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Disorder and chaos in quantum system: Anderson localization and its generalization (6 lectures) Igor Aleiner (Columbia) Boris Altshuler (Columbia) 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 Assume that all the states are localized DoS DoS DoS 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 Apply electric field: Thermal equilibrium Miller-Abrahams network (1960) Miller-Abrahams network (1960) 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: D=1 Qn: Find total conductance of the network Conductance is determined by the weakest link, are there is no way to bypass it one dimensions; Dependence on dimensionality: D=2,3 Qn: Find total conductance of the network 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 Two phase model: (Dykhne,1970) Duality in D=2 Nearest neighbor hopping (Dykhne,1970) 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 DoS 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!!!