A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *

A. Ramšak*J. MravljeT. Rejec*R. ŽitkoJ. Bonča*

The Kondo effect in multiple quantum dot

systems and deformable molecules

Department of PhysicsFaculty of Mathematics and Physics

University of Ljubljana

*

OutlineOutline

(1) Conductance (2) Kondo in a single quantum dot(3) Methods(4) Double quantum dots(5) Triple quantum dots(6) Deformable molecules(7) Center-of-mass motion(8) Summary

gV

12

n

~gateV

~sdV IA

( )gatesd

IG G VV

ConductanceConductance

Non-interacting systems: U=0

12

n

0.5 1 1.5 2 2.5 3

0

0.2

0.4

0.6

0.8

1( )dot

12

n

0.5 1 1.5 2 2.5 3

0

0.2

0.4

0.6

0.8

122( ) [ ]eGh

dot( ) ( )G

Non-interacting systems: U=0

The Anderson model: U > 0

d U

d

0 0.5 1 1.5 2 2.5 3

0

0.2

0.4

0.6

0.8

122( ) [ ]eGh

0U

0 0.5 1 1.5 2 2.5 3

0

0.2

0.4

0.6

0.8

1

( )

0 0.5 1 1.5 2 2.5 3

0

0.2

0.4

0.6

0.8

1

0U

U

or

0 0.5 1 1.5 2 2.5 3

0

0.2

0.4

0.6

0.8

122( ) [ ]eGh

0 0.5 1 1.5 2 2.5 3

0

0.2

0.4

0.6

0.8

1

0U

U

or

KT T

T

d U d

d

d U

( )

0

The Kondo effect in a quantum dotThe Kondo effect in a quantum dot

d U d

d

d U

( )


d U d

d

d U

( ) ( )1

2

d d UU

KT U e


/ 2d U


( / 2, )dA U

1

““Ring system” Ring system”

““OpenOpen system” system”

the GS energy of a the GS energy of a largelarge ring ring system is an universal function system is an universal function of fluxof flux

T. Rejec and A. Ramšak, PRB 68, 033306 (2003); 68 035342 (2003)

IF IF openopen system is Fermi liquid system is Fermi liquid

4 1( 3); elN n N even

1N

4 1( 3); elN n N odd

1N

0/g G G

Linear conductance from Linear conductance from the the ground-state energyground-state energy

0T

See also: J. Favand and F. Mila (Phys. J. 1998); O. Sushkov (PRB 2001); R. Molina et al. (PRB 2003)

Linear conductance from Linear conductance from the the ground-state energyground-state energy

0T

Aharonov – Bohm ringsAharonov – Bohm rings

Broken time-reversal symmetry

T. Rejec and A. Ramšak, PRB 68, 033306 (2003)


U=0

numerical tests…numerical tests…

The Kondo effect in a quantum dot: finite temperatureThe Kondo effect in a quantum dot: finite temperature

T

U=0 high T

low T

The Kondo effect in a quantum dot: finite temperatureThe Kondo effect in a quantum dot: finite temperature

T

high T

low T

loc max/M S S

21 /t t

( / 2, )dA U Fingerprints of Kondo…Fingerprints of Kondo…

/ 2d U

21 /t t


/ 2d U

loc max/M S S

21 /t t


/ 2d U

Chan et al, Nanotechnology 15, 609 (2004)

Vidan et al, Applied Phys. Lett. 85, 3602 (2004)

Electrostatic gates

QD

Elzerman et al, PRB 67, 16308 (2003)

QD

MultipleMultiple quantum dot systems quantum dot systems

Double quantum dotDouble quantum dot


2d t

2(b)

0t UV

2(a) t U

212

K(c) 4 2tJ TU

2 12t t t''

2RKKY KJ T


2d t

2(b)

0t UV

2(a) t U

212

K(c) 4 tJ TU

2 12t t t''

2(d) 0 & ~ : (4) Kondot V U SU

2d t

2(b)

0t UV

2(a) t U

212

K(c) 4 tJ TU

2(d) 0 & ~ : (4) Kondot V U SU

J. Mravlje, A. Ramšak, and T. Rejec, Phys. Rev. B 73, 241305(R) (2006)

Also for finite hibridization: [ (4)] ~ [ (2)]K KT SU U T SU

half filling: 2n

hibridization


2 Kondo2 Kondo

AFMAFM

1 Kondo1 Kondo

t t

12/ 3U t

12/ 50U t

12/ 60U t

1 2S S

Other topologies: local singlet vs the Kondo effect

A. Ramšak, J. Mravlje, R. Žitko, and J. Bonča, quant-ph/0608065.

Thermal equilibrium: A-B spin corelations

/B J/T J

2

4tJU

A BS S

2

4tJU

Zero magnetic field, thermal equilibrium

A

B

0, .B T const

2

4tJU

/J U

Zero magnetic field, temperature

A

B

2

4tJU

/J U

A

B


/J U2 cJ

A

B


/J U

3 ~cJ

A

B


Triple quantum dotTriple quantum dot

d

22d t

22d t

d

22d t

22d t

d U


d


2/ 1U t

10

20


2

5


1~ Kt T


1~ Kt T

~t J


1~ Kt T

~t J

1~ 2 KJ T

Deformable molecules Deformable molecules

e

Deformable molecules Deformable molecules

H. Park et al. Nature 407 (2000)

e

Change in:• local energy• hopping matrix elements

Modeling Modeling

))(1( aanMaanUnnH d

2eff (2) 2 (1) 2U E E U M

Isolated molecule:

d

d

molecule attached to the leads: Lang & Firsov transformation:

)1)((1 ,~

naaMeUHUUH

the result:

,M

,M

Reduction of U and narrowing of the level-widthA.C. Hewson and D.M. News J.Phys C 13 (1980)K. Schönhammer and O. Gunnarsson PRB 30 (1984)

Old knowledge …Old knowledge …

Udecrease

negative U:A. Taraphder and P. Coleman,PRL 66, 2814 (1991).

M

J. Mravlje, A. Ramšak, and T. Rejec, PRB 72, 121403(R) (2005);See also: P.S. Cornaglia, D.R. Grempel, and H. Ness, Phys. Rev. B 71, 075320 (2005), A. Mitra, I. Aleiner, and A.J. Millis, Phys. Rev. B 79, 245302 (2004).

Molecules with a center of mass motionMolecules with a center of mass motion

J. Mravlje, A. Ramšak, and T. Rejec, submitted to PRB


Friedel sum rule:


A

B

Summary Summary

• Linear conductance at T=0 can then be extracted from the GS energy:

• The Kondo effect: Low temperature destiny of quantum dots

U t

d

U

d

t0/g G G

ad summary…

Formulae are exact IF the system is Fermi liquid

note:

• linear conductance• zero temperature• non-interacting single-channel leads

Fisher – Lee relation …

Conductance formalismsConductance formalisms

non-equilibrium transport: T ≠ 0, V ≠ 0

U = 0

Landauer – Büttiker formula

linear response regime: T ≠ 0, V ~ 0

zero-temperature linear response: T = 0, V ~ 0

U ≠ 0

Meir – Wingreen formula

In Fermi liquid systems

Kubo formula

Proof of the method

Step 1. Conductance of a Fermi liquid system at T=0

Kubo

T=0

define (n.i.: Fisher-Lee)

‘Landauer’

Step 2. Quasiparticle hamiltonian (Landau Fermi liquid)

Step 3. Quasiparticles in a finite system

N

Step 4. Validity of the conductance formulas

A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *

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