Editing Quantum Hall effect (section)

=== Density of states ===
At zero field, the density of states per unit surface for the two-dimensional electron gas taking into account degeneration due to spin is independent of the energy
: <math>n_{\rm 2D}=\frac{m^*}{\pi \hbar^2}</math>.
As the field is turned on, the density of states collapses from the constant to a [[Dirac comb]], a series of Dirac <math>\delta</math> functions, corresponding to the Landau levels separated <math>\Delta\varepsilon_{xy}=\hbar \omega_{\rm c}</math>. At finite temperature, however, the Landau levels acquire a width <math display="inline">\Gamma=\frac{\hbar}{\tau_i}</math>  being <math>\tau_i</math>  the time between scattering events. Commonly it is assumed that the precise shape of Landau levels is a [[Gaussian distribution|Gaussian]] or [[Cauchy distribution|Lorentzian]] profile.

Another feature is that the wave functions form parallel strips in the <math>y</math>-direction spaced equally along the <math>x</math>-axis, along the lines of <math>\mathbf{A}</math>. Since there is nothing special about any direction in the <math>xy</math>-plane if the vector potential was differently chosen one should find circular symmetry.

Given a sample of dimensions <math>L_x \times L_y</math> and applying the periodic boundary conditions in the <math>y</math>-direction <math display="inline">k=\frac{2\pi}{L_y}j</math> being <math>j</math> an integer, one gets that each parabolic potential is placed at a value <math>x_k=l_B^2k</math>.
[[File:Potencialesparabólicos.jpg|alt=|thumb|270x270px|Parabolic potentials along the <math>x</math>-axis centered at <math>x_k</math> with the 1st wave functions corresponding to an infinite well confinement in the <math>z</math> direction. In the <math>y</math>-direction there are travelling plane waves.]]
The number of states for each Landau Level and <math>k</math> can be calculated from the ratio between the total magnetic flux that passes through the sample and the magnetic flux corresponding to a state.
: <math>N_B=\frac{\phi}{\phi_0}=\frac{BA}{BL_y\Delta x_k}=\frac{A}{2\pi l_B^2}\begin{array}{lcr}& l_B&\\&=&\\ &&\end{array}     
\frac{AeB}{2\pi \hbar}\begin{array}{lcr}& \omega_{\rm c}&\\&=&\\ &&\end{array} \frac{m^*\omega_{\rm c}A}{2\pi \hbar}</math>

Thus the density of states per unit surface is 
: <math>n_B=\frac{m^*\omega_{\rm c}}{2\pi \hbar}</math>.

Note the dependency of the density of states with the magnetic field. The larger the magnetic field is, the more states are in each Landau level. As a consequence, there is more confinement in the system since fewer energy levels are occupied.

Rewriting the last expression as <math display="inline">n_B=\frac{\hbar \omega_{\rm c}}{2} \frac{m^*}{\pi \hbar^2}</math> it is clear that each Landau level contains as many states as in a [[Two-dimensional electron gas|2DEG]] in a <math>\Delta\varepsilon=\hbar \omega_{\rm c}</math>.

Given the fact that electrons are [[fermions]], for each state available in the Landau levels it corresponds to two electrons, one electron with each value for the [[Spin (physics)|spin]] <math display="inline">s=\pm\frac{1}{2}</math>. However, if a large magnetic field is applied, the energies split into two levels due to the magnetic moment associated with the alignment of the spin with the magnetic field. The difference in the energies is <math display="inline">\Delta E = \pm \frac{1}{2}g\mu_{\rm B}B</math> being <math>g</math>  a factor which depends on the material (<math>g=2</math> for free electrons) and <math>\mu_{\rm B}</math> the [[Bohr magneton]]. The sign <math>+</math>  is taken when the spin is parallel to the field and <math>-</math>  when it is antiparallel. This fact called spin splitting implies that the [[density of states]] for each level is reduced by a half. Note that <math>\Delta E</math>  is proportional to the magnetic field so, the larger the magnetic field is, the more relevant is the split.
[[File:Densidadestadossinspin.jpg|alt=|thumb|263x263px|Density of states in a magnetic field, neglecting spin splitting. (a)The states in each range <math>\hbar \omega_{\rm c}</math> are squeezed into a <math>\delta</math>-function Landau level. (b) Landau levels have a non-zero width <math>\Gamma</math> in a more realistic picture and overlap if <math>\hbar \omega_{\rm c}<\Gamma</math>. (c) The levels become distinct when <math>\hbar \omega_{\rm c}>\Gamma</math>.]]

In order to get the number of occupied Landau levels, one defines the so-called filling factor <math>\nu</math> as the ratio between the density of states in a 2DEG and the density of states in the Landau levels.

: <math>\nu=\frac{n_{\rm 2D}}{n_B}=\frac{hn_{\rm 2D}}{eB}</math>

In general the filling factor <math>\nu</math> is not an integer. It happens to be an integer when there is an exact number of filled Landau levels. Instead, it becomes a non-integer when the top level is not fully occupied. In actual experiments, one varies the magnetic field and fixes electron density (and not the Fermi energy!) or varies the electron density and fixes the magnetic field. Both cases correspond to a continuous variation of the filling factor <math>\nu</math> and one cannot expect <math>\nu</math> to be an integer. Since <math>n_B\propto B</math>,  by increasing the magnetic field, the Landau levels move up in energy and the number of states in each level grow, so fewer electrons occupy the top level until it becomes empty. If the magnetic field keeps increasing, eventually, all electrons will be in the lowest Landau level (<math>\nu<1</math>)  and this is called the magnetic quantum limit.

[[File:NivelesLandausinspin.jpg|alt=|thumb|265x265px|Occupation of Landau levels in a magnetic field neglecting the spin splitting, showing how the [[Fermi level]] moves to maintain a constant density of electrons. The fields are in the ratio <math>2:3:4</math> and give <math>\nu=4,\frac{8}{3}</math> and <math>2</math>.]]