Scattering theory of electricalconduction

Markus BüttikerUniversity of Geneva

Ecole de Physique Mesoscopique, Cargese 2008

Nano = length scale

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

Nanophysics and Mesoscopic physics

Quantum scattering theory of electron transport

1 nm = 10 Angstroem =

Mesoscopic PhysicsWave nature of electrons becomes important

Webb et al. 1985

Yacoby et al. 1995

Molecular conductors

Kouwenhoven (2004)

Books Electronic Transport in Mesoscoic SystemsS. 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 ArticlesQuantum 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

Phase coherence length

Elastic scattering length

Inelastic scattering length

Geometrical dimension

Macroscopic conductor

Mesoscopic conductor

(size of conductor)

(distance an electron travels before suffering a phase change of

(mean free path between elastic scattering events)

(distance an electron travels before loosing an energy kT)

)

Lecture contentsConductance from scattering theory,

eigen channels, conductance quantizationFour 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

Thermal and shot noise Two-particle Aharonov-Bohm effectEntanglement

Noise

Dynamic conductance Quantum pumping

4

Conductance from Transmission1. Single channel conductors

Conductance from transmission

Fermi energy right contact

applied voltage transmission probability

reflection probability

Heuristic discussion Fermi energy left contact

incident current

density

density of states

⇒ independent of material !!⇒

6

Landauer formula

Drift and diffusion

⇒

⇒

at constant Einstein relation

for space dependent⇒

⇒

7

Scattering matrix

⇒

tr

8

scattering state

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 ⇒One dimensional localization:

arbitrary array of scatterers

localization length

but is normal distributed characterize the sample through its distribution

9

Conductance from transmission

conductance quantum resistance quantum

dissipation and irreversibility

boundary conditions

10

Persistent current(periodic boundary conditions)

Buttiker, Imry and Landauer, Phys. Lett. 96A, 365 (1983).

11

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

⇒

⇒

⇒

⇒

H-invariant if momenta and magnetic field are reversed

⇒ ⇒ ⇒

13

but ⇒is an even function of magnetic field

Tuneable wave splitterButtiker, Imry, Azbel, Phys. Rev. A30, 1982 (1984)

15

Aharonov-Bohm conductance oscillations14

Gefen, Imry, Azbel, PRL 2004

Buttiker, Imry, Azbel, Phys. Rev. A30, 1982 (1984)