Quantum Engineering of States and Devices: Theory and Experiments Obergurgl, Austria 2010

1. 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 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)

2. OUTLINE OF THIS TALK • NRG, Fermi Liquid description of the SIAM • Double quantum dot • Reservoir-mediated superexchange interaction • Conclusions

3. 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

4. Numerical Renormalization Group x0 x1 x2 V -1 N 0 1 2 3 . . . G L0 L-1/2 L-(N-1)/2 Ho Energy resolution H1 H2 H3 + HN

5. 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

6. 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

7. SIAM renormalized parameters

8. GL GR td TRANSPORT IN SERIAL DOUBLE QUANTUM DOTS L R 1 2 We consider two Kondo dots connected serially This is the artificial realization of the “Two-impurity Kondo problem”

9. Transport in double quantum dots in the Kondo regime Transport is governed byt=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. 85 1946 (2000)

10. Serial DQD, tC=0.5 J=25 x10-4 Two-impurity Kondo problem R. Lopez R. Aguado and G. Platero, Phys. Rev. Lett.89 136802 (2002)

11. 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

12. TRANSPORT IN SERIAL DOUBLE QUANTUM DOTS • We analyze three different cases: • Symmetric Case (ed=-U/2) • Infinity U Case • The transition from the finite U • to the infinity U Case

13. de do de do Symmetric Case: Phase Shifts • When td=0 both phase • shifts are equal top/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 4. Particle-hole symmetry: Average occupation is one Friedel-Langreth sum rule fullfilled

14. 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

15. Crossover: Scaling Function • The appearence of the unitary-limit-value conductance is explained in terms of a crossover between the Kondo phase and the AF phase • When J<<TK each QD forms a Kondo state and then G0 is very low (hopping between two Kondo resonances) • When J>>TK the dot spins are locked into a spin singlet state G0 decreases

16. Discrepancy for The Large U limit

17. 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

18. Linear Conductance The main peak is shifted toward larger td/G with increasing G and its width also increases with G Why the unitary-limit-value depends on G? Plateau of 2e2/h starting at ed : Spin Kondo in the even sector

19. Spin Kondo effect in the even sector • Plateau in G0: As td increases, the DD charge • decreases to one • The one-e- even-orbital state |N=1, S=1/2> • of isolated DD with energyed-tdis 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| • 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

20. 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

21. Renormalized parameters • Fermi liquid theories, like SBMFT, predicts td/2G= td*/2G* i.e., a universal behavior of G0 independently on the G value • 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?

22. vv Renormalize parameters: Symmetric U case 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

23. 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 ¡¡¡¡¡¡

24. Magnetic interactions • JU is the known direct coupling between the dots that vanishes for infinite U JU=4td2/U • 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

25. Magnetic correlations • Indeed the essential features of the system state should not change whatever value of Coulomb interaction U is • 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

26. Processes that generate JI

27. JI Reservoir-mediated superexchange interaction Final state JI S1 S2 Initial state

28. 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

29. 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 thentd/2G << td*/2G*This can explain the dependence on G of the peak position of the maximum in the linear conductance

30. From the Symmetric U to the Infinite-U Case

31. 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