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

Qq

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

Adding Coulomb interactions

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 charge relaxation resistance

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

Perturbation theory (second order)

Fermi liquid approach

)()(14

24

2xyF

U

e

hRq U

y d21

M. Lee, R. Lopez, M.-S. Choi, T. Jonckheere, T. Martin (PRB, 2010)

Filippone, Le Hur, Mora (PRL 2011)