1 Quantum Hall Effect
2 Relativistic Quantum Hall Effect
The Quantum Hall Effect makes possible a very precisely measurement of resistance, because the Hall resistance
only depends on nature constants. For derivation we consider an electrical conducting plate whith a thickness d,
a width b and a length l. Lengthwise a direct current Iy is flowing through the plate.
Perpendicularly the plate is penetrated by a magnetic field with a strength Bz.
As a result the Lorentz force
is exerted on a single electron (N = number of all electrons in the plate). The electrons which are retained and exhausted respectively at the boundary cause an electric field EH whose force
to the electrons compensates the Lorentz force. The result for Hall voltage is
Applies d<<l, d<<b to the plate dimensions there is a 2 dimensional electron gas. The number of electrons is
(h = Planck's constant). The quantisation is a result of a circle movement of the electrons in the magnetic field (cyclotron resonance). Quantum mechanically allowed along these circle curves are only standing waves. Then the result for Hall voltage is
with the Hall resistance
To describe the relativistic Quantum Hall Effect equation (1.1) must be generalised.
With the electromagnetic field tensor Fµν the covariant form of force is
is the 4-velocity. To speed of light applies the following relation:
To measure the Quantum Hall Effect in a gravity field, we consider a resting experiment facility e.g. on the surface of earth. Then we have
The result for uν is:
Then we obtain
or in a to (1.1) analogue component writing:
The value for g00 we get from Schwarzschild metrics
with the Schwarzschild radius
The result is a modification of the value for RH:
For M and r the mass of earth and the earth radius have to be inserted. For GM/rc2 we obtain a value of 7·10-10. If we could increase the precision of measurements of the Quantum Hall Effect up to 10-12 the gravity constant G could be determined.
Hermann Weyl, “Space, Time, Matter”, Dover Publications
Torsten Fließbach, “Allgemeine Relativitätstheorie”, 2nd edition, Spektrum Akademischer Verlag, 1995
Hajdu, Kramer, “Der Quanten-Hall-Effekt”, Phys. Bl., 41, (1985), 401-406
Title: The relativistic Quantum Hall Effect
Author: Tilmann Schneider
Rev. 3.0, 14.11.2009
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