Transcript Slide 1

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
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Phonons are DELOCALIZED
ABSORPTION
Phonon-induced hopping
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Emission
Phonon-induced hopping
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Master equation:
Probabilities for an electron to be
on corresponding levels
Phonon-induced hopping
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Thermal equilibrium
Phonon-induced hopping
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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)
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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!!!