Shubnikov-de Haas effect

The Shubnikov-de Haas effect is an oscillatory variation of electrical resistance with magnetic field caused by Landau levels moving through the Fermi energy. As the density of accessible states near that energy changes, the conductivity and scattering response oscillate.

A measurement sweeps field at low temperature while recording a longitudinal voltage for a controlled current. Oscillations are commonly approximately periodic in inverse field for a fixed closed Fermi-surface orbit. Their period probes the orbit area, while temperature-dependent amplitude constrains a cyclotron effective mass. Disorder and thermal broadening can wash out the oscillations even when a Fermi surface exists. The effect is a transport measurement, distinct from the oscillating magnetization measured in the thermodynamic quantum-oscillation counterpart.

A finite-temperature measurement averages the oscillatory response over the derivative of the Fermi-Dirac distribution. For the fundamental harmonic and a locally constant cyclotron effective mass, this gives $R_T=X/\sinh X$ with $X=2{\pi }^2{k}_{B}T/(\hslash {\omega }_c)$. The oscillations become small when thermal smearing substantially exceeds their Landau-level spacing.

The Dingle factor describes the damping of quantum oscillations by Landau-level broadening. For a Lorentzian half-width $\hslash /(2{\tau }_q)$, the fundamental harmonic has $R_D=\exp [-\pi /({\omega }_c{\tau }_q)]$. Since ${\omega }_c$ decreases with field, the amplitude decays at large inverse field. The lifetime here is the quantum lifetime, not automatically the mobility-derived transport time.