theoretical study of the phase evolution in a quantum dot in the presence of kondo correlations...

Theoretical study of the phase evolution in a quantum dot in the presence of Kondo

correlations Mireille LAVAGNA

Work done in collaboration with A. JEREZ (ILL Grenoble and Rutgers University) and P. VITUSHINSKY (CEA-Grenoble)

QD

Aharonov-Bohm oscillations of the conductance as a function of the magnetic flux

ref

source drain

the phase introduced by the QD is deduced from the shift of the oscillations with magnetic field

Experimental context: quantum dots studied by Aharonov-Bohm interferometry

allows to determine the phase and visibility of the QD

Quantum interferometry

Unitary limit

Kondo regime Coulomb blockade

Ji, Heiblum et Shtrikman PRL 88, 076601 (2002)

Evolution of the phase when reducing coupling strength

Uncomplete phase lapse

plateau

for the Kondo effect in QD

NRG and Bethe-Ansatz calculations

Theoretical context

Langreth PR 150, 516 (66) and Nozières JLTP 17, 31 (74)

for the Kondo effect in bulk metals

Gerland, von Delft, Costi, Oreg PRL 84, 3710 (2000)

2-reservoir Anderson model

Glazman and Raikh JETP Lett. 47, 452 (88)Ng and Lee PRL 61, 1768 (88)

where

1-reservoir Anderson model

Theoretical interpretation

In the case when there is no magnetic moment in the dot (for instance in the Kondo regime at T=0), spin-flip scattering cannot occur

incoming outgoing

Asymptotic solutions

Scattering theory in 1DScattering theory in 1D

(Friedel sum rule see Langreth Phys.Rev.’66)

Scattering theory

Using exact results on Fermi liquid at T=0, one can show that

For the symmetric QD, following Ng and Lee PRL ’88


^ ^ ^

Denoting the phase of by , one gets

Using trigonometric arguments

Using again exact results on Fermi liquid at T=0, one can show

Putting altogether, one gets

Scattering off a composite systemScattering off a composite systemGeneralized Levinson’s theorem

where is the number of bound states

is the number of states excluded by the Pauli principle

1s"+1s#1s"+1s"

HH + e

Example: scattering of an electron by an atom of hydrogenExample: scattering of an electron by an atom of hydrogen

Phase shift

“Singlet” scattering: Sztot=0“Triplet” scattering : Sz

tot=1

1s

1s2 00 1s"1s"

is the ground state of a hydrogen atom

Levinson’49Swan ’55Rosenberg and Spruch PRA’96

Quantum dot=

Artificial atom

Generalized Levinson’s theorem


The single level Anderson model (SLAM) is not sufficient to capture the whole physics contained in the experimental device which can be viewed as an artificial atom. One may try to start with a many level Anderson model (MLAM) description of the system. We have chosen another route and introduced the missing ingredients through an additional multiplicative factor in front of the S-matrix of the SLAM.

with

is chosen in order that satisfies the generalized Levinson theorem. It is easy to show that

Aharonov-Bohm interferometry

Landauer formula

Consequences (at T=0, H=0)

• Phase shift measured

• Conductance measured


P. Vitushinky, A.Jerez, M.Lavagna Quantum Information and Decoherence in Nanosystems, p.309 (2004)

Experimental check of the prediction


We have numerically solved the Bethe ansatz equations to derive n0 and hence / as a function of the parameters of the model (Wiegmann et al. JETP Lett. ’82 and Kawakami and Okiji, JPSJ ’82)

Particle-hole symmetry

Bethe-Ansatz solution at T=0Bethe-Ansatz solution at T=0

symmetric limit

A.Jerez, P.Vitushinsky, M.Lavagna

PRL 95, 127203 (2005)

Asymptotic behavior in the limit n0 0Universal behavior occurs when

The existence of both those universal and asymptotic behavior is of valuable help in fitting the experimental data

In the asymmetric regime, , n0 shows a universal behavior

as a function of the renormalized energy


Fit in the unitary limit and Kondo regimes

All the experimental curves are shifted in order to get = at the symmetric limit


(a) Unitary limit

(b) Kondo regime

Fit in the unitary limit and Kondo regimes

Very good agreement in presence of a single fitting parameter /U

(we consider linear correspondence between 0 and VG )

A.Jerez, P.Vitushinsky, M.Lavagna, PRL’05


Conclusions1. We have shown that there is a factor of 2 difference between the

phase of the S-matrix responsible for the shift in the AB oscillations and the phase controlling the conductance.

2. This result is beyond the simple single-level Anderson model (SLAM) description and supposes to consider the generalisation to the multi-level Anderson model (MLAM). Done here in a minimal way by introducing a multiplicative factor in front of the S-matrix in order to guarantee the generalized Levinson theorem.

3. Then the phase measured by A.B. experiments is related to the total occupation n0 of the dot which is exactly determined by Bethe-Ansatz calculations. We have obtained a quantitative agreement with the experimental data for the phase in two regimes.

4. We have also checked the prediction with experimental data on G(VG) and (VG) and also found a very good agreement.

theoretical study of the phase evolution in a quantum dot in the presence of kondo correlations...

Documents