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E�ect of �ux-dependent Friedel-oscillations uponthe e�ective
transmission of an interacting
nano-system
A. Freyn and J.-L. Pichard
DSM/DRECAM/SPEC - CEA Saclay
March 30, 2007
A. Freyn E�ective transmission of an interacting system
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Landauer-Approach for an interacting nano-system (T = 0)
The idea
Landauer-Approach without interaction: g = 2e2
h|t(EF )|2
Generalization with interaction: g = 2e2
h|te� (EF )|2
The e�ect
U = 0: Scatterer is localt
U 6= 0: Scatterer becomes non-localteff
effective
length
A. Freyn E�ective transmission of an interacting system
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The studied system
LC
L′C
RingΦ
Nano-system
tS
AB-scatterer
LR
Φ
−th−th −th−th
VG VG
0 1−td −tc−tc
U
Geometry: LC , L′C , LR
Hopping element: td
Local gate potential: VG
Nearest-neighbor interaction: U
A. Freyn E�ective transmission of an interacting system
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Hartree-Fock theory
Hamiltonian
H = Hnano-system + Hlead + Hcoupling
Hnano-system = −td (c+0 c1 + c+1 c0) + Un0n1 + VG (n0 + n1)
Hartree-Fock equations
V0 = VG + U〈c†1c1(v ,V0,V1)
〉V1 = VG + U
〈c†0c0(v ,V0,V1)
〉v = td + U
〈c†0c1(v ,V0,V1)
〉Transmission of the nano-system
|tS(kF )|2 =∣∣∣∣ (exp−2IkF −1)t2c v∏
i (Vi + 2 cos(kF )− t2c expIkF )− v2
∣∣∣∣2A. Freyn E�ective transmission of an interacting system
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e�ective transmission of the nano-system |tS |2 is non-local
Flux-dependence of |tS |2, kF = π2 for di�erent interactions
U
0.5
1
1.5
U = 2
2.5
3
3.5
4
0
0.2
0.4
0.6
0.8
1
|tS|2
|tS|2
−4 −3 −2 −1 0 1
VGVG
td = 0.1, LC = 2, cross: Φ = 0, circle: Φ/Φ0 = 0.5
A. Freyn E�ective transmission of an interacting system
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e�ective transmission of the nano-system |tS |2 is non-local
Dependence of the Distance LC between nano-system and
AB-scatterer upon |tS |2
0
0.25
0.5
0.75
|tS
|2|t
S|2
0 10 20 30 40 50
LCLC
U = 1, LC = 2, cross: Φ = 0, circle: Φ/Φ0 = 0.5
A. Freyn E�ective transmission of an interacting system
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e�ective transmission of the nano-system |tS |2 is non-local
Flux-dependence of |tS |2, kF = π8
U = 0.511.522.5
3
3.5
4
0
0.2
0.4
0.6
0.8
1
|tS|2
|tS|2
−4 −3 −2 −1 0
VGVG
td = 0.1, LC = 2, cross: Φ = 0, circle: Φ/Φ0 = 0.5
A. Freyn E�ective transmission of an interacting system
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An equivalent model
The Model
− tc√2
− tc√2
+ tc√2
− tc√2
−th−th −th−th −th−th
A
S
vAS
VA
VS
The Transformation
cS =1√2π
(c0 + c1)
cA =1√2π
(c0 − c1)
The Parameters
VS = VG − tdVA = VG + td
vAS = 0
A. Freyn E�ective transmission of an interacting system
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E�ect of external scatterers
Friedel oscillations of the density 〈c+p cp〉Friedel-like
oscillations of the correlation function 〈c+p cp+1〉
Flux-dependent correlation-function 〈c+p cp+1〉 of the
AB-scatterer
0.25
0.3
0.35
0.4
〈
c+ pc p
+1
〉〈
c+ pc p
+1
〉
0 5 10 15 20 25 30
pp
ΦΦ0
= 0
ΦΦ0
= 12
A. Freyn E�ective transmission of an interacting system
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The equivalent model is solvable in the trivial limit (td �
1)
In the trivial limit, 〈nS〉 ≈ 1 and 〈nA〉 ≈ 0 and |tS |2 can
becalculated analytically
only small dependence of external scatterers
Comparison with the exact results
0
0.25
0.5
0.75
1
|tS|2
|tS|2
−10 −5 0 5 10
VGVG
td = 4 3 2 1 .1
.1
A. Freyn E�ective transmission of an interacting system
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An equivalent model - interesting limit (td � 1)
large dependence of external scatterers
e�ective transmission |tS |2 of the nano-system
0.1
0.2
0.3
0.4
0.5
td = 1
0
0.2
0.4
0.6
0.8
1
|tS|2
|tS|2
−4 −3 −2 −1 0 1 2
VGVG
A. Freyn E�ective transmission of an interacting system
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Conductance of the complete system
Total conductance gT as function of �ux Φ
0
0.1
0.2
0.3
0.4
0.5
gT
gT
0 0.5 1 1.5 2 2.5 3
Φ/Φ0Φ/Φ0
U=1U=0
A. Freyn E�ective transmission of an interacting system
-
Conclusion & References
Conclusion
E�ective transmission of an interacting nano-system is
non-local
this non-locality can be very important
�Equivalent� model useful to understand the e�ect
References
R.A. Molina, D. Weinmann, J.-L. Pichard, EPJ. B 48, 243
(2005).
Y. Asada, A. Freyn, and J.-L. Pichard, EPJ. B 53, 109
(2006).
A. Freyn, and J.-L. Pichard, to be published in PRL.
A. Freyn, and J.-L. Pichard, in preparation.
A. Freyn E�ective transmission of an interacting system
