Dynamical response of nanoconductors: the example of the quantum RC circuit Christophe Mora...
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Transcript of Dynamical response of nanoconductors: the example of the quantum RC circuit Christophe Mora...
Dynamical response of nanoconductors: the
example of the quantum RC circuitChristophe Mora
Collaboration with Audrey Cottet, Takis Kontos, Michele Filippone, Karyn Le Hur
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
Mesoscopic and nanoscopic physics
I. Mesoscopic Capacitor (Quantum RC circuit)
II. Adding Coulomb interactions
III. Giant peak in the charge relaxation resistance
Outline of the talk
Mesoscopic capacitor or the quantum RC
circuit
Gwendal Fève, Thesis (2006)D. Darson
qg RCi
C
V
Q
0
0
1)(
)(
The Quantum RC circuit
Lead: Single-mode
Spin polarized
Gate
AC excitation
Dot
Classical circuit
)(11)(
)( 200
0
0
ORCiC
RCi
C
V
qg
Vg~0C
qR
Low frequency
response
)(gVB
QPC
The quantum RC circuit
)(1)(
)()( 2
022
Oie
V
Qe C
gC
Linear response theory
eQN
NtNtitC
/ˆ
)]0(ˆ),(ˆ[)()(
Gabelli et al. (Science, 2006) Fève et al. (Science, 2007)
Mesoscopic Capacitor
PV
PV
l < mm
e)(tI
GV
Gabelli et al. (Science, 2006)
Fève et al. (Science, 2007)
Meso, ENS
Quantum dot in a microwave resonator
dispersive shift of the resonance: capacitance
broadening (dissipation): resistance
Mesoscopic capacitor
D. Darson
Delbecq et al. (PRL, 2011) Chorley et al. (PRL, 2012) Frey et al. (PRL, 2012)
Mesoscopic CapacitorQuantum dot in a microwave resonator
dispersive shift of the resonance: capacitance
broadening (dissipation): resistance
Microwave Resonator
Delbecq et al. (PRL, 2011)Frey et al. (PRL, 2012)
Energy scales
GHz
psqqsmqq
m
v
lRC
fFCKL
v
KC
eE
Fq
F
gC
1
10.10
10
)1(2
12
150
2
Charging Energy
Level spacing
Excitation frequency
Dwell time
Experiment on the
meso. capacitor, LPA ENS
Energy scales
qg
RCiCV
Q00 1)(
)(
Differential capacitanceOpening of the QPC:
from Coulomb staircase
to classical behaviour
gV
QC
0
Cottet, Mora, Kontos (PRB, 2011)
Differential capacitance
Electron opticsSimilar to light propagation in a dispersive medium
Buttiker, Prêtre , Thomas (PRL, 1993)
)(tV
/2
/2)(
1)(
i
ii
re
ereS
)(
Ringel, Imry, Entin-Wohlman (PRB, 2008)
Electron optics
h
eC
CRq
2
0
02
22e
hRq
Wigner delay time
Experimental resultsGabelli et al. (Science, 2006)
Fève et al. (Science, 2007)
22e
hRq
Oscillations
Experimental results
Pertubative approaches
NchcctNEccH dkk
DkkLCDLk
kkkˆ..ˆ
','
2
/,
Weak tunneling
gd Ve
Strong tunneling (weak backscattering)
NchLLr
NEviH
dRR
CRxRF
ˆ..)()(
ˆ
*
2*
)(gVB
gC
gV1r
t
Hamamoto, Jonckheere,
Kato, T. Martin (PRB, 2010)
Mora, Le Hur (Nature Phys. 2010)
Universal resistancesResults for small frequencies
Small dot Large dot
22e
hRq 2e
hRq
Confirms result for finite dot, new result in the large dot case
Universal resistances
20)(Im CC
Hamamoto, Jonckheere,
Kato, T. Martin (PRB, 2010)
Mora, Le Hur (Nature Phys. 2010)
202)(Im CC
20
0
41
1
2 NDC
C
g
divergence for 2/10 N
Mapping to the Kondo hamiltonian (0 and 1 -> Sz = -1/2,1/2)
0 1
zSN 2
1ˆ
Charge states
)21( 0NEh c tJ )()( zzK
Correspondance
Matveev (JETP, 1991)
Kondo mapping
Korringa-Shiba relation
At low frequency
22 )0(Re2)(Im)0(Re2)(Im KKzzzz
and therefore 2e
hRq
Shiba (Prog. Theo. Phys., 1975)
Garst, Wolfle, Borda, von Delft, Glazman (PRB, 2005)
Korringa-Shiba relation
Energy conservation
at long times (low frequency)
Probability of inelastic scattering process small 2/ CE
)(
Usual Fermi liquid argument
of phase-space restriction
Even in the presence of strong Coulomb blockade
CEt /
Aleiner, Glazman (PRB,1998)
Dominant elastic scattering
Weak tunneling regimeFermi liquid approach
)(Im2
1 21 CP
Original model
220
212
1 CP
Low energy model
)cos()( 10 teVteV gg Power dissipated under AC drive
NH dˆ...
Linear response theory
',
')(kk
kkgk
kkk ccVKccH
related to through Friedel sum rule)( gVK 0C
22e
hRq
Filippone, Mora (PRB 2012)
Giant peak in the AC resistance
M. Lee, R. Lopez, M.-S. Choi, T. Jonckheere, T. Martin (PRB, 2010)
)()(14
24
2xyF
U
e
hRq
Uy d21
Fermi liquid approach
Peak comes from Kondo singlet breaking
Filippone, Le Hur, Mora (PRL 2011)
Perturbation theory (second order)
Fermi liquid approach
2
22
24 C
mCq e
hR
)(rC
n
n
)(rC
dC
n
d
C
n
Charge susceptibilities
22)(Im CCC
Charge-spin modes CCmCCC ,
Kondo limit
KTBfm /C remains small
charge frozen
ConclusionPrediction of scattering theory is recovered with an exact treatment of Coulomb interaction
Novel universal resistance is predicted for a large cavity
Peak in the charge relaxation resistance for the Anderson model
22e
hRq
2e
hRq
Mora, Le Hur (Nature Phys. 2010)
Conclusions
Filippone, Le Hur, Mora (PRL 2011)
Filippone, Mora (PRB 2012)
Perturbation theory (second order)
Anderson model
M. Lee, R. Lopez, M.-S. Choi, T. Jonckheere, T. Martin (PRB, 2010)
24e
hRq
At zero
magnetic field
Monte-Carlo calculation
Perturbation theory (second order)
Fermi liquid approach
2
22
24 C
mCq e
hR
)(rC
n
n
)(rC
dC
n
d
C
n
Charge susceptibilities
22)(Im CCC
Charge-spin modes CCmCCC ,
Kondo limit
KTBfm /C remains small
charge frozen