Mesoscopic Physics - Theory of Condensed Matter

69 Pages·2012·2.88 MB·English

Mesoscopic Physics
Smaller is different
1. Theories of Anderson localization
2. Weak localization: theory and experiment
3. Universality and Random Matrix Theory
4. Metal-Insulator Transitions
5. Mesoscopic physics beyond condensed matter
References:
1. Thouless, Phys. Rep. 13, 93 (1974).
2. Lee, Ramkrishnan, Rev. Mod. Phys. 57,387 (1985) .
3. Kramer, MacKinnon, Rep. Prog. Phys. 56, 1469 (1993).
4. Boris Altshuler, Boulder Colorado Lectures
http://boulder.research.yale.edu/Boulder-2005/Lectures/index.html
1. Interference\tunneling effects in a solid.
2. These effects usually occur at
What is mesoscopic
intermediate scales and at relatively low
physics?
temperatures.
3. Disorder plays a role in most materials.
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Why is mesoscopic
physics.
physics interesting? 2. Continuation of quantum mechanics.
3. Technological applications.
Are 4+1 lectures Problems are easy to understand but
difficult to solve rigorously.
enough?
Lecture I: From Anderson to Anderson: perturbative
formalism and scaling theory of localization
1. Intuition about quantum dynamics in a
disordered potential. Anderson localization
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a. Anderson 1957: “Absence of diffusion in certain
random lattices”
b. Anderson, Abou-Chacra, Thouless, 1973: “A self-
consistent theory of localization”
3. Abrahms, Anderson, et al., 1979: “Scaling
theory of localization”
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P. W. Anderson
∇ 2 r
ˆ
H = − + V (r )
ˆ
HΨ = E ψ 2 m
α α α
r r r r
2
V(r)V(r') =V δ(r −r')
0
Metal
2222
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2
r (t) ∝t 2
r (t) ∝const
Insulator
t→∞ t→∞
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Ψ (r) ∝1/ V Ψ (r) ∝ e−r/ξ
loc
α α
kk ll >>>>11 kk ll>>11
FF FF
P(t) →cons
P(t) →0
t→∞
t→∞
Theories of localization
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One parameter scaling theory
6203 citations!
What if I place a particle in a random potential and wait?
Tight binding model
1. (Locator) expansion around V=0
2. Probability distribution needed
3. At V=V perturbation breaks down → Metal
c
=0 Metal
Increase V until
perturbation
theory breaks
down
= ∞ Metal
1. Problem with small denominators
However:
2. Uncorrelated paths?
VV >> VV VV << VV
c c
Correctly predicts a metal-insulator transition in 3d and
localization in 1d
Interactions?
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theory
250 citations
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metal
insulator
Γ = ImS (E +iη)
i
> 0
metal
= 0
insulator
∼ η