Berry phase.ppt

1、Topological Aspects of the Spin Hall Effect,Yong-Shi Wu Dept. of Physics, University of Utah Collaborators: Xiao-Liang Qi and Shou-Cheng Zhang (XXIII International Conference on Differential Geometric Methods in Theoretical Physics Nankai Institute of Mathematics; August 21, 2005),Motivations,Electr

2、ons carry both charge and spin Charge transport has been exploited in Electric and Electronic Engineering: Numerous applications in modern technology Spin Transport of Electrons Theory: Spin-orbit coupling and spin transport Experiment: Induce and manipulate spin currents Spintronics and Quantum Inf

3、ormation processing Intrinsic Spin Hall Effect: Impurity-Independent Dissipation-less Current,Key advantages: Electric field manipulation, rather than magnetic field Dissipation-less response, since both spin current and electric field are even under time reversal Intrinsic SHE of topological origin

4、, due to Berrys phase in momentum space, similar to the QHE Very different from Ohmic current:,Electric field induces transverse spin current due to spin-orbit coupling,Family of Hall Effects,Classical Hall Effect Lorentz force deflecting like-charge carriers Quantum Hall Effect Lorentz force deflec

5、ting like-charge carriers (Quantum regime: Landau levels) Anomalous (Charge) Hall Effect Spin-orbit coupling deflecting like-spin carriers (measuring magnetization in ferromagnetic materials) Spin Hall Effect Spin-orbit coupling deflecting like-spin carriers (inducing and manipulating dissipation-le

6、ss spin currents without magnetic fields or ferromagnetic elements),Time Reversal Symmetry and Dissipative Transport,Microscopic laws in solid state physics are T invariant Most known transport processes break T invariance due to dissipative coupling to the environment Damped harmonic oscillator,(on

7、ly states close to the Fermi energy contribute!),Ohmic conductivity is dissipative: under T, electric field is even charge current is odd Charge supercurrent and Hall current are non-dissipative:,under T vector potential is odd, while magnetic field is odd,Spin-Orbit Coupling,Origin: Relativistic ef

8、fect in atomic, crystal, impurity or gate electric field = Momentum-dependent magnetic field Strength tunable in certain situations Theoretical Issues: Consequences of SOC in various situations? Interplay between SOC and other interactions? Practical challenge: Exploit SOC to generate,manipulate and

9、 transport spins,The Extrinsic Spin Hall effect,(due to impurity scattering with spin-orbit coupling),Dyakonov and Perel (1971) Hirsch (1999), Zhang (2000),The Intrinsic Spin Hall Effect Berry phase in momentum space Independent of impurities,impurity scattering = spin dependent (skew) Mott scatteri

10、ng plus side-jump effect,Berry Phase (Vector Potential) in Momentum Space from Band Structure,Wave-Packet Trajectory in Real Space,Chang and Niu (1995); P. Horvarth et al. (2000),Intrinsic Hall conductivity (Kubo Formula) Thouless, Kohmoto, Nightingale, den Nijs (1982) Kohmoto (1985),: field strengt

11、h; : band index,Field Theory Approach,Electron propagator in momentum space Ishikawas formula (1986): Hall Conductance in terms of momentum space topology,: odd under time reversal = dissipative response,Intrinsic spin Hall effect in p-type semiconductors,Valence band of GaAs,Luttinger Hamiltonian,(

12、 : spin-3/2 matrix, describing the P3/2 band),S,P,S,P3/2,P1/2,Luttinger model,Expressed in terms of the Dirac Gamma matrices:,Spin Hall Current (Generalizing TKNN),Of topological origin (Berry phase in momentum space) Dissipation-less All occupied state contribute,Spin Analog of the Quantum Hall Eff

13、ect At Room Temperature,(Sinova, Culcer, Niu, Sinitsyn,Jungwirth, MacDonald, PRL(2003),Rashba Hamiltonian,Intrinsic spin Hall effect for 2D n-type semiconductors in heterostructure,Kubo formula :,independent of,Effective magnetic field,SHE: Spin precession by “k-dependent Zeeman field” Note: is not

14、small even when the spin splitting is small. due to an interband effect,Spin Hall insulator,Motivation: Truly dissipationless transport Gapful band insulator (to get rid of Ohmic currents) Nonzero spin Hall effect in band insulators: - Murakami, Nagaosa, Zhang, PRL (2004) Topological quantization of

15、 spin Hall conductance: - Qi, YSW, Zhang, cond-mat/0505308 (PRL) Spin current and accumulation: - Onoda, Nagaosa, cond-mat/0505436 (PRL),Theoretical Approaches,Kubo Formula (Berry phase in Brillouin Zone) Thouless, Kohmoto, Nightingale, den Nijs (1982) Kohmoto (1985) Kubo Formula (Twisted Phases at

16、Boundaries) Niu, Thouless, Wu (1985) (No analog in SHE yet!) Cylindrical Geometry and Edge States Laughlin (1981) Hatsugai (1993) (convenient for numerical study),Cylindrical Geometry and Edge States,Laughlin Gauge Argument (1981):,Adiabatically changing flux Transport through edge states,Bulk-Edge

17、Relation:,(Spectral Flow of Edge States),(Hatsugai,1993),Topological Quantization of the AHE (I),Model Hamilatonian:,Topological Quantization of the AHE (II),Two bands:,Charge Hall conductance is quantized to be n/2p,Charge Hall effect of a filled band:,Band Insulator: a band gap, if V is large enou

18、gh, and only the lower band is filled,Topological Quantization of the AHE (III),Open boundary condition in x-direction Two arrows: gapless edge states The inset: density of (chiral) edge states at Fermi surface,Topological Quantization of Spin Hall Effect I,SHE is topologically quantized to be n/2p,

19、Paramagnetic semiconductors such as HgTe and a-Sn:,are Dirac 4x4 matrices (a=1,.,5) With symmetry z-z, d1=d2=0. Then, H becomes block-diagonal:,Topological Quantization of Spin Hall Effect II,For t/V small: A gap develops between LH and HH bands.,Conserved spin quantum number is,Topological Quantiza

20、tion of Spin Hall Effect III,Physical Understanding: Edge states I,In a finite spin Hall insulator system, mid-gap edge states emerge and the spin transport is carried by edge states,Energy spectrum for cylindrical geometry,Laughlins Gauge Argument: When turning on a flux threading a cylinder system

21、, the edge states will transfer from one edge to another,Topological Quantization of Spin Hall Effect IV,Physical Understanding: Edge states II,Apply an electric field n edge states with G12=+1(-1) transfer from left (right) to right (left).,G12 accumulation Spin accumulation,Rashba model: Intrinsic

22、 spin Hall conductivity (Sinova et al.,2004),+ Vertex correction in the clean limit (Inoue et al (2003), Mishchenko et al, Sheng et al (2005),Effect due to disorder,+ spinless impurities ( -function pot.),(Greens function method),Luttinger model: Intrinsic spin Hall conductivity (Murakami et al,2003

23、),+ spinless impurities ( -function pot.),Vertex correction vanishes identically! (Murakami (2004), Bernevig+Zhang (2004),Topological Orders in Insulators,Simple band insulators: trivial Superconductors: Helium 3 (vector order-parameter) Hall Insulators: Non-zero (charge) Hall conductance 2d electro

24、ns in magnetic field: TKNN (1982) 3d electrons in magnetic field: Kohmoto, Halperin, Wu (1991) Spin Hall Insulators: Non-zero spin Hall conductance 2d semiconductors: Qi, Wu, Zhang (2005) 2d graphite film: Kane and Mele (2005) Discrete Topological Numbers: in 2d systems Z_2: Kane and Mele (2005); Z_n: Hatsigai, Koh