Transcript 無投影片標題 - National Chiao Tung University

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
charge
• 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
• 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
Persistent charge current in a normal metal ring
Similar to a periodic
system with a large
lattice constant
L=2R
R
kL  kL  2
Phase coherence length
…

0
…
Persistent current
In  
E
e
e En
vn  
 n
L
L k

I
-1/2
1/2
/0
Smoothed by elastic scattering… etc
Metal ring in a textured B field (Loss et al, PRL 1990, PRB 1992)
2
1  p

H

eA
   B B  

2m  R

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”)
C
Ω(C)
Electron energy:
B
 n
2

A
 

n



   z  B B,
z
2 
2mR 
0
0 
2
 

 0 4
Persistent charge and spin current
(Loss et al, PRL 1990, PRB 1992)
1
e  n
I   n (e)v n
L n
Z
1 
F
=
ln Z  
  A
 A
  1/ kT , F  (1/  ) ln Z
1
e  n
I s   n sz v n
L n
Z

F
=
ln Z 
2e  
2e  
 

 0 4
Ferromagnetic Heisenberg ring in a non-uniform B field
(Schütz, Kollar, and Kopietz, PRL 2003)
Si  Si// mˆ i  Si
Si  Si1eˆi1  Si 2 eˆi2
(hi  g  B Bi )
Large spin limit, using
Holstein-Primakoff bosons:
Si//  S  bi bi
Si  2 Sbi ; Si  2 Sbi
1
J ij mˆ i  mˆ j Si// S //j   hi mˆ i Si// H //  O (S 2  S )

2
1
+  J ij Si  S j
H   O(S )
2
 H


    J ij S /j/ mˆ j  hi   Si
i  j

H '  O( S )
ˆ i (  Si ) with an error of order O(1)
// m
//
H //  H classical
Longitudinal part
to order S,

S
J ij mˆ i  mˆ j  bi†bi  b†j b j    hi mˆ i bi†bi

2
Transverse part
H 
1
J ij  Si1eˆi1  Si 2eˆi2    S j1eˆ1j  S j 2eˆ2j 

2
Choose the triads such that
Then,
ˆi m
ˆj
ei2 // e2j  m
(rule of “connection”)
ei1  e1j  mˆ i  mˆ j
ei2  e 2j  1
ei2 // ei21
e e  e e  0
1
i
or
eˆi  eˆi1  ieˆi2
2
j
2
i
1
j
1
ei  e j  ei  e j  mˆ i  mˆ j  1  O  
N
ei  e j  ei  e j  mˆ i  mˆ j  1
Im(ei  e j ' )=0  the "connection"
mi
ei1
mi+1 ei11
Local triad and parallel-transported triad
ˆi )
Let ij be the rotation angle (around m
ei2 // ei21
betwen these two, then
i
eˆi1  cos ij ei1  sin ij ei2 or, eˆi  e ij ei
eˆi2  sin ij ei1  cos ij ei2
 i
eˆi  e ij ei
mi
ei1
mi+1 ei11
1
einitial
 e1final
Anholonomy angle of parallel-transported e1
= solid angle traced out by m
N
   i i 1  i 1i 
Ω
i 1
N
or = Im log  eˆi  eˆi1
i 1
Gauge-invariant
expression
m
Hamiltonian for spin wave
(NN only, Ji.i+1 ≡－J)
Persistent spin current
F
I s ()   SW   J mˆ i  Si  Si1

Magnetization current
v
g B
g
Im 
I s =  B    h k/ kT
L k e k
1

//
H SW  H //  H classical
 H
 2 JS  bi†bi  h bi†bi
i


 JS  bi†1bi exp i i i 1  i 11    h.c.
i
= / N
Choose a gauge such that
Ω spreads out evenly
H SW     k  h bkbk ,  k  2 JS 1  cos ka 
k
2
where ka 
N


n



2 

ε(k)
• Im vanishes if T=0
(no zero-point fluctuation!)
• Im vanishes if N>>1
For a mesoscopic ring
 2 
with kBT    JS 
 ,
N


g
sin 
I m   B kT
cosh 2 h /   cos 
2


N
ka

For h  , I m  g B kT /

T
Schütz, Kollar, and Kopietz, PRL 2003
Experimental detection (from Kollar’s poster)
Peff  v  M
• measure voltage difference ΔV
at a distance L above and below
the ring
• magnetic field h  
• temperature
  T  J
Estimate:
L=100 nm
N=100
J=100 K
T=50 K
 (r ) 
1
4 0
r r '
 d r ' v (r ')  M (r ')  r  r '
3
 v M d r  I dr 
3
m
3
B=0.1 T
→ ΔV=0.2 nV
Antiferromagnetic Heisenberg ring in a textured B field
(Schütz, Kollar, and Kopietz, PRB 2004)
Large spin limit
• 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
v
(and H  0)
Ferrimagnetic Heisenberg chain,
two separate branches of spin wave:
(S. Yamamoto, PRB 2004)
• 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
  SB / S A  1
Using HP bosons, plus Bogolioubov transf.,
one has
where
Persistent spin current
  SB / S A  1
N  100,   0.8
(nearly sinusoidal)
T / JS A 
At T=0, the spin current remains non-zero

2 g  B JS A
N
Effective Haldane gap
1  2
 
4
0.020
0.015
0.010
0.005
System size, correlation length, and spin current (T=0)
1  2
 
 a /
4
AFM limit
  1, 
 : spin correlation length
2 / N
Magnon current due to
zero-point fluctuation
Clear crossover
between 2 regions
FM limit

1, 
2 / N
no magnon current
Magnetization current assisted by temperature
N  100
h0 / 2 JS A  0.01
  0.8
0.020
T / JS A 
0.015
0.010
0.005
h0 / 2JS A  0.05
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),
N
H  J  Sl  Sl 1
l 1
However,
Slz
 jlz  0
t
jl z  J  Slx Sly1  Sly Slx1   JS l S l1
• 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 3spin 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).
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 !