# Disorder and chaos in quantum system: Anderson localization and its generalization Boris Altshuler (Columbia) Igor Aleiner (Columbia) (6 lectures)

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Disorder and chaos in quantum system: Anderson localization and its generalization Boris Altshuler (Columbia) Igor Aleiner (Columbia) (6 lectures)

Lecture # 3 Inelastic 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 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 be on corresponding levels

Phonon-induced hopping   Thermal equilibrium

Phonon-induced hopping   Thermal equilibrium Apply electric field:

Miller-Abrahams network (1960)

Qn: Find total conductance of the network

Miller-Abrahams network (1960) Simplification: nearest neighbor hopping Qn: Find total conductance of the network

Dependence on dimensionality: Qn: Find total conductance of the network D=1 Conductance is determined by the weakest link, are there is no way to bypass it one dimensions;

Dependence on dimensionality: Qn: Find total conductance of the network 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 conductance is determined by typical configurations

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) DoS 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!!!

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