Transcript Syed Ali Raza Supervisor: Dr. Pervez Hoodbhoy  A brief Overview of Quantum Hall Effect  Spinning Disk  Spinning Disk with magnetic Field  Percolation  Future investigations.

Syed Ali Raza
Supervisor: Dr. Pervez Hoodbhoy
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A brief Overview of Quantum Hall Effect

Spinning Disk
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Spinning Disk with magnetic Field
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Percolation

Future investigations
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F=vxB
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2-D system, perpendicular
magnetic field
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Quantized values of Hall
Conductivity
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σ = ne2/h
Quantised Landau
Levels
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Enormous Precision
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Used as a standard of
resistance
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Does not depend on
material or impurities
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We first write our Hamiltonian
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Define a Vector Potential
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Solve it using many ways, e.g Operator
approach
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In terms of dimensionless variables
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Cyclotron Frequency
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Magnetic length
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We define the Hamiltonian in terms of some
operators
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m degenerate states in each Landau level
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the number of quantum states in a LL equals the number of flux
quanta threading the sample surface A, and each LL is
macroscopically degenerate.
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Lagrangian
Hamiltonian
Rotating Frame
Lab Frame
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Now we want to find the wavefunction for this Hamiltonian.
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This has the form of the Bessel Equations
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We take B = zero because otherwise there would be a singularity at
r = zero.
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where
Bessel function.
represents the nth root of the mth order
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Bessel functions are just decaying sines and cosines.
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We can also calculate the current for this spinning disk
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Hamiltonian
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where
We make our equations dimensionless and get
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Now we need to solve this to get the complete wavefunction.
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We solve for U(r) using the series solution method and solve it
exactly. After a lot of painful algebra, you get the following
recursion relation:
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And you can recover your energy relation from this recursion too
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For a single electron
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For more electrons
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Lagrangian
Hamiltonian
Rotating
Frame
Lab
Frame
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Making them dimensionless and applying the wavefunction.
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Applying the series solution method we get recursion relation
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We can get the energies from this too
As you can see, because of the spinning there are no more m
degenerate states in each landau level and now each m has an
energy
 The farther away from the centre, the more energetic they are
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The series solution is very messy and tedious, so we try to do it
with operators
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First we write our Hamiltonian
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We set up our change of coordinates and operators
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Substitute these in the Hamiltonian
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Looks horrifying but gladly most of the things cancel out and we
are left with
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Plug in operators
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We get our final Hamiltonian and energies