lunes, 28 de junio de 2010

Hall Effect

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 tex2html_wrap_inline506, 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
equation17
where tex2html_wrap_inline508 is the electric field and m is the electron mass. The current density is thus
equation23
where
equation30
and n is the electron density.

In the presence of a steady magnetic field, the conductivity and resistivity become tensors
equation35
and tex2html_wrap_inline510, tex2html_wrap_inline512. Still assuming that the relaxation time is tex2html_wrap_inline506, the Lorentz force must be added to the force from the electric field in Eq. (
1),
equation57
In the steady state, tex2html_wrap_inline516. We will always assume that the magnetic field is in z direction . Then in xy plane
equation70
where tex2html_wrap_inline518 is defined in Eq. (
3),
equation82
is the cyclotron frequency. From Eq. (
6), we can easily get
equation89
Eqs. (
8) directly leads to the relation between conductivity and resistivity
equation108
We can see that if tex2html_wrap_inline520, the conductivity tex2html_wrap_inline522 vanishes when the resistivity tex2html_wrap_inline524 vanishes. On the other hand,
equation122
Therefore when tex2html_wrap_inline526, tex2html_wrap_inline528, where tex2html_wrap_inline530 is given by the first term in Eq. (
10), i.e. Hall conductivity
equation134
In the experiment we can let Etex2html_wrap_inline532=0,
equation140

The above discussion is the classical result. In quantum mechanics, the Hamiltonian is ( E is along x direction)
equation147
For this problem it is convenient to choose the Landau gauge, in which the vector potential is independent of y coordinate
equation156
This choice allows us to choose a wavefunction which has a plane-wave dependence on the y coordinate
equation160
Substituting Eq. (
15) into Eq. (13), the Schrödinger equation becomes
equation167
where
equation179
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
equation187
where i=0,1,2,3, tex2html_wrap_inline534, and tex2html_wrap_inline536. 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
equation200
Thus tex2html_wrap_inline538=-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.

Fig. 1 Schematic dagram of the density of states of two and three-dimensional electron systems