1 Quantum Hall Effect

2 Relativistic Quantum Hall Effect

3 References

4 Imprint

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 I_{y} is flowing through the plate.
Perpendicularly the plate is penetrated by a magnetic field with a strength B_{z}.
As a result the Lorentz force

K_{L}=lB_{z}I_{y}/N

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
E_{H} whose force

K_{H}=-eE_{H} (1.1)

to the electrons compensates the Lorentz force.
The result for Hall voltage is

U_{H}=bE_{H}=-blB_{z}I_{y}/(Ne) .

Applies d<<l, d<<b to the plate dimensions there is a 2 dimensional electron gas.
The number of electrons is

N=eblB_{z}i/h, i=1,2,3,...

(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

U_{H}=-R_{H}I_{y}

with the Hall resistance

R_{H}=h/(ie^{2}) .

To describe the relativistic Quantum Hall Effect equation (1.1) must be generalised.
With the electromagnetic field tensor F_{µν} the covariant form of force is

K_{µ}=-eF_{µν}u^{ν}

where

(u^{ν})=(u^{0},u^{1},u^{2},u^{3})

is the 4-velocity. To speed of light applies the following relation:

c^{2}=u_{ν}u^{ν}=g_{µν}u^{µ}u^{ν} .

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

(u^{ν})=(u^{0},0,0,0)

and

c^{2}=g_{00}(u^{0})^{2} .

The result for u^{ν} is:

(u^{ν})=[c/(g_{00})^{1/2},0,0,0] .

Then we obtain

K_{µ}=-eF_{µ0}u^{0}

or in a to (1.1) analogue component writing:

K_{H}=-eE_{H}/(g_{00})^{1/2} .

The value for g_{00} we get from Schwarzschild metrics

(g_{µν})= |
1-r_{s}/r | 0 | 0 | 0 | |||

0 | -1/(1-r_{s}/r) | 0 | 0 | ||||

0 | 0 | -r^{2} | 0 | ||||

0 | 0 | 0 | -r^{2}(sinθ)^{2} |

with the Schwarzschild radius

r_{s}=2GM/c^{2} .

The result is a modification of the value for R_{H}:

R'_{H}=R_{H}(1-r_{s}/r)^{1/2}≈R_{H}(1-GM/rc^{2}) .

For M and r the mass of earth and the earth radius have to be inserted. For GM/rc^{2} 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

URL: http://www.relativistische-asynchronmaschine.de

E-Mail: admin@relativistische-asynchronmaschine.de

Rev. 3.0, 14.11.2009

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