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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 !