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

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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

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Temperature dependence of the conductivity DoS Assume that all the states are localized

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Phonon-induced hopping Phonons are DELOCALIZED ABSORPTION

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Phonon-induced hopping Emission

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Phonon-induced hopping Master equation: Probabilities for an electron to be on corresponding levels

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Phonon-induced hopping Thermal equilibrium

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Phonon-induced hopping Thermal equilibrium Apply electric field:

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Miller-Abrahams network (1960)

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Qn: Find total conductance of the network

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

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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;

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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

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Duality in D=2 (Dykhne,1970) Strongly fluctuating

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Duality in D=2 (Dykhne,1970) Change variables:

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Duality in D=2 (Dykhne,1970) For any realization of disorder: Not known

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Duality in D=2 (Dykhne,1970) For many interesting distributions

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Duality in D=2 (Dykhne,1970) Two phase model:

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Duality in D=2 (Dykhne,1970) Nearest neighbor hopping Observable conductance is determined by typical configurations

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Variable range hopping (Mott, 1968) Idea: Use hops much longer than to decrease the activation energy Optimal hop:

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Temperature dependence of the conductivity (some answers) DoS Phonon assisted hopping

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

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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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