Gauge Concepts in Theoretical Applied Physics

Transcription

1 Gauge Concepts in Theoretical Applied Physics Seng Ghee Tan, Mansoor BA Jalil Data Storage Institute, A*STAR (Agency for Science, Technology and Research) Computational Nanoelectronics and Nano-device Laboratory, Department of Electrical and Computer Engineering, National University of Singapore Conference on 60 Years of Yang-Mills Gauge Field Theories IAS, Nanyang Technological University, Singapore /25/ May

2 FRIENDS Tan, Seng Ghee ASTAR, NUS Mansoor BA Jalil NUS Fujita, Takashi Seagate Research Murakami, Shuichi, Tokyo Inst. Tech. Albert Liang, GC NUS Yang, Hyunsoo NUS Ho, Congson NUS Siu, Zhuobin NUS 5/25/2015 大道至简 2

3 Introduction to Applied & Electronic Physics General physics Condensed matter physics Mesoscopic physics Zero Dimension Universe, boundless vaccuum Periodic potential Many body interaction One Dimension Non-equilibrium device physics Electrode proximity effect Contact physics Two Dimension V Proximity effect Interface physics External EM force Local Charge Redistribution Conservation of Current 5/25/2015 大道至简 3

4 Introduction to Applied & Electronic Physics Metal Semiconductor Carbon (graphene, diamond, carbon nanostructures) Insulator Electron charge Resistors Capacitors Inductors Single electron tunneling device MOS-FET HEMT Quantum Hall Quantum Dot device CNT-FET Graphene Devices Molecular Device Topological conductance WAL physics Electron spin Spin-Valve (MR) MRAM Magnetometry Spin-FET Graphene spin-fet Nitrogen-vacancy magnetometry T.I. memory T.I. spin current source Photon Semiconductor Photonic Devices Graphene Display CNT display NV-single photon source Topological Faraday & Kerr 5/25/2015 大道至简 4

5 Introduction to Applied & Electronic Physics Sensor, Memory Data Writing Technology Conventional Field Writing Spin Current Writing Free Layer Spacer Pinned moment Numerous Implementations Charge Current Writing Metal Oxide Platinum Charge Current Writing Gd, Eu Charge Current Strain writing Graphene Ferromagnetic Ferromagnetic Rashba FM Rashba Rare Earth Phase Change Materials Data Reading / Sensing Technology Conventional MR Reading Free Layer Spacer Pinned moment Electrical Topological Anomalous, Spin Hall Reading Skyrmion with winding number 1 Topological Non-local Inverse Spin Hall Reading Spin current Topological Faraday / Kerr Reading Light Light Electric field Topological Insulator Topological QSHE Reading Edge Helical Pairs Topological Insulator Electrical Electrical Optical Electrical 5/25/2015 大道至简 5

6 Gauge physics underlies many useful phenomena Spin Orbit Gauge Murakami- Fujita Gauge The relation of gauge physics to the classical notion of forces and velocities. velocity force Spin orbit torque Spin current Torque Spin Hall Anomalous Hall MR Magnetic switching Resistance change Field sensing Data Writing in magnetic memory Data Reading in magnetic memory 5/25/2015 大道至简 6

7 Gauge physics: Classical notion of forces and velocities Classical notion of Lorentz force Quantum mechanics velocity spin-dependent magnetic field Heisenberg f = (p x ea x ) (A a A b ε abz ) n y Spin-transverse force spin-dependent magnetic field B z 5/25/2015 大道至简 7

8 Gauge physics: Classical notion of forces and velocities Simple spin orbit gauge Fig. Unification of Yang Mills force, Heisenberg spin force, SHE conductivity under the Heisenberg spin force equation. The simple spin orbit gauge is used in the above. References: [1] Seng Ghee Tan, Mansoor BA Jalil, J. Phys. Soc. Japan 82, (2013). [2] Takashi Fujita, MBA Jalil, SG Tan, Shuichi Murakami, J. Appl. Phys. (Appl. Phys. Rev.) 110, (2011); [3] Seng Ghee Tan, Mansoor BA Jalil, Introduction to the Physics of Nanoelectronics, Woodhead Publishing (2012). 5/25/2015 大道至简 8

9 Gauge physics: Classical notion of forces and velocities t-k space gauge & spin orbit gauge S.G. Tan et al., ArXiv: , 2015 More details on Murakami & Fujita gauge, see references: [1] S. Murakami, N. Nagaosa, S.C. Zhang, Science 301, 1348 (2003). [2] T. Fujita, M.B.A Jalil, and S.G. Tan, J. Phys. Soc. Jpn. 78, (2009). [3] T. Fujita, M.B.A Jalil, and S.G. Tan, New J. Phys. 12, (2010). [4] T. Fujita, M.B.A. Jalil, S.G.Tan, S. Murakami, J. Appl. Phys. 110, (2011). [Appl. Phys. Rev.] 5/25/2015 大道至简 9

10 Gauge physics: Classical notion of forces and velocities t-k space gauge & spin orbit gauge Fig. Unification of Yang Mills force, Heisenberg spin force, SHE conductivity under the Heisenberg spin force equation. The time-space spin orbit gauge is used in the above. First studied by S. Murakami in momentum space, T. Fujita in time space References: [1] Seng Ghee Tan, Mansoor BA Jalil, J. Phys. Soc. Japan 82, (2013). [2] Takashi 5/25/2015 Fujita, MBA Jalil, SG Tan, Shuichi Murakami, J. Appl. Phys. (Appl. 大道至简 Phys. Rev.) 110, (2011); 10 [3] Seng Ghee Tan, Mansoor BA Jalil, Introduction to the Physics of Nanoelectronics, Woodhead Publishing (2012).

11 Gauge physics underlies many useful phenomena Gauge Physics The relation of gauge physics to the classical notion of forces and velocities. velocity force Spin orbit torque Spin current Torque Spin Hall Anomalous Hall MR Magnetic switching Resistance change Field sensing 5/25/2015 大道至简 11

12 Concept of Spin Torque in Magnetic Memory Technological Importance Active element: magnetic tunnel junction One bit of MRAM cell MRAM cells integrated in a two-dimensional array 5/25/2015 大道至简 12

13 Spin Orbit Spin Torque Cartoon physics of SOC spin torque Electric field velocity In this picture the effective magnetic field applies strictly to spin, i.e. Electric field Spin (s) velocity Electric field Moment (d) s-d interaction Spin (s) velocity effective magnetic field effective magnetic field effective magnetic field s-rashba interaction Nanoscale device with strong spin orbit coupling Due to the Rasbha & Vasko physics in heterostructure E. Rashba 2005 Metal Oxide Electric field E field Mz Jx (1) Pt, Pd, heavy atom (2) Bi, BiSe, TI materials (1) Ferromagnetic Metal eg Co, NiFe (2) Rare earth eg Gd Free Co M Metal Oxide Platinum multilevel Metal Oxide z x-y plane Free Co M Electric field Platinum Spacer Pinned moment 5/25/2015 大道至简 13 AP to P P to AP Read Current

14 Spin Orbit Spin Torque Example in Real Space z z M M Spin orbit gauge Head Turning Math modification Spin orbit gauge (rotated) t-k gauge (Murakami- Fujita) 5/25/2015 大道至简 14

15 Spin Orbit Spin Torque What we have done so far is merely turning our head. H H z z H z M Instead of looking straight to see a tilted M M M We tilt our head so that we see a straight M New Physics was revealed? by simply performing a simple head rotation 5/25/2015 大道至简 15

16 Spin Orbit Spin Torque Modified LLG due to Spin Orbit Coupling Original LLG equation 1935 Modified LLG due to Spin Orbit Coupling Slonsweski et al. Berger et al (semiclassical physics) Bazaliy et al. Gen Tatara et al (gauge physics) S. Zhang et al. 2004, (semiclassical physics) Gen Tatara et al. 2005, (perturbative field theory) SG Tan et al. arxiv: (2007) Theoretical Predictions of the spin orbit torque [1] SG Tan, MBA. Jalil, and X.-J. Liu, arxiv: (2007) [2] SG Tan, MBA Jalil, X.-J. Liu, and T. Fujita, Ann. Phys. 326, 207 (2011). [3] A. Manchon, S Zhang 2008, PRB [4] K Obata, G Tatara, 2008, PRB Experimental Confirmation of the spin orbit torque [1] Ioan Mihai Miron et al. Nature Materials ). 9, 230 (2010) [2] JunYeon Kim et al., Nature Materials 12, 240 (2013) [3] JunYeon Kim et al., Phys. Rev. B 89, (2014 Nature Materials 12, 240 (2013) 5/25/2015 大道至简 16

17 Spin Orbit Spin Torque Recent Developments Nature Materials 12, 240 (2013) M z STT like (in-plane ) M H y Precession-like Spin Orbit Torque Damping like Spin Orbit Torque H y J x Nature Nanotechnology 9,211 (2014) Field like (perpendicular) RECENT work on spin orbit spin torque Kurebayashi et al 2014 SG Tan et al 2007 Hyunsoo Yang, NUS (1) Spin-Orbit Torques in Co/Pd Multilayer Nanowires, PRL 111, (2013). (2) Angular and temperature dependence of current induced spin-orbit effective fields in Ta/CoFeB/MgO nanowires, Sci Rep. 4, 4491 (2014). (3) Anomalous temperature dependence of current-induced torques in CoFeB/MgO heterostructures with Ta-based underlayers, PRB 89, (2014). (4) A new route to spin-orbit torque engineering via oxygen manipulation, Nature Nanotech (2014). (5) Magnetization switching through giant spin orbit torque in a magnetically doped topological insulator heterostructure, Nature Materials 13, 699 (2014). 5/25/2015 大道至简 17

18 Spin Hall: Review Spin Hall effect (SHE) has been discussed in the context of (1) Geometric & Gauge physics [1] (2) Kinetic & Kubo formulation [2] (3) Spin orbit gauge (Yang-Mills) [3] (4) Numerous semi-classical treatments Perel, 1976 Dyakonov 1976 It can be confusing if the different pictures have partial or overlapping claims of contribution to the SHE. We present a gauge-theoretic, time-momentum elucidation, which provides a general SHE equation of motion, that unifies under one theoretical framework, all contributions of SHE conductivity due to the (1) Kinetic, (2) Spin orbit force (Yang-Mills) (3) Geometric (Murakami-Fujita) effects. [1] S. Murakami, N. Nagaosa, S.C. Zhang, Science 301, 1348 (2003). [2] J. Sinova, D. Culcer, Q. Niu, N.A. Sinitsyn, T. Jungwirth et al., Phys. Rev. Lett. 92, (2004). [3] S-Q Shen, Phys. Rev. Lett. 95, (2005). 5/25/2015 大道至简 18

19 Spin Hall: Improved Theory v y z = v y z KE + v y z YM + v y z MF Kinetic v y z KE = ± p y m ħ γb Σ n n k x E x + B B Σ n. a z Yang-Mills v y z = e ħ n n E k y k x + n 2 n x k y k x Murakami-Fujita = ħ ( ) + E x + γ Bn k y. a z + +. Fig.1. Fermi sphere of a general electron gas system in the presence of spin orbit coupling shows a distribution of the momentum, band, and effective magnetic field projected along z (B z ). It is assumed that p = ħk. The shaded region encircled by the equator shows a specific system (Rashba 2D) where for + ( ) ( ) + + (a) the + band, B z changes sign over the Eastern and Western hemisphere, resulting in a positive kinetic spin velocity, (b) the band, B z changes sign in a similar manner, thus resulting in a negative kinetic spin velocity. The slender red arrow indicates spin polarization of ±n Σ. a j. 5/25/2015 大道至简 19

20 Spin Hall Prove & Illustrate Theory in a Simple Two-dimensional SOC System ħ ( ) Quantities Explicit 1. = (a) (b) Fig.2. (a) In the case of a 2D nanostructure, where n lies in the x y plane, the (n t n) term points along z, z thus one has v y KE = ± ħp y n γb t n ; (b) Table of quantities for the Rashba system that can be used to derive the SHE expression for the Rashba system. J y z = J y z = J y z = g dk xdk y 2π 2 v y z KE g dk xdk y 2π 2 v y z YM g dk xdk y 2π 2 v y z MF σ y z = e 8π σ y z = σ y z = + e 4π Total SHE Rashba 2DEG σ y z = + e 8π Total SHE Rashba Heavy Hole σ y z = + 9e 8π 5/25/2015 大道至简 20

21 Spin Hall Our work puts right an ambiguity surrounding previously partial treatments involving the Kubo, semiclassical, Berry curvatures, or the spin orbit force. Rashba 2DEG Rashba Heavy hole SHE Conductivity e 8π 9e 8π New SHE Conductivity + e 8π + 9e 8π 5/25/2015 大道至简 21

22 THANK YOU 大道至简 : 道勿在迩而求诸远, 事勿在易而求诸难 5/25/2015 大道至简 22