Transcript Theoretical study of the phase evolution in a quantum dot in the

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)
Experimental context: quantum dots studied
by Aharonov-Bohm interferometry
Aharonov-Bohm oscillations of
the conductance as a function of
the magnetic flux F
ref
source
F
drain
QD
the phase d introduced by the QD
is deduced from the shift of the
oscillations with magnetic field
Quantum interferometry
allows to determine the phase
and visibility of the QD
Evolution of the phase when reducing coupling
strength
Unitary limit
Kondo regime
Uncomplete
phase lapse
Coulomb blockade
plateau
Ji, Heiblum et Shtrikman PRL 88, 076601 (2002)
Theoretical context
for the Kondo effect in bulk
metals
Langreth PR 150, 516 (66) and
Nozières JLTP 17, 31 (74)
for the Kondo effect in QD
NRG and Bethe-Ansatz calculations
Gerland, von Delft, Costi, Oreg PRL 84,
3710 (2000)
Theoretical interpretation
2-reservoir Anderson model
Glazman and Raikh JETP Lett. 47, 452 (88)
Ng and Lee PRL 61, 1768 (88)
1-reservoir Anderson model
where
Scattering theory in 1D
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
Asymptotic
solutions
outgoing
Scattering theory in 1D
For the symmetric QD, following Ng and Lee PRL ’88
^
^ ^
Scattering theory
Using exact results on Fermi liquid at T=0, one can show that
Denoting the phase of
by
, one gets
(Friedel sum rule
see Langreth Phys.Rev.’66)
Using trigonometric arguments
Using again exact results on Fermi liquid at T=0, one can show
Putting altogether, one gets
Scattering off a composite system
Generalized Levinson’s theorem
where
is the number of bound states
Levinson’49
Swan ’55
Rosenberg and Spruch PRA’96
is the number of states excluded by the Pauli
principle
Example: scattering of an electron by an atom of hydrogen
“Triplet”
=1
“Singlet”scattering
scattering:: S
Szztot
tot=0
H+e
is the ground state of a hydrogen atom
1s
Phase shift
1s"+1s#
1s"+1s"
1s2
0
0
1s"1s"
Scattering theory in 1D
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.
is chosen in order that
satisfies the generalized Levinson
theorem. It is easy to show that
with
Scattering theory in 1D
Landauer formula
Aharonov-Bohm interferometry
Consequences (at T=0, H=0)
• Phase shift measured
• Conductance measured
Scattering theory in 1D
Experimental check of the prediction
P. Vitushinky, A.Jerez, M.Lavagna Quantum Information and Decoherence in
Nanosystems, p.309 (2004)
Bethe-Ansatz solution at T=0
We have numerically solved the Bethe ansatz equations to derive n0
and hence d/p as a function of the parameters of the model
(Wiegmann et al. JETP Lett. ’82 and Kawakami and Okiji, JPSJ ’82)
A.Jerez, P.Vitushinsky, M.Lavagna
PRL 95, 127203 (2005)
Particle-hole symmetry
symmetric limit
Bethe-Ansatz solution at T=0
In the asymmetric regime,
, n0 shows a universal behavior
as a function of the renormalized energy
Universal behavior occurs when
Asymptotic behavior in the limit n0
0
The existence of both those universal and asymptotic behavior is of
valuable help in fitting the experimental data
Bethe-Ansatz solution at T=0
Fit in the unitary limit and Kondo regimes
All the experimental curves are shifted in order to get d=p at the symmetric limit
Bethe-Ansatz solution at T=0
(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
Conclusions
1. 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 BetheAnsatz 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 d(VG) and also found a very good
agreement.