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Transcript 
Electronic transport
in semiconductor nanostructures
Thomas Ihn
ETH Zürich
FS 17
After this lecture you know
and understand…
• … quantum interference in diffusive wires
• ... why conductance flutuations in diffusive
systems have a universal amplitude of e2/h
• ... the weak localization phenomenon
• ... the qualitative importance of the phase
coherence length or time
Book: Chapters 14.8, 15.1-5
Landauer-Büttiker theory
Low-temperature
"Conductance Is Transmission"
at the Fermi energy
Zero temperature :
e2 N
G(E F ) =
Tab (E F )
å
h a ,b =1
Finite temperatures :
æ df ö
G(T) = ò dE G(E)ç - ÷
è dE ø
0
¥
Rolf Landauer (1927-1999)
Markus Büttiker (1950- 2013)
Classical vs. Quantum Probabilities
classical (incoherent)
Transmission probability
Reflection probability
quantum (coherent)
T
T = |t|2
t is probability amplitude
(complex number)
R = 1- T
R = 1 – T = |r|2
r is probability amplitude
(complex number)
Sequential transmission
(logical AND)
T = T1 T 2
t = t1 t2
Product rule of probability Product rule of quantum
theory
mechanics
Alternative paths
(logical OR)
T = T1 + T2
Sum rule of probability
theory
t = t1 + t2
Sum rule of quantum
mechanics
quantum interference
see Feynman Lectures on Physics III: Quantum mechanics
Altshuler-Aronov-Spivak oscillations
in metallic cylinders
Magnesium cylinders
evaporated on
quartz filaments
Sharvin and Sharvin, 1981
Aharonov-Bohm oscillations in metal
rings
resistance
Webb et al, PRL 1985
magnetic field
Aharonov-Bohm oscillations in a CNT
h/e
A. Bachtold et al, Nature 1999
Oscillatory magnetoresistance in
ballistic n-GaAs ring structures
A. Fuhrer, 2001
Oscillatory magnetoresistance in
ballistic p-GaAs ring structures
F. Nichele, 2011
raw data
h/e
h/2e
Oscillatory magnetoresistance in a
Bi2Se3 (topological insulator)
nanoribbon
H. Peng et al,
Nature Mat. 2009
Altshuler-Aronov-Spivak
oscillations in an array of InGaAs
rings
F. Nagasawa et al,
PRL 2012
1600 rings
r = 608 nm
Weak localization
Particle diffusion on time-reversed paths
Magnetic field dependent resistance
Logarithmic temperature dependence
of the resistance in diffusive thin films
Temperature dependent
phase coherence time
Reading
Chapter 14.8: Conductance fluctuations
Chapter 15.1-4: Weak localization effect
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