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

• Electron transport through non-uniform magnetization

S1 S2

Transmission amplitude modified by S1⋅S2 resistance This is not all

(S1⋅s) (S2⋅s)= S1⋅S2 +i(S1¥ S2)⋅s

amplitude of LfiR charge spin equilibrium spin current

(S1⋅s)(S2⋅s)- (S2⋅s)(S1⋅s) = 2i(S1¥ S2)⋅s

Konig,Bonsager & MacDonald (2001)

potential due to magnetization S

S⋅s

S

S⋅s

SU(2)

• Spin Josephson effect
• Superconducting junction

eif1 eif2

current

J µDsin(f1 - f2)

Josephson effect

• Ferromagnetic junction

S1⋅s S2⋅s

(S1⋅s) (S2⋅s)= S1⋅S2 +i(S1¥ S2)⋅s Spin current

J µ (S1¥ S2)⋅s

Spin Josephson effect Current driven by SU(2) phase spin and charge current

eip(S⋅s)/2

• Charge current by spin Josephson effect

tr[(S1⋅s) (S2⋅s) (S3⋅s)-(S3⋅s) (S2⋅s) (S1⋅s)]= 4i S1⋅(S2¥S3)

tr[si sj sk]=2ieijk

S1 S3 S2 (S1⋅s) (S3⋅s) (S2⋅s)

(S1⋅s) (S2⋅s) (S3⋅s)

S1 S3 S2

(S3⋅s) (S2⋅s) (S1⋅s)

These are not equal

• 3 spins
• spin chirality fi breaking of time reversal symmetry in orbital motion

Spontaneous charge current If electron is coherent

µS1⋅(S2¥S3) spin chirality

GT&Kohno, Phys. Rev. B67, 113316 (2003).

S1 S2 S3 JS2⋅ s JS3⋅ s JS1⋅ s ) ( ) cos( 2

3 2 1 3

S S S J L k L v e j

F F F

¥ ⋅ ˜ ˜ ¯ ˆ Á Á Ë Ê

• =

e

• current

GT&Kohno PRB (2003)

• Quantum dots
• Ferromagnets

Exchange interaction

• Persistent current in nanoscale ring

j = - heJ 3 m S1 ⋅ (S2 ¥ S3) S

X i dw 2p

Ú

f (w)—Im(gx1

r g12 r g23 r g3x' r ) |x'= x

) ( ) cos( 2

3 2 1 3

S S S J L k L v e j

F F F

¥ ⋅ ˜ ˜ ¯ ˆ Á Á Ë Ê

• =

e

spin chirality = non-adiabatic (perturbative) analog of Berry phase

S1⋅(S2¥S3) Úd2x S⋅(∂xS¥∂yS)= F

Continuum limit spin chirality spin Berry phase

• Loss, Goldbart& Balatsky PRL (1990)

Persistent current by spin Berry phase Only in adiabatic limit (strong coupling to a smooth spin structure)

F = Úd2x S⋅(∂xS¥∂yS)

• Our result
• Relation to the adiabatic case

Extension of Berry phase to non-adiabatic case

• Application to operation gates

j >0 j =0 j <0

| j |=XOR[S2, S3]

GT and Garcia, PRL 91, 076806 (2003).

• Unitary operation

R L R R+L

Superposition state

Ex jy

drift

s xy µ S1 ⋅ (S2 ¥ S3)

• Hall effect due to local persistent current

Electric field

• Frustrated magnets
• finite local spin chirality

Local orbital motion of electron Anomalous Hall effect

• Exchange interaction

¢ H = JS

X SX ⋅(c † r

s c)X

• Hall conductivity
• Kubo formula

s xy µ J 3t 2c0

c0 = 1

3N S

Xi

SX1 ⋅(SX2 ¥SX3) (a ¥ b)z ab ¢ I (a) ¢ I (b)I(c)

• total chirality c0

t: lifetime（impurity） 3rd order in J

I(r) = sinkFr kFr e-r / 2l

X1 X2 a X3 b c

geometrical weight RKKY

Finite c0 Hall effect

GT&Kawamura J.Phys. Soc.Jpn. (2002) cf：Kondo’62

• Adiabatic limit

Extension of Berry phase effect to non-adiabatic case

) (

2

S S S x d

y x xy

r r r ∂ ∂ ¥ ⋅ = F

Ú

Ye et al., PRL (1999).

• Magnetic vortex

x y

eiFxy

Berry phase

Hall effect

Adiabaticity was believed to be essential

• GT&Kawamura (2002)
• Quantiztion of Hall conductivity in the adiabatic limit

Ohgushi,Murakami & Nagaosa (2000)

s xy = e 2

h a

dx

Ú

1 4p S⋅ (∂xS ¥∂yS) = e 2 h na

aµ(Jt)3(l/kFa2)

If spin texture is smooth （compared with mean free path l ）

a: vortex size

• Hall conductivity as a topological invariant

n: integer

Coupling factor Topological invariant Quantization occurs even in the non-adiabatic case

GT,Yamanaka &Onoda (2003) Shinjo et al., (2000)

in the perturbative case

• experiment
• Frustrated ferromagnets

Nd2Mo2O7

Taguchi et al., Science (2001)

sxy

pyrochlore

• spin glass systems

La1.2Sr1.8Mn2O7 Chun et al. ('00) reentrant spin glass below Tgª40K

Rs

T

Tg

ferro spin glass new contribution in spin glass phase

T

spin Josephson effect Summary spin chirality

• spontaneous persistent current
• anomalous Hall effect
• 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