Basics of Hall Effect
In the Drude theory of the electrical conductivity of a metal, an electron is accelerated by the electric field for an average time 
, the relaxation or mean free time, before being scattered by impurities, lattice imperfections and phonons to a state which has average velocity zero. The average drift velocity of the electron is
where 
is the electric field and m is the electron mass. The current density is thus
where
and n is the electron density.
In the presence of a steady magnetic field, the conductivity and resistivity become tensors
and 
, 
. Still assuming that the relaxation time is , the Lorentz force must be added to the force from the electric field in Eq. (1),
In the steady state, 
. We will always assume that the magnetic field is in z direction . Then in xy plane
where 
is defined in Eq. (3),
is the cyclotron frequency. From Eq. (6), we can easily get
Eqs. (8) directly leads to the relation between conductivity and resistivity
We can see that if 
, the conductivity 
vanishes when the resistivity 
vanishes. On the other hand,
Therefore when 
, 
, where 
is given by the first term in Eq. (10), i.e. Hall conductivity
In the experiment we can let E
=0,
The above discussion is the classical result. In quantum mechanics, the Hamiltonian is ( E is along x direction)
For this problem it is convenient to choose the Landau gauge, in which the vector potential is independent of y coordinate
This choice allows us to choose a wavefunction which has a plane-wave dependence on the y coordinate
Substituting Eq. (15) into Eq. (13), the Schrödinger equation becomes
where
is the classical cyclotron orbit radius.
Eq. (16) can be easily solved by transformation to a familiar harmonic oscillator equation. The eigenvalues and eigenstates are
where i=0,1,2,3, 
, and 
. The different oscillator levels are also called Landau Levels. The electric field simply shifts the eigenvalues by a value without changing the structure of the energy spectrum. From Figure 1 we can see that in two-dimensional systems, the Landau energy levels are completely seperate while in three-dimensional systems the spectrum is continuous due to the free movement of electrons in the direction of the magnetic field.
From the wave functions, we can calculate the mean value of the velocities
Thus 
=-neEc/B, which is the same as Eq. (11) of the classical result. The current along the direction of electric field (x) is zero at Landau levels.
