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Conductance through coupled quantum dots
J. Bonča Physics Department, FMF, University of Ljubljana, J. Stefan Institute, Ljubljana, SLOVENIA
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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
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Double- and multiple- dot structures
Holleitner et el., Science 297, 70 (2002) Craig et el., Science 304, 565 (2004)
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Quantum Dot (Anderson single impurity problem)
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Quantum Dot U=1 ed+U ed d d=ed+U/2
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Quantum Dot U=1 d ed+U ed
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Quantum Dot U=1 d ed+U ed
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Quantum Dot U=1 d ed+U ed
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Quantum Dot U=1 d ed+U ed
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Quantum Dot U=1 d ed+U ed
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Quantum Dot U=1 d ed+U ed
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Quantum Dot U=1 ed+U ed d d=ed+U/2 Meir-Wingreen, PRL 68, 2512 (1992)
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Quantum Dot U=1 d ed+U ed d=ed+U/2
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Quantum Dot U=1 d ed+U ed d=ed+U/2
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Quantum Dot U=1 d ed+U ed d=ed+U/2
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Quantum Dot U=1 d ed+U ed d=ed+U/2
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Quantum Dot U=1 d ed+U ed d=ed+U/2
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Quantum Dot U=1 d ed+U ed d=ed+U/2 ~ gate voltage
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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).
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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)
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Comparison: CPMC,GS,NRG
GS-variational, Hartree-Fock: NRG: U<t; Wide-band Meir-Wingreen, PRL 68, 2512 (1992)
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Comparison: CPMC,GS,NRG
GS-variational, Hartree-Fock: NRG: U>>t; Narrow-band Meir-Wingreen, PRL 68, 2512 (1992)
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Side-coupled Double Quantum Dot
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Large td
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Large td – Widths of conductance plateaus:
Energies on isolated DQD: d1 d2
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Large td – Kondo temperatures:
Estimating TK using Scrieffer-Wolf:
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Large td – Kondo temperatures:
Estimating TK using Scrieffer-Wolf:
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Large td – Adding FM coupling
ES=1 ES=0 Large td – Adding FM coupling
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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
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Small td – Two-stage Kondo effect
Jeff>TK Jeff w 0.25 0.5
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Small td – Two-stage Kondo effect
Jeff~TK TK TK0 w 0.25 0.5
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Small td – Two-stage Kondo effect
TK0 Jeff<TK TK w 0.25 0.5
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Small td – Two-stage Kondo effect
Jeff<TK~T TK w 0.25 0.5
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Three coupled quantum dots
Using CPMC: NCPMC [100,180] Using GS – variational: NGS [1000,2000]
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Three coupled QDs 1 2 3
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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
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