A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča* The Kondo effect in multiple quantum dot...
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Transcript of A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča* The Kondo effect in multiple quantum dot...
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 V
V
ConductanceConductance
gV
12
n
~gateV
~sdV IA
( )gatesd
IG G V
V
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( ) [ ]
eG
h
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( ) [ ]
eG
h
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( ) [ ]
eG
h
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
( )
The Kondo effect in a quantum dotThe Kondo effect in a quantum dot
d U d
d
d U
( )
The Kondo effect in a quantum dotThe Kondo effect in a quantum dot
d U d
d
d U
( )
The Kondo effect in a quantum dotThe Kondo effect in a quantum dot
d U d
d
d U
( )
The Kondo effect in a quantum dotThe Kondo effect in a quantum dot
d U d
d
d U
( )
The Kondo effect in a quantum dotThe Kondo effect in a quantum dot
d U d
d
d U
( )
The Kondo effect in a quantum dotThe Kondo effect in a quantum dot
d U d
d
d U
( )
The Kondo effect in a quantum dotThe Kondo effect in a quantum dot
d U d
d
d U
( )
The Kondo effect in a quantum dotThe Kondo effect in a quantum dot
d U d
d
d U
( )
The Kondo effect in a quantum dotThe Kondo effect in a quantum dot
d U d
d
d U
( )
The Kondo effect in a quantum dotThe Kondo effect in a quantum dot
d U d
d
d U
( )
The Kondo effect in a quantum dotThe Kondo effect in a quantum dot
d U d
d
d U
( )
The Kondo effect in a quantum dotThe Kondo effect in a quantum dot
d U d
d
d U
( ) ( )1
2
d d U
UKT U e
The Kondo effect in a quantum dotThe Kondo effect in a quantum dot
The Kondo effect in a quantum dotThe Kondo effect in a quantum dot
The Kondo effect in a quantum dotThe Kondo effect in a quantum dot
The Kondo effect in a quantum dotThe Kondo effect in a quantum dot
/ 2d U
The Kondo effect in a quantum dotThe Kondo effect in a quantum dot
( / 2, )dA U
1
““Ring system” Ring system”
““OpenOpen system” 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
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)
The Kondo effect in a quantum dotThe Kondo effect in a quantum dot
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
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
loc max/M S S
21 /t t
( / 2, )dA U Fingerprints of Kondo…Fingerprints of Kondo…
/ 2d U
21 /t t
( / 2, )dA U Fingerprints of Kondo…Fingerprints of Kondo…
/ 2d U
loc max/M S S
21 /t t
( / 2, )dA U Fingerprints of Kondo…Fingerprints of Kondo…
/ 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
Double quantum dotDouble quantum dot
2d t
2(b)
0
t U
V
2(a) t U
212
K(c) 4 2t
J TU
2 12t t t''
2RKKY KJ T
Double quantum dotDouble quantum dot
2d t
2(b)
0
t U
V
2(a) t U
212
K(c) 4t
J TU
2 12t t t''
2(d) 0 & ~ : (4) Kondot V U SU
2d t
2(b)
0
t U
V
2(a) t U
212
K(c) 4t
J 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
Double quantum dotDouble quantum dot
2 Kondo2 Kondo
AFMAFM
1 Kondo1 Kondo
t t
12/ 3U t
12/ 50U t
12/ 60U t
1 2S S
Double quantum dotDouble quantum dot
2 Kondo2 Kondo
AFMAFM
1 Kondo1 Kondo
t t
12/ 3U t
12/ 50U t
12/ 60U t
1 2S S
Double quantum dotDouble quantum dot
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
4t
JU
A BS S
2
4t
JU
Zero magnetic field, thermal equilibrium
A
B
0, .B T const
2
4t
JU
/J U
Zero magnetic field, temperature
A
B
2
4t
JU
/J U
A
B
Zero magnetic field, temperature
/J U2 cJ
A
B
Zero magnetic field, temperature
/J U
3 ~cJ
A
B
Zero magnetic field, temperature
Triple quantum dotTriple quantum dot
d
22d t
22d t
d
22d t
22d t
d U
Triple quantum dotTriple quantum dot
d
Triple quantum dotTriple quantum dot
d
Triple quantum dotTriple quantum dot
Triple quantum dotTriple quantum dot
d
2/ 1U t
10
20
Triple quantum dotTriple quantum dot
2
5
Triple quantum dotTriple quantum dot
1~ Kt T
Triple quantum dotTriple quantum dot
1~ Kt T
Triple quantum dotTriple quantum dot
1~ Kt T
~t J
Triple quantum dotTriple quantum dot
1~ Kt T
~t J
1~ 2 KJ T
Triple quantum dotTriple quantum dot
Deformable molecules Deformable molecules
e
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
Molecules with a center of mass motionMolecules with a center of mass motion
Molecules with a center of mass motionMolecules with a center of mass motion
Molecules with a center of mass motionMolecules with a center of mass motion
Molecules with a center of mass motionMolecules with a center of mass motion
Friedel sum rule:
Molecules with a center of mass motionMolecules with a center of mass motion
Friedel sum rule:
Molecules with a center of mass motionMolecules with a center of mass motion
A
B
Molecules with a center of mass motionMolecules with a center of mass motion
Molecules with a center of mass motionMolecules with a center of mass motion
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