Spin-orbit interaction in the magnetization of two-dimensional electron systems

We review recent experimental and theoretical work on the quantum oscillations of the magnetization M, i.e., the de Haas-van Alphen (dHvA) effect, in two-dimensional electron systems (2DESs) with spin-orbit interaction (SOI). We focus first on a theoretical modeling by numerically solving the Hamiltonian including the Rashba (R) and Dresselhaus (D) SOI and the Zeeman term in an arbitrarily tilted magnetic field B. We second present experimental data on the SOI-modified quantum oscillations of M(B) in 2DESs formed in the InGaAs/InP and AlGaAs/GaAs material systems for different tilt angles between the 2DES normal and the direction of B. We find pronounced beating patterns in InGaAs/InP that are described quantitatively by assuming a dominant R-SOI except for a distinct frequency anomaly in M present in nearly perpendicular B. In AlGaAs/GaAs, beating patterns occur at large tilt angles. Here, anomalies in the dHvA wave form occur. The findings demonstrate that the understanding of the ground state energy of a 2DES is incomplete when SOI is present. Finally, we predict that the amplitude and anisotropy of specific dHvA oscillations with respect to the in-plane magnetic field component allow one to quantify the magnitude and relative signs of both R-SOI and D-SOI when simultaneously present. Calculated fan chart of Landau levels and oscillatory Fermi energy EF in a 2DES subject to SOI. The beating patterns in EF contain information on the SOI. They manifest itself in a variety of experimentally accessible observables. In this feature article, we focus on the magnetization M as a thermodynamic state variable. The inset sketches a micromechanical cantilever magnetometer used to measure M using the torque τ=M×B.