Shubnikov–de Haas effect
An oscillation in the conductivity of a material that occurs at low temperatures in the presence of very intense magnetic fields, the Shubnikov–de Haas effect (SdH) is a macroscopic manifestation of the inherent quantum mechanical nature of matter.
Thus, as the magnetic field increases, the spin-split Landau levels move to higher energy.
[citation needed] Consider a two-dimensional quantum gas of electrons confined in a sample with given width and with edges.
In the presence of a magnetic flux density B, the energy eigenvalues of this system are described by Landau levels.
The Landauer–Büttiker approach allows calculation of net currents Im flowing between a number of contacts 1 ≤ m ≤ n. In its simplified form, the net current Im of contact m with chemical potential μm reads where e denotes the electron charge, h denotes the Planck constant, and i stands for the number of edge channels.
That current equals the voltage μm / e of contact m multiplied with the Hall conductivity of 2e2 / h per edge channel.
the maximum number D of electrons with spin S = 1/2 per Landau level is Upon insertion of the expressions for the flux quantum Φ0 = h / e and for the magnetic flux Φ = BA relationship (2) reads Let N denote the maximum number of states per unit area, so D = NA and Now let each Landau level correspond to an edge channel of the above sample.
It follows that For a given sample, all factors including the electron density n on the right hand side of relationship (3) are constant.
When plotting the index i of an edge channel versus the reciprocal of its magnetic flux density 1/Bi, one obtains a straight line with slope 2e/(nh).
The slope of 0.00618/T as obtained from a linear fit yields the electron density n Shubnikov–de Haas oscillations can be used to map the Fermi surface of electrons in a sample, by determining the periods of oscillation for various applied field directions.
The signature of each effect is a periodic waveform when plotted as a function of inverse magnetic field.
