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Quantum conductance and indirect exchange interaction (RKKY interaction)
Conductance of nano-systems with interactions coupled via conduction electrons: Effect of indirect exchange interactions
cond-mat/0605756 to appear in Eur. Phys. J. B
Yoichi Asada (Tokyo Institute of Technology) Axel Freyn (SPEC), JLP (SPEC).
Interacting electron systems between Fermi leads:Effective one-body transmission and correlation clouds
Rafael Molina, Dietmar Weinmann, JLP Eur. Phys. J. B 48, 243 (2005)
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Scattering approach to quantum transport
22
,2
UEth
e
V
IG Feff
UEt Feff ,
SContact (Fermi) Contact (Fermi)
S(U)Fermi Fermi
1. Nano-system inside which the electrons do not interact
One body scatterer
Many body scatterer
effective one body scatterer
Value of ?Size of the effective one body scatterer?
Relation with Kondo problem
Carbon nanotubeMolecule,Break junctionQuantum dot of high rs Quantum point contact g<1YBaCuO…
2. Nano-system inside which the electrons do interact
222
FEth
e
V
IG
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How can we obtain the effective transmission coefficient?
The embedding method
How can we obtain ? Density Matrix Renormalization Group
Embedding + DMRG = exact numerical method.
Difficulty: Extension outside d=1
Permanent current of a ring embedding the nanosystem + limit of infinite ring size
2,UEtI Feff
I
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How can we obtain the size of the effective one body scatterer?
2 scatterers in series
• Are there corrections to the combination law of one body scatterers in series? Yes
• This phenomenon is reminiscent of the RKKY interaction between magnetic moments.
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Combination law for 2 one body scatterers in series
tt CF
T
SLST
Lcik
Lcik
L
S
Lk
tt
MMMM
e
eM
tt
rt
r
tM
F
F
42
42
*
*
*
22cos112
..
0
0
1
1
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Half-filling: Even-odd oscillations + correction
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The correction disappears when the length of the coupling lead increases with a power law
C
SC
L
LUALg
,
Correction:
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Magnitude of the correction
U=2 (Luttinger liquid – Mott insulator)
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RKKY interaction(S=spin of a magnetic ion or nuclear spin)
1S2S
).(1
'',
)'(kiikk
i kk
rkkieS
eSSe
SCaaeV
H
HHHH
i
4
)2sin()2cos(
.
ij
ijFijFijij
jii ij
ijRKKY
R
RkRkRJ
SSJH
R
RkJ
d
F )2cos(
1
Zener (1947)Frohlich-Nabarro (1940)Kasuya(1956)Yosida(1957)Ruderman-Kittel(1954)Van Vleck(1962)Friedel-Blandin(1956)
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The two problems are related: Electon-electron interactions (many body effects) are necessary.
The spins are not
SPINS:
Nano-systems with many body effects:
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Spinless fermions in an infinite chain with repulsion between two central sites.
L
iiiii cccctH
111
SL
iii VnVnU
21
)(v 01 vccUt
Mean field theory: Hartree-Fock approximation
1t2
1V (if half-filling)
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Reminder of Hartree-Fock approximation
The effect of the positive compensating potential cancels
the Hartree term. Only the exchange term remains
0''.'
'''.')(2
2
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rrrUdrrUrU
rErrrrrUdrrrUrm
h
jj
ion
iijijj
ii
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Hartree-Fock approximation for a 1d tight binding model
0,'1,1,'0,',
*',',
'',
'
.'.1.1.
pppppp
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HFpp
p
HFpp
UU
ppUt
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HF
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1,01,0
01*
1,0 .01.
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1 nanosystem inside the chain
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Hartree-Fock describes rather well a very short nanosystem
DMRG
Hartree-Fock
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2 nanosystems in series
CL
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The results can be simplified at half-filling in the limit
1/Lc correction with even-odd oscillations characteristic of half filling.
0U
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Conductance of 2 nanosystems in series
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Conductance for 2 scatterers
16
32
F
F
k
k
4.0U
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Hartree-Fock reproduces the exact results (embedding method, DMRG + extrapolation)
when U<t
2
Fk
DMRG
Hartree-Fock
Correction ),( FkUA
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Role of the temperature
• The effect disappears when
).2
.(Tk
hvL
LL
BFT
Tc
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How to detect the interaction enhanced non locality of the conductance ?
(Remember Wasburn et al)
U
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Ring-Dot system with tunable coupling (K. Ensslin et al, cond-mat/0602246)