Quantum Engineering of States and Devices: Theory and Experiments Obergurgl, Austria 2010
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Quantum Engineering of States and Devices: Theory and ExperimentsObergurgl, Austria 2010
The two impurity Anderson Model revisited: Competition between Kondo effect and
reservoir-mediated superexchange
in double quantum dotsRosa 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)
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OUTLINE OF THIS TALK
1. NRG, Fermi Liquid description of the SIAM
2. Double quantum dot 3. Reservoir-mediated
superexchange interaction4. Conclusions
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Numerical Renormalization Group
Spirit of NRG: Logarithmic discretization of the conduction band. The Anderson model is transformed into a Wilson chain
Example: Single impurity Anderson Model (SIAM)
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Numerical Renormalization Group
+
Ho
H1
H2
HN
H3
-1 0 1 2 3 N. . .
V
Energy resolution
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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 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
L Rtd
We consider two Kondo dots connected seriallyThis is the artificial realization of the “Two-impurity Kondo problem”
1 2
RL
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Transport in double quantum dots in the Kondo regime
For For GG002e2e22/h)/h)
For For GG00=2e=2e22/h,/h, For For GG00 decreases as decreases as
growsgrows
Transport is governed byTransport is governed by =t/=t/
R. Aguado and D.C Langreth, Phys. Rev. Lett. 85 1946 (2000)
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Two-impurity Kondo problem
R. Lopez R. Aguado and G. Platero, Phys. Rev. Lett.89 136802 R. Lopez R. Aguado and G. Platero, Phys. Rev. Lett.89 136802 (2002)(2002)
Serial DQD, Serial DQD, ttCC=0.5 =0.5
J=25 x10J=25 x10-4-4
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TRANSPORT IN SERIAL DOUBLE QUANTUM DOTS
We consider two Kondo dots connected seriallyThis 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 (d=-U/2)2.Infinity U Case3.The transition from the finite U to the infinity U Case
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Symmetric Case: Phase Shifts
1.When td=0 both phase shifts are equal to 2
2.For large td/we havee=,o=0 and the conductance vanishes
3. For certain value of td/
the conductance is unitary
e
o
e o 4. Particle-hole symmetry: Average occupation is oneFriedel-Langreth sum rulefullfilled
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Scaling function
The position of the main peak, td = tc1, is determined by thecondition = /2, which coincides with the condition that the exchange coupling J is comparable to TK, or J = Jc = 4tc1
2/U ~ 2.2 TK
The crossover fromthe Kondo state tothe AF phase isdescribed by a scaling function
Scaling function
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Crossover: Scaling Function
1. The appearence of the unitary-limit-value 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 1up to td/~ 1 the phase shifts show the same behavior as
the symmetric case. Finally for large td/the phase shift
difference saturates around /2
The phase shift difference shows nonmonotonic behavior
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Linear Conductance Why the unitary-limit-value depends on ?
The main peak is shifted toward larger td with increasing and its width also increases with
Plateau of 2e2/h starting at d : 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 one1.The one-e- even-orbital state |N=1, S=1/2> of isolated DD with energy d-td is lowered below the two-dots groundstate |N=2, S=0> and |N=2, S=1> with energy 2d as soon as td is increased beyond d
2.The conductance plateau is then attributed to the formation of a single-impurity Kondo state in the even channel, leading to e= The odd channel becomes empty with o~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
SBMFT marks the maximum for G0 when td
*/2td/This maximum is attributed to the formation of a
coherent superposition of Kondo states: bonding -antibonding Kondo states
R. Aguado and D.C Langreth,Phys. Rev. Lett. 85 1946 (2000)
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Renormalized parameters
1. Fermi liquid theories, like SBMFT, predicts td/2td
*/2*
i.e., a universal behavior of G0 independently on the value
2. However, NRG results indicate that the peak position of G0 depends strongly on This surprising result suggests that td/2flows to larger values, so that
td/2td*/2* Which is the origin of this discrepancy not
noticed before?
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Renormalize parameters: Symmetric U case
The unitary value of G0 coincides with <S1 . S2>=-1/4 denoting the formation ofa spin singlet state between the dots spinsdue to the direct exchange interaction
vv
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Renormalize parameters: Infinity U Case
Importantly: The unitary value of G0 coincideswith <S1 . S2>=-1/4 denoting the formation ofa spin singlet state between the dots spins. However, for infinite U there is no directexchange 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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Initial state
Final state
JI S1 S2
JI Reservoir-mediated superexchange interaction
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Using the Rayleigh-Shrödinger perturbation theory for the infinite U case (to sixth order) yields
For finite U case a more general expression can be obtained where the denominators in JI also depends on UIt is expected then a universal behavior of the linear conductance as a function of a scaling function given by
JI Reservoir-mediated superexchange interaction
..Remarkably: This high order tunnelingevent is able to affect the transport properties
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..
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 and then
td/2td*/2*
This can explain the dependence on of the peak position of the maximum in the linear conductance
J2 Reservoir-mediated superexchange interaction
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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