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Quantum Engineering of States and Devices:
Theory and Experiments
Obergurgl, Austria 2010
The two impurity Anderson Model revisited:
Competition between Kondo effect and
reservoir-mediated superexchange
in double quantum dots
Rosa López (Balearic Islands University,IFISC)
Collaborators
Minchul Lee (Kyung Hee University, Korea)
Mahn-Soo Choi (Korea University, Korea)
Rok Zitko (J. Stefan Institute, Slovenia)
Ramón Aguado (ICMM, Spain)
Jan Martinek (Institute of Molecular Physics, Poland)
http://ifisc.uib-csic.es - Mallorca - Spain
OUTLINE OF THIS TALK
1. NRG, Fermi Liquid
description of the SIAM
2. Double quantum dot
3. Reservoir-mediated
superexchange interaction
4. Conclusions
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Numerical Renormalization Group
Example: Single impurity Anderson Model (SIAM)
Spirit of NRG: Logarithmic discretization of the
conduction band. The Anderson model is
transformed into a Wilson chain
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Numerical Renormalization Group
x0
V
-1
x1
0
G
1
L0
Ho
x2
2
L-1/2
3
.
.
.
N
L-(N-1)/2
Energy resolution
H1
H
2
H3
HN
+
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Fermi liquid fixed point: SIAM renormalized
parameters
The low-temperature behavior of a impurity model can often
be described using an effective Hamiltonian which takes
exactly the same form as the original Hamiltonian but with
renormalized parameters
Example: SIAM, Linear conductance related with the
phase shift and this related with the renormalized paremeters
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Fermi liquid fixed point: SIAM renormalized
parameters
RENORMALIZED PARAMETERS
Ep(h) are the lowest particle and hole excitations
from the ground state.They are calculated from
the NRG output. g00(w) is the Green function at
the first site of the Wilson chain
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SIAM renormalized parameters
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TRANSPORT IN SERIAL DOUBLE
QUANTUM DOTS
GL
GR
td
L
R
1
2
We consider two Kondo dots connected serially
This is the artificial realization of the
“Two-impurity Kondo problem”
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Transport in double quantum
dots in the Kondo regime
Transport is governed by t=t/G
t<1
t>1
For t<1, G0 ~ (2e2/h)t2
For t=1, G0=2e2/h,
For t>1, G0 decreases as t grows
R. Aguado and D.C Langreth, Phys. Rev. Lett. 85http://ifisc.uib-csic.es
1946 (2000)
Two-impurity Kondo problem
Serial DQD,
tC=0.5
J=25 x10-4
R. Lopez R. Aguado and G. Platero, Phys. Rev. Lett.89 136802 (2002)
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TRANSPORT IN SERIAL DOUBLE
QUANTUM DOTS
We consider two Kondo dots connected serially
This is the artificial realization of the
“Two-impurity Kondo problem”
In the even-odd basis
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TRANSPORT IN SERIAL DOUBLE
QUANTUM DOTS
We analyze three different cases:
1.Symmetric Case (ed=-U/2)
2.Infinity U Case
3.The transition from the finite U
to the infinity U Case
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Symmetric Case: Phase Shifts
de
do
1. When td=0 both phase
shifts are equal to p/2
2. For large td/G we have
de=p,do=0 and the
conductance vanishes
3. For certain value of td/G
the conductance is unitary
de d o
4. Particle-hole symmetry:
Average occupation is one
Friedel-Langreth sum rule
fullfilled
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Scaling function
The crossover from
the Kondo state to
the AF phase is
described by a
scaling function
Scaling function
The position of the main peak, td = tc1, is determined by the
condition d= p/2, which coincides with the condition that
the exchange coupling J is comparable to TK, or J = Jc =
4tc12/U ~ 2.2 TK
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Crossover: Scaling Function
1. The appearence of the unitary-limitvalue conductance is explained in terms
of a crossover between the Kondo
phase and the AF phase
2. When J<<TK each QD forms a Kondo
state and then G0 is very low (hopping
between two Kondo resonances)
3. When J>>TK the dot spins are locked
into a spin singlet state G0 decreases
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Discrepancy for
The Large U limit
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Infinite-U Case
For td= 0 we have
Since U is very large, the dot occupation does not reach 1
up to td/G ~ 1 the phase shifts show the same behavior as
the symmetric case. Finally for large td/G the phase shift
difference saturates around p/2
The phase shift difference shows nonmonotonic behavior
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Linear Conductance
Why the unitary-limit-value depends on G?
The main peak is shifted toward larger td /G with
increasing G and its width also increases with G
Plateau of 2e2/h starting at ed :
Spin Kondo in the even sector
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Spin Kondo effect in the even sector
Plateau in G0: As td increases, the DD charge
decreases to one
1. The one-e- even-orbital state |N=1, S=1/2>
of isolated DD with energy ed-td is lowered
below the two-dots groundstate |N=2, S=0>
and |N=2, S=1> with energy 2ed as soon as td
is increased beyond |ed|
2.The conductance plateau is then attributed to
the formation of a single-impurity Kondo state
in the even channel, leading to de=p/2. The odd
channel becomes empty with do~0
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Linear conductance
• For the infinity U case the exchange interaction
vanishes. From Fermi Liquid theories (SBMFT, for
example) we know that
R. Aguado and D.C Langreth,Phys. Rev. Lett. 85 1946 (2000)
SBMFT marks the maximum for G0 when td*/2G* =td/2G
This maximum is attributed to the formation of a
coherent superposition of Kondo states:
bonding -antibonding Kondo states
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Renormalized parameters
1. Fermi liquid theories, like SBMFT, predicts
td/2G= td*/2G*
i.e., a universal behavior of G0 independently on the
G value
2. However, NRG results indicate that the peak position
of G0 depends strongly on G. This surprising result
suggests that td/2G flows to larger values, so that
td/2G << td*/2G*
Which is the origin of this discrepancy not noticed
before?
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Renormalize parameters:
Symmetric U case
vv
The unitary value of G0 coincides with
<S1 . S2>=-1/4 denoting the formation of
a spin singlet state between the dots spins
due to the direct exchange interaction
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Renormalize parameters: Infinity U Case
Importantly: The unitary value of G0 coincides
with <S1 . S2>=-1/4 denoting the formation of
a spin singlet state between the dots spins.
However, for infinite U there is no direct
exchange interaction ¡¡¡¡¡¡
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Magnetic interactions
1. JU is the known direct coupling between the
dots that vanishes for infinite U
JU=4td2/U
2. JI is a new exchange term that in general
depends on U but does not vanish when this
goes to infinity
JI(U=0) does not vanish
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Magnetic correlations
1. Indeed the essential features of the
system state should not change whatever
value of Coulomb interaction U is
2. The infinite U case is then also explained
in terms of competition between an
exchange coupling and the Kondo
correlations. Therefore, there must exist
two kinds of exchange couplings
J=JU+JI
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Processes that generate JI
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JI Reservoir-mediated superexchange interaction
Final state
JI S1 S2
Initial state
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JI Reservoir-mediated superexchange interaction
Using the Rayleigh-Shrödinger perturbation theory
for the infinite U case (to sixth order) yields
..
Remarkably: This high order tunneling
event is able to affect the transport properties
For finite U case a more general expression can be obtained
where the denominators in JI also depends on U
It is expected then a universal behavior of the linear
conductance as a function of a scaling function given by
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J2 Reservoir-mediated superexchange interaction
SB theories should be in agreement
with NRG calculations if ones
introduces by hand this new term JI.
This new term will renormalize td in a
different manner than it does for G and
.
.
then
*
*
td/2G << td /2G
This can explain the dependence on G
of the peak position of the maximum in
the linear conductance
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From the Symmetric U to the Infinite-U Case
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Conclusions
Our NRG results support the importance
of including magnetic interactions
mediated by the conduction band in the
theory in the Large-U limit. In this manner
we have a showed an unified physical
description for the DQD system when U
finite to U Inf
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