Persistent Current and Hall effect driven by Spin Chirality Gen - PowerPoint PPT Presentation

Persistent Current and Hall effect driven by Spin Chirality Gen Tatara 多々良 源 Osaka University Nico Garcia (Madrid) Hiroshi Kohno 河野浩(Osaka) Hikaru Kawamura (Osaka) spin Josephson effect

SU(2) This is not all Konig,Bonsager & MacDonald (2001) equilibrium spin current charge spin Transmission amplitude • Electron transport through non-uniform magnetization modified by S 1 ⋅ S 2 resistance S S 1 S 2 S ⋅s potential due to magnetization S S ⋅s ( S 1 ⋅s ) ( S 2 ⋅s )= S 1 ⋅ S 2 +i ( S 1 ¥ S 2 ) ⋅s amplitude of L fi R ( S 1 ⋅s ) ( S 2 ⋅s ) - ( S 2 ⋅s ) ( S 1 ⋅s ) = 2 i ( S 1 ¥ S 2 ) ⋅s

Josephson effect current Spin current Spin Josephson effect Current driven by SU(2) phase spin and charge current • Spin Josephson effect • Superconducting junction e i f 2 e i f 1 J µD sin ( f 1 - f 2 ) • Ferromagnetic junction ( S 1 ⋅s ) ( S 2 ⋅s )= S 1 ⋅ S 2 +i ( S 1 ¥ S 2 ) ⋅s J µ ( S 1 ¥ S 2 ) ⋅s S 1 ⋅s S 2 ⋅s e i p( S ⋅s)/2

GT&Kohno, Phys. Rev. B67, 113316 (2003). These are not equal spin chirality If electron is coherent Spontaneous charge current • Charge current by spin Josephson effect • 3 spins S 2 S 2 ( S 2 ⋅s) S 3 S 3 S 1 S 1 ( S 3 ⋅s) ( S 1 ⋅s) ( S 3 ⋅s) ( S 2 ⋅s) ( S 1 ⋅s) ( S 1 ⋅s) ( S 2 ⋅s) ( S 3 ⋅s) tr[ s i s j s k ]=2 i e ijk tr[( S 1 ⋅s) ( S 2 ⋅s) ( S 3 ⋅s)- ( S 3 ⋅s) ( S 2 ⋅s) ( S 1 ⋅s)]= 4 i S 1 ⋅ ( S 2 ¥ S 3 ) • spin chirality fi breaking of time reversal symmetry in orbital motion µ S 1 ⋅ ( S 2 ¥ S 3 )

GT&Kohno PRB (2003) Exchange interaction • Persistent current in nanoscale ring • Quantum dots S 1 • Ferromagnets S 3 J S 3 ⋅ s J S 1 ⋅ s J S 2 ⋅ s S 2 • current j = - h eJ 3 r g 12 r g 23 r g 3 x ' r ) | x ' = x Ú d w m S 1 ⋅ ( S 2 ¥ S 3 ) S f ( w ) — Im( g x 1 2 p X i 3 v J Ê ˆ j 2 e F cos( k L ) S ( S S ) Á ˜ = - ⋅ ¥ Á ˜ F 1 2 3 L e Ë ¯ F

Extension of Berry phase to non-adiabatic case (strong coupling to Continuum limit spin chirality spin Berry phase Persistent current by spin Berry phase Only in adiabatic limit a smooth spin structure) spin chirality = non-adiabatic (perturbative) analog of Berry phase • Relation to the adiabatic case • Loss, Goldbart& Balatsky PRL (1990) F = Ú d 2 x S ⋅ ( ∂ x S ¥∂ y S ) • Our result 3 v J Ê ˆ j 2 e F cos( k L ) S ( S S ) Á ˜ = - ⋅ ¥ F Á ˜ 1 2 3 L e Ë ¯ F Ú d 2 x S ⋅ ( ∂ x S ¥∂ y S )= F S 1 ⋅ ( S 2 ¥ S 3 )

GT and Garcia, PRL 91, 076806 (2003). R L R R+L Superposition state • Application to operation gates j >0 j <0 j =0 | j |=XOR[ S 2 , S 3 ] • Unitary operation

drift Electric field Local orbital motion of electron Anomalous Hall effect • Hall effect due to local persistent current • Frustrated magnets • finite local spin chirality j y E x s xy µ S 1 ⋅ ( S 2 ¥ S 3 )

RKKY GT&Kawamura J.Phys. Soc.Jpn. (2002) geometrical weight • Hall conductivity • Exchange interaction † r H = J S X S X ⋅ ( c c ) X ¢ s • Kubo formula 3rd order in J cf：Kondo ’ 62 s xy µ J 3 t 2 c 0 t : lifetime（impurity） • total chirality c 0 X 3 S X 1 ⋅ ( S X 2 ¥ S X 3 ) ( a ¥ b ) z c 0 = 1 3 N S I ( a ) ¢ I ( b ) I ( c ) ¢ ab c X i b X 1 a X 2 I ( r ) = sink F r k F r e - r / 2 l Finite c 0 Hall effect

Ohgushi,Murakami & Nagaosa (2000) Extension of Berry phase effect to non-adiabatic case Adiabaticity was believed to be essential Hall effect Berry phase Ye et al., PRL (1999). • Adiabatic limit • Magnetic vortex y x r r r 2 d x S ( S S ) F = Ú ⋅ ∂ ¥ ∂ e i F xy xy x y • Quantiztion of Hall conductivity in the adiabatic limit • GT&Kawamura (2002)

in the perturbative case (2000) Shinjo et al., GT,Yamanaka &Onoda (2003) Quantization occurs even in the non-adiabatic case Topological invariant Coupling factor • Hall conductivity as a topological invariant If spin texture is smooth （compared with mean free path l ） s xy = e 2 4 p S ⋅ ( ∂ x S ¥∂ y S ) = e 2 Ú 1 d x h n a h a n: integer a µ ( J t ) 3 ( l/k F a 2 ) a : vortex size

Nd 2 Mo 2 O 7 Taguchi et al., Science (2001) pyrochlore ferro spin glass new contribution in spin glass phase • experiment • Frustrated ferromagnets s xy T • spin glass systems R s La 1.2 Sr 1.8 Mn 2 O 7 Chun et al. ('00) reentrant spin glass below T g ª 40K T g T

spin Josephson effect Summary spin chirality G.Tatara and H. Kohno, Phys. Rev. B67, 113316 (2003). G.Tatara and N.Garcia, Phys. Rev. Lett. 91, 076806 (2003) G. Tatara and H. Kawamura J.Phys. Soc.Jpn. 71, 2613 (2002) G. Tatara,M. Yamanaka and M.Onoda (2003) References • spontaneous persistent current • anomalous Hall effect • • • •