Transcript Conductance through coupled quantum dots

Conductance through coupled
quantum dots
J. Bonča
Physics Department, FMF, University of Ljubljana,
J. Stefan Institute, Ljubljana, SLOVENIA
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Yalta, 2005
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)
d
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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
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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
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
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Three alternative methods:
 Constrained Path Monte Carlo method
(CPMC), Zhang, Carlson and Gubernatis, PRL 74
,3652 (1995);PRB 59, 12788 (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, 035342
(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 zerotemperature methods 
Ground state energy
Conditions: System exhibits
Fermi liquid properties
N-(noninteracting)
sites, N ∞
G0=2e2/h
Rejec, Ramšak, PRB 68, 035342 (2003)
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Comparison: CPMC,GS,NRG
•
•
•
U<t;
Wide-band
•
CPMC,
GS-variational,
Hartree-Fock:
NRG:
Meir-Wingreen, PRL
68, 2512 (1992)
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Comparison: CPMC,GS,NRG
•
•
•
U>>t;
Narrow-band
•
CPMC,
GS-variational,
Hartree-Fock:
NRG:
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 d 2
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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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ES=1
ES=0
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Large td –
Adding FM coupling
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Small td –
Two-stage Kondo effect
Cornaglia and Grempel, PRB 71, 075305 (2005).
Two energy scales:
Jeff=4td2/U,
TK
Jeff<TK:Two Kondo
temperatures:
TK and TK0
Jeff<TK
T K0
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TK
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Small td –
Two-stage Kondo effect
Jeff>TK
Jeff
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0
0.25
0.5
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Small td –
Two-stage Kondo effect
Jeff~TK
TK
TK0
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Small td –
Two-stage Kondo effect
T K0
Jeff<TK
TK
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Small td –
Two-stage Kondo effect
Jeff<TK~T
TK
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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
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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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