opposite spin-polarization along zˆ are deflected in opposite directions laterally. If there were no SOI, the spin degeneracy would be η η recovered, and σ⊥ = 0 for both spin states, as it should be. The expression for σ⊥ ,. according to Eq. (3.22), contains the difference of two terms, each is a summation over l, η but with l ≥ 0 and l ≤ 0, separately. If we have an amplification of kσ⊥ at the energy η η , we should have similar amplification, but of the opposite sign, at the energy En,−l , En,+l + , as according to Eq. (3.22). We plot individually the two terms that constitutes kσ⊥. shown in Fig. 3.3. We see from Fig. 3.3 that the positive-l curve, denoted by the red curve, has peak structures; the negative-l curve, as denoted by the blue curve, has peak structures as well but is shifted in energy relative to the red curve. The black curve + denotes kσ⊥ , which is the difference of the blue and the red curves.. 3.4.2. Resonant RRD in the presence of the SOI. The resonant dip structures of charge dipole strength are spin-split, as shown in Fig. 3.4. The resonant energy ε0nl is split into εσn,±l so that a dip structure in the RRD strength, or kσtr , becomes double dips. This is clearly seen in the (1,2), (1,3) and (1,4) structures in. the upper curve of Fig. 3.4. The split in energy εσn,+l − εσn,−l is larger for larger l. This is reasonable because the spin dependent term in Eq. (3.6) in creases with l. For the sake of simplifying the notation, we denote each spin-split resonant pair by (n, l), the resonant number for the mother resonant dip in kσtr .. 39.

(55) CHAPTER 3. RESIDUAL RESISTIVITY DIPOLE AND SPIN DIPOLE IN THE PRESENCE OF SPIN-ORBIT INTERACTION ARISING FROM IN-PLAIN POTENTIAL GRADIENT OF A MICROSTRUCTURE v0 = 1 , r0 = 6 , b = 9 4. 2. kσ+⊥. 0. −2. −4. −6. −8. 0.5. 1. 1.5. μ / E*. 2. 2.5. 3. + Figure 3.3: σ⊥ versus system energy: the red curve denotes the summation over posi∞ + + tive l, sin[2(δ l − δl+1 )]; the blue curve stands for the summation over negative l, l=0 ∞ + + + l=0 sin[2(δ−l − δ−(l+1) )]; the total σ⊥ is denoted by the black curve.. 3.4.3. Resonance of spin dipole. In this section, we present the resonant characteristics of the spin dipole. The spin dipole strength is shown to manifest large enhancement and sign reversal at resonance. From the lower curve of Fig. 3.4, the resonant features in the spin dipole strength, or kσ⊥ , is shown to carry a peak-dip structure. To show that this resonant feature is indeed remarkable, we plot, for comparison, the spin dipole strength due to a potential disc with the same outer radius a, but having the inner radius b = 0. Denoted by the red line in the kσ⊥ curve in Fig. 3.4, this line matches very well with the ring-shaped potential result in the non-resonant and low energy regions. From this we also see that quantum resonance has lead to very large enhancement in the spin dipole strength. Furthermore,. 40.