From Kondo problem to Transport
Through a Quantum Dot
Yupeng Wang
Institute of Physics, CAS, Beijing
2005-7-1, IOP
Collaborators:
Zhao-Tan Jiang, Ping Zhang,
Qing-Feng Sun,
X. C. Xie and Qikun Xue
Outline
I. Basic Issues
II. Dephasing problem through a dot
III. Spin-dependent transport through a dot
IV.Further considerations
I. Basic Issues
What is the Kondo problem?
Conduction electrons +magnetic impurity

H k  J  j  S
N

j 1
1
For a free moment  ~
T
Perturbation theory fails for Kondo
problem
Hk
n
~ n
Tk ~ E0 e
n 1
max  , T 
Tk
1

N 0  J
Tk is the energy scale distinguishing the
strong coupling regime and the weak
coupling regime
Theoretical methods developed from this problem
Poor man’s scaling J*=
Local Fermi-liquid theory
Ximp~Const
Wilson’s numerical RG
Slave boson approach
Gutzwiller variation
Exact solution with Bethe ansatz
Scalar potential in Luttinger liquids
[Kane-Fisher(92), Lee-Toner(90),
Furusaki-Nagaosa(94)]
J&V competing
PRL 77, 4934(96);79, 1901(97)
Some Basic issues of transport through a
quantum dot
A dot coupled to two leads
d  U
Artificial
d
eV
Kondo system!
A.Does the intra-dot Coulomb interaction
induce dephasing? How to test?
B.What’s the transport behavior of a
quantum dot with magnetic leads?
Dephasing is a basic problem in mesoscopic systems
High temperature,
Low temperature,
Macro system
，Mesoscopic
*
*
Which determines a system is macro or mesoscopic
and affects the application of quantum devices
Phonons, temperature and magnetic impurity may induce
dephasing but scattering with fixed phase shift does not.
Experiments showed partial coherence
R.Schuster, et.al. Nature 385, 417 (1997)
A. Yacoby, et.al. Phys.Rev.Lett. 74, 4047 (1995)
Former conclusion in AB-ring:
partial dephasing
incoherent：
coherent：
The direct physical picture for dephasing
,only 1 or 0 electron in the dot
Three second-order processes
coherent
coherent
dephasing
Theoretical result from the
Anderson impurity model
*Partial dephasing
*Asymmetric amplitude
Flux dependent part
of the conductance
Asymmetry
0 electron in the dot
1 electron in the dot
New experiment demonstrated the asymmetry
H. Aikawa, et.al., Phys. Rev. Lett. 92， 176802 (2004).
Now it seems that partial dephasing does exist!
(1)、A clear physical picture
(2)、A predicted asymmetric transmission amplitude
(3)、The asymmetry was
demonstrated in experiment
Our concern
(1)、Is the many-body effect unimportant？
(2)、A static transport consists of a sequential tunneling
processes which can be divided into many secondorder tunneling in different ways!
(1)
(2)
(3)
(4)
(5)
(6)
Coherent！
(1)
(2)
(3)
(4)
(5)
(6)
Incoherent！
(3)、Does the AB amplitude reflect dephasing？
The higher order processes have been discarded!
Reasonable？
*AB ring is a closed and limited system!
Higher-order tunneling important
even
is quite small
t ref
A
Dot
*
* Phase locking
invalid!
AB amplitude is irrelevant
to dephasing! Two-terminal
system is inappropriate to test
dephasing!
For U=0, AB amplitude is
zero but the process is
coherent!
The situation is not clear!
Geometry induces asymmetry?
A multi-terminal system
Z.T. Jiang et al, Phys. Rev. Lett. 93，076802（2004）
The basic idea is to use side-way effect to reduce
higher-order tunneling processes.
Coherence rate：
When higher order processes are unimportant
The model
Non-equilibrium Green’s function method
1、Equation of motion
for dot gr
2、 Dyson and Keldysh equations for
Gr and G<
3、Current and conductance：
4、Electron number in dot is determined self-consistently
Coherence rate
U 0
U 5
4 /   5
U 
Far away from the peak, r=1, coherent!
Close to the peak, higher order important!
(1)、In the limit
tend to 0.
，all higher order processes
For any value of
(2)、For finite
，the first order contains
while the higher orders contain
etc. Distinguishable in the formula!
we have
Multi-terminal to two-terminal:  4 /   0
We get the asymmetric conductance
G | tref ei  tco |2 +Tno
With magnetic field
Even
，
is less than1 !!!
U&B induce dephasing?
U=0 case must be coherent
An adequate description: spin-dependent rate
When
Our Conclusion
• Intra-dot Coulomb interaction
does not induce dephasing!
• The two-terminal AB-ring system is
inappropriate to test the dephasing
effect!
Spin dependent transport
P. Zhang et al, Phys. Rev. Lett. 89, 286803(2002)
Physics World Jan. 33 (2001) by L. Kouwenhoven and L. Glazman
The modified Anderson model
H

 
k , , L , R
 k ak ak    d d d  Ud d d d

 R(d d  d d ) 
Transformation：

 
k , , L , R
d (  ) 
(Vk ak d  H .c.)
1
(c  c )
2
Local density of states of the quantum dot
 ( ) ( )  
1


Im (Gcr )  (Gcr )  (Gcr )   (Gcr ) 
0.4
Parallel (a)
Spin-down
LDOS
0.3
0.2
0.1
0.0
0.2
(b)
LDOS
Antiparallel
0.1
0.0
-8
-6
-4
-2
0
Energy
2
4
6
8

Parallel Configuration，level splitting in the dot：
2
|
V
|
f ( k  )
~
k 
 d   d   ~
 d   k 
k
  2k BT
 1 ~d  
  d   ln
 Re    i
W
2k BT
 2 
2



Local density of states with spin flip process
Parallel configuration
(a)

0.2
0.1
0.0
Antiparallel configuration
(b)

0.2
0.1
0.0
-8
-6
-4
-2
0

2
4
6
8
Linear conductance
2.0
2
Antiparallel
G (e /h)
1.5
T=2
T=0.2
T=0.02
(a)
1.0
0.5
0.0
2.0
1.5
2.0
T=2
T=0.2
T=0.02
(b)
1.5
1.0
2
G (e /h)
Parallel

G↓
0.5
1.0
0.0
-8

-4
Spin-valve
G↑
0
4
8
0.5
0.0
2.0
2
Parallel
G (e )/h
1.5
T=2
T=0.2
T=0.02
(c)
R0
1.0
0.5
0.0
-8
-4
0
d
4
8
Conclusion
• In the mean-field framework, magnetic
resistance is insensitive to the spin
relaxation.
• For the parallel configuration, the spin
splitting of the Kondo resonance peak
can be controlled by the magnetization
and therefore induces spin valve effect
due to the correlation effect.
• The splitting of the Kondo resonance
peak is induced by the intra-dot spin
relaxation.
Further consideration
•The quantum dot array may simulate heavy
fermion systems
•Orbital degeneracy to multi-channel Kondo
effect: detect non-Fermi-liquid behavior with
transport
感谢叶企孙
奖励基金会
Thank You!