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Conductance through coupled quantum dots

Conductance through coupled quantum dots. J. Bonča Physics Department, FMF, University of Ljubljana, J. Stefan Institute, Ljubljana, SLOVENIA. Collaborators: R. Žitko , J. Stefan Inst., Ljubljana, Slovenia

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Conductance through coupled quantum dots

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  1. Conductance through coupled quantum dots J. Bonča Physics Department, FMF, University of Ljubljana, J. Stefan Institute, Ljubljana, SLOVENIA

  2. Collaborators: • R. Žitko, J. Stefan Inst., Ljubljana, Slovenia • A.Ramšak and T. Rejec,FMF, Physics dept., University of Ljubljana and J. Stefan Inst., Ljubljana, Slovenia

  3. Introduction • Experimental motivation • Single QD: using three different methods: NRG, CPMC and GS – accurate results in a wide parameter regime • DQD system: • Large td: Kondo regimes for odd DQD occupancy • Small td: Two-stage Kondo regime • Three QD’s: • Good agreement between CPMC and GS. • Two regimes • t’’>G: three peaks in G(d) due to 3 molecular levels • t’’<G: a single peak in G(d) of width ~ U • At t”<<D, two-stage Kodo effect is found with an unstable non-Fermi liquid fixed point • N-dot system in parallel: RKKY interaction  S=N/2 Kondo effect

  4. Double- and multiple- dot structures Holleitner et el., Science 297, 70 (2002) Craig et el., Science 304, 565 (2004)

  5. Quantum Dot(Anderson single impurity problem) d

  6. ed+U ed Quantum Dot U=1 d d=ed+U/2

  7. ed+U ed Quantum Dot U=1 d

  8. ed+U ed Quantum Dot U=1 d

  9. ed+U ed Quantum Dot U=1 d

  10. ed+U ed Quantum Dot U=1 d

  11. ed+U ed Quantum Dot U=1 d

  12. ed+U ed Quantum Dot U=1 d

  13. ed+U ed Quantum Dot U=1 d d=ed+U/2 Meir-Wingreen, PRL 68,2512 (1992)

  14. ed+U ed Quantum Dot U=1 d d=ed+U/2

  15. ed+U ed Quantum Dot U=1 d d=ed+U/2

  16. ed+U ed Quantum Dot U=1 d d=ed+U/2

  17. ed+U ed Quantum Dot U=1 d d=ed+U/2

  18. ed+U ed Quantum Dot U=1 d d=ed+U/2

  19. ed+U ed Quantum Dot U=1 d D=U>>G d=ed+U/2 ~ gate voltage

  20. Three alternative methods: • Constrained Path Monte Carlomethod(CPMC),Zhang, Carlson and Gubernatis, PRL 74 ,3652 (1995);PRB 59, 12788 (1999). • Projection – variational metod (GS), Schonhammer, Z. Phys. B 21, 389 (1975); PRB 13, 4336 (1976), Gunnarson and Shonhammer, PRB 31, 4185 (1985), Rejec and Ramšak, PRB 68, 035342 (2003). • Numerical Renormalization Group using Reduced Density Matrix (NRG), Krishna-murthy, Wilkins and Wilson, PRB 21, 1003 (1980); Costi, Hewson and Zlatić, J. Phys.: Condens. Matter 6, 2519, (1994); Hofstetter, PRL 85, 1508 (2000).

  21. How to obtain G from GS properties: • CPMC and GS are zero-temperature methods  Ground state energy • Conditions: System is a Fermi liquid ~ N-(noninteracting) sites, N ∞ ~ G0=2e2/h Rejec, Ramšak, PRB 68, 035342 (2003)

  22. Comparison: CPMC,GS,NRG • CPMC, • GS-variational, • Hartree-Fock: • NRG: U<t; Wide-band Meir-Wingreen, PRL 68,2512 (1992)

  23. Comparison: CPMC,GS,NRG • CPMC, • GS-variational, • Hartree-Fock: • NRG: U>>t; Narrow-band Meir-Wingreen, PRL 68,2512 (1992)

  24. Side-coupled Double Quantum DotPRB 73, 035332 (2006)

  25. Large td

  26. Large td – Kondo temperatures: Estimating TK using Scrieffer-Wolf:

  27. Large td – Kondo temperatures: Estimating TK using Scrieffer-Wolf:

  28. Small td – Two-stage Kondo effect Vojta et al., PRB 65, 140405 (2002); Hofstetter, Schoeller, PRL 88, 016803 (2002), Cornaglia and Grempel, PRB 71, 075305 (2005), Wiel et al., PRL 88, 126803 (2002). Jeff<TK:Two Kondo temperatures: TK and TK0 Two energy scales: Jeff=4td2/U, TK

  29. Small td – Two-stage Kondo effect Vojta et al., PRB 65, 140405 (2002); Hofstetter, Schoeller, PRL 88, 016803 (2002), Cornaglia and Grempel, PRB 71, 075305 (2005), Wiel et al., PRL 88, 126803 (2002). Jeff<TK:Two Kondo temperatures: TK and TK0 Two energy scales: Jeff=4td2/U, TK Jeff<TK TK0 TK

  30. Three coupled quantum dotsPRB 73, 153307 (2006) • Using CPMC: NCPMC [100,180] • Using GS – variational: NGS [1000,2000]

  31. Three coupled QDs 1 2 3 Oguri, Nisikawa,Hewson, cond-mat/0504771

  32. Three coupled QDs Non-Fermi-Liquid NRG Calculation Zitko & Bonca Condmat TK(1) MO AFM MO AFM TK(2) TSK TK(1) TK(2) TD NFL

  33. N - quantum dotsPRB 74, 045312 (2006) Schrieffer-Wolff Perturbation in Vk4-th order

  34. N - quantum dots • Three different time-scales: • Separation of time-scales: • Different temperature-regimes:

  35. Conclusions • Using three different methods: NRG, CPMC and GS – accurate results in a wide parameter regime • DQD system: • Large td: Kondo regimes for odd DQD occupancy • Small td: Two-stage Kondo regime • Three QD’s: • Good agreement between CPMC and GS. • Different phases exist: • t’’>G: three peaks in G(d) due to 3 molecular levels • t’’<G: a single peak in G(d) of width ~ U • Two-stage Kondo regime, when t’’<TK • NFL behavior is found • N-dot system in parallel: RKKY interaction

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