Download Document

Full Current Statistics in
Multiterminal Mesoscopic
Conductors
Dmitry A. Bagrets
Universität Karlsruhe
Collaboration: Yuli V. Nazarov
TU Delft
Technische Universiteit Delft
Outline
• What is Full Current Statistics? (N-terminal
• Circuit theory of FCS
(non-interacting regime)
• Master equation approach to FCS
(strong Coulomb blockade limit)
• Weak charge quantization and FCS
(weak Coulomb interaction)
• Conclusions
case)
Full Current Statistics
3-terminal case
N1 + N 2 + N 3 = 0
Current conservation
number of electrons transferred
Pt (N 1 , N 2 ) = full info
FCS in N-terminal case
• Generating function
exp{- S(  k )} =
 P (N
t
1
, N2 , N3 ) exp( i  k N k )
Nk
Action
k
Probability
• Currents
Ik
χ3
counting fields
e S
=
t  χk
• Noise correlations
e2  2 S
I k (t )I m (0) ~
t  χk  χm
• Higher order correlations
χ1
χ2
3-terminal circuit
Keldysh Method
• Effective Hamiltonian
Hˆ χ = Hˆ + 12  χ k Iˆk
+

χ(
τ
)
=
χ(τ
)
Keldysh path
n
• Generating function
exp{- S ( χ ) } = e
i Hˆ -χ t -i Hˆ χ t
e
( measuring device [spin] )
 Tτ exp[ 12 i  d  χ k (  )Iˆk ( )]
C
k
• Conventional Green function technique
τ
Circuit Theory
Non-interacting systems
Conductance
1
e2
R

• Semiclassical approach
-2f k 
 1 - 2f k
=

-2(
1
f
)
2
f
1
k
k


• Boundary conditions
G
• Gauge transform
Gk  e
(0)
k
i χ k τ 3 /2
(0) -i χ k τ 3 /2
k
G e
Minimal Action Principal
• Total action
S ( χ )   ij S ij ( χ )
• Action
2
i
G =1
Normalization condition
d
Sij ( χ )   
Tr ln[1 + 14 Tn(ij) ({Gi ,G j } - 2)]
2
n
• Saddle point
I
ik
0
( “1st Kirchhoff’s rule“ )
k
• Matrix current
( “2nd Kirchhoff’s rule“ )
Tn [Gi ,G j ]
d
I ij   
2 4 + Tn ({Gi ,G j } - 2)
n
Chaotic Quantum Dot
1
e2
R

• Connectors
1-tunnel
Tn << 1
2-diffusive
g0
1
ρ T  =
2 T 1 -T
3-ballistic
Tn = 1
I 1 = I 2 = -I 3 /2
Big current fluctuations!
Coulomb Blockade Dot
1
e
R
2

Coulomb blockade system
E
kT
e2
EC 
2C Σ
Markov process
• Master Equation
 t p(t) = - Lˆ p(t)
γ n   Γ m n
Lˆmn  δmn γ n  Γ mn ,
• “Orthodox” theory
(k)
Γ n±1
n
(
Rates
mn
Q = -e n
- charge quantized ! )
(k)
Δ En±1
1
n
 2
(k)
e Rk 1 - exp[- Δ En±
1n / k BT ]
• Limits of validity
1
e2
Rk

( No co-tunneling ! )
FCS + Master Equation
(1)  i χ1
ˆ
Γ e
Γˆ (k)
Time arrow
τn
τ
τ n+1
• Effective Master equation
 t p(t) = - Lˆ ( χ ) p(t)
• Generating function
(2)  i χ2
ˆ
Γ e
Γˆ (3)e i χ3
Lˆ ( χ )  γˆ   Γˆ (k)exp(i χ k )
k
S ( χ )   ln e -L( χ )t0 = t0 Λ min ( χ )
Coulomb Blockade Dot
Big current fluctuations!
Coulomb blockade
1 - U 1 = 1.25 e C Σ
2 - U1 = 2 e C Σ
3 - U1 = 4 e C Σ
4 - U 1 = 10 e C Σ
5 - no interaction
Weak Charge Quantization
Panyukov, Zaikin, ’91
Flensberg ’93, Matveev ’95
Nazarov ‘99
Free energy
F  F0 + EC 
n>0
CgVg 

fncos  2n

e


e2
G >>
πh
EC ~ ECexp  -  G GQ  - “Effective” charging energy
1/2
 2
 /8
-Tunnel junctions
- Diffusive contacts
Coulomb Island
g0   Tn[ k ] >> 1 - Conductance
n,k
EC  e 2 2C 
- Charging energy
ETh  g0
- Thouless energy
1/ RC  g0 EC - inverse RC-time
• Phase

= eV(t)
t
• Relations between energy scales
EC  ~ rs  L  F  >> 1
2
ETh  1  RC  min{ D/L2 , v F /L}
“Effective” Keldysh Action
exp[ S (  )]   D exp(   S[G,  ]d )
C
Interaction
-1
S =  Sk ({G[k]
,
G
})
+
iπ
δ
Tr{( i t -  )G} + S
χ
k
• Electrons
1
1 [k]


[k]
Sk = - Tr ln  1 + Tn {G χ , G} - 2 
2
4




• Electrostatic energy


i -1
S  Ec  d  2   2
2

G( t1 ,t 2 ) = TC ψ( t1 ) ψ + ( t 2 )
G2  1

Quantum corrections
• Weak localization correction:
( Random-matrix theory )
!
 g / g0 ~ 1/ g0
• Interaction correction
Inelastic ( eV  1/  RC )
2

 g / g0 ~ O  g0 EC /eV  


Elastic ( ETh << eV << 1/  RC )
g~
-2 n Tn (1 - Tn )

n
Tn
log  g0 EC /eV 
Divergent ! ?
 ~ 1/ | ω | g0
Dissipation!
Renormalization Group
• “Poor man’s” scaling
• RG Equations
“Running” cut-of
( 1-Loop order )
dTn
Tn (1 - Tn )
=2
d lnE
 m Tm
• RG diverges: Metallic saddle point is unstable !
At
E  EC , 1/ g (E )  
• Charging energy: Equivalence to instanton calculation !
 EC  1
[0]
ln 
~
ln(
1
-T


n )
 g0 EC  2 n
• Large phase fluctuations
Charge quantization !
[Q,f]=ie
Onset of Coulomb Blockade
• Critical conductance
 EC 
gC ~ α ln 

δ


-1
δ
Full Current Statistics
• Connectors
1-tunnel
Tn << 1
2-diffusive
g0
1
ρ T  =
2 T 1 -T
3-ballistic
Tn = 1
I 1 = I 2 = -I 3 /2
Conclusions
• The theory of FCS in N-terminal circuits:
- Non-interacting regime
- Coulomb blockade regime
- Weakly interacting regime
• Evaluation of probability of big current fluctuations
2
e
• Suppression of current fluctuation at 1
R 
• Renormalization of transmission eigenvalues
•Crossover to Coulomb blockade
regime at 1 << g < gC ~ α -1ln  EC 
 δ 