Quantum_Hall_Effect

daughterduckUrban and Civil

Nov 15, 2013 (3 years and 8 months ago)

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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


n


n


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