Quantum oscillation

A quantum oscillation is an oscillatory contribution to a metal's response as a magnetic field moves quantized orbital levels past the Fermi energy. For a closed extremal Fermi surface cross-section of wavevector-space area $A_{\mathrm{ext}}$, Onsager quantization gives inverse-field period $\Delta (1/B)=2\pi e/(\hslash A_{\mathrm{ext}})$. Oscillations of magnetization and electrical resistivity are respectively the de Haas-van Alphen effect and Shubnikov-de Haas effect. Low temperature and long quantum lifetime are needed so thermal smearing and disorder do not wash out the Landau levels.

At half filling the nearest-neighbor band $E-E_0=-2t(\cos x+\cos y)-t\cos z$ has an electron contour at $z=\pi$ and a hole contour at $z=0$. Translation $(x,y)\mapsto (x+\pi ,y+\pi )$ maps the enclosed pockets into each other, proving equal regular Onsager frequencies for a field along $z$. Comparing an occupied slice with the complement of a hole orbit is not a comparison of two different enclosed planar orbits. Additional band terms or changed filling can lift this equality.