conductance through coupled quantum dots

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J. BončaJ. BončaPhysics Department, FMF, University of Physics Department, FMF, University of

Ljubljana, Ljubljana,

J. Stefan Institute, Ljubljana, SLOVENIAJ. Stefan Institute, Ljubljana, SLOVENIA

Conductance through coupled quantum dots

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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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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 QD’s: Good agreement between CPMC and GS. Two regimes

t’’>: three peaks in G() due to 3 molecular levels t’’<: a single peak in G() of width ~ U

Introduction

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Double- and multiple- dot structures

Craig et el., Science 304, 565 (2004)

Holleitner et el., Science 297, 70 (2002)

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Quantum Dot (Anderson single impurity problem)

d

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Quantum Dot

d d+U

d

d+U/2

U

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Quantum Dot

dd+U

d

U

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Quantum Dot

d

d+U

d

U

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Quantum Dot

d

d+U

d

U

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Quantum Dot

d

d+U

d

U

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Quantum Dot

d

d+U

d

U

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Quantum Dot

d

d+U

d

U

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Quantum Dot

d

Meir-Wingreen, PRL 68, 2512 (1992)

d+U

d

d+U/2

U

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Quantum Dot

dd+U

d

d+U/2

U

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Quantum Dot

d

d+U

d

d+U/2

U

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Quantum Dot

d

d+U

d

d+U/2

U

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Quantum Dot

d

d+U

d

d+U/2

U

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Quantum Dot

d

d+U

d

d+U/2

U

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Quantum Dot

d

d+U

d

d+U/2 ~ gate voltage

U

=U>>

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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 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)

~

~

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Comparison: CPMC,GS,NRG

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

• NRG:

Meir-Wingreen, PRL 68, 2512 (1992)

U<t;Wide-band

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Comparison: CPMC,GS,NRG

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

• NRG:

Meir-Wingreen, PRL 68, 2512 (1992)

U>>t;Narrow-band

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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:

1 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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Large td –

Adding FM coupling

ES=1

ES=0

-Jad

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Small td – Two-stage Kondo effect

Jeff<TK:Two Kondo temperatures:

TK and TK0

TKTK0

Jeff<TK

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).

Two energy scales: Jeff=4td2/U, TK

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Small td –

Two-stage Kondo effect

Jeff>TK

Jeff

0.25 0.50

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Small td –

Two-stage Kondo effect

Jeff~TK

TK

0.25 0.50

TK0

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Small td –

Two-stage Kondo effect

Jeff<TK

TK

0.25 0.50

TK0

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Small td –

Two-stage Kondo effect

Jeff<TK~T

TK

0.25 0.50 Experimental evidence

Wiel et al., PRL 88, 126803 (2002).

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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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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()) Small td: Two-stage Kondo regime (analytical

expressions for TK0)

Three QD’s: Good agreement between CPMC and GS. Two regimes

t’’>: three peaks in G() due to 3 molecular levels t’’<: a single peak in G() of width ~ U

Conclusions