188. Integer and fractional quantum Hall effect – what is it?
COND-MAT — By Dmitry Podolsky on January 19, 2009 at 1:23 pm
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Fractional quantum Hall effect was and remains one of the biggest mysteries in condensed matter theory. To be as arrogant as usual, I will claim that theorist’s brain work necessary to explain fractional quantum Hall effect is roughly equal to the work necessary to explain confinement in QCD, although the former problem is much younger than the latter – the fractional Hall effect was first observed in 1982 by Tsui, Stormer and Gossard.
1. What is classical Hall effect?
Let us first recall what is the classical Hall effect. Suppose that an electric current through the material. If you apply a magnetic field perpendicular to the face of the material, the charge carriers get affected by the Lorentz force that curves their path. As a result, negatively and positively charged carriers accumulate on the opposite faces of material and create strong electric field that prevents further flow of the current. Quantitatively, you can describe the effect as appearance of effective resistivity for the electric current in the material.
2. What is quantum Hall effect?
Suppose that the magnetic field is directed along z-axis, electric current – along x-axis and electric field due to the Hall effect appears along y-axis. Basically, quantum Hall effect shows up as a fact that the Hall resistivity 
is quantized while 
vanishes.
In classical Hall effect, depends monotonically on 
and concentration of carriers 
:
In quantum Hall effect, is quantized according to the rule
,
where 
are integer or fractional numbers – that’s where the names “integer” and “fractional quantum Hall effect” come from.
The integer quantum Hall effect was discovered on Si metal-insulator-semiconductor devices in 1980 by the group of K. von Klitzing et al., while fractional one – a bit later, in 1982, when the states with 
heterostructures (later, also 
state was observed). Generally, heterostructures based on InP, GaSb, GaEs etc. are the main materials where the quantum Hall effect is studied nowadays on.
On practice, that’s what you basically measure and see:
As for the component, in both fractional and integer Hall effects it becomes vanishingly small at non-zero compared to its value at 
. When you increase the temperature 
, starts to grow.
That’s all for now, I hope that it was sufficiently clear – if not, please ask questions. We will discuss the theory of integer quantum Hall effect and some ideas that might help to understand the fractional Hall effect tomorrow 
6 Comments
Hi Dmitry
Thanks for your post.
Since when we apply the electric field in the x-direction, we want the current in x-direction, and vanishing of rho_{xx} means that it is superconducting in the x-direction. So we can conductor current with loss when the conductor exhibits quantum hall effect, right?
From the plot, to come up with a theory to explain it seems to me is as difficult as to explain the particle zoo. I guess the theory must be somehow related the BCS theory, right?
Keith
Hi Keith,
I would not call a material superconducting in the x-direction and not – in y-direction, especially taking into account that diagonal resistivity is an oscillating function of magnetic field (albeit its values are small compared to diagonal resistivity at
).
As for the particle zoo, let me turn to it in the tomorrow’s post
Cheers,
Dmitry.
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