Effective potential in 2D
V
g
V
g
= 0V
V
g
=

0.4V
2D Schroedinger equation
in the effective mass
approximation
V(x,y)

2D subband energy
The electrostatic aperture: a split gate
V(x,y)
x
y
SEM image of a typical surface

gate
pattern for a 1D system
y
x
Cyclotron Orbits
F=mv
.
Cyclotron Orbits
Cyclotron Orbits (Molenkamp)
Narrow
Wide
Collimation
High Potential
Low Potential
Imaging Collimation (Westervelt)
Imaging Cyclotron Orbits (Crook)
Cyclotron Orbits and Obstacles (Ford)
Skipping Orbits (Specter)
Chaotic Motion (Marcus)
Chaotic Motion (Marcus)
Imaging Chaotic Orbits (Crook)
n = 1
n > 1
Refractive index
gradient
Refraction in Optics
‘
d
d
q
q
d sin(
q
⤠㴠=⁴
搠d楮(
q
’) = v’ t
Classical Refraction
V
g
= 0
V
g
= 0
Graded Potential
‘
v
F
sin(
q
) = v’
F
sin(
q
’)
Conservation of momentum:
Classical Refraction (Specter)
Classical Lens (Specter)
Semi

classical transport theory in a magnetic field
Solution represents a shift of the Fermi surface by
D
k
Drude conductance relations
Drude conductance relations
r
xx
=
E
x
/
j
x
= (
V
AB
/
L
) / (
I
/
W
) =
V
AB
/
I
.
W/L
r
xy
=
E
y
/
j
x
= (
V
AC
/
W
) / (
I
/
W
) =
V
AC
/
I
=
R
xy
Hall and Shubnikov de Haas resistivities
x
y
J. Wakabayashi and S. Kawaji
K. von Kiltzing,G. Dorda and M. Pepper
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 e
2
/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 e
2
/h which corresponds well with the value accepted earlier. This is
the work that represents the discovery of the quantised Hall effect.
Schroedinger’s equation in a magnetic field
The symmetric gauge
1. Azimuthal symmetry:
2. Change variables:
3. Remove exponential dependence:
Energies:
Radial dependence:
Solution to B

field Schroedinger equation
Eigenstates
Radial States
Density of states in a Landau level
~
Maximum at:
By definition:
Radius of l
th
state:
Area of l
th
state:
Density of states enclosed by l
th
state:
Determining carrier density from
r
xx
n
2D
=
n
敂⽨e
n
=2
n
㴳
n
㴴
n
㴶
Determining carrier density from
r
xx
1/B =
(
e
/
hn
2D
)
n
1/B =
(
e/hn
2D
)
n
/2
Determining carrier density from
r
xx
Determining carrier density from
r
xy
r
xy
=
B
/
en
2D
J. Wakabayashi and S. Kawaji
K. von Kiltzing,G. Dorda and M. Pepper
Disorder Broadening of Landau Levels
Oscillation of the Fermi Energy
Oscillation of the Fermi Energy
Oscillation of the Fermi Energy
Oscillation of the Fermi Energy
Oscillation of the Fermi Energy
Oscillation of the 2DEG capacitance
Oscillation of 2DEG capacitance in a B

field
Conductivity of a two

dimensional system
Conductivity of a two

dimensional system
Two

dimensional disorder potential
A simple model
Oscillation of the Fermi Energy
Wide barrier bilayer

r
xx
Wide barrier bilayer

r
xx
Narrow barrier bilayer

r
xx
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