Transcript Part 4
Spintronics: How spin can act on charge carriers and vice versa Tomas Jungwirth Institute of Physics Prague University of Nottingham Two paradigms for spintronics “Mott“ non-relativistic two-spin-channel model of ferromagnets I I Mott, 1936 “Dirac“ relativistic spin-orbit coupling I Dirac, 1928 I SHE & STT switching SOT switching Ralph, Buhrman,et al., Science ‘12 Miron et al., Nature ‘11 -We see (anti)damping-like torque -We also see (anti)damping-like torque -SOT is field-like so we exclude it -SOT is field-like but maybe there is some (anti)damping-like SOT as well - non-relativistic STT in metals is dominated by the (anti)damping torque Ohmic “Dirac“ device: AMR Kelvin, 1857 Magnetization-orientation-dependent scattering Spin-orbit coupling Spin-orbit coupling Extraordinary magnetoresistance: AMR, AHE, SHE, SOT..... Ordinary magnetoresistance: response to external magnetic field Acting via classical Lorentz force Extraordinary magnetoresistance: response to internal quantum-relativistic spin-orbit field anisotropic magnetoresistance B Lord Kelvin 1857 _ _ _ _ _ _ _ _ _ _ _ FL +++++++++++++ I AMR s 1 2 __ M V ( ij ji ) ij ( M ) ji ( M ) FSO I ordinary Hall effect 1879 V AHE A 1 2 anomalous Hall effect 1881 ( ij ji ) k t df n , k dt E n , k k Classical Boltzmann equation f n , k f 0 ( E n , k ) e E v 0 n , k t E n ,k n d k d ( 2 ) d W n , k , n , k ( f n , k f n , k ) Non-equilibrium distribution function g n , k f n , k f 0 ( E n , k ) Steady-state current in linear response to applied electric field j i ij E j e n d d k ( 2 ) d i v 0 n , k g n , k ( E j ) Linear response: g linear in Ej Steady-state solution for elastic (impurity) scattering Steady-state solution for elastic (impurity) scattering if g(i,k)= Constant quasi-particle relaxation time solution Steady-state solution for elastic (impurity) scattering if is isotropic: depends on | - ’| g(i,k)= Transport relaxation time solution: back-scattering dominates Steady-state solution for elastic (impurity) scattering if is anisotropic: depends on k, k’ No relaxation time solution AMR in Rashba 2D system AMR s 1 2 Rashba Hamiltonian ( ij ji ) Eigenspinors AMR in Rashba 2D system QM: 1st order Born approximation i k r i k r dr e (r )e const . isotropic V 1 ( r ) V M x x (r ) anisotropic M AMR in Rashba 2D system Heuristic picture from back-scattering matrix elements V M y y (r ) V M x x (r ) Rashba SOI Rashba SOI current M M Back-scattering high resistivity No back-scattering low resistivity Anomalous Hall effect in FMs 1881 Polarimetry of electrons in FMs Spin Hall effect in PMs jc Mott, N. F. Proc. R. Soc. Lond. A 1929 Dyakonov and Perel 1971 Kato, Awschalom, et al., Science‘04 Electron spin-dependent scattering off Coulomb field of heavy atoms due to spin-orbit coupling Polarimetry of high-energy electron beams in accelerators Wunderlich, Kaestner, Sinova, TJ, PRL‘05 Electron spin-dependent scattering off Coulomb field of dopands in a semiconductor due to spin-orbit coupling Proposal for electrical spin injection by the spin Hall effect and electrical detection by the inverse spin Hall effect jc Hirsch PRL‘99 js Proposal for electrical spin injection by the spin Hall effect and electrical detection by the inverse spin Hall effect jc js - index Hirsch PNAS‘05 Intrinsic anoumalous Hall effect in (Ga,Mn)As FM (Ga,Mn)As TJ, Niu, MacDonald, PRL’02 Theoretical proposal of intrinsic spin Hall effect Non-magnetic GaAs Murakami, Nagaosa, & S.-C. Zhang, Science’03 Proposed detection by polarized electroluminescence Sinova, TJ, MacDonald, et al. PRL’04 Proposed detection by magneto-optical Kerr effect Magneto-optical Kerr microscopy Extrinsic SHE Kato, Awschalom, et al., Science‘04 Edge polarized electro-luminescence Intrinsic SHE Wunderlich, Kaestner, Sinova, TJ, PRL‘05 Optically generated spin current Optically detected charge accummulation due to iSHE fs pump-and-probe: iSHE generated before first scattering in the intrinsic GaAs intrinsic iSHE Zhao et al., PRL‘06 Werake et al., PRL‘11 AHE and SHE AHE A 1 2 ( ij ji ) AHE and SHE Skew scattering SHE Mott (skew) scattering SHE ll ' (3a ) ij ll ' ij (2) ji ji SHE AMR Skew scattering AHE (SHE) ll ' : not constant, not isotropic, not even symmetric no relaxation time solution (3a ) Approximation: Skew scattering AHE (SHE) Spin orbit torque M Ie Field-like SOT d k d si n ( 2 ) 0 n , k g n , k ( E j ) i d s E=Ex ^x Compare with AMR or skew-scattering SHE d k d ji e n ( 2 ) d i v 0 n , k g n , k ( E j ) Field-like SOT d k d si n ( 2 ) d 0 n , k g n , k ( E j ) i s E=Ex ^x (r) i k r i k r dr e (r )e const . isotropic Field-like SOT d k d si n ( 2 ) d 0 n , k g n , k ( E j ) i (r) i k r i k r dr e (r )e const . isotropic g(i,k)= Field-like SOT s E=Ex ^x s 1 me tr E x 2 3 yˆ H ex H ex H R dM dt J ex M s J ex M Intrinsic anoumalous Hall effect in FMs FM (Ga,Mn)As TJ, Niu, MacDonald, PRL’02 Intrinsic spin Hall effect in PMs Non-magnetic GaAs Murakami, Nagaosa, & S.-C. Zhang, Science’03 Sinova, TJ, MacDonald, et al. PRL’04 Wunderlich, Kaestner, Sinova, TJ, PRL‘05 Werake et al., PRL‘11 Linear response I. Boltzmann theory : non-equilibrium distribution function and equilibrium states Linear response II. Perturbation theory: equilibrium distribution function and non-equilibrium states H pˆ i t Ee 2 V ( rˆ ) 2m 1 A c t A cE i 2 e 2 pˆ e ( pˆ A ) A pˆ 2m c 2 m mc 1 vˆ 1 i [ rˆ , H ] 1 i [ rˆ , 2 pˆ ] 2m pˆ m e i t Uˆ ( t ) E vˆ e c .c i | l (t ) | l e i l t e i | l' l ' l i t l ' | E vˆ | l e l l' e i l t e i t Linear response II. Perturbation theory: equilibrium distribution function and non-equilibrium states H SO pˆ 2 [ zˆ pˆ ] V ( rˆ ) i t Ee 2m 1 A c t A cE i e i t 2 e 2 e pˆ e pˆ ( pˆ A ) [ zˆ ( pˆ A )] A [ ( zˆ )] 2m c c 2m c m 1 vˆ 1 i [ rˆ , H ] 1 i [ rˆ , ( 2 pˆ pˆ ˆ [ zˆ p ]] ( zˆ ) 2m m e i t Uˆ ( t ) E vˆ e c .c i | l (t ) | l e i l t e i | l' l ' l i t l ' | E vˆ | l e l l' e i l t Linear response II. Perturbation theory: equilibrium distribution function and non-equilibrium states Intrinsic SHE (AHE) ˆj z y Ex z z J y l ( t ) | ˆj y | l ( t ) f 0 ( l ) l | l (t ) | l e i l t 0 e i 0 | l' l ' l i t l ' | E vˆ | l e l l' e i l t Heuristic picture: Bloch equations ds y 1 dt eq ( s z B eff ) 2 d sy dt 2 0 B eff , y 0 py ds z 0 dt pz pz B eff , y ~ p x ~ E x t py px sz ~ E x px E=Ex ^x sz ~ E x Field-like SOT d k d si n ( 2 ) 0 n , k g n , k ( E j ) i d s E=Ex ^x Compare with AMR or skew-scattering SHE d k d ji e n ( 2 ) d i v 0 n , k g n , k ( E j ) Intrinsic antidamping SOT from linear response II. 0 0 Compare with intrinsic SHE 0 0 pz pz Hex=0 B eff , y ~ p x ~ E x t py Intrinsic SHE: transverse spin current pz py Hex >> HR px py px B eff , y ~ p x ~ E x t px Intrinsic SOT: spin polarization py sz ~ E x pz sz ~ E x sz ~ E x px sz ~ E x pz Intrinsic SHE: transverse spin current dB eff , y / dt s z , p s B eff , y ~ p x ~ E x t px py 2 2 p 2 2 eE x sin p pz equil ( B eff ) dB eff , y / dt s 2 sd 2J M 2 sz ~ E x 2 Intrinsic SOT: spin polarization sz sz ~ E x B eff , y ~ p x ~ E x t py sz ~ E x px eE x cos M (B equil eff ) 2 sz ~ E x Intrinsic SOT is antidamping-like pz M || xˆ B eff , y ~ p x ~ E x t px py pz M || yˆ px dM dt J ex px py B eff , y ~ p x ~ E x t py s z 0 py M s ~ M ([ E zˆ ] M ) s z 0 px SHE & STT switching SOT switching Ralph, Buhrman,et al., Science ‘12 Miron et al., Nature ‘11 -We see (anti)damping-like torque -We also see (anti)damping-like torque -SOT is field-like so we exclude it -SOT is field-like but maybe there is some (anti)damping-like SOT as well and maybe we found it intrinsic SOT analogous to intrinsic SHE - non-relativistic STT in metals is dominated by the (anti)damping torque