Transcript Conductance through coupled quantum dots

Conductance through coupled
quantum dots
J. Bonča
Physics Department, FMF, University of Ljubljana,
J. Stefan Institute, Ljubljana, SLOVENIA
www-f1.ijs.si/~bonca
Slonano 2007
Collaborators:
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
R. Žitko, J. Stefan Inst., Ljubljana, Slovenia
A.Ramšak and T. Rejec, FMF, Physics dept.,
University of Ljubljana and J. Stefan Inst.,
Ljubljana, Slovenia
www-f1.ijs.si/~bonca
Slonano 2007
Introduction
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Experimental motivation
Three QD’s:
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Good agreement between CPMC and GS and NRG
approaches
Many different regimes
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•
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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:
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d~0: S=N/2 Kondo effect
d~U/2: Quantum phase transitions
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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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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
(CPMC),
Monte Carlo method
Zhang, Carlson and Gubernatis, PRL 74 ,3652
(1995);PRB 59, 12788 (1999).
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How to obtain G from GS properties:
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CPMC and GS are zero-temperature
methods  Ground state energy
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Conditions: System is a Fermi
liquid
~
N-(noninteracting)
sites, N ∞
~
G0=2e2/h
Rejec, Ramšak, PRB 68, 035342 (2003)
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Comparison: CPMC,GS,NRG
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U<t;
Wide-band
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CPMC,
GS-variational,
Hartree-Fock:
Rejec, Ramšak, PRB 68, 035342
(2003)
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NRG:
Meir-Wingreen, PRL 68,
2512 (1992)
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Comparison: CPMC,GS,NRG
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U>>t;
Narrow-band
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CPMC,
GS-variational,
Hartree-Fock:
NRG:
Meir-Wingreen, PRL
68, 2512 (1992)
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Three coupled quantum dots
Žitko, Bonča, Rejec, Ramšak, PRB 73, 153307 (2006)
MO
AFM
TSK
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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
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Using NRG technique:
Using GS – variational: NGS [1000,2000]
Using CPMC: NCPMC [100,180]
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Three QDs Non-Fermi-Liquid:
Cv~T lnT ,
cs~lnT,
S(T0)=(1/2) ln 2
TK(1)
AFM
SU(2)spin x SU(2)izospin
MO
TK(2)
MO
AFM
TSK
Žitko & Bonča
PRL
98, 047203 Kuzmenko
TK(1)
TK(2)
et al.,Europhy.Lett. 64
218 2003
OBSERVATION
Potok et al.,
Cond-mat/0610721
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TD
ZOOM
NFL
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Three QDs Non-Fermi-Liquid:
Cv~T lnT , cs~ln T
Žitko & Bonča
PRL
98, 047203
TK(1)
MO
MO
AFM
ZOOM
AFM
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 QD’s
R.Žitko. & J.Bonča PRB 74, 045312 (2006)
Schrieffer-Wolf
Perturbation in Vk4-th order
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N - quantum dots
S=N/2-1
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Three different time-scales:
S=N/2
S(S+1)/3
N/4
N/8
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Separation of time-scales:
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Different temperature-regimes:
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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
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Quantum phase transitions in parallel
QD’s
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Conclusions
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Three QD’s in series:
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Good agreement between NRG, GS, and CPMC.
Different phases exist:
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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.
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Conclusions
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Three QD’s in series:
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Good agreement between NRG, GS, and
CPMC.
Different phases exist:
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•
•
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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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