Conductance through coupled quantum dots

www-f1.ijs.si/~boncawww-f1.ijs.si/~bonca RTN NANO Portoroz 2007RTN NANO Portoroz 2007

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

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

Conductance through coupled Conductance through coupled quantum dotsquantum dots


Collaborators:Collaborators:

R. R. ŽitkoŽitko, , J. Stefan Inst., Ljubljana, J. Stefan Inst., Ljubljana, SloveniaSlovenia

A.Ramšak and T. RejecA.Ramšak and T. Rejec,, FMF, Physics dept., FMF, Physics dept., University of Ljubljana and J.University of Ljubljana and J. Stefan Inst., Stefan Inst., Ljubljana, Ljubljana, SloveniaSlovenia


Experimental motivationExperimental motivation Three QD’s:Three QD’s:

Good agreement between Good agreement between CPMCCPMC and and GSGS and and NRG NRG approachesapproaches

Many different regimesMany different regimes• t’’>t’’>: three peaks in : three peaks in G(G()) due to 3 molecular levels due to 3 molecular levels• t’’<t’’<: a single peak in : a single peak in G(G()) of width ~ of width ~ UU

At t”<<D, in the At t”<<D, in the crossover regimecrossover regime an unstable non-Fermi an unstable non-Fermi liquid (NFL) fixed point exists liquid (NFL) fixed point exists

Two-stage Kodo effect is also followed by the NFL Two-stage Kodo effect is also followed by the NFL N-parallel QD’s: N-parallel QD’s:

~0: ~0: S=N/S=N/2 Kondo effect 2 Kondo effect ~~U/U/2: Quantum phase transitions2: Quantum phase transitions

IntroductionIntroduction


Double- and multiple- dot structuresDouble- and multiple- dot structures

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

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


Three alternative methods:Three alternative methods: Numerical Renormalization Group using Numerical Renormalization Group using

Reduced Density Matrix (NRG), Reduced Density Matrix (NRG), Krishna-murthy, Krishna-murthy, Wilkins and Wilson, PRB Wilkins and Wilson, PRB 2121, 1003 (1980); Costi, Hewson and , 1003 (1980); Costi, Hewson and ZlatiZlatićć, J. Phys.: Condens. Matter , J. Phys.: Condens. Matter 66, 2519, (1994); Hofstetter, , 2519, (1994); Hofstetter, PRLPRL 85, 1508 (2000).85, 1508 (2000).

Projection – variational metod (GS)Projection – variational metod (GS), , Schonhammer, Z. Phys. B Schonhammer, Z. Phys. B 2121, 389 (1975); PRB , 389 (1975); PRB 1313, 4336 (1976), , 4336 (1976), Gunnarson and Shonhammer, PRB Gunnarson and Shonhammer, PRB 3131, 4185 (1985), Rejec and , 4185 (1985), Rejec and RamRamššak, PRB 68, 035342 (2003).ak, PRB 68, 035342 (2003).

Constrained Path Monte CarloConstrained Path Monte Carlo methodmethod (CPMC),(CPMC), Zhang, Carlson and Gubernatis, PRL 74 ,3652 Zhang, Carlson and Gubernatis, PRL 74 ,3652

(1995);PRB (1995);PRB 5959, 12788 (1999)., 12788 (1999).


How to obtain G from GS properties:How to obtain G from GS properties: CPMCCPMC and and GSGS are zero-temperature are zero-temperature

methods methods Ground state energy Ground state energy Conditions: System is a Conditions: System is a Fermi Fermi

liquidliquid

N-(noninteracting) N-(noninteracting) sites, sites, N N ∞∞

GG00=2e=2e22/h/h

Rejec, RamRejec, Ramššak, PRB 68, 035342 (2003)ak, PRB 68, 035342 (2003)

~

~


Comparison: Comparison: CPMC,GS,NRGCPMC,GS,NRG• CPMC,CPMC, • GS-variational,GS-variational,• Hartree-Fock:Hartree-Fock:

• NRG:NRG:

Meir-Wingreen, PRL Meir-Wingreen, PRL 6868,, 2512 (1992)2512 (1992)

U<t;Wide-band

Rejec, RamRejec, Ramššak, PRB 68, 035342 ak, PRB 68, 035342 (2003)(2003)


Comparison: Comparison: CPMC,GS,NRGCPMC,GS,NRG• CPMC, • GS-variational,• Hartree-Fock:

• NRG:

Meir-Wingreen, PRL 68, 2512 (1992)

U>>t;Narrow-band


Three coupled quantum dotsThree coupled quantum dotsZitko, Bonca, Rejec, Ramsak, PRB Zitko, Bonca, Rejec, Ramsak, PRB 7373, 153307 (2006), 153307 (2006)

Using NRG technique:Using NRG technique: Using GS – variational: NUsing GS – variational: NGSGS [1000,2000] [1000,2000] Using CPMC: NUsing CPMC: NCPMCCPMC [100,180] [100,180]

MOMO

AFMAFM

TSKTSK


Three coupled quantum dotsThree coupled quantum dots

Using NRG technique:Using NRG technique: Using GS – variational: NUsing GS – variational: NGSGS [1000,2000] [1000,2000] Using CPMC: NUsing CPMC: NCPMCCPMC [100,180] [100,180]

MOMO

AFMAFM

TSKTSK

Half-filled case!Half-filled case!


Three QDs Three QDs NNon-on-FFermi-ermi-LLiquid: iquid:

Zitko & BoncaZitko & BoncaPRLPRL

98, 04720398, 047203 Kuzmenko Kuzmenko et al.,Europhy.Lett. et al.,Europhy.Lett. 6464

218 2003218 2003OBSERVATIONOBSERVATION

Potok et al.,Potok et al.,Cond-mat/0610721Cond-mat/0610721

TTKK(1)(1)

TTKK(2)(2)

T

TTKK(1)(1)

TTKK(2)(2)

MOMO

AFMAFM

TSKTSK

NFLNFL

MOMO

AFMAFM

ZOOMZOOM

Cv~T lnT , s~lnT, S(T0)=(1/2) ln 2

SU(2)SU(2)spinspin x SU(2) x SU(2)izospinizospin


Three QDs Three QDs NNon-on-FFermi-ermi-LLiquid: iquid:

CCvv~~TT ln lnTT , , ss~ln ~ln TT

TTKK(1)(1)

TTKK(2)(2)

T

TTKK(1)(1)

TTKK(2)(2)

MOMO

AFMAFM

TSKTSK

NFL

MOMO

AFMAFM

ZOOM

Zitko & BoncaPRL

98, 047203


Three coupled QDs Three coupled QDs NNon-on-FFermi-ermi-LLiquidiquid

MOMO

AFMAFM

TSKTSK

Affletck et al. PRB 45, 7918 (1992)


Three coupled QDs Three coupled QDs NNon-on-FFermi-ermi-LLiquidiquid

MOMO

AFMAFM

TSKTSK


Quantum phase transitions in parallel Quantum phase transitions in parallel QD’sQD’s


N - quantum dotsN - quantum dots

Three different time-scales:Three different time-scales:

Separation of time-scales:Separation of time-scales:

Different temperature-regimes:Different temperature-regimes:

S=N/2

N/8

N/4

S(S+1)/3

S=N/2-1



~0: ~0: SS==NN/2 Kondo effect/2 Kondo effect UU/2/2Discontinuities in Discontinuities in

GG Discontinuities in Discontinuities in G G

Quantum phase transitionsQuantum phase transitions




ConclusionsConclusions Three QD’s in series:Three QD’s in series:

Good agreement between Good agreement between NRG,NRG, GSGS, and , and CPMCCPMC.. Different phases exist:Different phases exist:

• t’’>t’’>: three peaks in : three peaks in G(G()) due to 3 molecular levels ( due to 3 molecular levels (MOMO), t’’<), t’’<: a : a single peak in single peak in G(G()) of width ~ of width ~ U U in the in the AFMAFM regime regime

• Two-stage Kondo Two-stage Kondo (TSK)(TSK) regime, when t’’<T regime, when t’’<TKK

• NFLNFL behavior is found in the behavior is found in the crossover regimecrossover regime. . A good A good candidate for the experimental observation. candidate for the experimental observation.


ConclusionsConclusions Three QD’s in series:Three QD’s in series:

Good agreement between Good agreement between NRG,NRG, GSGS, and , and CPMCCPMC..

Different phases exist:Different phases exist:• t’’>t’’>: three peaks in : three peaks in G(G()) due to 3 molecular levels ( due to 3 molecular levels (MOMO), ),

t’’<t’’<: a single peak in : a single peak in G(G()) of width ~ of width ~ U U in the in the AFMAFM regime regime• Two-stage Kondo Two-stage Kondo (TSK)(TSK) regime, when t’’<T regime, when t’’<TKK

• NFLNFL behavior is found in the behavior is found in the crossover regimecrossover regime. . A good A good candidate for the experimental observation. candidate for the experimental observation.

N-parallel QD’s: N-parallel QD’s: ~0: ~0: S=N/S=N/2 Kondo effect 2 Kondo effect ~~U/U/2: Quantum phase transitions2: Quantum phase transitions

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Conductance through coupled quantum dots

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