Persistent spin current

in

mesoscopic spin ring

Ming-Che Chang Dept of Physics Taiwan Normal Univ

Jing-Nuo Wu (NCTU)

Min-Fong Yang (Tunghai U.)

A brief history

• persistent current in a metal ring (Hund, Ann. Phys. 1934)

• related papers on superconducting ring• Byers and Yang, PRL 1961 (flux quantization)• Bloch, PRL 1968 (AC Josephson effect)

• persistent current in a metal ring

• Imry, J. Phys. 1982

• diffusive regime (Buttiker, Imry, and Landauer, Phys. Lett. 1983)

• inelastic scattering (Landauer and Buttiker, PRL 1985)

• the effect of lead and reservoir (Buttiker, PRB 1985 … etc)

• the effect of e-e interaction (Ambegaokar and Eckern, PRL 1990)

• experimental observations (Levy et al, PRL 1990; Chandrasekhar et al, PRL 1991)

• electron spin and spin current

• textured magnetic field (Loss, Goldbart, and Balatsky, PRL 1990)

• spin-orbit coupling (Meir et al, PRL 1989; Aronov et al, PRL 1993 … etc)

• FM ring (Schutz, Kollar, and Kopietz, PRL 2003)

• AFM ring (Schutz, Kollar, and Kopietz, PRB 2003)

• this work: ferrimagnetic ring

sp

inch

arge

Persistent charge current in a normal metal ring

n nn n

E Ee eI v

L L k

I

/01/2-1/2

Smoothed by elastic scattering… etc

Persistent current

Similar to a periodic system with a large lattice constant

RL=2R

Phase coherence length

0

2kL kL

… …

Metal ring in a textured B field (Loss et al, PRL 1990, PRB 1992)

R

21

2 B

pH eA B

m R

After circling once, an electron acquires

• an AB phase 2πΦ/Φ0 (from the magnetic flux)

• a Berry phase ± (1/2)Ω(C) (from the “texture”)

B

C

Ω(C) Electron energy:

22

20 0 0

,42

An z z Bn B

mR

Persistent charge and spin current (Loss et al, PRL 1990, PRB 1992)

1( )

1 = ln

n

n

A A

eI n e v n

L Z

FZ

1

= ln2 2

n

s zn

eI n s v n

L Z

FZ

e e

0 4

1/ , (1/ ) lnkT F Z

Ferromagnetic Heisenberg ring in a non-uniform B field

(Schütz, Kollar, and Kopietz, PRL 2003)

1 21

/

2

/

ˆ ˆ

ˆ

( )

i

i i i

i i i

i B i

i i

S S S

S S e S

h g B

e

m

// // // //

/

2

/

1ˆ ˆ ˆ ( )

2

1 + ( )

2

' ( )ˆ

ij i j i j i i i

ij

ij j j i

i j

ii j

H J m m S S h m S H O S S

J S S H O S

S H OJ S m h S

Large spin limit, using

Holstein-Primakoff bosons:

//

2 ; 2

i i i

i i i i

S S b b

S Sb S Sb

ˆ// ( ) with an error of order (1)i im S O

'

1 1

2 2

1 2 2 1

ˆ ˆ

1

0

1ˆ ˆor

Im( )=0 the "connection

1

ˆ ˆ 1

"

i j i j

i j

i j i j

i j i j i j

i j i

i j

j i j

e e m m

e e

e e e e

e e e e m m O

e e

N

e e e e m m

Transverse part

mi mi+1

2 21//i ie e

1ie 1

1ie

Longitudinal part to order S,

// //

† † †ˆ ˆ ˆ 2

classical

ij i i j j i ii j i ib b b b b b

H H

SJ m m h m

1 2 1 21 2 1 2

1ˆ ˆ ˆ ˆ

2 ij i i i i j j j jH J S e S e S e S e

Choose the triads such that Then,

2 2 ˆ ˆ//i j i je e m m (rule of “connection”)

1 2ˆ ˆ ˆi i ie e ie

Anholonomy angle of parallel-transported e1

= solid angle traced out by m

1 11

11

ˆ ˆor = Im log

N

i i i ii

N

i ii

e e

Ω

m

ˆLet be the rotation angle (around )

betwen these two, then

ij im

1 1 2

2 1 2

ˆ ˆcos sin or,

ˆ ˆsin cos

ij

ij

i

i ij i ij i i i

i

i ij i ij i i i

e e e e e e

e e e e e e

Local triad and parallel-transported triad

Gauge-invariant expression

1 1initial finale e

mi mi+1

2 21//i ie e

1ie 1

1ie

† †

1 1†

//

11

//

2

exp

.

= /

.

i i

SW classical

i

i i i

i i

ii

i

H H H H

JS b b b b

b b h c

h

JS i

N

Hamiltonian for spin wave (NN only, Ji.i+1 ≡ － J)

Choose a gauge such that Ω spreads out evenly

, 2 1 cos

wher 2

2e

SW k k k kk

H h b b JS ak

ka nN

Persistent spin current

Magnetization current

2

For a mesoscopic ring

2with ,Bk T JS

N

For , /m Bh I g kT T

Schütz, Kollar, and Kopietz, PRL 2003

1ˆ( ) SWs i i i

FI J m S S

• Im vanishes if T=0 (no zero-point fluctuation!)

• Im vanishes if N>>1

/=

1k

kB Bm s h kT

k

vg gI I

L e

sin

cosh 2 / cosB

m

gI kT

h

ka

ε(k)

N

Experimental detection (from Kollar’s poster)

• measure voltage difference ΔV at a distance L above and below the ring

• magnetic field

• temperature T J

h

Estimate:

L=100 nm

N=100

J=100 K

T=50 K

B=0.1 T

→ ΔV=0.2 nV

effP v M

33

0

3

1 '( ) ' ( ') ( ')

4 '

m

r rr d r v r M r

r r

v M d r I dr

Antiferromagnetic Heisenberg ring in a textured B field (Schütz, Kollar, and Kopietz, PRB 2004)

• half-integer-spin AFM ring has infrared divergence (low energy excitation is spinon, not spin wave)

• consider only integer-spin AFM ring. need to add staggered field to stabilize the “classical” configuration (modified SW)

for a field not too strong

Large spin limit

v

(and 0)H

(S. Yamamoto, PRB 2004)

Ferrimagnetic Heisenberg chain, two separate branches of spin wave:

• Gapless FM excitation well described by linear spin wave analysis

• Modified spin wave qualitatively good for the gapful excitation

Ferrimagnetic Heisenberg ring in a textured B field (Wu, Chang, and Yang, PRB 2005)

• no infrared divergence, therefore no need to introduce the self-consistent staggered field

• consider large spin limit, NN coupling only

Using HP bosons, plus Bogolioubov transf., one has

where

/ 1B AS S

Persistent spin current

At T=0, the spin current remains non-zero

/ 1B AS S

2 B Ag JS

N

Effective Haldane gap21

4

100, 0.8

(nearly sinusoidal

0.020

0.015

0

0.

)

/

.00

010

5

A

N

T JS

FM limit

1, 2 / N

AFM limit

1, 2 / N

Clear crossover between 2 regions

2

: spin correl

1/

4

ation length

a

System size, correlation length, and spin current (T=0)

no magnon current

Magnon current due to zero-point fluctuation

0.020

0.015

0.8

/

0.005

0.010AT JS

0

100

/ 2 0.01A

N

h JS

0 / 2 0.05Ah JS

Magnetization current assisted by temperature

Assisted by quantum fluctuation (similar to AFM spin ring)

• At low T, thermal energy < field-induced energy gap (activation behavior)

• At higher T, Imax(T) is proportional to T (similar to FM spin ring)

Issues on the spin current

• Charge is conserved, and charge current density operator J is defined through the continuity eq.

• The form of J is not changed for Hamiltonians with interactions.

• Spin current is defined in a similar way (if spin is conserved),

11

N

l ll

H J S S

1 1 1

0z

zll

z x y y xl l l l l l l

Sj

t

j J S S S S JS S

• Even in the Heisenberg model, Js is not unique when there is a non-uniform B field. (Schütz, Kollar, and Kopietz, E.Phys.J. B 2004).

• Also, spin current operator can be complicated when there are 3-spin interactions (P. Lou, W.C. Wu, and M.C. Chang, Phys. Rev. B 2004).

• Beware of background (equilibrium) spin current. There is no real transport of magnetization.

• Spin is not always conserved. Will have more serious problems in spin-orbital coupled systems (such as Rashba system).

However,

Other open issues:

• spin ring with smaller spins

• spin ring with anisotropic coupling

• diffusive transport

• leads and reservoir

• itinerant electrons (Kondo lattice model.. etc)

• connection with experiments

• methods of measurement

• any use for such a ring?

Thank You !