Dynamical response of
nanoconductors: the example of
the quantum RC circuit
Christophe Mora
Collaboration with Audrey Cottet, Takis Kontos,
Michele Filippone, Karyn Le Hur
Mesoscopic and nanoscopic physics
Outline of the talk
Three transverse concepts in mesoscopic physics
1) Quantum coherence (electrons are also waves)
2) Interactions (electrons are not social people)
3) Spin degree of freedom
Outline of the talk
I. Mesoscopic Capacitor (Quantum RC circuit)
II. Adding Coulomb interactions
III. Giant peak in the charge relaxation resistance
Q ( )
V g ( )

C0
1  i C 0 R q
Mesoscopic capacitor or the
quantum RC circuit
D. Darson
Gwendal Fève, Thesis (2006)
The quantum RC circuit
The Quantum RC circuit
Dot
Lead: Single-mode
B
Spin polarized
QPC
V g ( )
Linear response theory
e  C ( ) 
2
Q ( )
V g ( )
Q ( )
 C ( t )  i  ( t ) [ Nˆ ( t ), Nˆ ( 0 )]
 e  C 0 1  i     O ( )
2
Classical circuit
Gate
AC excitation
2
C0
Rq
~
Nˆ  Q / e
Vg
Low frequency

 C 0 1  i  C 0 R q   O ( )
response
V g ( ) 1  i  C 0 R q
C0
Gabelli et al. (Science, 2006)
2
Fève et al. (Science, 2007)
Mesoscopic capacitor
Mesoscopic Capacitor
VP
I (t )
e
Gabelli et al. (Science, 2006)

Fève et al. (Science, 2007)
l < mm
D. Darson
VG
VP
Meso, ENS
Quantum dot in a microwave resonator
dispersive shift of the resonance: capacitance
broadening (dissipation): resistance
Delbecq et al. (PRL, 2011)
Chorley et al. (PRL, 2012)
Frey et al. (PRL, 2012)
Microwave Resonator
Mesoscopic Capacitor
Quantum dot in a microwave resonator
dispersive shift of the resonance: capacitance
broadening (dissipation): resistance
Frey et al. (PRL, 2012)
Delbecq et al. (PRL, 2011)
Energy scales
Energy scales
Q ( )
Experiment on the
meso. capacitor, LPA ENS
Charging Energy
Level spacing
EC 
 
V g ( )
e
2
 1K
2Cg
 vF
 2K
( C   1 fF )
L
Dwell time
 C 0 1  i  C 0 R q 
  C 0 Rq 
Excitation frequency   1 GHz
l
vF

10 m m
5
qq 10 m . s
1
 qq 10 ps
Differential capacitance
Differential capacitance
Opening of the QPC:
from Coulomb staircase
to classical behaviour
C0 
Cottet, Mora, Kontos (PRB, 2011)
 Q
V g
Electron optics
Electron optics
Buttiker, Prêtre , Thomas (PRL, 1993)
Similar to light propagation in a dispersive medium
 ( )
Wigner delay time
 
V (t )
C0 
S ( )  e
i (  )

re
2 i / 
1  re
2 i / 

 2 RqC 0

e
2

h
Rq 
h
2e
2
Ringel, Imry, Entin-Wohlman (PRB, 2008)
Experimental results
Experimental results
Gabelli et al. (Science, 2006)
Fève et al. (Science, 2007)
Rq 
h
2e
Oscillations
2
Adding Coulomb interactions
Pertubative approaches
Weak tunneling
t
 d  eVg

H 
k
V g ( )
B

k
c c k
k ,  L / D


2

ˆ
 E C N  t   c kL c k ' D  h .c .    d Nˆ


 k ,k '

Strong tunneling (weak backscattering)
H   i v F R  x 
*
R
 E C Nˆ


2
*
 r  R (  L ) R ( L )  h .c .   d Nˆ
Cg
r  1
Vg
Hamamoto, Jonckheere,
Kato, T. Martin (PRB, 2010)
Mora, Le Hur (Nature Phys. 2010)
Universal resistances
Universal resistances
Results for small frequencies
Small dot
Im  C ( )     C 0 
Rq 
h
2e
2
Mora, Le Hur (Nature Phys. 2010)
Large dot
2
Im  C ( )  2    C 0 
Rq 
2
h
e
2
Confirms result for finite dot, new result in the large dot case
Hamamoto, Jonckheere,
Kato, T. Martin (PRB, 2010)
Kondo mapping
Matveev (JETP, 1991)
C0
2C g D

1
1 4N0
2
divergence for N 0  1 / 2
0
1
Charge states
Mapping to the Kondo hamiltonian (0 and 1 -> Sz = -1/2,1/2)
1
z
ˆ
N  S
2
J  t
h  E c (1  2 N 0 )
Correspondance
K ( )   zz ( )
Korringa-Shiba relation
Korringa-Shiba relation
Shiba (Prog. Theo. Phys., 1975)
At low frequency
Garst, Wolfle, Borda, von Delft, Glazman (PRB, 2005)
Im  zz ( )  2  Re  zz ( 0 )   Im K ( )  2  Re K ( 0 ) 
2
and therefore
2
Rq 
h
e
2
Dominant elastic scattering
Even in the presence of strong Coulomb blockade
t   / E C
 ( )
Energy conservation
at long times (low frequency)
Probability of inelastic scattering process small  / E C 
Usual Fermi liquid argument
of phase-space restriction
Aleiner, Glazman (PRB,1998)
2
Fermi liquid approach
Weak tunneling regime
0
Power dissipated under AC drive eV g ( t )  eV g   1 cos(  t )
Original model
H  ...   d Nˆ
P 
Linear response theory
Low energy model
H 
1
2
 1  Im  C ( )
2
  k c k c k  K (V g )  c k c k '


k
K (V g )
k ,k '
related to  C 0 through Friedel sum rule
Rq 
h
2e
2
P 
1
2
1  C0  
2
2
Filippone, Mora (PRB 2012)
2
Giant peak in the charge relaxation
resistance
Giant peak in the AC resistance
M. Lee, R. Lopez, M.-S. Choi, T. Jonckheere,
T. Martin (PRB, 2010)
Peak comes from Kondo singlet breaking
Fermi liquid approach
4

h 
U 

2

Rq 
1    F ( y ) ( x ) 
2 

4e 
 

y  1
2 d
U
Filippone, Le Hur, Mora (PRL 2011)
Fermi liquid approach
Perturbation theory (second order)

 C   (r )

 C   (r )
Charge susceptibilities
n
n
n
C  
n
Im  C ( )     C    C  
 d
2
C  C  C ,
Charge-spin modes
2
2
h  C  m

Rq 
2 
2
4e 
C
C  




2
 m  C  C
 C remains small
Kondo limit
m  f B / TK
 d

charge frozen
Conclusions
Conclusion
Prediction of scattering theory is recovered with an exact
treatment of Coulomb interaction
Rq 
h
2e
2
Novel universal resistance is predicted for a large cavity
Rq 
h
e
2
Peak in the charge relaxation resistance for the
Anderson model
Filippone, Mora (PRB 2012)
Mora, Le Hur (Nature Phys. 2010)
Filippone, Le Hur, Mora (PRL 2011)
Anderson model
Perturbation theory (second order)
M. Lee, R. Lopez, M.-S. Choi, T. Jonckheere, T. Martin (PRB, 2010)
Rq 
h
4e
2
At zero
magnetic field
Monte-Carlo calculation
Fermi liquid approach
Perturbation theory (second order)

 C   (r )

 C   (r )
Charge susceptibilities
n
n
n
C  
n
Im  C ( )     C    C  
 d
2
C  C  C ,
Charge-spin modes
2
2
h  C  m

Rq 
2 
2
4e 
C
C  




2
 m  C  C
 C remains small
Kondo limit
m  f B / TK
 d

charge frozen
Fermi liquid approach
Perturbation theory (second order)
4


h
U 

2
1    F ( y ) ( x ) 
Rq 
2

4 e 
 

y  1
M. Lee, R. Lopez, M.-S. Choi, T. Jonckheere, T. Martin (PRB, 2010)
Filippone, Le Hur, Mora (PRL 2011)
2 d
U