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

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)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

20. Three alternative methods:
Constrained Path Monte Carlo method (CPMC), Zhang, Carlson and Gubernatis, PRL 74 ,3652 (1995);PRB 59, (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, (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 exhibits Fermi liquid properties
N-(noninteracting)
sites, N ∞
G0=2e2/h
Rejec, Ramšak, PRB 68, (2003)

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

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

24. Side-coupled Double Quantum Dot

25. Large td

26. Large td – Widths of conductance plateaus:
Energies on isolated DQD: d1 d2

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

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

29. Large td – Adding FM coupling
ES=1
ES=0
Large td – Adding FM coupling

30. Small td – Two-stage Kondo effect
Cornaglia and Grempel, PRB 71, (2005).
Two energy scales: Jeff=4td2/U, TK
Jeff<TK:Two Kondo temperatures:
TK and TK0
Jeff<TK
TK0 TK

31. Small td – Two-stage Kondo effect
Jeff>TK
Jeff w
0.25
0.5

32. Small td – Two-stage Kondo effect
Jeff~TK TK
TK0 w
0.25
0.5

33. Small td – Two-stage Kondo effect
TK0
Jeff<TK TK w
0.25
0.5

34. Small td – Two-stage Kondo effect
Jeff<TK~T TK w
0.25
0.5

35. Three coupled quantum dots
Using CPMC: NCPMC [100,180]
Using GS – variational: NGS [1000,2000]

36. Three coupled QDs 1 2 3

37. 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 (analytical expressions for TK and widh G(d))
Small td: Two-stage Kondo regime (analytical expressions for TK0)
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