Conductance through coupled quantum dots

1
Conductance through coupled quantum dots
J. Bonca 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 QDs
• Good agreement between CPMC and GS and NRG
  approaches
• Many different regimes
• tgtG three peaks in G(d) due to 3 molecular
  levels
• tltG a single peak in G(d) of width U
• At tltltD, 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 QDs
• d0 SN/2 Kondo effect
• dU/2 Quantum phase transitions

4
Nature, 391 (1998)
Science, 281 (1998)
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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)
d
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Three alternative methods

• Numerical Renormalization Group using Reduced
  Density Matrix (NRG), Krishna-murthy, Wilkins and
  Wilson, PRB 21, 1003 (1980) Costi, Hewson and
  Zlatic, 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 Carlo method (CPMC),
  Zhang, Carlson and Gubernatis, PRL 74 ,3652
  (1995)PRB 59, 12788 (1999).

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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 ?8

G02e2/h
Rejec, Ramšak, PRB 68, 035342 (2003)
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Conductance formalisms
U 0
non-equilibrium transport T ? 0, V ? 0
Landauer Büttiker formula
In Fermi liquid systems
Fisher Lee relation
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Comparison CPMC,GS,NRG

• CPMC,
• GS-variational,
• Hartree-Fock

Rejec, Ramšak, PRB 68, 035342 (2003)
Ultt Wide-band

• NRG

Meir-Wingreen, PRL 68, 2512 (1992)
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Comparison CPMC,GS,NRG

• CPMC,
• GS-variational,
• Hartree-Fock

• NRG

Ugtgtt Narrow-band
Meir-Wingreen, PRL 68, 2512 (1992)
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Fermi-liquid E(f) is a universal
function of f
Number of electrons odd
Rejec, Ramšak, PRB 68, 035342 (2003)
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Zero-bias conductance
Rejec, Ramšak, PRB 68, 035342 (2003)
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Connection of G with charge stiffness
Rejec, Ramšak, PRB 68, 035342 (2003)
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GS variational method
Auxiliary Hamitonian
Variational wavefunctions
Projection operators
EH the lowest eigenvalue gives the
approximation to the GS of H
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GS variational method cont.
Using Wicks theorem
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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

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Three coupled quantum dots
Half-filled case!
MO
AFM
TSK

• Using NRG technique
• Using GS variational NGS 1000,2000
• Using CPMC NCPMC 100,180

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Three coupled QDs
Half-filled case!
Oguri, Nisikawa,Hewson
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Three coupled QDs
1
2
3
Oguri, Nisikawa,Hewson, cond-mat/0504771
Half-filled case!
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Three QDs Non-Fermi-Liquid
CvT lnT , cslnT, S(T?0)(1/2) ln 2
TK(1)
AFM
SU(2)spin x SU(2)izospin
MO
TK(2)
MO
AFM
TSK
Zitko Bonca PRL 98, 047203 Kuzmenko et
al.,Europhy.Lett. 64 218 2003 OBSERVATION Potok
et al., Cond-mat/0610721
TK(1)
TK(2)
TD
ZOOM
NFL
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Three QDs Non-Fermi-Liquid CvT lnT , csln T
Zitko Bonca PRL 98, 047203
TK(1)
MO
AFM
AFM
MO
ZOOM
TSK
TK(2)
TK(1)
TK(2)
TD
NFL
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Three coupled QDs Non-Fermi-Liquid
MO
AFM
TSK
Affletck et al. PRB 45, 7918 (1992)
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Three coupled QDs Non-Fermi-Liquid
MO
AFM
TSK
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Quantum phase transitions in parallel QDs
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N - quantum dots
SN/2
SN/2-1

• Three different time-scales

S(S1)/3
N/4
N/8

• Separation of time-scales

• Different temperature-regimes

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Quantum phase transitions in parallel QDs

• d0 SN/2 Kondo effect
• dU/2 ? Discontinuities in G
• Discontinuities in G ? Quantum phase transitions

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Quantum phase transitions in parallel QDs
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N - quantum dotsR.Z. J.B. PRB 74, 045312
(2006)
Schrieffer-Wolf
Perturbation in Vk4-th order
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Conclusions

• Three QDs in series
• Good agreement between NRG, GS, and CPMC.
• Different phases exist
• tgtG three peaks in G(d) due to 3 molecular
  levels (MO), tltG a single peak in G(d) of
  width U in the AFM regime
• Two-stage Kondo (TSK) regime, when tltTK
• NFL behavior is found in the crossover regime. A
  good candidate for the experimental observation.

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Conclusions

• Three QDs in series
• Good agreement between NRG, GS, and CPMC.
• Different phases exist
• tgtG three peaks in G(d) due to 3 molecular
  levels (MO), tltG a single peak in G(d) of
  width U in the AFM regime
• Two-stage Kondo (TSK) regime, when tltTK
• NFL behavior is found in the crossover regime. A
  good candidate for the experimental observation.
• N-parallel QDs
• d0 SN/2 Kondo effect
• dU/2 Quantum phase transitions