Theory of quantum transport in nanostructures

Theory of quantum transport
in nanostructures
Carlo Maria Canali
Linnaeus University
13 April 2012 – Lecture 5

Quantum Corrections to semiclassical transport
Outline
A. Quantum Interference
B. Phase shifts
C. Double tunnel junction
D. Aharonov-Bohm rings
E. Experimental evidence of quantum interference in
nanostructures
- Weak Localization (WL) corrections
- Universal Conductance Fluctuations (UCF)

A. Quantum Interference
2
P QM = | A1
+ A2
| = | A1
| + | A2
| + 2Re( A1
A2
)
2
=
P 1 +
P2
+
2 P1
P2
cos(
φ
) =
PCL
+
2
P1
P2
cos(
φ
)
2
*

B. Phase shifts

C. Double-tunnel junction

D. Aharonov-Bohm ring
χ :dynamical
phase

T-bean splitter
3
2 1
Lead 1 and 2 are symmetric
Electron coming from lead 3
has zero amplitude to be
reflected

Examples of interefering trajectories
in an AB ring

(b) ()
() (a)

Φ B
(d)
−Φ B
(e)

E. Experiments on quantum interference
Sharvin & Sharvin (1981):
weak localization li corrections
Periodicity:
Φ = Φ 0
300
L
/ L φ =
L
L φ

Weak localization in disordered conductors
2
2
2
*
| A1 + A2
| = | A1
| + | A2
| 2Re( A1
A2
P Q =
+
Quantum interference increase the time the electron
Q
spends in the loop decrease in conductance!
)

Universal conductance fluctuations
Webb et al, PRL 1985
20 X Φ 0
First fully coherent device!
Periodicity of MR:
Φ = 2Φ 0