Electronic States and Transport in Quantum dot
Ryosuke Yoshii
YITP
Hayakawa Laboratory
My research field: mesoscopic systems
Microscopic system
Macroscopic system
Mesoscopic system
Intermediate (Meson,Mesopotamia, etc)
~100nm
Why mesoscopic?
In recent years, we can fabricate
nano scale semiconductors.
Various tunable parameters
We can design quantum device
1.1 Small box for electron (Quantum dot)
Nano-fabrication of semiconductors
(1) Two-dimensional electron gas in semiconductor
hetero-structures: Epitaxial growth, layer by layer
You can make the sheet of electron gas
(2) Small metallic electrodes: Electron-beam lithography
InGaAs
2DEG
Quantum dot (Nanoscale semiconductor device)
Quantum dots are quantum-mechanical boxes
First layer
Second layer
Gate
Electron gas
Energy levels are discretized
Tunable by VG
We can control electro-statical
potential with gate voltage.
VG
Quantum dot (Nanoscale semiconductor device)
Quantum dots are quantum-mechanical boxes
First layer
Second layer
Gate
Electron gas
Negative voltage
Energy levels are discretized
Tunable by VG
We can control electro-statical
potential with gate voltage.
VG
Quantum dot (Nanoscale semiconductor device)
Quantum dots are quantum-mechanical boxes
Negative voltage
First layer
Second layer
Negative voltage
Gate
Energy levels are discretized
Tunable by VG
We can control electro-statical
potential with gate voltage.
Quantum dot
VG
Tunnel effect
source
drain
source
drain
Large potential for electrons
Gate voltage
VG
Energy of conduction electron < potential
Transport
lead
VG
When energy level in the quantum
dot lies between Fermi energy of
two leads.
This process is allowed.
lead
When energy level in the quantum
dot does not lie between Fermi
energy of two leads.
This process is forbidden.
lead
lead
source
drain source
drain source
drain
VG
“tunnel”
Coulomb oscilation
I
Spin of electron
Small magnet
VG
In the region between two peaks
Number of electrons in Quantum dot is fixed
“Tunable one by one” by VG
# of electrons is 2
Quantum point contact
The electron number in QD is countable as resistance
Gustavsson et al, PRL 2006.
We can detect the dynamics of # of electrons
Full counting statistics
S (  )  ln  Pt0 ( N )eiN
N
 k S  
Ck 
k
 i   0
2.Our research topics
2.1 Kondo effect
Coulomb valley
G
VG
Energy does not conserve
2nd order perturbation for tunnel process
“Cotunneling”: more than one electron participates.
2 0  U
Spin flip process
VR
0
1
VL

E
0
 0
VL
1 
VR
E
Result in screening effect
1 
2
2  1
Effective Hamiltonian J  VL  VR       0
E
E 

H   k a†k ak  2J  S　 (s)k ,k  Antiferromagnetic

k
k ,k 

dJ
 2J 2
d ln D
spin in quantum dot is screened by conduction electrons:
Kondo singlet state
(Many-body state)

Kondo temperature TK: binding energy of the
Kondo singlet state

T<<TK: Kondo singlet state is formed.
Resonant tunneling through the singlet state.
G
spin
spin
VG
Conductance increases with lowering the temperature
2.2 Quamtum pumping
Quamtum master equation with couting field
t system ( , t )  R(t ) system ( , t ) : parameter of system
Cumulant generating function
T. Sagawa and H. Hayakawa, PRE 84, 051110 ‘11




C
†
Total derivative with

Fsystem (  , t )    l , d
A, B  T rA B

l left

right
 surface
terms 
eigen vector for largest eigenvalue

R(t )
This geometrical representation
“adiabatic pump for physical quantity.?”
Can we make adiabatic pump with changing the observer’s
parameters?
Impossible for lead+dot in case of spinless fermion
T. Yuge, T. Sagawa, H. Hayakawa, unpublished
With localized spin? With many body effect?
3.Results
Quantum master equation for Kondo model
H   k a†k ak  J  S　σ
   a†k ak   in second order
k ,k 
k

 (t )
t system ( , t )  R system ( , t )
  (  , t ) 

 system (  , t )  


(

,
t
)
 

 a b

R  J   
b a
1 d
 i  
1  f   f    6  e
a 

4 2  ,  L, R
d
 i      
1  f   f   e
b

2  ,  L, R

 (t )
2



   

e
i       

Eigen vectors
b
2 a


R (t )  J   
b
a
  a | b |
b 

 (t )   
| b |
d
b
2

 (t )
l
b
1  f   f  e



,  L , R




|b|
i       

Fsystem (  , t )    l , d  surface terms 
C
Fsystem (  , t )
i
Remarks
 0
 l 
   
C  i



, d
 0 
 0

F is independent of J
F depend on b, but a
Case of
 L   R   ,  L   R   
 N N N N
  N L   N R   N L  N R 
L
R
L
R
1

l (t ) 
b | b |
2 |b|
1 b 
 


2 | b | | b |
d
 i  






b
1

f

f

e



2  ,  L, R
i (11) d
1  f   f   
be
 2  ,
 L , R
i (11)

 (t )

b / | b | e
Fsystem (  , t )
i
 0
 l 
   
C  i

 0

, d
 0

   


0


In general,


 (t )
1 b 
  is independent of 
2 | b |  | b |   0

Fsystem (  , t )
i
 0
 l 
   
C  i


0
, d
 0 
 0

If we add the potential scattering term
H   k a†k ak  J S　σ
   a†k ak    K
k ,k 
k

 (t )
R
 a b

 J   
b c 
2
†
a
 k ak
k ,k ,
ac
In progress
4.Summary
 We have examined the master equation for
Kondo model
 We have shown the quantum pumping is
independent of J
 We have shown that the geometric term of
the cumulant generating function only
depends on the off diagonal part of QME
 We have shown that the Kondo model
without potential scattering term doesn’t
produce the quantum pumping