Mesoscopic structures: Electronic transport

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Mesoscopic structures Electronic transport

• Heterostructures The 2-d electron gas
• Quantum point contact The conductance quantum
  and Landauer formula
• Different transport regimes typical
  dimensions
• Diffusive transport
• Ballistic transport
• Aharonov-Bohm effect
• Quantum dots
• Dominance of Coulomb Interactions The Coulomb
  Blockade Regime
• Many-body effects The Kondo regime and the
  increase of conductance

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Heterostructures 2-d electron gas

• Heterostructures evolved from vacuum tubes to
  sophisticated sandwiches of semiconducting
  alloys grown with atomic precision and near
  perfect purity

p-n junction
clever manipulation of band structures
http//hyperphysics.phy-astr.gsu.edu/hbase/solids/
pnjun.html
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2d electron gas (1)
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2d electron gas (2)
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2D electron gas (3)
Non-equilibrium
Donor dopped
Equilibrium re-established
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Band structure profile
triangular confining potential
Low T
movement in the confinement direction (z) is
suppressed
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Properties of a 2d electron gas (2DEG)
high electron mobility!!
long mean free path!!
long relaxation time!!
easy to change the density!!
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Quantum point comtact (1)
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Quantum point contacts (2)

• The fact that the electron gas stays 100 ?
  from the surface of the sample allows for
  interesting manipulations of the gas density
  through the use of metallic gates.

The constriction generates a 1-d electron gas
connected on both sides to 2DEGs
conductance quantization
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Electronic transport in a QC (1d)
Energy is quantized in the y direction, creating
channels for electron propagation
W
1d
2d
2d
EF
Current is transported through the quantized
channels between the two Fermi levels
x
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Conductance Quantization Experiment
W
W
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Landauer Formula
n
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Landauer Formula (2)
Where is the dissipation? Where is the resistance?
The resistance came from a non-equilibrium
process where the Fermi energy are not well
defined.
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Landauer Formula (3)
The Landauer formula can be generalized for Tgt0
and many leads.
Differences with Ohm law 1) independent of L
2)
increase with W (or M).
References 1) R. Landauer, Philos. Mag. 21, 863
(1970) 2) D.J Thouless, Phys.Rev. Lett. 47, 972
(1981) 3) A.D. Stone and A. Szafer, IBM J. Res.
Develop. 32, 384 (1988) What is measured
when you measure resistance -The Landauer
formula resisited.
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Typical Dimensions and different regimes

• Dimensions of the mesoscopic system
• Mean free path (elastic scattering)
• Phase coherence length (inelastic scattering)

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Ballistic transport
Localization in dissordered systems exist
localization (Anderson localization). The length
associated to this process S deppend if the
amount of dissorder.
Three different regimes of transport
1) Ballistic L,W ltlt l lt S
2) Diffusive l lt L,W lt S
3) Localized l lt S lt L,W
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References

• C. J. Beenakker and H. van Houten, Solid State
  Physics vol. 44, 1 (1991), Quantum Transport in
  Semiconductor Nanostructures.
• C. J. Beenakker and H. van Houten, Physics
  Today, July 1996, pg. 22 Quantum Point Contacts.
• Suprio Datta Electronic Transport in
  Mesoscopic Systems, Cambridge University Press
  (1995).
• Y. Alhassid, Rev. Mod. Phys. 72, 895 (2000)
  The Statistical Theory of Quantum Dots.

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Quantum dots (1)
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Quantum Dots (2)
Coupled Qds
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Quantum dots (3)
Quantum rings
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Quantum dots (4)
Quantum dot artificial atom
Different number of electrons, complex many body
problem