Disorder and chaos in quantum system:

Anderson localization and its generalization

Boris Altshuler (Columbia)Igor Aleiner (Columbia)

(6 lectures)

Lecture # 3Inelastic transport in insulators

(Hopping conductivity)

• Phonon assisted hopping• Miller-Abrahams random resistors network• How to find the resistance of a random resistor

network?• Mott variable range hopping• Phononless ac-conductivity

Temperature dependence of the conductivity

DoS DoSDoS

Assume that all the states are localized

Phonon-induced hopping

Phonons are DELOCALIZED

ABSORPTION

Phonon-induced hopping

Emission

Phonon-induced hopping

Master equation: Probabilities for an electron to beon corresponding levels

Phonon-induced hopping

Thermal equilibrium

Phonon-induced hopping

Thermal equilibrium

Apply electric field:

Miller-Abrahams network (1960)

Miller-Abrahams network (1960)

Miller-Abrahams network (1960)

Qn: Find totalconductance of thenetwork

Miller-Abrahams network (1960)Simplification: nearest neighborhopping

Qn: Find totalconductance of thenetwork

Dependence on dimensionality:

Qn: Find totalconductance of thenetwork

D=1

Conductance is determined by the weakest link, are there is no way to bypass it one dimensions;

Dependence on dimensionality:

Qn: Find totalconductance of thenetwork

D=2,3

One can always bypass the weakest link.

Rare configurations are not important

Duality in D=2 (Dykhne,1970)

Strongly fluctuating

Duality in D=2 (Dykhne,1970)

Change variables:

Duality in D=2 (Dykhne,1970)

For any realization of disorder:

Not known

Duality in D=2 (Dykhne,1970)

For many interesting distributions

Duality in D=2 (Dykhne,1970)

Two phase model:

Duality in D=2 (Dykhne,1970)

Nearest neighbor hopping

Observable conductanceis determined by typicalconfigurations

Variable range hopping (Mott, 1968)

Idea:

Use hops much longer than to decrease the activation energy

Optimal hop:

Temperature dependence of the conductivity (some answers)

DoSDoS

Phonon assisted hopping

Phonon-less a.c. conductivity (Mott,1970)

We have just learned • Electric transport in insulator are determined

by inelastic processes• Transport due to inelastic processes are

described by classical random networks• Results are often determined by optimal paths

Thank you very much!!!