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The Persistent Spin Helix Shou-Cheng Zhang, Stanford University Banff, Aug 2006 Credits Collaborators: • B. Andrei Bernevig (Stanford) • Joe Orenstein (Lawrence Berkeley Lab) • Chris Weber (Lawrence Berkeley Lab) Outline • Mechanisms of spin relaxation in solids • Exact SU(2) symmetry of spin-orbit coupling models • The Persistent Spin Helix (PSH) • Boltzmann equations • Optical spin grating experiments Spin Relaxation in Solids • Without SO coupling, particle diffusion is the only mechanism to relax the spin. 1 Dq 2 Spin Relaxation in Solids • With SO coupling, the dominant mechanism is the DP relaxation. S z S p : : The spin-orbit field Momentum relaxation time The 2D random walk problem: t 2 t , 2 2t 1 dS z 1 1 2 2 S z The effective S 1 cos t S dt S 2 reduction of Sz: S The Rashba+Dresselhaus Model k2 H k y x k x y k x x k y y 2m The Rashba spin-orbit coupling. Can be experimentally tuned via proper gating. The Dresselhauss spin-orbit coupling. Increase Dresselhauss The Rashba+Dresselhaus Model For α=β k2 H k x k y x y 2m Global spin U 1 1 i x y 2 2 rotation Coordinate change k 1 kx k y 2 2 2 k k H ReD U HU 2 k z 2m The Dresselhaus [110] Model Symmetric Quantum wells grown along the [110] direction: H 110 k x2 k y2 2m 2 k x z Fermi Surface and the Shifting Property • The shifting property: k k Q For the HReD model H ReD Q 4m , Q 0 For the H110 model Qx 4m , Qy 0 H110 The Exact SU(2) Symmetry • Finite wavevector spin components SQ k ckck Q , SQ k ckQck , S0z , SQ 2SQ , S0z k ckck ckck SQ , SQ S0z • Shifting property essential H ReD , c c k Q k c c 0 k Q k k Q k An exact SU(2) symmetry Only Sz, zero wavevector U(1) symmetry previously known: J. Schliemann, J. C. Egues, and D. Loss, Phys. Rev. Lett. 90, 146801 (2003). K. C. Hall et. al., Appl. Phys. Lett 83, 2937 (2003). The Exact SU(2) Symmetry • The SU(2) symmetry is robust against spin-independent disorder and Coulomb (or other many-body) interactions. q k ckq ck q , SQ q , S0z 0 Vq q , SQ Vq q , S0z 0, q q Vq q q , SQ Vq q q , S0z 0 q q x y • A S , S spin helix with wave vector Q has infinite life time Persistent Spin Helix Physical Picture: Persistent Spin Helix • Spin configurations do not depend on the particle initial momenta. • For the same x distance traveled, the spin precesses by exactly the same angle. • After a length L 2 Q the spins all return exactly to the original configuration. PSH for the H ReD Model and the H110 Model (a) PSH for the H ReD model. The spin-orbit magnetic field is in-plane (blue), where as the spin helix is in the x , z plane. (b) PSH for the H[110] model. spin The spin-orbit magnetic field Borbit , in blue, is out of plane, whereas the spin helix, in red, is in-plane. The Non-Abelian Gauge Transformation H ReD in the form of a background non-abelian gauge potential k2 1 2 H ReD k 2m z const. 2m 2m P. Q. Jin, Y. Q. Li, and F. C. Zhang, J. Phys. A 39, 7115 (2006) • Field strength vanishes; eliminate the vector potential by non-abelian gauge transf x , x exp i 2m x x , x , x , x exp i 2m x x , x H Re D k2 H 2m S x x x exp i 4m x S x S x x x exp i 4m x S x •Mathematically, the PSH is a direct manifestation of a non-abelian flux in the ground state of the models. 0 The Boltzmann Transport Equations For arbitrary α,β spin-charge transport equation is obtained for diffusive regime t n D n B1 x Sx B2 x Sx 2 i B1 2 kF2 2 , 2 B2 2 kF2 2 2 t Sx Di2Sx B1 x n C2 x Sz T2Sx C1 2 k F2 / m, t Sx Di2Sx B2 x n C1 x Sz T1Sx T1 2 kF2 , C2 2 k F2 / m 2 T2 2 kF2 2 t Sz Di2Sz C2 x Sx C1 x Sx T1 T2 Sz For propagation on [110], the equations decouple two by two For Dresselhauss = 0, the equations reduce to Burkov, Nunez and MacDonald, PRB 70, 155308 (2004); Mishchenko, Shytov, Halperin, PRL 93, 226602 (2004) The Boltzmann Transport Equations For α=β : k2 H k x k y x y 2m Gauge transformation S x S x iS y x x S x S x iS y x x k2 H 2m (Free Fermi gas) S x cos qx sin qx S x S sin qx cos qx S y y t S x D 2x 2y S x 2qD x S y Dq 2 S x 2 2 2 t S y D x y S y 2qD x S x Dq S y Simple diffusion equation S , S cos 4m x ,sin 4m x S x const. x y t Si D i2 Si The Boltzmann Transport Equations Along special directions the four equations decoupled to two by two blocks Propagation on [110] Propagation on [1ῑ0] qx q, qx 0 qx q, qx 0 n S x , S z S x i1,2 Dq 2 n S x , S z S 1 2T2 T1 T12 4q 2C22 2 2 2 At α=β i1 Dq T1 , i2 Dq The behavior of Sz is diffusive and exponentially decaying; this is the passive direction C1 2 k F2 / m, T1 2 kF2 , C2 2 k F2 / m T2 2 kF2 2 2 i1,2 Dq 2 At α=β 1 2T1 T2 T22 4q 2C12 2 i1,2 Dq2 T1 C1q At the shifting wave-vector Q i2 q 4m Q 0 An infinite spin life-time of the Persistent Spin Helix; this is the active direction The Optical Spin Grating Experiment C. P. Weber et. al., Nature 437, 1330 (2005) Interference of two orthogonally polarized beams The pump-probe technique: •The spatially modulation of spin or charge is first introduced by the ‘pump’ laser pulse. •The time evolution of the modulation is measured by the diffraction of a probe beam. •Spin transport and relaxation properties are probed. An optical helicity wave generates an electron spin polarization wave The Optical Spin Grating Experiment Measurements of the decay, at q close to the ‘magic’ shifting vector, at Rashba close, but not equal to Dresselhauss. Black is the active direction, red the passive. The Optical Spin Grating Experiment Fitting of experimental data to Boltzman transport equations, for Rashba/Dresselhauss ~ 0.2 - 0.3. Even though the Rashba and Dresselhauss are not yet equal, large enhancement of spin-lifetime for the spin helix is observed Generation of the PSH Current PSH associated with SU(2) charge – PSH current FM2 pulse delayed from FM1 pulse FM1 Two consecutive FM1 pulses delayed by FM2 [110] GaAs v E Generation of the PSH Current Optical detection of oscillating spin at given spatial point. Dresselhauss [110] For Rashba equal Dresselhauss: Optical detection Decay component: FM1 FM2 ReD GaAs Conclusions • Minimize spin-decoherence while keeping strong spin-orbit coupling • Shifted Fermi Surfaces: Fundamental property of some cond-mat systems, similar to nesting • Exact SU(2) symmetry of systems with Rashba equal to Dresselhauss or Dresselhauss [110]; finite wave-vector generators • Persistent Spin Helix • Experimental discovery