# Does Hall plateus require the existence of impurity in the sample?

While studying Hall conductivity with The Quantum Hall effect written by S.M.Girvin, I read a sentence

"We have shown that the random impurity potential (and by implication Anderson localization) is a necessary condition for Hall plateaux to occur"(page 7)

As you know, in QHE experiment, we can see regions where two conductivity $$\sigma_{xx}$$ and $$\sigma_{xy}$$ is constant, especially $$\sigma_{xx}=0$$ and $$\sigma_{xy}=\nu$$ $$e^2\over\hslash$$ where $$\nu$$ is the filling factor of Landau levels.

Many literature say that the plateus appear since there are localized states (which comes from impurities in real samples) between the extended states ( in DOE vs E graphs ).

I understand this statement in this way : When the Fermi energy is set to be in energy range of a localized state, there is no states carrying current about the fermi energy in the 'bulk' , but there are current carrying states on edge as Landau level's energy level vent upward due to edge confining field. Therefore the number of crossing between fermi level and energy level on the edge appears in the result $$\sigma_{xy}=\nue^2\over\hslash$$.

In this way of understanding, I concluded that there should be a region in which $$\sigma_{xy}$$ is constant whether the sample is perfectly pure or not, because the number of current channels on the edge is the only important one. (Is it right?)

Then why does the QHE require the existence of impurity in the sample?

• Your quote says "for the Hall plateau" to occur - not for the Hall effect to occur. You might want to edit your title accordingly - or clarify what it is you are asking (you are mixing the two). Mar 10, 2015 at 15:52
• I've changed. However, is there difference between quantum Hall effect and the appearence of Hall plateus? Mar 10, 2015 at 16:29