Integer Quantum Hall Effect

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Integer Quantum Hall Effect

• By
• Priyanka Milinda Rupasinghe

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Discovery of Hall Effect

• When an electric current passes through a metal
  strip with a perpendicular magnetic field, the
  electrons are deflected towards one edge and a
  potential difference is created across the strip.
  This phenomenon is termed the Hall Effect.

It was discovered in 1879 by an American
physicist E.H. Hall. And this effect has now been
thoroughly studied and well understood in common
metals and semiconductors.

• In Hall Effect experiments, It is observed that
  the Hall resistance (RH ) increases linearly with
  the applied magnetic field B.

• Entirely new phenomena appear when the Hall
  effect is studied in two-dimensional electron
  systems.

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Discovery of Quantum Hall Effect

• In the Spring of 1980 von Klitzing showed
  experimentally that the Hall conductivity has
  discrete values in two-Dimensional systems.
• As a result, in 1985, Klaus von Klitzing was
  awarded Nobel Prize in Physics for the discovery
  of quantum Hall effect.

Klaus von Klitzing
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The Landau Levels

• In a strong magnetic field the energy of the
  electron is quantized.
• These discrete energy values are given by

n 0,1,2,
Where is the Cyclotron frequency
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Longitudinal and Hall conductivities

• In the presence of a steady magnetic field, the
  conductivity and resistivity become tensors.

Here, and are called the
Longitudinal and Hall
conductivities
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Measurement of the Hall and Longitudinal
Resistivities
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The Filling Factor
Where e is the electronic charge and h is the
plank constant.
This indicates that for any fixed value of the
magnetic field , is the
appropriate unit for the electron concentration
(n). Here ? is called the filling factor.
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Experimental Results
According to an experiment done by V. Klitzing,
the Hall resistivity of silicon MOSFETs as a
function of the gate voltage (which is
proportional to the electron concentration) is a
constant within a certain range around each
integer value of the filling factor.
Simultaneously, in these regions, the
longitudinal conductivity was found to be
vanishing and the Hall conductivity remains a
constant.
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A schematic view of the quantum Hall effect in
terms of the conductivities as functions of the
filling factor
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Observed QHE in medium mobility ( 5200 cm2/Vs )
GaAs heterostructures at 60mK
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Explanation of IQHE

• When the Fermi energy is in a gap, i.e. between
  the fields (a) and (b) in the diagram, Hall
  resistance cannot change from the quantized value
  for the whole time, and so a plateau results.
• If the Fermi energy in the Landau level, i.e.
  the field (c) is reached in the diagram, it is
  possible to change the voltage and a finite value
  of resistance will be appeared. In this situation
  the step like behavior of the Hall conductivity
  is observed.

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In the presence of impurity, the density of
states will evolve from sharp Landau levels to a
broader spectrum of levels.

• There are two kinds of energy levels, namely
  localized and extended states
• As the density is increased (or the magnetic
  field is decreased), the localized states are
  gradually fill up without any change in
  occupation of the extended states, thus without
  any change in the Hall resistance. For these
  densities the Hall resistance is on a step in the
  Figure(right) while longitudinal resistance
  vanishes (at zero temperature).

• When the Fermi level passes through the
  extended state (Or the Fermi level is in the core
  of the extended state) the longitudinal
  resistance becomes appreciable and the hall
  resistance makes its transition from one plateau
  to the next as in figure(right)

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Schematic Picture of the Quantum Hall Effect
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Benefits of IQHE

• The integer quantum Hall effect is now used as
  the international standard of resistance.
• The incredibly accurate quantization of the Hall
  resistance to approximatelyone part in 108 makes
  this possible.
• Finding the structure constant
• The constant e2/h is proportional to the fine
  structure constant in electrodynamics, so the
  quantum Hall effect provides us an independent
  way of accurately measuring this constant.

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REFRENCES

• D. Yoshioka, The Quantum Hall Effect.
• M.JanBen, O. Viehweger,U. Fastenrath and J. Hajdu
  Introduction to the Theory of the Integer Quantum
  Hall Effect.
• T. Chakraborty and P.Pietilainen The Quantum Hall
  Effects (Fractional and Integral)
• www.pha.jhu.edu/qiuym/qhe/node2.html
• nobelprize.org/physics/laureates/1985/press.html
• www.warwick.ac.uk/phsbm/qhe.htm
• www.pha.jhu.edu/qiuym/qhe/node4.html

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THANK YOU