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Conductance in Coupled Quantum Dots: Insights from Various Computational Approaches

This study investigates the conductance properties of coupled quantum dots (QDs) using multiple theoretical approaches, including the Numerical Renormalization Group (NRG), Ground State (GS) methods, and Constrained Path Monte Carlo (CPMC) techniques. We analyze several regimes, highlighting the differences in conductance behavior as dependent on interdot coupling and molecular energy levels. Key findings reveal the presence of multiple phases, including non-Fermi liquid (NFL) behavior and significant quantum phase transitions, making these systems promising for experimental research.

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Conductance in Coupled Quantum Dots: Insights from Various Computational Approaches

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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 • Three QD’s: • Good agreement between CPMC and GS and NRG approaches • Many different 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, in the crossover regime an unstable non-Fermi liquid (NFL) fixed point exists • Two-stage Kodo effect is also followed by the NFL • N-parallel QD’s: • d~0: S=N/2 Kondo effect • d~U/2: Quantum phase transitions

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

  5. Three alternative methods: • 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). • 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). • Constrained Path Monte Carlomethod(CPMC),Zhang, Carlson and Gubernatis, PRL 74 ,3652 (1995);PRB 59, 12788 (1999).

  6. 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)

  7. Comparison: CPMC,GS,NRG • CPMC, • GS-variational, • Hartree-Fock: Rejec, Ramšak, PRB 68, 035342 (2003) U<t; Wide-band • NRG: Meir-Wingreen, PRL 68,2512 (1992)

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

  9. Three coupled quantum dotsZitko, Bonca, Rejec, Ramsak, PRB 73, 153307 (2006) MO AFM TSK • Using NRG technique: • Using GS – variational: NGS [1000,2000] • Using CPMC: NCPMC [100,180]

  10. Three coupled quantum dots Half-filled case! MO AFM TSK • Using NRG technique: • Using GS – variational: NGS [1000,2000] • Using CPMC: NCPMC [100,180]

  11. Three QDs Non-Fermi-Liquid: Cv~T lnT , cs~lnT, S(T0)=(1/2)ln2 TK(1) AFM SU(2)spin x SU(2)izospin MO TK(2) MO AFM TSK Zitko & Bonca PRL 98, 047203Kuzmenko et al.,Europhy.Lett. 64 218 2003 OBSERVATION Potok et al., Cond-mat/0610721 TK(1) TK(2) TD ZOOM NFL

  12. Three QDs Non-Fermi-Liquid: Cv~T lnT , cs~ln T Zitko & Bonca PRL 98, 047203 TK(1) MO AFM MO AFM ZOOM TSK TK(2) TK(1) TK(2) TD NFL

  13. Three coupled QDs Non-Fermi-Liquid MO AFM TSK Affletck et al. PRB 45, 7918 (1992)

  14. Three coupled QDs Non-Fermi-Liquid MO AFM TSK

  15. Quantum phase transitions in parallel QD’s

  16. N - quantum dots S=N/2-1 S=N/2 • Three different time-scales: S(S+1)/3 N/4 N/8 • Separation of time-scales: • Different temperature-regimes:

  17. Quantum phase transitions in parallel QD’s • d~0: S=N/2 Kondo effect • d~U/2 Discontinuities in G • Discontinuities in G  Quantum phase transitions

  18. Quantum phase transitions in parallel QD’s

  19. Conclusions • Three QD’s in series: • Good agreement between NRG,GS, and CPMC. • Different phases exist: • t’’>G: three peaks in G(d) due to 3 molecular levels (MO), t’’<G: a single peak in G(d) of width ~ U in the AFM regime • Two-stage Kondo (TSK) regime, when t’’<TK • NFL behavior is found in the crossover regime. A good candidate for the experimental observation.

  20. Conclusions • Three QD’s in series: • Good agreement between NRG,GS, and CPMC. • Different phases exist: • t’’>G: three peaks in G(d) due to 3 molecular levels (MO), t’’<G: a single peak in G(d) of width ~ U in the AFM regime • Two-stage Kondo (TSK) regime, when t’’<TK • NFL behavior is found in the crossover regime. A good candidate for the experimental observation. • N-parallel QD’s: • d~0: S=N/2 Kondo effect • d~U/2: Quantum phase transitions

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