Editing Quantum Hall effect (section)

=== Longitudinal resistivity ===
It is possible to relate the filling factor to the resistivity and hence, to the conductivity of the system. When <math>\nu</math>  is an integer, the [[Fermi energy]] lies in between Landau levels where there are no states available for carriers, so the conductivity becomes zero (it is considered that the magnetic field is big enough so that there is no overlap between Landau levels, otherwise there would be few electrons and the conductivity would be approximately <math>0</math>).  Consequently, the resistivity becomes zero too (At very high magnetic fields it is proven that longitudinal conductivity and resistivity are proportional).<ref>{{cite book |title=The physics of low-dimension|last=Davies J.H.|location=6.4 Uniform magnetic Field; 6.5 Magnetic Field in a Narrow Channel, 6.6 The Quantum Hall Effect|isbn=9780511819070}}</ref>

With the conductivity <math>\sigma=\rho^{-1} </math> one finds
: <math>\sigma=
\frac{1}{\det \rho}
\begin{pmatrix}
      \rho_{yy}&-\rho_{xy}\\
      -\rho_{yx}&\rho_{xx}
    \end{pmatrix} \; .</math>

If the longitudinal resistivity is zero and transversal is finite, then <math> \det \rho \neq 0 </math>. Thus both the longitudinal conductivity and resistivity become zero.

Instead, when <math>\nu</math>  is a half-integer, the Fermi energy is located at the peak of the density distribution of some Landau Level. This means that the conductivity will have a maximum .

This distribution of minimums and maximums corresponds to ¨quantum oscillations¨ called ''Shubnikov–de Haas oscillations'' which become more relevant as the magnetic field increases. Obviously, the height of the peaks are larger as the magnetic field increases since the density of states increases with the field, so there are more carriers which contribute to the resistivity.  It is interesting to notice that if the magnetic field is very small, the longitudinal resistivity is a constant which means that the classical result is reached.

[[File:Rhoxy.jpg|alt=|thumb|263x263px|Longitudinal and transverse (Hall) resistivity, <math>\rho_{xx}</math> and <math>\rho_{xy}</math>, of a two-dimensional electron gas as a function of magnetic field. Both vertical axes were divided by the quantum unit of conductance <math>e^2/h</math> (units are misleading). The filling factor <math>\nu</math> is displayed for the last 4 plateaus.]]