Theoretical study of spin dependent transport in nanoscale spintronic systems

THEORETICAL STUDY OF SPIN DEPENDENT TRANSPORT IN NANOSCALE SPINTRONIC SYSTEMS by Ma Minjie M.Eng., Beijing Jiao Tong University, Beijing, China A Thesis Submitted for the Degree of Doctor of Philosophy Department of Electrical and Computer Engineering National University of Singapore 2010 Acknowledgements Firstly, I’d like to thank for the financial support from Ph.D. Research Scholarship provided by National University of Singapore. It is great that the university offers friendly, convenient and excellent studying environment. Secondly, I am so grateful that I have been supervised by Prof. M. B. A. Jalil, who closely supervised me all the way through the four year study. Without him, I would not be able to finish my Ph.D He is passionate in doing research and patient in guiding students. He sets a good example for me and other students on how to be a researcher. Thanks should also go to my co-supervisors Dr. S.G. Tan and Dr. G. C. Han. They gave me valuable advice whenever I met with problems in my research. They are so helpful and always willing to help. Besides, I’d like to thank other teammates in our group, Guo Jie, Bala, Takashi, Chen Ji, Nyuk Leong, Zhuo Bin, Gabriel ., my friends Guangyu, Sun Nan, Ji Xin, Ho Pin, etc. and those in China. Lastly, I have to say “thank you” to my dear family in China. They gave me endless support in my life and study. i Contents Acknowledgements i Summary v List of Figures viii Publications, Conferences and Awards xv List of Abbreviations Introduction 1.1 Basic concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 3D, 2D, 1D and 0D systems . . . . . . . . . . . . . . . . 1.1.2 Density of states . . . . . . . . . . . . . . . . . . . . . . 1.1.3 Materials . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.3.1 Metal, semiconductor and insulator . . . . . . 1.1.3.2 Magnetic material and non-magnetic materials 1.1.4 Length scales . . . . . . . . . . . . . . . . . . . . . . . . 1.1.5 Basic spin concepts . . . . . . . . . . . . . . . . . . . . . 1.1.5.1 Pauli matrices . . . . . . . . . . . . . . . . . . 1.1.5.2 Spin polarization . . . . . . . . . . . . . . . . . 1.1.5.3 Spin relaxation . . . . . . . . . . . . . . . . . . 1.1.5.4 Zeeman splitting . . . . . . . . . . . . . . . . . 1.1.6 Transport regime . . . . . . . . . . . . . . . . . . . . . . 1.1.7 Magnetoresistive effect . . . . . . . . . . . . . . . . . . . 1.1.7.1 Giant magnetoresistance . . . . . . . . . . . . 1.1.7.2 Tunnel magnetoresistance . . . . . . . . . . . . 1.1.8 Spin-dependent single electron tunneling . . . . . . . . . 1.2 Motivations and objectives . . . . . . . . . . . . . . . . . . . . 1.3 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 7 11 15 15 16 17 18 19 20 21 25 26 28 29 ii CONTENTS Spin dependent transport in nanoscale CIP SVs 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 2.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Spin-flip and spin diffusive effects on GMR . . . . 2.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 32 33 39 42 Coherent spin dependent transport in QD-DTJ Systems 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Single energy level QD . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2.1 Hamiltonian . . . . . . . . . . . . . . . . . . . . . . 3.2.2.2 Tunneling current and tunnel magnetoresistance . . 3.2.2.3 Retarded Green’s function . . . . . . . . . . . . . . 3.2.3 Results and discussion . . . . . . . . . . . . . . . . . . . . . . 3.2.3.1 Spin-flip effects . . . . . . . . . . . . . . . . . . . . . 3.2.3.2 Coupling asymmetry effects . . . . . . . . . . . . . . 3.2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 QD with Zeeman splitting . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Results and discussion . . . . . . . . . . . . . . . . . . . . . . 3.3.3.1 Fully polarized current . . . . . . . . . . . . . . . . 3.3.3.2 Switching the polarization of the current . . . . . . 3.3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Diluted magnetic semiconductor QD system . . . . . . . . . . . . . . 3.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3 Non-collinearity dependence of the spin dependent transport 3.4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 44 46 46 49 50 52 55 57 57 67 70 72 72 74 77 77 80 83 84 84 86 90 93 94 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Incoherent spin dependent transport in a QD based DBMTJ System 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 General Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Transport regimes and master equation . . . . . . . . . . . . . . 4.1.2.1 Sequential tunneling . . . . . . . . . . . . . . . . . . . . 4.1.2.2 Cotunneling . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Collinear system . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1.1 I-V characteristics and TMR . . . . . . . . . . . . . . . 4.3.1.2 Occupancies and spin accumulation in the QD . . . . . 4.3.1.3 Spin-flip effects . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Noncollinear system . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 96 99 100 101 102 104 110 111 112 113 115 120 123 iii CONTENTS Spin dependent transport in a nanoscale SET system 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Tunneling current in sequential tunneling regime 5.2.2 Tunneling current in cotunneling regime . . . . . 5.2.3 Tunnel magnetoresistance . . . . . . . . . . . . . 5.3 Results and discussion . . . . . . . . . . . . . . . . . . . 5.3.1 Spin polarization effect . . . . . . . . . . . . . . 5.3.2 Spin-flip effects . . . . . . . . . . . . . . . . . . . 5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 127 130 130 134 136 136 137 140 143 Conclusion and future work 6.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Suggestions for future work . . . . . . . . . . . . . . . . . . . . 6.2.1 Study of spin transfer torque through DMSQD system . 6.2.2 Spin dependent transport through graphene . . . . . . . 6.2.3 Spin dependent transport through topological insulator . . . . . . . . . . . . . . . . . . . . . . . . . 145 145 150 150 151 152 . . . . . . . . . . . . . . . . . . . . . . . . . . . Bibliography 153 A Current in the DMSQD system 161 B Effective polarization of FM leads 167 iv Summary Spintronic devices make use of the spin properties of electrons besides its charge properties. They are novel devices which potentially have faster operation speeds, low energy consumption and smaller size. One of the widely used spin effects in spintronic devices are magnetoresistive (MR) effect, which includes giant magnetoresistance (GMR) and tunnel magnetoresistance (TMR) effect. One of the most interesting nanoscale spintronic devices is few-electron devices, such as quantum dot (QD) based spintronic devices. The theoretical work behind these applications has been attracting tremendously intensive interest since 1980s. One of the most important aspects of the theoretical study of spintronics is to model the spin dependent transport (SDT) through these devices. SDT naturally occur in spintronic devices where there exists an imbalance between different spin populations. In this thesis, we focus on the SDT through magnetoresistive devices and ferromagnetic single electron transistors (FM-SETs). Firstly, we studied the SDT and GMR effect in a current-in-plane spin-valve (CIP SV). We modeled the SDT through the CIP SV device using the well established Boltz- v SUMMARY mann equation method, systematically incorporating the effect of the spin-flip scattering which might occur in the bulk of each layer or at their interfaces. The combined effects of spin-flip scattering and interfacial diffusive scattering on GMR are investigated as well. Secondly, we analyzed QD based spintronic devices, in which a central QD is attached to two electrodes via tunnel junctions. These devices are referred to as QD based double tunnel junction (QD-DTJ) systems. The coherent and incoherent SDT through the QD-DTJ systems are theoretically studied based on the Keldysh nonequilibrium Green’s function (NEGF) approach and the master equation methods, respectively. For magnetoresistive QD-DTJs where the two electrodes are ferromagnetic (FM), the tunneling current and TMR are characterized with respect to the spin-flip events, the polarizations of the FM electrodes, the noncollinearity between the magnetization of the electrodes, and the coupling asymmetry between the two junctions. For non-magnetoresistive QDDTJs where the two electrodes are non-magnetic, we analyzed the effect of Zeeman splitting in the QD on the tunneling current. The tunneling current is found to be fully spin polarized when the system is under a proper bias voltage, and the polarization of the current can be switched via an applied gate potential to the QD. The use of a diluted magnetic semiconductor (DMS) in a QD-DTJ with FM electrodes is also studied, with a focus on the non-collinearity effect on the tunneling charge current, spin current and TMR. Finally, we investigate the SDT through nano-scale single electron transistors (SETs) with FM source and drain, employing the master equation formalism. The I − V characteristic is investigated in the cotuneling and sequential tunneling regimes. We found that the spin-flip scattering and the polarization of the FM electrodes maybe utilized vi SUMMARY to suppress the leakage current in the cotunneling regime. In conclusion, we have theoretically studied the SDT through CIP-SVs, QD based spintronic systems and SET system. Besides, a bi-polarized spin current generator and a optimized SET are proposed. The SDT models and investigations for those nanoscale spintronic systems are expected to provide basis for the future research in relevant systems. vii List of Figures 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11 1.12 1.13 1.14 Illustration of the Stern-Gerlach experiment. The graph is adopted from Theresa Knott. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spin up (left) and down (right) electrons. . . . . . . . . . . . . . . . . . Systems of different dimensions in real space and momentum space, and their DOS as a function of energy. . . . . . . . . . . . . . . . . . . . . . Fermi-Dirac distribution function. . . . . . . . . . . . . . . . . . . . . . Schematic diagram of the energy band structure of conductors, semiconductors and insulators. Ec , Ev , Eg and EF are the conductance band, valance band, energy gap and Fermi energy, respectively. . . . . . . . . Spin configurations of materials with different magnetic effect (zero magnetic field). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . DOS of spin-up (up arrow) and spin-down (down arrow) electrons in (a) ferromagnetic metal and (b) non-magnetic metal, respectively. EF is Fermi energy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Electrons (green) and holes (gray) in three types of semiconductors. Nonmagnetic semiconductor contains no magnetic ions. In the Paramagnetic DMS, the net spins (orange arrows) are with holes and show paramagnetic property. In the ferromagnetic DMS, the net spins (orange arrows) are showing ferromagnetic property. . . . . . . . . . . . . . . . . . . . . Elastic (blue line) and inelastic (red line) scattering of electrons in k space, where k is the initial momentum, and k is the momentum after scattering. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Illustration of Zeeman splitting effect. . . . . . . . . . . . . . . . . . . . Different transport regimes according to various length scales, where L is the size of the system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . GMR multilayer systems in parallel case (a) and antiparallel case (b), both for current-in-plane (CIP) and current-perpendicular-to-plane (CPP) geometries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Resistance as a function of magnetic field in GMR multilayers, where the arrows stand for the magnetization of FM layer. . . . . . . . . . . . . . A typical CIP SV structure. (Source: Hitatchi GST.) . . . . . . . . . . . 11 12 12 13 19 19 22 23 23 viii LIST OF FIGURES 1.15 Spin dependent transport in terms of DOS in the FM layers of a MTJ system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.16 A few electron box coupled to two electrodes via insulator. Vb is the bias voltage, under which the electrons tunnel through the electron box one by one. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.17 Topics covered in each Chapter of the thesis and their correlations. . . . 2.1 2.2 2.3 2.4 3.1 (a) Fe/Cr/Fe current-in-plane spin-valve (CIP SV) trilayers, with the two magnetization of the ferromagnetic layers (Fe layers) in parallel (top) and antiparallel (bottom) alignments, where the current flows along the in-plane direction of the layers, the arrows in the layers stand for the magnetization, the green lines and red lines show the less and greater scattering of electrons in the layers, respectively. (b) Schematic diagram for the axes employed to theoretically model the spin dependent transport in the CIP SV shown in (a). The current flow and the applied electric field E are along the x axis. The z axis is normal to the trilayer. The y axis is normal to both the x and z axes. Li (i=1 to 4) denotes the different layers of the SV structure. The arrows denote the magnetization directions of the Fe layers. The dotted x axis (at z =0) in the middle of the Cr layer marks the boundary where the reference spin axis is rotated by an angle θ. GMR as a function of the mean free path (MFP) of electrons in the Fe layer for varying (a) q values and (b) r values, where Ns = 6, D↑ = 0.5. The thickness of the Fe (Cr) layers is fixed at dFe = 10 nm (dCr = nm). GMR as a function of r and q values. The electron’s MFP in the FM layer λFe is set at 100 nm, while the other parameters take the same values as in Fig. 2.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (a) / (b) GMR as a function of the thickness (dFe ) / MFP (λFe ) of FM layer , for different Ns and D↑ values. The top (bottom) three curves in both (a) and (b) correspond to the case without (with) SF scattering, i.e., r = q = (r = 0.3, q = 0.2). In (a), λFe = 100 nm, while in (b) dFe =10 nm, the thickness of the nonmagnetic (NM) layer (Cr is nm. . . . . . . (a) Schematic diagram of the mesoscopic structure consisting of a QD sandwiched by two FM leads; (b) the schematic energy diagram for the QD-DTJ system in (a). In (a), the arrows in the leads indicate their magnetization directions, which can either be in parallel (solid) or antiparallel (dashed) alignment, Vb is the bias voltage between the two leads, λ characterizes the strength of the spin-flip in QD (SF-QD) from up-spin to down-spin, tLk↑,↓ characterize the spin-flip strength during tunneling the junction (SF-TJ) between the up-spin state in the left lead and the downspin state in the QD, tLk↑,↑ is the normal tunnel coupling strength between the lead and QD in the absence of spin-flip, and β = tRkσ,σ /tLkσ,σ denotes the coupling asymmetry between the two tunneling junctions. 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A.1, one may have the tunneling current through the junction between the left lead and the 161 CURRENT IN THE DMSQD SYSTEM QD, i.e., IL = ie [H, N ] , = ie [HL + Hd + HtL , N ] (A.2) = ie [HL , N ] + [Hd , N ] + [HtL , N ] in which [HL , N ] =    εLkσ a†Lkσ aLkσ , N  = 0, (A.3) k,σ εdσ a†dσ adσ , N [Hd , N ] = = 0, (A.4) σ 162 CURRENT IN THE DMSQD SYSTEM and [HtL , N ] =    θ θ t∗Lkσ cos a†dσ − σ sin a†d¯σ aLkσ + H.c, N  2  k,σ  θ θ t∗Lkσ cos c†dσ − σsin c†d¯σ cLkσ + H.c, 2  = k,σ θ = cos t∗Lkσ + cos k,σ tLkσ a†Lkσ adσ , a†dσ adσ σtLkσ a†Lkσ ad¯σ , a†dσ adσ k,σ k,σ σt∗Lkσ a†d¯σ aLkσ (δLkσdσ − δd¯σdσ ) k,σ θ θ + a†d¯σ aLkσ , a†dσ adσ k,σ + cos k,σ σt∗Lkσ t∗Lkσ a†dσ aLkσ (δLkσdσ − δdσdσ ) θ −sin = − a†dσ aLkσ , a†dσ adσ k,σ θ θ θ = cos −sin tLkσ a†Lkσ adσ (δdσdσ − δLkσdσ ) k,σ σtLkσ a†Lkσ ad¯σ (δd¯σdσ − δLkσdσ ) k,σ θ cos t∗Lkσ a†dσ aLkσ k,σ ,σ + k,σ θ cos tLkσ a†Lkσ adσ − Wdσ ,Lkσ a†Lkσ adσ = σ k,σ θ −sin −sin a†dσ adσ  k,σ θ σsin t∗Lkσ a†d¯σ aLkσ θ σsin tLkσ a†Lkσ ad¯σ † ∗ Wdσ ,Lkσ adσ aLkσ , − (A.5) k,σ ,σ where σ ∈ σ, σ ¯ , and ∗ Wdσ ,Lkσ   θ ∗ θ      cos tLkσ , ifσ = σ  cos tLkσ , ifσ = σ 2 = , Wdσ ,Lkσ = ,   θ θ    −σsin t∗ , ifσ = σ  −σsin t , ifσ = σ ¯ ¯ Lkσ Lkσ (A.6) 163 CURRENT IN THE DMSQD SYSTEM † † < If we define G< dσ ,Lkσ (t) = −i aLkσ adσ (t) , and GLkσ,dσ (t) = −i adσ aLkσ (t) , Eq. (A.5) becomes   = −i  [HtL , N ] ∗ <  Wdσ ,Lkσ GLkσ,dσ (t) . (A.7) Wdσ ,Lkσ G< dσ ,Lkσ (t) − k,σ ,σ k,σ ,σ One may now substitute Eqs. (A.3), (A.4) and (A.7) to Eq. A.2 to get the tunneling current through the left junction, in terms of the lesser Green’s functions, i.e., IL = e < ∗ Wdσ ,Lkσ G< dσ ,Lkσ (t) − Wdσ ,Lkσ GLkσ,dσ (t) . (A.8) k,σ ,σ We apply Fourier transform G< (t) = IL = +∞ e k,σ ,σ −∞ ∞ dω < iωt −∞ 2π G (ω)e to Eq. (A.8), i.e., dω ∗ < Wdσ ,Lkσ G< dσ ,Lkσ (ω) − Wdσ ,Lkσ GLkσ,dσ (ω) , 2π (A.9) for which one may apply the Dyson’s equations, i.e., G< Lkσ,dσ (ω) = Wdσ ,Lkσ t gLkσ,Lkσ (ω) G< dσ ,dσ < (ω) − gLkσ,αkσ (ω) Gtdσ ,dσ (ω) σ G< dσ ,Lkσ . ∗ Wdσ ,Lkσ (ω) = < gLkσ,Lkσ (ω) Gtdσ ,dσ (ω) − t gLkσ,αkσ (ω) G< dσ ,dσ (ω) σ (A.10) Eq. (A.9) becomes, IL = ∞ e k,σ ,σ,σ −∞ dω ∗ Wdσ ,Lkσ Wdσ 2π ,Lkσ < gLkσ,Lkσ (ω) Gtdσ ,dσ (ω) + Gtdσ ,dσ (ω) t t − G< dσ ,dσ (ω) gLkσ,Lkσ (ω) + gLkσ,Lkσ (ω) . (A.11) 164 CURRENT IN THE DMSQD SYSTEM Using the identity G> + G< = Gt + Gt , one may rewrite Eq. (A.11) as IL = ∞ e −∞ k,σ ,σ,σ dω ∗ Wdσ ,Lkσ Wdσ 2π ,Lkσ < < gLkσ,Lkσ (ω) G> dσ ,dσ (ω) − Gdσ ,dσ (ω) < > + G< dσ ,dσ (ω) gLkσ,Lkσ (ω) − gLkσ,Lkσ (ω) , (A.12) where we can apply the identity G> − G< = Gr − Ga , thus Eq. (A.13) becomes, IL = ∞ e dω ∗ Wdσ ,Lkσ Wdσ 2π −∞ k,σ ,σ,σ ,Lkσ < gLkσ,Lkσ (ω) Grdσ ,dσ (ω) − Gadσ ,dσ (ω) < > + G< dσ ,dσ (ω) gLkσ,Lkσ (ω) − gLkσ,Lkσ (ω) . (A.13) If we assume the two ferromagnetic leads are noninteracting, the lesser and greater < Green’s functions of the leads in Eq. (A.13), have the form of gLkσ,Lkσ (ω) = 2πifLσ δ (ω − Lσ ) > and gLkσ,Lkσ (ω) = −2πi [1 − fLσ ] δ (ω − Lσ ). By substituting those identi- ties into Eq. (A.13), one may have IL = ie ∗ Wdσ ,Lkσ Wdσ ,Lkσ fLσ Grdσ ,dσ ( Lσ ) − Gadσ ,dσ ( Lσ ) k,σ ,σ,σ + G< dσ ,dσ ( We now replace Lσ Lσ ) . (A.14) with , and take the approximation that → d ρLσ ( ) ,where k ρLσ is the density of states for electrons the left lead respectively. Eq. (A.14) becomes IL = ie ∗ d ρLσ ( )Wdσ ,Lkσ ( )Wdσ ,Lkσ ( ) σ ,σ,σ fLσ Grdσ ,dσ ( Lσ ) − Gadσ ,dσ ( Lσ ) + G< dσ ,dσ ( Lσ ) , (A.15) 165 CURRENT IN THE DMSQD SYSTEM ∗ in which we can adopt the low-bias approximation, 2πρLσ ( )Wdσ ,Lkσ ( )Wdσ ,Lkσ ( )= Lσ ΓLσ σ ,σ , where Γσ ,σ is taken to be constant (zero) within (beyond) the energy range close to the lead’s electrochemical potential around which most of the transport occurs, i.e., ∈ [µα − D, µα + D], where D is some constant. [44] Eq. (A.15) can be rewritten as IL = ie d r fLσ ΓLσ σ ,σ Gdσ ,dσ ( Lσ ) a − ΓLσ σ ,σ Gdσ ,dσ ( Lσ ) σ ,σ,σ < + ΓLσ σ ,σ Gdσ ,dσ ( Lσ ) . (A.16) Eq. (A.16) is the final form of the tunneling current in the DMSQD system shown in Fig. 3.16. 166 APPENDIX B Effective polarization of FM leads The effective polarization is shown here explicitly for the QD-DBMTJ system in Chapter with the presence of the spin-flip events during tunneling the junction between the lead and the dot (SF-TJ). The effective polarization of the lead ν in the presence of SF-TJ effect is given by pν = = = ρν↑ − ρν↓ ρν↑ + ρν↓ (B.1) |[(1 − ην↑ ) ρν↑ + ην↓ ρν↓ ] − [(1 − ην↓ ) ρν↓ + ην↑ ρν↑ ]| [(1 − ην↑ ) ρν↑ + ην↓ ρν↓ ] + [(1 − ην↓ ) ρν↓ + ην↑ ρν↑ ] |(1 − 2ην↑ ) ρν↑ − (1 − 2ην↓ ) ρν↓ | , ρν↑ + ρν↓ where ρνσ (ρνσ ) is the real (effective) density of states (DOS) for the electrons of spin state σ (σ = {↑, ↓}) in the lead ν. Applying the simplifying assumptions of (i) spin and lead independent SF-TJ rate, ην↑ = ην↓ = η, and (ii) symmetric leads, pν = p, Eq. (A1) 167 EFFECTIVE POLARIZATION OF FM LEADS becomes p = pν |(1 − 2ην↑ ) ρν↑ − (1 − 2ην↓ ) ρν↓ | ρν↑ + ρν↓ |ρν↑ − ρν↓ | = (1 − 2η) ρν↑ + ρν↓ = (B.2) = (1 − 2η) pν = (1 − 2η) p. In this case, by applying the two current model, one obtains TMR ∝ (p )2 = (1 − 2η)2 p2 . For a fixed p, we obtain TMR ∝ (1 − 2η)2 . 168 [...]... current of spin- up and spin- down electrons, respectively 16 INTRODUCTION 1.1.5.3 Spin relaxation Spin relaxation is the process through which spin relaxes towards equilibrium The relaxing processes usually involve spin- orbit coupling which provides the spin- dependent potential, and momentum scattering which provides a randomizing force The typical time scales for spin relaxation in electronic systems. .. Numerous theoretical and experimental studies have been performed on spintronic devices over the last few decades [1, 2, 5, 11, 16–18] As more and more interesting experimental results are presented for spintronic devices, theoretical understanding of the mechanisms behind these results began to attract ever-growing interest In spintronic devices, the electron -transport is spin- dependent The theoretical study. .. spin- flip scatterings An electron might undergo a thousand to a million scattering events before its spin is flipped 2 Dyakonov−Perel mechanism This type of spin relaxation operates in systems with no inversion symmetry In these systems the momentum of spin doublet is split into up -spin and down -spin singlets The energy difference between the two split states is proportional to the spin- orbit coupling... theoretical study of spin- dependent transport (SDT) is to set up theoretical models for the SDT in spintronic devices SDT models are essential for the theoretical study of spintronic devices, since the models provide the platform for analyzing the many SDT properties which include I −V characteristics for spin and charge current, the GMR or TMR and other spin effects The analysis of the SDT properties... in nanoseconds, and the spin relaxation time range is from pico- to microseconds Here are four principal mechanisms of spin relaxation in semiconductors [12] 1 Elliott−Yafet mechanism In this mechanism, the spin relaxes by momentum scattering Bloch states with spin- orbit coupling are an admixture of up -spin and down -spin Pauli states The scatterings which connect different momentum states includes spin- flip... applications, in particular when semiconductor and magnetic technologies become more integrated [12] One of these possible spintronic devices are ferromagnetic single electron transistors [13–15], which may find applications in single spin logic and quantum computing which uses spin in a quantum dot to encode qubits [2] 2 INTRODUCTION z e- Sz e- 1 ! 2 Spin up Sz 1 ! 2 Spin down Figure 1.2: Spin up (left)... examined or verified In this thesis, we theoretically study the SDT through several types of nanoscale 3 INTRODUCTION spintronic devices These nanoscale spintronic devices are: current -in- plane (CIP) spin valves (SVs), quantum dot based double tunnel junctions (QD-DTJ) and ferromagnetic single electron transistors (FM-SETs) The rest of this chapter is organized as follows In the Section 1.1, we introduce... the spin- down” electron is the one with Sz = − 1 2 The employment of the electron spin in electronics gives rise to a new type of field, referred to as “spintronics” [1] Spintronics combines the advantages of both conventional 1 INTRODUCTION Figure 1.1: Illustration of the Stern-Gerlach experiment The graph is adopted from Theresa Knott electronics and spin, such that it may offer higher degrees of integration,... an inhomogeneous magnetic field (schematically shown in Fig 1.1) This observation suggested that electrons have a quantized intrinsic magnet momentum The intrinsic magnetic property of an electron is referred to as the electron spin, which has two discrete levels, i.e., spin- up and spindown, as shown in Fig 1.2 The spin- up” electron is the electron with the z-component of the spin- angular momentum of. .. and spin of sub-atomic particles, but these effects are also in uenced by the electronic configuration of different elements and the way they combine chemically In matter, the greatest magnetic effects 8 INTRODUCTION are due to the spins of electrons rather than their orbital moments In molecules, the uncompensated spins also play an overwhelming role The filling of the shells is governed by Schr¨dinger’s . THEORETICAL STUDY OF SPIN DEPENDENT TRANSPORT IN NANOSCALE SPINTRONIC SYSTEMS by Ma Minjie M.Eng., Beijing Jiao Tong University, Beijing, China A Thesis Submitted for the Degree of Doctor of. been attracting tremendously intensive interest since 1980s. One of the most important aspects of the theoretical study of spintronics is to model the spin dependent transport (SDT) through these. effect of the spin- flip scattering which might occur in the bulk of each layer or at their interfaces. The combined effects of spin- flip scattering and interfacial diffusive scattering on GMR are investigated