What causes the Shubnikov-de Haas Oscillations?

Question

If I have a 2DEG with a voltage in the $x$-direction and a $B$-Field in the $z$-direction (so I also get a hall-voltage in the $y$-direction (classicaly)). But if I do this stuff at low temperatures I can observe some quantum effects like Shubnikov-de Haas Oscillations and the Quantum Hall Effect.

I understand how the Quantum Hall Effect emerges but I can't seem to understand why the Shubnikov-de Haas Oscialltions happen.

As far as I know, if the Fermi-Energy (or chemical potential) starts to cross the density of states (as a function of energy) (which represents one Landau Level) it causes all the electrons to conduct a current. At that region the resistence rises. If the Landau Level is passed (the Fermi-Energy is between two Landau Levels) the resistence seems to drop again, but I don't know why.

What causes the Shubnikov-de Haas Oscillations? Why do Shubnikov-de Haas Oscillations occur?

Answer 1

Roughly speaking, the conductivity is inversely proportional to the scattering rate $1/\tau$ (in the Drude model, the longitudinal conductivity is $\sigma_0=\frac{n e^2 \tau}{m}$). From Fermi's golden rule, $1/\tau$ is proportional to the number of density of states a quasiparticle can scatter into, which is proportional to the density of states near the chemical potential. In the presence of a magnetic field, the energy levels are quantized into equally spaced Landau levels. Therefore the density of states at the chemical potential(or Fermi energy) $N(\mu)$ oscillates in $1/B$ periodically, where $B$ is the magnetic field, as the chemical potential passes each Landau level by varying the magnetic field. Therefore $1/\tau$ oscillates as well and the oscillation of $\sigma$ follows. That gives the SdH oscillation.

Note that this is different from the van-Alphen de-Haas oscillation, where thermodyanimc quantity such as magnetization is directly measured and the magnetization quantity is directly proportional to the DoS at the chemical potential. In the current case, the oscillation of DoS comes in indirectly via the scattering rate.

More detail discussions can be found in D. Schoenberg, Magnetic oscillations in metals.