2D electron gas review

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2D electron gas (review)
Non-equilibrium
Donor dopped
Equilibrium re-established
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Quantum point comtact (review)
Landauer Formula
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Landauer Formula (review)
Where is the dissipation? Where is the resistance?
The resistance came from a non-equilibrium
process where the Fermi energy are not well
defined.
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).
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Mesoscopic structures Electronic transport

• Quick review of relevant ideas
• Gradual quantization of movement in all three
  directions
• Quantum dots
• Dominance of Coulomb Interactions The Coulomb
  Blockade Regime
• Vertical quantum dots Man-made artificial atoms.

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Quantization of movement in x, y and z

• 2D electron gas electrons constrained to move
  in the xy plane
• increased mobility and quantum effects
• Quantum point contacts electrons are free to
  move only along one direction
• quantization of conductance Landauer formula
• Final step electrons are constrained in all
  three directions (quantum dot)
• charge quantization Coulomb blockade
• Kondo effect many-body effects lead to an
  increase in conductance

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Quantum dots (1)
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Quantum dots
If parameters are just right current through
QD will carry one electron at a time
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Quantum Dots (2)
Quantum rings
Coupled Qds
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Modeling of a quantum dot
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Further modeling

• Quantum well (movement constrained in all 3
  directions)
• Contacts are modeled as controllable potential
  barriers

Level quantization
Position of the bottom of the well is
controlled by Vg (gate potential)
By varying Vg the levels can move up or down!!
2DEG
2DEG
To add an electron to the dot one has to pay
charging energy
Vg
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Coulomb blockade experimental

• How cold and how small should a conductor be so
  that adding a single electron to it will have a
  measurable effect?
• Keep Vsd 0 and constant, control Vg, measure
  the current

charging energy
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Strongly Correlated Model


Anderson Impurity Model
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Anderson model

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Kondo model
When n 1 the impurity Anderson model can be
mapped in the Kondo impurity model
(Schrieffer-Wolff transformation).
Important A resonant state appears at EF New
many body effects.
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Conductance
Vg
The conductance is defined as (0 bias)
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The effect of the temperature
Fermi distribution when T increase
Vg
Meir et al. Phys.Rev.B 23, 3048 (1991)
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Lateral vs. vertical quantum dots
Confining potential is not symmetric and the
irregular boundaries can lead to
single-particle dynamics which is mostly chaotic
Confining potential is regular and symmetric.
Hartree-Fock calculations of single-particle
wave functions can explain most of the
spectroscopic measurements
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Vertical QD structure
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Experimental results Magic Numbers
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Modelling 2d harmonic oscillator
Constant Interaction Model
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Wave functions
Fock-Darwin states
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References

• H. Tamura, K. Shiraishi and H. Takayanagi,
  Quantum dots and nanowires, ed. By S. Banyopahyay
  (2003)
• L. Kouwenhoven and C. Marcus, Quantum Dots,
  Physics World, 35 (June 1998)
• M. A. Kastner, The single electron transistor,
  Rev. Mod. Phys. 64, 849 (1992) Artificial Atoms,
  Physics Today, 24 (January 1993)
• L. Kouwenhoven et. al, Electron Transport in
  Quantum Dots, in Mesoscopic Electron Transport
  http//qt.tn.tudelft.nl/yuki/leo_publi/pub99.htm
• S. Reimann and M. Manninen, Electronic
  Structure of Quantum Dots, Rev. Mod. Phys. 74,
  1283 (2002)
• L. Kouwenhoven et al., Few-electron quantum
  dots, Rep. Prog. Phys. 64, 701 (2001)

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Coulomb Diamond
Wider peaks in the conductance
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Coulomb Blockade model
Adjust Vg and C so that
current can flow!!