quantum engineering of states and devices: theory and experiments obergurgl, austria 2010

http://ifisc.uib-csic.es - Mallorca - Spain

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)

http://ifisc.uib-csic.es

OUTLINE OF THIS TALK

1. NRG, Fermi Liquid description of the SIAM

2. Double quantum dot 3. Reservoir-mediated

superexchange interaction4. Conclusions


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)


Numerical Renormalization Group

+

Ho

H1

H2

HN

H3

-1 0 1 2 3 N. . .

V

Energy resolution


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


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


SIAM renormalized parameters


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


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)


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



We consider two Kondo dots connected seriallyThis is the artificial realization of the

“Two-impurity Kondo problem”

In the even-odd basis



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


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


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


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


Discrepancy for The Large U limit


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


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


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


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)


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?


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


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 ¡¡¡¡¡¡


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


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


Processes that generate JI


Initial state

Final state

JI S1 S2

JI Reservoir-mediated superexchange interaction


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


..

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


From the Symmetric U to the Infinite-U Case


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

quantum engineering of states and devices: theory and experiments obergurgl, austria 2010

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