Effective potential in 2D

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Effective potential in 2D
Vg 0V
Vg -0.4V
Vg
V(x,y) - 2D subband energy
2D Schroedinger equation in the effective mass
approximation
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The electrostatic aperture a split gate
x
y
V(x,y)
SEM image of a typical surface-gate pattern for a
1D system
x
y
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Cyclotron Orbits
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Cyclotron Orbits
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Cyclotron Orbits (Molenkamp)
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Collimation
High Potential
Narrow
Wide
Low Potential
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Imaging Collimation (Westervelt)
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Imaging Cyclotron Orbits (Crook)
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Cyclotron Orbits and Obstacles (Ford)
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Skipping Orbits (Specter)
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Chaotic Motion (Marcus)
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Chaotic Motion (Marcus)
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Imaging Chaotic Orbits (Crook)
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Refraction in Optics

n gt 1
q
d
Refractive index gradient
d
q
n 1
d sin(q) v t
d sin(q) v t
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Classical Refraction
Vg 0

Graded Potential
Conservation of momentum
vF sin(q) vF sin(q)
Vg 0
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Classical Refraction (Specter)
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Classical Lens (Specter)
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Semi-classical transport theory in a magnetic
field
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Solution represents a shift of the Fermi surface
by Dk
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Drude conductance relations
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Drude conductance relations
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Hall and Shubnikov de Haas resistivities
y
x
rxx Ex / jx (VAB / L) / ( I/W) VAB/I
. W/L
rxy Ey /jx (VAC / W) / (I/W) VAC / I
Rxy
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J. Wakabayashi and S. Kawaji K. von Kiltzing,G.
Dorda and M. Pepper
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The Quantised Hall Effect An important step in
the direction of the experimental discovery was
taken in a theoretical study by the Japanese
physicist T. Ando. Together with his co-workers
he calculated that conductivity could at special
points assume values that are integer multiples
of e2 /h, where e is the electron charge and h is
Planck's constant. It could scarcely be expected,
however, that the theory would apply with great
accuracy.During the years 1975 to 1981 many
Japanese researchers published experimental
papers dealing with Hall conductivity. They
obtained results corresponding to Ando's at
special points, but they made no attempt to
determine the accuracy. Nor was their method
specially suitable for achieving great
accuracy.A considerably better method was
developed in 1978 by Th. Englert and K. von
Klitzing. Their experimental curve exhibits well
defined plateaux, but the authors did not comment
upon these results. The quantised Hall effect
could in fact have been discovered then.The
crucial experiment was carried out by Klaus von
Klitzing in the spring of 1980 at the
Hochfelt-Magnet-Labor in Grenoble, and published
as a joint paper with G. Dorda and M. Pepper.
Dorda and Pepper had developed methods of
producing the samples used in the experiment.
These samples had extremely high electron
mobility, which was a prerequisite for the
discovery.The experiment clearly demonstrated
the existence of plateaux with values that are
quantised with extraordinarily great precision.
One also calculated a value for the constant e2
/h which corresponds well with the value accepted
earlier. This is the work that represents the
discovery of the quantised Hall effect.
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Schroedingers equation in a magnetic field
The symmetric gauge
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Solution to B-field Schroedinger equation
1. Azimuthal symmetry
2. Change variables
3. Remove exponential dependence
Radial dependence
Energies
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Eigenstates
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Radial States
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Density of states in a Landau level

Maximum at
Radius of lth state
By definition
Area of lth state
Density of states enclosed by lth state
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Determining carrier density from rxx
n2D n eB/h
n2
n3
n4
n6
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Determining carrier density from rxx
1/B (e/hn2D ) n
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Determining carrier density from rxx
1/B (e/hn2D) n
/2
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Determining carrier density from rxy
rxy B/en2D
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J. Wakabayashi and S. Kawaji K. von Kiltzing,G.
Dorda and M. Pepper
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Disorder Broadening of Landau Levels
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Oscillation of the Fermi Energy
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Oscillation of the Fermi Energy
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Oscillation of the Fermi Energy
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Oscillation of the Fermi Energy
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Oscillation of the Fermi Energy
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Oscillation of the 2DEG capacitance
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Oscillation of 2DEG capacitance in a B-field
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Conductivity of a two-dimensional system
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Conductivity of a two-dimensional system
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Two-dimensional disorder potential
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A simple model
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Oscillation of the Fermi Energy
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Wide barrier bilayer - rxx
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Wide barrier bilayer - rxx
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Narrow barrier bilayer - rxx