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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!!!