Scattering theory of electrical
conduction
Markus Büttiker
University of Geneva
Ecole de Physique Mesoscopique, Cargese 2008
Nanophysics and Mesoscopic physics
Nano = length scale
1 nm = 10 Angstroem =
Nano physics = widely used expression for physics on the small
atomistic length scale
Mesoscopic physics = in between the atomic scale and
the macroscopic scale
Common ground:
Wave nature of electrons becomes important
Quantum scattering theory of electron transport
Mesoscopic Physics
Wave nature of electrons becomes important
Webb et al. 1985
Yacoby et al. 1995
Molecular conductors
Kouwenhoven (2004)
Books
Electronic Transport in Mesoscoic Systems
S. Datta, Cambridge Unversity Press, 1995
Introduction to Mesoscopic Physics,
Y. Imry, Oxford University Press, 1997.
Mesoscopic Physics of Electrons and Photons
E. Akkermans and G. Montambaux, Cambridge University Press, 2007
Review Articles
Quantum Transport in Semiconductor Nanostructures
C.W. J. Beenakker , H. van Houten, Solid State Physics 44, 1 (1991)
Shot Noise in Mesoscopic Conductors
Ya. M. Blanter, M. Buttiker , Phys. Rep. 336, 1 (2000).
Length scales
Geometrical dimension
(size of conductor)
Phase coherence length
(distance an electron travels before suffering a phase change of
Elastic scattering length
(mean free path between elastic scattering events)
Inelastic scattering length
(distance an electron travels before loosing an energy kT)
Macroscopic conductor
Mesoscopic conductor
)
Lecture contents
Conductance from scattering theory,
eigen channels, conductance quantization
Four probe resistances,
Reciprocity and Onsager relations,
Edge states and quantum Hall effect
Voltage probes,
From coherent to incoherent transport,
local, global and partial density of states
Point contact measurements.
electrochemical and electrostatic potentials
Dynamic conductance
Quantum pumping
Noise
Thermal and shot noise
Two-particle Aharonov-Bohm effect
Entanglement
4
Conductance from Transmission
1. Single channel conductors
Conductance from transmission
Heuristic discussion
Fermi energy left contact
Fermi energy right contact
applied voltage
transmission probability
reflection probability
incident current
density
density of states
⇒
independent of material !!
⇒
Landauer formula
6
Drift and diffusion
at constant
Einstein relation
⇒
for space dependent
⇒
⇒
⇒
7
Scattering matrix
8
scattering state
r
t
scattering matrix
current conservation
⇒
S is a unitray matrix
In the absence of a magnetic field S is an orthogonal matrix
Transfer matrix
Transfer matrix is muliplicative
⇒
9
arbitrary array of scatterers
One dimensional localization:
localization length
is normal distributed
but
characterize the sample through its distribution
10
Conductance from transmission
conductance quantum
resistance quantum
dissipation and irreversibility
boundary conditions
Persistent current
11
(periodic boundary conditions)
Buttiker, Imry and Landauer, Phys. Lett. 96A, 365 (1983).
Mapping of ring on crystal
Particle in a periodic potential
fonction de Bloch , Brillouin zone of width
note that V(x) and u(x) have the same period
Eigenvalues come in “bands” E(k) with
Particle on a ring
Comparison with particle in a periodic potential
shows that
Current conservation
Scattering matrix is a unitary matrix
⇒
⇒
⇒
⇒
⇒
12
Magnetic field symmetry
13
H-invariant if momenta and magnetic field are reversed
⇒
⇒
⇒
⇒
⇒
⇒
⇒
but
⇒
is an even function of magnetic field
Tuneable wave splitter
Buttiker, Imry, Azbel, Phys. Rev. A30, 1982 (1984)
15
14
Aharonov-Bohm conductance oscillations
Gefen, Imry, Azbel, PRL 2004
Buttiker, Imry, Azbel, Phys. Rev. A30, 1982 (1984)