Intrinsic Hall Effects of Electrons and Photons – Geometrical Phenomena in Linear Response Theory Masaru Onoda (CERC-AIST) The 21 st Century COE International Symposium ~07 Linear Response Theory in Commemoration of its 50 th Anniversary

Collaborators –S. Murakami (TIT) N. Nagaosa (Univ. of Tokyo) Special thanks to –Y. Tokura (Univ. of Tokyo) H. Aoki (Univ. of Tokyo)

Outline Motivation Intrinsic mechanism of Hall effects Semiclassical Interpretation Optical Hall Effect Summary

Mission of the theory team of CERC New functionalities based on geometrical phase of electron systems Anomalous Hall effect (AHE)  Quantum AHE Spin Hall effect (SHE)  Quantum SHE Optical Hall effect (OHE) AHE  high-sensitive Hall element QAHE  resistance standard without external magnetic field Edge states of QSHE  spin filter, control of nuclear spin OHE + tunable photonic crystal (PX)  optical switch Beam with internal rotation in PX  optical mixer without fin

Anomalous Hall Effect Current M M B Quantization Our contribution not presented today Disorder induced quantization of AHE : M. Onoda and N. Nagaosa, PRL 90, (2003) V

Spin Hall effect Current Quantization Our contributions not presented today Real space simulation: M. Onoda and N. Nagaosa, PRB 72, (R) (2005);PRL 95, (2005) Disorder effect: M. Onoda, Y.Avishai, N. Nagaosa, PRL 98, (2007)

Optical Hall effect

Mission of the theory team of CERC New functionalities based on geometrical phase of electron systems Anomalous Hall effect (AHE)  Quantum AHE Spin Hall effect (SHE)  Quantum SHE Optical Hall effect (OHE) AHE  high-sensitive Hall element QAHE  resistance standard without external magnetic field Edge states of QSHE  spin filter, control of nuclear spin OHE + tunable photonic crystal (PX)  optical switch Beam with internal rotation in PX  optical mixer without fin

Skew scattering Side jump L.Berger, PRB 2, 4559 (1970) J.Smith, Physica 24, 39 (1958) Conventional mechanisms of AHE and SHE  xy in Kubo formalism + diagrammatic technique Spin dependent scattering M. I. Dyakonov and V. I. Perel, Phys. Lett. A 35, 459 (1971) J. E. Hirsch, PRL 83, 1834 (1999) S. Zhang, PRL 85, 393 (2000) Extrinsic spin Hall effect

B HH B HH Conventional HE  Hall element Semiconductor with high resistivity and high mobility AHE  high sensitive Hall element (patent No ) Material search and design are needed for optimization.  Research on intrinsic mechanism

Intrinsic Mechanism Quantum Hall effect K. v. Klitzing, G. Dorda, M. Pepper, PRL 45, 494 (1980) TKNN, PRL 49, 405 (1982) H. Aoki and T. Ando, PRL 57, 3039 (1987) Intrinsic anomalous Hall effect due to chiral spin order K. Ohgushi, S. Murakami, N. Ngaosa, PRB 62, R6065 (2000) Y. Taguchi et al., Science 291, 2573 (2001) Intrinsic spin Hall effect S. Murakami, N. Nagaosa., S.-C. Zhang, Science 301, 1348 (2003) J. Sinova et al., Phys. Rev. Lett. 92, (2004) Multi-band effect R. Karplus and J. M. Luttinger, Phys. Rev. 95, 1154 (1954) J. M. Luttinger, Phys. Rev. 112, 739 (1958)  1 2 3

Berry curvature ~ magnetic field in k-space EFEF kxkx kyky Geometrical aspect only in special situations?

t 2g model d xy d yz d zx pxpx pypy M.Onoda and N. Nagaosa, JPSJ 71, 19 (2002)

Nearly degenerate (resonant) point  Large  Topological/Geometrical and Resonant aspects Small change of a parameter, e.g., M z  Drastic change of   Drastic change of  xy Topological transition in QHE J. E. Avron, R. Seiler, B. Simon, PRL 51, 51 (1983) B. Simon, PRL 51, 2167 (1983) Y. Hatsugai and M. Kohmoto, PRB 42, 8282 (1990) M. Oshikawa, PRB 50, (1994)

Berry curvature of a t 2g band Z. Fang et al., Science 302, 92 (2003) SrRuO 3, Sr 0.8 Ca 0.2 RuO 3 Sr 1-x Ca x RuO 3 R. Mathieu et al., PRL 93, (2004)

x: electric field y: spin current z: spin direction S. Murakami, N. Nagaosa., S.-C. Zhang, Science 301, 1348 (2003) Intrinsic spin Hall effect in p-type GaAs GaAs

momentum spin J. Sinova et al., PRL 92, (2004) Intrinsic spin Hall effect in n-type GaAs

Y. K. Kato et al., Science 306,1910 (2004) Spin Hall Effect in n-type GaAs J. Wunderlich et al., PRL 94, (2005) Spin Hall Effect in p-type GaAs Extrinsic (Conventional) Intrinsic

S. Murakami, PRL 97, (2006) Bi bilayer B. A. Bernevig, Science 314, 1757 (2006) Graphene C. L. Kane and E. J. Mele, PRL 95, (2005) Candidates of QSHE CdTe/HgTe/CdTe QW

M. König, et al., Science 318, 766 (2007) HgTe/Hg 0.3 Cd 0.7 Te QW

Anomalous velocity  QHE Effective Lorentz force Spin-dependent  Intrinsic spin Hall effect M.-C. Chang and Q. Niu, PRL 75, 1348 (1995). Semiclassical Interpretation Equations of motion of a wave-packet in magnetic flux commensurate with lattice Magnetic Bloch bands

M.-C. Chang, Q. Niu, PRB 53, 7010 (1996) 1 st level 2 nd level 3 rd level Berry curvatureInternal rotation

EE

Multi-band effect  Projection due to  nk Resonant enhancement Internal Rotation Spin-orbit coupling in a broad sense Wave-particle duality

Wave optics  Eikonal  Fermat’s principle  Geometrical optics Quantum mechanics  Path integral  least-action principle  Classical mechanics Semiclassical interpretation

Equations of motion of optical packet Anomalous velocity Neglecting the spin, i.e., polarization →Conventional equation of geometrical optics M. Onoda, S. Murakami, N. Nagaosa, PRL 93, (2004)

Solid (transmission) and broken (reflection) lines: conservation of angular momentum per photon ● ■ ： Maxwell’s equations Linear polarization Imbert-Fedorov shift Elliptical polarization M. Onoda, S. Murakami, N. Nagaosa, PRE 74, (2006)

Internal Angular momenta of light Linear S=0Right circular S=1Left circular S=-1 Spin angular momentum Orbital angular momentum L=0L=1L=2L=3

Transverse shift of Laguerre-Gaussian beam Experiment, H. Okuda, H. Sasada, Opt. Exp. 14, 8393 (2006) Experiment, R. Dasgupta, P. K. Gupta, Opt. Comm. 257, 91 (2006) Theory, V. G. Fedoseyev, Opt. Comm. 193, 9 (2001)

Optical Hall effect in photonic crystals Multi-band  Resonant enhancement

Berry curvature in PX

Trajectory of optical wave-packet in PX Overhead view of the 2D PX. The crystal structure is not shown.

Berry curvature and internal rotation (TE mode)

A. T. O’Neil et al., PRL 88, (2002) Applications of optical torque (Extensions of the optical tweezer technology) H. Ukita, 精密工学会誌 72, 977 (2006) (in Japanese) 3  x 1.5  calcite  dielectric

Beam with large angular momentum by photonic crystal  optical mixer without fin

Summary Intrinsic mechanism of Hall effects (IHE) in electron systems –Suitable for material search and design –Multi-band effect –Geometrical/topological –Resonant enhancement –Internal rotation + spin-orbit interaction in a broad sense Optical Hall effect (OHE) –Counterpart of electronic IHE in optical systems –Imbert-Fedorov shift as a simple example of OHE –Resonant enhancement of OHE in photonic crystals –Optical state with large angular momentum in PX