Conductance through coupled quantum dots

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Conductance through coupled quantum dots
J. Bonca 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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Introduction

• Experimental motivation
• Single QD 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
• Small td Two-stage Kondo regime
• Adding FM coupling
• Three QDs
• Good agreement between CPMC and GS.
• Two regimes
• tgtG three peaks in G(d) due to 3 molecular
  levels
• tltG a single peak in G(d) of width U

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(No Transcript)
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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
U1
d
dedU/2
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Quantum Dot
U1
d
9
Quantum Dot
U1
d
10
Quantum Dot
U1
d
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Quantum Dot
U1
d
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Quantum Dot
U1
d
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Quantum Dot
U1
d
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Quantum Dot
U1
d
dedU/2
Meir-Wingreen, PRL 68, 2512 (1992)
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Quantum Dot
U1
d
dedU/2
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Quantum Dot
U1
d
dedU/2
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Quantum Dot
U1
d
dedU/2
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Quantum Dot
U1
d
dedU/2
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Quantum Dot
U1
d
dedU/2
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Quantum Dot
U1
d
DUgtgtG
dedU/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, 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
  Zlatic, 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 is a Fermi liquid

N-(noninteracting) sites, N ?8

G02e2/h
Rejec, Ramšak, PRB 68, 035342 (2003)
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Comparison CPMC,GS,NRG

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

• NRG

Ultt Wide-band
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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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
-Jad
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Small td Two-stage Kondo effect
Vojta et al., PRB 65, 140405 (2002) Hofstetter,
Schoeller, PRL 88, 016803 (2002), Cornaglia
and Grempel, PRB 71, 075305 (2005), Wiel et al.,
PRL 88, 126803 (2002).
JeffltTKTwo Kondo temperatures TK and TK0
Two energy scales Jeff4td2/U, TK
JeffltTK
TK
TK0
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Small td Two-stage Kondo effect
JeffgtTK
Jeff
w
0.25
0.5
0
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Small td Two-stage Kondo effect
JeffTK
TK
TK0
w
0.25
0.5
0
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Small td Two-stage Kondo effect
TK0
JeffltTK
TK
w
0.25
0.5
0
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Small td Two-stage Kondo effect
JeffltTKT
TK
Experimental evidence Wiel et al., PRL 88, 126803
(2002).
w
0.25
0.5
0
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Large td Adding FM coupling
Two-stage Kondo effect?
Voja et al., PRB 65, 140405 (2002), Hofstetter,
Schoeller, PRL 88, 016803 (2002),
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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
Oguri, Nisikawa,Hewson, cond-mat/0504771
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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 QDs
• Good agreement between CPMC and GS.
• Two regimes
• tgtG three peaks in G(d) due to 3 molecular
  levels
• tltG a single peak in G(d) of width U